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
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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
CONTENTS O F PREVIOUS VOLUMES
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
CONTENTS O F PREVIOUS VOLUMES
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
CONTENTS OF PREVIOUS VOLUMES
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
CONTENTS OF PREVIOUS VOLUMES
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.
CONTENTS OF PREVIOUS VOLUMES
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
V O L U M E XXV (1988) I
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.
1-86
87-179
181-270 271-359 361-416
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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
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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
WAVEGUIDE DIFFRACTION GRATINGS
PLANAR OPTICAL WAVEGUIDE
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
USES FOR WAVEGUIDE GRATINGS
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
WAVEGUIDE DIFFRACTION GRATINGS
[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
USES FOR WAVEGUIDE GRATINGS
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
WAVEGUIDE DIFFRACTION GRATINGS
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
PLANAR OPTICAL WAVEGUIDE
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
USES FOR WAVEGUIDE GRATINGS
I
PLANAR OPTICAL WAVEGUIDE
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
WAVEGUIDE DIFFRACTION GRATINGS
1
PLANAR OPTICAL WAVEGUIDE
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
USES FOR WAVEGUIDE GRATINGS
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
=
Es exp( - Y S X )
x
(6)
where E,, E,, and Es are constants. The remaining parameters satisfy yt
=
(p2 - n?k:)ll2
(i = c, s) ,
kf‘ = (n,Zk: - f12)’12 ,
(7) (8)
with k, = o / c = 2 4 2 , and $Js is just the TIR phase-shift angle associated with the lower interface; +s is defined by
Yi
tan$Ji= - (i = c, s) , kf
(9)
for the case of TE modes. The mode function for the TM modes has the sirnilar form H,(x)= H, exp[ - y,(x - h ) ] x > h , =
Hfcos(kfx - +tm)
0 <x
=
H, exP(- YSX)
x
(10)
where H,, H,, and Hsare constants, and the parameters are defined in the same way for TE and TM modes. The phase-shift angle for the TM polarization is slightly different from that in eq. (9),
16
WAVEGUIDE DIFFRACrION GRATINGS
[I. B 3
The requirement that the wave must be localized in or near the higher-index layer (n,> n,, n,) determines that pis restricted to the range n, k, < p < n,k,. It further requires that the mode functions E,,,(x) and H J x ) exhibit exponential decay with increasing distance from each interface. The application of the boundary conditions on the tangential components of E and H produces a dispersion relation given, for TE modes, by kfh -
4=- 4s= m n .
(12)
The corresponding result for TM modes is k,h -
$trn- 4irn= m n ,
(13)
which differs from eq. (12) only in that the correct phase shifts must be used for each polarization. The presence of the mode integer m is of central importance in these dispersion relations. Equations (7)-(9) and (1 1) show that for a given wavelength and set of refractive indices, kf and the phase shifts are functions of the propagation constant 8. Each value of m in either eq. (12) or (13), therefore, leads to a new value of 8. The allowed values of fl thus form a discrete set, not a continuum. Figure 10 shows the electric field profiles E,(x) associated with the three lowest order T E modes of a typical planar waveguide. It is clear that the mode integer m determines the number of zero crossings that each mode exhibits. The same general behavior occurs for TM modes. The careful reader might have noticed that the wave equations in eqs. (3) and (4) do not contain the V E terms ( E is the permittivity; E = con2, with E, = 8.85 x 10- ” F/m) that appear for a medium in which the refractive index n depends on the coordinates [recall, n’(x) = E ( x ) / E , ] . Clearly, n = n ( x ) for the planar waveguide. This dependence can be made explicit by writing E ( X ) in the form E ( X ) = E,[H?
+ (n,2 - n:)
O(X)
+ (n,Z - n:)
O(x - h ) ] ,
(14)
where O(x - a ) is the unit step function, defined according to O(x - a ) = 0
for x t a ,
O(x - a ) = 1 for
x> a
,
(15)
which shows explicitly the abrupt changes in E, and hence n’, at the boundaries of the waveguide. The more general wave equations are
V2E
+ ~ ’ ( x )(w/c)’ E =
-V
I“;&)
-
9
17
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
2 5 ~ '
-0.5
I
*
'
I
0.0
I
'
1
I
"
0.5
I
'
'
1
1 .o
1.5
1 .o
1.5
1 .o
1.5
.
x ( microns)
20 10
i i
._ Y + V al
o
w -10 -20 -0.5
0.0
0.5
x ( microns)
"
w -10
-0.5
0.0
0.5
x ( microns ) Fig. 10. Electric field distributions for the three lowest order TE modes for a planar optical waveguide of thickness h = 1.5 pm and n. > n,.
18
WAVEGUIDE DIFFRACTION GRATINGS
and
V 2 H + n 2 ( x )(w/c)’H
=
iw(E x V E ).
(17)
When E m ( x ) and Hm(x) in eqs. ( 5 ) and ( 6 ) are written in the style of eq. (14), it is not difficult to show that the previous solutions do indeed satisfy eqs. (16) and (17) as well as eqs. (3) and (4), as long as the boundary conditions on the tangential components of E and H are satisfied. More specifically, the deltafunction terms generated by the V2 operator and by V Ecan be made either to cancel or to vanish separately by applying the boundary conditions. The right-hand side of eq. (16), for example, vanishes for the TE modes of the planar waveguide, since the dot product E * V E = 0. It is conventional to introduce the effective index of refraction N, defined according to N
=
B/ko7
(18)
where, again, k,, = w/c = 2 4 A . One of the central properties of an optical waveguide is its ability to transport energy in a given direction, chosen to be the z-direction here. Since is the z-component of the propagation constant, guided waves with field profiles such as those shown in fig. 10 can be said to propagate along z, treating the waveguide as a medium of refractive index N . It is easy to show that N is restricted to the range
n, < N G n,,
(19)
for bound modes, where it has been assumed that the substrate has the larger refractive index of the two outer media in fig. la: n, 2 n,. This refractive index convention will be adopted throughout this chapter. The effective index can be related to the propagation angle 8,defined in a ray-optics model (see fig. la) as the angle between the ray and the normal, by the relation N = n, sin 8,from which it is clear that the lower limit in eq. (19) represents the minimum value of 8 that provides total internal reflection at both interfaces. The upper limit in eq. (19) represents the natural limit 8 = f n. The dispersion relations in eqs. (12) and (13) can now be regarded as transcendental equations that determine N for guided waves of the TE and TM polarizations. Figure 11 shows illustrative plots of the effective index N as a function of the film thickness h for both the m = 0 and m = 1 TE modes for a typical asymmetric geometry (n,# n,). The parameters used for the plot are given in the figure caption. Typically, for afixed wavelength and choice of refractive indices, there is a minimum thickness required to support a given mode of order m.For TE modes, for example, it is easy to show from eq. (12) that the minimum value
19
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
I
I 0.0
.
.
.
.
I . 0.5
.
.
.
I
1.o
.
.
.
.
I 1.5
.
.
.
.
I I 2.0
h ( microns ) Fig. 11. Plots of the effective index of refraction N as a mnction of the layer thickness h for the m = 0 and m = 1 modes of a sample optical waveguide, for which n, = 1.46, n f = 1.7, and no = 1.
of the ratio h / l , the so-called cutoff value, for a guided wave of order m is given by
where
Inspection of eq. (20) shows that higher-order modes require thicker waveguides for propagation. Equation (20) also shows that for the case of a symmetric waveguide, for which n, = n, and a = 0, the ratio = 0. This means that there is no nontrivial cutoff for the m = 0 mode of the symmetric optical waveguide. Equation (20) holds only for TE modes, but one obtains the same conclusion for TM modes. A symmetric waveguide of arbitrary nonzero thickness will support the m = 0 mode of both polarizations. This is analogous to the case of an optical fiber, a structure with refractive index confinement in
20
WAVEGUIDE DIFFRACIION GRATINGS
[I, 8 3
both transverse dimensions (x, y ) and cylindrical symmetry, for which the lowest order mode also has no nontrivial cutoff. One of the most important features to note in fig. 11 is that for a given waveguide with a given thickness h, the effective index N is different for the m = 0 and the m = 1 modes, even though the wavelength A is the same for both modes. This is of central interest for efforts in the field of integrated optics, which attempts to define optical components such as lenses, gratings, switches, and modulators in or on optical waveguides. The effective index N determines the way in which a guided wave interacts with a component. If some of the incident energy in the waveguide is carried in each of the m = 0 and m = 1 modes, then each mode will interact with the component in a different way to produce two different effects. In the case of a lens, for example, this means that there will be two different focal lengths. Other components have similar problems. It is for this reason that integrated optics is usually considered to be restricted to the use of single-modewaveguides. The presence of only one mode (of each polarization) in the waveguide permits a more precise definition of the operation of the components that make up an integrated optical or optoelectronic system. A convenient normalization for the bound modes of the planar waveguide makes use of a power normalization. For the time dependence assumed here, exp( - iwt), the time-averaged Poynting vector S can be written as S
=
iRe{E x H * } ,
(22)
where Re designates the real part of the bracketed quantity and the asterisk designates the complex conjugate. The standard normalization sets to unity the power per unit width carried by the guided wave:
where the hat ( " ) designates a unit vector. For TE modes this reduces to OD
2pow
E,(x) Ez(x) d x
=
1 (TE) ,
-a,
whereas for TM modes.
where E ( X ) is as in eq. (14). The size of the planar waveguide is assumed to be
1.5 31
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
21
very large along the y-direction so that its size places no restrictions on the fieId distributions that can propagate. For this reason the integration in eq. (23) is carried out over only the x-dimension, and the result is referred to as the “power per unit width (along y)”. In reality, eq. (23) is just a normalization condition. The boundary conditions lead to relationships between the amplitude constants that appear in eqs. (6) and (10). For the TE case, for example, one obtains the following formulas
and
These reduce the number of amplitude constants in eq. (6) from three to one, i.e. E,. When eq. (6) is inserted into the normalization integral in eq. (24), E, is then obtained in terms of the various waveguide parameters
where
heff = h t
1 ~
Yc
+
1 -
(TE)
Ys
is termed the effective waveguide thickness. Equation (28) determines the value of E, subject to the normalization condition in eq. (23). The TM result is a bit more complicated, but of the same form. The boundary conditions give
and
22
WAVEGUIDE DIFFRAmION GRATINGS
[I. 8 3
where qi = (N/n,)2
+ (N/ni)" +1
(i
=
c, s) ,
and the preceding notation follows that of KOGELNIK[ 19751. The normalization integral, in turn, gives
where hes = h
1 1 ++ __ 'Ycqs
YS9S
(TM)
(34)
The mode functions also satisfy a useful orthogonality relation. For real refractive indices this relation is given by
s-
al
{E,,(x) x H;,(x)} co
- 2 dx
=
0 for m # n ,
(35)
where the subscript t designates the transverse component of the vector field. Equation (35) can be applied to electric and magnetic fields of the forms given in eqs. (1) and (2), as long as one of them vanishes at x = f 00. If one or both of them is a bound mode, this is certainly the case, not only for modes of the step-index planar waveguide, but also for more complicated structures such as planar, graded-index optical waveguides. It is only necessary that eqs. (1) and (2) describe the fields and that they behave properly at x = f 03.
3.2. BOUND MODES OF THE GRADED-INDEX OPTICAL WAVEGUIDE
One often encounters optical waveguides for which the refractive index is a continuous function of at least one of the spatial coordinates. Such waveguides are called graded-index waveguides. A sketch of the simplest type of a planar, graded-index waveguide, for which n, depends on the single coordinate x, appears in fig. 12. Note that x increases downward from the upper waveguide surface in fig. 12, in contrast to fig. 1. In this structure the refractive index n,(x) is greatest at x = 0 and diminishes with increasing x until it reaches some constant value. In practice, waveguides of this type are often made by diffusing some species into a host crystal. The result is a region typically a few microns thick for which the average refractive index exceeds the ambient value in the
1, I 31
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
x=o
ns
REFRACTIVE INDEX
n g + An
HOST MATERIAL OF REFRACTIVE INDEX
23
ns
Fig. 12. The geometry for a typical graded-index optical waveguide. The refractive index is highest at the surface and decreases with depth toward the ambient value of the most material.
host crystal. This higher-index region can confine light in much the same way as the step-index structure. The qualitative features of the bound modes of the graded-index waveguide are not significantly different from those of the step-index waveguide. Electric and magnetic fields of the form given in eqs. (1) and (2) solve the wave equations in eqs. (16) and (17), subject to the appropriate boundary conditions. The graded-index waveguide supports both T E and TM modes, but the mode functions Em@)and H J x ) are more complicated than those for the step-index waveguide. It is usually the case that both the mode functions and the dispersion relations can only be determined by numerical techniques. A discussion of these techniques is outside the scope of this chapter, but the reader can consult the published literature for further information. (CONWELL [ 19731, KOGELNIK[ 19751, KOROTKYand ALFERNESS [ 19871, HOCKERand BURNS [ 19751).
3.3. BOUND MODES OF THE NONLINEAR OPTICAL WAVEGUIDE
Certain materials exhibit a type of nonlinear response that leads to a refractive index that depends on the intensity of the optical wave propagating in the medium. If the wave has a nonuniform spatial profile, it produces an index gradient that, in turn, modifies the properties of the propagating wave. The self-focusing of a laser beam in a nonlinear medium is perhaps the most familiar example of this process. A layer of this nonlinear material, bounded by linear media, is capable of supporting guided waves. An exhaustive discussion of this subject is beyond
24
WAVEGUIDE DIFFRACTION GRATINGS
[I, § 3
the scope of this chapter, but one example will be examined here. Equation ( 3 6 ) gives the refractive index configuration for this example, n2(x)= n,’
x>h,
=n,Z+A+n,IEI2 =
n,2
O<x
(36)
Here, n2 is the nonlinear coefficient that describes the (real) magnitude of the nonlinear term in the refractive index of the layer and A is a small, negative (real) number. For E = 0, the refractive index of the layer is smailer than that of the substrate (n,), which means that the layer cannot serve as an optical waveguide in the conventional sense, since the total internal reflection condition cannot be satisfied at both interfaces. As E grows, the nonlinear term will first equal, then exceed A, and one expects bound modes of some sort to appear. The wave equation for TE modes can be solved using the familiar form
E
=
G ( ~ei(/3. ) -W
(37)
where G(x)= E, e-Yc(x-h)
x>h,
=
E, sech{k,(x - x,)}
0<x
=
E, eySx
x
(38)
is the mode profile. Application of the boundary conditions on the tangential components of the electric and magnetic fields leads to the dispersion relation tanh(k,h)
=
k,(Yc + YS) k f + Yc Ys
(39)
It is convenient at this point to introduce a few new parameters in terms of which to discuss some interesting features of these nonlinear guided waves. The parameters V, D,and a are defined according to
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
1,s 31
25
The original parameters can be expressed in terms of the new ones as:
k f = k , { l d l (1 + D)}1’2,
(43)
The dispersion relation can therefore be written in the form
The exponential decay rates in eqs. (44) and (45) must be positive to assure proper behavior at x = + 00. This is only possible for D 2 0, which means that the minimum power the wave must carry is determined from the condition
which is a condition that ultimately determines the required minimum intensity of the source used to excite the nonlinear guided wave. The larger the nonlinear coefficient, the smaller the required intensity. The effective index of refraction N = /?/(o/c), introduced earlier, is “power dependent” for the nonlinear wave and is given by N
=
(n:
+ (dlD)’l2.
(48)
As in the discussion of the step-index waveguide, N must exceed ns, which follows immediately from eq. (48) and the condition D 2 0. This suggests a connection with the normal internal reflection mechanism that is at work in the step-index case. The dispersion relation provides the allowed values of N for a given structure and choice of wavelength or frequency. Figure 13 shows a plot of the dispersion relation from eq. (46), plotted as N versus V, since D determines N according to eq. (48), for a symmetric structure (a = 0). It is significant that for a given V, i.e. a given structure, there are two distinct solutions for N, and hence for D,for a given value of V. This result means that two different nonlinear guided waves can, in principle, be supported by a particular nonlinear waveguide at two different power levels. There are other structures that support nonlinear guided waves. For example, a nonlinear waveguide can be formed by depositing a layer of a linear material onto a nonlinear substrate, or by sandwiching a linear layer between two
26
[I,§ 3
WAVEGUIDE DlFFRACTlON GRATINGS
t'
I
0.0
0.2
1
0.4
I
I
0.6
0.8
v
I
1.o
I
1.2
I
1.4
Fig. 13. Plot of the effective index N as a function of the parameter V for the nonlinear optical waveguide.
nonlinear media. Furthermore, important issues such as the stability of the nonlinear waves and the means of exciting these waves remain the subjects of very active investigation. The reader is referred to the extensive literature on nonlinear waveguides for more exhaustive treatments of the subject (STEGEMAN, BURKE and SEATON[ 19871).
3.4. RADIATION MODES OF THE STEP-INDEX WAVEGUIDE
The previous sections described the bound modes that are supported by planar optical waveguide structures of various kinds, with emphasis on the step-index waveguide. The TE and TM modes were described by electric and magnetic fields of the form
E
=
f+E,(x)ei(flz - a t ) and H
=
f+H,,(x)ei(flz-ot),
(49)
which vanishes at f 00, and for which n,(w/c)< B < n,(w/c). The wave equations also admit solutions of the form given in eq. (49) for other values of (real) fl, but they are not localized near the waveguide layer. These solutions are termed radiation modes and correspond to oscillating fields in at least two
1,s 31
MODES SUPPORTED BY PLANAR OPTICAL WAVEGUIDES
21
of the three media in fig. la. Since they have the form of eq. (49), the radiation modes of the step-index waveguide are orthogonal to the bound modes of the same waveguide, according to eq. ( 3 9 , although they cannot be normalized by the simple prescription in eq. (23). Only the essential characteristics of the radiation modes will be described here. The two types of radiation modes for the geometry in fig. l a are substrate radiation modes and substrate-cover radiation modes. Retaining the usual convention that n, 2 n,, for the substrate radiation modes, pis restricted to the range n, k, < fi < n, k,. The field profiles for the TE and TM substrate modes are
and
where each is in a form consistent with that used by KOGELNIK[ 19751. It is clear from the field profiles and the range of p considered that a substrate radiation mode has total internal reflection at the cover/film boundary (x = h), but not at the substrate/film boundary ( x = 0). In fact, eqs. (50) and (51) are precisely the fields one obtains by solving the Fresnel reflection problem for the case of a plane wave incident on the film layer from the substrate at an angle greater than or equal to 8 = arcsin (nc/ns),measured with respect to the normal. islcontinuous within its prescribed range for It is important to note that # radiation modes, as one might expect given the analogy with the Fresnel reflection problem. The situation is more complicated for the substrate-cover radiation modes, for which 0 < < n, k,. There are two sets of these radiation modes, which can be understood by noting that for a symmetrical waveguide the field profiles will possess even or odd symmetry. For the asymmetric waveguide one set of radiation modes must possess even symmetry and the other, odd symmetry, in the limit n, + n,. The field profiles for the TE and TM substrate-cover modes
28
WAVEGUIDE DIFFRACTION GRATINGS
[I, § 3
are of the form E,,(x) = E:
cos {qc(X - h ) + d : ' } sin
X
>h
9
=
E:
cos {qfx - d e ) } sin
O<x
=
E:
cos (qsX - d;)) sin
x
and HJX)
=
H:
cos { d x - h) + c',"'} sin
x>h ,
=
Hf'
cos {qfx - dm) 1 sin
O<x
where, in each expression, e and m label TE- and TM-related quantities, + and - designate even and odd modes, respectively, and cosine is used for even ( + ) modes, sine is used for odd ( - ) modes. Physically, these modes correspond to spatially oscillating fields in all three media. They can be viewed as properly phased superpositions of the solutions to the Fresnel reflection problem for the cases of plane waves incident upon the layer of thickness h from above and below. Although this will not be discussed in detail at this point, the bound and radiation modes of the step-index planar optical waveguide constitute a complete set of orthogonal functions. The bound modes make up the discrete spectrum, and the radiation modes make up the continuous spectrum. It is often convenient to use a mode expansion based on these functions when analyzing interactions in optical waveguides that have been perturbed in some fashion. A typical example is a waveguide with one corrugated surface. The periodic surface perturbation can provide a mechanism for converting forwardgoing waves into backward-going waves, or for coupling the bound modes with the radiation field, as might be exploited for input or output coupling. Since the major emphasis in this chapter, the interaction between bound
1, I 41
NONPLANAR OPTICAL WAVEGUIDES
29
modes and waveguide gratings, does not require a deep understanding of the radiation modes, no further details about them are given here. The interested reader is referred to the works of KOGELNIK[ 19751 and MARCUSE[ 19741.
0 4. Nonplanar Optical Waveguides The waveguide geometries so far considered in this chapter provide confinement along only a single coordinate axis. Waveguides that provide confinement along two axes are required for several applications. The circularly cylindrical optical fiber is the most familiar example, but other types are in common use as well. Unfortunately, numerical techniques are needed to analyze these structures in detail, but their qualitative features are easy to infer from those obtained earlier for planar waveguides. The fields associated with the bound modes of the three-dimensional waveguide are of the general form E(x, y, z, t ) = $+Em,(x,y ) ei(P--Z-w‘),
(54)
and
for assumed propagation along the z-direction. Note that the propagation constant p now depends on the two mode integers m and n, and that the field amplitudes depend on both the x- and y-coordinates. When the size of the waveguide is sufficiently small along each of the two transverse directions, a standing wave is set up within the guiding region much like the case for one-dimensional confinement. This leads to field distributions much like those shown in fig. 10 along each transverse direction. Only one integer label was needed in the case of the planar waveguide, an integer that specifies the number of zeros of the field. Two such integers are needed for confinement along two axes, since the number of zero-crossings need not be the same in both directions. Strictly speaking, the x- and y-dependences in eqs. (54) and ( 5 5 ) are not separable, although it is sometimes a useful approximation to write
and likewise for the magnetic field. The modes satisfy a dispersion relation that involves the two mode integers m and n. They also satisfy an orthogonality relation given by
-
{Emqt(x)x H,*,,(x)} 1 d x dy = 0 for m # n and q # s ,
(57)
30
WAVEGUIDE DIFFRACTION GRATINGS
[I, 8 5
where the subscript t, again, designates the transverse component(s) of the field, just as in eq. (35) for planar waveguides. The notation (three subscripts) is somewhat cumbersome, but eq. (57) is just the natural extension of the earlier result. This chapter does not make use of the detailed forms of the modes supported by the two-dimensional waveguide, so no further discussion of them is included here. The interested reader can consult the references for further information (ADAMS[ 19811, KOGELNIK[ 19881).
8 5. Coupling Between Guided Waves The problem of a guided-wave propagating in a periodic medium can be formulated in a variety of ways. Of particular interest is the case for which the propagation constant p (along z ) very nearly satisfies the Bragg condition. The most popular theoretical technique develops a pair of coupled-mode equations that connect the amplitudes of the forward- and backward-propagating waves. These equations can be extracted directly from the one-dimensional wave equation, as demonstrated below. Consider the following one-dimensional differential equation:
[$ 8’1 +
f ( z ) = - 2BK(z) f ( z ) ,
where the specific choice of constants on the right-hand side has been chosen for convenience. The important feature of the right-hand side of the equation is the product form; K ( z ) is simply some function of z. A particular solution of eq. (58)can be written in terms of a Green function g(z, z ’ ) for the onedimensional Helmholtz equation according to
where g(z, z f) for the one-dimensional Helmholtz equation is known to be ,ifllz-z’l
g(z, z ’ ) = ~,
2iB
(60)
subject to the requirement that only outgoing waves appear at z + + co. [Recall the time dependence used here is exp( - iwt).]
I , $ 51
31
COUPLING BETWEEN GUIDED WAVES
To simplify what follows it is convenient to define the right-hand side of eq. ( 5 8 ) as Q(z>,
Q(4=
- 2BK(z) f(z) .
(61)
The absolute value in eq. (60) makes it clear that eq. (59) can be written as the sum of two terms, one for z' > 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
WAVEGUIDE DIFFRACTION GRATINGS
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
WAVEGUIDE DIFFRACTION GRATINGS
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)
COUPLING BETWEEN GUIDED WAVES
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
WAVEGUIDE DIFFRACTION GRATINGS
[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.
COUPLING BETWEEN GUIDED WAVES
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
COUPLING BETWEEN GUIDED WAVES
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
WAVEGUIDE DIFFRACTION GRATINGS
[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
COUPLING BETWEEN GUIDED WAVES
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
WAVEGUIDE DIFFRACTION GRATINGS
[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
otherwise,
as in fig. 15. The boundary conditions we consider are such that for perfect Bragg matching, 6 = 0, A'(z)= 1 z < O , A-(z)=O z = L .
(116)
The conditions at z = 0 and z = L yield the solutions A (z) = +
- i6 sinh [ a(L - z)] exp( - i6z) , acosh(aL) - i6 sinh(aL)
a cosh [a(L - z)]
(1 17)
1.31
43
COUPLING BETWEEN GUIDED WAVES
51
and A - (z) =
i @) sinh [ a(L - z)] exp ( + i6z) , a cosh(aL) - i6 sinh(aL)
with
{ ~ ( 8 ) ’- 6’)’’.
(119)
We now examine the characteristics of these solutions. Figures 16 and 17 show plots of IA +(z)I2 and lA-(z)12 for rc(8)L = 2 for perfect Bragg matching (6 = 0) and an illustrative detuning (6 = 1.95/L), respectively. The increased detuning in fig. 17 results in a reduction in the amplitude of the reflected wave at z = 0 to approximately 80% and an accompanying increase in the forward wave amplitude at z = L in comparison with the perfectly Bragg-matched case of fig. 16. The grating reflectivity R is defined as
’
=
A - ( z = 0) l A + ( z = Od ’
6=0 0.8 ’.O/
R 0.6 v
s N
7
0.4
-
0.2 -
0.0
0.2
0.4
0.6
0.8
1 .o
Z/L Fig. 16. Plots ofA(z)A*(z) for forward-going ( t ) and backward-going( - )waves for rc(0)L = 2 and perfect Bragg matching, S = 0.
44
[I, § 5
WAVEGUIDE DIFFRACTION GRATINGS
0.0
0.2
0.6
0.4
0.8
1.o
Z/L Fig. 17. Plots ofA(z)A*(z) for forward-going ( + ) and backward-going ( - )waves for K ( @ L= 2 and sample detuning 6 = 1.95/L.
so that
Note that the form of eq. (121) must be changed to one expressed in terms of sines and cosines for ~ / K ( B> ) 1. Figure 18 shows a plot of the reflectivity R versus 6 / K ( e ) for the two cases K(B)L= 1 (dashed line) and K(B)L= 2 (solid line). The larger coupling strength produces the higher peak reflectivity, greater than 90% in this example. ) decreases the maximum reflectivity Reducing the coupling coefficient ~ ( 0both and broadens the spectral response, as one would expect. The detuning, for a fixed grating period, is a measure of the wavelength (or frequency) deviation from the Bragg-matched value. The Bragg-matched reflectivity R takes on the very simple form
R
=
tanh2(K(8)L) (6 = O),
(122)
plotted in fig. 19. The reflectivity for 6 = 0 saturates at unity for products of the coupling coefficient and the grating length greater than - 3 . The coupling coefficient,of course, depends on the angle of incidence Oaccording to eq. (1 15)
1, § 5J
45
COUPLING BETWEEN GUIDED WAVES
1.o
0.8
>.
t > F
0.6
!!L L $
0.4
0
0.2
6I
K(e)
Fig. 18. Plots of the grating reflectivity R , eq. (121), as a function of 6 / x ( 0 ) for tc(0)L = 1 (dashed line) and tc(0)L = 2 (solid line).
0
2
6
4 K(e)
8
L
Fig. 19. Grating reflectivity R as a function of the coupling strength K(0)L for the case ot pertect Bragg matching, S = 0.
46
WAVEGUIDE DIFFRACTION GRATINGS
[I, § 5
4
1
ANGLE 8 ( degrees ) Fig. 20. Normalized absolute value of the coupling coefficient I K(O)( as a function of the angle 0 (see fig. 15) for TE-polarized incident and diffracted guided waves (TE-TE). Note that ~ ( 0 ) has been set to unity for simplicity.
for TE-guided waves, a dependence that is illustrated in fig. 20 for the normalization IC = 1 at 8 = 0. The coupling coefficient vanishes at 8 = 45" and rises rapidly as 8- 90". Divergence of the coupling coefficient for 8 = 90" is expected, since this corresponds to grazing incidence for which the reflectivity should approach unity.
5.4. COUPLING BETWEEN TM-GUIDED WAVES
The coupling coefficient K ( 8)contains the essential information regarding the strength of the interaction between the guided wave and the surface corrugation. As mentioned earlier, the various theoretical approaches generally agree on the matter of the TE-coupling coefficient, eqs. (93) and (1 15). This was not the case until very recently for the TM-coupling coefficient. It is becoming clear, in fact, that eq. (94), the coupling coefficient obtained from the ideal-mode expansion, is incorrect. STREIFER,SCIFRESand BURNHAM[ 19761 were the first to recognize that calculations based on the ideal-mode expansion for T M modes in a waveguide with a corrugated surface led to certain difficulties. They found that various
COUPLING BETWEEN GUIDED WAVES
41
formulations of the problem led to very different values of K for the case of a large refractive-index difference at the corrugated boundary. SIPE and STEGEMAN [ 19791 then reported the results of a comparison (for B = 0) of the coupling coefficients obtained by the ideal-mode version of coupled-mode theory and by “total field analysis”, a theory that attempts to satisfy the boundary conditions at the corrugated surface in an explicit way. They found agreement for the TE case but disagreement for the TM case. STEGEMAN, SARID,BURKEand HALL[1981] generalized the “total field analysis” to arbitrary B and found general disagreement with earlier extensions of the ideal-mode analysis to arbitrary 8 for the TM case. GRUHLKEand HALL [ 19841 examined (for 13= 0) both the grating-reflection problem and the related problem of the radiation pattern produced by a guided wave interacting with a surface grating. They used a boundary perturbation technique that satisfies the boundary conditions at the corrugated surface to first order in the grating height. Again, the results agreed perfectly with those of the ideal-mode version of coupled-mode theory for T E polarization but disagreed for both problems for the TM polarization. MARCUSE[ 19741 describes two different formulations of coupled-mode theory, one of which is based on the ideal-mode expansion we have already discussed, and the other based on the so-called local normal mode expansion. Whereas the former expands the fields of the corrugated waveguide in terms of the fields of the uncorrugated mean waveguide, the latter expands them in terms of the fields for an unperturbed waveguide with the local thickness. The difference between the two is that the location (but not the slope) of the boundary of the perturbed waveguide coincides with that of the unperturbed waveguide for the local normal mode expansion, but not for the ideal-mode expansion. MARCUSE’S [ 19741 analysis for 8 = 0 shows that these two formulations predict different coupling coefficients for the TM polarization. MARCUSE’S[1974] local normal mode (LNM) analysis was recently generalized to arbitrary angle 6 by WELLER-BROPHY and HALL[ 19881. The predicted coupling coefficient ~ ( 6 (in ) pm- ’) appears as the solid curve in fig. 21 for an illustrative choice of parameters (n, = 1.0, n, = 1.56, n, = 1.47, h = 0.9 pm, L = 0.8 pm). The dashed curve, shown for comparison, is the prediction of the ideal-mode theory of WAGATSUMA, SAKAKIand SAITO [ 19791. The LNM analysis is in complete agreement with those theories that satisfy the boundary conditions (to first order in the corrugation height). The most striking feature in fig. 2 1 is the zero-crossing that occurs near 8 = 20” for the LNM theory, but does not occur for the ideal-mode theory. This suggests that an experiment that examines the grating reflectivity in the vicinity of the
48
[I, 8 5
WAVEGUIDE DIFFRA(JTI0N GRATINGS
e Fig. 21. Angular dependence ofthe absolute value of the coupling coefficient 1 K ( B ) I (in pm-') for TM-polarized incident and diffracted guided waves (TM-TM), as predicted using the local normal mode approximation (solid curve) and the ideal-mode approximation (dashed curve). The parameters used to make the plots are given in the text. The former shows a distinctive zerocrossing that is absent in the latter.
zero-crossing will be a good test of the two theories, since one theory predicts a very small value compared with the other [see eq. (122) for the relation between the reflectivity R and the coupling coefficient ~ ( e ) ] . WELLER-BROPHY and HALL[ 19871 reported the results of such an experiment. The waveguide and grating parameters were chosen so that, for the experimental conditions, the ideal-mode theory predicted a 100% reflectivity for both the TE and the TM cases, whereas the LNM theory predicted a 100% reflectivity for TE and a 13% reflectivity for TM. The comparison with experiment is summarized in table 1. TABLE 1 Comparison of measured and predicted grating reflectivities (in % ). ~
Theory
TE-TE TM-TM
Experiment
Ideal mode
Local normal mode
100
100
75
100
13
9
1,s 51
COUPLING BETWEEN GUIDED WAVES
49
The measured reflectivities agree better with the LNM result. Neither calculation included the effects of propagation losses in the waveguide, so the difference between the measured and calculated LNM values is not considered significant.The large difference between the measuredTE and TM reflectivities, however, lends strong support to the LNM version of coupled-mode theory as the more correct analysis. It appears, not surprisingly, that the boundary conditions on the corrugated surface must be handled carefully.
5.5. LOCAL NORMAL MODE EXPANSION AND COUPLED-MODE EQUATIONS
(TM)
The success of the local normal mode (LNM) expansion over the ideal-mode expansion in predicting the results of the measurement described in the previous section for the TM polarization raises the question of the essential difference between the two approaches to the grating reflection problem. Both Marcuse's original derivation for 0 = 0 (MARCUSE[ 19741) and the extension of this work to arbitrary 0 (WELLER-BROPHY and HALL[ 19881) are rather cumbersome, however. More importantly, the derivations are sufficiently different so that the connection with that for the ideal-mode approach can be difficult to make. Here, we make use of a new treatment of the LNM approximation that, hopefully, makes the comparison easier. The theoretical development in this section parallels that for the TE polarization presented in eqs. (loo)-( 115). We begin with the wave equation in eq. (loo), repeated here for convenience,
and consider the same geometry that appears in fig. 15. As before, A Edescribes the perturbation [see eq. (91)] introduced into the structure of the mean (or ideal) waveguide, as in fig. 14. Motivated by the earlier treatment of the TE problem, we write the field E in the form E ( x , y , z, t ) = f(z) e'fl1.YE,(x, z ) e-'"',
where E,(x, z ) gives the x-dependence of the electric field profile at a given position z in the perturbed waveguide. In the LNM approximation, E,(x, z ) is taken to be the field profile for the mode of an uncorrugated waveguide with the local (L) thickness, i.e. that for a given z. As before, we rearrange the wave
50
WAVEGUlDE DIFFRACTION GRATINGS
[I, § 5
equation so that the perturbation term appears on the right
(V2
+
$ tki)
k,Z E(x, y, Z, t ) .
E(x, y, z, t ) = -~
(125)
60
At this point, eq. (125) still contains the spirit of the ideal-mode expansion, since the permittivity E ( X , z) = E ( X ) + As(x, z) has been split into two parts that for the ideal waveguide [ E ( x ) ] and that for the perturbation [ A E ( x z)]. , The chosen form for the field in eq. (124), however, does not make use of the field profile appropriate for the ideal, uncorrugated waveguide. The essence of the LNM approximation, as treated here, is that the field E is treated differently on the left- and right-hand sides of eq. (125). In particular, the z-dependence in E,(x, z ) is neglected on the left-hand side, but is retained on the right-hand side. The right-hand side of eq. (125) drives the differential equation, and so great care must be taken to model it as well as possible. This means that the approximate fields in the perturbed regions must be handled carefully. We will return to this point soon. Since it is assumed from the beginning that the perturbation is relatively small, the propagation of the forward- and backwardpropagating waves should not be very different for the corrugated and uncorrugated waveguides. Therefore, we treat E,(x, z) on the left as independent of z, and indistinguishable from the field profile for the ideal [superscript (i)] waveguide. With this approximation on the left, eq. (125) becomes E"'(x)
[7 + p,' ] :z2
f ( z )x
-k :
r?)
f(z)EL(x, z ) ,
(126)
-
where we have assumed that E,(x, z ) E(')(x)satisfies eq. (102), consistent with the approximation. Next, we integrate the x-dependence out of eq. (126) by first forming the field
The quantities E$)(x) and @ ) ( x ) are the components of E(')(x)for a forwardgoing wave propagating along z (& = 0); designates a unit vector. The dot product in eq. (127) is positive (negative) for forward- (backward-) going waves. The amplitudef(z) consists of forward- and backward-going waves, as we have seen earlier in this chapter. The construction in eq. (127) allows us to project the forward-going field onto a backward-going ideal mode, and the backwardgoing field onto a forward-going ideal mode, to examine their mutual coupling.
51
COUPLING BETWEEN GUIDED WAVES
We accomplish this by forming the dot product of eq. (126) with the complex conjugate of eq. (127) and then integrating over x, and obtain
where K(z) has the familiar form K(z) = ;iw
’
S_a_
A E ( x ,Z) E,(x, Z)* {E$’(x)}* d x ,
(129)
and IN is a normalization integral given by
1
OCI
IN =
E“’(x) {E$’(x)}* d x .
(130)
--uo
-
The normalization integral I, is similar to that for TE modes, eq. (24). In fact, 2p0w/fi the normalization for TM modes given in eq. (25) implies that I, for the lowest-order TM mode. (For most cases IN rarely differs from 2p0w/B by more than 1% for the lowest-order TM mode, primarily because the integral is dominated by the term involving the transverse components of the fields.) Equation (128) becomes
which has the same form as that in eq. (104) for TE modes. From this point, the analysis proceeds just as before. The Green function technique is used to develop coupled-mode equations, which are then solved using the synchronous approximation [see above eq. (1 13)]. The all-important coupling coefficient is obtained by writing K ( z ) = 2 K cos (Koz), so that
where we recall that the factor l/cos(8) is contributed by the Green function, since it is proportional to 1//?,. K ( z ) must be evaluated carefully. E,(x, z) is the model field of an uncorrugated waveguide with the local thickness, whereas E(’)(x)is described in terms of the modes of the ideal (mean) waveguide. This means that each field must be placed in the correct medium, a point made in the paper by STEGEMAN, SARID,BURKE and HALL[ 19811. Table 2 attempts to make the distinction
52
[I, 8 5
WAVEGUIDE DIFFRACTION GRATINGS
TABLE 2 Comparison of ideal-mode and local normal mode treatments for the TM-TM interaction (P = perturbed; U = unperturbed).
Ideal mode*
Local normal mode*
A& = &,(n: - n:)
h
n,,
Ept(x, Z)
n
=
E&,
zf
n
= ncr
Ept(x, Z )
n
E p A Z)
n = n,,
= nr.
Ep,(x, z ) x E?(h)
n = n,,
E,,(x, z ) x E f ) ( h )
Ep,(x, z) x (n,/n,)’E$)(h)
n
n,,
E,,(x, z ) x Ej“(h)
E,,(x, z ) PZ E l i ) ( h )
n = no. E p t ( x ,z ) x E!’)(h)
E,,(x,z)
n = n,,
PZ
(n,/n,)*E$)(h)
=
E,,(x, z ) x E$)(h)
t and z designate the transverse and z-components of the vector fields.
between the ideal-mode and local normal mode approaches clear. For regions d > h, for which the actual surface of the perturbed waveguide extends beyond the mean surface at x = h, the field is approximated by that of a waveguide with constant thickness dusing n = n,. The same is true ford < h, except that n = n,. The correct refractive index is assigned to each perturbation region for the purpose of determining the field. The ratios n,/n, and n,/n, that appear in the expressions for the z-components of the fields in table 2 for the ideal mode case are the result of assigning a refractive index other than the actual one to each perturbation region. When evaluated properly, eq. ( 132) becomes n Ah n , Z - N 2 1
_____-
.(O)
=
1 he*
N
40
cos 0
(TM - TM) ,
COUPLING BETWEEN GUIDED WAVES
53
for the cosine surface grating of eq. (90). Note that qc was defined in eq. (32), and all quantities should be evaluated for the lowest-order TM mode. Equation (133) is in complete agreement with that obtained by WELLERBROPHYand HALL[1988], and is also in agreement with that obtained by STEGEMAN. SARID,BURKEand HALL[ 19811 after a few minor algebraic corrections are made. A comparison with Kogelnik’s result, eq. (94), for 8 = 0 shows agreement, provided the quantity in curly brackets in eq. (94) is set to unity, as mentioned earlier. Indeed, it is eq. (133) that is plotted in fig. 21 (solid line) and has so far shown good agreement with experimental results (WELLERBROPHY and HALL [ 19871). Equation (1 15) is obtained when the method discussed in this section is applied to the TE-TE Bragg reflection problem; thus, the method is both straightforward and reliable. The ideal mode approach to the grating reflection problem differs from the treatment given in this section in two ways. First, the LNM approach does not weight the z-component of E differently from the transverse components of E, as is the case for the ideal-mode approximation. This follows as a natural consequence of the assumption that E,(x, z ) is evaluated as the mode of an ideal waveguide with the local thickness. Second, the net etrect of treating E , ( x , z) this way is that products of the field components in eq. (129) contain one field for each medium, the cover and the film, a feature not present in the ideal-mode approach. The success of the various versions of the LNM approximation highlights the importance of treating the field very carefully in the perturbation term in the wave equation.
5.6. SUMMARY OF COUPLED-MODE TREATMENTS
The previous sections reveal several important points about coupled-mode formulations. First, derivations of the coupled-mode equations, eq. (1 13) and (1 14), often make use of a slowly varying envelope approximation, in which the second derivatives of both the forward- and backward-wave amplitudes with respect to z are neglected. This is a completely unnecessary approximation. Both the Green function approach and other formulations arrive at the proper equations without invoking such an approximation. Second, some researchers have asserted that the case of non-normal incidence to the grating “rulings” cannot be treated by solving the wave equation directly (POPOVand MASHEV [ 1985a,b]). They argue that one must begin the analysis at a low level, so to speak, with Maxwell’s curl equations. The Green function method discussed here demonstrates that this statement is incorrect. Third, the flaw in the
54
WAVEGUIDE DIFFRACTION GRATINGS
[I, 0 5
reasoning behind the ideal-mode expansion treatment of the grating-reflection problem was revealed to be an improper treatment of the fields in the region of the surface corrugation. The wave equation contains a perturbation term proportional to ( A E E, ) in which E must be considered to be a field in the actual medium for the perturbed structure, one for which the permitivity is E ( X ) t AE(x,z). The ideal-mode expansion violates this to produce two errors, an improper treatment of the z-component of the field [see eq. (84)] and an improper evaluation of the field within each perturbation region in the coupling coefficient. The LNM expansion suffers from neither of these difficulties, since the boundary for the local normal mode E,(x, z ) occurs at the same location as, but with different slope than, that for the actual waveguide field. Previous derivations of the LNM results (MARCUSE[ 19741,WELLER-BROPHY and HALL [ 19881) obtain the correct coupled-mode equations and coupling coefficients for both the TE-TE and TM-TM cases, but the Green function technique obtains the same results for the lowest-order (m = 0) modes with much less labor and in a more direct manner from the wave equation.
5.7. PERTURBATIVE TREATMENT
The previous treatments of the problem of guided waves interacting with a surface-corrugation grating have considered the coupling per unit length between the incident and Bragg-reflected waves to be weak. This is implied in the restriction that the ratio Ah/h or Ah/A is small, which allows the coupling coefficient ~ ( 0 to ) be expressed in a relatively compact form. These same treatments, however, allow the total interaction to be large so that the depletion of the incident wave cannot be neglected. The weak coupling between the forward- and backward-going waves that occurs within any single period of the corrugated waveguide can build up coherently when the Bragg condition is at least nearly satisfied. In this way, even a weak interaction can produce a nearly 100% conversion between the two waves in a finite, but sufficiently long, structure. Thus we see that the coupling coefficient describes the interaction per unit length, whereas the coupled amplitude equations describe the relative amplitudes of the forward- and backward-going waves. A few authors have attempted to reduce the complexity of the waveguide grating problem by separating the two main parts of the problem (STEGEMAN, SARID,BURKEand HALL[ 19811). First, the coupling coefficient is determined in the weak-scattering limit in which one ignores the depletion of the incident wave. The coupling coefficient is subsequently inserted into a pair of coupled-
1, § 51
COUPLING BETWEEN GUIDED WAVES
55
mode equations to obtain the full solution including depletion. This approach offers the advantage that ~ ( 6can ) be determined, in principle, to arbitrary precision by using a power-series expansion in, e.g., Ahlh. It has the disadvantage, however, that the coupled-mode equations must be obtained separately, a process that has led to errors in the past when these equations have been obtained rather intuitively (STEGEMAN, SARID,BURKEand HALL[ 19811). The technique for obtaining the coupling coefficient for one such perturbation approach is illustrated in this section. TUAN[ 19731, TUANand O u [ 19731, and TSAIand TUAN[ 19741 used a surface-perturbation theory formulated by CHEN [ 19681 to examine the scattering of guided waves by a single groove or deformation in the surface of an otherwise unperturbed planar optical waveguide. The technique is based on an expansion of the scattered field and the boundary conditions in power series in the parameter Ahlh, which is presumed to be small. Their theoretical work does not treat the grating problem explicitly, although it also applies to that case. This discussion will assume that the surface-corrugation waveguide grating in fig. 14 is the structure of ultimate interest. HALL[ 19801 has shown that for the problem of a guided wave that radiates due to the interaction with a surface structure, the first-order (in Ah/h) boundary perturbation method leads to the same result as the coupled-mode theory of MARCUSE[ 19741 for the TE polarization, but to different results for the TM polarization (GRUHLKE and HALL[ 19841). Again, the disagreement for the TM case is due to the same shortcoming on the part of the ideal-mode expansion, discussed in the previous section. The boundary-perturbation method gives the correct result. The boundary-perturbation theory was originally formulated in two dimensions, but was later extended to three dimensions (HALL[ 19811). Since this section merely aims to outline the methodology, the simpler two-dimensional case will be considered; i.e., we will examine the Bragg reflection of a guided wave incident on a grating at normal incidence (0 = 0 in fig. 15). The grating length L is taken to be sufficiently small so that depletion of the incident wave can be neglected. The more difficult TM-TM case will be considered here to make a strong connection with the content of the previous section, although the theory also works well for the TE polarization. The basic geometry is shown in fig. 9. The upper surface x = d of the waveguide is taken to be of the form d=h{l
+ qp(z)}
forO
(134)
and uncorrugated with d = h, otherwise. The parameter is small, and p(z) describes the surface perturbation. The magnetic field for the TM polarization
56
WAVEGUIDE DIFFRACTION GRATINGS
is oriented along the y-direction,
H
=
yHy.
(135)
The field Hy is expanded as the sum of two parts, Hy
=
(136)
Hy, inc + Hy. scatt *
The incident fields, labeled "inc", are taken to be those given in eq. (lo), the fields for the unperturbed waveguide of thickness h, for a forward-going wave according to
Hy,inc = Hm(x) ~ X [P~ ( P z-
(1 37)
The scattered field in medium j is expanded in a power series in the small parameter q,
where H,, is the nth-order scattered field, and j = s, f, or c in the substrate, film, or cover regions, respectively. For a surface grating of the form of interest here, p(z) = cos(K,z) and = Ah/h. The boundary conditions on the tangential components of E and H can both be expressed in terms of H by using the unit normal un to the corrugated surface. Both H,, and ( ~/E)u,* VHy must be continuous across the corrugated interface between the cover and film media. This is difficult to accomplish exactly, but the boundary conditions can be expanded in a power series in q and satisfied up to a specified order in q. The operator u, V can be written in terms of the surface profile as
When the boundary conditions are satisfied through first order in q, we obtain
Hs'(h, Z)- Hif!(h, Z) = h p ( z )
{H$!)iinc(~,Z) - H:),.(X,
z)l]lx=h
3
M,(z)
3
(140)
and
o
COUPLING BETWEEN GUIDED WAVES
1, 51
51
where
As before, the superscripts (c) and (f) on the incident fields designate the medium in which eq. (137) will be evaluated. The bottom surface of the waveguide at x = 0 is taken to be unconugated, of course, so that the first-order boundary conditions produce the much simpler requirements
Hi?(O, z ) = H??(O, z ) ,
(143)
and
Equations ( 140)-( 144) can be solved for the unknown first-order scattered fields by introducing plane-wave expansions for H,, in all three media. We write
m
Hi?(x,
2) = 2A
[V1(t)ei'rx
+ V2(()e-iCfX]e"'dt,
(146)
--Q)
where
These reduce eqs. (140)-( 144) to a set of algebraic equations that can be solved for the amplitudes U,V , , V,, and W. The procedure is rather tedious, but the integral expressions for the fields are versatile. The radiation fields can be determined very easily in the far-field by applying the method of steepest descents. The field for the reflected field in the
58
WAVEGUIDE DIFFRACTION GRATINGS
11.5 5
waveguide can be obtained by noting that the integrals contain simple poles at 5 = p. The pole at = - p gives the field for the “backscattered” guided wave. Again, the process is rather tedious, but the result is quite simple. For example, the first-order scattered field within the waveguide layer is obtained from eqs. (138) and (146) to be
<
(149)
where H,,,(x) refers to the field in the film region 0 c x < h [see eq. (lo)]. A comparison with eq. (133) reveals that the quantity in square brackets is just ) for O = 0, as required by the twothe coupling coefficient ~ ( 0 evaluated dimensional geometry (implied normal incidence) considered here. The field is, then, just
This means that the back-reflected wave has the same spatial profile as the incident wave, and has strength proportional to K(O)L. ~ ( 0can, ) therefore, be interpreted as the fraction per unit length of the incident field coupled from the forward-going incident wave into the backward-going reflected wave, consistent with the interpretation earlier in the chapter. The agreement between ~ ( 0in) eq. (150) and eq. (133) is illuminating. The former was obtained by satisfying the boundary conditions to first order in the presumed small parameter Ah/h. The latter was obtained from coupled-mode theory, which typically makes no explicit use of the boundary conditions. The afore-mentioned agreement gives some confidence that eq. (133) can be relied upon, provided Ah/h is small.
5.8. TE-TM MODE CONVERSION
The previous sections focused on the issue of a correct formulation of the TE-TE and TM-TM Bragg interactions in a corrugated optical waveguide. These are not the only possibilities, however, since for non-normal incidence TE-TM mode conversion also occurs in an optical waveguide. That is, for incident angle fIi # 0, a TE- or TM-guided wave incident on a corrugated section of an optical waveguide can, in first order, generate a TM- or TE-
I , § 61
59
SUMMARY
reflected guided wave if the B r a g condition
is at least nearly satisfied. In this case the incident and reflected guided waves will propagate at the different angles Bi and 0, with respect to the grating normal, since the effective index of refraction (NTE)for the TE polarization differs from that (NTM)for the TM polarization in a given waveguide, even though both are of the same vacuum wavelength 1.The local normal mode approximation also can be used to give a satisfactory result for this case (WELLER-BROPHY and HALL[ 19881). Only the result for the coupling coefficient is given here (note the dependence on two angles),
where we have used the notation of WELLER-BROPHY and HALL[ 19881so that
There is no TE-TM mode conversion for normal incidence
ei = 0, = 0.
5 6. Summary This chapter has considered the interaction between the modes of a planar optical waveguide and a periodic surface corrugation, which is an important interaction in many applications. The specific case of the grating-induced coupling between two guided waves in a planar waveguide structure received the principal emphasis here, since it has been the subject of some controversy over the last ten years, a controversy that has only recently been resolved. The physical nature of the interaction is well understood, namely, as being due to Bragg scattering. The quantitative details are complicated, however, by the mode structure characteristic of even an elementary planar optical waveguide. Approximations that work very well and lead to excellent agreement with each other for TE-polarized guided waves disagree with each other significantly for TM-polarized guided waves. A recent experiment was able to distinguish between classes of theories. Those theories based on the familiar ideal-mode
60
WAVEGUIDE DIFFRACTION GRATINGS
[I
expansion discussed in several textbooks and monographs disagree with the measured results for the TM polarization. Those theories based on either the local normal mode expansion or boundary perturbation techniques agree well with each other and with the experiments. The local normal mode theory makes no explicit consideration of the boundary conditions at the corrugated interface. The boundary perturbation techniques, in contrast, satisfy the boundary conditions up to a desired order in the grating height. It is interesting that two such different approaches should agree so well. After examining the fundamental principles of importance for a variety of optical waveguides, discussion turned to the various techniques used to attack the problem of a guided wave interacting with a waveguide diffraction grating. An attempt was made to formulate the various approaches to enable a comparison among the various treatments. It emerged that the deficiency in theories based on the ideal-mode expansion can be attributed to an improper treatment of the electric field in the perturbation regions. In essence, these theories ascribe the wrong polarization (in the dipole sense) to the perturbation regions by consistently embedding the approximate fields in the wrong media. A new, very direct formulation of the local normal mode approximation avoided the complexities of earlier versions of the theory and make it relatively easy to identify the principal features of the approximation that make it so successful. We now appear to have at our disposal a theoretical description of the guided-wave Bragg-reflection problem that can be relied upon, at least for the case of shallow surface corrugations. Equally importantly, our understanding of the way in which the problem must be treated has improved. This will likely be of benefit in future treatments of scattering problems in optical waveguides, particularly those that involve TM-polarized guided waves.
List of Symbols
asymmetry parameter forward-wave amplitude backward-wave amplitude speed of light in vacuum power parameter electric field electric field profile transverse component of E,(x) field profile for ideal waveguide
11
LIST OF SYMBOLS
local normal mode E,(x) at x = h E,(x) at x = 0 maximum value of E,(x) unit vectors nonlinear field profile Green function waveguide thickness effective waveguide thickness magnetic field magnetic field profile transverse component of H,(x) H,(x) at x = h H,,(x) at x = 0 maximum value of H,(x) field profile for ideal waveguide normalization integral coupling integral transverse- and z-parts of K,,,(z) grating constant ( 2 4 A ) field profile parameter length of corrugated region mode integer or polarization index effective index of refraction substrate refractive index cover refractive index film refractive index x-dependent refractive index nonlinear coefficient TE mode effective index TM mode effective index TM mode parameters reflectivity time-averaged Poynting vector unit surface normal nonlinear waveguide parameter
B Y c , Ys
propagation constant field decay constants
61
WAVEGUIDE DIFFRACTION GRATINGS
detuning parameter index offset parameter permittivity perturbation surface grating amplitude x-dependent permittivity permittivity of free space surface height parameter propagation angle designates unit vector coupling coefficient wavelength in vacuum grating period permeability of free space surface profile TIR half-phase shifts (TE) TIR half-phase shifts (TM) (angular) optical frequency
References ADAMS,M. J., 1981, An Introduction to Optical Waveguides (Wiley, New York). CHEN,Y. M., 1968, J. Math. Phys. 9, 439. CONWELL, E. M., 1973, Appl. Phys. Lett. 23, 328. DAKSS,M.L., L. KUHN,P. F. HEIDRICH and B. A. Scorr, 1970, Appl. Phys. Lett. 16, 523. DALGOUTTE, D. C., 1973, Opt. Commun. 8, 124. EVANS, G. A,, J. M. HAMMER, N. W. CARLSON, F. R. ELIA,E. A. JAMESand J. B. KIRK,1986, Appl. Phys. Lett. 49, 314. FLANDERS, D. C., H. KOGELNIK, R. V. SCHMIDT and C. V. SHANK,1974, Appl. Phys. Lett. 24, 194. FUKUZAWA, T., and M. NAKAMURA, 1979, Opt. Lett. 4, 343. GRUHLKE, R. W., and D. G. HALL,1984, Appl. Opt. 23, 127. GRUSS,A,, K. T. TAMand T. TAMIR,1980, Appl. Phys. Lett. 36,523. HALL,D. G., 1980, Appl. Opt. 19, 1732. HALL,D. G., 1981, Opt. Lett. 6, 601. HALL,D. G., 1987, in: Integrated Optical Circuits and Components, ed. L. D. Hutcheson (Dekker, New York). HANDA, Y., T. SUHARA, H. NISHIHARA and J. KOYAMA, 1980, Opt. Lett. 5, 309. HATAKOSHI. G., and S. TANAKA, 1978, Opt. Lett. 2, 142. HOCKER, G.B., and W. K. BURNS, 1975, IEEE J. Quantum Electron. QE-11, 270. KOGELNIK, H., and C. V. SHANK,1971, Appl. Phys. Lett. 18, 152. KOGELNIK, H., and C. V. SHANK,1972, J. Appl. Phys. 43, 2327. KOGELNIK, H. G., 1975, in: Applied Physics, Integrated Optics, Vol. 7, ed. T. Tamir (Springer, New York).
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REFERENCES
63
K O G ~ L N IH.K G., , 1988, in: Electronics and Photonics, Guided-Wave Optoelectronics, Vol. 26. ed. T. Tamir (Springer, New York). KOHWI'KY, S. K., and R. C. ALFERNESs, 1987, in: Integrated Optical Circuits and Components, ed. L. D. Hutcheson (Dekker, New York). KURMER, J. P., and C. L. TANG,1983, Appl. Phys. Lett. 42, 146. and J. M. DELAVAUX, 1981, IEEE Trans. LIN,Z., S. ZHOU.W. S. C. CHANG,S. FOUOUHAR Microwave Theory and Tech. MTT-29, 88 I . LIVANOS, A. C., A. KATZIR,A. YARIVand C. S. HONG,1977, Appl. Phys. Lett. 30, 519. S.H., J. S. M o m , R. J. NOLL,G. M. GALLATIN, E. J. GRATRIX, S. L. OIDWYER and MACOMBER, S. A. LAMBERT, 1987, Appl. Phys. Lett. 51, 472. MARCUSE, D., 1974, Theory of Dielectric Optical Waveguides (Academic Press, New York). MILLER, S. E., 1969, Bell Syst. Techn. J. 48(7), 2059. NAKAMURA, M., A. YARIV,H. W. YEN,S. SOMEKH and H. L. GARVIN, 1973, Appl. Phys. Lett. 22, 5 15. PENNINGTON, K. S., and L. KUHN,1971, Opt. Commun. 3, 357. 1985a, Opt. Acta 32, 265. POPOV,E., and L. MASHEV, POPOV,E., and L. MASHEV,1985b, Opt. Acta 32, 635. J . B., C. S. HONG and A. YARIV,1977, Opt. Commun. 23, 298. SHELLAN, SIPE,J. E., and G. I. STEGEMAN, 1979, J. Opt. SOC.Am. 69, 1676. SOMEKH, S., and A. YARIV, 1972, Opt. Commun. 6, 301. G . I., D. SARID,J. J. BURKEand D. G. HALL,1981, J. Opt. SOC.Am. 71, 1497. STEGEMAN, STEGEMAN, G. 1.. J . J. BURKEand C. T. SEATON,1987, in: Integrated Optical Circuits and Components, ed. L. D. Hutcheson (Dekker, New York) and references therein. STONE,F. T., and S. AUSTIN,1976, IEEE J. Quantum Electron. QE-12, 727. 1975, IEEE J. Quantum Electron. QE-11, STREIFER, W., D. R. SCIFRESand R. D. BURNHAM, 867. STREIFER, W., D. R. SCIFRES and R. D. BURNHAM, 1976, IEEE J. Quantum Electron. QE-12, 74. SUHARA, T., and H. NISHIHARA, 1986, IEEE J. Quantum Electron. QE-22, 845. TSAI,T. L., and H. S. TUAN,1974, IEEE J. Quantum Electron. QE-10, 326. TUAN,H. S., 1973, IEEE Trans. Antennas & Propag AP-21, 351. TUAN,H. S., and C. H. OU, 1973, J. Appl. Phys. 44, 5522. K., H. SAKAKIand S. SAITO,1979, IEEE J. Quantum Electron. QE-15, 632. WAGATSUMA, WELLER-BROPHY, L. A,, and D. G. HALL,1987, Opt. Lett. 12,756. WELLER-BROPHY, L. A., and D. G. HALL,1988, J. Lightwave Tech. LT-6,1069. YAMAMOTO, Y., T. KAMIYA and H. YANAI,1978, IEEE J. Quantum Electron. QE-14, 245. YARIV,A., 1973, IEEE J. Quantum Electron. QE-9, 919. YI-YAN, A., C. D. WILK~NSON and P. I. R. LAYBOURN, 1980, IEEE J. Quantum Electron. QE-16, 1089.
This Page Intentionally Left Blank
E. WOLF, PROGRESS IN OPTICS XXIX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991
I1
ENHANCED BACKSCATTERING IN OPTICS BY
Yu. N. BARABANENKOV Center for Surjhce and Vacuum Studies 117313 Moscow. USSR
Yu. A. KRAVTSOV Academy of Sciences of the USSR General Physics Institute, GROT Vavilov St. 38 117942 Moscow, USSR
V. D. OZRIN Academy of Sciences of the USSR Nuclear Safety Institute B . Tulskaya 52 113191 Moscow, USSR
A. I. SAICHEV Lobachevski State University 603600 Nizhni Novgorod, USSR
CONTENTS PAGE
§ 1.
INTRODUCTION
. . . . . . . . . . . .
.
.
. .
. . . 67
$ 2 . ENHANCED BACKSCATTER FROM SOLIDS IMMERSED IN A TURBULENT MEDIUM. . . . . . . . . . . . . .
69
$ 3 . ENHANCED BACKSCATTERING BY A RANDOM MED I U M . . . . . . . . . . . . . . . . . . . . . . . . . 123
5 4. MULTIPATH COHERENT EFFECTS IN SCATTERING FROM A LIMITED CLUSTER OF SCATTERERS . . . . 168 $ 5 . ENHANCED BACKSCATTERING BY ROUGH SURFACES . . . . . . . . . . . . . . . . . . . . . . . . 183 § 6.
RELATED EFFECTS IN ALLIED FIELDS OF PHYSICS
. . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . $ 7 . CONCLUSION.
. 186
. . . .
.
. .
.
. . .
.
.
. .
189
. 190
. . . . .
190
6 1. Introduction Propagation of light in a turbulent atmosphere or another random medium gives rise to a variety of fluctuation effects caused by the random inhomogeneities in the medium. As a rule, these effects degrade the radiation by corrupting its coherence, broadening the beam, and decreasing the intensity. In the last two decades, a qualitatively new class of fluctuation effects has been observed, caused by the fine coherence effects that arise in a double passage of waves through the same inhomogeneities of the medium. These effects result in some ordering of the scattered radiation rather than its degradation. The manifestation of coherent effects in multiple scattering has been suggested by WATSON[ 19691 with reference to a private communication from Ruffine. The Ruffine-Watson coherence effects arise in the multiple scattering of a wave from a large number of discrete scatterers. We shall refer to this class of phenomena as (multipath) coherent effects. According to Watson and Ruffine, to any closed scattering path (Os, s2, . .. s,O in fig. 1.la) connecting the source at point 0 with the receiver placed at the same point 0 there corresponds an opposite path Os, . . . s2s10, such that the fields in the direct path, uo,2 .. and reverse path, uOn .,, 210, are coherent for arbitrarily located scattering centers s , , s2, . . ., s,. When the source 0 and the receiver 0’ are separated (fig. l.lb), the fields uo,2 and uOn 210, are no longer identical and lose coherence for sufficiently distant 0 and 0‘. Watson treats points s,s2 . . . s, as centers of infinitesimal elements of the random medium, which are summarized in integrating over the entire volume of the medium. This line of reasoning has been continued by DE WOLF [ 19711, who applied it to the description of waves backscattered from small-scale inhomogeneities of a turbulent medium. Although this effect proved to be exceedingly weak in a turbulent medium, de Wolfs’ theory has served as a jumping-off place for considering a more realistic problem of backscattering from solids embedded in such media (BELENKIIand MIRONOV[1972], VINOGRADOV, KRAVTSOVand TATARSKI~ [ 19731). This effect is observed as a higher intensity of a wave scattered strictly backwards in a turbulent medium rather than that of a wave backscattered in ,,,
61
_,_
68
ENHANCED BACKSCATTERING IN OPTICS
Fig. 1.1. (a) When the locations of the transmitter and receiver coincide, the forward and reverse scattering paths are identical and the respective fields are coherent. (b) Separation of the transmitter and receiver breaks down the coherence between the paths Os,s, ... s,O’ and 0s”
... S,S,O’.
a homogeneous medium. This was the first of a series of effects in which a random medium produced an enhanced rather than a degraded intensity. BARABANENKOV[1973, 19751 has interpreted the coherent paths in a medium with small-scale inhomogeneities by means of scattering diagrams. When the point of observation, 0’,is brought into the location 0 of the source, the contribution of cyclic diagrams becomes equal to that of ladder diagrams ; therefore, the intensity peak scattered backwards by a random inhomogeneous medium is about twice as large as the intensity of backscatter. Cyclic diagrams have proved to be a very useful interpretation tool that has been widely accepted for the analysis of enhancement in multiple scattering of waves. More recently, the effect of backscatter enhancement has been observed experimently by KUGAand ISHIMARU [ 19841, WOLFand MARET[ 19851, and some other workers. It has been reported under the name of weak localization, borrowed from the theory of electron scattering in metals. It is worth noting that in metals the weak-localization effect has a very small magnitude because of the strong Coulomb interaction of electrons. This effect is stronger for photons, which do not interact with one another, and it leads to a marked enhancement of backscattering. In one-dimensional, random-inhomogeneous media, coherent effects are of major importance for propagation of waves and lead to phenomena similar to the strong-localization regime of electrons in solids predicted by ANDERSON [ 19581. Specifically, GAZARYAN [ 19691 and other workers (see, e.g., KLYATSKIN [ 19751) have found that the transparency of a one-dimensional, randomly inhomogeneous slab falls off exponentially with thickness in much the same way as the wave function of electrons does under strong localization, whereas the phenomenological transport theory predicts a much weaker power law.
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
69
The effects of backscatter enhancement and weak localization head the list of phenomena in which coherent paths of the Watson-Ruffine type are essential. In this survey we intend to describe multiple new manifestations of coherent effects in backscattering. These include the effects of long-distance correlations, the partial reversal of the phase front in a random medium, the magic cap effect, antispecular scattering by very rough random surfaces, and bsckscattering involving surface waves. This survey will replace our previous review papers (KRAVTSOVand SAICHEV [ 1982b, 19851) and a recent monograph by BANAKHand MIRONOV [ 19871 on lidar sounding of a turbulent atmosphere, which unfortunately have become obsolete and need to be updated. We hope that this publication will be useful not only for researchers in optics but also for workers in other fields of wave physics. We would like to use it to acquaint researchers in the West with relevant studies made in the Soviet Union and not known to our western colleagues. We became aware of the need for updated information on the subject during the Tallinn workshop in 1988, which was organized by V. I. Tatarskii and A. Ishirnaru. Personal contacts in this workshop stimulated us to prepare this review.
8 2.
Enhanced Backscatter from Solids Immersed in a Turbulent Medium
2.1. ABSOLUTE EFFECT OF ENHANCED BACKSCATTER: A POINT
TRANSMIM'ER A N D A POINT SCATTERER IN A TURBULENT MEDIUM
2.1.1. Pure effect of enhanced backscatter Consider monochromatic waves propagating in a medium with dielectric constant E = 1 + 1. Assume that the inhomogeneities are weak enough, i.e., ] 5 I 4 1, statistically uniform and isotropic, and the characteristic scale I, of inhomogeneities is large compared with the wavelength I , i.e., 1, % 1. These conditions are typical of light waves propagating in a turbulent atmosphere. In such a medium, scattering occurs predominantly in the direction of propagation. Therefore, a description of such waves may be based on a scalar approximation. An optical wave backscattered from solids immersed in a medium with large-scale inhomogeneities passes through the same inhomogeneities which it has passed through in the forward direction (fig. 2.1). The double passage of a wave through the same inhomogeneities gives rise to the effect of backscatter enhancement (BSE). This effect occurs in media whose inhomogeneities
70
ENHANCED BACKSCATTERING IN OPTICS
Fig. 2.1. For a point ofobservation rplaced near the transmitter r,, the scattered radiation travels back through the same inhomogeneities through which the forward wave has passed.
remain virtually unchanged during the wave travelling from the transmitter to the scatterer and back. The effect of time-dependent variations of the inhomogeneities upon the field of the reflected wave will be discussed here; at the moment we shall assume that they are time-invariant, i.e., 5 = 5(r). In this case the Green function G ( r l ,r 2 )describing the field of a scalar monochromatic wave obeys the path reciprocity theorem
w,,r2)
=
G(r2, r1) *
(2.1)
This relationship is of principal significance in describing the BSE and other effects of double passage of waves. Let a point transmitter at point r = r, produce a primary field u ( r ) = G ( r t ,r). When this wave is incident on a point scatterer at point r,, it gives rise to the scattered field us(4 = f G (rt, r s ) G (rs,r ) ,
where f is the amplitude of scattering. Thus, from eq. (2. l), u,(r) = fG(rs9 r,) G(r,, 4
.
At an arbitrary point r the intensity of the scattered field is
Is(r) = aI(r,, r t ) I ( r s rr ) ,
(2.2)
where a = 1 f I is the scattering cross section in vacuum, and I(rs, r ) is the intensity of a point source at ro observed at point r, I(r,, r ) =
IW,,dI2.
We normalize the intensity I,(r) to the intensity I,O(r) of the scattered wave in a homogeneous medium, =
oIo(lrt - r s l ) I o ( l r - r s l ) ,
1 1 9 5 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
71
where J ( r , , r ) = I ( r , , r)/Io(I r - r, I ) is the relative intensity of the wave emitted from the location of the scatterer. The effect of enhanced backscattering can be derived from elementary considerations. We note that from the conservation of full flux of energy for the average intensity of a wave emitted by a point isotropic source, it follows that ( I @ , , r ) ) = M l r - r,I), so that
( J ( r , ,4 )
=
1.
(2.4)
From eq. ( 2 . 3 )the average relative intensity of a backscattered wave observed at the location of the transmitter, r = rt, is (J,(rt))
=
( J 2 ( r , ,rt)> .
The mean square, i.e., (J’(r,, rt)), always exceeds the square of the mean, i.e., ( J ( r s , rt))*, which is unity in view of eq. (2.4). Therefore, the backscatter enhancement factor Kbsc is always larger than unity, Kbsc
=
Wt)/CYrt)
=
(J&t))
=
( J 2 ( r s ,rt)) >
(W,,rt))2 = 1 .
(2.5)
For a turbulent medium the BSE effect was first predicted by BELENKIIand MIRONOV[ 19721 and VINOGRADOV, KRAVTSOVand TATARSKII [ 19731. A typical dependence of K upon the distance between the transmitter and scatterer in a turbulent medium is given in fig. 2.2 with three regions indicated for weak, strong, and saturated fluctuations of intensity. Mechanisms leading to these regions have been discussed by ISHIMARU[ 19781, RYTOV,KRAVTSOV and TATARSKII [ 1989a,b], YAKUSHKIN[ 19851, BANAKH and MIRONOV [ 19871, and MARTINand FLATTE [ 19881. As L = I Y, - r,l tends to infinity, the enhancement factor asymptotically approaches K ( c o ) = 2, and in the region of focusing K can somewhat exceed this value.
0
L=lr,-r,J
Fig. 2.2. Dependence of the backscatteringenhancementfactor Kbsc = K(rt, rs) on the distance between the transmitter and scatterer in a turbulent medium. (a) Region of weak fluctuations of intensity; (b) region of strong fluctuations caused by random focusings; (c) region of saturated fluctuations.
72
ENHANCED BACKSCATTERING IN OPTICS
111, § 2
2.1.2. A phase screen Random inhomogeneities of a medium inducing the fluctuations in intensity should not necessarily fill all the path from source to scatterer. They may fill, e.g., a narrow layer in this path. The effect of this layer on a wave can be described by the phase-screen approximation (VINOGRADOV, KRAVTSOV and TATARSKII [ 19731). Fluctuations in the intensity of plane waves having passed through a random phase screen distorting the phase of a passing wave have been studied by SALPETER[1967], ALIMOVand ERUKHIMOV [1973], and SHISHOV [ 19741. A recalculation ofthese results for the case of spherical waves is straightforward (ISHIMARU [ 19781). If a point scatterer is in the focusing region of a wave passing through the screen, the enhancement factor is calculated to be considerably above the asymptotic value K = 2 characteristic of a medium with three-dimensional inhomogeneities. A rough estimate for this case may be obtained as K z In o,; where o i is the variance of the phase distortions caused by the screen. 2.1.3. Spatial redistribution of the scattered intensity
Enhancement of average backscatter intensity does not contradict the conservation laws because it is accompanied by the reduction of scattering sideways. In other words, a spatial redistribution of the average intensity takes place. Consider the average relative intensity ( J , ( r ) ) of a scattered wave at points r = r, + p of a sphere with radius L = 1 r, - r, I centered on a scatterer, as shown in fig. 2.3a. By definition, ( J , ( r ) ) represents the enhancement factor en route r, -+ r, -+ r = r, + p, so that on the sphere we have K(p, L ) = ( J , ( r ) ) =
( J ( r , , r t ) J ( r s ,4 )
.
(2.6)
From the conservation of the full flux of energy of the scattered wave it follows that the average of (J,) = K over the sphere is (4nL2)-
$
K (p , L ) ds = 1 .
(2.7)
For a wave scattered strictly in the backward direction, K ( 0 , L ) = Kbsc > 1; therefore, at some distance from the transmitter K - 1 must be negative. A typical plot of K (p, L ) as a function of the angle 0 reckoned from the specular direction is shown in fig. 2.3b.
SOLIDS IMMERSED IN A TURBULENT MEDIUM
13
n b
Fig. 2.3. (a) Placing a point of observation r on the sphere of radius L = I r, - r s1; (b) profile of the enhancement factor K as a function of the angle 0.
This redistribution of intensity of scattered radiation is no longer evident for 8,L is the distance from source to detector at which the emitted and scattered waves actually propagate through different, statistically independent inhomogeneities of the medium. In fact, pc is the intensity transverse correlation radius of a wave that has travelled a distance from r, to r, in one direction. This can be readily verified by representing the “single-passage” relative intensity J in the form ( J ) + AJ = 1 + AJ, where A J = J - ( J ) represents fluctuations of the relative intensity. Now, the enhancement factor (2.6) can be written as
p > pc, where pc x
K(p, L ) = 1 + B,,(p, L )
9
(2.8)
where B,,(p, L ) =
is the correlation function of fluctuations AJ at the adjacent points r, and Y, + p for a single passage of a wave from Y, to Y,. Because of the small angular dimensions of the backscatter cone, the photodetector that will record the backscatter-enhancement effect should be of small angular dimensions. If the detector aperture is larger than pc, the intensity averaged over the aperture will be almost equal to the intensity that would be obtained in the absence of inhomogeneities. This averaging effect of the detector aperture has been noted by VINOGRADOV,KRAVTSOV and TATARSKII [ 19731.
74
ENHANCED BACKSCATTERING IN OPTICS
[II, 8 2
2.1.4. Backscatter enhancement under weak fluctuations of intensity
If the scatterer is in the region of weak fluctuations of the intensity of the emitted wave, Rytov's approximation may be used for analysis of backscattering enhancement (see, e.g., TATARSKII[ 19671, ISHIMARU[ 19781, RYTOV,KRAVTSOV and TATARSKII [ 1989a,b]). In this approximation,
where k = 2x/A is the wavenumber and @Je(x)the spatial spectrum of permittivity fluctuations. In media with single-scale inhomogeneities the correlation of the intensity fluctuations breaks down at pc FZ OcL 1,. Therefore, a redistribution of the average backscatter intensity occurs within a cone with half-included angle 0, z 1JL. In a turbulent medium the correlation radius of level fluctuations is given by the Fresnel scale pf z (AL)'I2.Quantitative data on enhanced backscattering under various illuminating conditions and for different scatterers may be found in the monograph by BANAKHand MIRONOV[ 19871.
-
2. I .5. Saturated fluctuations of intensity
If we assume that the scatterer is in the region of saturated fluctuations of the intensity of an emitted wave, the field of a wave incident on the scatterer is the sum of a large number of statistically independent waves passing from transmitter to scatterer through different paths. In light of the central limit theorem, near the scatterer the field of the emitted wave will be asymptotically Gaussian (ZAVOROTNYI, KLYATSKINand TATARSKII [ 19771, DASHEN [ 19791, YAKUSHKIN [ 19781). Its statistical properties are completely defined by the average ( G ( r s ,r ) ) = 0 and the coherence function r(p,L ) = ( G(rs, r ) G*(r,, r , ) ) with r(0,L ) = ro(L). The principal physical characteristic of a wave in the saturation regime is the radius of coherence pc(L) of the spherical wave, which is defined by 1 r(pc,L ) I / I o @ ) E l/e. A saturability condition of intensity fluctuations of a wave that has travelled a distance L (ZAVOROTNYI,KLYATSKINand TATARSKII [ 19771) has the form y(L) = L/kp,2(L)P 1 ,
(2.10)
which allows for a simple geometrical interpretation. The quantity y(L) is the
1138 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
15
ratio of the characteristic side shift of rays (in the random inhomogeneous medium and in the homogeneous medium)
.,(Q= ~ / k P , ( L )
(2.11)
to the coherence radius pc(L). In agreement with eq. (2.10) the intensity fluctuations become saturated when the side shift of the rays exceeds pc. Applying the laws of Gaussian statistics to the complex amplitude of the emitted wave yields for the average in eq. (2.6) K(p, L ) = 1 + ! r 2 ( p , L ) ! /ML).
(2.12)
This expression suggests in particular that, in the saturation regime, K = K (0, L ) = 2. Backscatter enhancement can be observed in a small neighborhood of the transmitter confined by the radius &.). The Gaussian approximation used in deriving eq. (2.12) fails to observe that for p 2 p&) enhancement gives way to reduction. However, there are reasons to believe that in the regime of saturated fluctuations of intensity the reduction is small, 1 1 - K ( p , L ) < 1is valid in the wide angular region pc < p < ap.Useful results on backscatter enhancement under the conditions of saturated fluctuations are given in the book by BANAKHand MIRONOV[ 19871. 2.1.6. A lens interpretation of backscatter enhancement An easily tractable lens model of the backscatter-enhancement effect is based on representing a random medium as a collection of focusing and defocusing lenses, i.e., biconvex and biconcave lenses shown in fig. 2.4. Assume that the scatterer is a sphere of radius a. In the geometric optics approximation the scattering cross section of the sphere is cr = nu2. If the sphere is in the focus of a lens of radius R 4 a which is placed between the source and the sphere, the effective cross section of this system, ofoc= nR2, is many times the scattering cross section of the sphere alone, afoc/a= (R/a)2% 1. If the lens defocuses the incident radiation, then, assuming the same focal length as that for the biconvex lens, the effective scattering cross section will be only of 0; namely, ~ J ~z (F/2F)’ ~ ~ ~= ~ / o If we assume that the probability of encountering a focusing or defocusing element is the same, than for R/a 4 1 the mean effective cross section
a.
will exceed the “vacuum” value cr. Thus, the backscatter-enhancement effect
16
ENHANCED BACKSCAlTERlNG IN OPTICS
Fig. 2.4. Scattering by a sphere of radius n in a homogeneous medium (top), scattering after passing through a focusing lens (middle), and defocusing lens (bottom).
may be treated as a result of a strong asymmetry in the action of the focusing and defocusing inhomogeneities. 2.1.7. Backscatter-enhancement relying on multipath coherent efects
The effect of backscattering enhancement allows an interpretation with the aid of multipath coherent effects of Watson-Ruffine that suggests a common cause for a wide variety of phenomena. Imagine an opaque screen with two widely spaced pinholes placed midway between the transmitter and scatterer, as shown in fig. 2.5. A wave arrives at the scatterer along two different paths, so that we can separate the Green function accordingly, i.e.,
Fig. 2.5. Coherent effects occur when radiation from a source arrives at a scatterer through a few paths.
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
I1
Assume that random inhomogeneities in front of and behind the screen give rise to strong phase fluctuations so that the two waves incident on the scatterer are mutually incoherent; i.e., ( G I G t ) = 0. The field of the scattered wave at the point of the transmitter is proportional to
This field may also be represented as the sum of two waves reaching the transmitter through different pinholes us = U:
+ U:
=.fGC,
+ fGG,
Unlike the waves of G I and G, incident upon the scatterer, the terms ti,’ and uf always contain identical (coherent) components, G,G, and GIG,, corresponding to the scattered waves that have passed the same route, transmitter - hole 1 - and scatterer - hole 2, but in opposite directions. Because of the mutually coherent addends, the average intensity of the scattered wave at the transmitter is
which exceeds by 21,, the average intensity corresponding to the wave intensities u,’ and u,’ added incoherently, viz.,
where
I ~ ~ . =, , o ( I12
=
1 ~ ~ 1 )
a(lG:I>
(IG2’1).
It is worth noting that ( I , ) inc& equals the average intensity of the scattered wave measured at a point r, at a distance from the transmitter where u,’ and uf are no longer coherent. We introduce a coefficient K
= (Is(rt))/(Is)incoh
(‘bsc)/(Isep)
that characterizes the enhancement of a backscattered wave over the intensity at points far separated from the transmitter. If I,,,,, % I,,, then K z :. It should be evident that as the number of pinholes M in the screen increases, the enhancement factor grows as K z 2 - l/M and approaches two as M tends to infinity. In this imaginary experiment the effect of enhanced backscattering is caused
78
ENHANCED BACKSCATTERING IN OPTICS
[II, § 2
by the artificially provided multipath (two-path for the two-pinhole case) coherent effects of the Watson-Ruffine type. In other words the effect near the transmitter is caused by the interference of waves that have passed in opposite directions through the same random inhomogeneities of the medium. A similar process takes place in propagation of waves in a turbulent medium. However, unlike the preceding experiment with the screen, here the multipath propagation of a wave incident upon a scatterer occurs as a result of random walk and entanglement of rays in a turbulent medium. In what follows we picture a ray pattern of backscatter enhancement in a turbulent medium, which although rough, yields correct quantitative estimates of the effects of double passage. We begin with estimating the distance between two rays emanated from a source at an angle $. In a turbulent medium, at a distance x from the source, each of the rays experiences fluctuations of the angle of propagation in the order of 8 = l/kp,(x). If the rays pass through different random inhornogeneities, the distance between them will be JoL [ O O +
m dx =
Po@)
+ cr,(L)
9
where cr,(L) is given by eq. (2.1 1) and po(L) = 60 L is the distance between the rays in vacuum. The smallest angle O0 at which the rays may still be thought of independent random walks compares with the coherence angle &(L) = p,(L)/L. The dimensionless ratio of cr,(L) and po(L)w OcL = pc, denoted by y(L) in eq. (2. lo), characterizes the degree of random broadening of ray tubes in the turbulent medium. As long as y(L) < 1, the ray tubes emerge almost undistorted and the fluctuations of intensity 61 y 2 caused by random compression and expansion of the ray tubes are s m d and may be treated with Rytov’s approximation. At y I, random focusing phenomena appear (random caustics) that are responsible for the strong fluctuations of the wave intensity (KRAVTSOV[ 19681). Further away from the transmitter, where y(L)B 1, rays are entangled in a random manner and many almost independent rays meet at every point, as shown in fig. 2.6; the number of such rays may be estimated as M w y 2 % 1. The stochastic interference of independent waves arriving through various rays now becomes the principal mechanism of intensity fluctuations, rather than the compression and expansion of ray tubes as is the case for y 4 1. Accordingly, the field of a wave incident upon a scatterer may be represented as a sum of a large number M , of statistically independent components, namely, N
5
M W t ,
rs)
=
1 ffl=
1
Gffl(rt9 rs).
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
19
Fig. 2.6. Multipath coherent effects in the range of saturated tluctuations in a turbulent medium.
From the reciprocity theorem it follows that the scattered waves return to the transmitter through the same M rays (paths). The fields of all rays incident upon the source will be partially coherent, because to every pair of rays incident on the scatterer, say m and n in fig. 2.6, there correspond a pair of scattered waves propagating along these rays in an opposite direction. Multiple coherent scattering paths that occur in a turbulent medium result, as in the preceding experiment with a screen, in an enhancement of the average backscattered intensity, the enhancement factor K approaching two. When the point of observation is shifted a distance p 2 pc(L)from the transmitterscatterer line, the coherence paths break down and the mutual coherence of waves scattered in various rays disappears. The coherence paths also break down when the observation point moves along the transmitter-scatterer line over a distance of the order of the longitudinal radius of coherence estimated as I,, x pC/& = kpf(L) (VINOGRADOV [ 19741). The domain of observability of enhanced backscattering is illustrated in fig. 2.7. 2.1.8. Experimental evidence
GURVICH and KASHKAROV [ 19771were the first to observe the “pure” effect of enhanced backscattering in optics. A point source of light was simulated by a laser beam focused with a lens system (fig. 2.8). The receiving aperture was a small opening in a blackened face of a prism, enabling the point of observation of the scattered field r to be brought to within 0.5 mm of the effective trans-
I
I
2Pll
4
Fig. 2.7. Region of observation of enhanced backscattering is defined by the transverse ( p , ) and longitudinal (pil) radii of correlation of the spherical wave reflected by the scatterer.
80
[II, § 2
ENHANCED BACKSCATTERING IN OPTICS
1 4 )
T Fig. 2.8. Schematic diagram of a laboratory set-up for observation of backscatter enhancement (GURVICHand KASHKAROV[1977]). 1, Laser; 2, receiver prism; 3, photomultiplying tube; 4, heater; 5 , fan; 6, turbulent air flow; 7, scatterer (spherical mirror).
mitter, r,. A fan and heater produced a turbulent flow of air. The scatterer was a sheet of paper or a 1 cm diameter spherical mirror with curvature radius a w 0.5 m. The Fresnel spot of radius was larger than the coherence radius pc E 5 mm, so that the convex mirror actually played the role of a point scatterer. Figure 2.9 shows the experimental dependence of the enhancement factor K upon the distance p = I r, - r I from transmitter to receiver. Enhancement is of the order of 1.4 near the source with a gradual decrease at a distance of about 3 mm, which is comparable with the coherence radius pc.
fi
Lj.
r. 2 1. I
1.0
Q
I
I
I
I
1
2
3
4
I
I P
y
Fig. 2.9. Enhancement factor K versus distance between the transmitter and receiver in a laboratory experiment. Experimental points are shown as open circles with bars representing standard deviation. The cross-hatched area corresponds to the values of K(p) calculated through the measured values of B,,(p).
11, !21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
81
A field experiment was carried out in a 1300 m long route (KASHKAROV [ 19831). The source and receiver were the same as in the laboratory experiment. To increase the intensity of backscattered light, the scatterer was made as a set of 4000 spherical scatterers placed over 1 m2 of a screen. In addition to the average backscattered intensity, the correlation function of the field intensity B,, in the rectilinear path was measured. The plot of K ( p , L ) - 1 as a function of B,,(p) is represented in fig. 2.10. The experimental points lie fairly close to the bisector, in agreement with the theoretical predictions (see 5 2.1.3). The value of these experiments has been not only that they confirmed the theoretical predictions but also that they tested a new technique for monitoring turbulent media. Instead of a correlation function, this method measures the average backscattered intensity, which is much easier to do. Another practical advantage of the method is that the transmitter and detector can be placed in the immediate vicinity of one another rather than being separated by a considerable distance, as with the traditional monitoring techniques.
Fig. 2.10. Enhancement K ( p ) - 1 plotted versus measured values of the intensity correlation coefficient Bhl(p) for several experimental runs shows small deviations from the bisector (dashed line).
82
ENHANCED BACKSCATTERING IN OPTICS
III, § 2
2.1.9. Enhancement of backscattered intensityjuctuations: Residual correlation of the intensity An enhancement of the average intensity of a backscattered wave detected near the transmitter is accompanied by an enhancement of fluctuations of the intensity compared with the intensity scattered sideways (VINOGRADOV, KRAVTSOV and TATARSKII [ 19731, BELENKIIand MIRONOV[ 19741 and BANAKHand MIRONOV[ 19871). The variance of the relative intensity fluctuations of a scattered wave is
((AJ)2>= ( J 3 r ) ) =
-
(Js(r))2
( J 2 ( r , ) J 2 ( r ) )- ( J ( r t ) J ( r ) > 2 .
(2.13)
Here we assume, as before, that the observation point r is at a distance - rtl from the transmitter (see fig. 2.3). For the wave scattered strictly backwards the variance of the relative intensity fluctuations is
p = Ir
((AJbsc)2>
= (J4) - ( J 2 > ’ .
(2.14)
It would be natural to compare this quantity with the variance of relative intensity fluctuations for a wave scattered sideways, i.e., at a distance p % pc, where it may be safely taken that the inhomogeneities encountered by the emitted and scattered waves were almost independent. The average terms in eq. (2.13) break down into the products of the means to give
((A5sep)2)
3
(J2)’ - 1.
We introduce the enhancement factor of intensity fluctuations as the ratio of intensity variances for waves scattered backwards and sideways, i.e., K A J = ((AJbsc)2)
=
/ ((AJsep)2>
[ (54) - (J”>’]/[ ( 5 ’ ) Z - 11.
(2.15)
If the fluctuations are weak, the averages in eq. (2.15) may be computed in Rytov’s approximation. In this approximation 5 = exp (2x), where x is the level of amplitude obeying a Gaussian distribution with variance CT,’ = B,(O, L ) of eq. (2.9) and mean ( x ) = - CT,’. From the normal distribution of x it follows that (J“) = exp[2n(n - I)a,’],
so that the enhancement factor of the fluctuations of backscattered intensity
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
83
is KAJ =
exp(24o;) - exp(8o;) exp(8o;) - 1
>2.
For the case of saturated fluctuations of intensity, it follows from the fact that the field of the forward wave is asymptotically Gaussian, that the probability density of the normalized intensity J(r) asymptotically approaches the exponential law ( J > 0)
W ( J ) = exp( - J ) . The moments of intensity are then ( J " ) obtain KAj =
=
n ! , so that from eq. (2.15) we
[4! - (2!)2]/[(2!)2 - 11 = 6.67
Experimental studies into fluctuation effects have been reported by BELENKII, MAKAROV, MIRONOV and POKASOV[ 19781,PATRUSHEV, PETROV and POKASOV [ 19831, and KASHKAROV, NESTEROVA and SMIRNOV[ 19841. NESTEROVA and SMIRNOV[ 19841 measured the Specifically, KASHKAROV, intensity moments of the forward wave, (J 2 ) and (J4), and the variances of intensity fluctuations of a wave scattered backwards and sideways. Results of these experiments agree satisfactory with the theory outlined above. Detailed discussions of intensity fluctuations for scattered waves may be found in the [ 19811, BANAKHand MIRONOV[ 19871 and ZUEV, monographs by MIRONOV BANAKHand POKASOV[ 19881. One more effect is noteworthy for fluctuations of intensity of scattered waves, namely, the effect of a residual correlation of intensity for arbitrarily separated points of observation. According to eq. (2.3) the correlation function of the relative intensity of a scattered wave J, is given by
where p,2 = r1,2- r,. If the points of transmission and observation are sufficiently far from one another so that pl,2 B p,, then B,(Pl.,B P O L )
= (J2> (J>2= (J2)
.
(2.16)
This quantity exceeds unity, since ( J 2 ) = Kbsc> 1. If J,(rl) and Js(r2)were uncorrelated, then for B, we should obtain = ((J))4 = 1. The differ-
84
ENHANCED BACKSCAITERING IN OPTICS
111.5 2
ence B , ( p , , , % pc, L ) - 1 = ( 5 , ) - 1 = Kbsc - 1 > 0 is exactly the quantity that characterizes the residual correlation of the backscattered intensity for distant points. This difference is caused by the fact that the wave incident upon the scatterer is amplitude modulated because of the fluctuations in between the transmitter and the scatterer. This modulation, identical for all points of observation, is responsible for the residual correlation [ 19741. effects, which was discussed first by BELENKIIand MIRONOV 2.1.10. Scattering from small inhomogeneities in a turbulent medium A hybrid approach An enhanced backscattering may be obtained not only for bodies embedded in a turbulent medium, but also for the small-scale component of the spectrum of inhomogeneities. DE WOLF [ 19711 has analyzed this problem on the basis of a selective summation of series in perturbation theory, and VINOGRADOV and KRAVTSOV[ 19731have tackled it in the framework of a hybrid approach. In this approach the zeroth order approximation is the field that has been distorted already by large inhomogeneities and the effect of the small-scale component is taken into account with the aid of perturbation theory. This is essentially a statistical version of the distorted wave Born approximation (DWBA) method. The hybrid approach has advantages over the selective summation technique in that it leads to an objective faster and, what is more important, in a more consistent manner. Indeed, whereas de Wolf has taken small inhomogeneities into account twice (in the propagator and inhomogeneity spectrum), the hybrid approach handles small-scale and large-scale inhomogeneities independently. Let B,(p) = ( Z ( r + p ) E ( r ) ) be the correlation function and @&(x) the spectrum of random and isotropic fluctuations of medium permittivity. Represent the spectrum Q e ( x ) as a sum of two nonnegative components, i.e., @&)
= @&>
+ @,(4 9
(2.17)
then the correlation function B,(p) is B & b )= B , ( d + B,(P).
(2.18)
We think of v(r) as the large-scale component and p ( r ) as the small-scale component, the spectrum @ & ( x )being divided by the boundary wavenumber x * , as shown in fig. 2.1 1.
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
85
Fig. 2.11. One version of dividing the spectrum of fluctuations of permittivity into small-scale ( x > x‘) and large-scale ( x < x ‘ ) components.
Partitioning the fluctuations of E into two parts v and p, we observe that in view of eqs. (2.17) and (2.18) these parts are uncorrelated, i.e., B,, = 0. Now, we write the wave equation as AU + k2(1 t
V)U =
-k2pu,
(2.19)
and as a zeroth approximation we take the solution u , caused by the passage of the wave through large inhomogeneities and satisfying
A M ”+ k2(1 +
V)U, =
0.
(2.20)
The respective Green function will be G,. To a first approximation, from eq. (2.19) we obtain the once-scattered field (2.21) The average intensity of this field, I;’), can be calculated by carrying out independent averaging over v and p, namely ?:)(r)
=
k4
JJ
B,(r’
-
r”)
x ( G , ( r , r ’ ) G,*(r,r ” ) u , ( r ’ ) u , ( r ” ) d3r’ d3r” . Assume that the coherence radius p, of the fields u , and G , is large compared with the correlation radius of the small-scale component l,, and that the fields u , and G , differ from the respective values in vacuum, u,,(r’) and G,(r, r ‘ ) = - exp(ik I r - r’ 1)/4n I r - r ’ 1, by random factors whose squared moduli equal, respectively, J ( r ’ , r t ) and J(r, r ’ ) with r, being the emanation point of the primary wave. Then the mixed moment in the integrand for 7;’)
86
ENHANCED BACKSCATTERING IN OPTICS
can be replaced with a simpler expression
where Z,(R) is the primary field intensity at the point R = i ( r ’ + r”), q = k (ni - n,) = k [ ( R - ro)/ I R - r, 1 - ( R - r)/ I R - r I ] is the scattering vector, and K ( p , R ) = (IJ(R,rt)12 IJ(RJ)I2) is the enhancement factor that we have already considered. Now, the intensity Zi’) may be represented as
(2.22) which is close to the traditional formula of Born. Here, o;(q) = i n / ~ ~ @is~ ( q ) the small-scale part of Born’s scattering cross section per unit volume. According to eq. (2.22), the effective scattering cross section Ka,, differs from Born’s quantity ap by a factor K , which has a peak of enhancement toward the source (point rt) and is close to unity in all other directions. It is an easy matter to demonstrate (VINOGRADOVand KRAVTSOV[ 19731) that eq. (2.22)is invariant with respect to small variations of the boundary value x * , if the small-scale component p(r) does not cause a marked extinction over a distance L, i.e.,
2 a p L = Laps,,,
=
L
s
a,,(q)dOQ 1.
(2.23)
This inequality is much weaker than the usual condition of applicability of Born’s approximation, i.e., 2ueL = 2 ( a , + a p ) LQ 1, because the extinction coefficient for the large-scale component a, exceeds considerably that of the small-scale component ap. Thus the hybrid approach succeeds in not only revealing an enhanced backscattering, but also in considerably expanding the limits of applicability of Born’s approximation. For this purpose it suffices to replace the total scattering cross section a, with its small-scale part a,,. Moreover, the potentialities of this approach seem to be far from exhausted.
11.8 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
87
2.1.1 1. Polarization egects Media with large-scale inhomogeneities (1, g A) propagate electromagnetic waves almost without changing their polarization. This follows from the estimates obtained by TATARSKII [ 19671 for “diffraction” depolarization (described by Rytov’s method or by the method of parabolic equation) and by KRAVTSOV [ 19701 for “geometric” depolarization caused by the rotation of field vectors due to torsion of rays. Both diffraction and geometric depolarizatior, of a light wave are small, and therefore a consideration of light effects may be limited to a scalar approximation.
2.2. EXTENDED TRANSMITTERS, SCATTERERS, AND RECEIVERS
2.2.1. Wave description within the parabolic equation framework A wave propagating through a medium with large-scale random inhomogeneities of permittivity suffers multiple scattering, which is predominantly directed forward in a narrow cone of angle B N All, 4 1. As a result of the multiple scattering, the wave propagates practically in the emanant direction. It is convenient to describe this wave with the aid of the parabolic equation philosophy (see, e.g., the book of RYTOV, KRAVTSOVand TATARSKII [1989a,b]. This philosophy also may be applied to the description of the propagation of waves scattered from a body embedded in a turbulent medium. Let a wave propagating along the x-axis have a complex amplitude uo ( p ) in the plane of the transmitter, x = 0 ( p represents the transverse coordinates). Then the complex amplitude of the wave in a cross plane passing on x is given by U ( P ? x) =
1
uo(p’)g(p’, p, x) d2P’ .
(2.24)
The Green function g(p’, p, x) describing the field of a spherical wave emanated from point p ’ , x = 0 satisfies the parabolic equation of quasi-optics ag 2ik + A,g
ax
+ k2 E(p’, x)g = 0 ,
where A I is a two-dimensional Laplacian in a plane orthogonal to the x-axis.
88
[II, § 2
ENHANCED BACKSCATTERING IN OPTICS
Let a scatterer with a local reflection factor f(p), recalculated for a plane x = L, be placed at point x = L. Using reversibility of light paths, the complex amplitude of the wave scattered backwards can be written (in the plane x = 0) as (GELFGAT[ 19761, SAICHEV[ 19781) us (PI =
=
s
u (P‘ 9
L ) f ( P ’ 1g(P7 P‘ L ) d2P’ 7
uO(p’~f(p’’)g(P p ”, , L ) g ( p ’ , P ” , L ) d2p‘ d’p”
.
(2.25)
In the following description we need a theory of wave propagation without reflection in a random inhomogeneous medium. Valuable information about waves propagating in randomly inhomogeneous media may be obtained with the aid of moment functions. In the Markov approximation, these functions [ 19781, RYTOV,KRAVTSOV and satisfy closed equations (see, e.g., ISHIMARU TATARSKII [ 1989a,b], GURBATOV, MALAKHOV and SAICHEV[ 19911). The equations for the average field ( u (p, x)) and for a coherence function have an exact solution. The average field of optical waves in a turbulent atmosphere is almost always zero. Therefore, we confine ourselves to the coherence function for the Green function g(p,, p, x),
rg(Ro,P O , R, P,X ) = ( g ( R o + ;PO, R + $p*x)g*(R,-
R - ;P*XI)
(2.26) The effect of random inhomogeneities of the medium is taken into account by the function (2.27) which is equal to the mean squared random phase difference calculated in the geometric optics approximation along two straight rays. The initial distance between these rays (in the x = 0 plane) is p,, and the final distance, at z = x, is p . If p is in the inertial interval, I, < p < Lo, then (ISHIMARU [ 19781) D ( p ) = 5.83C2p5/’,
(2.28)
where C; is the structural characteristic of fluctuations of the refractive index of the turbulent medium.
11.8 21
89
SOLIDS IMMERSED IN A TURBULENT MEDIUM
If the coherence function (2.26) is known, it is not hard to compute the coherence function of the wave field u, m , p , x)
=
( u ( R + i p , x ) u*(R - $ p , x ) )
=
J f u,(R’ + $ p ’ ) u ; ( R ’
-
i p ’ )TJR’, p ’ , R, p, x ) dZR‘ d2p . (2.29)
From this expression it follows that, in particular, for a spherical wave emanated from the origin ( p = 0, x = 0)
Let us determine the coherence radius pc ( x ) of a spherical wave as the value of p at which the modulus of coherence function reduces by a factor of lie to give d(0,pc, x ) = 2. If p,(x) lies in the inertial interval, i.e., if (2.28) is valid, then p,, sph (x) = 1.44(k2C,’X)-
’”,
(2.30)
and the coherence function of the spherical wave takes the form (2.3 1)
The
following coherence
function
corresponds
to
a
plane
wave
uo(p) = uo = const.
q h x ) = IoexP[ - $ d ( P , P 7 X I 1
I
1, =
/u0l2.
(2.32)
Accordingly, the coherence radius of the plane wave is p c . p , ( ~=) 0.8(k2C,”)-’’’
.
An important physical characteristic of a random wave field is the radiant intensity 2 ( R , 8 , x )=
(51s
T(R,p,x)e-ik(e’p)d2 P.
(2.33)
Substituting T(R,p , x ) from eq. (2.29), we obtain (VINOGRADOV,KOSTERIN, MEDOVIKOVand SAICHEV[ 19851)
f ( R , 8, x)
=
sf
Y,(R’,8’, x) W ( R - R ’ , 8 - 8’, x)d*R’ d28’ ,
(2.34)
90
PI, § 2
ENHANCED BACKSCATTERING IN OPTICS
where yo(R,0, x) is the radiant intensity of a wave propagating in vacuum ( Z = 0), and the function
1
x l e x p [ E R ( p - p ’ ) - i k ( B . p ) - ~ d ( p ’ , p , x ) d2p’d2p X
(2.35) may be given a simple geometrical interpretation; namely, it is the probability density of lateral shifts and angular deviations of rays propagating in a turbulent medium from the positions and directions of the respective rays in vacuum. Averaging eq. (2.27) over the angles 8 yields the average wave intensity
(2.36) which is the intensity of the wave in vacuum, I o ( p , x), convoluted with the probability density of lateral shifts of rays in a turbulent medium
If pc (x) lies in the inertial interval, then (2.38) where op(x)is the characteristic lateral shift of the rays of eq. (2.11) and
(2.39) Let a ( x ) be the radius of a wave bundle propagating in vacuum as measured in the cross section at x. Equation (2.36) suggests that if the lateral shift of rays ap is much less than the beam radius, then (I@, x)) = I,,(p, x). Expressed differently, we can say that in this case random inhomogeneities do not affect the profile of the average beam intensity. Conversely, if up(x)$- u ( x ) , then ( I @ , x)) = Pa W,(p, x); i.e., the average intensity follows the profile of the
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
91
probability density of lateral shifts of rays. (Here, P = I, d2p is the total energy flux of the beam.) Integrating eq. (2.34) with respect to R, we obtain the angular distribution of the average energy flux of the wave passing through the cross section at x
p ( e , x)
=
s
p o ( e ’ ,X)
w,(e - e’, x) do‘ .
(2.40)
This expression includes the probability density of fluctuations of oncoming angles for rays in a turbulent medium; for I, < P ~ , ~ ~ (<XLo, ) it equals
where %(X> = 1/b%*&) (2.41) is the characteristic angular deviation of rays from their directions in vacuum. Useful results on backscatter enhancement of laser radiation have been reported by AKSENOVand MIRONOV[ 19791 and AKSENOV,BANAKHand MIRONOV [ 19841. 2.2.2. Statistical description of backscattered waves in the region of saturated fluctuations of intensity
The effects of double passage of scattered waves through the same inhomogeneities of a medium are more clear cut in the case of saturated fluctuations of radiated intensity when the moment functions of random fields may be handled with the aid of Gaussian statistics. As an example, consider the coherence function of a backscattered wave in the plane of the transmitter. According to eq. (2.25), it is
r,(RP)= ( u , ( R + i P ) U W - $PI) =
11
f ( p ’ ) f*(P”) ( U ( P ’ 9
+ iP, P’?L )
x u*(p”,L)g*(R - i p , p ” , L ) ) d2p’ d2p” . A general solution of the equation for the fourth-order moment function entering this expression is yet to be found. However, given that the condition of saturated fluctuations of emanant intensity, eq. (2. lo), is satisfied, the fields u and g can be deemed to be Gaussian with zero means and the last equality can be rewritten as
r,(R, P) = rl( R ,P,L ) + r2w,P,Q
(2.42)
92
ENHANCED BACKSCATTERING IN OPTICS
where
r , ( R ,P , L ) =
Jf(R'
+ f q ' ) f *(R' - f q ' )
x ( u * ( p " , L ) g ( R + f p , p ' , L ) ) d2p' d2p" , The first term on the right-hand side of eq. (2.42) describes the coherence function of a backscattered wave assuming that the random inhomogeneities encountered by the radiated and scattered waves are statistically independent, whereas T, takes into account the double passage of the wave through the same inhomogeneities. The function r,(R,p , L ) may be interpreted as the coherence of a wave that has passed, in a random inhomogeneous medium, a path of length 2L, bisected by a semitransparent screen (at x = L ) of transmittance fl(p). For an ideal mirror, when f , ( p ) = 1, T I ( R , p L , ) = I'(R,p, 2L). In the case of a rough scatterer with small-scale inhomogeneities, we can assume
( f , ( R+ f P ) f * ( R- f P ) >
=
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)>
=
=
j f ( R +:P)f*(R-fP) x ( g ( O , R + f p , L ) g ( p , R+ i p J ) g * ( O , R - $ p 3 L ) x g*(p, R - i p , L ) ) d2R d2p.
(2.47)
11, § 21
S O L I D S I M M E R S E D IN A TURBLILENT M E D I U M
93
Making use of the small-angle approximation, we represent the random Green function in the form (2.48) where A (pa,p, x) is a function statistically uniform in both po and p introduced to take into account the effect of the random inhomogeneities of the medium. Substituting eqs. (2.48) and (2.46) into (2.47) yields
X
A*(O, R - f p , L ) A*@, R - ;/I, L)) d2R d 2 p .
(2.49)
Since A (pa,p, x) is statistically uniform, the preceding average depends only on p and p , i.e., ( A A A *A *) = B,(p, p , L ) . In the circumstance, integration in (2.49) with respect to R gives (I,(P))
=
CYO) K(p3 L)
3
where
is the intensity of a wave reflected in a vacuum, f2 = ka2/L, and
(2.50) is a coefficient describing backscatter enhancement and the redistribution of the average intensity of the reflected wave in the plane of the transmitter. The function
indicates that the reflected wave is formed not by the whole mirror but, rather,
94
ENHANCED BACKSCATTERING IN OPTICS
[II, § 2
by an area with effective radius a,, = a [ 1
+ (kU2/L)*]
-
112
.
(2.5 1)
Some properties ofE,(p, p , L ) are discussed below. It should be obvious that EA(p,0, L ) = ( J ( p , L)J(O,L ) ) and the value of E,(O, 0, L ) equals the mean square of relative intensity of a spherical wave ( J 2 ( 0 ,L ) ) at a distance L from the source. At higherp, the function E,(O, p , L ) tends to zero at a characteristic rate pc ( L ) . Specifically, if the mirror is in the range of saturated fluctuations of intensity of an emanating wave, the laws of Gaussian statistics may be applied to the averages (2.47) and (2.49) to give N
In agreement with eq. (2.43), the backscatter enhancement factor is given by Rbsc = K(O, L ) =
s
B,(o, p , L ) p ( p , L ) d2p .
(2.52)
Clearly, the absolute effect of enhanced backscatter will be realized in the reflection from a mirror if p ( p , L ) singles out the value of EA(0, 0, L ) corresponding to p = 0. This is feasible if the size of the bright spot on the mirror is smaller than the coherence radius, i.e., a,,
(2.53)
4 pc ( L ).
Then the absolute enhancement factor will be given by the familiar expression of eq. (2.5), namely Kbsc = ( J 2 ) > 1. Condition (2.53) is not always valid. In the range of saturated fluctuations of intensity, the absolute effect of enhanced backscattering will be observed only for small (a < p,) and substantially large (a % a,) mirrors. Moreover, in the interval pc < a < op enhanced backscatter gives way to reduction of scattered intensity and K 1. The situation is illustrated in fig. 2.12 by a,,/pc and K plotted as functions of log(a/p,) for y = aJpC E 10. A reduction of the average backscattered intensity (compared with that in vacuum) in the interval pc < a < apmay be explained by analogy with eq. (2.36) as a reduction of the average intensity of the reflected wave due to random walk of the rays. Furthermore, it is easy to demonstrate that for any size of the mirror a relative effect of enhanced backscattering takes place, namely,
-=
K , ( L ) = (IS(O))/Il(O,
L)> 1
(2.54)
Here, I , (p, L ) = T1(p,0, L ) is the average intensity of the reflected wave calculated under the assumption that random inhomogeneities of the medium encountered by the emanant and reflected waves are statistically independent.
11, § 21
95
SOLIDS IMMERSED IN A TURBULENT MEDIUM
*
WQlp,) Fig. 2.12 Transition from enhancement (Kbsc> 1 ) to attenuation (Kbrc c 1 ) for greater ratios of mirror radius a to coherence radius pc. -I
0
7
2
For a spherical wave GOCHELASHVILI and SHISHOV [ 19811 have noted the existence of enhanced backscattering whatever the size of the scattering body. Enhanced backscatter from extended bodies is also discussed in the monographs of MIRONOV[1981], BANAKHand MIRONOV[1987] and ZUEV, BANAKHand POKASOV[ 19881. Recently, AGROVSKII, BOGATOV,GURVICH, KIREEVand MYAKININ [ 19911 and BOGATOV,GURVICH, KASHKAROV and MYAKININ [ 19911have carried out a more accurate theoretical and experimental analysis and demonstrated that the backscatter coefficient K reduces when the size of the body is comparable with the coherence radius of the prime wave. Finally, mention should be made about the spatial redistribution of the average intensity of a wave reflected from an unlimited mirror. In this case, p( p , L ) = S( p ) and from eq. (2.50) it follows that K ( p , L ) = B,(p, 0, L ) = ( J ( p , L)J(O,L ) ) .
As in the case of the point scatterer defined by eq. (2.7), the quantity K ( p , L ) repeats the profile of the correlation function of the relative intensity of the spherical wave at a distance L from the source. 2.2.4. Eflect of long-distance correlations and partial reversal of the wa vefront The effect of enhanced backscattering can be realized if the aperture b of the source does not exceed pc(L).When b > pc, the enhancement effect gives way to the effect of long correlations (KRUPNIKand SAICHEV[ 19811, KRAVTSOV and SAICHEV[1982a,b]). With this effect, in addition to a narrow peak of radius - p , ( L ) , the coherence function of the reflected wave acquires in the plane of the transmitter x = 0 a low but wide pedestal corresponding to long
96
ENHANCED BACKSCATTERING IN OPTICS
[II, § 2
correlations. It is displayed in full measure when the intensity fluctuations of an emitted wave are saturated. In what follows we confine ourselves to the analysis of this situation. To be more specific we shall assume that the reflector is pointsize, i.e., ,f(p) = 6(p), and that a collimated wave pencil of radius b is radiated in the plane x = 0. The coherence function of the reflected wave is given by eq. (2.42), where
(2.55) and the coherence-function component responsible for the effects of double passage is
rZ(RP,L) = U * ( P , )
(2.56)
O(f2).
Here, u(p) =
(2 n L )’1
u0 ( p - p ) exp
[ (:>”’ -
-
ikL (p - p)] d2p ,
(2.57)
and pI = R + $ p and p2 = R - $ p are the coordinates of points in the plane x = 0, in which the mutual coherence of the field of the reflected wave is determined. Suppose that the radius of the radiated beam is sufficiently large, b % p,, then U(P)
= UO(P)
Wp
where W,(p, L)is given by eqs. (2.38) and (2.39). Thus, Wp(PI3L) W,(p2, L)*
r 2 = Uo*(Pl) UO(P*)
(2.58)
Simple estimates obtained with the aid of eqs. (2.55), (2.58), and (2.36) indicate that forb 9 pc(L) we have r2(0, 0, L ) Q rl(0, 0, L ) and the backscatter enhancement effect is virtually indistinguishable. It is worth noting that the coherence function of the reflected wave considered as a function of distance between points of observation p = Ipl - p 2 ) consists of a narrow peak TI, decreasing with p at a rate pc (L), and a wide pedestal r,. A typical plot of r, as a function of p for R = 0 and b % pc(L,) is shown in fig. 2.13. We note that the component r2of the coherence function is proportional to the product of the primary fields, r 2
G ( P l ) U(P2)
9
91
SOLIDS IMMERSED IN A TURBULENT MEDIUM
11, § 21
Fig. 2.13. Typical profile r, of the coherence of backscattered field as a function of the distance between observation points p = Ip, - pzJ.
which may be treated as the complex conjugate function of coherence of the emitted wave. As a result, the effect of long correlations may be thought of as coming from a partial reversal of the wavefront of the reflected wave with and SAICHEV[ 1982a,b, 19851). respect to the prime wave (KRAVTSOV We interpret the mechanism of partial reversal by way of a simple example of a wave produced by two mutually coherent point sources at points rI and r,. Let such a wave be incident upon a scatterer at point rs in a random inhomogeneous medium. If cp, and cp2 are the phases of the emitted waves, the complex amplitude of the scattered wave at an arbitrary point r is u,(r) = [e'"' G ( r , ,r,) t eim G ( r 2 ,r,)] G ( r s ,r ) .
If we assume that the point of observation coincides either with Y, or with r2, two opposite paths occur in the random inhomogeneous medium, rl + r, + r, and r2 + r, r l . The mutually coherent components of the scattered wave are
By the reciprocity theorem we have G I , so that
=
G,, , cplz
=
q2,and I GI, I
=
I G,, 1,
(rl) = 1 GI, I ei(rp2" I 2 ) , +
(r,)
=
1 GI, 1 ei("'
+
"I2)
.
Denoting cp = cpl t (p2 t tp12 and rewriting cpl t qI2= - (p2 + cp and cp2 t cpI2 = - cp, + cp, we note that the phases of the mutually coherent com-
98
"I, § 2
ENHANCED BACKSCATTERING IN OPTICS
ponents of the scattered wave are reversed with respect to the phases of the emitted waves accurate to within the additive phase cp, U , , ~ ~ ~= ( YJG,,I ~ )
~,.,,~(r~)
=
e-iql+iq,
lG121 e - i m + i q .
A similar semiquantitative explanation of the long correlation effect may be extended to the case of a collimated wide beam of radius b $ pc(L). Imagine that the aperture of the source is divided into partial beams of radius pc. Let pm and p,, be the centers of mth and nth partial beams separated by a large distance [ 1 pm - p,, 1 $= pc(L)]. Since the phases propagating in the opposite coherent paths pm t-* rs -p,, are reversed, the coherence function of the scattered waves at points p, = pm and p2 = p,, is proportional to u8(pl) u0(p2). A coherence path occurs, provided that the beams emanating from points pm and p,, can arrive at the scatterer as the result of a random walk in the random inhomogeneous medium. Therefore, the coherence function defined by eq. (2.58) is proportional also to the product of the probability densities of lateral shifts of the beams. In the field of the scattered wave long correlations occur when a(,!,) 9 pc(L); i.e., when conditions (2.10) and (2.11) of saturated fluctuations are satisfied, y = o,,/pc9 1.
-
2.2.5. Enhanced backscattering in the focal plane of a lens As has been pointed out for a wide beam with b 9 p,(L), no enhancement of backscattering is observed. However, a partial reversal of the wavefront in the field of the scattered wave gives rise to a highly coherent component that can be focused by a lens. This leads to another effect because of the partial reversal of the wavefront, namely, to an enhancement of the average intensity in the focal plane of the lens. Let the scattered wave be incident upon a lens placed in the x = 0 plane. The lens aperture coincides with that of the source and is described by the function uo(p). The field in the focal plane of the lens is given by
For a plane-emitted wave [ b $= a,(,!,)], when the common enhancement effect is negligible, from eqs. (2.42), (2.55), and (2.58), it follows that the average
1 4 8 21
99
SOLIDS IMMERSED IN A TURBULENT MEDIUM
intensity of the field in the focal plane is proportional to
where
In the middle of the focal plane ( p = 0) the average intensity of the scattered wave is twice the intensity in the homogeneous medium, { JF(0)) = 2 1 uo(0) 14. This implies the absolute effect of backscatter enhancement with K F = 2. The component JF1( p ) corresponds to a wide pedestal with radius bF/L, which is about the size of a spot produced by a spherical wave from a point scatterer in a homogeneous medium. The intensity J F 2 ( p )forms a sharp peak corresponding to the quasi-plane (resulting from a reversal of the wavefront) component of the scattered wave with coherence radius a,(L). This enhancement effect in the focal plane may be treated as a common backscatter enhancement, if the primary field is assumed to be that of a point scatterer in the focal plane rather than the field of the plane-emitted wave. In turn, the ordinary backscatter enhancement, which was developed in the common scheme of a point source, namely, random inhomogeneous medium, point scatter, may be explained as the effect of focusing the reversed component of the scattered wave near the source (KRUPNIK[ 19851). The reversed component of the scattered wave is present in any plane between the source and scatterer. Near the scatterer it has a random character and a small coherence radius of about pc ( L ) ,similar to the wave incident upon the scatterer. Closer to the source the reversed component acquires a higher spatial coherence with simultaneous focusing on the source. Despite the random character of this component, it can be detected as follows. Let u ( p , x) be the complex amplitude of the prime wave at distance x = L from both source and scatterer, and let u,(p, x) be the complex amplitude of the scattered wave in the plane with this coordinate. Recognizing that the reversed component of the scattered wave is proportional to u*(p, x), one may verify (KRUPNIK[ 19851) that in the presence of the reversed wave the
-
4
I00
ENHANCED BACKSCATTERING IN OPTICS
moment function (~,(PI,~)~(P,,X)~,*(P,,X)~*(P2,X))
= (U(P1,
-
x) U ( P , , x) U*(P2?x) U*(P2, x)>
(I(PI3 x)> ((l(P2r x)>
remains almost invariable when the points pI and p2 are separated. Thus, the enhancement of average backscattered intensity and the long correlation effect are two sides of the same coin, namely, the partial reversal of the wavefront of the scattered wave. Using the concept of partial reversal as a point of departure provides an insight into the nature of enhanced backscattering in the conditions where the sourcein an extended a n t e n n a ( Y [~19831, ~ KRAVTSOVand SAICHEV [ 19851). Suppose that uo(p)describes the distribution of current in an antenna in the transmitting mode. Then, in view of eq. (2.25) the scattered signal received by the antenna is proportional to
(2.59)
Specifically, for a rough scatterer, eq. (2.44), we obtain P
(2.60)
Let us compare this quantity with the intensity of an oncoming signal
r which has been calculated on the assumption that the random inhomogeneities in forward and reverse paths are statistically independent. In a random inhomogeneous medium ( 1 ' ) > ( I ) ' , therefore, the relative backscatter enhancement effect with K , = J J J , > 1 can be observed independently of antenna size.
11,s 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
101
If the radiated wave is such that the average intensity ( I ( p , L ) ) coincides with the intensity in the homogeneous medium Io(p, L), then we obtain an absolute enhancement effect. Assume that the antenna radiates a collimated beam of radius b. From eq. (2.36) it follows that ( I ) = I, for sufficiently small radii (b < p,) and sufficiently large ( b > op) radii. The first case corresponds to the ordinary backscatter enhancement for essentially a point source, and the second case corresponds to an absolute effect of enhanced backscattering in the middle of a focal plane of a large lens. In scattering from solids embedded in a turbulent medium the partial reversal of the wavefront also may be considered as an echo effect observed in various physical systems. 2.2.6. Enhancement of radiant intensity The partial reversal of a backscattered wave also produces the effect of enhancement of radiant intensity allied to the effect of backscatter enhancement in the middle of a focal plane of the lens. Representing the radiant intensity of a backscattered wave in the plane of radiation x = 0 in the form (2.61) As an example, let us discuss the case of a plane wave u(p, 0) = 1 emitted in the plane x = 0, with the reflector being a phase screen, i.e., a statistically uniform rough surface which causes the phase of the reflected wave to be changed by a random quantity cp(p), i.e., a surface with f ( p ) = exp[icp(p)]. Denoting the structural function of phase distortions ( [ q ( r + s) - ( ~ ( p ) ] ~ ) , introduced by the reflecting surface, by d,(s) we obtain
We assume that d,(s) increases monotonically with s, whereas B,(s) falls off to zero at a characteristic rate pp After straightforward transformations exploiting the statistical homogeneity of the radiated wave u ( p , L ) with respect to p, the coherence function of the scattered wave can be rewritten (SAICHEV [ 19801) T,(P) =
s
B,(s) @(P. s, L ) d2s
9
(2.62)
102
ENHANCED BACKSCATTERING IN OPTICS
where
@ ( p ,s, L ) =
s
(g(p
+ p , s, L ) g * ( p , 0, L ) u(s, L ) U * @ , L ) ) d2P *
From the orthogonality of the Green function in the parabolic equation formalism, i.e.,
s
g(p, PI, x)g*(p, P2, x) d2P = W P ,
- Pz)
9
it follows that @(O, s, L ) = ( l ( 0 , L ) ) b(s) = 6(s) .
(2.63)
By virtue of eq. (2.62) the average reflected intensity
rs(o)= ( 4 ( P ) >
=
B,(O)
=
1
coincides with the intensity of a plane wave reflected in a homogeneous medium from an ideal mirror off = 1. This implies that the ordinary enhancement effect is completely absent in the case under consideration. The situation is completely different for the radiant intensity of the reflected wave. Substituting eq. (2.55) into eq. (2.54) yields (2.64)
is the moment function of plane waves propagating at an angle 0 to one another,
The radiant intensity of a wave reflected strictly backwards is (2.66) The quantity B,(s, L ) = ( u Z ( s , L ) u*2(0, L ) )
11, 5 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
103
is the coherence function of the squared wave function incident upon a reflecting surface. This function varies in s at a rate of pc (L).If the reflecting surface is constituted by small-scale inhomogeneities, then pf 6 pc ( L )and Bf(s) singles out in eq. (2.66) the value B,(O, L ) = (Z*(O, L ) ) = K > 1 , so that (0) is Kbsctimes the radiant intensity J;(O) of the wave backscattered in a vacuum. For an arbitrary angle 8 and a small-scale reflector with pf 4 pc (L), from eq. (2.66) it follows that
where
is the radiant intensity of the wave reflected in a vacuum. Its characteristic angular spread is 8, l/kp,. The coefficient K ( 6 ) = ( I ( 0 , L ) Io(O, L ) ) with I&, L ) = 1 u i ( p , L ) 1 describes the angular distribution of the radiant intensity resulting from the double passage of the wave through the same random inhomogeneities. From eq. (2.63) it follows that
-
I/(6)d28=
r,(O)=
s
&0)(6)d28,
in addition to the enhancement of the backscattered radiant intensity, the coefficient K ( 6 ) describes some reduction in the intensity ( K < 1) of the backscatter compared with y:(fl), as depicted in fig. 2 . 2 ~ . Finally, let us determine the radiant intensity of a reflected wave for the case of saturated fluctuations of the incident wave. Applying the laws of Gaussian statistics to the average in eq. (2.65) yields
X(e)=$,(e,L)+$,(e,L). The quantity $,(6, L ) =
s
(2.67)
A(#)Wo(6 - 6',2L) d28'
represents the radiant intensity of an initial plane wave that has passed in a
104
“I, 3 2
ENHANCED BACKSCATTERING IN OPTICS
random inhomogeneous medium a distance 2L without reflections, subject to the condition that a phase screen is placed midway, thus introducing a phase distortion rp(p). The expression for yl also includes the probability density of oncoming angles W,(B, 2L), whose characteristic scale in 8 is a,(2L), as outlined in eq. (2.41). The second component of the radjant intensity,
x exp [ - $d(s + BL, S, L ) - ;d(s - BL, s,L ) - ik(B.s)] d2s,
takes into account the double passage of the wave through the random inhomogeneous medium. This component is responsible for the backward enhancement of radiant intensity. The radiant intensity of a wave reflected from an ideal mirror (f= 1) in a random inhomogeneous medium is plotted in fig. 2.14. It consists of a narrow peak of h ( B , L ) with semiangular width 8, p,/L and a wide pedestal of L ) = W,(B, 2L) with characteristic angular scale a, l/kp,(L). There is no absolute enhancement of radiant intensity scattered strictly backwards in this case, but there is a relative enhancement with coefficient K, = 2s (0)/yl (0) = 2. This effect may be interpreted as being due to a partial reversal of the reflected wavefront in the random inhomogeneous medium. Some applications ofthis formalism are worth noting. LUCHININ[ 19791has established a radiant-intensity enhancement for the light reflected from the bottom which, thus, passed twice through the disturbed surface of the ocean. SAICHEV[ 19801 has described an enhancement of the radiant intensity backscattered from a system of discrete scatterers randomly dispersed in a random inhomogeneous medium. JAKEMAN [ 19881 has described in depth the effect of enhancement of the radiant intensity that has twice passed through a random phase screen, before and after reflection from a mirror. N
-
Fig. 2.14. Radiant intensity profile for a seatterer embedded in a turbulent medium.
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
105
2.2.1. Giunt hackscatter enhancement in laser sounding of the ocean HOGEand SWIFT [ 1983a,b] have recorded strong fluctuations of echo pulses in laser sounding of scattering layers through the rippled surface of the ocean (fig. 2.15). They associated the strong fluctuations with a nonuniform distribution of scatterers in the horizontal. VLASOV [ 19851 has given a more plausible explanation by pointing out the important role of the double passage of the light through the wavy surface of the ocean, which has the same effect as a random phase screen. (Random focusing of light caused by a disturbed ocean surface also has been discussed by GEHLHEAR [ 19821.) The theory of enhanced backscattering in the presence of a phase screen outlined in 5 2.1.2 is not well suited to handle laser sounding of the ocean because at the air-water interface the width b of a laser beam is usually small compared with the characteristic scale of surface roughness, 1,. More extensive work has been done by BUNKIN, VLASOV and MIRKAMILOV [1987] and APRESYAN and VLASOV [ 19881, and, therefore, our discussion will be in line with these studies. Suppose that the ocean surface is irradiated with a Gaussian beam u = uo exp ( - p 2 / b 2 )of width b. Given b 4 I,, the distortions of the eikonal $ ( p ) of the incident wave may be described by second-order polynomials $(P)
= $0
+ 0 . P + +(w: + a2P,2).
Here, the directions of p, and p2 are taken to coincide with those of the main lines of curvature of +(p). The vector 8 = (e,, 8,) describes the angular deviations of the beam from the vertical x-axis, whereas = l/R1,*, where R , , 2
-
- - -- -- - 5 = L
J.
IX
Fig 2 15 Backscattering of a narrow laser beam from scatterers under a rough sea surface
106
[II, § 2
ENHANCED BACKSCATTERING IN OPTICS
are the main curvature radii of the phase front immediately behind the phase screen. In this approximation the beam remains Gaussian at any distance from the rough surface, its intensity in a plane at x = L being I(p,L)=
IU0l2YI(P1 -
(2.68)
4 L . )Y Z b Z - & L ) ,
where the quantities Yl.Z(Y)
= exP[ - Y 2 / b 2 ~ : , 2 ( L ) l / ~ , , 2 ( L )
and
/?= 2L/kb2, characterize the compression or expansion of the beam in p, and p2. Let F(p, L ) be proportional to the cross section of backscattering by a unit volume of water at depth x. Then the intensity Z,(L) of the echo signal from a depth of L in monostatic measurement is given by (2.69) Unlike eq. (2.60), this expression has not been averaged over all of the random inhomogeneities of the surface (over random values of slope 01,2and curvature al,2in the case of a narrow beam) because we wish to know not only the averages Z, but also the deviations from these values (fluctuations). Substituting eq. (2.68) into eq. (2.69) and assuming that the scattering layer is statistically uniform, F = const., we obtain for K(L)= 4 ( L ) I W )
9
where I,O(L)is the intensity for the case of a plane interface
=
0),
where bxo( L ) = b J m is the radius of the diffraction-broadened, undisturbed, Gaussian beam in the plane of scatterers at x = L. From eq. (2.70) it follows that if the rough surface focuses the beam in the plane of scatterers (a, = t12 = l / L ) , the intensity of the received signal
11, § 21
107
SOLIDS IMMERSED IN A TURBULENT MEDIUM
experiences a giant enhancement, namely, it becomes K,,, = (1 + P2)/P2 times and VLASOV [1988]). In a typical that for the plane interface (APRESYAN sounding situation of an upper layer of the ocean with k = lo7 m-’, b = 0.01 m, and L = 10 m, we have K,,, = 2.5 x lo3. Of course, practical intensity peaks are smaller because the focusing in one coordinate is often accompanied by an expansion of the beam in the other coordinate. In particular, when the random eikonal $ ( p ) is a statistically isotropic Gaussian field with correlation function
the joint probability density of curvatures a1 and a2 (assuming a2 > a l ) is
It is obvious that wa vanishes for identical curvature radii of the wavefront, so that the compression of the beam is essentially different in various coordinates. In a case more typical of the ocean surface, when the eikonal depends only upon one coordinate p,, the enhancement factor of the received intensity
is caused by the one-dimensional focusing of the sounding beam. Complete information for the frequency of unusually large values of K may be obtained from the probability density wK(K).Let $(pJ be a random Gaussian function for which the probability density of curvature a of the wavefront is w,(a) =
1
f i oa
~
exp(-4). 20,
In this case, the probability density of K is given by (2.7 1)
108
ENHANCED BACKSCATTERING IN OPTICS
where
=o
for n > 1 .
Also, K ,= J(1 + fl; z'>/fl; z 2 is the maximum feasible value of K corresponding to the focusing of the beam in the plane of scatterers, z = a, L = L / R * is the dimensionless thickness of the scattering slab, R = l/a, is the typical depth of focusing, and /I,= 2R ,/kb2 is the wave parameter of the beam at depth L = R , . Exceedingly large values of K can be observed for K ,, % 1 or, equivalently, at p, z 4 1. For moderate values of K that are far from K , 1//3, z the probability density (2.64) is described by a relatively simple expression, corresponding to the geometrical optics approximation (p, + 0),
,
-
The plots of wK ( K )for z = 0.5, 1, and 2 are given in fig. 2.16. The maximum has shifted toward values of K that do not exceed unity. This implies that detection of a reduction, rather than an enhancement, of the signal, compared with that due to the plane interface, is more probable. For example, at z = 1 the probability of obtaining K = 1 is 0.523. At the same time, from eq. (2.72) it is clear that for 1 4 K 4 K the probability density falls at a comparatively
,
Fig. 2.16. Probability density ofthe backscatter enhancement factor for a surface with Gaussian statistics of the curvature.
It,! 21
109
S O L I D S I M M E R S E D IN A T U R B U L E N T M E D I U M
slow rate, approximately as K - ’. This means that fadings of the signal, which correspond to K < 1, alternate with giant surges of K %- 1. These rare but strong intensity surges can raise the average enhancement factor
significantly above unity. The plots of ( K ( L ) ) versus dimensionless depth z = L / R , for j3, = 0.01 and 0.05 are given in fig. 2.17. A sizeable rise of ( K ) above unity at z 1 and 8, 4 1 will result in an equal measure of reduction for z %- 1 (APRESYANand VLASOV[ 19881). Indeed, at a great distance where z 9 1, the rough surface always causes a defocusing of the beam, even where there is a focusing action, as in the initial segment 0 < L < R,. The beam width increases in proportion to OfL, with Of = 2/cr,b = 2R ,/b being a typical angle of divergence of the beam. On the other hand, the unperturbed beam expands at large depths as O,L, where do = l / k b is the angle of beam divergence due to diffraction. At large depths of the scattering layer the competing defocusing and diffraction divergence of the beam leads to
-
which becomes small for P, 4 1. In the case of Gaussian curvature of the rough surface a, more accurate developments yield the ultimate value as
Specifically, at P,
=
0.01, ( K ( m ) ) = 0.22.
I
I
0
1.0
I
2.0
I
3.0
1
4.0
*
z-L/U*
Fig. 2.17. Enhancement factor ( K ) as a function of the dimensionless depth z
= L/R,
.
110
ENHANCED BACKSCATTERING IN OPTICS
2.2.8. Backscattering of pulse signals A backscattered echo pulse is expanded in time because of multiple paths of backscattering; moreover, its arrival time becomes a random variable. For very short pulses the echo signals corresponding to various coherent channels diverge in time and no backscatter enhancement will be observed. This leads us to the following condition for observation of enhanced backscattering of pulse signals. The pulse duration T should greatly exceed the mean square time of random lag, i.e., T 4 a7,
(2.73)
The quantity a: consists of two terms. The first, a:o, is the variance of time it takes to propagate along a straightened ray -
1 2c
z= -
IoL
2 ( p , x) dx .
This quantity is related to the variance of the eikonal a; as a,'o = a$/c2. The second part, a:,, is due to the elongation ofthe ray caused by the random inhomogeneous medium. It may be estimated by
where 0,' is the mean square of lateral deviation of a ray defined by eq. (2.1 1). Condition (2.73) for enhanced backscattering also allows a spectral treatment. The bandwidth 62 = 2n/T of the signal should not exceed the band of coherence = 2n/az. Since the variance of phase increases twice on the way from source to scatterer and back when compared with the variance of phase on the rectilinear path of length 2L (see $ 6.3), one might expect doubling of the variance of the arrival time. As far as our information goes, this effect has not yet been subjected to experimental verification in laser sounding applications. 2.2.9. Moving random inhomogeneities of the medium The reciprocity theorem, eq. (2.1), holds true only in media whose inhomogeneitiesare time invariant. Time-dependent fluctuations of permittivity
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
111
can lead to a reduction and even disappearance of the enhancement effect because the wave has to pass through different inhomogeneities on its forward and backward paths. To illustrate the effects feasible, let us calculate the average intensity of a wave emitted by a point source uo(p) = 6 ( p ) and backscattered by a point scatterer of f ( p ) = 6(p). Given saturated fluctuations and uniform “frozen” motion of inhomogeneities in the atmosphere with wind velocity V, then (2.74) where 0, = V , / c , V , is transverse to the ray, and d(p,, p , x) is the mean square of the phase difference over a distance x when the points of radiation are spaced by pa and the points of observation by p. [This quantity is defined by eq. (2.27).] From eq. (2.74) it follows that a uniform drift of the inhomogeneities at a velocity V shifts the enhancement effect of K = 2 from the point of emittance a distance pv = 20,L by an amount equal to the transverse travel of the inhomogeneities in time At = 2L/c. This effect of the shift of the enhancement BUNKIN,VLASOVand KRAVTSOV [ 19841 region has been noted by AKHUNOV, and KRAVTSOVand SAICHEV[1985]. It should be obvious that when the magnitude and direction of the drift change markedly along the path, the enhancement effect weakens until it vanishes altogether. The effect also disappears when the condition that the inhomogeneities travel as a “frozen” cluster is no longer velid, i.e., when the inequality At = 2L/c 4 z, is violated, where z, is the lifetime of an inhomogeneity in the system where it is at rest. This effect of vanishing enhancement, i.e., the transition from (Is) = 21: to ( 1 , ) = Z:, may be employed for monitoring the atmosphere.
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 rejected wave The effect of double passage of a wave through the same random inhomogeneities of the medium is essential in evaluating the efficiency of adaptive systems and phase-conjugation systems, i.e., mirrors reversing the wavefront (WFR mirrors). Such systems are being designed to compensate the effect of
I12
“I, $ 2
ENHANCED BACKSCATTERING IN OPTICS
random inhomogeneities in the medium and to attain the diffraction limit in focusing the reflected intensity into the neighborhood of the source. In considering the properties of a wave reflected in a random inhomogeneous medium, we shall assume an ideal reversal of the wavefront within the aperture of the WFR mirror and suppose also that in a time At = 2L/c, while the wave travels from source to mirror and back, the inhomogeneities will remain almost unchanged. If a monochromatic wave with complex amplitude u0 ( p ) emitted in the plane x = 0 toward a WFR mirror with reflection factor f ( p ) placed at x = L, then the complex amplitude of the reflected wave measured in the source plane x = 0 is Us@)
=
s
=
~ ~ l ( p ~ ) f ( p ’ ~ ) gp *” ,(Lp)’g,( p ’ ,p ” , L ) d2p‘ d2p“ .
f ( p ’ ) U * W , L ) g ( p , p ’ , L ) d2P‘
(2.75)
We look for a condition which would ensure that the WFR mirror makes up for the effect of random inhomogeneities of the medium on the field of the reflected wave. For this purpose it suffices to consider the expression derived and SAICHEV[ 198 11 for the average field. by POLOVINKIN From eqs. (2.75) and (2.26) it follows that (Us ( P I ) =
‘S
1 L
UC?
xeXP[-
( P - PI F( P l L )
(k)
513
ik
+ -L ( p * p ) - 2-ikLp 2
1
d2p,
(2.76)
where (2.77) is the scattering diagram of the mirror with an angular width of A0 = l/ka, where a is the mirror radius. If p
-A8L 4
P,(L)
1
(2.78)
from eq. (2.76) it is obvious that the average field of a wave reflected by a WFR mirror in a turbulent medium coincides with that reflected in vacuum. There-
11, § 21
113
SOLIDS IMMERSED IN A TURBULENT MEDIUM
fore, under this condition a WFR mirror compensates completely for the effect of random inhomogeneities on the field of the reflected wave in the plane x = 0. Condition (2.78) allows for a dual interpretation. On the one hand, in order to achieve compensation it is required that the random phase increment along any backscattered ray should almost coincide with that along the respective incident ray. Backscattered rays lie in a cone with an included angle dB l / k a . Clearly, the phase increments acquired along the incident and reflected rays will differ little from one another when the characteristic distance between them, L AO, in the plane of the source is smaller than pc(L),which leads to eq. (2.78). On the other hand, compensation in a turbulent medium requires that the rays incident on the WFR mirror should be almost the same as in vacuum. This requirement will be met if the characteristic lateral shift ap of the rays due to the random walk in the turbulent medium is within the mirror aperture, i.e., a,,( L )5 a . Clearly, this inequality is also equivalent to condition (2.78) (SAICHEV[ 19821).
-
2.3.2. Average intensity of a wave re8ected from a WFR mirror: Effect of superfocusing In applications the major interest is focused on the distribution of the average power flux of the reflected wave in the vicinity of the source. The total flux is given by (2.79) We limit our analysis to a typical situation where the prime field u, is due to a point source. Then, P = W j lf(P)I2d2P,
(2.80)
-
Z,(L) 1/L2 being the intensity of a spherical wave incident on the mirror in vacuum. We represent the average intensity of a wave reflected from the WFR mirror as a sum of the coherent and noncoherent components
(L(P)>
=
4(P) +L(P)
with
L(P)
I
= (U,(P))
I2
9
(2.81)
I14
ENHANCED BACKSCATTERING IN OPTICS
and I ” ( P ) =
The coherent component Zc(p) is the lower bound of the average intensity of the reflected wave, since always I c ( p ) 6 ( Z s ( p ) ) . For a point source, 1 , (P) =
m-4exp [
- 2 (P/Pc)5’3
1
(2.82)
9
where I,”(P) = 10 ( L ) IF(P/L)I 2/L2
is the intensity of the wave reflected by the WFR mirror in vacuum. It is remarkable that Z,(O) = 1,”(0), which implies that the average intensity of the reflected wave, as measured near the source, cannot be lower than the intensity Z,”(O) of the wave reflected in vacuum. In other words, a turbulent medium cannot reduce the wave intensity at the light spot (AKHUNOV,BUNKIN, VLASOVand KRAVTSOV [ 19841). Once condition (2.78) is satisfied, then I , ( p ) = I;(p), so that the radius of the focal spot in the plane of the source is L/ka as it is in vacuum. However, when condition (2.78) is violated, the focal spot of the reflected wave defined by eq. (2.82) contracts to about a size of pc L/ka. We shall refer to such focusing of a coherent field improved over that in vacuum as superfocusing. The effect of superfocusing is explained by a contraction of the effective radiation pattern of the WFR mirror, which forms the coherent component, N
Fee@)
=
F ( 8 ) exp [ - (8/8c)5/3].
(2.83)
In agreement with this formula only those rays which are reflected within the coherence cone 8,< Oc(L) ?: pc(L)/Lcontribute to the coherent component of the reflected field. The superfocusing of the coherent component may also be treated as an effective increase of WFR mirror size in a turbulent medium. We determine the effective reflection coefficient for the coherent component as
LAP)= =
s
FeR(8) exp(ikp. 8) d28
1
f ( p + P’) ~ J P ‘ L , ) d2p’ ,
where W,(p, L ) is, as before, the probability density of the lateral shifts of rays determined by eq. (2.37).
11, § 21
SOLIDS IMMERSED IN A TURBULENT MEDIUM
I15
phase
conjiiqated rni.rrbr
Fig. 2.18. Effective rise of WFR mirror size in a turbulent medium due to random walk of rays (effect of superfocusing).
The effective enlargement of the WFR mirror may be readily explained by a random walk of rays in the turbulent medium. Indeed, random walks can carry to the mirror even those rays which missed the mirror in vacuum (fig. 2.18). Accordingly, for a coherent field the size of the mirror increases to a + o,(L). Physically, the reasons for obtaining a narrow region of focusing for Zc(p) are different for pc > a, and pc < op The first inequality, p, > op,holds true in the range of weak fluctuations of intensity incident on a wavefront-reversing mirror. Excluding amplitude fluctuations, which are insignificant in this case, the field of a reflected wave in the plane of the source may be represented as
where u,”(p) is the complex amplitude of the reflected wave in vacuum, and rp(p,L) is the random phase increment (with account for phase inversion) acquired along the paths of the incident and reflected waves. The mean square of q ( p , L ) is comparable to unity only at p - pc(L). Consequently, if pc(L)> a&), then ( I , ( p ) ) N I:(p), irrespective of the WFR mirror size, and for a < op the superfocusing is caused by purely phase effects, i.e., by an incomplete compensation achieved with the WFR mirror of the random phase increments en route from the source (i.e., WFR mirror) point of observation. Now we turn to long paths along which o,(L) > pc(L). If we separate the correlations with the rules appropriate for a Gaussian field, under saturated fluctuations we obtain for the average intensity
1 I6
ENHANCED BACKSCATTERING IN OPTICS
where Iind(p) =
s
I:(P+ P I )
wp(p1,
L)d2p1
(2.85)
is the intensity of the reflected wave calculated on the assumption that the fluctuations are statistically independent in the forward and backward paths, W,(p) is defined by eq. (2.37), and for the turbulent medium d ( p ) = 4(p/p,)s'3. From eq. (2.84) it follows that Zind(p)N I,O(p) for sufficiently small mirrors only (a < p,), when the diffraction divergence of the rays reflected from the mirror is about L/ka, which exceeds the lateral shift np, i.e., when the WFR mirror is nowhere better than an ordinary reflector of the same size a < p,. The simple approximation (2.84) does not always lead to physically correct conclusions. For example, the full power flux of the reflected wave is P + P, rather than P, i.e., exceeds the true average flux (2.80) by
P, =
s
Z,(p)d*p.
Specifically, when condition (2.79) of compensation of the effect of inhomogeneities is satisfied, then P, = P so that the average flux computed with eq. (2.84) is twice the true value. Nevertheless, the easy-tractable ray considerations allow to modify the expression for the noncoherent component I,,(p) so as to keep the advantages of eq. (2.84) but to eliminate the energy paradoxes peculiar to this expression. As already mentioned, the coherent part of the field of the reflected wave is formed by rays whose angle of reflection differs from the angles of incidence by a value smaller than the angle of coherence, 8 5 O,(L). The other rays do not contribute to the coherent part of the reflected field and form an incoherent component. If we suppress the coherent part responsible for scattering within the cone 0 5 0, in the scattering pattern of the mirror (2.77), we obtain an effective pattern for the noncoherent component of the reflected field, e.g., in the form
which secures the energy balance
iw)t2.
IF,(~)I~=
If we calculate I,, with this F,, directional pattern, assuming statistical independence of random inhomogeneities of the medium in the forward and back-
1 I7
SOLIDS I M M E R S E D IN A T U R B U L E N T M E D I U M
11, § 21
(2.86)
(2.87) and yields correct results in all the preceding situations that have been considered. In particular, in the saturated fluctuation regime (p, < op) eq. (2.87) yields the expression (‘s(p)>
=
I c ( p ) + Iind(p) -
pc w p < p ?
L, *
which at a 4 crp almost coincides with eq. (2.84) in view of the smallness of the last term on the right-hand side. For strong fluctuations, pc 6 o,,,this formula suggests that the envelope of the average intensity of the reflected wave will have a different structure, depending on the WFR mirror size. When a > ap, the mirror completely compensates the effect of turbulence on the reflected field, so that ( I , ( p ) ) = I,”@) as in the case of weak intensity fluctuations. For op> a > pc the average intensity consists of a high, sharp peak of I , ( p ) of radius pc, rising up to I,O(O)over a wide and low pedestal Iind(p), which, however, carries a considerable proportion of the energy flux. When a < p c , the average intensity of the reflected wave does not differ from the average intensity of the wave reflected from a common reflector of the same size. Certain aspects of the spatial flux distribution of a wave reflected from a WFR mirror have been described by AKHUNOVand KRAVTSOV[ 1983a1, SAICHEV[ 19831, KRAVTSOVand SAICHEV[ 19851, and MALAKHOV, POLOVINKIN and SAICHEV[ 1983, 19871.
-
-
2.3.3. EfSect of a drft of random inhomogeneities on the eficiency of WFR mirrors A time variability of the random inhomogeneities of the medium and even a transverse drift of inhomogeneities can result in a degraded focusing of the wave reflected from a WFR mirror in the neighborhood of the source (MALAKHOVand SAICHEV[ 19811, AKHUNOV,BUNKIN,VLASOV and KRAVTSOV [ 1982, 19841, and POLOVINKIN and SAICHEV[ 19841). In this
118
ENHANCED BACKSCATTERING IN OPTICS
[II, § 2
section we limit consideration to a case of turbulent inhomogeneities that drift across the path with a constant velocity V. Using reasoning similar to that employed earlier leads us to conclude that the average intensity of the reflected wave is given by eq. (2.87) in the vicinity ofthe point source, Iind( p ) is defined by eq. (2.85),and the coherent component of the reflected field is given by
(2.88) where 6, = V , / c . It is obvious that the motion of inhomogeneities alters the distribution of the average reflected intensity if the angle of the drift 20, exceeds the angle of coherence 0, p J L ; i.e., 20, > 8., However paradoxical it may seem at first glance, in a turbulent medium the efficiency of a WFR mirror is least for large mirrors, a > max {p,, a,}. In this case, from eqs. (2.85)-(2.88) we have
-
where v
=
exp[ - 2 ( 8 , / e ~ ~ / ~ ]
is introduced to reflect a reduction of the coherent component of the reflected wave. The low efficiency of focusing by a large WFR mirror in a drifting turbulent medium is attributed to a narrow directional pattern of the large mirror (A0 = l/ka < &). Therefore, rays reflected from the mirror pass through random inhomogeneities and emerge shifted in the transverse direction by a distance 20,L from the locations of the inhomogeneities through which the incident rays have passed. In contrast to the situation in vacuum, the reduction of the WFR mirror size in a turbulent medium may result in a relative improvement of the focusing (POLOVINKIN and SAICHEV [ 19841) because in the cone of reflected rays with the included angle A 0 > 2 4 , there always exist rays passing through the same inhomogeneities through which the respective incident rays have passed. As a result, the superfocusing spot of the coherent component shifts, with respect to the source, a distance 20,L rather than disappears as is the case for a large mirror.
11, § 21
119
SOLIDS IMMERSED IN A TURBULENT MEDIUM
2.3 A. Magic-cap ejfect : Compensation of backscattering from small-scale inhomogeneities by a WFR mirror With an ideal, boundless WFR mirror of reflectivity 1 f I = 1, one more very interesting effect may be observed that will be referred to as the effect of the magic cap. This effect occurs because the ideal WFR mirror compensates not only the phase fluctuations of the field caused by large-scale inhomogeneities [ 1981, but also the backscattering from small-scale inhomogeneities (SAICHEV 19821). A closely related effect of compensation of phase distortions in wavefront inversion in random acoustic waveguides has been pointed out by GELFGAT[ 19811. We illustrate the magic-cap effect by the example of a scalar monochromatic wave satisfying the Helmholtz equation A u + k 2 [ 1 + .F(p,x)]u=O.
Assume that inhomogeneities are present only in a final layer 0 < x < L bounded at x = L by a wavefront-reversing mirror of reflectivity f. Following MALAKHOVand SAICHEV [1979], we represent the field as u ( p , x ) = T ( p , x) + R ( p , x), where T ( p , x) is the field of the forward wave propagating along the x-axis, and R ( p , x) is a wave in the backward direction. These waves satisfy the following equations,
Here, the operator M =m /,i and A = 112. Let a wave equal to uo(p) at x = 0 be incident on the inhomogeneous layer from the side of negative x-values. Then eqs. (2.89) should be solved with the boundary conditions (2.90)
If we suppose that &? and I? are purely imaginary, which corresponds to neglecting the total internal reflection, then with (2.89) and (2.90) we can verify
120
ENHANCED BACKSCATTERING IN OPTICS
that the difference Q
=
R ( p , x)
- fT*(p,
x) satisfies the equation
3 + dQ= +PAI ( Q - fQ*)+ (1 - /f12)$k2fiIT ax
(2.91)
with the boundary condition Q(p, 15)= 0. If 1 f 1 = 1, i.e., ideal reflection, then the last term of eq. (2.91) equals zero, the equation becomes closed for Q, and its solution is identically zero, Q ( p , x) = 0. Hence, for If I = 1, the relation R ( p , x) = f T * ( P , x)
holds true at any distance from the mirror. In particular, at x = 0 we have R ( p , 0) = &&), which implies that even under backscattering from inhomogeneities of the medium the ideal WFR mirror provides a complete compensation of the distortions introduced by these inhomogeneities in the reflected wave. Actually, the WFR mirror sets up such phase relations between the fields of T and R that they extinguish the backscattering from small-scale disseminations, making them invisible for the observer outside the inhomogeneous layer, i.e., at x 0. Let us also examine the behavior of the field inside a random inhomogeneous layer confined by an ideal WFR mirror. Assume for simplicity that both the wave incident on the inhomogeneous layer and the inhomogeneities of the layer proper change smoothly along the transverse coordinates in the scale of the wavelength. It is convenient to represent the field inside the layer in the form
-=
u ( p , x) = eik.r
v ( ~x) ,+ e W L - ~ f v(P, 4
9
where the complex amplitudes U and V of the forward and reflected waves satisfy the system of parabolic equations following from eq. (2.89) upon substituting ik(1 + +Al) for and - ik for fi, viz., ~
av ax
2ik - + AI U + k2 I U + k2fi V = 0 ,
-2ik
av + A , V +
-
ax
k21 V + k2ji*U = 0 , (2.92)
11, I 21
121
SOLIDS IMMERSED IN A TURBULENT MEDIUM
If we represent the complex amplitude of the reflected wave as the sum
the equations for U ( p , x ) and W ( p ,x) follow from eq. (2.92) in the form 2ik
au + A I U + k 2 B U + k2jlfU* + k 2 j l W = 0 ,
-
ax
- 2ik
a w + A L W + k2 E W - k2j l *f W* + k2(1 - I f
~
ax
U(P9 0) =
1' ) p* U = 0 , (2.94)
UO(P)
1
W(p, t)= 0 . If the backscattered intensity is rather weak in a random inhomogeneous medium, in the equation in W we may neglect the two last terms responsible for backscattering (in formal terms this corresponds to j l = 0). In this case W = 0 for any J and eq. (2.93) describes the familiar effect of compensation of the effect of large-scale inhomogeneities on the wave reflected from a WFR mirror of a sufficiently large size. If the backscattering from small-scale inhomogeneities of the medium is significant, then, as noted for the more general case, for x < 0 the inhomogeneities do not influence the profile of the reflected wave only if If I = 1. An interesting effect of localization (accumulation) of the energy of a wave in the small-scale random inhomogeneous layer positioned in front of the WFR mirror occurs where a compensation of backscattering takes place. To demonstrate this effect, we assume that the random inhomogeneities in the layer 0 < x < L are such that the diffusion approximation (see, e.g., the monograph by KLYATSKIN[ 19801) is justified. This approximation is equivalent to replacing small-scale fluctuations B (p, x ) and j l (p, x) with Gaussian &correlated random fields with correlation functions
(EG) =o, in which for clarity we assume that A ( s ) 2 0 and a ( s ) 2 0.
122
ENHANCED BACKSCATTERING IN OPTICS
For
If1
E
1 the field U ( p , x) satisfies the equation
au
2ik - + A L U + k2 ?. U + k2GfU* = 0 ,
ax
U(P, 0) = % ( P )
9
which follows from eq. (2.94). Using standard closing procedures for closing [ 1980]), it is not hard to transform this equation averages (see, e.g., KLYATSKIN to the one for the coherence function of field U ( p , x), namely
ar ax
i (V,V,)r k
t ik2[D(p) -
a]r=$ k 2 a ( s ) r , (2.95)
D(p)=A -A(p), A =A(O), a=a(O). Although a general solution of this equation is yet to be found, it is not hard to determine the flux P ( p , x) =
w,p, x) d2R .
Assuming for simplicity that P ( p , 0) is a real function ofp, which is valid, e.g., for the normal incidence of a collimated beam on a nonhomogeneous medium, we obtain from eq. (2.95) P(p, x )
=
P ( p , 0) exp [ - ak2D(p)x + bk2(a
+ a ( p ) ) x ].
It should be obvious that in the presence of small-scale inhomogeneities of the medium (a > 0), the average flux of the incident (as well as the reflected) wave increases exponentially as it approaches the WFR mirror, P(x)
=
s
( I ( p , x)) d2p = P ( 0 ) exp($k2ax).
Thus the small-scale, random inhomogeneous layer in front of the I f 1 = 1 mirror accumulates, on average, the energy of the incident wave. Inversion of the phase of a wave reflected from the WFR mirror actually converts the inhomogeneous layer into a resonator. This seems to prove indirectly that the effects of partial reversal of backscattered wavefronts caused by multiple coherence paths may be one of the mechanisms of wave localization in multidimensional, small-scale, random inhomogeneous media.
I ] , § 31
RANDOM MEDIA
123
4 3. Enhanced Backscattering by a Random Medium 3.1. OVERVIEW
In the preceding section we have considered backscatter enhancement in light scattered from a body embedded in a random medium. A similar phenomenon occurs in multiple scattering of light, or other types of waves, in a randomly inhomogeneous medium where there are no clear-cut scatterers that stand out against a background. In this case, the intensity of the backscattered radiation is found to be almost twice as large as that predicted by transport theory. This effect, called coherent backscattering, has been extensively discussed in the literature. It seems to have been first noted by RUFFINEand DE WOLF [ 19651 and WATSON[ 19691 in a theoretical analysis of radio waves backscattered by the turbulent ionospheric plasma (see also DE WOLF[ 19711). BARABANENKOV [ 1973, 19751 had discussed the effect in connection with a microscopic basis for the phenomenological theory of radiative transport that ignores the previously mentioned interference processes. The salient features of the effect have been demonstrated experimentally by [ 19841 and KUGA,TSANGand ISHIMARU [ 19851, who KUGAand ISHIMARU observed an enhanced backscattering from a medium composed of a suspension of latex particles. The earlier enhancement was less significant that that in later experiments. TSANGand ISHIMARU [ 1984, 19851 have interpreted the results by considering first double and then multiple scattering. A recent upsurge of interest in the effect is attributed primarily to the recognition that the processes underlying coherent backscattering are intimately related to the similar processes that occur when electrons interact with [ 19831,AKKERMANS and impurities in disordered metals (JOHNand STEPHEN MAYNARD [ 1985l). Effects of the type of coherent backscattering, commonly referred to as weak localization (ABRAHAMS, ANDERSON, LICCARDELLO and RAMAKRISHNAN [ 19791, GORKOV, LARKINand KHMELNITSKII [ 19791, BERGMANN [ 1984]), lead to a decreased conductivity at low temperatures when compared with that predicted by the classical kinetic theory. Simultaneously with the interference effects the behavior of conductivity is considerably influenced by the Coulomb interaction of electrons (ALTSHULER, ARONOV,KHMELNITSKII and LARKIN [ 19821). Optical experiments make it possible to observe directly how the pure effects of weak localization enhance backscattering of light from dense water suspensions of submicron particles of latex or polystyrene (WOLF and MARET
I24
ENHANCED BACKSCATTERING IN OPTICS
III,§ 3
[ 19851, VAN ALBADAand LAGENDIJK [ 19851, LAGENDIJK, VAN ALBADAand VAN DER MARK [ 19861, VAN DER MARK,VAN ALBADAand LAGENDIJK [ 19881, WOLF, MARET, AKKERMANSand MAYNARD [ 19881, ETEMAD [1988], DAINTY,Q u and Xu [1988], Q u and DAINTY[1988]). Similar experiments with solid disordered media have been reported by ETEMAD, THOMPSON and ANDREJCO[1986] and KAVEH,ROSENBLUH,EDREIand FREUND[ 19861. These studies have demonstrated that the angular distribution of scattered intensity is an almost triangular peak pointed backwards, the relative enhancement being close to two. The form of the peak as a function of depth of scattering layer, extinction, size of scatterers, and polarization of incident and detected radiation has been also investigated. A theoretical treatment of the effect has been carried out by AKKERMANS, WOLFand MAYNARD[ 19861 with the scalar theory formalism. They calculated the contribution to the beam intensity of cyclical (or maximally crossed) diagrams and their sum for the case of point scatterers. The sum was computed with the aid of the diffusion approximation of radiation transport theory, which [ 19731. This approximahas been devised for this purpose by BARABANENKOV tion has been explored by STEPHENand CWILICH[ 19861 and CWILICHand STEPHEN[ 19871 to investigate polarization effects in coherent backscattering, and by BARABANENKOV and OZRIN[ 19881 and ISHIMARUand TSANG[ 19881 to study the effect of the size of the scatterers on the angular distribution of intensity. The characteristic features of the angular distribution have been elucidated by SCHMELTZER and KAVEH[ 19871 on the basis offield-theoretical methods. Extensive research efforts have been devoted to various factors leading to a smoothing of the coherent backscattering peak. These are primarily the finite and TSANG extension of the scattering medium and its extension (ISHIMARU [ 19881, ETEMAD,THOMPSON, ANDREJCO, JOHN and MACKINTOSH[ 19871, EDREIand KAVEH[ 19871, AKKERMANS,WOLF, MAYNARDand MARET [ 1988]), the motion of scatterers (GOLUBENTSEV [ 1984a]), and the presence of a magnetic field in the case of a gyrotropic medium (GOLUBENTSEV [ 1984b] and MACKINTOSHand JOHN [1988]). The two last factors result in a breakdown of the symmetry relative to the inversion of time and violate the reciprocity condition. This brief overview indicates that various aspects of coherent backscattering and weak localization of light have been well documented recently. In electron transfer theory the weak localization is thought to foretell the strong or Anderson localization. In the optical range the possibility of strong localization has also been considered (see, e.g., the review papers of KAVEH[ 19871 and JOHN [ 1988]), but so far no observation has been reported for this regime.
11. § 31
RANDOM MEDIA
125
3.2. GENERAL THEORY OF MULTIPLE SCATTERING: LADDER A N D MAXIMALLY CROSSED DIAGRAMS
A simple case of a scalar wave field u(r) satisfying the Helmholtz equation will be considered initially. Note in passing that the solution of the scalar problem is applicable when the illumination and detection beams are copolarized. Let a plane monochromatic wave uo(so,r) = exp(ik,s,. r), k, = 2 n / l , propagating along the direction of the unit vector so be incident on a confined random inhomogeneous medium. Then the field obeys the equation
(3.1) where I ( r ) is the random part of dielectric permittivity responsible for multiple scattering. For a source at infinity the solution of Helmholtz equation (3.1) is represented with the aid of the scattering operator T(r,r ’ ) as (LAX [ 195 11, FRISCH [ 19651 and GOLDBERGER and WATSON[ 19641)
Here and below it is assumed that the integration is taken over the entire volume occupied by the scattering medium, and G,(Y) = ( - 47~)-I exp(ik,r) is the Green function for free space where I = 0. According to eq. (3.2),in the Fraunhofer zone, the wavefield u(r) is the sum of the incident wave and a spherical scattered wave whose amplitude is the Fourier transform of the scattering operator accurate to a constant multiplier. We assume, for clarity, that the scattering medium is constituted by discrete scatterers. The preceding enhanced backscattering experiments used dense discrete media in which the average interscatter distance was comparable with the size of the scatterer. In such media, mutual correlations of scatterers may play a significant role. Let us expand the scattering operator T into the orders of multiple scattering from individual scatterers, and substitute the result into eq. (3.2). Averaging the respective expansions of u(r) and u ( r I )u*(rz)over the ensemble of realizations of the scatterers yields the Dyson and Bethe-Salpeter equations for average field and coherence functions (FRISCH[ 19651 and FINKELBERG [ 19671
( ~ ( r ) =) uo(r)+
d3r1d3r; G,(Y - r l ) W r l , r i ) ( u ( r ; ) ) ,
(3.3)
126
ENHANCED BACKSCATTERING IN OPTICS
W,§3
(u(r1)u*(y2)) = ( u ( r 1 ) ) ( u * ( r 2 ) ) t
d3r; d3r[ d3r; d’r,” ( G ( r l ,r ; ) ) (G*(r,, r ; ) )
x K ( r i , r ; ; r y , r i )( u ( r ; ) u ( r , ” ) ) ,
(3.4)
where G ( r l ,r ; ) is the Green function of Helmholtz equation (3.1) having the average ( G ( r l ,r ; )), which satisfies the Dyson equation, which differs from eq. (3.3) by having Go(r - r ‘ ) and ( G ( r , r ‘ ) ) in place of uo(r) and ( u ( r ) ) , respectively. The mass operator M(r, r ’ )and the intensity operator K ( r , , r,; r ; , r ; ) of the Dyson and Bethe-Salpeter equations characterize the optical properties of the effective inhomogeneities for coherent and incoherent (or partially coherent) fields. These operators are evaluated by the diagram technique for the respective discrete medium. Examples of diagrams for both A4 and K are given in fig. 3.1. The solution of the Bethe-Salpeter equation may be represented with the aid of the correlation function of the scattering operator (LAX [ 19511, BARABANENKOV [ 19751) W 1 , r 2 ; G , r 9= ( W l , r ; ) T * ( r z , G ) )- ( T ( r l , r ; ) ) ( T * ( r 2 , r ; ) ) , as follows, (u(R1) u*(R2))
=
(u(R1))
+
(u*(R2))
d3r, d’r, d’r; d’r; Go(R,,r l ) G*(R2,r2) x ~(rl,r2;r~,r;)uo(r~)u*(r;).
(3.5)
Fig. 3.1. Diagrams for the mass operator M , and intensity operator K , in the single-group approximation. Solid lines correspond to the Green function Go, horizontal dashed lines to GX. and wavy lines to the correlation function of scatterers. Nonhorizontal dashed lines correspond to the interaction operator or the scattering operator for an isolated scatterer, but in the last case the first diagram for&f, is absent.
127
RANDOM MEDIA
11, § 31
This representation indicates that the correlation function of the field is expressed directly through the correlation function of the scattering operator. Denote by ~ , , , , , ( r )the average energy flux of the wavefield less the energy flux of the average field. Incorporating the concept of albedo (see, e.g., AKKERMANS, WOLFand MAYNARD [ 1986]), a(s, so), which characterizes the intensity of the fluctuation component of the scattered field propagating along the direction of the unit vector s,into the representation of eq. ( 3 . 9 , we obtain for the Fraunhofer zone of the medium
(3.6) where s = R / R , C is the cross section of the medium, and c
O(p, q ; p ' , q ' ) =
J
d3r1d3r2d3r;d3r; V(rl, r2; r ; , r ; )
x exp ( - ip * rl
+ iq - r2 + ip'
-
r ; - iq' r ; )
is the Fourier transform of the correlation function of the scattering operator. In addition to eq. (3.5) actual fluctuations of the scattering operator are calculated with another representation of the Bethe-Salpeter equation ( 4 r I ) u * ( r 2 ) )= (u(rl)) ( u * ( r 2 ) )
+
d3r; d3r; d3r: d3r; ( G ( r , , r ; ) ) ( G*(r2,r ; ) )
x r(ri,r;;ry,ri) ( u ( r ; ) ) ) ( u * ( r i ) ) .
(3.7)
The vertex function r(rl,r,; r ; , r ; ) satisfies an equation obtained by substituting eq. (3.7) into the Bethe-Salpeter equation (3.4), which, written symbolically, has the form
r =K + K
( G ) (G*) r ,
(3.8)
where for brevity we have dropped the arguments of functions and integrations like (3.4). Comparison of the two representations (3.5) and (3.7) of the solution to the Bethe-Salpeter equation yields a relation between the correlation function of
128
ENHANCED BACKSCAlTERING IN OPTICS
111,s 3
the scattering operator and the vertex function, G,G;UG~G; = ( G ) ( G )
r (G)
(G*),
which is resolved for the correlation function of the scattering operator
u =xx*ryy* with the aid of two auxiliary operators X=l+(T)Go
and
Y=l+Go(T),
whose origin may be easily traced by representing the field (3.2) in terms of the scattering operator of the medium. In view of the reciprocity of the scattering operator (FRISCH[ 19651)
T(r, r ’ ) = T ( r ’ ,r ) ,
(3.9)
the operators X(r, r ’ ) and Y(r, r ’ ) are related by
X(r, r ’ ) = Y(r’,r ) , Finally, from the definition of Y and eq. (3.2) it follows that
s
d’r’ Y(r, r ‘ ) uo(r’)= ( u ( r ) ) .
Now, we have arrived at the fundamental formula for the albedo
(4n)2 z
d’r, d3r2d’r; d3r; x ( u( - s, r, )> ( u * ( - s, r 2 ) ) U r , ,r2;r ; , r ; ) x (u(s0, r ; ) ) (u*(so, r;>>
?
(3.10)
where (4% r ) ) represents the solution to eq. (3.3) with the boundary condition involving uo(s, r). Formula (3.10) is exact and rather general. It is commonly used under some simplifying assumptions made with respect to the medium and the wavefield. The model of a medium constituted by independent (uncorrelated) scatterers ofcharacteristic size ro (FOLDY [ 19451, LAX[ 195 11, DOLGINOV, GNEDIN and SILANTYEV [ 19791) is appropriate at a low density n of scatterers distributed in space so that nr: 4 1. For dense scattering media where nr; is not small any longer, a model should take into account the correlations of scatterers. The single-group approximation (FINKELBERG [ 19671, BARABANENKOV and FINKELBERG [ 19671,
11, I 31
RANDOM MEDIA
129
BARABANENKOV [ 19751) takes into account the correlation functions of scatterers of all orders. In this approximation the mass operator M and intensity operator K equal the sums M , and K , of all possible diagrams linear in the correlation functions of scatterers. Some of these diagrams are depicted in fig. 3.1. The kernels M I and K , are put in correspondence to inhomogeneities of the medium of a spatial scale r, that is of the same order as the maximum of either the size of scatterers or their correlation radius. The representation of localized inhomogeneities forms the basis of a physically appealing approach to coherent and incoherent scattering of waves. This approach is based on the assumption that the main contribution to wave scattering is due to the far configurations of inhomogeneities when these are in the Fraunhofer zone with respect to one another (BARABANENKOV and FINKELBERG [ 19671). For coherent scattering the Fraunhofer approximation is equivalent to neglecting the spatial dispersion, and for incoherent scattering it is equivalent to a description in terms of radiative transfer theory. Neglecting the spatial dispersion reduces the Dyson equation for the average Green function in an unbounded scattering medium to the Helmholtz equation with an effective compiex wavenumber k , related to the Fourier transform of the mass operator M , ( p ) by the relation
k:
ki
- Ml(ko).
The expression for the average function GI satisfying the Dyson equation with M I is then GI@,r ' ) x -(4nR)-' exp(ikR - R/21,,), (3.11) where R = 1 r - r' 1, and l,>' involves the contributions due to elastic scattering, 1/1, and due to true absorption, 1/Iab, namely, le;'
=
- Im{M,(k,)}/k,
=
1/1+ l/fab.
(3.12)
In eq. (3.11) it is assumed that k , / k o is close to unity,
4 I&.
(3.13)
For a bounded scattering medium the question of whether or not it is possible to neglect dispersion near the boundary requires a separate consideration. Digressing from this problem for a moment, the average field in the medium on neglecting the reflection and refraction of waves at the interface can be written as ( u ( s , r ) ) zexp(ik,s*r- r/2feE),
where the origin is also placed inside the medium.
(3.14)
130
ENHANCED BACKSCA'TTERING IN OPTICS
PI, § 3
(3.15) and consider the solution QCL)to the approximate Bethe-Salpeter equation for an unbounded medium, which is derived from eq. (3.4) by substituting eqs. (3.15) and (3.11) for the inhomogeneous term and K , for the intensity operator. In this "ladder" approximation, in agreement with eq. (3.8), the vertex function becomes
r
I'z K ,
= K 1 QjCL'K I ' + r(L), rCL)
(3.16)
Expanding @ ( I - ) in scattering orders, i.e., in powers of K , ,yields a representation of in the form of a series whose terms can be represented by ladder diagrams, one of which is shown in fig. 3.2, along with the respective patterns of disturbance travel. The condition r, 6 leE, k, r; G (3.17) assumes that inhomogeneities of the medium are in the Fraunhofer zone with respect to one another (BARABANENKOV [ 19751). The intensity operator K , then can be represented as
K , ( R + i r , R - i r ; R'
+ i r ' , R'
- '2 r ' )
and the following relations holds for both @(L) and Q0 (BARABANENKOV [ 1969]), the latter being defined as f ( R + $r, R - i r ; R' + i r ' , R' - 'r') 2
s
d3rd3r' exp(-ik-r
z ( 2 1 ~k,4 ) ~ b(k
+ i k * r ' ) ( @ ( L )a0) ;
- k,)
b(k' - k,)
(F, &),
(3.19)
Fig. 3.2. (a) Typical ladder diagram:the solid line correspondsto G, = Go t G,M, G , ,horizontal dashed line to G y , and vertical dashed lines to K , . (b)The path y corresponding to the ladder diagram in real space.
RANDOM MEDIA
131
where s = k / k , s’ = k ’ / k ,
$JR,
R ‘ , s’)
S;
=
6(s - s’) 6(s - s R R . ) x (4njR - R‘ I)-‘exp( - / R - R‘ j/ZeK),
(3.20)
( R - R ’ ) /I R - R‘ 1, and the quantity 9 is the Green function of the transport equation (CASE and ZWEIFEL[ 1967]), which follows from the Bethe-Salpeter equation for c P ( ~ namely, ), SRR. =
9 ( r , s; r ’ , s’) = c%(r, s; r ’ , s’) P
+ (47~)’J
d3rl d’s, d2s; &(r, s; r l , sl) x na(s,, s;) F(r,,s;;r ’ , s‘).
(3.21)
Here and below d2s is an elementary solid angle. The coefficient of elastic scattering of a unit volume of the medium is defined as n C ( S , s’)
=
W(kos, k o s ’ ).
(3.22)
For uncorrelated scatterers, a(s, s’) is the cross section of scattering of an individual particle. The mean free path 1 in elastic scattering can be obtained by
I-
’ =n
s
ds a(s, s’)
(3.23)
Now we return to eq. (3.10) for the albedo of the medium. We assume the scattering medium under consideration is a plane parallel slab perpendicular to the z-axis as shown in fig. 3.3. The transverse size of the slab is supposed
Fig. 3.3. Geometry of the problem for albedo calculation [eqs. (3.10) and (3.24)]. The arrowed lines represent the average fields ( u ) and ( u * ) . The quadrilateral represents a vertex function f.
132
ENHANCED BACKSCATTERING IN OPTICS
111, § 3
to be rather wide so that we may safely neglect edge effects and make use of the translation invariance of the vertex function r with respect to shifts along the slab. The final expression for the albedo, with taking into account eq. (3.14), becomes
r
where r, = (p,, z,), the integration is over the entire slab 0 < z < L, I p, I < L, -+ a,and for backscattering s, < 0 for soz > 0. Substitution of eqs. (3.16), (3.18), and (3.19) into eq. (3.24) yields the albedo in the form of a sum, a z a ( l )+ dL), where a ( ’ )stands for the first term on the right-hand side of eq. (3.16), i.e., corresponds to single scattering, and the computation of the ladder part dL)of the albedo is a matter of solving the transfer equation, eq. (3.21). The ladder approximation and transfer theory are known to provide a satisfactory description of incoherent scattering for almost all directions s (see, e.g., BARABANENKOV [ 19751, ISHIMARU [ 19781) except for the narrow cone around the retroreflection direction with an included angle of A 0 5 l/kol,where f3 is the angle between s and -so, i.e. cos0 = - s - s o . At these angles the contribution of the interference processes that fail to be explained by the ladder approximation is insignificant. A qualitative insight into the matter may be provided by referring to the reasoning of AKKERMANS, WOLFand MAYNARD [ 19861 (see also MACKINTOSHand JOHN [ 19881). Consider a half-space filled by randomly spaced scatterers. Denote by u , the contribution to the scattered field due to the passage of the wave along a path y including, in that order, the points R , , . . . , R,, locating the position of scatterers, which are assumed for simplicity to be point-like and uncorrelated. A typical path y is shown in figs. 3.2 and 3.4 by a continuous line. The overall field u is the sum of u, over all possible paths ,y, and this is readily recognized to be an expansion of u in a series of multiple scattering devised by GOLDBERGER and WATSON[ 19643. The analysis can additionally be simplified by considering, at each section, R,, - R, instead of a diverging wave represented by the function Go(R,+ - R,), one of its plane components exp[ik,(R,+ - R,)],and by assuming that u y is defined not only by the positions of the scatterers R , , . . . , R, but also by a fixed set of wave vectors k , = k,s,, . . ., k , , k , k , = kos. The intensity I u I will, of course, contain cross terms u y u ; , due to
,
’
,,
133
R A N D O M MEDIA
--. /', /
,/
/
I%,+?
x;:
/
+L+L-&A--
3,x,23 \'\ \
\
n
Fig. 3.4. (a) Typical maximally crossed (cyclical) diagram; (b) corresponding paths y and which interfere coherently at small 8.
- y,
the interference of different paths, but the ladder approximation disregards these cross terms, retaining only the terms like IuYI2. It is worth noting that the scalar waves passing through R , , . . . ,R N ,and the amplitudes of scattering at these points are the same for the path y and the time-reserved path - y (dashed line in fig. 3.4), in which these points are traversed in reverse order. In the plane-wave representation the set of wave vectors k,, k , , . . . , and kN is replaced by the set ko, - k,. .., - k i * kN. Consequently, ul,and u can only differ due to the sections from the medium boundary to the first and to the last scatterer. If we neglect extinction in this section, the difference will consist of the phase multiplier, i.e., u,u? = 1 u y /* exp[ik,(s + so).(RI- ItN)]. Thus, the contribution of the sum of these two paths to the intensity takes the form
,,
I u y + u - ),I
=
2
2 1 uy 1 + 2 I u y
I cos [k,(s + so> ' ( R1 - R N ) I
9
where the first term on the right-hand side corresponds to the contribution of ladder diagrams (see fig. 3.2), and the second oscillating term, due to the interference of y and - y paths, can be pictured as a cyclical, or maximally crossed, diagram such as the one shown in fig. 3.4. Clearly, for backscattering at zero angle, when s = - so, these quantities coincide for any path y. In this case, their contributions to the albedo are equal; i.e., a(L) (
- so, so) = dC)( - so, so)
(3.25)
and the sum is twice the classical value of dL). We can estimate the angular width of the backscatter peak by assuming that the vector R , - R , is parallel to the medium boundary and that the incident light is normal to this boundary so that so= = 1. Then, the interference (cyclical) term will give a sizeable contribution in the intensity provided that the phase
134
[It § 3
ENHANCED BACKSCATTERING IN OPTICS
increment is small; i.e., ko(s + so)*(Rl- R,) = ko8* IR, - RNI 4 1. The average value of the distance I R , - R , I for the minimum number of scattering events (N = 2) is the mean free path of the travelling light, i.e., the extinction length 1. Therefore, an enhancement will be observed in a cone with included angle A 8 5 l/koI. From this result one may conclude, by the way, that the wings of the coherent backscatter peak (0 > l/kol)are mainly due to the contribution from double scattering (BARABANENKOV [ 19731, ISHIMARUand TSANG [ 19881, GORODNICHEV, DUDAREVand ROGOZKIN[ 19891). When N grows large, we may assume that the path y consists of random segments and make good use of the diffusion approximation, which yields IR, - RNI2= D(rN - t , ) = IS,/3, where the path length S, = c(r, - r l ) is estimated as IN. This implies that a path of length S , contributes to the interference part of the intensity only within the cone 8 < 0, z l/ko(ISN)1’2; consequently, the top part of the backscattering peak is mainly formed by high multiplicity scattering. If path lengths are limited by a value of S,, say, due L (or to a finite thickness (or transverse dimension) of the slab, when S,,, S,,, L l ) , or due to absorption when S,,, la,,, then we can expect that the peak will be rounded off at angles 8 < em,, 5 l/k,,(lSmax)”2. A similar rounding off, associated with the suppression of the contribution of high multiplicity scattering, must be observed in the presence of factors breaking down the time-reversal invariance (MACKINTOSHand JOHN[ 19881). Proceeding along the lines of this qualitative reasoning, one can verify that the interference terms u,u? corresponding to different paths y’ # & 7, will contain, even at s = -so, uncompensated factors of the type exp [ikos, (Ri - R,)], which depend not only on the random locations R,, Rj, but also on the random direction s,. The averaging of such quantities will introduce (for l/koZ 4 1) only insignificant corrections to the intensity. Therefore, in describing coherent backscattering, we may safely limit ourselves to consider only the ladder and maximally crossed (cyclical) diagrams. Now we turn to a more rigorous substantiation of the preceding results, specifically, equality (3.25) (BARABANENKOV [ 1973,19751). The key point here is the property of reciprocity of the scattering operator (Green function) formulated by eq. (3.9) and the respective property of the intensity operator in the single-group approximation
-
-
-
-
_-
KI(rl,r2;r ; , G I
=
K,(rl, G ; G , r2)
=
K , ( r ; , r2; r l , GI.
(3.26)
Based on the symmetry of the intensity operator K , and the reciprocity
RANDOM MEDIA
11, § 31
135
Fig. 3.5. Equivalence of ladder diagrams and maximally crossed diagrams. Inversion of points on the upper (lower) row in a maximally crossed diagram transforms the diagram into the corresponding ladder diagram.
condition for the average Green function G , , eq. (3.1 l), satisfying the Dyson equation with mass operator M , , we can establish a type of equivalency of maximally crossed diagrams and ladder diagrams under inversion of the points on the upper or lower row in a diagram (fig. 3.5). This brings us to a fundamental conclusion for backscatter enhancement theory. Let K ( c ) ( r l ,r,; r ; , r ; ) represent the sum of all maximally crossed diagrams, then K ( c ) ( r , ,r,, r ; , r ; )
= =
P L ) ( r , ,r i ; r ; , r,)
P L ) ( r ; ,r,; r l , r ; ) ,
(3.27)
This relation represents the contribution of the sum of all maximally crossed diagrams in the intensity operator in terms of the vertex function in the ladder approximation, eq. (3.16). The quantity Kcc) is also the contribution of the sum of maximally crossed diagrams in the vertex function,
P C ) ( r , , r 2 ; r ; , r ;=) K(c)(rl,r2;ri,ri),
(3.28)
and is an approximate solution to eq. (3.8) with K w K , + K(c). Substituting PL) or rCc) for the vertex function r in the general formula for the albedo [eq. (3. lo)] yields, respectively, the expressions for the ladder part, dL)(s,so), or the maximally crossed (cyclical) part, dC)(s,so), of the total albedo. If in these integrals we let s = - so and make use of eqs. (3.27) and (3.28), we obtain equality (3.25).
3.3. TRANSFER EQUATION AND ENHANCED BACKSCATTERING
The qualitative consideration in the preceding section indicated that enhanced backscattering can be explained by choosing the vertex function in the form
136
ENHANCED BACKSCATTERING IN OPTICS
[II, 8 3
We recall that the random inhomogeneous media under consideration are those with parameters satisfying conditions (3.13) and (3.17). This means that both the ladder F L )and the associated maximally crossed r(c) vertex function can be expressed through the solution of the transfer equation. Thus, the formulas (3.16)-(3.22) and (3.28) form a computational basis for the albedo. Substituting eqs. (3.29) to (3.24), we obtain after some algebra a = a(’)
+ a’
,
(3.30)
where a ( ’ )is the contribution of single scattering, a(”(s, so) = nfe,a(s, so) (3.31)
(3.32)
P
=
ko leff(Sr + s o z ) + i(soz +
Isz
I )/2 I s z I soz
*
Thus the sum a‘ depends on the parameters q and p and the albedo due to scattering from an isolated inhomogeneity o = leff/l. The function F(q, p , p ’ ) may be written as the integral transform F(q, Pl P’)=
jOLnm
dz d z f eiPZ + iP‘z F(q. z ’ ) 1
9
(3.33)
of the function F(q; z, z ’ ) , which is expressed through the Green function of the transfer equation 9 ( r l s; r ’ , s ’ ) = 9 ( p - p ’ , z, s;z‘, s ‘ ) ,
namely,
s
F(q, z, z’) = 4 7 r o ( 4 ~ n Z 1 , ~ ) d2pexp(iq.p) ~ x
ds, ds2 4- s o , s,) 4 s 2 ,so) (3.34)
11, B 31
RANDOM MEDIA
137
The first term on the right-hand side of eq. (3.32) is the contribution of the ladder diagrams, i.e., the result of classical transport theory, and the second term is the contribution of maximally crossed diagrams that take into account the interference effects. In deriving (3.32)-(3.34), we assumed that the angle 0 between s and -so is small; indeed, in coherent backscattering experiments this angle never exceeded 100 mrad. If we proceeded without this assumption, in the expression for dL)instead of the cross section a( - so,sl) we would put a(s, s,), and in the expression for dC)instead of a( & so,s’) we would use the function W ( k;, k,s‘) with k; = fko(so - s) and j kbl = k, cos($0), which is defined in eqs. (3.18) and (3.22), and at k; z k, can be approximated by the cross section a( i s;, s‘) with s; = (so - s)/lso - s J (BARABANENKOV and OZRIN [ 19881). The assumptions made in (3.32) and (3.34) are justified for Q41 when the sharp maximum of the differential cross section a(s, s‘), if any, points forward, and s s’ = 1, as is usually the case. A similar approximation has been employed by ISHIMARU and TSANG[ 19881 in considering the bistatic scattering coefficient y = 4na/ 1 s, 1. The condition Q 4 1 makes it possible to neglect in the exponent of eq. (3.33) some insignificant differences to the result obtained by AKKERMANS, WOLF, MAYNARD and MARET[ 19881 in the framework of a heuristic approach. It should be emphasized that for random inhomogeneous media for which 1/koleE-g 1, the condition 0 4 1, in fact, has no effect on the experimental line shapes of coherent backscattering confined within a cone such that k,Ie,AO 5 1. Thus, the calculation of the angular dependence of the albedo has been reduced to the evaluation of the Green function of the transfer equation for the slab. First, let us consider the simplified situation of a medium consisting of point-like uncorrelated scatterers with isotropic scattering cross section ~ ( s s‘) , = 1/(4d). (3.35) In this case, transfer equation (3.21) simplifies appreciably, and for the function F(q; z , z ’ ) defined in eq. (3.34) it takes the form
-
Jo
where FoTo(qz, ) = w
with
r =
[
Jp’+zz.
d2p
exp (iq p - r) 4nr2
(3.37)
138
ENHANCED BACKSCATTERING IN OPTICS
§3
RYBICKI [ 19711 has investigated eq. (3.36) in depth, both for finite optical depths L/lcKand for the half-space with L/leK+ 00. The last case is of particular interest because for L tending to infinity, eq. (3.36) can be solved explicitly. This solution may be used as the point of departure for subsequent studies of the angular profile of the albedo under more complicated, and realistic, conditions. Indeed, as L + 00 from eq. (3.36), it follows (SOBOLEV [ 19631)
(iZ+ a3 -
- F(q; z, zl) =
&; z, 0) F(q;0, z’) .
After the transformation (3.33) (at L expression becomes
= 00
(3.38)
this is the Laplace transform), this
where the condition of symmetry F(q, z, z ’ ) = F(q; z’, z ) following from eq. (3.36) has been used, and F(q, p ) stands for the Laplace transform of &; z, 0). For this function eq. (3.36) is an integral equation of the type of convolution on a half-axis with the kernel Fo(q, z) that exponentially decays as IzI + 00. Therefore, this equation can be solved by the standard Wiener-Hopf approach, which operates with the Laplace transform F(q, p ) analytical on the half-plane 00. Im{p} > 0 and vanishing as [PI-, The Wiener-Hopf method produces an auxiliary function A(q, p ) related to the kernel of the integral equation by dz eiprFo(q, z ) (3.40)
Jm.
with k = This function is analytic on the complex plane cut along the imaginary axis Re { p } = 0,l Im { p } I > and has two simple zeros at p = & ipo(q), po(q) < where for q < 1 and 0 < 1 - w < 1,
d m
PO(d
d m ,
= JF-a=-s.
The analytic strip commonly considered in the Wiener-Hopf method lies along the real axis I Im { p } I < po(q). In this strip, the function A(q, p ) can be easily factorized, and the solution becomes F(q, P) = H ( 4 . UP) - 1
3
(3.41)
I I , 31 ~
139
RANDOM MEDIA
where H(4, w ) is the generalized Chandrasekhar function introduced by RYBICKI[ 19711
(3.42) where Re { w } > 0. This function satisfies the nonlinear integral equation
'*
= (1 + 4 2 ) - ' I 2 , At 4 = 0, this function coincides with the ordinary where function from CHANDRASEKHAR [ 19601. Formulas (3.41) and (3.39) yield a solution to the albedo problem (3.32) for point-like scatterers occupying a half-space. Summing up the preceding results, we obtain (GORODNICHEV, DUDAREVand ROGOZKIN[ 19891)
(3.43) with the parameters p and 4 given in eq. (3.32).
3.4. ANGULAR DISTRIBUTION OF BACKSCATTERED INTENSITY
Let us analyze the angular behavior of the albedo for the rather frequently encountered experimental situation of almost normal incidence of light upon an interface, i.e., so= z 1, with a small angle of backscattering 8 6 1, so that s, = cos 8 x - 1 and p x i in eq. (3.43), whereas 4 = kale, sin 8 x kole,B being not necessarily small by virtue of k0leR4 1. The first term in braces in eq. (3.43) represents the contribution of ladder diagrams and single scattering in the albedo; i.e., it corresponds to the result of the classical transfer theory. This quantity, denoted by a,,, remains almost constant for angle 8 varying in the range A 8 l/kole,. On the other hand, the second, interference (cyclic) term almost coincides with the first at 8 = 0 or 4 = 0 and decays for 4 4 1 as q - ' . We introduce the backscatter enhancement factor K ( 8 ) as the ratio of the albedo a(s, so) = a(8) to its value on the plateau, i.e. for 4 > 1, where it
-
140
ENHANCED RACKSCATTERING IN OPTICS
[I[, § 3
coincides with the classical value acl, K(@ = a ( W c , .
(3.44)
On substitution of a(s, so) from eq. (3.43), we obtain
K(e)= 2
+ [ H 2 ( q , 1) - H
~ O1), - 1 1 / ~ 2 ( 0 , i ) .
(3.45)
According to eq. (3.42), at weak absorption, i.e., 1 - o -4 1, and a small value of q, the function H(q, w) behaves like
It decreases monotonously in the region of high q's and for q & 1 becomes H ( g , 1) x 1 + no/4q.
(3.47)
Thus, for elastic scatterers, K ( 8 ) of the angular distribution of coherent backscattering has a triangular line shape peaked about the backward direction with
q e ) - K ( o ) x -2q,
q=koio-4i,
a=1.
(3.48)
The halfwidth of this peak calculated at the level K(8,,,) = 1.44 amounts to = 0.36; or 1 3 , ,= ~ 0.36 k,I. The maximum value of the enhancement factor K ( 0 ) = 2 - W 2 ( 0 ,1) x 1.88 is somewhat less than two as a result of the contribution of single scattering; indeed, from eqs. (3.25) and (3.30), ~ ( ~ ' ( =0 d) C ) ( 0 )and q,,2
K ( 0 ) = ( a ( ' ) + 2a'L')/(a"'
+ a(=') = 2 - u(l)/uc,.
For large q, K(6)x 1 +
con 9-', 2H2(0, 1)
q>l
(3.49)
This result can be obtained by direct computation by including the contribution of only one maximally crossed diagram of second order in the interference term dc).(In eq. (3.36) this corresponds to neglecting the integral term.) Consequently, the background of the coherent backscattering line shape is determined mainly by double scattering events, as predicted by BARABANENKOV[ 19731. As can be seen from eq. (3.46), weak absorption, 1 - (041, leads to a reduced albedo a(0) and to a reduced peak of K ( 6 ) in the range q 6 J-), which confirms the estimates derived earlier. A similar alteration of the K ( 8 ) line shape is observed for slabs offinite optical depth z = L/lem.This conclusion has been drawn by TSANGand ISHIMARU [ 19851 and VAN DER MARK,VAN
141
R A N D O M MEDIA
Albada and LAGENDIJK[ 19881 on the basis of numerical solutions of eq. (3.36) for various z and w. These solutions indicate in particular that at z = 32 (w = l), a(@ amounts to 95% of the albedo at z = 03. We note in passing that for q @ 1 a transition from the linear dependence of a(0) for a semi-infinite medium with z = GO to a quadratic dependence for r < 03 can be demonstrated with an equation for the derivative of F(q; z, 2') with respect to q, which follows from eq. (3.36) and at q = 0 and z < co has only a trivial solution. An analysis of eq. (3.36), performed by GORODNICHEV, DUDAREV and ROGOZKIN[ 1990b1, indicates that for 1 -3 z co, equality (3.48) for K ( 0 ) - K ( 0 ) holds true if z% q -4 1, and in the region of 4% @ 1 its righthand side must be replaced with - 2q2z/3. One more circumstance is worth noting. In formulating a theory describing coherent backscattering, we have considered a situation when a plane wave or a beam of width Lo far exceeding the lateral size of an illuminated region, L , , is incident on the region occupied by a scattering medium. On the other hand, in a standard experiment generally the diameter of a spot illuminated by the laser beam is such that L, B L I.Thus, we may assume for simplicity that the wave incident along the normal to the surface of the slab under consideration has the form
-=
'"
u0(r) z exp(ik,z - p2/2Li).
Then in eq. (3.10) the incident average fields Z
21,,
2L;
(3.50)
and the outcoming average fields ( u ( - s, r ) ) obey eq. (3.14). Incorporating the respective modifications in eq. (3.24) and observing (3.28) and (3.29) yield for the classical part of the albedo, a,, = a ' ' ) + dL), the following simple expression a,,(&Lo) = (L,/L,
)2
WI(0)
Y
(3.51)
where aCl(0)corresponds to Lo = 00. Now, the interference part of the albedo has the form of a convolution and the enhancement factor is given by (3.52) Given I,,
+ Lo, the enhancement factor decreases by a factor of I/L, because
142
[II, I 3
ENHANCED BACKSCATTERING IN OPTICS
of the finite size of the beam, and the line shape peak rounds off in the range of angles ko16 < JI/L,.
3.5. DIFFUSION APPROXIMATION
The scalar version of the theory developed for point-like scatterers described the salient features of coherent backscattering well. However, to attain a quantitative agreement with experiment, this simple version should incorporate several additional factors. The first factor is, beyond doubt, the finite dimensions of scatterers, which are frequently of the order of the wavelength, and also the anisotropy of the scattering properties when the extinction length I is a fraction of the transport length I, = I/( 1 - p), where p is the average cosine of the angle of scattering. Unfortunately, no explicit solution of the transport equation for an arbitrary cross section of scattering a(s, s ’ ) is known. Nevertheless, there exists a simple method of approximate calculation of the albedo a(6) for small angles, which also allows a useful physical interpretation. From eqs. (3.32)-(3.34), it follows that the deviation of the interference part of the albedo a(6) = a(s, so),soz = 1 from the value at 6 = 0, has the structure a(6) - a(0) =
s
d2p [eiq.P - 11 f ( P )
9
(3.53)
where f ( p ) is expressed through an integral with the Green function of the transfer equation for the slab. An asymptotic expansion of the integral in eq. (3.53) for q = kOl6+ 0 is governed by the behavior of f ( p ) as p+ co; the This is the expansion begins with a term linear in q only if f(p) asymptotic diffusion expansion of the solution of the transfer equation for a half-space. Thus, to evaluate the difference a(6) - a(0) for small angles, we may represent the Green function of the transfer equation in the form F ( p - p ’ ; z, s; z’, s’) = [ 1 - 3 D ( s * V - s‘ . V ’ ) ] F ( p - p ’ ; 2, z ’ ) ,
(3.54) where F(p;z, z ‘ ) obeys the stationary diffusion equation
~ v ~ F -( p’,p z, z ’ ) =
- 6(r -
r’)/(4n)’,
(3.55)
RANDOM MEDIA
143
with the diffusion coefficient
D
=
iltr,
(3.56)
where ltr = f/(l - p), and p = ni
s
s * s’ a(s, s‘) d2s .
The diffusion approximation is faced with a major difficulty due to the approximate condition for 9 ( p ; z, z ’ ) at the boundary ofthe scattering medium. This approximation must be used in place of the exact conditions on the Green function of the transfer equation 9 ( p ; z, s; z‘, s‘) = 0, s, > 0, z = 0 or s, < 0, z = L. One version of the boundary conditions used by ISHIMARU and TSANG [ 19881 and BARABANENKOV and OZRIN[ 19881 corresponds to the integral flux from a source inside the layer vanishing at the boundary z = 0 or z = L. Then, (3.57a) and on the other boundary, z = L, the respective equality differs from this by the sign of the derivative. The parameter y is usually unity; it has been incorporated to compare the results with another popular version of the boundary condition, that of the “absorbing plane” type, when 9(p;2
= - z o* , z ‘ ) = O ,
9 ( p ; 2, z‘
=
L
(3.57 b)
+ zo*) = 0 ,
where z8 w 0.7 1 ltr, and at p = 0 this is the extrapolated length in Milne’s problem (CASEand ZWEIFEL[ 19671). It is not hard to verify that the asymptotic expansions of the solutions of eq. (3.55) with the boundary condition (3.57a) obtained for y = 2 8 / 2 0 and with the boundary condition (3.57b) for p-+ 00 coincide. The boundary conditions (3.57a) were first used by BARABANENKOV [ 19731 to estimate the contribution of maximally crossed diagrams in the backscattering intensity. Based on this approximation, AKKERMANS, WOLF and MAYNARD [ 19861 investigated the angular profile of the intensity in coherent backscattering for both isotropic and anisotropic situations. A number of workers have used this approximation to estimate how various factors affect the peak line shape (EDREI and KAVEH[1987], FREUND,ROSENBLUH,
144
[II, § 3
ENHANCED BACKSCATTERING IN OPTICS
BERKOVITSand KAVEH [1988], VAN DER MARK, VAN ALBADA and LAGENDIJK [ 19881, AKKERMANS, WOLF, MAYNARDand MARET[ 19881). It should be emphasized that none of the afore-mentioned boundary conditions is perfectly appropriate for albedo calcuIations. Their derivation is based on a study of how the solution to the transfer equation behaves for z’ $- 1 or z, z’ 9 1, whereas the main contribution to integrals (3.33) and (3.34) is due to the region 0 < z, z’ ;5 1 near the boundary. This disadvantage is, of course, irrelevant for the exact solution for isotropic scatterers occupying the halfspace, which has been outlined in the preceding section. The asymptotic expansion of the exact solution coincides with the solution to the diffusion problem subject to eq. (3.57a) obtained at y = z1/2D or subject to eq. (3.57b) after the substitution of z1 for zd, where zl/l = - 1 + H(0, 1)/$ x 0.68. Since no exact solution is known for the general case of anisotropic scatterers, we can only assume that y = zf/2D with zf x 0.681,, is a suitable choice in this case. In fact, the different listed boundary conditions lead to almost identical results for the albedo in the range of small angles 8. It is worth noting that there exist different ways of expressing the diffusion formula eq. (3.54) for the Green function of the transport equation. Sometimes, the gradient terms are dropped, being treated as small corrections (ISHIMARU and TSANG[ 19881 ). These terms are small for an infinite medium, or for z, z‘ % it, in the case of a half-space. However, in the situation under consideration the key role is played by the region 0 < z, z’ ;5 I, where from eq. (3.56) it follows that l l V 9 1 9so that all terms on the right-hand side of eq. (3.54) have the same order of magnitude. After substitution in eq. (3.54) and then in eqs. (3.32)-(3.34) the solution of the problem (3.55) subject to eq. (3.57) is an explicit expression for the albedo (BARABANENKOV and OZRIN[ 19881). We give a simplified formula for the enhancement factor by omitting the contribution due to single scattering. For normal incidence of a wave on the interface of a medium occupying a halfspace, we have for small angles 0,
-
I?
where q
=
co\
k,l8. In the limit of small q, this expression yields (3.59)
11.8 31
145
RANDOM MEDIA
At 6 = 0, K ( 6 ) has a triangular line shape of halfwidth q,/2 dependent on p. In the case of isotropic scatterers, p e 1, eq. (3.59) coincides with the result obtained by AKKERMANS, WOLF and MAYNARD [ 19861, accurate to within the substitution y = 3z0/1. For large-scale inhomogeneities, 1 - p < 1, 44,,2-3(1 - p ) ( l + 2 ~ > [ 2 y (+l $ r ) I - ’ . In general, the line shape of K ( 8 ) depends appreciably on the average cosine ofp. For p < pcr = $, K ( 8 )monotonously decreases as q increases. For p > per, the contribution of maximally crossed diagrams becomes negative at sufficiently large angles of scattering and K ( 8 ) crosses the axis K = 1 at q = qo, peaks at q = q,, and asymptotically approaches K = 1 as q - The value qo and a rough estimate of q, may be obtained from
’.
40 4m
2(1 - 11) [ I
4-
3(1 + p)/47/(3p - l>1
3
= [(3 + 7) (1 - P)/2YI’/’
9
where 1 - p 6 1. The behavior of K ( 8 ) as a function of q for different p is illustrated in fig. 3.6. A noticeable feature of K ( 6 ) behavior is the presence of a minimum in the case of large-scale scatterers. This result is usually consistent with the general representations of the interference pattern and with the results quoted in the previous section. However, in deriving eq. (3.58), we have used the boundary condition (3.57a), which is, of course, an approximation. Moreover, if we solve the problem subject to the boundary condition (3.57b), the enhancement factor
-2
-1
0
I
L
Fig. 3.6. Enhancement factor K ( 0 ) calculated for coherent backscattering: (a), (b), and (c) show the diffusion approximation (3.58) with y = 1 computed for p = 0, 0.93 and 0.67, respectively; (d) exact solution for isotropic scatterers ( p = 0) represented for K(0)/0.94; (e) diffusion approximation with boundary condition (3.57b) of the absorbing-plane type calculated for p = 0.67.
146
[II,8 3
ENHANCED BACKSCATTERING IN OPTICS
becomes
(3.60) where ~6 z 0.71. This K ( 0 ) is seen to be a monotonous function of q. Its difference from the results of AKKERMANS, WOLF, MAYNARDand MARET [ 19881 and VAN DER MARK,VAN ALBADAand LAGENDIJK [ 19881 may be attributed to the fact that in considering the diffusion approximation with the boundary condition (3.57b), these authors failed to account for the gradient terms in eq. (3.54) and also substituted I, for I in the exponents of (3.32) and (3.33). It is noteworthy that the position of the minimum belongs to the range of angles and parameter q in which, strictly speaking, the diffusion approximation (accurate for q --t 0) is inapplicable. Therefore, the question of whether or not the minimum exists can be answered only with the aid of an exact solution of the transfer equation for a half-space with nonisotropic scatterers. Another fact worthy of note is that the diffusion approximation gives a lowered value of the albedo in the maximum 0 = 0 and a distorted pattern of K ( 6 ) at large q. Actually, as in the isotropic case, K(@ q - ' , q %- 1. This aspect has been treated in detail by ISHIMARUand TSANG[ 19881. However, in the range of small q, q 5 1, where the diffusion approximation is operable, results (3.58) and (3.60) agree well with the experimental evidence (fig. 3.7).
-
-. h
9 1. a
Fig. 3.7. (a) Experimental enhanced backscattering from a 10% water suspension of 0.46 pm diameter polystyrene spheres, where I,, = 20 Im, the incident and detected wave vectors are co-polarized so that I e . e, I = 1, and the plane of scanning is perpendicular to e, (WOLF,MARET, AKKERMANS and MAYNARD [1988]); (b) diffusion approximation curve obtained with a boundary condition of the absorbing-plane type.
147
RANDOM MEDIA 1.0
(e)
n QCl
0.5
0
Fig. 3.8. Relative intensity of coherent backscattering calculated for scalar waves in the diffusion approximation with 5 = (MACKINTOSHand JOHN [1988]). Similar K ( 0 ) profiles have been obtained for: (1) a layer offinite thickness (< = L/I)in the scalar problem or the co-polarized configuration, or identical helicities of circularly polarized waves, and (2) Faraday rotation for identical circular polarizations (( = l/gk,J).
Jl,bi31
The diffusion approximation is well suited for an account of the effects of absorption. For this purpose in the diffusion equation, eq. (3.55), one should replace the operator DV2 with [DV2 - fib']], with lab = o f / ( l - 0). Then in the final formulas (3.58) or (3.60) the parameter q is replaced by Jq2 + 3(1 - w), and the backscattering peak finds itself rounded off as for isotropic scatterers (fig. 3.8). For a slab of finite thickness, like in the presence of absorption, the asymptotic behavior to the diffusion equation changes as p+m, and it decays exponentially rather than by a power law ( P - ~ ) .For example, for boundary condition (3.57b), L %- z,, and p+m, we have
Therefore, near the maximum the enhancement factor becomes 21 K(O) z 2 - - (k,LO)2 (1 + $ 7 ) ( 1 3L ji =
0,
+ $y)-'
,
(3.61)
k 0 L 0 4 1,
and at larger angles 0, K ( 0 ) again becomes a linear function. Similar estimates have been obtained by EDREIand KAVEH[ 19871.
148
ENHANCED BACKSCATTERING IN OPTICS
[II, § 3
3.6. POLARIZATION EFFECTS
The vectorial nature of the electromagnetic field is another factor that appreciably affects the coherent backscattering. Polarization effects have been studied in the experiments of WOLF and MARET[ 19851, VAN ALBADAand LACENDIJK [ 19851, ETEMAD, THOMPSONand ANDREJCO [ 19861, ROSENBLUH,EDREI, KAVEHand FREUND [ 19871, ETEMAD[ 19881, and [ 19881. They have demonstrated WOLF,MARET,AKKERMANS and MAYNARD that the backscattering pattern is strongly dependent on the angle between the polarizations of the incident and detected radiation. Before we begin discussing polarization effects, we wish to introduce necessary corrections to the statement of the problem formulated in fi 3.2. The medium under consideration is nonmagnetic, is isotropic on the average, and exhibits a fluctuating permittivity or refractive index. Let a linearly polarized plane monochromatic wave E(O)(s,, r)
= e,
exp(ik,s,r) ,
with e, so = 0 and 1 e, I = 1 be incident on the space occupied by the medium in the direction of so.The field in the space obeys the Maxwell equations, which may be written in the form
(3.62) E,(eo, so, 4 IS(, . r + - 00
=
eo, exp (ikosor)
9
where a and fl label the Cartesian projections of the vectors, and summation is assumed over the repeating indices. For the average field (E,(r)) and the mutual coherence function ( E , ( r ) E J ( r ) ) , tensor analogs of eqs. (3.3) and (3.7) are formulated, which involve a tensor mass operator Mas(r, r’), average Green function ( GUg(r,r’)) of eq. (3.62), and vertex function T,p;a3p’(rltr2; r;, G). For free space and r + 00, the Green function becomes Giy(r) x - (4nr)- ‘P,,(s) exp(ik,r), where s
=
P,&)
r/r, and the polarization matrix =
h,,
- SaSp
(3.63)
secures the transversality condition. Similarly to eq. (3.6), we introduce the albedo
(3.64)
149
R A N D O M MEDIA
where s = R / R , and I ( R , e ) = ( I e E ( R )I ’) is the intensity corresponding to the component of scattered field polarized along unit vector e ; i.e., e s = 0. Making use of the Dyson equation and the reciprocity condition, which is now written as +
G,,j(r, r ‘ ) = Gp,(r’, r )
(3.65)
3
we may generalize (3.10)-(3.14) and (3.24) for the case of the vector field. If we take for the average field in the medium the effective wavenumber approximation ( E , ( e , s, r ) ) x e , exp(ik,s * r - r/2/eff), then the expression for the albedo is (s, so; e, eo) =
r
J+
(4xL,)-,
x exp[
dr, dr, dr; dr; e,ege,,,eo8.
-
z,
z2
~
21effSz
+
z ; z; - __ 21eRSoz
-
rap: a.B.(rI, r2; r ; , r ; )
ik,s ( r , - r 2 ) + ik,s,
*
1
(r; - r;)
,
(3.66) This expression “automatically” takes into account the transversality of the incident and scattered waves. We will again consider the contribution to the vertex function from singlescattering events and ladder and maximally crossed diagrams. As in the case of a scalar field, the qualitative reasoning underlying this constraint is based on the analysis of the phase relationships (see 0 3.2). For the vector field, however, these relationships are not enough to ensure that the contributions of the ladder and maximally crossed diagrams coincide. The rotation that the polarization vector undergoes in the processes of scattering should also be taken into account. Following AKKERMANS, WOLF and MAYNARD[ 19861 (see also for a more detailed discussion and estimates, AKKERMANS, WOLF, MAYNARDand MARET[ 1988]), we consider a simple case of Rayleigh scattering by an isolated scatterer, in which the polarization vector el of the scattered field has the form e l = P ( s , )e,, where P ( s , ) is the polarization matrix (3.63). Along a trajectory y defined by fixed positions of scatterers, R , , . . . ,R,, and a sequence of wave vectors k , = sok,, k , , ..., k,-,, k , = sk,, the polarization vector of the outcoming wave is e, = M,eo, where M,, = P(s,,,) P(s,- ,) . .. P ( s , ) P(s,), and sj = kj/kj.For the time-reversed path
150
ENHANCED BACKSCATTERING IN OPTICS
-y with the sequence of wave vectors k,, - k,_
[II, 5 3
,, . ., - k , , k,,
the polarization vector of the outcoming wave is e - = M-ye,. The matrices P(s,) are symmetrical and independent of the sign of k,. Therefore, for backscattering at zero angles when k , = - k,, we have M - = It is conceivable that in this case the projections of e, and e - on the direction of e = e, coincide [the case of the parallel (co-polarized) polarization of incident and detected radiation]. The coherence is conserved, and an enhancement of backscattering may be expected which is similar to that of the scalar case with K ( 0 ) z 2. For orthogonal polarization, e Ie,, the projections e .e , and e * e - differ for all N except N = 2. This leads to the suppression of the interference part of the intensity, which peaks to about one half of its classical value. To obtain a more detailed description of the polarization effects, we consider a simple model of a medium constituted by a point-like, nonabsorbing, isotropically polarized species. The reciprocity condition, eq. (3.65), leads to a relation combining the contributions of the ladder and maximally crossed diagrams into the vertex function
,
,
,
,
(3.67) which differs from (3.27) and (3.28) by an additional transposition of the polarization indices. To calculate P L )and r(c), we may resort to the scheme of transfer theory, outlined in $ 3.2 and 0 3.3, as generalized for an electromagnetic field. The intensity operator of a pointline scatterer will be written as
(3.68) Then for the medium occupying the space, we obtain
where the contribution due to single scattering is 3i (eae,)’ a‘” = ~
(3.69)
811 p - p * ’ and the contribution due to the ladder and maximally crossed (cyclical)
c
11, 31
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151
diagrams are
(3.70)
(3.71)
is related to the Fourier transform Fin the difference of transverse coordinates p - p’ of the Green function Yap; a,p ( p - p’ ; z, z’) of the transfer equation for an electromagnetic field in the medium occupying a half-space (DOLGINOV, GNEDIN and SILANTYEV [ 19791). The equation for F“ (more accurately, the system of equations) has the form
FaS;,,B’(q;
2,
z’) = F$$ a’B’ (4;z - z ’ )
P x
where
(3.74)
Jm,
with r = and s = rjr. From eq. (3.73) it follows that we generalized relationship (3.38), which after the Laplace transformation (3.72) becomes
152
ENHANCED BACKSCATTERING IN OPTICS
[It, § 3
For functions pap;a,p ( q , p ) , analytic in the upper half-plane of Im ( 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
ENHANCED BACKSCATTERING IN OPTICS
[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
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-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
ENHANCED BACKSCATTERING IN OPTICS
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
ENHANCED BACKSCATTERING IN OPTICS
[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
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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
ENHANCED BACKSCATTERING IN OPTICS
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
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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
ENHANCED BACKSCATTERING IN OPTICS
[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
LIMITED CLUSTER OF SCATTERERS
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
ENHANCED BACKSCATTERING IN OPTICS
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
LIMITED CLUSTER OF SCATTERERS
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
ENHANCED BACKSCATTERING IN OPTICS
[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
LIMITED CLUSTER OF SCATTERERS
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
ENHANCED BACKSCATTERING IN OPTICS
[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
ENHANCED BACKSCATTERING IN OPTICS
[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).
LIMITED CLUSTER OF SCATTERERS
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
ENHANCED BACKSCATTERING IN OPTICS
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
LIMITED CLUSTER OF SCATTERERS
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
ENHANCED BACKSCATTERING IN OPTICS
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
ENHANCED BACKSCATTERING IN OPTICS
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
ENHANCED BACKSCATTERING IN OPTICS
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
ENHANCED BACKSCATTERING IN OPTICS
[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
ENHANCED BACKSCATTERING IN OPTICS
[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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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
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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
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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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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<).
(2.40a)
This solution represents the so-called “bright soliton” (shown in fig. 2, solid line), whose amplitude decreases at infinity (see SATSUMAand YAJIMA [ 19741). Another important soliton solution may be derived in the case of normal dispersion (k2 > 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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
THEORETICAL BACKGROUND
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’(<) = 1
+ (0.5R-’
- 1) sin2(,/%
0,
(2.49b)
and
R=
T2k,n;Io
(2.49~)
n,lk,l For a small distance l,eq. (2.49b) coincides with eq. (2.36~).From eq. (2.43) one finds for the coherence function
w,,0 z27
=
(dz,,
5)4*(72,<)>
where
is the coherence function of the input pulse. Then, from eqs. (2.44) and (2.49) it follows that
where
220
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
(111, $ 3
In 0 4.4.4 we shall apply this formalism to describe some features accompanying the propagation of a pulse with partial coherence in a nonlinear medium. Finally, we note that an alternative analytical approach for an investigation of the NSE, based on a variational procedure, was proposed by ANDERSON [1983] and was further developed by ANDERSON,LISAKand REICHEL [ 19881.
0 3. Generation of Femtosecond Optical Pulses This section will demonstrate how the simultaneous action of both lasing medium and resonator leads to the formation of subpicosecond optical pulses. We also show some ways in which the once generated pulse can be further shortened and amplified. Several excellent and more detailed reviews and texts are available (see HERRMANNand WILHELMI [ 19871, AKHMANOV, VYSLOUKH and CHIRKIN[ 19881).
3.1. BROADBAND MEDIA
As we noted in 0 2, the short pulse is considered as a superposition of a large number of properly phased spectral components. Thus, the first problem occurring from efforts to generate short optical pulses is seeking media with a gain contour as broad as possible. The organic dyes have become the most widely used generating media in a wide range, from the ultraviolet to the near-infrared (see, e.g., TELLE, HUFFERand BASTING [ 19811). A typical resident is Rh6G, whose tuning curve covers the spectral region from 560 to 640 nm (see also SCHAFER[ 19731). Taking into account that the duration of a transform-limited pulse is inversely proportional to its spectral bandwidth, we can estimate that the simultaneous generation of such a broad spectrum could give a pulse as short as 10 fs. In fact, there are some circumstances limiting the output pulse duration, so that the generated pulses are always longer than this value. It should be noted that the technical advancement in recent years has allowed the synthesis of some new materials possessing a ultrabroadband gain profile. For example, the titanium-doped sapphire (Ti : A1,0,) has a tuning range in excess of 200 nm, centered at 800 nm (see, e.g., SANCHEZ,STRAUSS, AGARWAL and FAHEY [ 1988]), and alexandrite (BeAI,O, :Cr3 ) allows a continuously tunable generation in a range from 700 to 800 nm (e.g., WALLING, PETERSON,JENSSEN,MORRISand O’DELL [ 19801). Suitable broadband +
111, 8 31
GENERATION OF FEMTOSECOND OPTICAL PULSES
22 1
sources, especially in the far-infrared, are the color centers-doped ionic crystals, tunable in the range of 0.8 to 4 pm (see, e.g., MOLLENAUER[ 19871). The second problem arising from the generation of ultrashort pulses is the necessity for ensuring proper phase relations between the delivered frequencies. In the next section we shall consider some of the most successfully used methods for generating a broadband spectrum with a phase-locking of the spectral components. 3.2. MODE-LOCKING TECHNIQUES
An important parameter of the laser cavity is the round-trip time T, = 2L/c, L being the cavity length. In continuously operating lasers or in lasers delivering pulses whose duration is much greater than T, and a bandwidth of the active medium Sv much larger than l / T c ,there are random space-time fluctuations of the output intensity (see, e.g., KHANIN[ 19751). The average duration of a single fluctuation peak is in the order of 1/Sv. The mode-locking regime means that the energy in the cavity is concentrated into a single burst, which runs back and forth between the mirrors at the velocity of light. Because of the partial transmission through the output mirror, the laser radiation consists of a train of pulses separated in time by T,. This corresponds to the so-called “temporal description” of the laser operation in a mode-locked regime. It is based on the representation of the electric field in the form E ( t ) = E ( t ) exp[io,t
+ i@(t)],
(3.1)
where a,,is the central (carrier) frequency, and E“(t) and @(t) are the slowly varying amplitude and phase, respectively. In this approach the pulse formation is considered to be the result of predominant amplification of the most intense burst of the initial noise field into the cavity. There is also a “spectral description”, where the pulse is determined by specifying the amplitudes En and phases cp, of the participating frequency components
E,, exp(iw,t
E(t)=
+ icp,),
(3.2)
n
where w,, = w, + n Sw, with 6w = 2n/Tcbeing the frequency shift between two adjacent longitudinal modes in the cavity. In a mode-locking regime we have cp, = (po + n6, where 6 is a constant phase shift. For a Gaussian distribution El, = E,, exp( - En’), the output intensity is given by (NEW[ 19831)
(3.3)
222
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 3
which corresponds to a pulse with duration 6 z = 2 ln2/(n6v), 6v being the full width at half maximum (FWHM) of the spectral intensity profile. In general, the techniques for mode-locking fall into two classes. In the case of active mode-locking the cavity losses are modulated by applying an external sinusoidal RF signal to the acousto-optic element placed into the cavity. However, since the RF modulation is slowly varying, the duration of the output pulses is several tens of picoseconds. Considerably shorter pulses are produced when the gain of the laser is modulated by pumping with another mode-locked laser of a properly matched cavity length (synchronous pumping scheme, see 0 3.2.2). Another class of techniques uses passive mode-locking, which is achieved by inserting a suitable absorber into the cavity (see § 3.2.1). As a rule, this is an organic dye in solution, which has a great absorption cross section and a relaxation time T, -4 T,. The passive mode-locked lasers are the simplest and cheapest devices, producing a train of stable tunable subpicosecond pulses in a wide spectral region. There are also hybrid mode-locking techniques embodying some of the advantages of both the active and the passive schemes. Various regimes and some theoretical models of mode-locking were reviewed by SIEGMAN and KUIZENGA [ 19741, IPPENand SHANK[ 19771,NEW[ 19831,and PENZKOFER [ 19881. 3.2.1. Passive mode-locking Passive mode-locking (PML) of dye lasers was observed first by SCHMIDT and SCHAFER[ 19681by using a flash-lamp pumped Rh6G laser with DODCI as an absorber. However, the most important results in this field were obtained by continuous pumping. There are two main reasons. On the one hand, since PML systems are very sensitive to the pump parameters (power level, pump stability, etc.), the continuous-wave (CW) pump source makes it possible to adjust the necessary regime with high precision. On the other hand, when the CW pump is applied, the output radiation appears as a continuous train of pulses. Despite their low energy (- 10- l 2 J), due to the high repetition rate ( 100 MHz), it is possible to store and average the signal in order to perform each measurement with a high degree of accuracy. We should note that a common feature of the continuous-train dye lasers is the necessity of avoiding some detrimental effects (e.g., the degradation of the dye solutions due to heating by the pump). Therefore, these solutions are formed as free jets or flowing into thin cells.
-
111,s31
GENERATION OF FEMTOSECOND OPTICAL PULSES
223
The basic mechanism leading to generation of ultrashort pulses by PML is the saturation of both the absorption and the amplification of the gain medium. The role of the absorber saturation is to separate the most intense burst from the initial noise field. Obviously, the absorber may perform this successfully if it restores its absorption after the round-trip time of the cavity (ie., T,,b < T,, where T,,b is the relaxation time of the absorber). However, in the case of a “slow” absorber, when the burst duration is smaller than the relaxation time, the absorber may shorten only the leading front of the running pulse. The trailing front is sharpened due to the gain saturation of the amplifying dye. That is why the proper choice of the combination active/passive dye is of primary importance for producing ultrashort pulses. Although the regime of PML has been investigated by using various combinations of dyes, the best results so far were achieved by a combination of Rh6G as the amplifier and DODCI as the absorber. This pair operates in a spectral range from 615 to 635 nm. A first important step in generating sub-100 fs pulses was the development of a colliding-pulse mode-locking (CPML) technique. In this regime two counter-propagating pulses in the cavity collide in the absorber. The interference between these pulses induces a nonstationary modulation of the inversion. When the pulse “length” is smaller than the absorber thickness, this interference leads to an increase in the effective absorption cross section and hence to pulse shortening and broadening of the stability region (see HERRMANN, WEIDNERand WILHELMI[ 19811, STIX and IPPEN [ 19831). Although the PML regime was initially realized in a linear resonator (see e.g., RUDDOCKand BRADLEY [ 1976]), for CPML performance the ring scheme proposed by FORK,GREENand SHANK[ 19811 was found to be more suitable. An important advantage of the ring cavity is the increase of the energy saturating the absorber with respect to the energy saturating the gain medium. This occurs because the two counter-propagating pulses saturate the absorber simultaneously but are amplified separately. By using semiclassical considerations (see 0 2. l), KUHLKE,RUDOLPHand WILHELMI [ 19831 showed that, due to the CPML regime, the pulse can be expected to shorten three times. The second important step in generating optical pulses shorter than 100 fs was realizing the important role played by the combined action of both the self-phase modulation (SPM, see 2.3.1) and the group velocity dispersion (GVD, see § 2.2.1) experienced by the pulse into the cavity components. DIETEL,D ~ P E LKUHLKE , and WILHELM~ [ 19821 observed evidence of downchirp in the output (85 fs) pulses delivered by a CPML dye laser. This effect is attributed to the resonant interaction between the pulse and the resonance media (amplifying and absorbing dyes). It has been demonstrated that this
224
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[]I], $ 3
chirp may be compensated for by proper GVD (e.g., into glass, see $ 4.2.l), so that an intracavity balance between SPM and G V D takes place (DIETEL, DIETEL and DIELS[ 19831). That is FONTAINand DIELS[ 19831, FONTAIN, why, by inserting a glass prism into the cavity, shortening of the output pulse to 55 fs has been observed. Various sources of G V D into the cavity include glass components, jets, multilayer dielectric mirrors, unsaturated gain of the active medium, unsaturated absorption of the absorber, and angular dispersion of the resonator prisms. The interaction with the resonant amplifier and the absorber inserts both down-chirp and up-chirp into the pulse (DE SILVESTRI, LAPORTAand SVELTO[ 19841). The glass components insert an up-chirp (k2> 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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
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,
-
-
GENERATION OF FEMTOSECOND OPTICAL PULSES
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
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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
GENERATION OF FEMTOSECOND OPTICAL PULSES
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
GENERATION OF FEMTOSECOND OPTICAL PULSES
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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GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
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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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GENERATION OF FEMTOSECOND OPTICAL PULSES
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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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GENERATION A N D PROPAGATION OF ULTRASHORT OPTICAL PULSES
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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
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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]).
240
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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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GENERATION OF FEMTOSECOND OPTICAL PULSES
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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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-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.
244
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
P LL
!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
GENERATION A N D PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[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
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
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
negative group-velocity dispersion, inserting down-chirp into the transmitted pulse. Here, we will discuss in more detail some effects that arise when an ultrashort pulse passes through optical elements such as a glass slab, single prism, pair of prisms, single grating, or a lens. The main feature accompanying the propagation of an optical pulse is the difference between the phase velocity and the group velocity of light in the material. This dispersive effect leads to a considerable distortion of the pulse after passing through optical components, especially in the femtosecond time scale. 4.2.1. Ray-optics approach The simplest example is shown in fig. 14, where the pulse travels through a refractive surface. According to the Fermat principle, if AB and C D are wave fronts before and behind the boundary, respectively, then the phase times for the paths AC and BD must be equal; i.e., A C / c = nBD/c, where n is the refractive index. However, due to the dispersion in the medium, there is a delay At between the pulse arrival at point E and the phase front arrival at point D, which is equal to At = BD/vg - nBD/c, where vg is the group velocity. Following TOPPand ORNER[ 19751, we calculate that for a liquid medium (benzene) and for a beam diameter of about 0.1 cm, the delay time is as short as 200 fs (for I = 550 nm and 0 z 45"). Let u s now consider the case of a plane wave refracted by a slab (fig. 15a). If 1 is the distance between points A and B, the
25 1
PROPAGATION EFFECTS
A
C/
AIR
MEDIUM
d
/ Fig. 14. Transmission of a short optical pulse through a refracting surface.
phase delay along this segment is equal to (MARTINEZ,GORDONand FORK [ 19841)
$(a)=
w -
n(w)l cos O(w).
(4.13)
C
The second-order derivative of $ with respect to w determines the group velocity dispersion (GVD, see 8 2.2.1) (4.14) where it is assumed that the central wavelength 1, propagates along& (0 = 0). The physical meaning of eq. (4.14) is that the field at any point B is a superposition of angularly dispersed waves emitted by different points on the input
L1
Fig. 15. Refraction by slab (a) and by prism pair (b). The main (carrier) frequency propagates along AB.
252
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, § 4
surface (AA ’). Thus, the geometrical delay of the different wavelengths arriving at point B results in GVD. The role of the second surface is to recollimate the beam so that the transverse delay at the slab output is eliminated in contrast with the previous case. Thus, the action of the slab is similar to that of a grating pair. It can be seen from eq. (4.14) that the angular dispersion (d0ldA) yields negative GVD, but for a slab it is much smaller than the positive GVD introduced by the material (d2n/dA2), so that this configuration possesses a positive total GVD. The contribution of the angular contribution to the GVD increases considerably when additional surfaces are introduced in a pair of prisms scheme (see MARTINEZ, GORDON and FORK[ 19841, and fig. 15b). In this case, GVD is determined by (4.15) Provided the angle 0 is chosen so that sin 0 -4 cos 0, the prism pair will have a well-defined negative GVD. The derivatives d0/d;l and d20/dA2 can be calculated using simple geometrical calculations (GORDON and FORK[ 19841). If a Brewster angle of incidence at the first prism is used, the following formula for the GVD is valid (FORK,MARTINEZ and GORDON [ 1984]),
(4.16) In the typical case of a quartz prism, we have for A. = 620 nm, n = 1.457, and d2n/dA2 = 0.1267 p m - 2 (see MARCUSE dn/dl = - 0.03059 pm[ 19801). Then, if 1 sin tl= 2 mm, for eq. (4.16) it follows that when the separation of the prisms l > 138.4 mm, the prism pair will have a negative GVD. It should be noted also that provided whether or not a second pair of prisms conjugated to the first one is introduced, the dispersion doubles and the lateral “walk-off’’ of the rays after the system is eliminated. This is especially useful for intracavity applications (see 0 3.2). Some experiments of the group-velocity effects for picosecond pulses in a single prism were reported by BOR and RACZ [ 19851. Another interesting case, using a simple geometrical consideration, is the distortion of an ultrashort pulse by a lens. Because of the difference between the phase and the group velocity, the phase front is delayed with respect to the
’,
s
111, 41
PROPAGATION EFFECTS
253
pulse front (a surface coinciding with the peak of the pulse) after the lens. By using the Fermat principle, BOR [ 19891 has shown that this delay is given by At,=-
r,' - r 2
A($).
2cf 2
(4.17)
where ,f is the focal length of the lens, and r and r, are the input paraxial radii of an arbitrary ray and the marginal ray, respectively. Equation (4.17) is valid for both a lens and a zone plate. Insofar as the derivative df/d1 represents the longitudinal chromatic aberration of the lens, it can be seen that this is the physical reason responsible for the pulse distortions. Moreover, in contrast with the phase front, the pulse front changes its shape from convex to concave (i.e., it is flat), not at the lens focus but at a distance L = f/(n - 1) ( - 1dnldl) behind the focus (here n(1) is the refractive index of the material). In addition, it can be seen from eq. (4.17) that the delay is different for different points along the lens radius. This effect can be avoided by the use of an achromatic doublet. The effect of pulse broadening due to GVD of the lens material has been calculated by COHEN and LIN [1977] who used the formula Ar, = (1/c) (d2n/dA2)A l l(r), where l(r) is the path length of the ray in the lens and A 1 = 1*/2cT is the bandwidth of a transform-limited input pulse with duration T. For example, for a fused-silica lens it follows that ALtJr) = 64 A t , (r),which shows that the delay between the phase and the pulse fronts is considerably greater than the pulse broadening due to GVD.For an achromat the pulse broadening due to GVD is independent of the radius (BOR [ 19891).
,
4.2.2. Wuve-optics approach An analysis based on the wave theory should be accomplished to consider more precisely the effects accompanying the transmission of short optical pulses through optical components. Here we discuss gratings and prisms in the light of scalar diffraction theory (Kirchhoff-Fresnel integral). The distortion of an ultrashort pulse with finite beam size after passing the diffraction grating was studied by MARTINEZ[ 1986a,b]. Let 8 = O(y, Q) be the diffraction angle (fig. 16), depending on both the angle of incidence y and the frequency shift s2 = w - o, by the grating formula sin y
+ sin 0 = 2 lccm , ~
wd
(4.18)
where d is the grating drove spacing and m is the diffraction order. To estimate
254
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 8 4
Fig. 16. Reflection by grating. The angle $represents the mismatch between the phase and the pulse front.
the influence of the beam divergence (different angles y), as well as the finite bandwidth of the pulse (different w), we expand the function O(y, w) in a truncated power series (4.19) which is valid both for low divergence of the beam and for a relatively narrow spectrum. Then, from eqs. (4.18) and (4.19) it follows that a = (aelay), = - cos Y,/COS e,, and p = (ae/aw), = - 2Kcm/(w,z d cos $1, where the angles yo and Oo correspond to the carrier frequency w,. Let us denote by k, and k, the angular frequencies of the incident and diffracted waves, respectively. Thus, in one dimension we can write ki
=
koAy
k,
=
k o A 8 = ak,
(4.20a)
9
+ kJ8.
(4.20b)
Then the spatial transmittance function of a grating is equal to exp(ik,x,), where x, is the transverse coordinate of the output beam. Thus, the spatial dependence of the spectral amplitude after the grating is given by
(4.2 1)
111. § 41
255
PROPAGATION EFFECTS
where E,(k,, Sa) is the spectral amplitude of the incident pulse. After a Fourier transform of eq. (4.21), we obtain for the transmitted pulse profile E(x2, t ) = E0(ax2,t - kBx2), which repeats the input pulse profile but with a delay At = kpx,, depending on the coordinate. This corresponds to a tilt angle rl/ = arctan(kpc) (see fig. 16). The field distribution at a distance z after the grating may be obtained substituting eq. (4.2 l), written for a Gaussian beam, into the well-known Kirchhoff-Fresnel integral (MARTINEZ[ 1986a1) ~ ( xz,, t ) = b exp
[
- ika2x2
] [
t + kBx,q(d’)/q(d’
+ a2z)
2q(d’ + a2z) exp T 2 - 2ikB2zq(d’)/q(d’+ a2z)
(4.22) where T is the input pulse duration, q ( z ) = z + i m / l is the complex parameter of the beam, a is the e - radius of the beam, and d’ is the position of the beam waist, defined as positive if it is located before the grating. Under typical experimental conditions, when T / k p 4 a and for a wide beam, z/ka2 4 1, eq. (4.22) yields for the pulse duration after the grating
’
(4.23) which corresponds with evidence of GVD [see eqs. (2.25)], arising from the lateral walk-off (angular dispersion) of the different spectral components after the grating. A similar treatment is valid also for dispersion of a short pulse with finite beam size by means of a prism. For the most convenient “minimumdeviation” configuration and for Brewster-angle incidence, MARTINEZ[ 1986bl found for the parameters a = 1 and = - (A;/nc)(dn/dA),, . Finally, we should mention that the focusing of a short optical pulse by a lens can also be considered in terms of the wave optics. When a Gaussian pulse with duration T and beam radius a is focused by a lens that acts only as a phase corrector (the material dispersion is neglected), the focal radius a, is given by (CHRISTOV [ 1985aj) (4.24)
where af“ = 2fc/aco, is the focal radius of a focused CW Gaussian beam (see, e.g., YARIV[ 19751). Equation (4.24) can be compared with eq. (4.4), corresponding with the far-field approximation. Obviously, with decreasing T, the focal spot size increases due to the transverse chromatic aberrations of the lens.
256
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 8 4
Some features accompanying the total internal reflection of ultrashort pulses were considered by CHEUNGand AUSTON[ 19851. The case of reflection of an ultrashort pulse by a multilayer dielectric mirror was studied by CHRISTODOULIDES, BOURKOFF,JOSEPHand SIMOS[ 19861. 4.3. PROPAGATION THROUGH DISPERSIVE SYSTEMS
The interest in propagation of short optical pulses in dispersive systems is related to the possibility of achieving considerable reshaping of the pulse profile. Despite its linear nature, the dispersive system can act strongly on the passing pulse. The wide class of optical dispersive systems could be divided into two subclasses, the first consisting of continuous media and the second of systems with discrete components. Typical residents of the first class are the dispersive substances. Early in the 1970s some authors demonstrated that if an optical pulse with nanosecond or picosecond duration passes through an atomic vapor, considerable reshaping of its time profile occurs. GRISCHKOWSKY [ 19741 showed that when a preliminary chirped pulse passes through a cell containing dilute Rb vapor, the output pulse exhibits strong modulations and even breaks down to a series of subpulses. In a closely related investigation, LOY [ 19751 reported a dispersive modulator where the resonant atomic frequency was modulated, instead of modulation of the input pulse. Similar systems have also been used for pulse compression in the nanosecond domain. The theoretical works of GRISCHKOWSKY [ 1973J and GRISCHKOWSKY and LOY [ 19751 have explained the observed effects caused by the strong dispersion near the atomic resonance. By passing frequencymodulated CW radiation through sodium vapor, BJORKHOLM, TURNERand PEARSON [ 19751 observed that when the carrier frequency is within about 10 GHz of the resonance frequency, conversion of the CW light into a train of pulses as short as 240 ps takes place. The second class of dispersive systems uses transformation of the radiation by a sequence of discrete optical components (e.g., a confocal lens pair between two conjugated gratings), which reshapes the initial pulse into a pulse with a new envelope and phase. Here we will show that both groups of dispersive systems can be described in terms of so-called “temporal modes”. 4.3.1. Temporal modes representation of a propagating pulse Our consideration is based on the integral representation given by eqs. (2.18) and (2.21a). We noted in !j 2.2.1 that for a continuous medium the spectral
PROPAGATION EFFECTS
251
transmittance function H ( R ) is determined by eq. (2.20a), whereas for a discrete system it represents the transmittance of a mask placed in an optical processor (see 0 4.3.3). Let us expand H ( R ) in a Taylor series as follows
(4.25) where currently is the shift from the carrier frequency coo. Then, substituting eq. (4.25) into eq. (2.21a), we obtain Ix,
Eout(l> =
1 anEn(l)
(4.26)
3
n=O
where
, (dR") d" H ( R )
a,,=%
and
(4.27a)
R= 0
-
En@)=
d"Ei,,(t) . d t"
~
(4.27b)
The functions En(t)appearing in eqs. (4.26) and (4.27b) are called "temporal modes" (CHRISTOV [ 1988a,b], CHRISTOV and DANAILOV [ 19891). Thus the temporal modes arise in a natural way as the time derivatives of the input pulse profile. From eqs. (4.26) and (4.27a,b) it follows that by choosing a suitable function H(SZ), one can control the weights of the different temporal modes in the output pulse shape. In fact, the temporal modes E,,(t) comprise a complete but not orthogonal set of functions in time. This is the main difference between them and the well-known orthogonal set of spatial modes, where the boundary conditions in the laser cavity play a crucial role (see, e.g., YARIV[1975]). However, we will show that by using a simple orthogonalization procedure, a system of time orthogonal envelopes can easily be obtained. 4.3.2. Propagation of a short pulse in a dispersive medium Let us consider the off-resonance propagation of a Gaussian pulse in a continuous dispersive substance. CHRISTOV[ 1988a,b] showed that, in this case, the series (4.26) may be written as (4.28)
258
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
where the coefficients a,,(z) are given by
and the corresponding temporal modes are modified Gauss-Hermite functions (4.30) In eqs. (4.29) and (4.30), T is the half-duration of the input pulse, and Li- ‘)(x) is a Laguerre polynomial,
c I1
L y ) ( x )=
(;--Il)
(-XI” ~
m= I
m!
.,
L 0( - I ) = 1
(4.3 1)
From eqs. (4.28)-(4.30) one can obtain the “weight” of the mth temporal mode in the output pulse shape,
8,
=
~ ~ J2”smt!a,(z).
(4.32)
Hence, when either T or w, - oodecrease, more terms in eq. (4.28) should be considered. This may be interpreted as an excitation of higher-order temporal modes due to the propagation. Since the higher-order Hermite polynomials H,,(x)have a complicated oscillating nature, temporal distortions and oscillations in the output pulse will appear. In fact, this approach is equivalent to accounting for the full series expansion of the dispersive phase over the frequency [see eq. (2.23)]. Figure 17 shows the first four temporal modes E,,(v); n = 0, 1,2, 3. Obviously, this picture will be similar for all symmetrical input pulses (e.g., sech, Lorentzian, etc.). It is also similar to the picture representing the first four spatial resonator modes (see, e.g., YARIV[1975]), but here the exponential decreases more rapidly. Figure 18 shows the time behavior of a Gaussian input pulse with T = 100 fs and carrier frequency oo= 3.4 x l O I 5 Hz, propagating overadistancez = 0.5 CminagaswithdensityN = 1.35 x l O I 9 ~ m - ~ ( 0 atm). .5 Figures 18a-d are produced taking into account the temporal modes up to the 5th, loth, 15th, and 20th, respectively. It can be seen that the inclusion of higher-order temporal modes leads to fast oscillations in the output pulse shape. For shorter distances the contribution of these modes is smaller and the output pulse is weakly disturbed (CHRISTOV [ 1988bl). To explain this behavior we should note that for an excitation of an opticaIIy thin medium, the radiated
IIL§ 41
259
;..:pq ;:;f-jq PROPAGATION EFFECTS
^r
-
-
0.00
>
t;
-24-12 0
12 24
-24 -12 0
12 24
-24-12 0
12 24
-24-12 0
12 24
TIME (.lo-” s)
Fig. 17. Time dependence of the first four temporal modes: (a) n (d)n = 3 (CHRISTOV [1988b]).
=
0; (b) n = 1; (c) n
=
2;
field is proportional to the time derivative of the incident field [see eq. (15) from CHRISTOV [ 1988al for small 21. Thus, if the incident pulse represents the nth temporal mode [Ei,(t) = E,(t)], the radiated field will be the (n + i)th temporal mode [Erad= E,,+ i( l) ; i = 1,2, . . .]. Moreover, introducing the pulse area as 0 = J E ( t ) dt, from eq. (4.27b) we find that all temporal modes [except E,(t)] are zero-area pulses (0 = 0). Consequently, the propagation of a short optical pulse in a resonant medium can be interpreted in terms of temporal modes as
-24-12 0
>
t, w z W F-
e
12 24
O : . p 1 0.00
-24-12 0
;:i;Wl
12 24 TIME
-24-12 0
12 24
-24-12 0
12 24
S)
Fig. 18. Intensity profiles of a pulse propagating in a dispersive medium as a sum of temporal modes: (a) up to the 5th mode; (b) up to the 10th mode; (c) up to the 15th mode; (d) up to the 20th mode (CHRISTOV [1988b]).
260
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 5 4
follows. The atoms absorb the input pulse and emit a set of temporal modes corresponding to that pulse [see eq. (4.27)]. Inasmuch as the temporal modes are zero-area pulses, the area of the propagating pulse decreases and its profile develops oscillations. We should note that similar behavior is inherent also for an on-resonance case with finite T2 (see eq. (2.20) and CRISP[ 1970]), and it was experimentally observed by ROTHENBERG, GRISCHKOWSKY and BALANT [ 19841 for picosecond pulses. In contrast, in the case of on-resonance propagation with infinite T, + 00, the radiated (by a thin medium) field is proportional to an integral from the incident pulse (see eq. (25) from CHRISTOV [ 1988al and SARGENT, SCULLYand LAMB[ 19741). Thus, if the input pulse is E,(t), the radiated pulse will be [from eq. (2.27b)I equal to E n - , ( t ) . From eq. (4.27b) it follows that the spectral intensity of the n th temporal mode is given by
Figure 19 shows the spectral intensities of an input Gaussian pulse and of the 5th and 20th temporal modes. It is easy to calculate that the function Z,(Q) has maxima at R = &/T, so that the frequency shift between the peaks in fig. 19 increases for a shorter input pulse. A more complicated formalism describing the optical pulse propagation in a linear causally dispersive medium was developed by OUGHSTUNand SHERMAN [ 19901.
1.00 -
-3
CI
>-
t,
0.50-
u)
z w €-
z *
0.00I
I
I
I
Fig. 19. Frequency dependence of the spectral intensities of the initial pulse -( 5th temporal mode (-------); and of the 20th temporal mode (----------) [ 1988bl).
I
); of the
(CHRISTOV
PROPAGATION EFFECTS
26 1
4.3.3. Pulse shaping As we noted in the preceding section, the transmission of ultrashort optical pulses through a dispersive medium can result in considerable reshaping of its profile. However, precise control of the pulse shape is unlikely. Using the concept of temporal modes, we can say that it is difficult to govern the weights of single temporal modes adequately only by varying the parameters of a continuous medium. A discrete dispersive system has a crucial advantage in this respect. The reason is simple: the continuous dispersive medium can only change the phase relations between the spectral components, whereas the discrete system operates on both the phases and amplitudes of these components. The design of a discrete dispersive system requires spatial dispersion of the input spectrum (e.g., by grating). After some manipulations of the amplitude and phase, the spectrum is again transferred into the temporal domain as a pulse with a new envelope and phase (see also 8 4.5.2). DEBOIS, GIRES and TOURNOIS[1973] and AGOSTINELLI, HARVEY,STONE and GABEL[ 19791 demonstrated pulse-shaping based on a filtering of the spectral components spatially dispersed by a grating pair. Some early results from the shaping of picosecond pulses were reviewed by FROEHLY,COLOMBEAU and VAMPOUILLE[ 19831. A considerable progress in the shaping technique was achieved when the spatial dispersion of the grating pair was combined with its action as a compressor for preliminary chirped pulses. A simple demonstration of the technique was achieved by HERITAGE, THURSTON, TOMLINSON and WEINER[ 19851, who applied spectral windowing of the spatially dispersed spectrum after the first pass in a double-pass grating compressor to suppress the undesirable high- and low-frequency components. This results in the nearly complete elimination of the wings of the compressed pulse. Further experiments with chirped input pulses were reported by HERITAGE,WEINERand THURSTON[ 19851. By introducing simple amplitude and phase filters at an appropriate location in the pulse compressor, a variety of picosecond pulse shapes are produced. A computational model that accurately predicts the output pulse shapes as a function of the mask transmittance was developed by THURSTON, HERITAGE, WEINERand TOMLINSON [ 19861. It was shown that the output pulse profile is given by
s
EOut(r) = &(a)exp( - ia,Q2 + iQt) S(Q)dQ, where
d& 5
(4.34)
I
s(fl> =
H(x) exp[ -2(x - a,G?)2/~,2] dx,
(4.35)
262
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 9 4
where H ( x ) is the mask function, a, is the chirp parameter of the input pulse, and the GVD of the grating pair for a double passage is a' = 4n2cb'/(oid2 cos2 7') [see eq. (3.5)]; b' is the slant distance between the gratings for the main wavelength A,, y' is the angle of reflection, and d is the gratings grove spacing. In eq. (4.39, w, is the beam radius at the mask. By using eqs. (4.34) and (4.35) it is easy to calculate the mask function necessary to synthesize a given output pulse. If the input pulse is preliminarily chirped (e.g., in an optical fiber) and the grating pair is adjusted to compensate precisely for this chirp, then in the high-resolution limit one finds
EOut(t) z
/-!2nw;
E,(61) H(a,61) eiRrdSZ.
(4.36)
Hence, if the mask function H ( a , 62) varies rapidly compared with the input spectrum, the output pulse envelope will be nearly a Fourier transform of H ( a , 61). For example, if the mask represents a sinc function, the output pulse is square (THURSTON, HERITAGE, WEINERand TOMLINSON [ 19861. This has been experimentally verified by WEINER,HERITAGE and THURSTON [ 19861by manufacturing a special mask. A considerable flat-topped pulse as short as 5.7 ps is synthesized. These authors reported a generation of an odd pulse by shifting by n the phase of the half of the originally symmetrical spectrum. However, it is difficult to perform shaping in the femtosecond domain using similar apparatus because the grating pair causes considerable stretching of the output pulse. An alternative method for generating arbitrarily shaped femtosecond pulses was proposed by HANERand WARREN[ 19881. Pulses delivered by a synchronously pumped mode-locked dye laser (see § 3.2.2) pass through a fiber, after which they are transmitted through a high-speed Ti : BaTiO, electro-optic waveguide modulator. The modulator is driven by a waveform generator and acts as a time-domain filter, contrary to the previous case. Furthermore, the pulses pass through a grating pair in a double-pass scheme, resulting in a shape determined by the driving signal. A time resolution of about 100 fs has been achieved. The dispersion-free design shown in fig. 20 (see also FROEHLY, COLOMBEAU and VAMPOUILLE[ 19831 is most suitable to use for all-optical shaping with femtosecond pulses. Note that due to the lenses the temporal resolution of the system increases considerably. A detailed theoretical description of this device was presented by DANAILOV and CHRISTOV[ 1989bl. Let the processed light be factorized at the system input (4.37)
111. § 41
263
PROPAGATION EFFECTS
X
X,
Fig. 20. Scheme of an optical system for pulse processing. When the gratings are aside the focal planes of the lenses, the system acts as a compressor whose dispersion is greatest for the gratings position at points 0, and O,, separated from the lenses by twice the focal distance (DANAILOV and CHRISTOV [1989b], MARTINEZ 119871).
where x is the coordinate transverse to the beam direction z . Then, taking into account the transmittance functions of the successive optical components, we find for the spatial and temporal dependence after the processor
where u is the radius of the Gaussian input beam, a = cos Oo/cos yo, p = 2nc/(o,2 d cos yo), k, = o o / c , and x = k,x/f. In eq. (4.38), H ( x ) is the mask transmittance function, and b is a parameter including some slowly varying variables, as well as the reflection and absorption losses. To perform the intergration over R in eq. (4.38), we assume that the input pulse is Gaussian with half-duration T. Then, substituting &,(a)= exp ( - R2 T 2 ) into eq. (4.38), we obtain
(4.39) where s = auqa2
+ (kopu/T)2]-',
r
=
( a x 3 + k o f i u 2 t / ~ 2[ a) 2 + (koSu/T)21-',
p
=
$ T 2 t ( k , p ~ / 2 a .) ~
(4.40)
264
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[Ill, $ 4
In general, it is difficult to calculate the mask transmittance function necessary to synthesize a preliminary specified pulse shape directly from eq. (4.39). Therefore, we will apply the approach used in 0 4.3.2. After expanding H ( x ) in a power series,
c m
H(X) =
C,X"
(4.41)
3
n=O
and substituting eq. (4.41) into eq. (4.39), we find (4.42) where H,(y) is a Hermite polynomial, and the following abbreviations are used
d,
=
b($i)n s - ( n +
gx
=
a a [ a2 + (kofla/T)']'/'
(4.43a)
1)/2cn ;
3
g,
=
a0 ko Pa
[a'
.
+ (k,fla/T)2]1/2
(4.43b)
It can be seen from eq. (4.42) that the output radiation is presented as a superposition of modified Gauss-Hermite functions, whose argument is a linear combination of the spatial and temporal variables. To estimate the contribution of these variables, we take the ratio Q = g,t/g,x for t = T and x = a. Then eqs. (4,43a,b) yield
(4.44) Three cases for the Q value are of interest. (i) Q 1. This corresponds to a very short input pulse and/or high spectral resolution of the system (large fl). Then, from eq. (4.42) it follows:
+
(4.45) i.e., in this case we obtain a pure expansion over temporal modes [compare with eq. (4.30)]. Now, using the orthogonality of the Gauss-Hemite functions, one can find the coefficients c, participating in eq. (4.41),
x exp [ - t2(T-' - g:)] d t .
(4.46)
111,s41
PROPAGATION EFFECTS
265
(ii) Q 4 1. This means that the input pulse is long, or the resolution of the system is low [e.g., when the gratings are replaced by mirrors (/3 fi: O)], and pure spatial Fourier processing is performed. Then, we have from eq. (4.42) (4.47) i.e., in this case we obtain an expansion over a set of spatial modes similar to those arising in the theory of the laser cavity. (iii) Q x 1. This case is not suitable for practical purposes because the output radiation is not spectrally pure. As an example, let w,, = 3.14 x l O I 5 Hz ( I , = 600 nm), a = 1 cm, a x 1, d = 5.6 x l o p 5cm (1800 l/mm). Then, /3 = 3.5 x 10- l6 s, and we obtain the following typical values of the Q parameter for the preceding three cases: (i) T = lOOfs, Q = 350. (ii) T = 10ns, Q = 0.003. (iii) T = l o p s , Q = 3.5. In fact, the parameter Q determines the resolution power of the system. The spot size in the mask plane is given by (DANAILOV and CHRISTOV[ 1989b]), (4.48) which for large Q is Q times greater than the “elementary” spot size a x O , corresponding to the focused CW beam (Sx, = 2af/koa). We will see in $4.5.2 that the larger Q, the greater the amount of information that can be transferred from the mask to the light pulse. In addition, we should note that the developed formalism can be successfully applied for analyzing the system of THURSTON, HERITAGE,WEINERand TOMLINSON [ 19861. Moreover, for producing the desired mask transmittance, the large amount of experience gained in the field of conventional Fourier optics and holography may be used (see, e.g., [ 19781). CASSASENT Let us now consider in more detail the high-resolution case (Q % 1). Then E,(SZ) varies slowly compared with the real exponential under the integral in eq. (4.38), and, therefore, we may approximate this exponential by 6(Q - ~ / k , p ) where , 6is the 6-function. Thus, performing the integration over SZ in eq. (4.38), we obtain (CHRISTOVand DANAILOV [ 19891) (4.49)
266
GENERATION A N D PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
where the relation between the variable x ‘ and the physical coordinate x ’ in the Fourier plane F in fig. 20 is x ’ = x’/k,fi. In fact, eq. (4.49) is similar to eq. (4.36), and it allows us to obtain the mask function H ( x ’ ) in a closed form instead of a series expansion. Taking the inverse Fourier transform of eq. (4.49), we find b‘ H ( x ’ ) = - &;‘(XI) 2n
s
(4.50)
E,,,,(t)exp(-ix’t)dt.
Therefore, if we want, e.g., to synthesize a sech2(t/T,) pulse, we need to insert into the processor a mask with transmittance (4.5 1) On the other hand, the mask transmittance necessary to produce the n th temporal mode is a simple power function H ( x ’ ) x‘”. The shaping of femtosecond pulses was experimentally demonstrated by WEINERand HERITAGE[ 19871. A CPML ring dye laser (see 8 3.2.1), delivering pulses as short as 75 fs as a source, is used. By means of the system shown in fig. 20 with relatively high resolution (d- I = 1700 l/mm, f = 15 cm, 1, = 620 nm), two spectral components separated by 2.6 THz have been selected. These two frequencies interfere, causing a beat at the system output. Another example of the exquisite control attainable by this system is demonstrated by placing a pseudo-random phase mask, which leads to the appearance of pseudo-noise bursts with picosecond duration at the output. It was shown that a reconstruction of the initial femtosecond pulse is attainable by passing through a second phase-conjugated mask adjacent to the first one. A synthesis of a variety of shaped femtosecond pulses was reported by WEINER,HERITAGE [ 19881. An odd pulse with full duration of about 1 ps, and a and KIRSCHNER square pulse as short as 2 ps but with rise and fall times of the order of 100 fs is produced. Moreover, in 0 4.4.2 it will be shown that such a shaping system also enables us to synthesize both bright and dark solitons. We should note that this shaping technique provides new opportunities for investigating some fast coherent transient processes, as well as new opportunities for communication purposes (see 5 4.5). A special kind of shaping represents the “magnification” of light pulses. The idea is to design a device acting in the time domain similarly to the conventional lens in space. The principles of such a system were proposed by TELEGINand CHIRKIN [ 19851 and further clarified by KOLNER and NAZARATHY [ 19891. A design of a “time telescope” was introduced by [ 1990bl. CHRISTOV
-
111.8 41
PROPAGATION EFFECTS
267
As we noted earlier, the temporal modes comprise a complete but not an orthogonal set of functions in time. Here we will consider the possibility of obtaining a set of time orthogonal functions (CHRISTOV [ 1990aj). Let us use the temporal modes &(t) given by eq. (4.27b) as a base set of functions in the functional space. Then we can apply the well-known Schmidt procedure (see, e.g., ELLIOTand DAWBER [ 19791) in order to obtain an orthogonal set of functions corresponding to E,,(t). Since E,(t) = Ein(t),we take &(t) = 60 Eo(t) as an initial “vector” in the functional space. Then, for any symmetrical input pulse the first several orthogonal envelopes will be =
b,E,((t),
E ; ( t )=
b,E,(t),
&(t)
(4.52)
Ei(t) = b2[E2(t)+ Eo(t)w,/w,], etc., where 4, are scaling factors ensuring a unit “length” of l?;(t)
in the functional space, and W,, is the energy of the nth temporal mode defined as W,, = E:(t) dt. The set of functions l?;(t) is unique for every input pulse. A simple calculation shows that for a Gaussian input pulse the functions introduced by eq. (4.52) are exactly the standard Gauss-Hermite functions E;?(t)= b,,H,(t) exp( - t 2 / 2 T 2 )(see, e.g., HOCHSTRASSER [1970]). It is very easy to find W,, in the case of a simple input pulse. For example, if Eo(t)= exp ( - t 2 / T 2 ) we , have W , = W,, W , = 3 W,, W , = 15 W,, . . . , and if E,](t)= sech(t), we obtain W , = 0.67W0, W, = 0.47W0, W, = 1.48 W,, .... Using eqs. (4.26) and (4.52), we can write the envelope of the output pulse as a series expansion over the new set of functions &’(I), (4.53) The time orthogonality of ,!?;(t) means that j l?;(t) E;(t) dt = i.e., the fields E;l(f)do not interfere with each other. Then the energy of the output pulse will be the sum of the energies of all fields I?:(?) contributing to eq. (4.53). On theotherhand,theenergycorrespondingtoE;(t)is Ic;12 = I J,!?“:o,t(t)E;(t) dtI2. From eq. (4.52) we obtain the dispersive functions necessary to synthesize
E;,01,
268
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
where the dl, are constants. As we noted earlier, such functions could in practice be realized as a set of transparencies with variable transmittance. One simple method to test the temporal orthogonality of the envelopes given by eq. (4.52) is to measure their cross correlation in a Michelson interferometer (see, e.g., BRADLEY and NEW[ 19741). As is already known, a detector placed after the interferometer registers the transmitted energy f(r) as a function of the delay z between the input pulses. Thus, the signal from the detector will be proportional to cos (ooz) f(T) =
4+
s
&(t)
E;(r + T) dt
w:,+ w:,
(4.55)
Figure 2 1 shows the linear interferometric autocorrelation of an input pulse Eh(t) consisting of a few optical cycles. The oscillations in the signal arise from the cos(ooz) term in eq. (4.55). Figure 22 shows a cross correlation between Eh(t) and E;(t). Near zero delay a region where the interference vanishes because of the time orthogonality of the envelopes can be clearly seen. Finally, we should note that light pulses with time-orthogonal envelopes have potentially interesting applications in the research of linear and nonlinear transient phenomena. For example, let us consider a coherent propagation of a light pulse in a two-photon resonant medium (BELENOVand POLUEKTOV [ 19691). If the pulse consists of two orthogonal modes, the two-photon pulse area will be proportional to the simple sum of their energies (without the crossed term). This would enable a more flexible control of the propagation effects.
DELAY (ARBITRARY)
Fig. 21. Interferometric autocorrelationfunction of an input Gaussian pulse consisting of a few optical cycles (CHRISTOV [ 1990al).
111, § 41
PROPAGATION EFFECTS
2.0
269
,
0.0
DELAY (ARBITRARY)
Fig. 22. Interferometric cross correlation between the zero- and fourth-order orthogonal envel[199Oa]). opes (see eq. (4.52) and CHRISTOV
4.4. PROPAGATION IN A NONLINEAR MEDIUM
We noted in 0 2.3.1 that by using the nonlinear Schrddinger equation, HASEGAWAand TAPPERT [ 1973al theoretically predicted the possibility of producing optical solitons by mutual compensation of the dispersive broadening and nonlinear self-phase modulation in an optical fiber. The optical fiber is highly suitable for soliton experiments, since it is a stable and simple medium possessing high optical fidelity and low losses. Another important feature of the silica-glass-based fibers is the dependence of the GVD on the wavelength. As is already known, the GVD changes its sign at about A = 1.3 pm (MARCUSE [ 1980]), so that for A > 1.3 pm the GVD is negative (anomalous dispersion), I< 1.3 pm it is positive (normal dispersion). We should note that whereas for , by varying the fiber profile design, the zero-dispersion wavelength can be shifted to 1.6pm (AINSLIEand DAY [1986]). HASEGAWA and TAPPERT [ 1973a,b] also showed that the nonlinear propagation with negative GVD enables the observation of bright optical solitons, whereas in the opposite case (positive GVD) it is possible to produce dark solitons (see 0 2.3.1). Moreover, due to the Raman effect the propagation of both the bright and the dark solitons is accompanied by so-called “self-frequency shift”, which we shall discuss in 0 4.4.3. 4.4.1. Formation of bright solitons The first experimental observation of bright solitons in an optical fiber was connected with the development of mode-locked color-center lasers tunable in the infrared (IR, A > 1.3 pm). MOLLENAUER, STOLENand GORDON[ 19801
270
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
used a synchronously pumped, mode-locked laser utilizing F-j+centers in NaCl with tunability in the range of 1.35 to 1.75 pm (see § 3.2.2). At input parameters A. = 1.55 pm, D = 2nck,/A; = - 16 ps/(nm*km), S,, = cm2, T x 4 ps, and n; = 3.2 x 10- l6 cm2/W, the theoretical formula (2.37) yields for the critical power P, x 1 W. At an input power of about 1.24 W a typical soliton behavior of IR pulses with a duration of several picoseconds in a 700 m long fiber has been observed. Some findings about picosecond solitons were reviewed by VYSLOUKH[ 19821. It should be noted that the soliton formation becomes easier when the carrier frequency of the pulse is near the zero of the GVD, because the critical power P, is proportional to the dispersive parameter I k, I [see eq. (2.37)]. In fact, the nonlinear propagation in an optical fiber near zero GVD may be accompanied by some interesting effects that arise because of the third-order material dispersion (BLOW, DORANand COMMINS[1983]). By the use of the NSE, including the losses and the third-order dispersive parameter k, = d3k/dw3, WAI, MENYUK,LEE and CHEN[ 19861 investigated such a regime of propagation for the fundamental soliton, as well as for the second- and third-order solitons (breathers). By numerical simulations it was shown that the principal effect of the third-order GVD is to stimulate a resonance peak at a frequency o = oot 1/(2p,T), where p, = k3/6k,T. When the losses are neglected, the fundamental soliton is slightly affected, but even when A - A, B 0.001 pm, the dispersion causes a considerable decrease of the pulse amplitude. In the case of breathers the magnitude of the resonance peak increases fast and, in addition, there is a threshold of k, above which the breathers are broken into their constituent solitons. This picture is valid also for the third-order breather. Moreover, it was shown that for pulses with femtosecond duration the breathers are significantly affected by the third-order dispersion when A - A. x 0.01 pm. Another important prediction of the theory is that when the carrier frequency coincides precisely with the zero-dispersion wavelength, a splitting of the spectrum accompanied by a formation of two distinct fragments takes place, the first one in the anomalous dispersion regime (soliton), and the second in the normal dispersion regime (dispersive wave) (see WAI, MENYUK, LEE and CHEN[ 19871). The generation of femtosecond solitons in a single-mode fiber with a zeroGVD wavelength A. = 1.38 pm was reported by GOUVEIA-NETO, FALDON and TAYLOR [ 19881. The pulse source used is a Nd: YAG laser-pumped self-frequency shifter, which delivers femtosecond pulses as short as 170 fs, tunable in the spectral region of 1.32 to 1.45 pm (GOUVEIA-NETO,SOMBRA and TAYLOR [ 19881). A 50 m long optical fiber, single mode at 1.32 pm, is
111, § 41
PROPAGATION EFFECTS
27 1
used. The spectral splitting effect noted earlier, followed by the formation of a soliton and dispersive wave, is observed experimentally. A soliton as short as 140 fs down-shifted at 1.41 pm, has been produced, accompanied by a dispersive wave with a duration of 1.4 ps centered at about 1.35 pm. The increase in the pump amplitude leads to an increase in the frequency shift between the soliton and the dispersive wave. We should note that the ideal soliton regime exists only in a loss-less fiber. If the losses are taken into account, the soliton duration increases and its amplitude decreases during the propagation until the pulse decays completely. Furthermore, we shall see that this effect can be compensated for by continuous energy pumping of the soliton due to the stimulated Raman scattering process. 4.4.2. Formation of dark solitons The production of dark solitons enables us to realize a soliton regime in the visible, where the fiber GVD is positive (normal). The first experimental observation of optical dark solitons was made by WEINER,HERITAGE,HAWKINS, THURSTON,KIRSCHNER,LEAIRDand TOMLINSON [ 19881. The main difficulty concerning the synthesis of the initial dark pulse is overcome by the use of a powerful shaping technique, which, in principle, allows the production of light pulses with an arbitrary time profile. The design of such a device is discussed in detail in Q 4.3.3. A preliminary synthesized, odd-symmetry dark pulse as short as 185 fs is passed through a 1.4 m-long, single-mode optical fiber. When the input power reaches the critical power for a dark soliton (about 300 W), the pulse leaves the fiber without broadening. The dark soliton that is produced has a duration of 100-200 fs and 1-7 ps Gaussian background (fig. 23). The same authors have also synthesized an even-symmetry dark pulse, which splits, in the fiber, into two holes, separated by 2.3 ps (fig. 24). Some theoretical studies on the properties of propagating dark solitons are also of interest. By the use of the NSE, including the losses, ZHAO and BOURKOFF[ 1989al examined the propagation of a single dark pulse in a fiber. As we noted in 8 2.3.1, when the amplitude of the input dark pulse is equal to N - a, the pulse evolves into a fundamental soliton and 2(N - 1) secondary pulses which are excited during the propagation. They are symmetrically located on each side of the main pulse and move away from it with increasing distance (fig. 25). The total energy of the secondary solitons is W = - 2(n - a)’, which is always negative. An additional amount of energy equal to W , = 2 4 1 - a) is transferred into a generated nonsoliton part of radiation. It can be seen that W , is positive, which is attributed to the effect
212
GENERATION A N D PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III. 5 4
TIME(psec) Fig. 23. Measured (dotted lines) and calculated (solid lines) cross-correlation data for an oddsymmetry dark pulse. (a) Input dark pulse; (b)-(e) pulses emerging from the fiber for peak input power of (b) 1.5 W; (c) 52.5 W; (d) 150 W; (e) 300 W (WEINER, HERITAGE,HAWKINS, THURSTON, KIRSCHNER, LEAIRDand TOMLINSON [ 19881).
of bright pulses on the background. In contrast with the bright solitons, where (for N > 1) the picture of propagation is periodical (breathers), here the secondary dark solitons spread in a simple way. This can be explained by the repulsive character of the potential represented in the NSE [the term - I q I 2q in eq. (2.38)]. In the presence of losses in the fiber, the energy of a propagating and bright soliton decreases exponentiallyand its width increases (HASEGAWA KODAMA[ 19811). However, the spreading of the dark pulse is about two times slower and, moreover, more insensitive to the presence of background noise. This makes the dark solitons very promising as information carriers in optical communication systems(see § 4.5.1). Another important advantageof the dark solitons in this respect is their behavior during mutual interactions. By the use
111, § 41
213
PROPAGATION EFFECTS
"
-4
0
-2
4
2
TIME(psec)
Fig. 24. Measured (dotted lines) and calculated (solid lines) cross-correlation data for an evensymmetry dark pulse. (a) Input dark pulse; (b)-(e) pulses emerging from the fiber for a peak input power of(b) 2.5 W; (c) 50 W; (d) 150 W; (e) 285 W (WEINER,HERITAGE, HAWKINS, THURSTON, KIRSCHNER, LEAIRD and TOMLINSON [ 19881).
I
-10
-5
0
5
10
time Fig. 25. Propagation of a dark pulse with amplitude qo = 2 [see eq. (2.41)]and a = 0.5. The pulse shapes correspond to different distances (ZHAOand BOURKOFF[ 1989a1).
214
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 5 4
ofthe inverse scattering method, GORDON[ 19831 and DESEMand CHU[ 19871 showed that the interaction force between two bright solitons (i.e., a soliton pair) causes oscillations of its separation 6 with the distance 5, according to exp [2(6 - So)] = + { 1 + cos [45 exp ( - So)]} ,
(4.56)
where 6, is the initial distance between the solitons. It can be seen that the distance varies periodically along the fiber with a period $ 7 1 exp (h0). In the attractive phase the solitons may interfere provided 6, is not too large. On the other hand, the increase of b, limits the transmission rate of the fiber. By means [ 1989~1investiof a numerical solution of the NSE, ZHAOand BOURKOFF gated the interaction between a pair of dark solitons. It was shown that with the decrease of the initial distance 6, between the solitons, the repulsive force pushes them away from each other, which prevents a detrimental interaction. The following empirical dependence for the intersoliton separation is found, exp[2(6- So>]= $ { I + e x p [ 4 ~ e x p ( - 2 b o ) 1 } .
(4.57)
Thus, the distance between the dark solitons increases monotonically in contrast with the oscillating behavior given by eq. (4.56). 4.4.3. The soliton seFfrequency shgt
In 1985 DIANOV, KARASIK, MAMYSHEV, PROKHOROV, SERKIN,STELMKH and FOMICHEV observed a spectral asymmetry to the low-frequency side of the mean soliton frequency when multi-soliton pulses propagate into a fiber. They attributed this effect to the stimulated Raman conversion into a pulse spectrum. By experimental investigation of the propagation of femtosecond pulses in a single-mode, polarization-preserving fiber, MITSCHKEand MOLLENAUER [ 19861 registered the same effect and explained it as arising as a result of the flow of energy from the higher to the lower-frequency spectral components of the soliton. This effect is also called “the soliton self-frequency shift”. When the input power exceeds the critical power P, corresponding to the fundamental soliton, the propagating pulse splits into two satellites, whose separation increases with the increase in input power. This is accompanied by the formation of a long-wavelength shifted spectral peak; for Pin= 3P0 this shift is about 8 THz for a soliton of duration of 260 fs. This effect can be explained by bearing in mind that the higher-frequency components of the pulse act as a Raman pump and thus they amplify the lower frequencies. It is interesting to note that in spite of this energy transfer, the soliton remains as a stable entity.
111, § 41
PROPAGATION EFFECTS
215
The theory of the soliton self-frequency shift was developed by GORDON [ 19861. It is based on the NSE, in which the Raman effect is introduced in an elegant way by modifying the nonlinear term in order to describe a delayed response: 1 q 1 2q -,q J f ( s ) 1 q(t - s) I ds, where f ( s ) is a real function in the loss-less case. The primary result is the formula for the Raman-aided shift of the main soliton frequency (provided that this shift is a linear function of the distance): dv,/dz (THz/km) = 0.0436/T4, where T is the soliton duration in picoseconds, D is the time-of-flight dispersion constant, and both A and c are in units of centimeters and picoseconds. For example, if the full width at half maximum of the soliton is 1 ps, then after 23 km of propagation, the spectrum will be shifted by about 1 THz due to the Raman self-frequency shift, which is on the same order as the experimentally observed vaiues (MITSCHKEand MOLLENAUER [1986]). We should note that TAI, HASEGAWAand BEKKI [ 19881 predicted a quadratic dependence of the shift on the distance, which further justifies the preceding calculations. The effect of a self-shift in the case of dark solitons was observed by WEINER,THURSTON,TOMLINSON, HERITAGE,LEAIRD,KIRSCHNERand HAWKINS[ 19891. Dark solitons as short as 200 fs exhibit a quadratic length dependence of the self-shift. 4.4.4, Nonlinear propagation of chirped and noise pulses As we already showed in 8 3, there are many sources that deliver ultrashort pulses with a sweep of frequency (chirp). Therefore, it is interesting to study some effects that arise when such pulses propagate in a dispersive nonlinear medium. By the use of the prescribed channel approximation (see 9 2.3.2), FATTAKHOV and CHIRKIN[ 19841 analyzed the influence of a regular linear chirp on the propagation of an input Gaussian pulse q(0, z) = exp [( - 2’1 2T2) + iaoz2]. For the normalized pulse intensity one obtains the same relation as in eq. (2.36a,c), but with a’(<) = 1 - 2a,T2 sin(25) [compare with eq. (2.49b)l. Parenthetically, we should note that this equation is a result of a simplification of the exact equation in which it is assumed that R = 0.5 (see 8 2.3.2). It can be seen that when a, > 0 the pulse duration initially decreases, but when a, < 0 it increases. Therefore, evidence of a frequency modulation always causes an obstacle to soliton formation. However, DESEMand CHU [ 19861 have shown that for a given input chirp parameter aorthere is a certain value of the input power at which the pulse evolves into a soliton, which is always broader than in the nonchirped case. It is followed by a nonsoliton dispersive wave decaying asymptotically as 1/$. We shall use the results for the coherence function given by eqs. (2.50)-(2.52) to describe the nonlinear propagation of a noise pulse. Let the input pulse be
216
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[]]I, 8 4
Gaussian,q(O, T ) = exp [( - t 2 / 2 T 2 )+ icp(~)],wherethephasecpisaGaussian random process whose time of correlation and dispersion are T, and a, respectively. Then, from eqs. (2.50)-(2.52) it follows that in the case of strong phase fluctuations ( a 8 l), when the equality R = ;( 1 + 4a2/T;) is fulfilled, the pulse exhibits a soliton-like propagation. This is in contrast to the case of regular phase modulation (chirp) because of the statistical averaging of the contributions of the phase modulations. If there is a noise pulse given by q(0, T ) = exp ( - T ~ / ~F ( T ) at ~ the ) input, F ( T ) being a Gaussian random process with a coherence time T,, then at a short distance ( z 4 L D ) from eqs. (2.50) and (2.52) it follows that the averaged time duration T(5)are given by T(5) = 40 T ,
(4.58a) (4.58b)
and where a 2 ( r )= (1 - 2 R r 2 ) 2 + 5’ + (1 + 4R2t2)T2/T,2t2. FATTAKHOV C H ~ R K[I19851 N showed that an interesting situation exists in which the pulse duration increases whereas the coherence time decreases, which leads to an irregularization of the pulse due to the nonlinear propagation. However, when the distance of propagation considerably exceeds L,, both the averaged time duration and coherence time increase.
4.5. FEMTOSECOND PULSES IN INFORMATION SYSTEMS
4.5.1. Soliton-based communication systems Because of their distortionless propagation, the optical solitons are highly suitable as a carrier in the design of an extra-high bit rate transmission system. However, the losses in the fiber hinder the soliton transmission over a long distance. HASEGAWA [ 19831showed that the use of stimulated Raman scattering as a supporting process may allow reshaping of the propagating soliton. The idea is to inject a continuous pump wave at a shorter wavelength periodically in both directions into the fiber where the soliton train propagates, so that the energy transfer from the pump wave to the soliton can compensate for the losses (see also BLOW,DORANand WOOD [ 19881). By means of numerical simulations, HASEGAWA[ 19841 showed that a stable transmission of 2 10 Gbit over a distance of = 5000 km can be achieved by the proper choice
111, § 41
PROPAGATION EFFECTS
211
of system parameters. This idea was realized experimentally in the picosecond domain by MOLLENAUER and SMITH [ 19881. In this experiment 55 ps solitons (A = 1.6 pm) recirculate many times in a closed 42 km loop with losses compensated for by the Raman gain (Apump = 1.497 pm). The demonstrated transmission is over a distance of 4000 km, and it corresponds to a rate-length product of about 11 000 GHz * krn. However, as we have already discussed in $ 4.4, the dark solitons are more stable against fiber losses and noise than the bright solitons. ZHAOand BOURKOFF[ 1989bl theoretically predicted that an all-optical stable system with a transmission rate of 20 GHz can be designed by using dark solitons. The losses can, in this case, be compensated for by stimulated Raman gain. In addition to the losses and background noise in the fiber, there are some circumstances limiting the transmission rate. The first effect is attributed to the soliton self-frequency shift (see 0 4.4.3). Whereas the interpulse Raman scattering developed self-frequency shift causes a down-shift of the mean frequency, the stimulated Raman scattering from an external pump pulse gives rise to preferential amplification at the peak of the Stokes band. Thus, the external Raman amplificatjon can suppress the soliton self-frequency shift. This has been observed experimentally by GOUVEIA-NETO, GOMESand TAYLOR [ 19891, who produced a stable soliton operation (T = 450 fs) near 1.44 prn, which is slightly down-shifted from the peak of the Raman gain. The second effect limiting the performance of a soliton-based communication system is the mutual interaction between the solitons. As we showed in $4.4, the attractive force between two adjacent solitons decreases exponentially with their mutual distance. Therefore, the separation between the solitons should be sufficiently large to avoid undesirable interference effects. By using an analytical perturbative approach, ANDERSON and LISAK[ 19861 found that the interaction is highly sensitive to the initial difference between the phases of the individual solitons. In fact, the distance between two equal solitons that are at rest initially is given by
a(()
=
b,
1 +{f[cosh(4vm() + cos(4vn()l), 2v
(4.59)
where bo is the initial distance between the solitons, v is a parameter, and m and n are given by (4.60a) (4.60b)
where $(O) is the initial relative phase of the solitons. It can be seen from
278
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, 8 4
eqs. (4.59) and (4.60) that for $ A < $(O) < A the two solitons separate monotonously, whereas for $(O) < A the interaction is attractive. This means that by proper phase shifting (larger than $ n) of consecutive pulses, the solitons will separate during the propagation, which prevents the bit-rate degradation due to interpulse interaction [compare with eq. (4.56)]. In fact, the use of solitons as an information carrier has a crucial advantage. Because of the balance between dispersion and nonlinearity, the soliton does not exhibit the influence of so-called “modulation instability (MI)”. On the other hand, an MI can be successfully used to produce a train of optical solitons, and, therefore, we shall consider it in some detail. The MI is caused by the interplay between the nonlinearity and dispersion in the fiber, when the pulse is with an amplitude and/or a phase modulation (natural or induced). Then the pulse exhibits an exponential growth of the initial modulations, becomes unstable, and breaks down to a series of pulses during the propagation (HASEGAWA and BRINKMAN [ 19801). It should be noted that MI appears only in the anomalous dispersion regime (k, < 0). The frequency of the modulation of the pulse profile due to MI has an upper limit given by a, = 2/T, where T is the bright soliton duration given by eq. (2.37) (ANDERSON and LISAK[ 19841). For example, for a typical set of parameters no = 1.45, n2 = 1.1 x 10- l 3 esu, A0 = 1.55 pm, and input intensity of lo6 W/cmZ,one finds that T z 4 ps, which corresponds to a critical frequency a = 5 x 10” s - l . Hence, most of the available sources fall into the region of the MI. The first experimental observation of MI without an external modulation was made by TAI, HASEGAWA and [ 19861 by using a Nd: YAG source ( A = 1.319 pm) delivering 100 ps TOMITA pulses at a 100 MHz repetition rate. The period of the observed MI modulations is of the order of 2 ps, and it decreases with the increase of the input power. The generation of a train of solitons by an externally induced modulation was studied theoretically by HASEGAWA[ 19841. It was shown that an initial sine modulation of CW radiation can be reshaped in the fiber as a train of picosecond solitons. An interesting demonstration of this transformation in the femtosecond domain was performed by ISLAM, SUCHA,BAR-JOSEPH, WEGENER, GORDON and CHEMLA[ 19891. They observed that the combined action of both the soliton self-frequency shift and the modulation instability in the fiber cause a multi-soliton collision process, which results in a narrow fundamental soliton as short as 1OOfs over a spectral range 1.55 pm < I < 1.85 pm. The pump source was a NaCl color-center laser (Ipump = 1.5 pm) delivering pulses with a duration of 14 ps. Such tunable femtosecond pulses are especially suitable as information carrier.
+
111, § 41
PROPAGATION EFFECTS
219
Until now we have considered the noise in the fiber phenomenologically. It is known, however, that the noise has a well-established quantum origin. Here, we shall briefly discuss the influence of the quantum noise on the performance of a soliton-based information systems. The effect is related to the squeezing phenomenon that arises when a light wave passes through a nonlinear medium (see, e.g., YUEN [ 19761, LOUDONand KNIGHT [ 19871). Accordingly, the nonlinear interaction modifies the noise properties of the radiation, transmitting the larger fluctuations and suppressing the weaker ones. Speaking in terms of quantum theory, the squeezing implies a reduction of the fluctuations of a certain variable below the fundamental quantum limit. This is always accompanied by an increase in the fluctuations of the complementary variable. An example in optics is the squeezing of the mean photon number of the radiation mode, i.e., the reduction of the amplitude fluctuations accompanied by a corresponding increase in the phase fluctuations. Inasmuch as the propagation of light through an optical fiber exhibits nonlinear properties, the presence of squeezing can easily be observed by monitoring the degree of the second-order coherence in an interferometer, in whose arm the fiber is inserted (AGARWAL [ 19891). The theory treating the squeezing in solitons is ofgreat interest because of their promising applications in optical communications. By studying the stochastic nonlinear Schrddinger equation, DRUMMOND and CARTER[ 19871 showed that the propagation of solitons in an optical fiber over a large distance leads to the appearance of quantum solitons whose fluctuations are considerably below the vacuum level. In this case, a large squeezing takes place over the entire soliton spectrum. Another limitation that can be overcome by squeezing is attributed to the shot noise in the detector for a weak input signal. CARTER and SHELBY [ 19891 showed, under certain circumAs DRUMMOND, stances (e.g., in the presence of a so-called “time-dependent local oscillator”) considerable squeezing localized near the soliton peak may be induced. This causes a great decrease in the measured quantum noise. 4.5.2. Image processing by oplical pulses Standard image processing is based on certain manipulations of the radiation spatial structure (wave front) or, equivalently, of its angular spectrum (see, e.g., GOODMAN [ 19681). This is described in terms of Fourier optics, whose main ideas can be successfully transferred into the time domain (see 0 4.3). However, the temporal processing requires a spectrum of the input radiation as broad as possible in contrast to the spatial case, where a plane initial wave front is suitable for many applications. Moreover, it is desirable for the input spectral components to be regularly phased in time to achieve sensitivity to phase
280
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[III, $ 4
information. Time-domain Fourier optics uses all-optical schemes similar to that shown in fig. 20 and described in 0 4.3.3. In recent years, the development of sources delivering femtosecond pulses made it possible to design some practical schemes embodying the preceding ideas. WEINERand HERITAGE [ 19871 proposed a configuration (similar to that shown in fig. 20) acting as an ultrafast self-routing switch. It is based on coding a mask pattern into the time structure of the processed pulse in a transmitter. In the receiver the information carried by the pulse can be decoded by a scheme similar to the transmitter. In fact, the output signal in the receiver is proportional to the cross correlation between the receiver mask and the transmitter mask. That is why a high-rate recognition of one-dimension patterns transmitted over a long distance through many channels (a code-division, multiple-access communication network) can be performed. DANAILOV, CHRISTOVand MICHAILOV [ 19891 analyzed and experimentally verified a similar Fourier-processing system for the purposes of an all-optical image transfer (see fig. 26). The information is coded into the temporal domain by a mask with transmittance function H(k). The pulse leaving the transmitter propagates in free space and falls onto the receiver, which restores the image at the input of the image sensor. It should be noted
r
Transmitter -----
Input
1
r
Receiver ---_ -
1
pulse
I Y-
Fig. 26. Optical scheme of the image transfer system (DANAILOV, CHRISTOV and MICHAILOV [ 19891).
111, I 41
PROPAGATION EFFECTS
28 1
that, although this device is particularly suitable for the transmission of amplitude information, it could also be used successfully for the reconstruction of phase information by applying conventional methods for phase retrieval, e.g., interference with a reference beam in the sensor plane or using the phasecontrast method (BORN and WOLF [ 19681, DANAILOV and CHRISTOV [ 199Ocl). The analytical description of the system performance is based on the formalism developed in § 4.3.3. For a configuration near to Littrow ( a % 1) the spectral distribution in the plane of the image sensor is given by E,(x) = E(X) H(X)
7
where
E(x)=
(4.6 1a)
c
J &(a) exp [ - a(x - k,PR)2a2]d a
(4.6 1b)
is the spatial distribution in both the Fourier plane F and in the plane F; of the receiver. Here x is a modified coordinate in the sensor plane x = oox/Jc, and k, = w,,/c, j3 = 2 m/o,’d cos yo, where J is the focal distance of the lenses, and a is the radius of the Gaussian input beam. We should note that the Gaussian beam is especially convenient for the transmission in free space because it remains shape-invariant during propagation. Thus, from eq. (4.61b) it follows that the field in the plane F can be considered as a superposition of various frequency components, each having a Gaussian spatial profile with a width 6xo = 2/a (in punits), centered at x = koPn (k, corresponds to the direction of the mean frequency 0,). Then, performing the integration in eq. (4.61b) for a Gaussian input pulse with half-duration T, we find (4.62) where the parameter Q = k/3a/T describes the ratio of the total spot size in the plane F(which is equal to 6xtot = 2( 1 + Q2)’12/a)to the “monochromatic” spot size 6xo [see eq. (4.48)]. It should be noted that the parameter Q, in practice, determines the resolution of the system and it increases for a broader input spectrum (shorter pulse). An essential requirement for a good performance of the system is that the beam leaves the transmitter nearly Gaussian. This requirement will be fulfilled provided that the function H( x ) varies slightly over the elementary size 6 x o . This is not valid, e.g., when H ( x ) is a binary image because in this case the diffraction on the edges will lead to high diffraction orders, and a single edge in the original image will appear smeared in the
282
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
(111,s 4
W
c " 1 + ._ L
5 m
i
+ 0.5 Y m u1
0
-20
a x
!I
-
z
-10
1 -
0
10
20
c
u1
W C
+
.c _ 'p
w
-
N ._
0.5
.
m
E
L 0
z
0 -*
Fig. 27. (a) Transmittance H(x)of a mask with period 1.5 x0; (b) intensity distribution at the CHRISTOV and MICHAILOV [ 19891). input of the image sensor (DANAILOV,
-20
b
-10
0
10
20
30
%t-coordinate
Fig. 28. (a) Transmittance H ( x ) of a mask with period 4xo; (b) intensity distribution at the input of the image sensor (DANAILOV, CHRISTOV and MICHAILOV [1989]).
1 1 4 8 41
283
PROPAGATION EFFECTS
restored image. This is demonstrated in figs. 27 and 28 for two binary amplitude gratings with different periods. The intensity profiles in the image sensor plane are calculated using the Kirchhoff integral (far zone). It can be seen that when the period of the image is equal to 1.5 xo, the transmitted image differs considerably from the mask function. When the period grows to 4 xo, the reconstruction becomes more successive. Hence, when a binary image is transferred, the minimum size of the image pixel should be at least equal to 2 x,,. Inasmuch as the full size of the transferred image cannot exceed 6xtot, the information capacity of the system is about f ( 1 + Q2)1/2 x f Q bits. In the experimental set-up demonstrated in fig. 26, a CPML rhodamine 6 G dye laser (see § 3.2.1), delivering pulses as short as 110 fs at a repetition rate of 100 MHz, was used
SPECTRUM
-3
-L
a
-2
-1
0
1
2
FREQUENCY x E + 13 Hz
Fig. 29. (a) Plot of the input spectrum; (b) magnified photo of the original image, hole diameter 70 pm; (c) a reconstructed image (DANAILOV, CHRISTOV and MICHAILOV [1989]).
284
GENERATION AND PROPAGATION OF ULTRASHORT OPTICAL PULSES
[111
as the source. The distance between the transmitter and receiver is about 1 m. All gratings in the system are holographic with d - = 1800 l/mm; two equal Fourier objectives with a focal distance f = 18 cm are used in the transmitter, and another Fourier objective with f = 25 cm is used in the receiver. These parameters give a value of about 340 for the parameter Q. Figure 29 presents the spectrum of the input pulses and the result from the transfer of a mask consisting of several holes.
'
8 5. Conclusion We have reviewed the principles and characteristics of some devices that use femtosecond optical pulses. They allow us to achieve a deeper understanding of the dynamic nature of light-matter interaction. In fact, this is a rapidly developing field, and new interesting results in generation techniques can be expected. It seems difficult to overcome the few-optical-cycles limit of pulse duration produced so far. However, the spectral enriching of generated radiation due to high intense interactions has the potential to eliminate the carrier optical cycle as a measure of the pulse duration. There is no doubt that the various phenomena of pulse propagation effects discussed in this review will be an object of further pure and applied research areas, such as optical communications, dynamic spectroscopy, and ultrafast processes in solid state.
Acknowledgments The author would like to thank Professor S. M. Saltiel and Drs. N. I. Michailov, V. P. Petrov, N. I. Minkovski, and M. B. Danailov for their helpful comments in the course of preparing this review.
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E. WOLF, PROGRESS IN OPTICS XXIX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991
IV
TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY* BY
GERDWEIGELT Max-Planck-Instituifur Radioasironomie Auf dem Huge169 0-5300Bonn 1 . Germany
* Based on data collected at the European Southern Observatory, La Silla, Chile.
CONTENTS PAGE
$ 1 . INTRODUCTION
. . . . . . . . . . . . . . . . . . . 295
$ 2. SPECKLE MASKING: BISPECTRUM OR TRIPLE CORRELATION PROCESSING . . . . . . . . . . . . . . . . 296 $ 3 . OBJECTIVE PRISM SPECKLE SPECTROSCOPY . . . . . 309 $ 4. WIDEBAND PROJECTION SPECKLE SPECTROSCOPY . 309 $ 5 . OPTICAL LONG-BASELINE INTERFEROMETRY AND APERTURE SYNTHESIS . . . . . . . . . . . . . . . . 311 $ 6. CONCLUDING REMARKS . . . . . . . . . . . . . . . 315
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 316 APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . .
316
. . . . . . . . . . . . . . . . . . . . . . .
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5
1. Introduction
The refractive index variations in the atmosphere of the earth restrict the angular resolution of large, ground-based telescopes to about 0.5 arcsec. This resolution limit, called the “seeing limit”, is frustrating since the theoretical diffraction limit of a 5 m telescope is about 0.02 arcsec at a wavelength of 500 nm. Fortunately, it is possible to overcome atmospheric image degradation by various interferometric methods. The speckle interferometry method [ 19701) can reconstruct the diffraction-limited autocorrelation (LABEYRIE of astronomical objects. True images with diffraction-limited resolution (e.g., 0.02 arcsec) can be reconstructed by the Knox-Thompson method (KNOXand THOMPSON [ 1974]), by the nonredundant mask technique (see e.g., BALDWIN, PEARSON, HANIFF,MACKAYand WARNER[ 19861, READHEAD,NAKAJIMA, NEUGEBAUER,OKE and SARGENT [ 19881, NAKAJIMA,KULKARNI, GORHAM, GHEZ,NEUGEBAUER, OKE,PRINCEand READHEAD [ 1989]), and by the speckle masking method (WEIGELT[ 19771, WEIGELTand WIRNITZER [ 19831, LOHMANN, WEIGELTand WIRNITZER[ 1983]), which is based on hispectrum or triple correlation processing of astronomical speckle interferograms. More references to image reconstruction methods can be found in the review articles on interferometric imaging by DAINTY[ 19841 and RODDIER [ 19881. The nonredundant mask technique and speckle masking can measure the so-called “closure phases”, which are very important for optical longbaseline interferometry. The great advantage of optical long-baseline interferometry with three or more telescopes is that it can yield images and spectra with fantastic angular resolution. For example, with a baseline of 150 m a resolution of 0.001 arcsec can be obtained. The biggest planned optical longbaseline interferometer is the Very Large Telescope (VLT) of the European Southern Observatory (ESO). It will consist of four 8 m telescopes and at least two 2 m telescopes. We will discuss the theory and astronomical applications of the speckle masking method, two different speckle spectroscopy methods, and optical long-baseline interferometry. The speckle spectroscopy methods have the advantage that they yield the spectrum of each resolution element of the object in addition to the high-resolution image. This additional spectral information 295
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is, of course, highly important for the study of the physical nature of astronomical objects.
8 2.
Speckle Masking: Bispectrum or Triple Correlation Processing
In speckle masking a diffraction-limited image of the object is reconstructed from a large number of speckle interferograms recorded with a large telescope. Speckle interferograms are photographs recorded with a short exposure time of about 0.05 s or less. This short exposure time is necessary to “freeze” the atmosphere during the exposure. Figures 1-3 show that speckle interferograms consist of many small, bright dots, called speckles. Speckles are interference maxima with an average diameter of about A/D 0.02 arcsec for wavelength A 500 nm and telescope diameter D 5 m. In other words, speckles are as small as the theoretical Airy pattern of the aberration-free telescope. Therefore speckle interferograms contain high-resolution object information. High-contrast speckle interferograms can be recorded with speckle cameras consisting of the following components: (1) Shutter. The optimum exposure time depends on the atmospheric conditions (speed of the turbulence cells). Typical exposure times are in the range of 0.05 to 0.002 s. (2) Image intensijer. The required gain depends on the brightness of the object. For objects fainter than astronomical magnitude (brightness) 10, the gain of the image intensifier has to be large enough (- lo6)to record individual photon events. ( 3 ) Microscope. The speckle interferograms in the focal plane of the telescope have to be magnlfied by a microscope in front of the image intensifier because of the poor resolution of high-gain image intensifiers. The size of individual speckles is about AF, where F is the f-ratio of the telescope and A is the wavelength of the light. For F = 8 and I = 0.5 pm the speckle size is only about 4 pm, which is much too small for high-gain image intensifiers. Therefore a microscope with, for example, magnification 32 in front of the image intensifier is used to obtain a speckle size of -0.1 mm. (4) Znteferencefilter. A white-light speckle interferogram consists of radial interference structures instead of sharp speckles. Radial structures are obtained since different wavelengths produce similar speckle interferograms, but with a different scale, which is proportional to the wavelength. Typical filter bandwidths are in the range of 10 to 50 nm (quasi-monochromatic light).
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Fig. 1. Time sequence of three speckle interferograms of the double star Beta Mon (separation -2.9 arcsec; 1.5m Danish/ESO telescope). The speckle interferograms were recorded at a frame rate of 32 frames/s. One can see that the speckle fine structure has partially changed after 1/32 s and that the speckle interferograms of the two components of the double star are similar (partial isoplanicity).
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( 5 ) Prism system for compensating atmospheric dispersion. Atmospheric dispersion elongates the speckles. The elongation increases with the zenith distance. One can compensate for atmospheric dispersion by inserting nondeviating prisms into the beam. A prism system with the required variable
-
Fig. 2. Speckle interferograms of different types of objects. (a) Single star Alpha Hydrae (2.2m ESO/MPG telescope). (b) Double star 95 Herculi (separation 6 arcsec). The speckle interferogram consists of a small number of large speckles since it was recorded with a small 60cm telescope (Bamberg observatory), (c) Double star Zeta Aquarii (separation 1.7 arcsec; 3.6m ESO telescope). (d) Gravitational lens Quasar PG 11 15 + 080 (each dot is an individual photon event; 1.5 m Danish/ESO telescope).
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TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY
Fig. 3. Speckle masking observation of the close spectroscopic double star Psi Sagittarii (epoch 1982.378). (a) One of 150 speckle interferograms recorded with the 3.6m ESO telescope. (b) High-resolutionimage reconstructedfrom the 150 speckle interferograrns by speckle masking. The separation of the two stars is about 0.18arcsec. (From WEIGELTand WIRNITZER [ 19831.)
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dispersion can be obtained by using two nondeviating prisms that can be rotated relative to each other. ( 6 ) Digital CCD-camera and computer for data recording. The intensified image on the phosphor screen of the image intensifier can be recorded, for example, with a CCD camera, or real-time photon-counting detectors can be used instead of an intensified CCD camera. The intensity distribution i,(x) of the n th recorded speckle interferogram can be described by the incoherent, space-invariant imaging equation L(x) = 4 x ) @ P J X ) =
1
o(x')p,(x - x')dx' , n
=
1 , 2 , 3 , . . . ,N
( N - 103-105), (2.1)
where o(x) is the object intensity distribution, 8 denotes the convolution operator, p,(x) is the point spread function of the atmosphere/telescope, x is a two-dimensional space vector, and N is the total number of recorded speckle interferograms. In the following, the index n of the random functions i,(x) and p,(x) and of their Fourier transforms will be omitted. The goal of speckle imaging is to reconstruct a diffraction-limited image of the object o(x) from a sequence of typically 103-105 speckle interferograms i(x). Speckle masking consists of the following four image processing steps: Image processing step I : This step includes two possibilities: calculation of the ensemble average triple correlation
( T ( x ,y ) )
=
(s
i ( x ' ) i(x'
+ x ) i(x' + y ) dx' ,
)
or calculation of the ensemble average bispectrum
of a large number of speckle interferograms i(x). The angle brackets denote ensemble average over all N speckle interferograms, the asterisk denotes complex conjugation, and u and u are two-dimensional coordinate vectors in Fourier space. I(u), I(v), and I*(u + v ) are Fourier transforms of i(x); i.e. I(u) =
I(v) =
I
i(x) exp ( - 2 lciu * x) dx ,
s
I*(u + u )
i(x) exp ( - 2 lciv x) dx , =
s
i(n) exp [2 xi(u
+ u ) - x ] dx .
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The bispectrum B(u, u) is the four-dimensional Fourier transform of the triple correlation T(x,y ) ,
B(u,u)=
11
T(x,y)exp[ - 2 n i ( u * x + u * y ) l d x d y (see Appendix A) .
(2.5)
Since x, y , u, and u are two-dimensional vectors, the triple correlation T(x, y ) and the bispectrum B(u, u ) are four-dimensional functions. If the speckle interferograms are one-dimensional, the bispectrum and the triple correlation are two-dimensional. The dimension of the triple correlation and of the bispectrum is always twice the dimension of the speckle interferograms. In many applications it is sufficient to work with small subsets of the average bispectrum. The data processing then can easily be performed with a small computer. An alternative is to evaluate many different one-dimensional projections of each two-dimensional speckle interferogram. In this case the bispectra obtained are two-dimensional. A one-dimensional image of the object can be reconstructed for each projection direction. From many different onedimensional projections of the two-dimensional object, a two-dimensional image can be reconstructed by tomographic techniques. This method is highly instructive since the modulus of the two-dimensional bispectra can easily be displayed on the computer monitor. In most applications it is advantageous to use bispectrum processing instead of triple correlation processing. The advantages of the triple correlation are that it can be easily visualized and that it can be used for photon-counting triple correlation techniques. In the following, we will discuss the theory of bispectrum processing. Image processing step 2 : Compensation of the photon bias in the ensemble average bispectrum ( B ( u , u ) ) (WIRNITZER [ 19851, HOFMANNand WEIGELT [ 19871). Image processing step 3 : Compensation of the speckle masking transfer function. From i(x) = o ( x ) @ p ( x ) it follows for the Fourier transform I(u) of i(x):
where I(u), O(u), and P(u) are the Fourier transforms of i(x), o(x), and p ( x ) , respectively. If we insert eq. (2.6) into eq. (2.3), we obtain for the ensemble
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average bispectrum ( B ( u , u ) ) , (B(u,u ) )
+ u) P*(u + 0 ) ) = O(u) O(u) O*(u + u) ( P ( u )P(0) P*(u + u ) ) . = ( O ( u )P(u) O(u) P(u) O*(u
(2.7)
( P ( u ) P(u) P*(u + u ) ) is called the speckle masking transfer function. It can be derived from the speckle interferograms of a point source, or it can be calculated theoretically (LOHMANN, WEIGELTand WIRNITZER[ 19831, VON DER LUHE [ 19851). Since the speckle masking transfer function is greater than zero up to the diffraction cutoff frequency, eq. (2.7) can be divided by the speckle masking transfer function, and we obtain for the bispectrum B,(u, u) of the object o(x), Bo(u, u)= O(u) O(u) O*(u t =
(I@)
I ( 0 ) Z*(u
(2.8)
0)
+ u ) ) /( P(u) P ( 0 ) P*(u + 0 ) ) .
(2.9)
Illustrative examples of two-dimensional average bispectra (modulus) of onedimensional speckle interferograms, speckle masking transfer functions, and the modulus of two-dimensional object bispectra are described in the literature. Image processing step 4 : Derivation of modulus and phase of the object Fourier transform O(u) from the object bispectrum B,(u, u). We denote the phase of the Fourier transform of the object by q(u) and the phase of the (2.10) (2.11)
(2.12) iq(u
+ u)] ,
(2.13) (2.14) (2.15)
Equation (2.15) is a recursive equation for calculating the phase of the object Fourier transform at coordinate w = u + u if the phase of the object Fourier
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TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY
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transform is known at coordinates u and u. The phase B(u, u) is known from eq. (2.9). Equation (2.14) is called a "closure phase relation". In other words we have shown that speckle masking is a speckle method which can reconstruct closure phases. Closure phases play an important role in radio interferometry (JENNISON [ 19581, CORNWELL [ 19891). Since speckle masking can measure closure phases, it can be applied to both single-dish telescopes and optical long-baseline interferometers. Image reconstruction is possible even if the optical transfer function consists of isolated patches with large gaps between the patches (as in the case of the VLT). In this case the Knox-Thompson method cannot be applied. For the recursive calculation of the phase q(w) = cp(u + u) we need in addition to the bispectrum phase P(u, u) the starting values cp(0, 0), q(0, l), and cp(1,O). Since o(x) is real, O(u) is Hermitian. Therefore O(u) = 0*( - u), O(0,O) = O*(O, 0), and therefore cp(0,O) = 0. Since we are not interested in the absolute position of the reconstructed image, q(0, 1) and cp(1,O) can be set to zero. With these starting values we obtain, e.g.,
d o , 2) = d o , 1) + 440, 1) - B [ ( O , 11, (0, 111 d o , 3) = d o , 2) + C p W , 1) - B [ ( O , 21, (0, 111 d o , 4) = cP(O93) + Cp(0, 1) - 8[(0,3), (0,1)1 ...
9
9 9
(P(290) = cp(L 0) + d l , 0) - B [ ( 1 , 0 ) , (1,O)l (P(3,O) = rP(Z 0) + d l t 0) - "29 01, (1,O)I (P(470) = (P(390) + d l , 0) - 8[(3,0), (1,O)l 9
1
7
...
Cp(L 1) = rp(L 0) + d o , 1) - 8[(1,0), (0, 1)1 (P(29 1) = (P(2, 0) + d o , 1) - 8[(2,0), (0, 1)1 , 443, 1) = (P(330) + d o , 1) - 8[(3,0), (0, 111 9 9
...
44392) = ~ ( 1 -, 1) + ~ ( 2 ~-3B)[ ( l , - 11%(2,3)1* (P(3-2) = cp(L 0) + 4 G 2 ) - B [ ( L O), (2,211 (P(392) = (P(290) + rp(L 2) - 8[(2,0), (1,2)1 d 3 9 2) = Cp(0, 1) + d 3 , 1) - "0, 11, (3, 111 9 (P(392) = rp(L 1) + d 2 , 1) - B [ ( L 11, (2, 111 (P(392) = cP(O92) + (P(3,O) - BE(0, 21, (3,011 9 1
1
9
...
(2.16)
The advantage of this recursive phase calculation is that for each element of the object Fourier phase q(w) = rp(u + u) there are many different recursion paths and that it is possible to average over all cp(w)-values to improve the
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signal-to-noise ratio. For example, for the element 9(3,2) there are 8 recursion paths and for ( ~ ( 6 ~there 4 ) are 64 paths. Averaging over all paths yields
where A4 is the number of recursion paths. In actual applications the phase calculation is performed with complex exponential functions,
Not all recursion paths for the same cp(w)-value yield the same signal-to-noise ratio. Therefore, different weight functions have to be chosen for different paths. Another possibility is to use only a few short u-vectors (HOFMANNand WEIGELT[ 1986a1). The modulus of the object Fourier transform can be derived from the object bispectrum in two different ways. From eq. (2.8) it follows for u = 0, B,(O, u) = O(0) O(u) O*(O + u) = const. I O(u)I2.
(2.19)
The second way is the recursive calculation. From eq. (2.12) it follows
(2.20) f o r q u w i t h IO(u)I IO(u)I 2 0 . Modifications or extensions of the version of speckle masking that has just been described are photon-counting triple correlation processing (SCHERTL, FLEISCHMANN, HOFMANNand WEIGELT[ 19871, AYERS,NORTHCOTTand DAINTY[ 1988]), cross-triple correlation processing (HOFMANNand WEIGELT [ 1987]), tomographic speckle masking (SCHERTL,FLEISCHMANN, HOFMANN [ 1987]), and alternatives for the reconstruction of the image of and WEIGELT the object from the bispectrum (LOHMANN[ 19861, LANNES [ 19881, GIANNAKIS [ 19891 and references herein, HOFMANNand WEIGELT[ 19901, GORHAM,GHEZ, KULKARNI, NAKAJIMA, NEUGEBAUER, OKE and PRINCE
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[ 19901). Pupil plane bispectrum methods were described by HOFMANNand WEIGELT[ 1986b] and RODDIERand RODDIER[ 19861. References to bispectrum applications in various research areas were described by LOHMANN, WEIGELTand WIRNITZER [ 19831. A simple special case, where speckle masking has been applied, is the case where there is no atmospheric image degrada[ 19851). tion but only random image motion (see, e.g., BARTELTand WIRNITZER In normal astronomical speckle masking observations all the following types of image degradation exist simultaneously: atmospheric turbulence (speckle noise), image motion (e.g., tracking errors of the telescope in addition to atmospheric image motion of the speckle interferograms, which is always present), stationary or slowly changing telescope aberrations, photon noise, and other types of noise (e.g., dark current). Figures 3-6 show speckle masking observations of the spectroscopic double star Psi Sagittarii, the very massive variable object Eta Carinae, the central object in the H I1 region NGC 3603, and of the Seyfert galaxy NGC 1068 (HOFMANN, MAUDERand WEIGELT[ 19891). Highly interesting speckle mask-
Fig. 4. Diffraction-limited image of Eta Carinae reconstructed from 300 speckle interferograms by speckle masking. The reconstructed image shows that Eta Carinae consists of a dominant
star and three close objects at separations of 0.11 arcsec, 0.18 arcsec, and 0.21 arcsec. (From HOFMANNand WEIGELT[1988].)
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Fig. 5. Diffraction-limited image ofthe central object in the H I1 region NGC 3603 reconstructed from 300 speckle interferograms by speckle masking. The image shows that the object is a star cluster consisting of six stars. The stars have astronomical magnitudes (brightness) in the range of 12 to 15. The separation of the closest pair at the bottom is -0.09 arcsec. (From BAIER, ECKERT, HOFMANN, MAUDER,SCHERTL, WEGHORNand WEIGELT[ 19881.)
ing observations were, for example, also made of the following objects: multiple star R 136a in the 30 Doradus nebula (NERIand GREWING [1988]), solar granulation (PEHLEMANNand VON DER LOHE [1989]), IR double stars (LEINERTand HAAS[ 1987]), IR object Red Rectangle (LEINERTand HAAS [ 1989]), infrared object HL Tauri (BECKWITH,SARGENT,KORESKOand WEINTRAUB [ 1989]), infrared double stars (FREEMAN, CHRISTOU, MCCARTHY and COBB [ 19871, CHRISTOU,FREEMAN,RODDIER, MCCARTHY,COBBand SHAKLAN [ 19871, FREEMAN, CHRISTOU, RODDIER, MCCARTHYand COBB[ 1988]), visible double stars (MENG, AITKEN,HEGE and MORGAN[ 1990]), and computer simulations of Neptune (BELETIC [ 19881). In the last five publications interesting comparisons between speckle masking and the Knox-Thompson method and extensions of speckle masking are reported in addition to the experiments. The comparisons illustrate the advantages of speckle masking in the experiments performed.
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TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY
NGC 1OEB. 0111. Speckle-Masking Reconstruction
Fig. 6. Speckle masking observation ofthe Seyfert galaxy NGC 1068. (a)One of 10000 photon-counting speckle interferograms recorded with the 1.5m Danish/ESO telescope. Each black dot is a photon event. (b) Long-exposure image of NGC 1068 calculated by averaging all 10000 re-centered speckle interferograms. (c) High-resolution image of NGC 1068 reconstructed from the same 10000 speckle interferograms by speckle masking. The image shows that NGC 1068 consists of three clouds (separation about 0.5 arcsec). One of the clouds is elongated or double. (From HOFMANN,MAUDERand WEIGELT[1989, 19901.)
WIDEBAND PROJECTION SPECKLE SPECTROSCOPY
309
4 3. Objective Prism Speckle Spectroscopy Imaging spectroscopy plays an important role in astronomy. The methods of objective prism speckle spectroscopy (WEIGELT[ 19811, WEIGELT,BAIER, FLEISCHMANN, HOFMANNand LADEBECK[ 19861) and EBERSBERGER, projection speckle spectroscopy Q 4) can yield both high-resolution images and the spectrum of each resolution element. The raw data for objective prism speckle spectroscopy are objective prism speckle spectrograms is(x), which are obtained by inserting a prism or grating into a pupil plane in the speckle camera, and thus dispersing each speckle to a linear spectrum. The instantaneous intensity distribution i,(x) of an objective prism speckle spectrogram can be described by (3.1) where om(x - x,) denotes the m th resolution element of the object, x is a two-dimensional vector in image space, sm(x) is the spectrum of the mth resolution element, p ( x ) is the instantaneous point spread function of the atmosphere/telescope, and Q is the convolution operator; p ( x ) is wavelength independent in large wavelength bands. From a sequence of speckle spectrograms i s @ ) the desired high-resolution objective prism spectrum
can be reconstructed by speckle masking. Laboratory simulations of objective prism speckle spectroscopy are described in the literature.
6 4. Wideband Projection Speckle Spectroscopy Projection speckle spectroscopy (GRIEGER,FLEISCHMANN and WEIGELT [ 19881) has the additional advantages that (1) it can be applied to general
objects and (2)the whole spectrum from 350 to 850nm can be obtained simultaneously (with S25 cathodes). The principle of projection speckle spectroscopy is summarized in fig. 7. The figure shows from top to bottom: Image 1 : two-dimensional object, a triple star; Image 2 : two-dimensional speckle interferogram of the object ; Image 3 : one-dimensional projection of the two-dimensional speckle inter-
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Projection Speckle Spectroscopy
2-D object
r----l
I
**.
I cylindrical lenses for 1-D projection
speckle lnterferogram
1-D projection of t h e 2-D speckle interferogram
I
I
I--)
projection speckle spectrogram
a object, s p e c t r u m reconstruction
speckle masking
c,
Fig. 7. Principle of projection speckle spectroscopy.
ferogram. The projection can be performed by an anamorphic imaging system of two crossed cylindrical lenses (COOKE [ 19561, LABEYRIE[ 19811, KINGSLAKE [ 19831). The one-dimensional projection of the two-dimensional speckle interferogram is equal to the convolution of the one-dimensional projection of the object intensity distribution with the one-dimensional projection of the atmospheric point-spread function. Image 4 : Spectrally dispersed version d ( x , A) ofthe one-dimensional speckle interferogram. The spectral dispersion can, e.g., be performed by a non-devia-
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311
ting prism. The same projection and dispersion technique was used by LABEYRIE [ 19811 to obtain spectrally dispersed one-dimensional projections of Michelson interferograms. The projection speckle spectrograms d(x, A ) are the raw data for projection speckle spectroscopy. Image 5 : From the speckle spectrograms d(x, L) the objectlspectrum reconstruction o’(x, L) can be obtained by the application of one-dimensional speckle masking to all lines of d(x,A). The object/spectrum reconstruction o‘(x, A) is a dispersed version of a one-dimensional projection of the twodimensional object. High-resolution spatial object information is contained in the horizontal x-direction. The spectral information is found in the y-direction. Three-dimensional data cubes o”(x, y, A) can be obtained if many twodimensional object/spectrum reconstructions are made using different projection and dispersion directions and if tomographic techniques are applied. A laboratory simulation of the projection speckle spectroscopy method is shown in fig. 8. A first astronomical application was reported by GRIEGER and WEIGELT[ 19901).
6 5. Optical Long-Baseline Interferometry and Aperture Synthesis The great advantage of optical long-baseline interferometry with three or more telescopes is that it can yield images and spectra with excellent angular resolution. For example, at L 600 nm and with baselines of 150 m a resolution of 0.001 arcsec can be obtained. Possible image reconstruction methods are the Radio phase closure or nonredundant mask method (JENNISON[ 19581, RHODESand GOODMAN [ 19731, BALDWIN,HANIFF,MACKAYand WARNER [ 19861; READHEAD,NAKAJIMA,PEARSON, NEUGEBAUER,OKE and SARGENT[ 19881, NAKAJIMA, KULKARNI,GORHAM,GHEZ,NEUGEBAUER, OKE, PRINCEand READHEAD[ 19891) and the speckle masking method. The largest planned optical interferometer is that of the European Southern Observatory’s VLT (ENARD [ 19901, BECKERS, ENARD, FAUCHERRE, MERKLE,DI BENEDETTO, BRAUN,FOY,GENZEL,KOECHLINand WEIGELT [ 19901). Its construction started in 1990, and it will consist of four 8 m telescopes and at least two 2 m telescopes. The longest baseline will have a length of about 100-150 m. At visible wavelengths the VLT interferograms will consist of many “fringed speckles”, since there will be many turbulence cells in front of each 8 m telescope (so-called “multi-speckle case”). The size of each speckle will be about 0.01 arcsec, and the width of each fringe will be about 0.00 1 arcsec. Smaller optical long-baseline interferometers have already been
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Fig. 8. Laboratory simulation of the projection speckle spectroscopy method. (a) Laboratory object, a triple star. (b) Spectrally dispersed one-dimensional (1-D) projection o’(x, A) of the object. Each horizontal sequence of dots is a 1-D image of the object. The wavelengths of the monochromatic 1-D images are, from top to bottom, 644 nm, 578 nm, 546 nm, 509 nm, 480 nm, and 467nm. The three emission line spectra of the three stars were produced with a mercury-cadmium lamp and an edge filter in front of the third star. (c) One of the 2000 recorded speckle spectrograms d(x, A). (d) Diffraction-limited objectfspectrum image o’(x, A) reconstructed by 1-D speckle masking. (From GRIEGER, FLEISCHMANN and WEIGELT[ 19881.)
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314 TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY
_--_
-"--"_ _-
..._
- -D. _. ----_-=--- - - - ------~ I-------I--,. \-, - -r- _-I-_.-I. --=- .,_-.. -. "I..,"-Y telescopes. (b) Object, a triple star. (c) One of the 34000 calculated interferograms.(d) Same interferogram after injection of photon noise corresponding to a mean count number of 6 000 counts per interferogram.(e) Diffraction-limited image reconstructedby speckle masking and the image processing and WEIGELT method CLEAN from the 34000 interferograms with photon noise of 6000 photon events per interferogram. (From REINHEIMER I ----I
--I-------
[19901.)
I
I
Y _
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315
built at the CERGA observatory (LABEYRIE [ 19761, LABEYRIE, SCHUMACHER, DUGUE,THOM,BOURLON, FOY,BONNEAU and FOY[ 19861, GAYand MEKARNIA [ 1988]), Steward observatory (BECKERS[ 1988]), Mt. Wilson observatory (SHAO, COLAVITA, HINES, STAELIN, HUTTER, JOHNSTON, MOZURKEWICH, SIMON,HERSHEYand KAPLAN[ 1988]), University of California at Berkeley (DANCHI,BESTERand TOWNES[ 1988]), and University of Sydney (DAVIS[ 19881). A complete computer simulation of optical long-baseline interferometry and aperture synthesis is shown in fig. 9. Figure 9a shows the pupil function of the computer experiment: four 8 m telescopes with a distance of 25 m between them. Earth rotation and 12 hours’ observing time were simulated (aperture synthesis by earth rotation) to obtain a two-dimensional, synthetic optical transfer function. Figure 9c shows one of the 34 000 calculated interferograms. The interferograms consist of “fringed” speckles since many turbulence cells were simulated in front of each telescope. Figure 9d shows the same interferogram after the injection of photon noise corresponding to a mean count number of 6000 photon events per interferogram. Figure 9e is the diffractionlimited image reconstructed from the 34 000 interferograms (with the aforementioned photon noise) by speckle masking and the image processing method CLEAN. This computer simulation and other computer simulations (HOFMANNand WEIGELT [ 1986~1,REINHEIMERand WEIGELT [ 19871, REINHEIMER, FLEISCHMANN, GRIEGER and WEIGELT[ 19881) show that speckle masking can be applied to optical long-baseline interferometers of large telescopes despite the fact that the instantaneous optical transfer function consists of a few isolated patches with large gaps between the patches.
5 6. Concluding Remarks We have discussed the high-resolution imaging methods of speckle masking (bispectrum or triple correlation processing), speckle spectroscopy, and optical long-baseline interferometry with arrays of large telescopes. The astronomical results show that a resolution of 0.03 arcsec (30 milli-arcsec) has been achieved already, which is more than ten times the resolution of the best conventional images made with ground-based telescopes. Speckle masking observations with one of the 8 m VLT telescopes will yield a resolution of about 10 milliarcsec at A 400 nm. Since the VLT mirrors are about three times larger than the mirror of the Hubble Space Telescope, speckle masking observations with single VLT telescopes will yield a resolution three times higher than the Hubble Space Telescope at visible wavelengths. On the other hand, the Hubble Space Telescope can observe much fainter objects with high resolution, and at ultra-
-
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TRIPLE-CORRELATION IMAGING IN OPTICAL ASTRONOMY
[IV,5 5
violet wavelengths (A 140 nm) it can yield a resolution of about 15 milliarcsec if speckle deconvolution methods are applied (LOHMANN and WEIGELT [ 19793, MULLERand WEIGELT[ 19871). A much higher resolution than that of the Hubble Space Telescope can be achieved by optical long-baseline interferometry. An extremely high resolution of 1 milli-arcsec can be obtained with a 100m optical array. Speckle masking seems to be useful in the so-called multi-speckle case, in experiments at visible wavelengths and with interferometers consisting of large telescopes, as, e.g., the VLT of the European Southern Observatory. N
Acknowledgement We thank the European Southern Observatory for observing time. The results shown in figs. 1-6 are based on datacollected at the European Southern Observatory, La Silla, Chile.
Appendix A We will show that the Fourier transform of the triple correlation T(x, y) = i ( x ’ ) i(x’ + x) i(x’ + y) dx’ is equal to the bispectrum B(u, u) = I(u) I(u) I*(u + u): Fourier transform of the triple correlation
1
i(x’)i(x’
+ x ) i ( x ’ + y)dx‘
=
i(x’)i(x‘
=
=
ss1 ss 1s
+ x ) i ( x ’ + y)exp[ - 2 n i ( u * x + u * y ) ] dx’ d x d y
s
i(x’ + y) exp( - 2niu-y) dy dx’
i ( x ’ )i(x‘
+ x) exp[ - 2niu*x]dx
i ( x ’ )i ( x r
+ x) exp [ - 27th ax] dx Z(u) exp (2niu.x’) dx’
(shift theorem) = si(x’){i(x’
=
=
=
+x)exp[-2ni~~x]dxI(u)exp(2niu*x‘)dx’
1
-
i ( x ’ ) I(u) exp (2niu x ’ ) I(u) exp (2niu- x ’ ) dx’ (shift theorem) i ( x ’ ) exp [2nix’ (u
I*(u + u) I(u) I(u) .
+ u)] dx‘ I(u) I(u)
317
REFERENCES
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BECKWITH,S. V. W., A. I. SARGENT, C. D. KORESKO and D. A. WEINTRAUB, 1989, Astrophys. J. 343, 393. BELETIC,J. W., 1988, Comparison of Knox-Thompson and Bispectrum Algorithms for Reconstructing Phase of Complex Extended Objects, in: Proc. Conf. High-resolution Imaging by Interferometry, Garching, 1988, ed. F. Merkle (ESO, Garching) p. 357. CHRISTOU, J. C., J. D. FREEMAN,F. RODDIER,D. W. MCCARTHY,M. L. COBBand S. B. 1987, Application of Bispectrum Analysis to Infrared Speckle Data, in: Proc. Conf. SHAKLAN, Digital Image Recovery and Synthesis, San Diego, 1987, ed. P. S. Idell (SPIE The International Society for Optical Engineering, Bellington) SPIE 828, 32. COOKE,G. H., 1956, J. SOC.Motion Pict. & Telev. Eng. 65, 151. CORNWELL, T. J., 1989, Science 245, 263. DAINTY, J. C., 1984, Stellar Speckle Interferometry, in: Laser Speckle and Related Phenomena, Topics in Applied Physics, Vol. 9, ed. J. C. Dainty (Springer, Berlin) ch. 7. DANCHI,W. C., M. BESTERand C. H. TOWNES,1988, The U.C. Berkeley Infrared Heterodyne Interferometer, in: Proc. Conf. High-resolution Imaging by Interferometry, Garching, 1988, ed. F. Merkle (ESO, Garching) p. 867. DAVIS, J., 1988, The Sydney University Stellar Interferometer, in: Proc. Conf. High-resolution Imaging by Interferometry, Garching, 1988, ed. F. Merkle (ESO, Garching) p, 817. ENARD,D., 1990, ESO VLT Project: I. A Status Report, in: Proc. Conf. Advanced Technology Optical Telescopes IV, Tucson, 1990, ed. L. D. Barr (SPIE - The International Society for Optical Engineering, Bellington) in press. J. D., J. C. CHRISTOU, D. W. MCCARTHY and M. L. COBB,1987, A Comparison of FREEMAN, Phase Retrieval Algorithms Applied to Infrared Astronomical Speckle Data, in: Proc. Conf. Digital Image Recovery and Synthesis, San Diego, 1987, ed. P. S. Idell (SPIE - The International Society for Optical Engineering, Bellington) SPIE 828, 40. FREEMAN, J. D., J. C. CHRISTOU, F. RODDIER, D. W. MCCARTHY and M. L. COBB,1988, J. Opt. SOC.Am. A 5, 406. GAY,J., and D. MEKARNIA, 1988, Infrared Interferometry at CERGA, in: Proc. Conf. Highresolution Imaging by Interferometry, Garching, 1988, ed. F. Merkle (ESO, Garching) p. 81 1. GIANNAKIS, G. B., 1989, J. Opt. SOC.Am. A 6, 682. GORHAM, P. W., A. M. GHEZ,S. R. KULKARNI, T. NAKAJIMA, G. NEUGEBAUER, J. B. OKEand T. A. PRINCE,1990, Astron. J. 98, 1783. GRIEGER,F., and G. WEIGELT,1990, Projection Speckle Spectroscopy and Objective Prism Speckle Spectroscopy, in: Proc. Conf. 15th Congress of the International Commission for Optics, eds F. L a d , H.J. Preuss and G. Weigelt, Garmisch-Partenkirchen, 1990, SPIE 1319, 440.
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REINHEIMER, T., and G. WEIGELT,1987, Astron. & Astrophys. 176, L17. REINHEIMER, T., and G. WEIGELT,1990, Optical Long-Baseline Interferometry in Astronomy, in: Proc. Conf. 15th Congress of the International Commission for Optics, eds. F. Land, H.-J. Preuss and G. Weigelt, Garmisch-Partenkirchen, 1990, SPIE 1319, 678. REINHEIMER, T., F. FLEISCHMANN, F. GRIEGER and G. WEIGELT,1988, in: Proc. Conf. Highresolution Imaging by Interferometry, Garching, 1988, ed. F. Merkle (ESO, Garching) p. 581. RHODES,W. T., and J. W. GOODMAN, 1973, J. Opt. SOC.Am. 63, 647. RODDIER, F., 1988, Phys. Rep. 170, 97. RODDIER, F., and C. RODDIER,1986, Optics Commun. 66, 350. SCHERTL,D., F. FLEISCHMANN, K.-H. HOFMANN and G. WEIGELT,1987, High-resolution Astronomical Imaging by Photon-counting Speckle Masking, in: Inverse Problems in Optics, The Hague 1987 (Society Photo-Optical Instruments Engineering, Bellingham) SPIE 808,38. SHAO,M., M. M. COLAVITA, B. E. HINES,D. H. STAELIN, D. J. HUTTER,K. J. JOHNSTON, D. J. L. HERSHEY and G. H. KAPLAN, 1988, Astron. & Astrophys. MOZURKEWICH, R. S. SIMON, 193, 357. VON DER LOHE,o., 1985, Astron. & Astrophys. 150, 229. WEIGELT, G., 1977, Opt. Commun. 21, 55. WEIGELT,G., 1981. Speckle Interferometry, Speckle Holography, Speckle Spectroscopy and Reconstruction of High-Resolution Images from HST Data, in: Proc. Conf. Scientific Importance of High-Angular Resolution at Infrared and Optical Wavelengths, Garching, 1981, eds. M. H. Ulrich and K. Kjar (ESO, Garching) p. 95. WEIGELT, G., and B. WIRNITZER, 1983, Opt. Lett. 8, 389. WEIGELT,G., G. BAIER,J. EBERSBERGER, F. FLEISCHMANN, K.-H. HOFMANNand R. LADEBECK, 1986, Opt. Eng. 25, 706. WIRNITZER, B., 1985, J. Opt. SOC.Am. A 2, 14.
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E. WOLF, PROGRESS IN OPTICS XXIX 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1991
V
NONLINEAR OPTICS IN COMPOSITE MATERIALS 1. Semiconductor and Metal Crystallites in Dielectrics
BY
C. FLYTZANIS, F. HACHE, M. C. KLEIN, D. RICARD and PH. ROUSSIGNOL Laboratoire d'Optique Quantique du C.N.R.S. Ecoie Polytechnique 91 128 Palaireau CEDEX, France
CONTENTS PAGE
INTRODUCTION
. . . . . . . . . . . . . . .
. . .
,
323
FABRICATION AND CHARACTERIZATION TECHNIQ U E S . . . . . . . . . . . . . . . . . . . . . . . . . 325 CONFINEMENT EFFECTS . . . . . . . . . . .
. . . .
338
NONLINEAR OPTICAL PROPERTIES OF METAL COMPOSITES . . . . . . . . . . . . . . . . . . . . . . . 368 NONLINEAR OPTICAL PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY . . . . . . . . , , . . . 375 NONLINEAR OPTICAL PROPERTIES OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES . . . . 383
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONCLUSIONS AND EXTENSIONS REFERENCES
404 406
6 1. Introduction In recent years, the nonlinear optical properties of quantum-confined semiconductor and, to a lesser degree, metal crystallites have received considerable attention, and their study has become a major research activity, reactivating interest in their linear optical properties. This interest is motivated and justified on both fundamental and technological grounds. The technological aspect is related (see, e.g., FLYTZANIS and OUDAR[ 19861 to the need for materials that, in the near future, will allow implementation of several nonlinear optical effects in devices with performances that either will surpass those of electronic devices or open new possibilities in information processing and transmission that are inaccessible or inconceivable with present day electronics technology. The homogeneous organics and inorganics have not yet produced the expected breakthrough, and current attention is directed towards artificial heterogeneous semiconductor-dielectric or metaldielectric microstructures where quantum confinement plays a central role. The fundamental aspect is related to the long-standing question of the validity range of crystal solid-state concepts (and behavior) and their relation to molecular and atomic properties. This issue is essential in the understanding and prediction of properties in crystals with extended electronic states, like metals, semiconductors, or conjugated polymers, where the electronic density distribution of the constituent elements undergoes drastic modifications in the course of forming the crystal. In one dimension such confinement effects have been extensively studied in quantum wells, because their fabrication techniques (see e.g., KELLY and WEISBUCH[1986]) have reached a high degree of precision and sophistication. The study of the confinement effects in three dimensions, on the other hand, which can be more readily contrasted with the behavior of the (usually optically isotropic) bulk metal or semiconductor crystal, did not keep up the same pace because of the lack of fabrication techniques that would allow one to obtain microcrystals and nanocrystals with a narrow size distribution and a well-defined interface with the transparent dielectric in which they are embedded. Our present understanding of the global confinement effects only relies on investigations in composite materials, i.e., semiconductor or metal uniformly 323
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NONLINEAR OPTICS IN COMPOSITE MATERIALS
IV,S 1
doped with glasses or their colloids, where the crystallites grow by a thermal diffusion controlled process and, consequently, show a wide size distribution. Several of the experimental investigations initially concerned commercial semiconductor- or metal-doped glasses, since they are available for other industrial purposes, or cut-off optical filters, or artistic stained glass, in which quantumconfinement effects are severely masked or nonexistent. Recently, however, better controlled laboratory-made samples have been used that show distinct quantum-confinement features and give a meaningful comparison with the theoretical models being developed. This review will assess the different experimental and theoretical results relating to nonlinear optical properties of these materials and will place them in a context that will lead to a unifying and realistic description of the electron states, their dynamics, and their response to intense optical fields. Although the emphasis is on nonlinear optical properties, nonlinear optical techniques, in particular the time-resolved ones, are of more general interest, since they provide additional and extremely important information when compared with linear techniques, and their use allows one to examine previously inaccessible questions in these materials. Furthermore, they provide direct information concerning the nonlinear optical effects that may find applications with these materials in the future. We believe that this review, although premature in several respects, may also help to assess the potential use of these materials for applications in optoelectronics. The material is organized in several sections. Section 2 briefly describes the preparation techniques of these materials, which is followed by a discussion of the linear and nonlinear optical characterization techniques at our disposal. In 0 3 we outline electronic motion and its coupling to other degrees of freedom. Section 4 is devoted to the nonlinear optical properties of metal-doped glasses and colloids, and 0 5 and § 6 describe those of the semiconductor-doped glasses and colloids. The final section contains a general assessment of the nonlinear composite materials and possible extensions. Although metals and semiconductors have distinct features in the bulk and this apparently persists, but for different reasons, in quantum-confined crystallites, in several places and in particular 0 2 and 0 3, we have adopted a common description for the electronic motion. In the discussion of their nonlinear optical properties, however, the separation was necessary because these bear the signature of two distinct confinement mechanisms: the dielectric one in the case of metal particles and the quantum mechanical one in the case of semiconductor particles.
FABRICATION AND CHARACTERIZATION TECHNIQUES
325
6 2. Fabrication and Characterization Techniques 2.1. FABRICATION TECHNIQUES
2.1.1. Metal crystallites Metal-doped glasses have a very long history; they have been used for [ 18571 was centuries to fabricate the stained glasses for cathedrals. FARADAY one of the first to relate their color to the metal particles, and this culminated in MIE’Stheory [ 19081of the optical properties of these particles. The preparation, however, of well-defined metal crystallites in controlled and reproducible conditions dates back only a few decades and received a strong impetus with the theoretical work of KUBO[ 19621 and GORKOV and ELIASHBERG [ 19651 on quantum-confinement effects. This triggered much work for the fabrication of metal crystallites along with intense theoretical and experimental investigations. (For a detailed and up-to-date discussion of metal particles in noncrystalline media, see PERENBOOM, WYDERand MEIER[ 19811, and in crystalline solids see HUGHESand JAIN[ 19791; see also HALPERIN [ 19861.) Here we give a short account of the preparation techniques of metal crystallites. These include colloidal suspensions of metals, particles in glasses or other solid matrices, crystallites inserted in porous materials, and free clusters in gases or beams. In all cases the purpose is to produce particles with variable average size but with a very narrow size distribution and well characterized in shape and chemical constitution, since these features strongly influence the electronic motion. The procedure for making colloidal suspensions of noble-metal particles consists of four main steps: (1) acid solution of the appropriate metal element, (2) reduction with an agent (usually sodium citrate) to produce the suspension of particles, (3)dialysis to remove remaining ions, and (4)coating with a protective agent to hinder aggregation or other deterioration. Colloidal suspensions of noble metals or hydrosols can be easily prepared with the TURKEVICH, STEVENSON and HILLIER[ 19551 technique. For example, in the case of gold, one adds 15 ml of a 1% sodium citrate solution to a well-stirred, 300-ml boiling tetrachlorauric acid (HAuCl,) solution obtained by dissolving 35 mg of salt in doubly distilled de-ionized water. After a minute or so the mixture turns dark blue-grey and in a few minutes fades to a deep wine-red color; the reaction continues for half an hour, and the solution is then allowed to cool down to room temperature. A hydrosol can be stabilized by adding a protective agent, e.g., gelatin. The average particle size is d w 14.7 f 1.3 nm,
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NONLINEAR OPTICS IN COMPOSITE MATERIALS
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and the spread in size is related to the spread in time for the formation of nuclei, whereas the average size is strongly affected by the concentration. For even larger particles the seeding technique of ZSIGMONDY119061 can be used, whereas for smaller ones the solution is reduced with phosphorus. A similar procedure can be used for the other noble metals, silver and platinum in particular (WILENZICK,RUSSELL, MORRIS and MARSCHALL[ 19671, MARZKE and GLAUNSINGER [ 19831);mercury hydrosols can also be prepared (FEICK[ 1925]), but their size cannot be kept constant because of coagulation. Although this is of no direct concern to us here, we wish to point out that extended studies have been made on granular metal films, usually obtained by [ 19741). vacuum deposition on cleavage planes of ionic crystals (SCHMEISSER The preparation of crystallites in glasses (STOOKEY[ 19491, MAURER [ 19581, MErER and WYDER[ 19731) or other matrices usually proceeds as follows: small amounts of gold (0.1 to 1% by weight in the form of HAuCI, nH,O), are added to the components of a glass consisting of 71.5% SiO,, 23% Na,O, 4% Al,O,, 1% ZnO, 0.13% CeO,, and 0.3% Sb,O,. This mixture is heated to 1400°C for 8 h, and after cooling down, a colorless and transparent glass is obtained. If this glass is irradiated with ultraviolet light or y-rays, the photosensitive agent CeO, reduces some of the gold ions, which become nucleation sites. When the sample is heated to the softening temperature of the glass (= 530"C), gold diffuses through the glass lattice to these nuclei and clusters are formed, leading to the coloration of the sample to ruby red; the uniform distribution and the density of the nuclei, and hence of the clusters, can be controlled with the ultraviolet or y-radiation dose. As new nuclei are formed spontaneously during the annealing process, the size distribution cannot be kept narrow. The same technique can be used for silver and platinum (DUPREEand SMITHARD [ 1972]), as well as other metal particles. Along with these techniques, other techniques (BOREL,BOREL-NARBEL and MONOT[1974], CALLEJAand AGULLO-LOPEZ [1974], HUGHESand JAIN [ 19791, PERENBOOM, WYDERand MEIER [ 19811, HALPERIN[ 19861) have also been developed that allow one to form metal clusters in crystals, in particular alkali halide crystals. Finally, simple impregnation techniques have been devised to prepare small metal particles in gels and porous glasses (LINDQUIST, CONSTABARIS, KUNDIGand PORTIS[ 19681, WATSON[ 1966, 19701). The latter typically contain 96% SiO,, 3% B,O,, and small amounts of Na,O, A1,0,, and other oxides. By removing the boron-rich phase, a porous body is left that is immersed in a bath of molten metal. The metal is thus forced into the pores and because of the high uniformity of the pore diameter, particles with a very narrow size distribution are obtained. However, there are indica-
v>8 21
FABRICATION A N D CHARACTERIZATION TECHNIQUES
327
tions from X-ray studies that in porous glasses these particles are not completely independent of each other. Finally, we indicate that there are techniques all based on the method devised by PFUND[ 19331 that allow one to obtain finely divided metal particles by evaporation into a background gas (KIMOTO,KAMIYA,NONOYAMA and UYEDA[ 19631, GEN, ZISKINand PETROV[ 19721) or in a beam geometry (HAGEMAand OBERT[ 19721, LARSON,NEOHand HERSCHBACH [ 19741).
2.1.2. Semiconductor crystallites Semiconductor crystallites can also be produced in different solid dielectric matrices (glass, ionic crystals, gel, polymer) or in liquids (semiconductor colloids). The preparation of semiconductor colloids (LUCAS[ 18861, EWAN [ 19091, BERRY[ 19671, PAPAVASSILIOU [ 19811, 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) and doped dielectrics (REMITZ,NEUROTH and SPEIT [ 19891, BUMFORD [ 19771, EKIMOV,ONUSHCHENKOand TSEKHOMSKII [ 19801, ITOH and KURHAVA[ 19841, ITOH, IWABUCHI and KATOKA [ 19881, WANG and HERRON [ 1987a,b], WANG, SUNA, MAHLER and KASOWSKII[ 19871, PARISE,MACDOUGALL, HERRON,FARLEE,SLEIGHT, WANG,BEIN,MOLLER and MORONEY [ 19881) also has a long history, although much less so than that of metal ones. The last two decades have witnessed a major improvement over the existing essentially diffusion-controlled techniques and the appearance of new ones that may allow the preparation of semiconductor crystallites in a relatively well-defined environment and with reproducible characteristics. The average size of crystallites grown by the diffusion-controlled techniques can cover a wide range of values, from several micrometers down to a few angstroms, but the size distribution and quality of the interface with the supporting dielectric differ substantially from case to case. As will be discussed later, not all semiconductor compounds can be grown by this technique inside a dielectric medium; growth depends on several factors related to the constituent chemical elements and the supporting dielectric. At present, only crystallites of II-VI and, to a much lesser degree, I-VII semiconductors have been grown (EKIMOV, ONUSHCHENKO and TSEKHOMSKII [ 19801) in glasses; attempts to do so with III-V or elemental JV-IV semiconductors have failed. The situation may change, however, with the introduction of molecular epitaxy techniques (WORLOCK [ 19891, ZINKE-ALLMANG, FELDMANand NAKAHARA [ 19881).
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In the solid glass matrices, where most efforts until now have been reported (REMITZ, NEUROTHand SPEIT [ 19891; EKIMOV,ONUSHCHENKOand TSEKHOMSKII [ 1980]), the semiconductor crystallites are produced by a moreor-less controlled thermal diffusion process. The so-called batch material is formed by heating up to 1400°C a silicate glass (20% ZnO, 20% K20, 50% SiO,) together with chemicals containing the elements required to form the semiconductor, for instance Cd, S, and Se for the most commonly studied CdS, Se, - ,ternary semiconductors or Cu and C1 for CuC1. During the heating process, the bonds that bind these elements in the chemicals are broken, and the elements diffuse in atomic or ionic form inside the soft glass matrix. In addition to these primary chemicals, one also dissolves other elements to improve other properties needed for the current applications of these composite materials. Thus, in the case of the CdS, Se, - ,crystallites, one dissolves Te in order to improve the mechanical compatibility of the glass and the semiconductor. During the heating-up and melting process, the semiconductor constituent elements are uniformly distributed inside the liquid glass through a thermal diffusion process, which must counterbalance the evaporation process that is also speeded up as the batch material is heated to the melting point. In addition, the mobility of the semiconductor constituent elements should not be hindered by chemical binding or accommodation in sites other than the ones compatible with the semiconductor crystallite formation. It seems that these and other conditions in silicate glasses, and presumably also in borate glasses, are most easily satisfied for 11-VI and, to a lesser degree, for the I-VII compounds but not for the 111-V or IV-IV ones, whose constituent elements are highly volatile or possess extremely low or nonexistent mobilities in solid or liquid matrices. Subsequently, the material is cooled and heated again to less than 500°C to relax mechanical constraints. Up to this stage the material is colorless like the initial undoped glass, but the nucleation centers for the subsequent crystallite growth process have been formed. These nucleation centers are clusters with few atoms, of the order of lo2, which are chemically compatible precursors to the semiconductor crystallites but do not yet completely possess the solid-state characteristics of the latter, and in particular, their surface energy is dominant over, or close to, their volume energy; in these clusters the number of “surface” atoms is larger than that of “volume” atoms. After the nucleation stage the crystallites are formed through the striking process (REMITZ,NEUROTHand SPEIT[ 19891, BUMFORD[ 1977]), and the material also acquires the color. The striking process consists of heating the nucleated glass in the range of 500 to 750°C for a limited time, extending from
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a few minutes to several hours. During this striking stage the clusters grow further, and true crystallites are formed to an average size and size distribution that depend on the striking temperature and duration of heating. It is believed that, at this stage, a coalescence process goes on where the larger nuclei increase further at the expense of the smaller ones and reach sizes where volume properties become dominant over surface properties and the solid-state-like features are acquired; the color is also fixed at this stage, as will be discussed in 3 3. The growth of these crystallites in a solid saturated solution has been assumed (EKIMOV,ONUSHCHENKO and TSEKHOMSKII [ 19801) to follow the process theoretically described by LIFSHITZand SLEZOV[ 19591; see also LIFSHITZand PITAEVSKII [ 198 1 1. There one assumes spherical particles throughout the growth process, and their average radius is given by -
a
=
($oDcz)'/3,
(2.1)
where o is the surface tension in the interface, D is the compound diffusion constant, c is a constant that exponentially depends on the striking temperature T,, and z is the duration of the striking process. This law is derived from a model for the diffuse decomposition of a supersaturated solid solution for growth at a constant temperature. This model also predicts that the size distribution is given by
where u = a / Z ; it is asymmetrical with a faster fall-off for a > 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
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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
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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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
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[ 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
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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
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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.
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NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
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[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
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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.
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NONLINEAR OPTICS IN COMPOSITE MATERIALS
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)
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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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
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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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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)
<
370
470
570
670
A (nm)
,088 0 370
470
570
67OA (nm)
Fig. 7. Absorption spectra for six gold-doped glass samples showing the broadening of the surface plasmon resonance when the radius (indicated in nanometers in the parentheses) is reduced.
v, 8 31
CONFINEMENT EFFECTS
359
3.3.3.2. Semiconductor crystallites In quantum-confined semiconductor crystallites the situation concerning the broadening of the optical transitions is more complex. For the ideally spherical and isotropic semiconductor crystallites we assume here, we may single out as as most important, two broadening mechanisms (SCHMITT-RINK, MILLER and CHEMLA[ 19871) one from each type: an intrinsic one, the electron-phonon coupling (HUANGand RHYS[1950], DUKE and MAHAN[1965], MERLIN, G~NTHERODT,HUMPHREYS, CARDONA, SURYANARAYANAN and HOLTZBERC [ 1978]), which introduces a homogeneous broadening, and an extrinsic one, the size distribution of the crystallites in the dielectric (EFROSand EFROS[ 1982]), which introduces an inhomogeneous broadening. The impact of the latter is now more conspicuous than with the metal particles because of the minimum gap E, and the much lighter electron masses, which shift the quantum-confinement effects in the optical spectrum range. Before we discuss these two mechanisms, we wish to point out that in real semiconductor crystallites, there are several other perturbations that randomly modify the electron structure and lead to additional and substantial broadening, including the following: - Deviations from sphericity, which lead to different confinement lengths in different directions for a given crystallite. - Lack of inversion symmetry, which leads to a breakdown of the selection rules of eq. (3.49) and mixing of transitions. - Overlap of spin-orbit split-off states. - Random orientation of the crystallites which, in the case of the polar semiconductors, introduces a distribution of E , in eq. (3.46). - Stoichiometry fluctuations, which introduce potential fluctuations. - Level shifts caused by random fields and impurities in the interface. - Auger processes. Their impact is still not fully appreciated and is not easy to evaluate. Here, we shall limit our discussion to the electron-phonon and size distribution broadening mechanisms. Homogeneous broadening. The absorption spectrum of eq. (3.52) of a single spherical semiconductor crystallite is intrinsically broadened by the latticeinduced modification of the electronic structure in the crystallite; this can be introduced through an electron-phonon coupling term Veepin the Hamiltonian (HUANGand RHYS [1950] , DUKEand MAHAN[1965], NEUMARK and KOSAI[ 1983]), which is a function of the nuclear displacement Q; for most purposes it suffices to assume a single phonon branch and keep the linear term
360
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V,§ 3
in the phonon amplitude Q in the Taylor development of Vep. The physical origins to this coupling include the Frtrhlich coupling of electrons to the polar optic phonons, the deformation potential, and the piezoelectric coupling (DUKEand MAHAN[ 19651). The last two mechanisms involve acoustic phonons as well. In the case of bulk crystals, electron-phonon coupling is well documented, particularly in connection with the impurity levels in semiconductors. It has been argued that in polar semiconductors, the Frtrhlich mechanism is the dominant mechanism. Because of the similarity in the approaches to find the eigenstates of an impurity and those in a crystallite, which are both based on the effective-mass approximation, we can presume that the same mechanism is dominant in the latter case as well, the relevant modes now being the internal longitudinal optic (LO) modes and the surface optic (SO) modes (MORIand ANDO[ 19891, KLEIN, [ 19901.) HACHE,RICARDand FLYTZANIS Concentrating our attention on the lowest 1s-1 s transition (KLEIN,HACHE, RICARDand FLYTANIS[ 1990]), following HUANGand RHYS [ 19501 and DUKEand MAHAN[ 19651, the problem is treated by Fourier expanding the electron-phonon Hamiltonian term Hd,
$(4 p(r) d r ,
= sphere
where $(r) is the electric potential due to the polar optic phonon modes and p(r) is the electronic charge density distribution, the sum of the electron and hole distribution each in the 1s quantum-confined state, i.e., p(r> = 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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[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
-
<
0
PROPERTIES OF METAL COMPOSITES
D e w (psec)
Backward Pump Delay ( psec)
313
Fig. 13. Time response of the nonlinearityin a gold colloid. At low fluence (a) it is very fast; at higher fluence (b) a weaker and slower thermal component is visible.
4
w
314
NONLINEAR OPTICS IN COMPOSITE MATERIALS
t
I
D
550
500
Wavelength
600
( nm )
Fig. 14. The surface plasmon resonance enhancement. ( 0 ) experimental results, (-) calculated curve of If, I '.
be roughly size independent when the diameter 2a was varied from 28 to 300 A and equal to about 5 x 10- * esu; this indicates that the intraband contribution is not important in eq. (4.4). The determination of the phase of for gold ruby glass with respect to that of C S , was made using an interferometric technique and yielded the value I cpI z 80°, implying that is essentially imaginary. This finding, together with the saturation behavior of x(') observed at high laser intensities, indicated that Im f 3 ) is positive and thus establishes the hotelectron contribution as the dominant one in the xxxx-component, which is the only component in the nonlinear polarization when all three input beams are co-polarized. With half-wave plates one can cross the polarization of any of the three input beams and access x$,Ly, x!:$y, or xi;$,.From the absolute values of these components with respect to x!3;?, and the sum rule (3)
xxxxx
(3) - xxyyx
(3)
+ xxyxy +
(3)
xxxyy
9
(4.91
it was inferred that the sign of xi;;y is opposite that of xL?ix and x!3,?,. These results should now be confronted with characteristic trends of eqs. (4.5), (4.7), and (4.8). We recall that the intraband term is size dependent,
v, I 51
PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY
315
whereas the other two are not; furthermore, in the interband contribution, whose imaginary part is negative, all components are expected to be non-zero, whereas for the hot-electron contribution, whose imaginary part is positive, the xyxy-component vanishes. We thus reach the conclusion that, in the co-polarized beam configuration the hot-electron contribution dominates, whereas in the cross-polarized configuration only the interband contribution is present. The preceding conclusions can also be substantiated quantitatively (HACHE [ 19881, HACHE, RICARD, FLYTZANIS and KREIBIG[ 19881) by inserting plausible values for the unknown physical quantities in eqs. (4.9, (4.7), and (4.8). Thus, for the intraband term (4.5) we find ImxCjr,
= -
10-
lo
esu ,
and a, = 136 A with E , = 5.5 eV for gold, T2 = 2 x s, T , NN s, o = 3.55 x loL5s - I , and a = 50 A. For the interband term (4.7) we find Imxf&
NN -
1.7 x
esu,
withA, = Ti = 2 x 10-l4 s, Ti = 2 x s,and ( P I 2= 3.4 x Finally, for the hot-electron contribution (4.8) we find
esu.
Imxht) = 1.1 x io-’esu, with ELz, x 5.1 x 10- l 8 s and y = 66 J m - 3 K-*. These results only pertain to the gold composites, but there is no a priori reason to expect significant differences in the other noble-metal composites; their study currently is hampered by having no short-pulse tunable sources in the frequency domain where their surface plasrnon resonances appear.
4 5. Nonlinear Optical Properties of Semiconductor Composites: Theory In contrast to the case of the metal composites (0 4), the quantum confinement has a profound influence on the optical properties of composites consisting of a suspension of small semiconductor crystallites in a transparent dielectric. In particular, the transitions between quantum-confined levels as discussed in 0 3, notwithstanding their substantial broadening, dominate the optical absorption spectrum of semiconductor composites. On the other hand, even close to these quantum-confined resonances, the dielectric confinement, so important in metal composites, plays a less important role in semiconductor particles, and we may disregard it momentarily. Indeed the real part of the semiconductor dielectric constant usually shows too weak a frequency depen-
316
NONLINEAR OPTICS IN COMPOSITE MATERIALS
tv9g
5
dence to satisfy E ' ( w )t 20, = 0 in eq. (3.8), the local-field factor f,remains constant and close to unity; without other consequences it can be incorporated by renormalizing the oscillator strengths if not otherwise stated. The nonlinear optical properties of the semiconductor composites and particularly those with glass as the host transparent dielectric have been extensively studied since 1983, since they show, among all composites, the most promising potential use in nonlinear optical devices in addition to their current multiple uses as efficient optical filters. We are only interested in their optical properties close to or above the onset of optical absorption in these crystallites; below this onset the optical properties are primarily those of the host dielectric, since the volume concentration of the semiconductor crystallites in all the investigated materials is very small, 0.1% or less. The essential features of the optical absorption spectrum of the semiconductor composites were introduced and discussed in 5 3. Here, we expand that description to infer the behavior of the linear and nonlinear optical susceptibilities f l ) ( w ) and ~(~'(0, - w, a), respectively, and compare it with the measured ones. We shall begin with theoretical considerations concerning the optical nonlinearities and, in particular, that of the optical Kerr effect in quantum-confined crystallites. As was hinted in connection with the previous discussion of the broadening mechanisms of the resonances in these crystallites, the optical Kerr effect may result from three distinct mechanisms. (1) Saturation nonlinearity (SCHMITT-RINK,MILLER and CHEMLA [ 19871): If the transition between the hole and electron states, say the 1 s-1 s transition, is isolated enough, each particle behaves as a two-level system that may be bleached and contributes to the degenerate Kerr susceptibility. (2) Coulomb interaction-mediated nonlinearity (TAKAGAHARA[ 19871, HANAMURA [1988], BANYAI,Hu, LINDBERGand KOCH [1988]): if two electron-hole pairs are excited per crystallite, transitions between one-pair and two-pair states lead to induced absorption. This latter transition is shifted in frequency compared with the previously discussed wls due to the Coulomb interaction, and, therefore, we expect this second mechanism to contribute only weakly to the degenerate Kerr susceptibility. (3) Impurity-mediated nonlinearity (HENGLEIN,KUMAR,JANATAand JEDJU,WILSON,BAWENDI, STEIGERWALD and WELLER[ 19861, ROTHBERG, BRUS[ 19901): The photocreated carriers are assumed to be rapidly trapped at the surface of the particle, and the static electric field they create modifies the absorption spectrum as in the case of the confined Franz-Keldysh effect (MILLER,CHEMLAand SCHMITT-RINK [ 1986, 19881, HACHE,RICARDand FLYTZANIS [ 19891). It is difficult at the present stage of knowledge to make
PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY
311
a quantitative estimate for such a mechanism, and we limit ourselves here only to the first two mechanisms. If we restrict ourselves close to the 1s-1s transition, where the impact of quantum confinement is most visible, in a strongly confined crystallite one can use the analytical treatment of a two-level system in an intense optical field of frequency o to calculate the susceptibility at this frequency for an arbitrary field intensity as
1 + (wlS - w)’T,Z
1
olS- o - i/T2 1 + (ols- o ) ~ T $+ I/I, ’
(5.1)
with
where T I and T2 are the energy lifetime and dephasing time introduced in 5 3 and V is the crystallite volume. At low intensity a Taylor development of eq. (5.1) around I = 0 gives
1
(5.5)
This expression can also be obtained using the conventional time-dependent perturbation approach. A similar behavior is expected around each quantumconfined resonance and, because of the selection rules of eq. (3.49), the I ),or xC3) can be rigorously expressed as a sum of contribususceptibilities ~ ( o tions (5.1) or (5.4), respectively, each multiplied with the appropriate degeneracy factor gnl.Actually, for large nl the spacing between levels decreases and the spectrum reverts to a continuum with a density of states close to that of the bulk semiconductor. In the intermediate quantum-confinement regime the situation may become complicated by Coulomb interaction if more than one electron-hole pair is
378
NONLINEAR OPTlCS IN COMPOSITE MATERIALS
[V.§ 5
photoexcited; this interaction, which is the germ of the bi-exciton state in certain bulk semiconductors like CuCl, is small with respect to the energy spacing in the strong quantum-confinement regime and was neglected in deriving eqs. (5.1) and (5.4). Its impact in the intermediate confinement regime cannot be assessed analytically, but using certain simplifications (BRUS [ 1984, 19861, BAWENDI,STEIGERWALD and BRUS [ 19901) concerning the Coulomb interaction between electron-hole pairs, BANYAI,Hu, LINDBERGand KOCH [ 19881 by extending the approach of TAKAGAHARA [ 19871, derived the expression of ~ ( ~ ' ( -wo, , w) for w close to the 1 s electronic transition in the intermediate quantum-confinement regime in the form
4A2 1 1 + i ( q S - w)z 1 + (als - w)2z2 2
1 1 + i ( 0 2 - ols - o)z 1 + (wls - o)2z2
+
1
[ 1 + i ( q S - w)z] [ 1 + i(w2 - 2w)zI
1 1
+ i(02 - w l s - o)z
-
1 + i ( q S- o ) ~
whereas ~ ( " ( ois)the same as in eq. ( 5 . 5 ) ; w l s and o, are the transition frequencies for one and two electron-hole pairs, respectively, A and B are the corresponding dipole moment matrix elements, A
=
I(0lPI 1 s ) l 2 ,
B
=
I ( 0 IPI 1 s ) ( 1 s IPI 2 )
(5.7)
I,
(5.8)
and z is a phenomenological relaxation time irrespective of the nature of the damping processes. The latter is a very drastic simplification, which actually unduly exaggerates the impact of the Coulomb interaction, which, as can be inferred from eq. (5.6), introduces an asymmetry above and below the w l s resonance (fig. 15). Furthermore, the parameters B and o2cannot be easily evaluated or extracted from an experiment where the size distribution in actual samples averages out most of these effects. For B = 0 or 0,= 2wl,, eq. (5.6) reduces to eq. (5.4)after redefinition of certain parameters. In the weak-confinement regime, where the electron-hole interaction cannot be neglected with respect to the confinement energy (EFROS and EFROS
v, I 51
PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY
319
Fig. 15. Real and imaginary parts of I(’)in the strong confinement (b) and intermediate confinement (a) regimes showing the influence of Coulomb effects. E , is the Rydberg energy of the and KOCH [1988].) exciton. (From BANYAI,Hu, LINDBERG
[ 1982]), one may have exciton states like those in the bulk semiconductor that will contribute to the dielectric constant together with the band-to-band transitions with due attention to the confinement effects, which are essentially slight shifts of the transitions with respect to those of the bulk. Since the actual crystallite shape becomes less relevant with increasing crystallite size, one may include these effects by considering a cube instead of a sphere, which leads to much simpler expressions analytically. HANAMURA [ 19881derived the expression of ~ ( for~ a large 1 crystallite using this simplification and also including the Coulomb interaction between electron-hole pairs as previously discussed; he also derived the expression for a spherical crystallite, as did BANYAI,Hu, LINDBERGand KOCH [ 19881. Their expression of x ( ~ is ) similar in form to eq. (5.6), with appropriate redefinition of the physical parameters, since now ol stands for the exciton transition frequency and o,for that of the bi-exciton. HANAMURA [ 19881 also discussed the case when the crystallites are in a superradiant state and predicted large nonlinearities for crystallites. However, there seems to be some controversy in the interpretation of the experimental results with this model (MASUMOTO,YAMAZAKIand SUGAWARA[ 19881, NAKAMURA, TOKIZAKI, KATAOKA,SUGIMOTO and MANABE[ 19901). More recently, careful analysis of these expressions unambiguously showed that such giant nonlinearities cannot occur (ISHIHARAand CHO [1990], SPANOand MUKAMEL [ 19891). For very large crystallites, where the confinement effects are minor, one may assume bulklike behavior (ROUSSIGNOL, RICARDand FLYTZANIS [ 19871). Restricting ourselves for simplicity to a two-band model (MILLER,SEATON,
380
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V,8 5
PRISEand SMITH [ 19811, WHERRETT and HIGGINS[ 19821, DE ROUGEMONT and FREY[ 19881) with parabolic shapes and assuming only direct transitions that conserve the wave vector k, one obtains
where we take into account the spin degeneracy in counting the density of states and k wcv(k)= Eg +, h2k2/2p . (5.10) The intensity dependence in eq. (5.9) is implicit in the Fermi-Dirac occupation and FREY[ 19881) probabilities (DE ROUGEMONT (5.11) and (5.12) for electrons and holes, respectively, through the Fermi quasi-levels $v and $c, which are related to the population of the photoexcited carriers and hence depend on the light intensity I. Their position is fixed by the condition of charge neutrality JOm
pc(k)k 2 d k
=
[ 1 - pv(k)] k2 d k
=
n2 N ,
(5.13)
where (5.14) with a(w) = 4 n w Im ~ ( wZ)/nc, , the absorption coefficient; in eqs. (5.11) and (5.12), p = l/k, T. The preceding assumptions and expressions pertain to the so-called bandfilling model (MILLER,SEATON,PRISEand SMITH[ 1981]), which reflects the fact that, as electrons are excited in the conduction band and quickly relax to the bottom of this band, the states there are excluded from further occupation during a time interval which is of the order of the electron-hole recombination or impurity trapping times. Thus, the transition spectrum during this time lapse
v, 5 51
381
PROPERTIES OF SEMICONDUCTOR COMPOSITES: THEORY
is blue-shifted. Coulomb effects (HAUG and SCHMITT-RINK [ 19841) have not been included in the preceding derivation. These have two main influences: first, they renormalize the bands and actually lead to a red-shift (SHANK,FORK, LEHENYand SHAH [1979]), and, second, they modify the recombination times because of the Auger effect (see, e.g., BLAKEMORE [ 19621, PETERSON [ 198 1 I); we shall discuss these aspects in connection with the experimental results. Returning to eq. (5.9), we wish to make two important observations by referring to figs. 16 and 17. First, the intensity-dependent ~ ( wI) , saturates at
xi---
Fig. 16. Intensity dependence of the effective Kerr susceptibility for "large" particles (bulklike case) for three different detunings, A = (E,/h - o)T,.
Re
x
_-----___---, ,
-2 -4 \
'"....I m X
(3)
_______.._ -------.__ --. --__.._...._._........---------
w
-2
o
2
4
6
a
10
12
14
16
18 (W - u p )
T
for a bulk semiconductor from the Fig. 17. Real part, imaginary part, and modulus of band-filling model.
382
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V.§ 5
an intensity (ROUSSIGNOL, RICARDand FLYTZANIS [ 19871) (5.15) and A x = ~ ( o I ),- x(w, 0) remains constant above I, while it is linear with I below I,, allowing a third-order susceptibility x(3) to be defined there; and second, this f 3 ) scales as x(')(o)= ~ ( w0), for a wide frequency domain that includes the absorption threshold. Closely related to the optical Kerr effect is the electroabsorption (see, e.g., CARDONA [ 1966]), i.e., the change in the absorption spectrum due to a static electric field E,; for a single atom this is the static Stark effect, whereas for a bulk semiconductor this is the Franz-Keldysh effect (FRANZ [ 19581, KELDYSH[ 19581). In fact, it was shown that the same mechanism underlies the two phenomena (MILLER,CHEMLAand SCHMITT-RINK [ 1986,19881); the discussion includes a term H'
=
e(r, - rh) E, ,
(5.16)
in eq. (3.32) which can be treated as a perturbation as long as IeaE,IE,I < 1 ,
(5.17)
where Ec is a characteristic energy of the confinement, e.g., the average level spacing (one usually neglects the electron-hole interaction in calculating the effect of the static field E,). In the strong-confinement regime the problem reduces to the solution of the SchrBdinger equation for the envelope with Hamiltonian (5.18) where i = e, h. To the extent that eq. (5.17) applies, one can perform a calculation within the Rayleigh- Schrodinger perturbation up to the second order. The main features are: (1) any nl state is now mixed with all n'(l 1) ones, and (2)its energy is shifted by an amount proportional to E;; in particular, its 21 + 1 degeneracy is partially broken and gives rise to 1 + 1 levels. As a consequence, new transitions now appear with oscillator strengths borrowed from the initially allowed transitions in the absence of the static electric field; the oscillator strength of the latter is reduced in the presence of E, because of the incomplete overlap of the electron and hole envelopes. The compound effect of this rearrangement is the appearance of oscillations in the
V, $61
PROPERTIES OF SEMICONDUCTOR COMPOSITES. EXPERIMENTAL STUDIES
383
differential absorption coefficient 6a = a(w; E,) - a(w; 0), whose period should reflect the confined level spacing. All these features are the same as those expected in the static Stark effect in atoms or molecules. One can also write 6 a = 12nw Im ~ ' ~ ' ( 0 0,, w ) E:/nc
and obtain the value of the static Kerr effect susceptibility x(')(O,O, 0). As the crystallite size increases, condition (5.17) will eventually cease to be valid, and one must resort to a nonperturbative solution of the problem along the lines of the treatment used for the bulk semiconductor; this is the Franz-Keldysh effect (FRANZ[ 19581, KELDYSH[ 19581). The theoretical treatment actually shows that the two extreme cases merge, and that features evolve from those of the one model to those of the other model as a is continuously changed from a ah to a > a,. Before discussing the experimental results in Q 6, we note that the restriction to the two-band model can actually be relaxed by also introducing the spin-orbit split-off valence band. In addition to the two-band contributions already mentioned, one may also have three-band contributions, which lead to complicated expressions. As electrons are photocreated, one must also include a free-electron contribution to ~ ( wI,) , which in particular, is responsible for a residual absorption at large intensities. Furthermore, one must be concerned about two-photon transitions at high intensities that may mask the saturation trend predicted by eq. (5.15) and shown in fig. 16.
-=
8 6.
Nonlinear Optical Properties of Semiconductor Composites: Experimental Studies
The first study of a nonlinear optical effect in a semiconductor composite dates back to 1964 with the observation of absorption saturation in commercial Schott fitters and its use to Q-switch a laser (BRETand GIRES[ 19641). The real impetus to the systematic study of the nonlinear optical properties of the semiconductor-doped glasses came, however, with the work of JAIN and LIND [ 19831. Using the optical phase-conjugation technique in the DFWM configuration, they measured values of the optical Kerr coefficients i ( 3in ) the range 10- 8-10-9 esu for frequencies above the absorption onset of these commercial filters. Although there was no doubt that these high nonlinearities were entirely due to the photocarriers in the microcrystallites (ROUSSIGNOL, and PEYGHAMBARIAN RICARD,RUSTAGIand FLYTZANIS [ 19851,OLBRIGHT
384
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V, § 6
[1986]), the identification of the actual mechanisms, and in particular the temporal evolution of the effect, required further theoretical and experimental study. For “large” crystallites, as is the case in commercial filters, the nonlinear mechanism was identified as being band filling (ROUSSIGNOL,RICARD, LUKASIKand FLYTZANIS [ 19871, OLBRIGHT, PEYGHAMBARIAN, KOCHand BANYAI[1987]), whereas the impact of the quantum confinement on the optical Kerr effect in small crystallites was evidenced both by direct measurement of the frequency behavior of xC3)as a function of the average crystallite radii (ROUSSIGNOL, RICARDand FLYTZANIS [ 19901) and by spectral holeburning spectroscopy (ALIVISATOS, HARRIS,LEVINOS,STEIGERWALD and BRUS [ 19881, ROUSSIGNOL,RICARD,FLYTZANIS and NEUROTH[ 19891, PEYGHAMBARIAN, FLUEGEL, HULIN,MIGUS,JOFFRE,ANTONETTI,KOCH and LINDBERG [ 19891, ROTHBERG, JEDJU, WILSON, BAWENDI, and BRUS[ 19901). The controversy concerning the different STEIGERWALD decay times reported by different groups (JAIN and LIND [1983], YAO, KARAGULEFF, GABEL,FORTENBERRY, SEATONand STEGEMAN[ 19851, COTTER[ 1986b3, PEYGHAMBARIAN, OLBRIGHTand FLUEGEL [ 19861) was resolved only with the observation of the photodarkening effect 6rst reported by ROUSSIGNOL, RICARD,LUKASIK and FLYTZANIS [ 19871.The hole burning HARRIS,LEVINOS,STEIGERWALD and BRUS [ 19881, studies (ALIVISATOS, ROUSSIGNOL, RICARD,FLYTZANIS and NEUROTH[ 19891,PEYGHAMBARIAN, FLUEGEL,HULIN, MIGUS, JOFFRE, ANTONETTI,KOCH and LINDBERG [ 19891, ROTHBERG,JEDJU,WILSON,BAWENDI,STEIGERWALD and BRUS [ 19901) also allowed the extraction of information about the character of the broadening mechanisms of the quantum-confined resonances. More recently, electroabsorption studies gave additional confirmation about the validity of the primary aspects of the quantum confinement in small crystallites (MILLER, CHEMLAand SCHMITT-RINK [ 1986,19881, HACHE,RICARDand FLYTZANIS [ 19891). We shall summarize the main results and conclusions of these experimental studies, beginning with the large crystallites whose nonlinear optical properties were investigated first, and proceed to the more recently studied smaller crystallites with quantum confinement. 6.1. LARGE SEMICONDUCTOR CRYSTALLITES
6.1.1. Frequency and intensity dependence of optical nonlinearities
As stated earlier, we are concerned with the light-induced changes of the absorption and refractive index or, equivalently, the optical Kerr effect. The
v, $61
PROPERTIES OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES
385
coefficient n2 or equivalently ~("(0, - o,o) related to this effect is most conveniently measured with the optical phase-conjugation technique (FISHER [ 19831, ZELDOVICH, PILIPETSKY and SHKUNOV [ 19851) in the DFWM configuration briefly outlined in $ 2. We note here that with all three input beams co-polarized and coincident in time, one measures the magnitude of I x ( ~ ) ( o ) I and also its frequency dependence if the wavelength of the light source can be tuned. In the co-polarized beam configuration, if the beams are appropriately delayed with respect to one another, one also measures the temporal evolution of x ( ~ ) ,namely, the decay time z in eq. (3.19), whereas in the cross-polarized beam configuration one has access to the same information concerning x$,. By using interferometric techniques one also has access to the phase of x ( ~ ) . As the pump intensity is increased, nonlinear contributions higher than third order become involved in the process, and the effective nonlinear coefficient becomes intensity dependent and eventually reaches a saturation regime. As stated in $ 2, for the optimization of the phase-conjugated signal, one uses thin samples with aL x 1. We shall not dwell here on the absolute magnitude of i(3), which, as initially reported by JAIN and LIND [ 19831 and confirmed by several groups (ROUSSIGNOL, RICARD,RUSTAGIand FLYTZANIS[ 19851, OLBRIGHTand PEYGHAMBARIAN [ 19861, COTTER [ 1986a, 1988]), is certainly high, 10-8-10-9 esu in commercial CdS, Se, -,-doped glasses close to their absorption edge. Actually, these values should be handled with caution, particularly when a comparison with those of the corresponding bulk semiconductor is attempted. Indeed, the fabrication process of these semiconductor-doped filters introduces many uncertainties in their size, shape, surface, and stoichiometry of the crystallites as well as nonidentified defects that make difficult any comparison with the bulk. Thus, because of the weak quantum confinement and the concomitant absorption edge shift, comparison of such values reported at the same laser frequency for the bulk and the doped glasses is misleading. The value of x in the crystallites may not be known with accuracy, information concerning the impurity content and the semiconductor/glass interface provided by the manufacturer is scarce or nonexistent, and the laser pulse characteristics play an important role in assessing the values of the optical Kerr effect coefficient (HACHE,ROUSSIGNOL, RICARDand FLYTZANIS[ 19873). The magnitude and frequency dependence of x(3) for a Corning CS 2.61 sample at room temperature are shown (ROUSSIGNOL,RICARD and FLYTZANIS [ 19871) in fig. 18, together with those of the absorption coefficient a ; similar results were also obtained for a Schott RG 610 sample. The measurements were done with pulses of a few nanoseconds in duration and were also
386
IV, f 6
NONLINEAR OPTICS IN COMPOSITE MATERIALS
4
3 6
2
....................
1
-
0
5
...I.....
-
4
Id
3
a
-1
v
2
-1
1
. moo
18400
wisoo
17200
.
0
17800
Fig. 18. Semilog plot of the frequency dependence of u (solid line) and of zc3)(dotted line) for “large” CdSSe particles.
repeated at other sample temperatures. The most striking conclusions drawn from this study are (1) the absence of any discontinuity near the band gap and (2) the almost complete proportionality between ~ ( ~ ’ and (0 ~) ( 0throughout ) the frequency domain covered by the tunable source, a dye laser pumped with a Q-switched, frequency-doubled Nd : YAG laser. This behavior implies that the ratio ~ ( ~ ) / zdefined a, in 3 as the figure of merit of an absorbing nonlinear material, is a constant and in the case of the Schott RG 630 sample, which contains CdS,,, Se.5 crystallites, was measured to be the same as in the bulk semiconductor with the same value of x . Theoretically, as discussed in 5 5, we also expect this to be the case within the band-filling model (ROUSSIGNOL, RICARDand FLYTZANIS[ 19871) for all direct gap semiconductors, and the preceding result actually is an+ind.irectconfirmation of the validity of this model. A direct confirmation of the validity of the band-filling model was obtained (ROUSSIGNOL,RICARD and FLYTZANIS[ 19871, OLBRIGHT, PEYGHAMBARIAN, KOCHand BANYAI[ 19871) by measuring the absorption spectrum changes shortly after the sample was excited with a short light pulse above its absorption edge. Figure 19 shows this absorption change in a Schott OG 570 0.6 mm thick sample after being excited with a 25 ps pulse at wavelength of 532 nm which lies 780 cm- above the absorption edge of the spectrum (ROUSSIGNOL, RICARDand FLYTZANIS[ 19871); similar results
V, 61
PROPERTIES OF SEMICONDUCTORCOMPOSITES: EXPERIMENTAL STUDIES
560
520 Wavelength (nrn)
387
480
Fig. 19. Transmission spectra for a Schott OG 570 glass before ( 1 ) and immediately after (2) picosecond excitation at t = 532 nm. The blue-shift indicates band filling.
were obtained by OLBRIGHT,PEYGHAMBARIAN, KOCH and BANYAI[ 19871. As one sees in fig. 19, the absorption curve is shifted to the blue. As also explained in 0 5, this is because the photocarriers generated with the short excitation pulse after thermalization quickly fill the states at the bottom of the conduction band and because of the Fermi statistics, which make them inaccessible for the probe until they decay, either by impurity trapping or band-to-band recombination ; thus, the apparent absorption edge is shifted to the blue. For much shorter pulses, in the femtosecond duration range or so, one also expects to see a red-shift of the absorption edge because of the band-gap renormalization by many-body effects '(SHANK,FORK,LEHENYand SHAH [ 19791, HAUGand SCHMITT-RINK[ 19841). There is no conclusive evidence of such an effect, which seems to be completely dominated by band filling in the crystallites; on the other hand, there is evidence for the related Auger effect FREY,ROUSSIGNOL,RICARDand FLYTZANIS [ 19871, (DE ROUGEMONT,
388
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V, 8 6
ROUSSIGNOL, KULL, RICARD, DE ROUGEMONT, FREYand FLYTZANIS [ 1987, 1989]), as will be discussed later. The previous measurements concerned [ f 3 ) I , and in order to assess the validity of the band-filling model, one also needs the phase of f 3 ) and, in particular, its real and imaginary parts that can be related to the light-induced changes in the refractive index and absorption. This was done by an interference technique (ROUSSIGNOL,RICARD and FLYTZANIS[ 19871, ROUSSIGNOL [ 19891) by comparing the nonlinearity of the sample with that of liquid CS,, which at these wavelengths is essentially positive and due to the reorientation of the CS, molecules in the liquid; one can, in addition, compare two semiconductor-doped glass samples. In these phase determination measurements of f 3 ) the phase of the laser pulse must be carefully controlled and, in particular, the pulse self-phase modulation be suppressed. Indicative results of this phase determination obtained at wavelength 1 = 532 nm, the second harmonic of the Nd: YAG, are as follows: for an OG 530 sample, hence below the absorption edge, x(3) is real negative, which is also confirmed with a more elaborate interferometric technique, whereas for an OG 570 sample, hence just above the absorption edge, ~ ( is ~essentially 1 imaginary with a small real negative part; finally, for an RG 610, hence well above the absorption edge, the imaginary part is still dominant but the real part is now positive and relatively more important. These results are in agreement with the theoretical calculations using the band-filling model (fig. 17) (ROUSSIGNOL, RICARDand FLYTZANIS [ 19871). The light-induced change of the index of refraction n, can also be measured and PEYGHAMBARIAN [ 19861, by interferometric techniques (OLBRIGHT COTTER[ 1986a1). In this respect the Mach-Zehnder interferometer and its variants have been extensively used to measure different features of n, (NATTERMANN, DANIELZIK and VON DER LINDE [1987], SALTIEL, VAN WONTERGHEM and RENTZEPIS [ 19881, ABBATE, BERNINI, MADDALENA, DE NICOLA,MORMILE and PIERATTINI [ 19891). In the semiconductor-doped glasses one can easily observe nonlinear effects of a higher order than the third one (ACIOLI, GOMESand RIOSLEITE[ 19881, BLOUIN,GALARNEAU and DENARIEZ-ROBERGE [ 19891). Actually, as anticipated for large pump intensities, the optical nonlinear coefficient measured by the optical phase-conjugation techniques becomes intensity dependent (RUSTAGIand FLYTZANIS [ 19841, KOUSSIGNOL, RICARDand FLYTZANIS [ 19871) and, in particular, exhibits saturation (fig. 20). The observation of saturation of the absorption in the optical filters was first reported in the experiments of BRET and GIRES[ 19641, and in their optical phase conjugation
v, 8 61
PROPERTIES OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES
0.01
0.1
1
to
389
100
Fhmce (mJ/cm*)
Fig. 20. Intensity dependence of the phase conjugate reflectivity for a CdSSe-doped glass with "large" particles.
experiments JAIN and LIND [ 19831 reported limitation of the optical phase conjugation reflectivity efficiency to x 10% as the pump intensity was increased. A careful study of the saturation behavior revealed the existence of a saturation threshold (ROUSSIGNOL,RICARD,LUKASIKand FLYTZANIS [ 19871) in the nonlinear coefficient that was essentially the same as for the absorption saturation. Furthermore, the behavior of the nonlinear coefficient and of the optical phase-conjugated reflectivity below or close to this saturation intensity is similar to that expected for a broadened two-level system (ABRAMS and LIND [ 1978a,b], SILBERBERG and BAR-JOSEPH[ 1981]), but for higher pump intensities it deviates significantly from this behavior and for two reasons: as will be shown, first, the recombination time that corresponds to T , in the two-level system becomes intensity dependent because of the Auger effect (DE ROUGEMONT,KULL, RICARD, DE ROUGEMONT,FREY, ROUSSIGNOL, RICARDand FLYTZANIS [ 19871, ROUSSIGNOL, FREYand FLYTZANIS[ 1987, 1989]), which asymmetrically broadens the saturation regime close to I,, and second, at still higher pump intensities two-photon absorption (CANTO, MIEZAK,HAGAN,SOILEAUand VAN STRYLAND[ 19881; see also NATHAN and GUENTHER [ 19851) sets in and modifies the initial process (RoussrGNoL
390
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V,§ 6
[ 19891, DELONG, GABEL, SEATONand STEGEMAN[ 19891). The saturation intensity I, differs for different samples from different manufacturers. For a CS 3.68 sample using a frequency-doubled, Q-switched Nd : YAG laser, with pulse duration of 10 ns, the saturation starts at a fluence of 1 mJ/cm2, and the reflectivity varies only slightly up to 100 mJ/cm2, the highest fluences attainable with this laser. We also point out that in all measurements, care was taken to make sure that the samples did not get photodarkened. 6 .I .2. Temporal evolution of optical processes: Photodarkening
The knowledge of the time evolution of the nonlinear optical coefficients is essential in assessing the potential use of these materials in nonlinear optical devices but also for fundamental reasons. Thus the decay time t in eq. (2.4) enters the expression of the figure of merit eq. (2.7) for an absorbing nonlinear medium and sets an upper bound on the repetition rate of the signal in information-processing techniques ;furthermore, it measures the strength of the energy decay processes that play an essential role in any microscopic description of the optical processes. Hence, the intensive study of the temporal evolution of n, and, in particular, the determination of t is understandable, but the initial reports by the different groups (JAINand LIND[ 19831, ROUSSIGNOL, RICARD,RUSTAGIand FLYTZANIS[ 19851, YAO, KARAGULEFF,GABEL, FORTENBERRY, SEATON and STEGEMAN[ 19851) initiated controversy because of the widely different values of t and temporal evolutions observed there. This point was settled with the observation that the temporal evolution depends on the previous integrated photo-exposure of the samples (ROUSSIGNOL, RICARD,LUKASIKand FLYTZANIS[ 19871) and is markedly different for low fluences and high fluences. In between these two regimes the RICARD, sample undergoes a process termed photodarkening (ROUSSIGNOL, LUKASIKand FLYTZANIS[1987]), and hence, a distinction must be made between fresh samples that have been exposed to low fluences and photodarkened samples that have been exposed to high fluences. The precise mechanism of this process is still under investigation (MITSUNAGA, SHINOJIMA and KUBODERA[ 19881, VAN WONTERGHEM, SALTIEL,DUTTON and RENTZEPIS [ 19891, HENGLEIN[ 1987]), so we will content ourselves only with some conjectures and a qualitative discussion of its implications on the temporal behavior of the optical Kerr effect and the luminescence. Photodarkening. A signature of the photodarkening effect is the dramatic change that the luminescence spectrum and its global temporal evolution
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undergoes as the sample undergoes the photodarkening process (ROUSSIGNOL, RICARD,LUKASIKand FLYTZANIS[ 19871). A luminescence spectrum for a fresh commercial Schott OG 570 sample is shown in fig. 21 (ROUSSIGNOL, RICARD,LUKASIKand FLYTZANIS [ 19871) and for the photodarkened one in fig. 22. The most striking difference is the disappearance of the broad feature present at long wavelengths in the fresh samples while the narrower one at shorter wavelengths remains roughly unaffected. Since the latter seems to be largely due to band-to-band recombination processes, the conjecture was made that the broad feature was caused by impurity states (ROUSSIGNOL,RICARD, LUKASIK and FLYTZANIS [ 1987]), presumably situated at the crystallite/glass interface, which act as luminescence traps and are quenched at high fluences. That this broad feature corresponds to different processes (ROUSSIGNOL, RICARD,LUKASIKand FLYTZANIS[ 19871, MITSUNAGA,SHINOJIMA and KUBODERA[ 19881) than those responsible for the narrower feature is also inferred by the markedly different decay evolution of the luminescence of a fresh sample measured at two wavelengths, at the centers of the broad and narrow features, and found to be slow and fast, respectively. For the darkened sample the fast decay persists with a roughly unchanged time constant. More careful measurements of the luminescence spectrum in fresh samples actually revealed a more complex structure of the narrower feature (HACHE,
I
I
I
I
0
Fig. 21. Luminescence spectrum of a fresh Schott OG 570 glass. Typical temporal behaviors are shown in the insets.
392
NONLINEAR OPTICS IN COMPOSITE MATERIALS
0
800
700
600
A(nm)
Fig. 22. Luminescence spectrum of a photodarkened OG 570 sample.
KLEIN, RICARDand FLYTZANIS [ 1991]), that we attributed to the band-toband recombination. It appears that this feature consists of two components, which behave differently as the intensity of the laser excitation is varied and correlated differently with the excitation intensity dependence of the broad feature. As stated earlier, the microscopic mechanism of the photodarkening has not yet been identified. From physicochemical studies in CdS colloids (HENGLEIN [ 1988]), it was inferred that the CdS crystallites are charged and injection of other charges drastically modified their absorption and bleaching in a glass. Such a crystallite may introduce substantial “distortions” in its neighborhood and, in the interface with the glass; the distortions can either be induced by the electric field of the charges or by charge transfer complexes between crystallite and glass. Under high laser fluences these undergo a change above a threshold and their nonradiative lifetime becomes much shorter. This change is permanent as long as the sample temperature of the glass is kept below the softening temperature of the glass ( x 500 “C);once the photodarkened sample is heated close to or above this temperature, it recovers the features of a fresh sample - an indication that supports the association of the broad feature and the photodarkening process with the crystallite/glass interface. Clearly, more work is needed to understand the role of the different defects in these crystaland STEEL[ 19881) are scarce, lites; spectroscopic studies (REMILLARD Temporal evolution of n,. In mapping out the temporal evolution of n2, we should keep in mind that this arises from the photocreated carriers at a given energy state through real band-to-band transitions in the crystallites and, accordingly, the temporal evolution of n2 will correlate with the energy decay of this state. In general, one can distinguish three time scales for this decay: The thermalization decay to the bottom of the conduction band occurs in the
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subpicosecond-to-picosecond time range, the band-to-band recombination occurs in the 1 ns time range, and the impurity-dominated radiative decay is longer than this. As energy is absorbed in the resonant nonlinear optical processes and subsequently deposited as heat into the lattice of the crystallite, there will also be a thermal contribution to n, (THIBAULT and DENARIEZ-ROBERGE [ 19851, BERTOLOTTI, FERRARI, GUAPPI,MONTENERO, SIBILIAand SUBER[ 19881); this contribution, which occurs on a much longer time scale, will not be considered further. All three regimes have been observed either by measuring the evolution of the absorption time by an excite-and-probe technique or by introducing a delay between the backward pump and the other two input beams in the optical phase-conjugation technique. For excitation energies well above the gap energy the intraband thermalization regime was observed by Nuss, ZINTH and KAISER[ 19861 and, later, by PEYGHAMBARIAN, OLBRIGHTand FLUEGEL [ 19861, in the absorption changes by using laser pulses in the 60 fs range. The thermalization time depends on the position of the excitation energy above the gap energy and on the number of available traps. Since the thermalization takes place in the subpicosecond time range, with pulses longer than a few picoseconds, this process cannot be observed. For excitation above but close to the gap, only the recombination and impurity channels remain, whereas for excitation close to but below the gap, only the impurity channel remains, which is particularly true when excitation and probe pulses are a few picoseconds in duration. This was shown in the measurement of the temporal evolution of the absorption as well as the optical phase-conjugated signal. The time evolution of the latter is shown in fig. 23 for a fresh O G 530 sample and in fig. 24 for a photodarkened one of the same make; the difference of the time constants is substantial as was anticipated. With appropriate care, indications about the decay times can also be obtained from photoluminescence studies (WARNOCK and AWSCHALOM [ 19851, SHUM,TANG,JUNNARKAR and ALFANO[ 19871, ZHENG,SHI, CHOA, LIN and KWOK [1988], TOMITA,MATSUMOTOand MATSUOKA[ 19891). In the previous experiments the laser intensity was kept sufficiently low to assume that the electron-hole pair density is very low and interactions between them do not matter. When the laser intensity is increased, this assumption is no longer valid, since the density of electron-hole pairs increases and their mutual interactions must be reckoned with. In particular, the Auger process [ 1962]), where the strong Coulomb interaction of two electrons (BLAKEMORE in close contact in the conduction band accelerates the recombination of one of the electrons with a hole and at the same time promotes the other to a
394
[V,5 6
NONLINEAR OPTICS IN COMPOSITE MATERIALS
1.:
-a .-m Y)
m A 0
1
1.3
0
. 0
( b)
1
.
0.4~s
a
Delay (ns) 1
2
3
4
Fig. 23. Semilog plot of the dependence of the phase-conjugated signal as a function of the backward-pump pulse delay for fresh samples. (a) Corning CS 3.68, (b) Schott OG 530.
correspondingly higher energy in the conduction band, becomes effective and has been experimentally demonstrated (DE ROUGEMONT, FREY, ROUSSIGNOL, RICARD and FLYTZANIS[ 19871, ROUSSIGNOL, KULL, [ 1987, 1989]), as shown in RICARD,DE ROUGEMONT,FREYand FLYTZANIS fig. 25. There the absorption decay of a commercial Schott OG 570 sample for three different excitation fluences is depicted as a function of the probe delay. The sample was kept as fresh as possible: after the cycle of these measurements the absorption decay at a low fluence was roughly the same as that observed at the beginning of the cycle. At a low fluence the decay shows the two characteristic regimes previously discussed = the fast one due to recombination and the slower one due to the impurities. As the fluence is increased, the time constant of the fast regime becomes shorter, whereas that of the slow regime remains essentially constant; at very high fluences the latter becomes weaker as photodarkening sets in. The shortening of the time constant of the fast component can be traced back to the Auger process, which hastens the recom-
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PROPERTIES OF SEMICONDUCTOR COMPOSITES. EXPERIMENTAL STUDIES
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t'
' 0 .
Delay ( n s )
Fig. 24. Dependence of the phase-conjugated signal as a function of the backward-pump pulse delay for photodarkened glasses. (a) Corning CS 3.68, (b) Schott OG 530.
bination process. This effect can be phenomenologically described by writing dn
1
-=
--
dt
5
n - Bn2 - A n 3 ,
where n is the number of free photocarriers, the third term on the right-hand side stands for the Auger recombination process (BLAKEMORE [ 1962]), and the second term stands for the geminate recombination (STREET[ 19811, SAITO and GOBEL[ 1985]), which in fitting the results was found to be negligible.
396
NONLINEAR OPTICS IN COMPOSITE MATERIALS
0
*\
Bo
4020
-
400
800
c)
Fluewe ~
P/an2 J
. . -._
Thus, for the absorption decay one can write d dt
1
- 6a = - - &a - 7(sa)3, z
since n x 6a and y is related to A if the semiconductor volume concentration p is known. By fitting the experimental results in fig. 25 or fig. 26, one finds a value of a few cm' s - for the Auger constant A, quite close to the one measured for the bulk semiconductor (BLAKEMORE [ 19621, PETERSON [1981]). Similar values were obtained by other groups (NATTERMANN, DANIELZIK and VON DER LINDE[ 19871, ZHENG and KWOK[ 19891, KULL [ 19891, KULL,COUTAZ,MANNEBERG and GRIVICKAS [ 19891).
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PROPERTIES OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES
397
16-.
(4 E = 1920pJ/cm2
.. t(ns)
.
5
L
E = 64pJ/cmz v
j
.25
1
2
3
4
t
(
~
Delay (ns)
Fig. 26. Time dependence of the absorption change 6a for various energy fluences showing evidence both of Auger recombination and of photodarkening (d).
The Auger effect and the two-photon absorption that appears in still higher light intensities can significantly modify the intensity dependence of the absorption coefficient and the nonlinear susceptibility, as expected from a simple broadened two-level system in the presence of intense pump beams (ABRAMS and LIND[ 1978a,b], SILBERBERG and BAR-JOSEPH [ 19813) or from the band-filling model (ROUSSIGNOL, RICARDand FLYTZANIS [ 19871). Both systems predict that
398
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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and K = -
I I, (1
1
(6.4)
+ 41/1,)3'*
for the absorption and optical phase-conjugated reflectivity, respectively. It is not possible to incorporate the Auger and two-photon processes rigorously and derive simple analytical expressions for these two coefficients. One can grasp the main trends however, by using some plausible simplifications(ROUSSIGNOL [ 19891). Thus, from eq. (6.1) with B x 0, one can define an effective recombination time zA
I (1
'
+ zAn2),
and also set
a(o)Iz hop
n=-+-
PI' 2hwp
where p is the two-photon absorption coefficient and p the volume fraction. Assuming eqs. (6.3) and (6.4) to be formally still valid but with z replaced by zA and n by eq. (6.6) and using an iterative procedure, one finds the curve
-6
t /
-1
-
- 88 V
\ \\
'" '
"
'
"'
" '
''
'b '
'
\~, b
Fig. 27. The influence of Auger recombination (solid line) and two-photon absorption (dashed line) on the intensity dependence of the phase-conjugatedreflectivity for semiconductor-doped glasses. The inset shows the experimental results for an OG 530 glass.
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depicted in fig. 27 (ROUSSIGNOL[ 19891) for the reflectivity together with the experimental results for the Schott OG 570 sample.
6.2. QUANTUM-CONFINED CRY STALLITES
The experimental studies of the nonlinear optical properties involved the optical Kerr effect, the spectral hole burning and saturation, and the electroabsorption. Since spectral hole burning and absorption saturation studies were reported in 8 3 in connection with the discussion of the broadening mechanisms of the quantum-confined resonances, we will focus on the other two nonlinear effects. 6.2.1. Enhancement of the optical Kerr efect As stated in 5 5, close to the quantum-confined resonances of semiconductor
crystallites, we expect the optical Kerr coefficient of a strongly quantumconfined semiconductor crystallite to behave similarly to that of a homogeneously broadened two-level system. In assessing the impact of the quantum confinement on the optical Kerr effect, it is important to perform the measurements on samples where only the average radius of the crystallites changes from sample to sample while all other parameters remain the same. This can be partly achieved by performing the striking process in the fabrication of these glasses on a nucleated glass rod that is kept in a constant temperature gradient (REMITZ,NEUROTH and SPEIT[ 19891). The frequency dependency and magnitude of the nonlinear optical susceptibility ~ ( ~ ' ( w were ) measured for a series of samples of different average crystallite radii (ROUSSIGNOL, RICARD and FLYTZANIS[ 19901) prepared as described earlier. The results for the susceptibility x ( ~ are ) depicted in fig. 28. They were obtained by the optical phase-conjugation technique with all beams co-polarized using liquid CS, as a reference. The source was a picosecond distributed feedback dye laser pumped by the second harmonic of a single transverse mode, mode-locked and Q-switched Nd: YAG laser. The results were taken with sufficiently low laser energies to avoid any photodarkening of the sample or any saturation. As one can see, xc3) as a function of frequency for each sample peaks around the corresponding quantum-confined resonance, a behavior that contrasts with the insensitivity on w of the same ratio that was observed in large crystallites ; furthermore, this enhancement increases as the average radius decreases in the average size range studied. This is in agreement
400
NONLINEAR OPTICS IN COMPOSITE MATERIALS
Wavelength ( n m )
600
590
580
Wavelength
570
5w
(nm)
Fig. 28. Semilog plot of the frequency dependence of 1 f 3 ) ( 0 )for ( CdSe particles of various sizes whose absorptionspectra are shown on top. The mean radius is: (1) 3 nm, (2) 2.8 nm, (3) 2.6 nm, and (4) 2.4 nm.
with the increased oscillator strength concentration as the size is reduced as predicted by the simple model of 5 3. The enhancement factors of x ( ~ ) / o ! reported in fig. 28 can reach a value of 2 or more with respect to that of the bulk. The theoretical results obtained by computing expressions (5.4) and (5.5) are also shown in fig. 28 and in general confirm the validity of the quantum confinement and broadening mechanisms discussed in 3 3. The best fit was obtained with T2 = 15 fs, which corresponds to a width of 88 meV, in agree0 at low temperature, and a ment with the experimentally measured ~ 6 meV size distribution width A = 10% for all samples. The size distribution, for simplicity, was taken to be
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PROPERTIES OF SEMICONDUCTOR COMPOSITES: EXPERIMENTAL STUDIES
40 1
instead of the LIFSHITZ-SLEZOV [ 19591 distribution; with eq. (6.7), for small A, one has ( a ) x a,. The temporal evolution of the optical Kerr coefficient was also studied by delaying the backward pump with respect to the other two input beams in the optical phase conjugation setup. Here, too, one observes both a fast and a slow regime, but their attribution to specific mechanisms needs additional study.
6.2.2. Electroabsorption: Static Stark shijl and Franz-Keldysh effect As pointed out in
3 5,
a static electric field E, induces changes in the absorption spectrum that can be related to two processes taking place simultaneously in a quantum-confined crystallite (MILLER, CHEMLA and SCHMITT-RINK [ 1986,19881,HACHE,RICARDand FLYTZANIS [ 19891). First, the displacement of the energy levels induces a shift of the elementary absorption peaks, which is also accompanied by a decrease of the oscillator strength
1
Fig. 29. Differential absorption spectra induced by a static field E , for particles of various sizes. The mean radius is: (a) 5 nm,(b) 3 nm, and (c) 1.5 nm.
402
6a
NONLINEAR OPTICS IN COMPOSITE MATERIALS
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PROPERTIES OF SEMICONDUCTORCOMPOSITES:EXPERIMENTAL STUDIES
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as the overlap between hole and electron wave functions is decreased. Second, new transitions appear due to the breakdown of inversion symmetry and mixing of states. As a consequence the absorption change Fa = a(E,,) - a(0) shows oscillations with spacings that depend on the crystallite size; furthermore, this modification varies as E,Z. In fig. 29a-c, we show the representative electroabsorption measurement series (HACHE,RICARDand FLYTZANIS [ 19891) for three different sizes of radius 50, 30, and x 15 A, respectively, which corroborate these predictions. As expected from quantum confinement, the whole structure broadens and blue-shifts as the radius is decreased. These observations clearly indicate a static Stark effect. For the “large” crystallites a replica of the oscillations due to the spin-orbit split-off valence band can be seen. This is also visible for the intermediate-size crystallites but disappears in the smallest size particles because of the increased broadening or the valence band mixing. The experimental behavior can also be quantitatively reproduced as in fig. 30, where the results of the calculation of the absorption change 6cl and the
Fig. 30. Comparison between measurements(dashed lines) and perturbation theory calculations (solid lines) for the sample corresponding to fig.29b.
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NONLINEAR OPTICS IN COMPOSITE MATERIALS
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absorption coefficient 01 for a small crystallite (a c 30 A) are shown; the width of the transition was taken to be 60 meV and the size dispersion 12%. These results confirm the interpretation of the electroabsorption as a static Stark effect for quantum-confined crystallites ; as the crystallite size increases and the band states are recovered, one expects that the electroabsorption spectrum will bear the signature of the Franz-Keldysh effect. This was also confirmed experimentally by CO~TER and GIRDLESTONE [ 19901. These observations also confirm that the same microscopic mechanisms underlie the two effects.
8 7. Conclusions and Extensions In the previous sections we surveyed the nonlinear optical properties of semiconductor and metal crystallites in dielectric matrices and summarized a description that emerged from the experimental and theoretical investigations. The field will certainly evolve and some aspects of the proposed description may need re-examination, but others now seem well established and we can risk reaffirming them here. Thus, the impact of the dielectric confinement is certainly important in metal crystallites in a transparent dielectric and enhances the optical Kerr effect through the local-field resonant behavior close to the surface plasmon resonance. On the other hand, the quantum confinement has no influence here as it does not affect the mechanism of the nonlinearity, which was found to be the hot-electron contribution. The situation is totally different in semiconductor crystallites in dielectrics, glass or liquid. The dielectric confinement here is dominated by the quantum confinement, which substantially modifies the spectrum, redistributes the density of states and oscillator strengths, and alters the relaxation processes. Its impact decreases with increasing crystallite size, and for large sizes the semiconductor bulk behavior can be recovered. As a consequence, the nonlinearity for strongly confined semiconductor crystallites may be ascribed to the saturation of a two-level system whose broadening, homogeneous and inhomogeneous, is still unclear. Coulomb effects may play a role, but we believe that the surface states and defects play a more important role in modifying the simple two-level behavior, both with respect to the magnitude of the nonlinearity, its frequency dependence, and decay time. For large semiconductor crystallites, larger than the exciton radius, the band-filling model, the same model that also applies in the bulk, gives a satisfactory description of the behavior of the optical nonlinearities. Freshly prepared semiconductor-doped glasses at high fluence
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405
undergo an irreversible process called photodarkening, which radically modifies the relaxation processes and the figure of merit of these samples. This effect, which was attributed to a modification of the surface states and defects, appears to be particularly important in commercial samples but has also been demonstrated in artificial ones that show quantum confinement. The quantum confinement has also been confirmed by electroabsorption studies, where one observed a continuous evolution from a static Stark effect to a Franz-Keldysh effect as the size of the crystallite increases. Although the previous description is largely coherent and allows extraction of the salient features of the optical nonlinearities and the underlying mechanisms, it is still too early to provide a secure basis for predicting the nonlinearities in the presently fabricated materials or the ones that might be fabricated. These fabrication techniques are rather crude and introduce substantial uncertainties, which preclude any meaningful comparison between the crystallite and the bulk of a given semiconductor or between crystallites in different dielectrics or any other comparison. But we can give certain indications. Whatever the mechanism or the material, any enhancement of the optical nonlinearity near a resonance also causes an increase of the absorption losses, and in certain cases such as in large semiconductor crystallites, there is a complete proportionality between the optical Kerr and absorption coefficients over a very extended spectral range below and above the energy gap. The proportionality factor, however, differs from compound to compound, and simple arguments point to the prediction that CdTe will be the most favorable of the 11-VI compounds, both with respect to the figure of merit, eq. (2.7), and the decay time close to the quantum-confined transitions. The occurrence of the photodarkening effect also indicates,+hatsubstantial improvement can be expected from judicious application of this effect or by proper treatment of the crystallite interface with the dielectric. On the other hand, the metal crystallites in solid matrices do not compare favorably with the semiconductor-doped ones in view of the mechanisms that operate there and the difficulty in increasing the metal volume concentration. Pending improvement of the characteristics and fabrication techniques of the semiconductor crystallites, several groups (CULLEN,IRONSIDE,SEATONand STEGEMAN [ 19861, PATELA,JEROMINEK, DELISLEand TREMBLAY [ 19861, AINSLIE,GIRDLESTONE and COTTER[ 19871, ASSANTO,GABEL,SEATON, STEGEMAN, IRONSIDE and CULLEN[ 19871, IRONSIDE,CULLEN,BHUMBRA, BELL, BANYAI,FINLAYSON, SEATONand STEGEMAN[ 19881, ASSANTO, SEATONand STEGEMAN [ 19881, FINLAYSON, BANYAI,WRIGHT,SEATON,
406
NONLINEAR OPTICS IN COMPOSITE MATERIALS
[V
STEGEMAN, CULLENand IRONSIDE [ 19881, JEROMINEK, PIGEON,PATELA, JAKUBCZYK, DELISLEand TREMBLAY [ 19881, JEROMINEK, PATELA, PIGEON, [ 19881, ASSANTO[ 19891, BANYAI, JAKUBCZYK, DELISLEand TREMBLAY FINLAYSON, SEATON,STEGEMAN, O’NEILL,CULLENand IRONSIDE [ 19891, FINLAY SON,BANYAI, SEATON, STEGEMAN, O’NEILL,CULLENand IRONSIDE [ 19891)have moved into designing semiconductor-doped glass waveguides and undertaken efforts to implement nonlinear operations (DANIELZIK, and VON DER LINDE[1985], YUMOTO,FUKUSHIMA and NATTERMANN KUBODERA [ 19871, YUMOTOand USESUGI[ 19881) in these materials that become prototypes for nonlinear optical devices. Thus, techniques such as laser heating (PAVLOPOULOS and CRABTREE[ 1974]), ion-exchange (CULLEN, IRONSIDE,SEATONand STEGEMAN [ 19861, ASSANTO,GABEL,SEATON, STEGEMAN, IRONSIDE and CULLEN[ 19871, IRONSIDE, CULLEN,BHUMBRA, BELL,BANYAI,FINLAYSON, SEATONand STEGEMAN[ 19881, ASSANTO, SEATONand STEGEMAN [ 19881, FINLAYSON, BANYAI,WRIGHT,SEATON, STEGEMAN, CULLENand IRONSIDE [ 19881, ASSANTO[ 19891, BANYAI, FINLAYSON, SEATON,STEGEMAN, O’NEILL,CULLENand IRONSIDE[ 19891, FINLAYSON, BANYAI, SEATON, STEGEMAN, O’NEILL,CULLENand IRONSIDE [ 1989]), sputtering (PATELA,JEROMINEK, DELISLEand TREMBLAY [ 19861, JEROMINEK, PATELA, PIGEON, JAKUBCZYK, DELISLEand TREMBLAY [ 19881, JEROMINEK,PATELA,PIGEON, JAKUBCZYK, DELISLE and TREMBLAY [ 19881) and direct pulling (AINSLIE,GIRDLESTONE and COTTER[ 19871) from semiconductor-doped glass melts have already been used with mixed success, but this effort is highly important for the future evolution of the field. Notwithstanding the several drawbacks of these composite materials at this stage, they still compare favorably with the homogeneous nonlinear materials organics or inorganics as far as the nonlinear optical properties are concerned. In addition, they show properties not shared by the homogeneous materials, such as robustness, adaptability, thermal resistance, and many others, and most importantly their properties can be artificially tailored through the confinement effects to meet many demands.
References ABBATE, G.,U. BERNINI, P. MADDALENA, s.DE NICOLA,P. MORMILE and G. PIERATTINI, 1989, Opt. Commun. 70, 502. ABRAMS, R. L., and R. C. LIND, 1978a, Opt. Lett. 2, 94. ABRAMS, R. L., and R. C. LIND,197813, Opt. Lett. 3, 205. ACIOLI, L. H., A. S.L. GOMESand J. R. RIOSLEITE,1988, Appl. Phys. Lett. 53, 1788.
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AUTHOR INDEX
A ABBATE, G., 388,406 E., 123, 190 ABRAHAMS, ABRAMS, R. L., 389, 397, 406 ACIOLI,L. H., 388, 406 ADAMS,M.C., 229, 234,286 ADAMS,M.J., 14, 30, 62 AGARWAL,G. S., 281,286, 343,407 R. L., 222,292 AGARWAL, AGOSTINELLI, J., 263,286 AGROVSKII, B. S.,95, 190 F., 326,407 AGULLO-LOPEZ, AINSLIE,B. J., 271,286, 405, 406, 407 AITKEN,G. J. M.,307,318 S. A., 210, 211, 222, 247, 251, AKHMANOV, 286,287 KH.G.,111, 114, 117. 168, AKHUNOV, 170-172, 187, 190 E., 123, 124, 127, 132, 137, AKKERMANS, 143-146, 148, 149, 162, 166, 167, 191, 197 V. P., 91, 191 AKSENOV, ALBERSHEIM, W. J., 241,290 ALFANO,R.R.,393,411 R. C., 23,63 ALFERNESS, ALIMOV, V. A,, 72, 191 ALIVISATOS, A. P., 363, 364, 366, 384,407 ALLAIS,M.,331, 332,407 ALLEN,L., 208,287 ALTSHULER, B. L., 123, 191 AHANO,K., 244,290 D., 222,279, 280,287 ANDERSON, P. W., 68, 123, 190, 191 ANDERSON, ANDO,T., 360, 361,410 ANDRUCO,M.J., 124, 148, 161, 162, 192 ANDRIANASOLO, B., 331,407 ANGEL,G., 231,287 ANTONETTI, A., 238,287, 363~366,367, 384, 410 APRESYAN, L. A., 105, 107, 109, 191 G., 235,289 ARJAVALINGAM, 413
ARONOV,A. G., 123,191 ARYA,K., 186, 191 ASHCROFT, W., 345,346,407 ASHLEY,P. R.,371, 407 ASSANTO,G., 405,406,407 AUSTIN,S., 14, 63 AUSTON,D. H., 258,287 H., 229,287 AVRAMOPOULOS, D. D., 393,411 AWSCHALOM, AYERS,G. R., 305,317
B BADO,P., 239,291 BAER,T., 229,290 BAHAR,E., 184, 185, 191 BAIER,G., 307, 309,317,319 BALANT,A. C., 214, 243, 262,289,291,292 BALDWIN, J. E., 295, 31 1.31 7 V. A., 69, 71, 74, 75, 82, 83, 91, BANAKH, 95, 191, 197 BANFI,G. P., 331, 333, 407 L., 366, 369, 376, 378, 319, 384, BANYAI, 386, 387,407,410 W. C., 405,406,407409 BANYAI, I., 280,290,389,397,411 BAR-JOSEPH, Yu.N., 68, 123, 124, 126, BARABANENKOV, 128-130, 132, 134, 137, 140, 143, 144, 191 BARAL,S., 327, 330, 333, 334,411 BARANOV, A. V.,364,407 BARTELT,H., 306,317 BASTING,D., 222,292 M.G., 351, 353-355, 363, 366, BAWENDI, 376, 378, 384,407,410 BECKER.P. C., 238,242,287,289 BECKERS,J. M., 311,313,317 BECKWITH, S. V. W., 307,317 BEIN,T., 327, 330, 334,410 BEKKI,N., 277,292 M. S., 67, 71, 82-84, 95, 191 BELENKII,
414
AUTHOR INDEX
BELENOV, E. M., 270,287 J. W., 307, 317 BELETIC, BELL,J., 405, 406, 409 BERCMANN, G., 123, 189, 191 R., 143, 189, 192 BERKOVITS, BERNINI, U., 388,406 BERRY, C. B., 327,407 M., 393,407 BERTOLOTTI, BESTER,M., 316, 317 BHUMBRA, B. S., 405,406,409 J. L., 186, 191 BIRMAN, BIRNBOIM, M. H., 343,410 J. E., 258,287 BJORKHOLM, J. S., 381, 393,395, 396, 407 BLAKEMORE, M., 343,408 BLOEMER, BLOEMER, M. J., 371,407 BLOUIN, A,, 388,407 BLOW,K. J., 235,212, 278,287 BOBOVlCH, Y.S., 364,407 BOGATOV, A. N., 95, 190, 192 BOGCESS,T. F., 231, 234, 288 BONNEAU, D., 315,318 BOR,Z., 254, 255, 287 BOREL,J. P.,326, 407 BOREL-NARBEL, C., 326,407 BORN,M., 212,248,283,287 BORRELLI, N. F., 333,407 BOSENBERG, W. R., 236,288 B ~ T T C H EC. R ,J., 340,407 A., 331,408 BOUKENTER, E., 217, 258, 273, 275. 276, 279, BOURKOFF, 287,293 BOURLON, P., 315,318 BOUWKAMP, C. J., 211,287 BOWDEN, C. M., 343,408 BOYER,G., 238,287 BRADLEY, D. J., 225,229, 234, 270, 286, 287,292 BRAUN, R.,311,317 BRET,G., 334,383, 388,407 W. F., 280,289 BRINKMAN, BRIT0 CRUZ, C. H., 226, 227, 238, 242,287, 289,291 S. D., 242,287 BRORSON, D. A. G., 341,407 BRUCGEMAN, BRUN,A,, 228,230,238,287,289,292 BRUS.L. E., 327, 330, 333, 334, 348, 351, 353-355, 363, 364, 366, 316, 378, 384,407, 410 BUMFORD, C. R., 327, 329, 407
BUNKIN, A. F., 105, 192 BUNKIN, F. v., 111, 114, 117, 190 BURKE,J. J., 26, 47, 51, 53-55, 63 R. D., 32,46, 63 BURNHAM, BURNS,W. K., 23, 62 BUTKOVSKII, 0.YA., 176, 187, 192
C CALLWA, J. M., 326,407 CANTO,E., 389,407 M., 331, 359-361, 382, 407, 409 CARDONA, CARLSON, N. W., 14.62 CARROLL, P.J., 364,407 CARTER,S. J., 281,288 W. H., 212,293 CARTER, CASE,K. M., 131, 143, 192 CASPERSON, L. W., 208,232,287,290 D., 267,287 CASSASENT, CATHERALL, J. M., 231,232,287,291 CELLI,V., 186, 192, 195, 196 J. P., 238,287 CHAMBARET, CHAMPAGNON, B., 331,407, 408 S., 139, 192 CHANDRASEKHAR, CHANG,W. S. C., 32, 63 D. S., 343, 359, 361, 363, 368, 376, CHEMLA, 382, 384,401,407,410,411 0. S., 280,290 CHEMLA, CHEN,G., 228,230, 288 CHEN,H. H., 212,292 CHEN,Y. M.. 55, 62 CHENG,L. K., 236,288 CHESNOY, J., 23 1, 234,287 CHEUNG,K. P., 258,287 CHIRKIN, A. S., 210, 211,218, 221, 222, 247, 251,268,277, 218,286-288,292 CHO, K.,319,409 CHOA,F. S., 393,411 CHRISTODOULIDES, D. N.,258,287 CHRISTOU, J. C., 301,317 CHRISTOV, I. P., 209-21 1,229, 230, 239, 241-247,249-252,257,259-262, 264, 265, 261-211, 282-285,287,288,291 R.W., 358,409 CHRISTY, CHU, P.L., 216, 217,288 CHUNG,W. H., 327, 330, 333, 334,411 COBB,M. L., 307,317 COHEN,L. G., 255,288 M. M., 315,319 COLAVITA, B., 263,264,289 COLOMBEAU, E., 212,287 COMMINS,
AUTHOR INDEX
G., 326,409 CONSTABARIS, CONWELL, E. M., 23, 62 COOKE,G. H., 310,317 COOPER, J., 21 1, 247,288 CORCUM, P.B., 236,288,291 T. J., 304,317 CORNWELL, COTTER,D., 384, 385, 388,404-406,407 COUTAZ, J. L., 396, 409 K., 406,410 CRABTREE, CRISP, M. D., 209, 262,288 T. J., 405,406,407-409 CULLEN, G., 124, 152, 158-160, 162, 192, CWILICH, I96 D DA SILVA, V. L., 233,288 J. C., 124, 184, 192, 194-196, 295, DAINTY, 305, 317 DAKSS,M. L., 13, 62 DALGOUTTE, D. C., 13,62 DANAILOV, M. B., 239, 244-246, 259, 264, 265, 267, 282-285.287, 288 W. C., 316,317 DANCHI, B., 388, 396, 406,407, 410 DANIELZIK, S., 241, 290 DARLINGTON, DASHEN, R., 74, 192 J., 316, 317 DAVIS, L., 236, 288 DAVIS, DAWBER, P. G., 269,288 DAWSON,M. D., 230, 231, 234,288,289 DAY,C. R., 271,286 DE GIORGIO, v., 331, 333, 407 DE NICOLA,s., 388,406 DE ROUGEMONT, F., 380, 387-389, 394, 407, 410 DE SILVESTRI, S., 226, 286, 288 DE VRIES, P., 168, 194 DE WOLF,D. A., 67, 84, 123, 192, 196 J., 263, 288 DEBOIS, J. M., 32, 63 DELAVAUX. DELIGEORGIEV, T. G., 230,234,291 DELISLE, C., 405, 406, 409, 410 DELONG, K. W., 390, 408 D E M T R ~ D EW., R , 337, 363,408 DENARIEZ-ROBERGE, M. M., 388,393,407, 411 DESEM,C., 276, 277,288 DI BENEDERO,G. P., 31 1,317 E. M., 233, 276, 288 DIANOV, T., 232, 234,290 DICKSON,
415
DIELS,J.-C., 226, 228, 288 DIEMAN, E., 327. 330, 333, 334, 411 DIETEL,W., 225, 226, 228, 288 DOBLER, J., 231, 234,288 A. Z., 128, 151, 192 DOLGINOV, DOPEL,E., 225, 226,288 DORAN,N. J., 272,278,287 DOREMUS, R. H., 358,362, 408 M. C., 236, 238, 242, 244,288, DOWNER, 290 DOYLE,W. J., 358, 362,408 DRABOVICH, K. N., 211,286 DRUMMOND, P. D., 281,288 DUDAREV, S. L., 134, 139, 141, 167, 189, 192, 193 M. A., 244,289 DUGUAY, DUGUE,M., 315, 318 DUKE,C. B., 359-361,408 I. N., 231.292 DULING, DUMMER, R. S., 186, 193 DUPREE,R., 326,408 S., 343,407 D U ~ GUPTA, A DUTTON,T. E., 390.41 1 DUVAL,E., 331,408 F., 331,407 DUVAL, DYAKOV, Y., 251,287
E EBERLY, J. H., 208,239,244,287,288 J., 309,319 EBERSBERGER, ECKERT,J., 307,317 EDELSTEIN, D. C., 236,288 EDREI,I., 124, 143, 147, 148, 160, 166, 192, 194, 196 EESLEY, G. L., 370,411 EFIMOV,A,, 188, 192 EFROS,A. L., 346, 348, 349, 353, 359, 362, 379, 408 EFROS,AL. L., 346, 348, 349, 353, 359, 362, 379,408 EIMERL,D., 236,288 EKIMOV, A. I., 327-329, 333,408 ELIA,F. R., 14, 62 ELIASHBERG, G. M., 325, 338, 346,408 ELLIOT,J. P., 269,288 ENARD,D., 311, 317 ERUKHIMOV, L. U., 72, 191 ETEMAD,S., 124, 148, 161, 162, 192 EVANS,G. A,, 14, 62 EWAN,T., 327,408
416
AUTHOR INDEX
F FAHEY, R.E., 222,292 M. E., 272,289 FALDON, M., 325,408 FARADAY, R., 327, 330, 334, 410 FARLEE, FAWAKHOV, A. M., 218,221, 277, 278,288 M., 31 1,317 FAUCHERRE, FEICK,R.,326, 408 L. C., 327,411 FELDMAN, A., 393,407 FERRARI, FERRIERE, R.,248,289 FINCH,A,, 228, 230,288 FINI,L., 231, 234,287 V. M., 125, 128, 129, 191, 192 FINKELBERG, N., 405,406,407-409 FINLAYSON, R.A,, 335, 371, 385, 408 FISHER, M. A., 184, 185, 191 FITZWATER, D. C., 8, 62 FLANDERS, FLAITE, S. M., 71, 195 FLEISCHMANN, F., 305, 309, 312, 316, 318, 319 B., 363, 366, 367,384,410 FLUEGEL, B. D., 384, 393,410 FLUEGEL, FLYTZANIS, C., 323, 334, 341-344, 351, 352, 351, 360-365, 369-372, 375, 376, 379, 382-392, 394, 397, 399,401,403,407-411 FOCHT,G., 236,288 A,, 327, 330, 333, 334,411 FOJTIK, L. L., 128, 192 FOLDY, A. A., 233,276,288 FOMICHEV, J. J., 226, 228,288 FONTAIN, FORK,R. L., 225,227,228,237-240, 242, 244,253, 254, 286,287-292,381, 387,411 FORTENBERRY, R.,384,390,411 S., 32,63 FOUOUHAR, FOY,F., 315,318 FOY,R.,311, 315,317,318 C. V., 362,409 FRAGSTEIN, M., 238,287 FRANCO, W., 382,383,408 FRANZ, FREEMAN, J. D., 307,317 FRENCH, P. M. W., 229,230,287,289,293 FREUND, I., 124, 143, 148, 160, 166, 192, 194, 196 FREY,R.,380, 387-389, 394,407,410 A. T., 184, 194, 211, 249,289 FRIBERG, U., 125, 128, 192 FRISCH, FROEHLY, C., 263, 264,289 FUJIMOTO, J. G., 226,293, 370.41 I FUKUSHIMA, S., 406,411
FUKUSHIMA, Y., 244,290 T., 10, 62 FUKUZAWA, G
GABEL, A., 384,390,405,406,407,408, 411 GABEL,C., 231, 263,286,292 GAGEL,R.,231,287 P., 388,407 GALARNEAU, G. M., 14,63 GALLATIN, M., 331, 332, 407 GANDAIS, U., 206, 237, 240,289 GANIEL, D. W., 231, 234,288 GARVEY, H. L., 11.63 GARVIN, GAY,J., 315,317 Yu. L., 68, 192 GAZARYAN, U., 105,193 GEHLHEAR, V. I., 88, 119, 193 GELFGAT, GEN,M. Y., 327,408 GENZEL,L., 357, 358, 362, 369,408, 409 GENZEL,R.,311,317 P., 230,238,287,289 GEORGES, GERSTL,S. A. W., 183,193 GHEZ,A.M., 295, 305, 306,311,317, 318 GIANNAKIS, G. B., 306,317 GIBSON,G. N., 236,290 GILLIE,J. K., 363, 408 I. V., 187, 193 GINDLER, J. A., 244,289 GIORDMAINE, H. P., 404-406,407 GIRDLESTONE, GIRES,F., 243, 263,288,289, 334, 383, 388, 407 GLAUNSINGER, W. S., 326,409 J. H., 235,236,238,289 GLOWNIA, GNEDIN, Yu. N., 128, 151, 192 GOBEL,E. O., 395,411 K. S., 95, 193 GOCHELASHVILI, GOEDGEBUER, J. P., 248,289 GOLDBERGER, M. L., 125, 132,193 GOLDSMITH, H., 230,293 A.A., 124,164,166,193 GOLUBENTSEV, GOMES,A. S. L., 233, 235,279,288,289, 388,406 J. W., 210,281,289, 311,319 GOODMAN, J. P., 227, 228, 242, 253, 254, 271, GORDON, 276, 277, 280,286,289-291 P. W., 295, 305,306, 3 11,317, GORHAM, 318 GORI, F., 213,289 L. P., 123, 193, 325, 338, 346,408 GORKOV,
AUTHOR INDEX
E. E., 134, 139, 141, 167, GORODNICHEV, 189, 193 GOUVEIA-NETO, A. S., 233, 235, 272, 279, 289 E. K., 236,288 GRAHAM, G R A N G I EP., R , 228,292 G R A l R I X , E.J., 14, 63 GREEN,B. I., 225,288 GRELLA, F., 213, 289 GREWING, M., 307,318 F., 309, 311, 312, 316,317-319 GRIEGER, GRISCHKOWSKY, D., 214, 238, 243, 258, 262,289,291, 292 V., 396,409 GRIVICKAS, GRUHLKE, R. w., 47, 5 5 , 62 GRUSS, A,, 13, 62 G u , Z.-H., 186, 193 GUAPPI,G., 393,407 GUENTHER, A. H., 389,410 GCJNTHERODT, G., 359-361,409 S. N., 88, 193 GURBATOV, GIJRVICH, A. S., 79, 80, 95, 190, 192, 193 H HAAS,M., 307,318 HACHE,F., 343, 351, 352, 357, 360-362, 364, 365, 369-372, 375, 376, 384, 385, 391, 392, 401,403, 408,409 HAGAN,D. J., 389,407 0. F., 327,408 HAGEMA, HAHN,E. L., 208,291 HALROUT, J. M., 238,289 HALL,D.G., 14.47-49, 51, 53-55, 59, 62, 63 HALL,D. W., 333, 407 HALPERIN, W.P., 325, 326, 329, 331, 332, 334, 338, 341, 343, 371,408 HAMMER, J. M., 14, 62 E., 376, 379, 408 HANAMURA, Y., 8, 62 HANDA, M., 264,289 HANER, HANIFF,C. A,, 295, 311, 317 HANSEN, J. W., 244,289 HARDY, A., 206, 237,240,289 HARRIS, A. L., 363, 364, 366, 384,407 T. D., 364, 407 HARRIS, HARRISON, W. A,, 345, 346,408 D., 239,291 HARTER, HARVEY, G., 263,286
417
HASEGAWA, A., 214, 271, 274, 277, 278, 280,289,292 HATAKOSHI, G., 11,62 HAUG,H., 381, 387,408 HAWS,H. A., 228,235, 242, 287,289 HAUS,J. W., 343, 371,407,408 HAWKINS, R.J., 273-275, 277,293 HAYES,J. M., 363,408 HEGE,E. K., 307, 318 HEIDRICH, P. F., 13, 62 E. J., 357, 358, 362, 408 HEILWEIL, A., 327,330, 333, 334, 355, 366, HENGLEIN, 376, 390,392,408,409,411 HEPPNER, J., 243,290 HERITAGE, J. P., 229, 263, 264, 267, 268, 273-275,277,282,289,290,292,293 HERRMANN, J., 207,222,225, 228,290 HERRON, N., 327, 330, 334, 410,411 HERSCHBACH, D. R., 327,409 J. L., 315,319 HERSHEY, HIGGINS,N. A,, 380,411 HILINSKI, E. F., 363,409 HILLIER,J., 325,411 HINES,B. E., 315,319 HIRLIMAN, C., 237,289 HNILO,A. A,, 240,290 HOCHSTRASSER, R.M., 357,358, 362,408 HOCHSTRASSER, U. W., 269,290 HOCKER,G . B., 23,62 HOFMANN, K.-H., 302, 305-309, 316, 317-319 HOGE,F. E., 105, 193 HOLLAND, H. J., 333,407 HOLTZBERG, F., 359-361,409 HONG,C. S., 10,63 Hu, Y.Z., 366, 369, 376, 378, 379,407 HUANG,K., 359-361,409 HUFFER,W., 222,292 HUGHES,A. E., 325,326,329, 331-333, 338, 341, 343, 371,409 HWI,P. M., 343,411 HULIN,D., 363, 366, 367, 384,410 HUMPHREYS, R., 359-361,409 HUTCHINSON, M. V., 236,290 HUTTER,D. J., 315,319 HUTTER,J., 351,408 HWONG,D. M., 327, 330, 333, 334,411
I IGARASHI, J., 188. 193
418
AUTHOR INDEX
INGUVA, R., 343.408 IPPEN,E. P., 224-226, 237, 239, 240, 290-293 C. N., 405,406,407-409 IRONSIDE, ISHIDA,Y.,234,290 ISHIHARA, H., 379,409 A., 68, 71, 72, 74, 88, 123, 124, ISHIMARU, 132, 134, 137, 140, 143, 144, 146, 193, 194, 196 ISLAM,M. N., 233,235,280,289,290 ITOH,T., 327, 330, 409 IWABUCHI, Y.,327, 330,409
J JACKSON, J. D., 340,409 JACOBOVITZ, J. R., 226,227,291 JAIN, R.K., 229,289,290, 334, 383-385, 389, 390,409 JAIN, S. C., 325, 326, 329, 331-333, 338, 341, 343,371,409 JAISWAL, A. K., 212,290 JAKEMAN, E., 104, 193 Z., 406,409 JAKUBCZYK, JAMES,E. A,, 14, 62 JANATA, E., 366,376,409 JANNSON, T., 243,290 JEDJU,T. M., 363, 366, 376,384,410 R. C., 304, 311, 318 JENNISON, JENSSEN, H. P., 222,292 JEROMINEK, H.,405, 406,409, 410 JOFFRE,M., 363, 366,367, 384,410 JOHN,S . , 123, 124, 132, 134, 147, 161-164, 166, 192, 194, 195 JOHNSON, A. M., 229, 231,232,290 P. B., 358,409 JOHNSON, JOHNSTON, K. J., 315,319 JOSEPH,R. I., 258,287 M. R., 393,411 JUNNARKAR,
K KACHOYAN, B. J., 184, 195 KAFKA,J. D., 229,231,290,292 KAISER, w., 393,410 KALYANIVALLA, N., 343,408 KAMIYA, T., 32, 63 KAMIYA, Y., 327,409 KAPLAN,G. H., 315,319 C., 384, 390.411 KARAGULEFF, KARASIK, A. YA.. 233,276,288 S. S., 79-81, 83,95, 192-194 KASHKAROV,
KASOWSKII, R.,327, 330, 334, 41 I KATAOKA, T., 379,410 KATOKA,M., 327,330,409 KATZIR,A., 10.63 KAVEH,M., 124, 143, 147, 148, 160, 166, 189, 192, 194, 196 KAWABATA, A., 357,362, 369,409 KAYANUMA, Y., 351,409 KELDYSH,L. V., 382, 383,409 KELLER,J. B., 181, 194 KELLY,J. C., 323,343,409 KENNEY-WALLACE, G. A,, 232, 234,290 KHANIN,YA.~..223, 290 D. E., 123, 191, 193 KHMELNITSKII, R. V., 211,286 KHOKHLOV, KHOROSHKOV, J. V., 251,290 KIM, M.-J., 184, 194, 196 KIMOTO,K., 327,409 KINGSLAKE, R.,310,318 KIREEV,S. V., 95, 190 KIRK,J. B., 14,62 E. M.,268,273-275,277,293 KIRSCHNER, KISYL,A. V., 251,290 KLAUDER, J.R., 241,290 KLEIN,M.C., 360-362, 364, 365, 391, 392, 408,409 KLYATSKIN, V. I., 68, 74, 121, 122, 194, 197 KNIGHT,P. L., 281,290 KNOX,K. T., 295,318 KNOX,w. H., 238,242,244,290 KOBAYASHI, T., 244,290 KOCH,S . W., 363, 366, 367, 369, 376, 378, 379, 384, 386, 381,407,410 KOCH,U., 327,330, 333, 334,411 KODAMA, Y.,274,289 KOECHLIN, L., 311,317 KOGELNIK, H., 8, 11, 62. 247,290 KOGELNIK,H. G., 14,22,23,27,29,30, 32, 33, 36, 42, 62, 63 KOLNER,B. H., 268,290 KORESKO,C. D., 307,317 S. K., 23,63 KOROTKY, KOSAI,K., 359-361,410 KOSTERIN, A. G., 89,197 KOVRIGIN, A. I., 211,232,286,290 J., 8, 62 KOYAMA, KRAVTSOV, Yu. A., 67, 69, 71-74, 78, 82, 84, 86-88, 95, 97, 100, 111, 114, 117, 168, 170-172, 176,178,180,183, 185, 187, 188, 190, 192-194, 196, 197
AUTHOR INDEX
KREIBIG, U., 357, 358, 362, 369-372, 375, 408,409 A. 9.. 95.99, 194 KRUPNIK, KUBO,R.,325, 338, 346, 357, 362, 369, 409 K., 390, 391, 406,410,411 KUBODERA, KUBOTA,H., 231, 234,290 KUGA,Y., 68, 123, 194 KUHL,J., 243, 290 KOHLKE,D., 225,226,238,288,290,291 KUHN, L., 8, 13, 62, 63 KUIZENGA, D. J., 224,292 S. R.,295, 305, 306, 31 I , 317, KULKARNI, 318 KULL,M., 388, 389, 394, 396, 409,410 KUMAR,A,, 366, 376,409 W., 327,330, 333, 334,41 I KUNATH, KONDIG,W., 326, 409 KURASHOV, V. N., 251,290 KURHAVA, T., 327,330,409 J. P., 14, 63 KURMER, KUROKAWA, K., 231, 234, 290 KUZKIN,V. M., 172, 190 KWOK,H. S., 393, 396,411
L LABEYRIE, A., 295, 310, 311, 315, 318 LADEBECK, R.,309, 319 LAGENDIJK, A,, 124, 140, 141, 144, 146, 148, 161, 168, 176, 194, 197 LAMBJR, W. E., 202, 262,292 LAMBERT, S. A., 14,63 LANG,R. H., 188, 195 LANNES,A,, 305,318 LAPORTA,P., 226, 286,288,290 A. I., 123, 191, 193 LARKIN, LARSON,R.A,, 327,409 LAUBEREAU, A,, 231,287 LAX,M., 125, 126, 128, 195 LAYBOURN, P. J. R., 11, 63 LEAIRD. D. E., 273-275, 277,293 LEE, Y.C., 272,292 R.F., 381, 387,411 LEHENY, LEINERT, C., 307, 318 LEUNG,K. M., 343,409 LEVINOS, N. J., 363. 364, 366, 384,407 LI, T., 247,290 D. C., 123. 190 LICCARDELLO, LIFSHITZ,I. M., 329, 333, 362, 401,409 LIN,C., 214,243, 255,288,292 LIN,P. L., 393. 411
419
LIN, W. Z., 370,411 LIN,Z., 32, 63 LIND,R.C., 334, 383-385, 389, 390, 397, 406,409 LINDBERG, M., 363,366, 367, 369, 376, 378, 379, 384,407,410 R.H., 326,409 LINDQUIST, LINN,J. W., 229,292 LISAK,M., 222, 279, 280,287 LIVANOS,A. C., 10, 63 LOHMANN, A. W., 295,303,305,306,317,318 LOTSHAW,W. T., 232, 234, 290 LOUOON,R.,28 I , 290 Lou, M. M. T., 258,289,290 LUCAS,M., 327,409 LUCAS,P. A., 363,409 LUCHININ, A. G., 104, 195 J., 384, 389-391, 410 LUKASIK,
M MACASKILL, C., 184, 195 MACDOUGALL, J., 321, 330, 334,410 MACFARLANE, D. L., 208,290 C. D., 295, 311,317 MACKAY, MACKINTOSH, F. C., 124, 132, 134, 147, 161-164, 166, 192, 195 MACOMBER, S. H., 14, 63 MADDALENA, P., 388,406 MAGNI,V., 226, 286,290 MAHAN,G. D., 359-361.408 MAHLER,W., 327, 330, 334,41 I MAINE,P., 239,288, 291 MAKAROV, A. A., 83, 191 MALAKHOV, A. N., 88, 117, 119, 193, 195 P. V., 233,216, 288 MAMISHEV, MANABE, T., 379, 410 MANDEL,L., 212,213,251, 291 MANNEBERG, 0. G., 396,409 MARADUDIN, A. A., 186, 192, 193, 195 MARCUSE, D., 14,29, 32, 47, 49, 54, 55, 63, 216, 254, 271,291 MARET,G., 68, 123, 124, 137, 144, 146, 148, 149, 162, 166, 167, 191, 195, 197 MARINOV, M. S., 218,291 S. W., 326,411 MARSCHALL, A. L., 183, 196 MARSHAK, MARTIN,J. M., 71, 195 MARTIN,T. P., 351, 362, 369,408 0. E., 221,228, 239, 240, 242, MARTINEZ, 243,253-255, 251, 265, 286,289-291
420
AUTHOR INDEX
MARVIN, A. M.. 186, 192 MARX,E., 21 1,247,288 MARZKE.R. F., 326,409 MASHEV, L., 53, 63 MASUMOTO,Y.,379,409 T., 393,411 MATSUMOTO, MATSUOKA, M., 393,411 MAUDER, W., 306-308,317,318 R. D., 326,409 MAURER, J. C., 339,409 MAXWELL-GARNEIT, R., 123, 124, 127, 132, 137, MAYNARD, 143-146, 148, 149, 162, 166, 167, 191, 197 MCCALL,S. L., 208, 291 D. W., 307,317 MCCARTHY, MCGURN,A. R., 186, 192, 193, 195 MCINTYRE, I. A., 236,290 MCMORROW, D., 232,234,290 A. S., 89, 197 MEDOVIKOV, MEHTA,C. L., 212,290 MEIER,F., 325, 326, 329, 331-334, 338, 341, 343, 371,409,410 MEKARNIA, D., 315,317 MENDEZ,E. R., 184, 186, 195 MENG,J., 307,318 MENYUK, C. R., 272,292 MERKLE,F., 311,317 MERLIN,R., 359-361.409 MERMIN,N. D., 345,346,407 MICHAILOV, N. I., 229, 230, 234, 282, 284, 285,288,291 MICHEL,T., 186, 195 MICIC,O., 327, 330, 333, 334, 410 MIE,G., 325, 339, 409 MIEZAK,E., 389, 407 MIGUS,A., 238-240,287,291, 363, 366, 367, 384,410 MILLER,D. A. B., 343, 359, 361, 363, 368, 316, 379, 380,382, 384,401, 407, 409411 MILLER,S. E., 1, 63 R. S., 226, 227, 291 MIRANDA, MIRKAMILOV, D. M., 105, 192 V. L., 67, 69, 71, 74, 75, 82-84, MIRONOV, 91,95, 191, 195 J., 236,238,289 MISEWICH, F. M., 276, 277,291 MITSCHKE, MITSUNAGA, M., 390, 391,410 L. F., 223, 233, 235, 271, 276, MOLLENAUER, 277,279,290,291 MOLLER,K., 327, 330, 334,410 MONOT,R., 326,407
A.. 393,407 MONTENERO, MOORE,D. S., 236,291 J. S.,307,318 MORGAN, MORI,N., 360, 361,410 MORIMOTO,A., 244,290 MORMILE,P., 388, 406 L. M., 327, 330, 334, 410 MORONEY, MORRIS,R. C., 222,292 MORRISS,R. H., 326,411 M o m , J. S., 14,63 MOUROU,G., 231,234,238,239,288,291, 292 MOZURKEWICH, D., 315,319 W., 236,238,292 MUCKENHEIM, S., 379,411 MUKAMEL, MOLLER,M., 317,318 E., 236,238,292 MOLLER-HORSCHE, V. A,, 95, 190 MYAKININ, MYAKININ, V. Z., 95, 192
N NABATOV, A. S., 188, 192 K., 234,290 NAGANUMA, NAIR,S. V., 351,410 S., 327, 330, 333, 334,410,411 NAKAHARA, T., 295, 305, 306, 311,317,318 NAKAJIMA, A., 379,410 NAKAMURA, M., 10, 11,62,63 NAKAMURA, H., 331,411 NAKANO, NAKATSUKA, H., 214,291 NAKAZAWA, M., 231,234,290 S. A., 168, 187, 194 NAMAZOV, V., 389,410 NATHAN, K., 388, 396,406,407,410 NATTERMANN, NAZARATHY, M., 268,290 NEEVES,A. E., 343,410 NEKHAENKO, V. A., 231,232,290,291 NENADOVIC, M. T., 327, 330, 333, 334,410 NEOH,S. K., 327,409 NERI,R., 307,318 T. N., 83, 194 NESTEROVA, G., 295,305, 306, 31 I , 31 7, NEUGEBAUER, 318 NEUMANN, G., 206,237,240,289 G. F., 359-361,410 NEUMARK, NEUROTH,N., 327-329, 363-365, 384, 399, 410 NEW,G. H. C., 223,224,227,229, 231, 232, 270,287,291 NICKEL,D., 238,291
AUTHOR INDEX
M., 184, 195, 196 NIETO-VESPERINAS, H., 8, 14,62, 63 NISHIHARA, NOLL,R. J., 14, 63 NONOYAMA, N., 327,409 NORRIS,T., 234,291 M. J., 305,317 NORTHCOTT, NOZIK,A. J., 327,330, 333,334,410 Nuss, M. C., 393,410 H. M., 182, 195 NUSSENZVEIG,
42 1
PEYGHAMBARIAN, N., 363, 366, 367, 383-388, 393,410 PFUND,A. H., 326,410 PIERATTINI, G., 388,406 PIGEON,M., 406,409 N. T., 335, 371, 385, 411 PILIPETSKY, PITAEVSKII, I. P., 329,409 PODSHIVALOV, A. A., 231,291 POKASOV, V. V., 83,95, 191, 195, 197 POLOVINKIN, A.V., 112, 117, 118, 195 POLUEKTOV, I. A., 270,287 POPOV.E., 53,63 PORTIS,A. M., 326,409 POTTER,B. G., 331, 333,410 POWERS,B. J., 183, 193 PRICE,A. C., 241,290 PRINCE,T. A., 295, 305, 306, 311, 317, 318 PRISE,M. E., 319, 380,409 PROKHOROV, A. M..233,276,288 PUTHOFF,H. E., 202,205,291
0 OBERT,W., 327,408 ODELL,E. W., 222,292 ODONNELL, K. A,, 184, 195 ODWYER,S.L., 14, 63 OKE,J. B., 295, 305, 306, 31 1, 317, 318 OLBRIGHT, G. R., 383-388, 393,410 O’NEILL,M., 406,407,408 ONUSHCHENKO, A. A,, 327-329, 333,408 M. M., 229,230,287,289 OPALINSKA, ORNER,G. C., 252,292 V. E., 183, 184, 197 OSTASHEV, OU, C. H., 55, 63 OUDAR,J. L., 323,408 K.E., 262,291 OUGHSTUN, OZRIN,V. D., 124, 137, 143, 144, 191
R
P PANTELL, R. H., 202, 205,291 PAPAVASSILIOU, G. C., 327,410 PARISE, J. B., 327, 330, 334,410 PATELA,S.,405,406,409,410 PATRUSHEV, G. YA., 83, 195 T. G., 406,410 PAVLOPOULOS, PEARSON, D. B., 258,287 T. J., 295, 311, 318 PEARSON, PEHLEMANN, E., 307,318 PENNINGTON, K. S., 8, 63 A,, 224,291 PENZKOFER, J. A. A. J., 325, 326, 329, PERENBOOM, 331-334, 338, 341, 343, 371,410 PERSHIN, S.M., 231,232,290, 291 PESSOT,M., 239,291 PETERSON, 0. G., 222,292 PETERSON, P. E., 381, 396.410 PETIAU,J., 331, 410 PETROV, A. I., 83, 195 PETROV, V., 208, 233, 234,291 PETROV,Y. I., 327,364,407,408
RACZ,B., 236,254,287,292 RADMORE,P. M., 231,287 RAJH,T., 327, 330, 333, 334,410 RNI, M., 248,289 T. V., 123, 190 RAMAKRISHNAN, READHEAD, A. C. S., 295,311,318 REICHEL,T., 222, 287 T., 314, 316, 319 REINHEIMER, J. T., 392,410 REMILLARD, REMITZ,K. E., 321-329, 399,410 RENNIE,A., 331, 333,407 RENTZEPIS, P. M., 388, 390,411 RHODES,C. K., 236,290 RHODES,W.T., 311,319 RHYS,A., 359-361.409 RICARD,D., 334,342,351, 352, 357, 360-365, 369-312, 315, 316, 319, 382-392, 394, 391, 399,401,403,407-411 RIGHINI,G., 331, 333,407 RIOS LEITE,J. R., 388,406 RODDIER, C., 306,319 RODDIER, F., 295, 306, 301,317, 319 ROGER,G.. 228,292
Q Qu, D. N., 124, 192, 195 QUEL,E. J., 240, 290
422
AUTHOR INDEX
ROGOZKIN, D. B., 134, 139, 141, 167, 189, 193 ROLLAND, C., 236,291 ROSENBLUH, M., 124, 143, 148, 160, 166, 192, 194, 196 Ross, J. K., 183, 196 ROSSETTI,R., 327, 330, 333, 334,410 ROTHBERG, L., 363,366,376, 384,410 J. E., 235,262,289,292 ROTHENBERG, P., 331, 334, 342, 363-365, ROUSSIGNOL, 371, 379, 382-391, 394, 397-399, 407,408 410,411 S. N., 188, 192 RUBTSOV, RUDDOCK, J. S., 225, 292 RUDOLPH, W., 208,225, 226, 228, 233,288, 290-292 RUFFINE, R. S., 123, 196 RUPPIN,R., 357,362, 369, 411 RUSSELL,D. C., 326,41 1 RUSTAGI, K. C., 334,342, 351, 383, 385, 388, 390,410, 411 RYABYKIN, V.V.,176, 185, 187, 192-194 M. I., 167, 189, 193 RYAZANOV, RYBICKI, G. B., 138, 139, 196 RYTOV,S. M., 71, 74, 87, 88, 178, 196 S SAICHEV, A. I., 69, 88, 89, 95, 97, 100, 101, 104, 111-113, 117-119, 183, 185, 188, 193-197 SAITO, H., 395, 41 I SAITO,S., 32, 47, 63 SAKAKI, H., 32,47, 63 SALIN,F., 228, 230, 238,287,289,292 E. E., 72, 196 SALPETER, SALTIEL, S. M., 388, 390, 41 1 SANCHEZ, A,, 222,292 SANDROFF, C. J., 327,330, 333, 334,41 I SANT,A. J., 184,194, 196 SARACHIK, E. S., 244,292 A. I., 307,317 SARGENT, SARGENT 111, M., 202,262,292 SARGENT, W. L. W., 295, 31 1,318 SARID, D., 47, 51, 53-55, 63 SASAKI, Y.,331,411 J., 216, 217,292 SATSUMA, SCARPARO, M. A. F., 226,227,291 SCHAFER, F. P., 205,222, 224, 236, 238,292 D., 305, 307,317,319 SCHERTL, SCHMEISSER, H., 326, 41 1
SCHMELTZER, D., 124, 196 SCHMIDT,H. M., 327,330,333, 334,411 SCHMIDT, R. V., 8, 62 SCHMIDT,S. C., 236,291 W., 224,292 SCHMIDT, SCHMITT-RINK, S., 343, 359, 361, 363, 368, 376, 381, 382, 384, 387,401,408,410,411 SCHOENLEIN, R. W., 370,411 SCHUBERT, D., 232,292 SCHULZ,H. H., 231,234,288 G., 315,318 SCHUMACHER, SCIFRES,D. R., 32.46, 63 Scorn, B. A., 13, 62 M. O., 202,262,292 SCULLY, SEATON,C. T., 26, 63, 379, 380, 384, 390, 405,406,407-409,41 1 SERKIN, V. N., 233,276,288,292 SHABAT, A. B., 216,293 SHAH, J., 381, 387,411 S. B., 307,317 SHAKLAN, SHANK,C. V., 8, 11,62, 224, 225, 237-240, 242-244,287-292, 381,387,411 SHAO,M., 315,319 G. T., 244,292 SHAPPERT, SHELBY, R. M., 281,288 J. B., 10, 63 SHELLAN, SHEN,Y. R., 213,214,292, 335, 341, 343, 344,411 SHERMAN, G. C., 262,291 SHEVERDYAEV, A. D., 188, 192 SHI, L., 393, 411 SHIMIZU,F., 214, 243,292 SHINOJIMA, H., 390, 391,410 V. I., 72,95, 193, 196 SHISHOV, V. V., 335, 371, 385,411 SHKUNOV, SHUM,K., 393,411 W., 228-230,234,286,288 SIBBETT, SIBILIA, C., 393, 407 SIDHU,J. S., 188, 195 A. E., 224,292 SIEGMAN, N. A,, 128, 151, 192 SILANTYEV. SILBERBERG, Y.,389,397,411 C., 183, 193 SIMMER, J. H., 331, 333,410 SIMMONS, SIMON,R. S., 315,319 SIMOS,T., 258,287 SIMPSON,W. M., 229,231,232,290 SINHA, S., 351,410 SJPE,J. E., 47, 63 SIZER11, T., 231, 234,291, 292
AUTHOR INDEX
SLEAT,W., 228, 230,288 SLEIGHT, A. W., 327, 330, 334, 410 SLEZOV, V. V., 329, 333, 362,401,409 SMALL,G. J., 363,408 SMIRL, A. L., 231,234,288 SMIRNOV, A. S., 83, 194 SMITH,D. W., 333,407 S M I T HK., , 279, 291 SMITH,R. G., 244,292 SMITH,S. D., 379, 380, 409 SMITHARD, M. A., 326,408 V. V., 138, 196 SOBOLEV, SOFFER,B. H., 229,292 SOILEAU, M. J., 389,407 A. S. B., 272,289 SOMBRA, SOMEKH, S., 11, 12,63 P. P., 235,236, 238,289 SOROKIN, J. M., 184, 195, 196 SOTO-CRESPO, SPANO,F. C., 379,411 SPEIT, B., 327-329, 399, 410 SQUIER, J., 239,291 STAELIN, D. H., 315,319 U., 207, 232, 233,291,292 STAMM, STEEL, D. G., 392,410 G. I., 26.47, 51, 53-55, 63, 384, STEGEMAN, 390,405,406,407-409, 411 STEIGERWALD, M. L., 351, 353-355, 363, 364,366, 316, 378, 384, 407, 410 M. F., 233,276,288 STELMAKH, M. J., 123, 124, 152, 158-160, 162, STEPHEN, 192, 194, 196 P. C., 325,411 STEVENSON, STIX,M. S., 225,292 STOLEN,R. H., 214,233, 235, 242-244, 271, 290-292 STONE,F. T., 14, 63 STONE,T., 263, 286 STOOKEY, S. D., 326,411 STRAUSS, A. J., 222,292 STREET,R.A,, 395.41 I STREIFER, W., 32, 46, 63 STRICKLAND, D., 238,239,288,292 STROUD,D., 343, 41 I Su, Z. B., 186, 191 SUBER,G., 393, 407 SUCHA,G., 280,290 SUDOL,R. J., 249,289 SUETA,T., 244,290 H., 379,409 SUGAWARA, N., 379,410 SUGIMOTO,
423
SUHARA, T., 8, 14, 62, 63 A. P., 211,247,286,287 SUKHORUKOV, SUNA,A,, 327, 330, 334,411 SURYANARAYANAN, R., 359-361.409 SVELTO,O., 226,286,288 SWIFT,R.N., 105, 193 SZATMARI, S., 236, 238,292
T TAI, K., 277,280.292 TAKAGAHARA, T., 376,378,411 TAM,K. T., 13,62 TAMIR,T., 13, 62 TANAKA, S., 11, 62 TANG,C. L., 14,63, 229, 236,288,293 TANG,D., 363,408 TANG,G. C., 393,41 I TAPPERT, F., 214, 271,289 TATARSKII, V. I., 67, 71-74, 82, 87, 88, 170, 178, 196, 197 TAYLOR, J. R., 229, 230, 233-235, 272, 279, 286-289,293 TELEGIN, L. S., 268,292 TELLE,H., 222,292 231,293 TESCHKE,0.. THIBAULT, G., 393,41 I THOM,C., 315,318 THOMPSON, B. J., 295,318 R.,124, 148, 161, 162, 192 THOMPSON, THURSTON, R. N., 263,264,267, 273-275, 271,289,290,292,293 TOKIZAKI, T., 379,410 TOMITA,A., 280,292 TOMITA,M., 393,411 W. J., 243,263, 264, 267, TOMLINSON, 273-275,277,289,292,293 TOMOV,I. V., 229, 230, 234,241, 242,288, 291 TOPP,M. R., 252,292 P., 243, 263,288, 289 TOURNOIS, TOWNES,C. H., 316,317 TRAN,P., 186, 196 TREACY, E. B.,241,292 R., 405,406,409,410 TREMBLAY, TREVES,D., 206, 237,240, 289 TSAI,T. L., 55, 63 TSANG,L., 123, 124, 134, 137, 140, 143, 144, 146, 193, 194, 196 TSEKHOMSKII, V. A., 327-329,333,408 TUAN,H. S., 55, 63
424
AUTHOR INDEX
TURKEVICH, J., 325,411 E. H., 258,287 TURNER,
U UFIMTSEV, P. YA., 181, 196 USESUGI,N., 406,411 UYEDA,R., 327,409
v VALDMANIS, J. A,, 228, 242,244,290,292 VAMPOUILLE,M., 263,264,289 VAN ALBADA, M. P., 124, 140, 141, 144, 146, 148, 161, 176, 194, 197 V A N DER MARK,M. B., 124, 140, 141, 144, 146, 161, 194, 197 VAN STRYLAND, E. w., 389,407 VAN WONTERGHEM, B., 388,390,41 I VELSKO,S., 236,288 A. G., 67, 71-73, 79, 82, 84, VINOGRADOV, 86,89, 187, 197 VLASOV,D.V., 105, 107, 109. 111, 114, 117, 190-192, 197 VON DER LINDE,D., 238,291, 388, 396, 406, 407,410 VON DER LOHE,o., 303, 307,318,319 VREEKER, R., 168, 194 VYSHILOV,A. S., 188, 192 VYSLOUKH, V. A., 210, 222, 233, 272,287, 292
WEINTRAUB, D. A., 307, 31 7 C., 323, 343,409 WEISBLJCH, WEISS, K., 327, 330, 333, 334, 41 I WELLER,H., 327, 330, 333,334, 366, 316, 409,41 I WELLER-BROPHY, L. A., 47-49, 53, 54, 59, 63 WHERRETT,B. S., 380,411 R. M., 326,411 WILENZICK, B., 208,222, 225, 226, 228, 232, WILHELMI, 233,288,290-292 C. D., 11,63 WILKINSON, F., 327,330, 333, 334,410 WILLIAMS, J. A. R., 229, 230,287,293 WILLIAMS, WILSON,W. L., 363, 366, 376, 384, 410 WIRNITZER, B., 295, 300, 302, 303, 306, 317-319 WISE, F. w., 229,293 WOLF, E., 211, 212, 248, 251, 283, 287, 289, 291,293 WOLF, P. E., 68, 123, 124, 127, 132, 137, 143-146, 148, 149, 162, 166, 167, 191, 195, 197 WOOD, D., 235,278,287 WOOD, V. E., 343,411 WORLOCK,J., 327,411 WRIGHT,E. M., 405,406,408 WYDER,J., 325, 326, 329, 331-334, 338, 341, 343, 371,410 WYDER,P., 326, 409
W WACHMAN, E. S., 236,288 WAGATSUMA, K., 32,47,63 WAI,P. K. A., 272,292 WALLING, J. C., 222,292 WALMSLEY, I. A., 229,293 WANG,Y., 327, 330, 334, 363,409-41 I WARNER, P. J., 295, 311, 317 WARNOCK, J., 393,411 WARREN, W. S., 264,289 J. H. P., 326,411 WATSON, WATSON,K. M., 61, 123, 125, 132, 168, 173, 193, 197 WEGENER, M., 280,290 WEGHORN, H., 307,317 F., 207,225, 228, 233,290, 292 WEIDNER, WEIGELT,G., 295, 300, 302, 303, 305-309, 311, 312,314, 316, 317,317-319 WEINER,A.M., 226, 263, 264, 267, 268, 273-275,277, 282,289,290,292,293
X XU, S., 124, 192
Y YAJIMA,N., 216,217,292 YAJIMA,T., 234, 290 0. I., 188, 192 YAKOVLEV, YAKUSHKIN, 1. G., 71, 74, 197 YAMAMOTO, Y., 32,63 M., 379,409 YAMAZAKI, YANAGAWA, T., 331,411 YANAI,H., 32,63 YANG,C. C., 187, 197 YAO, H.,244,290 YAO, s. s., 384, 390, 411 YAW, A., 10-12, 32, 63, 211, 248, 257, 259, 260,293 YASA,Z. A., 231,293 YATOM,H., 357, 362,369,411
AUTHOR INDEX
YEH,K. C., 100, 187, 197 YEN, H. W., 1 1 , 63 YEN, R., 237,288, 289 YI-YAN, A., I I , 63 YUEN,H., 281,293 YUMOTO, J., 406,411 2
ZAKHAROV, V. E., 216,293 Z A L K I NA., , 236,288 ZAVOROTNYI, V . U . , 74, 170, 183, 184, 197
B. Y., 335, 371, 385, 411 ZELDOVICH, ZHAO,W., 217, 273, 275, 216, 279,293 ZHENG,J. P., 393, 396,411 ZHOU, s.,32,63 ZINKE-ALLMANG, M., 327, 41 I 231, 234,288, 393,410 ZINTH, w., ZISKIN,M., 327,408 R., 326, 411 ZSIGMONDY, ZUEV, V. E.,83, 95, 197 P. F., 131, 143, 192 ZWEIFEL,
425
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SUBJECT INDEX
A amplified spontaneous emission, 235 Anderson localization, 124 angular spectrum approximation, 21 1 Auger effect, 381, 387, 389, 397, 398
diffusion approximation, 142, 146, 158 - coefficient, 143 distributed feedback, I 1 Dyson equation, 125, 126, 129, 149
B backscatter enhancement, 74, 75, 79, 139 benzene, 372 Bethe-Salpeter equation, 125-127, 130 131 Bloch band state, 345 - function, 353 Bohr radius, 348 Born approximation, 86 - distorted wave, 84 Bragg condition, 39 - reflection, 8, 33 reflector, 4, 1 I Brewster angle, 255 Brillouin zone, 345, 350
electron-phonon coupling, 359, 361, 362 electro-optic modulator, 242 ethanol, 372 Euler equation, 218
E
F Faraday rotation, 164 femtosecond optical pulse, 220 Fermat principle, 250, 253 Fermi-Dirac distribution, 370 - level, 346, 347 statistics, 387 Feynman path integral approach, 218 fiber optics, 3, 19 Floquet theorem, 345 Fourier optics, 210, 265, 279 transform, 125, 209, 212,266, 301-303, 305 Franz-Keldysh effect, 376, 382, 383, 401, 404 Fraunhofer approximation, 129 - zone, 125, 127, 130 frequency-doubling, 233, 234 Frahlich coupling, 360 mechanism, 360
C
-
Chandrasekhar’s function, 153, 154 Clausius-Massotti approximation, 339 coherence function, 125, 212, 219 - spatial, 249 temporal, 249 time, 247 colloid, semiconductor, 327 correlation function, 121, 126 cross-spectrally pure, 212 crystallite, metal, 325. 332, 346, 351, 356 - quantum-confined, 399 semiconductor, 327, 328, 331, 332, 353, 356, 359, 362, 363, 384, 405
-
G Gauss-Hermite function, 264,267 glass, metal-doped, 325 porous, 327 silicate, 328 graded-index waveguide, 22 grating formula, 253
-
D diffraction grating, 4 - X-ray, 331 427
428
SUBJECT INDEX
Green function, 53, 70, 76, 85, 87, 88, 93 126, 129, 131, 134-137, 142, 148, 151, 157-159, 164, 173 group velocity dispersion, 21 I, 251 H Helmholtz equation, 30, 119, 126 Helmholtz-Kirchhoff integral, 246 Hermite polynomial, 258 holography, 265 homogeneous broadening, 359 Huang-Rhys parameter, 360 Hubble telescope, 315, 316 hydrosol, 325
I inhomogeneous broadening, 362 integrated optics, 3, 10, 20 intensity fluctuations, 82 interferogram, Michelson, 311 - speckle, 295, 296, 303, 306 interferometer, dielectric multilayer, 24 1 - Mach-Zehnder, 388 Michelson, 268 interferometry, optical long-baseline, 3 1 1, 315 interferometry, speckle, 295 inversion symmetry, 359 ionospheric plasma, 123
K Keller diffraction rays, 181, 182 Kerr coefficient, 335 - effect, 224, 335, 336, 342, 363, 368, 316, 382-385,390,399,404 Kettler effect, 171 Kirchhoff-Fresnel integral, 253, 255 Knox-Thompson method, 295, 304, 307
L Laplace transform, 151, 153, 154 lithography, electron-beam, 6 laser, Ar -ion, 242 - color-center, 231 copper-vapor, 236 Nd : YAG, 23 1, 399 soliton, 233 synchronously pumped mode-locked, 227 +
M magneto-optical effect, 337 material dispersion, 224 Maxwell equations, 39, 148, 164, 202 Maxwell-Garnett theory, 341 Mie's theory, 325, 339 mode-locking, 201 - colliding-pulse, 223 hybrid, 229 passive, 222 modulation instability, 278
N nitrobenzene, 372 nonlinear optics, 201 0 optical fiber, 233, 241, 269, 270, 279 - phase conjugation, 335,371, 388
P parabolic equation, 87 paraxial approximation, 245 partially coherent pulse, 212 phase-contrast method, 281 - matching, 4, 335 screen approximation, 72 photolithography, 6 photon echo, 337 polymer, 332 polystyrene, 123 Poynting vector, 20
Q Q-switching, 201 quantum well, 323, 343
R radiant intensity, 101-104 Raman gain, 277 - scattering, 242, 277 spectroscopy, 334 random medium, 67.69 rate equation approximation, 205, 207 Rayleigh scattering, 149 ray-optics, 250 Rytov's approximation, 74, 82, 87 S saturable absorber, 225
SUBJECT INDEX
Schell-model source, 247 SchrBdinger equation, 202, 279, 338, 350, 382 - nonlinear, 214 self-focusing, 236 - phase modulation, 214 soliton-based communication system, 276 - bright, 216 dark, 216, 271, 272, 274, 275 picosecond, 270 pulse, 231 squeezing in, 279 speckle spectroscopy, 309-31 1, 315 spectral purity, 212 squeezing, 279 Stark effect, static, 383, 403 - shift, dynamic, 401 step-index waveguide, 5, 25, 26 susceptibility, third order, 335, 342, 344, 372 synchronous approximation, 38,41, 51
429
T Thomson scattering, 243 total internal reflection, 119 turbulent medium, 67,90 two-level system, 205 - phonon absorption, 391, 398
V van Cittert-Zernike theorem, 249
W Watson equations, 173, 175 waveguide grating, 5 - nonlinear optical, 29 Wiener-Hopf equations, 152, 158, 159, 165 - method, 138
Z zeolite, 330
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CUMULATIVE INDEX
- VOLUMES I-XXIX
ABELES,F., Methods for Determining Optical Parameters of Thin Films 11, 249 ABELLA,I. D., Echoes at Optical Frequencies VII, 139 ABITBOL, C. I., see J. J. Clair XVI, 71 Dynamical Instabilities and ABRAHAM, N. B., P. MANDEL,L. M. NARDUCCI, Pulsations in Lasers xxv, 1 G. S., Master Equation Methods in Quantum Optics AGARWAL, XI, 1 G. P., Single-longitudinal-mode Semiconductor Lasers AGRAWAL, XXVI, 163 Crystal Optics with Spatial Dispersion AGRANOVICH, V. M., V. L. GINZBURG, IX, 235 ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers IX, 179 E. 0.. Synthesis of Optical Birefringent Networks IX, 123 AMMANN, ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, 21 1 ARNAUD, J. A., Hamiltonian Theory of Beam Mode Propagation XI, 247 BALTES,H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body XIII, 1 in a Nonequilibrium Environment BARABANENKOV, Yu. N., Yu. A. KRAVTSOV, V. D. OZRINand A. I. SAICHEV, XXIX, 65 Enhanced Backscattering in Optics BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images I, 67 BARRETT, H. H., The Radon Transform and its Applications XXI, 217 S., Beam-Foil Spectroscopy XII. 287 BASHKIN, BASSETT,I. M., W. T. WELFORD,R. WINSTON,Nonimaging Optics for Flux Concentration XXVII, 161 BECKMANN, P., Scattering of Light by Rough Surfaces VI, 53 BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns XVIII, 259 BERTOLOTII, M., see D. Mihalache XXVII, 227 BEVERLY 111, R. E., Light Emission from High-Current Surface-Spark Discharges XVI, 357 BJORK,G., see Y. Yamamoto XXVIII, 87 BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements IX, 1 BOWMAN, M. A,, W. A. VANDE GRIND,P. ZUIDEMA, Quantum Fluctuations in Vision XXII, 77 BOUSQUET, P., see P. Rouard IV, 145 BROWN, G . S., see J. A. DeSanto XXIII, 1
43 I
432
CUMULATIVE INDEX
- VOLUMES
I-XXIX
BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation xv, 1 BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 XI, 167 BRYNGDAHL, O., Evanescent Waves in Optical Imaging O., F. WYROWSKI, Digital holography - Computer-generated BRYNGDAHL, holograms XXVIII, 1 11, 73 BURCH,J. M., The Meteorological Applications of Diffraction Gratings XIX, 21 1 BUTTERWECK, H. J., Principles of Optical Data-Processing CAGNAC,B., see E. Giacobino XVII, 85 CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition CEGLIO,N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications CHRISTENSEN, J. L., see W. M. Rosenblum CLAIR,J. J., C. I. ABITBOL,Recent Advances in Phase Profiles Generation CLARRICOATS, P. J. B., Optical Fibre Waveguides - A Review
XVI, 289 XXI, 287 XIII, 69 XVI, 71 XIV, 327
v, 1 COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, 187 COLOMBEAU, B., see C. Froehly XX, 63 COOK,R. J., Quantum Jumps XXVIII, 361 C O U R T ~G., S , P. CRUVELLIER, M. DETAILLE,M. SAYSSE,Some New Optical Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects XX, 1 CREATH,K., Phase-Measurement Interferometry Techniques XXVI, 349 CREWE,A. V., Production of Electron Probes Using a Field Emission Source XI, 223 XXIX, 199 1. P., Generation and Propagation of Ultrashort Optical Pulses CHRISTOV, xx, 1 CRUVELLIER, P., see C. G. Courtes VIII, 133 CUMMINS, H. Z., H. L., SWINNEY, Light Beating Spectroscopy DAINTY, J. C., The Statistics of Speckle Patterns XIV, 1 DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DECKER Jr., J. A., see M. Harwit XII, 101 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 DEMARIA, A. J., Picosecond Laser Pulses IX, 31 DESANTO,J. A., G. S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, 1 DETAILLE, M., see G. Courtbs xx, 1 DEXTER,D. L., see D. Y. Smith X, 165 DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 DUGUAY, M. A., The Ultrafast Optical Kerr Shutter XIV, 161 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons VII, 359 ENGLUND,J. C.. R. R. SNAPP,W. C. SCHIEVE, Fluctuations, Instabilities and XXI, 355 Chaos in the Laser-Driven Nonlinear Ring Cavity XVI, 233 ENNOS,A. E., Speckle Interferometry
CUMULATIVE INDEX
- VOLUMES I-XXIX
433
FANTE,R. L., Wave Propagation in Random Media: A Systems Approach XXII, 341 I, 253 FIORENTINI, A., Dynamic Characteristics of Visual Processes FLYTZANIS, C., F. HACHE,M.C. KLEIN,D. RICARDand PH. ROUSSIGNOL, XXIX, 321 Nonlinear Optics in Composite Materials IV, 1 FOCKE,J., Higher Order Aberration Theory Measurement of the Second Order Degree of CoheFRANCON, M., S. MALLICK, VI, 71 rence FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical SigIX, 311 nals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of PicoXX, 63 second Light Pulses FRY,G. A., The Optical Performance of the Human Eye VIII, 51 I, 109 GABOR,D., Light and Information 111, 187 GAMO,H., Matrix Treatment of Partial Coherence XIII, 169 A. K., see M. S. Sodha GHATAK, GHATAK, A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVIII, 1 XVII, 85 GIACOBINO, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy IX, 235 V. L., see V. M. Agranovich GINZBURG, 11, 109 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media XXIV, 389 GLASER,I., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction IX, 281 Theory of Elastic Waves VIII, 1 GOODMAN, J. W., Synthetic-Aperture Optics XII, 233 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XXIX, 321 HACHE,F., see C. Flytzanis HALL,D. G., Optical Waveguide Diffraction Gratings: Coupling Between Guided XXIX, 1 Modes XX, 263 HARIHARAN, P., Colour Holography XXIV, 103 HARIHARAN, P., Interferometry with Lasers XII, 101 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry X, 289 HELSTROM, C. W., Quantum Detection Theory VI, 171 HERRIOTT,D. R., Some Applications of Lasers to Interferometry HUANG,T. S., Bandwidth Compression of Optical Images x, 1 IMOTO,N., see Y.Yamamoto XXVIII, 87 JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive V, 247 Index JACQUINOT, P., B. R'OIZEN-DOSSIER, Apodisation 111, 29 JAMROZ, W., B. P. STOICHEFF,Generation of Tunable Coherent Vacuum-Ultraviolet Radiation XX, 325 IX, 179 JONES,D. G . C., see L. Allen v, 1 KASTLER,A., see C. Cohen-Tannoudji XXVI, 105 KHOO,I. C., Nonlinear Optics of Liquid Crystals
434
CUMULATIVE INDEX
- VOLUMES I-XXIX
KIELICH, S., Multi-Photon Scattering Molecular Spectroscopy XX, 155 IV, 85 KINOSITA, K., Surface Deterioration of Optical Glasses XXVIII, 87 KITAGAWA, M., see Y. Yamamoto XXIX, 321 KLEIN,M. C., see C. Flytzanis KOPPELMANN, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, 1 KOTTLER,F., The Elements of Radiative Transfer 111, 1 IV, 281 KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoff's Theory KOTTLER, F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory VI, 331 XXVI, 227 KRAVTSOV, Yu. A., Rays and Caustics as Physical Objects Yu. A., see Yu. N. Barabanenkov KRAVTSOV, XXIX, 65 I, 211 KUBOTA,H., Interference Color A,. High-Resolution Techniques in Optical Astronomy LABEYRIE, XIV, 47 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XI, 123 LEE,W.-H., Computer-Generated Holograms: Techniques and Applications XVI, 119 Recent Advances in Holography LEITH,E. N., J. UPATNIEKS, VI, 1 V. S., Laser Selective Photophysics and Photochemistry LETOKHOV, XVI, 1 VIII, 343 LEVI,L., Vision in Communication X-Ray Crystal-Structure Determination as a Branch LIPSON,H., C. A. TAYLOR, of Physical Optics V, 281 LUGIATO, L. A., Theory of Optical Bistability XXI, 69 MACHIDA, M., see Y. Yamamoto XXVIII, 87 MALACARA, D., Optical and Electronic Processing of Medical Images XXII, 1 MALLICK, L., see M. FranGon VI, 71 MANDEL, L., Fluctuations of Light Beams 11, 181 MANDEL, L., The Case for and against Semiclassical Radiation Theory XIII, 27 MANDEL, P., see N. B. Abraham xxv, I MARCHAND, E. W., Gradient Index Lenses XI, 305 MARTIN,P. J., R. P. NETTERFIELD, Optical Films Produced by Ion-Based Techniques XXIII, 113 MASALOV,A. V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation XXII, 145 MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings XXI, 1 XV, 17 MEESSEN,A., see P. Rouard VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting MIHALACHE, D., M. Bertolotti, C. Sibilia, Nonlinear wave propagation in planar structures XXVII, 227 MIKAELIAN, A. L., M. I. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation VII, 231 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction XVII, 279 MILLS,D. L., K. R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids XIX, 43
CUMULATIVE INDEX
- VOLUMES I-XXIX
435
I, 31 MIYAMOTO. K., Wave Optics and Geometrical Optics in Optical Design MOLLOW,B. R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence XIX, 1 K., Instruments for the Measuring of Optical Transfer Functions V, 199 MURATA, VIII, 201 MUSSET,A., A. THELEN, Multilayer Antireflection Coatings L. M., see N. B. Abraham xxv, 1 NARDUCCI, XXIII, 113 NETTERFIELD, R. P., see P. J. Martin XXIV, 1 NISHIHARA, H., T. SUHARA, Micro Fresnel Lenses XXV, 191 OHTSIJ,M., T. TAKO,Coherence in Semiconductor Lasers OKOSHI, T., Projection-Type Holography XV, 139 VII, 299 OOUE,S., The Photographic Image G . V., Yu.I. OSTROVSKY, Holographic Methods in Plasma OSTROVSKAYA, Diagnostics XXII, 197 XXII, 197 Yu.I., see G. V. Ostrovskaya OSTROVSKY, XXIX, 321 OZRIN,V. D., see Yu.N. Barabanenkov K. E., Unstable Resonator Modes OUGHSTUN, XXIV, 165 K. P., The Self-Imaging Phenomenon and its Applications PATORSKI, XXVII, 1 PAUL,H., see W. Brunner xv, 1 PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 PEGIS,R. J., see E. Delano VII, 67 J.,, Photocount Statistics of Radiation Propagating through Random and PERINA Nonlinear Media XVIII, 129 P. S., Non-Linear Optics PERSHAN, V, 83 J., see K. Gniadek PETYKIEWICZ, IX, 281 PICHT,J., The Wave of a Moving Classical Electron V, 351 PORTER,R. P., Generalized Holography with Application to Inverse Scattering and Inverse Source Problems XXVII, 315 PSALTIS,D., see D. Casasent XVI. 289 RAYMER, M. G., I. A. WALMSLEY, The Quantum Coherence Properties of Stimulated Raman Scattering XXVIII, 181 RICARD,D., see C. Flytzanis XXIX, 321 RISEBERG, L. A., M. J. WEBER,Relaxation Phenomena in Rare-Earth Luminescen ce XIV, 89 RISKEN,H., Statistical Properties of Laser Light VIII, 239 RODDIER, F., The Effects of Atmospheric Turbulence in Optical Astronomy XIX, 281 ROIZEN-DOSSIER, B., see P. Jacquinot 111, 29 RONCHI,L., see Wang Shaomin XXV, 279 ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye XIII, 69 ROUSSIGNOL, PH., see c. Flytzanis XXIX, 321 ROTHBERG, L., Dephasing-Induced Coherent Phenomena XXIV, 39 ROUARD,P., P. BOUSQUET,Optical Constants of Thin Films IV, 145
436
CUMULATIVE INDEX
- VOLUMES 1-XXIX
ROUARD, P., A. MEESSEN,Optical Properties of Thin Metal Films RuBlNowIcz, A., The Miyamoto-Wolf Diffraction Wave RUDOLPH, D., see G. Schmahl SAICHEV, A. I., see Yu. N. Barabanenkov SAYSSE, M., see G. Courtes SAiTa, S., see Y. Yamamoto H., see G. A. Vanasse SAKAI, SALEH,B. E. A., see M. C. Teich SCHIEVE, W. C., see J. C. Englund G., D. RUDOLPH,Holographic Diffraction Gratings SCHMAHL, The Mutual Dependence between Coherence SCHUBERT, M., B. WILHELMI, Properties of Light and Nonlinear Optical Processes SCHULZ,G., J. ScHwiDER, Interferometric Testing of Smooth Surfaces SCHULZ, G., Aspheric Surfaces J., see G. Schulz SCHWIDER, SCHWIDER, J., Advanced Evaluation Techniques in Interferometry Tools of Theoretical Quantum Optics SCULLY, M. O., K. G. WHITNEY, SENITZKY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical Framework SIBILIA, C., see D. Mihalache SIPE,J. E., see J. Van Kranendonk SITTIG, E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self-Induced Transparency SMITH,A. W., see J. A. Armstrong SMITH,D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in Insulators SMITH,R. W., The Use of Image Tubes as Shutters SNAPP,R. R., see J. C. Englund SODHA,M. S.,A. K. GHATAK, V. K. TRIPATHI, Self Focusing of Laser Beams in Plasmas and Semiconductors SOROKO,L. M., Axicons and Meso-Optical Imaging Devices STEEL,W. H., Two-Beam Interferometry B. P., see W. Jamroz STOICHEFF, STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE,G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY, K. R., see D. L. Mills SUHARA, T., see H. Nishihara SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams SWEENEY, D. W., see N. M. Ceglio H. H., see H. Z. Cummins SWINNEY, TAKO,T., see M. Ohtsu
xv,
I1 IV, 199 XIV, 195 XXIX, 65
xx,
1
XXVIII, 87 VI, 259 XXVI, 1 XXI, 355 XIV, 195 XVII, 163 XIII, 93 xxv, 349 XIII, 93 XXVIII, 271 X, 89 XVI, 413 XXVII, 227 XV, 245 X, 229 XII, 53 VI, 211 X, 165 x, 45 XXI, 355 XIII, XXVII, V, XX, IX,
169 109 145
325 73
11, I XIX, 43 XXIV, 1
XII, 1 XXI, 287 VIII, 133 XXV, 191
CUMULATIVE INDEX
- VOLUMES I-XXIX
TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANGO,W. J., R. Q.TWISS,Michelson Stellar Interferometry TATARSKII, V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation in a Randomly Inhomogeneous Medium TAYLOR, C. A., see H. Lipson TEICH, M. C., B. E. A. SALEH,Photon Bunching and Antibunching TER-MIKAELIAN, M. L., see A. L. Mikaelian THELEN, A., see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see A. Ghatak TONOMURA, A., Electron Holography TRIPATHI, V. K., see M. S. Sodha TSUIIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering Twiss, R. Q., see W. J. Tango UPATNIEKS, J., see E. N. Leith UPSTILL,C., see M. V. Berry USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids VAMPOUILLE,M., see C. Froehly VANASSE, G. A., H. SAKAI,Fourier Spectroscopy VAN DE GRIND,W. A., see M. A. Bouman VAN HEEL,A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE,Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media VERNIER, P., Photoemission WALMSLEY, I. A., see M. G. Raymer WANG,SHAOMIN, L. RONCHI,Principles and Design of Optical Arrays WEBER,M. J., see L. A. Riseberg WEicELT, G., Triple-correlation Imaging in Optical Astronomy WELFORD, W. T., Aberration Theory of Gratings and Grating Mountings WELFORD,W. T., Aplanatism and Isoplanatism WELFORD,W. T., see I. M. Bassett WILHELMI, B., see M. Schubert WINSTON, R., see I. M. Bassett WITNEY,K. G., see M. 0. Scully WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information WYNNE,C. G., Field Correctors for Astronomical Telescopes WYROWSKI, F., see 0. Bryngdahl YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements Using Laser Light
437
XXIII, 63 XVII, 239 XVIII, 207 V, 287 XXVI, 1 VII, 231 VIII, 201 VII, 169 XVIII, 1 XXIII, 183 XIII, 169 11, 131 XVII, 239
VI, 1 XVIII, 259 XIX, 139 XX, 63 VI, 259 XXII, 77 I, 289 XV, 245 XIV, 245 XXVIII, 181 XXV, 279 XIV, 89 XXIX, 293 IV, 241 XIII, 267 XXVII, 161 XVII, 163 XXVII, 161 X, 89
I, 155 X, 137 XXVIII, 1 XXII, 271
438
CUMULATIVE INDEX
- VOLUMES I-XXIX
YAMAMOTO,Y., S. MACHIDA, S. SAITO,N. IMOTO,T. YAMAGAWA, M. KITAGAWA,G. BJBRK,Quantum Mechanical Limit in Optical Precision XXVIII, 87 Measurement and Communication YAMAJI,K., Design of Zoom Lenses VI, 105 YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference VIII, 295 Microscopy YANAGAWA,T., see Y. Yamamoto XXVIII, 87 H., Recent Developments in Far Infrared Spectroscopic Techniques XI, 77 YOSHINAGA, Yu, F. T. S., Principles of Optical Processing with Partially Coherent Light XXIII, 227 ZAVOROTNYI, V. U., see V. I. Tatarskii XVIII, 207 XXII, 77 ZUIDEMA, P., see M. A. Bouman
