Progress in Optics, Vol. 24

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

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PROGRESS IN OPTICS VOLUME XXIV

EDITORIAL ADVISORY BOARD L. ALLEN,

London, England

M. FRANCON,

Paris, France

F. GORI,

Rome, Italy

E. INGELSTAM,

Stockholm, Sweden

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, F.R.G.

M. MOVSESSIAN,

Armenia, U.S.S.R.

G . SCHULZ,

Berlin, G.D.R .

J. TSUJIUCHI,

Tokyo, Japan

W. T. WELFORD,

London, England

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

EDITED BY

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

Contributors H. NISHIHARA, T. SUHARA, L.ROTHBERG P. HARIHARAN, K. E. OUGHSTUN, I. GLASER

1987

NORTH-HOLLAND AMSTERDAM. OXFORD. NEW YORK .TOKYO

0 ELSEVIER SCIENCE PUBLISHERS B.v., 1987

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical.photocopying, recording or otherwise, without the prior permission of the publisher, Elsevier Science Publishers B. V.(North-Holland Physics Publishing Division), P.O. Box 103. 1000 A C Amsterdam, The Netherland. Special regulations for readers in the U.S.A. :Thispublication has been regktered with the Copyright Clearance Center Inc. (CCC). Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. AN other copyright questions, including photocopying outside of the U.S.A. should be referred to the publisher. LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297 ISBN: 0 444 87050 4

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C O N T E N T S O F V O L U M E 1(1961) I.

THEMODERNDEVELOPMENT OF HAMILTONIAN OPTICS,R. J. PEGIS . . 1-29 WAVE OPTICS AND GEOMETRICAL OPTICS IN OPTICAL DESIGN, K. . . . . . . . . . . . . . . . . . . . . . . . . . . . , MIYAMOTO 3 1-66 AND TOTALILLUMINATION OF ABERRATION111. THEINTENSITY DISTRIBUTION FREEDIFFRACTION IMAGES,R. BARAKAT. . . . . . . . . . . . . . 67- 108 IV. LIGHTAND INFORMATION, D. GABOR . . . . . . . . . . . . . . . . 109- 153 ON BASIC ANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN OPTICAL V. AND ELECTRONIC INFORMATION, H. WOLTER. . . . . . . . . . . . . 155-2 10 COLOR,H. KUBOTA. . . . . . . . . . . . . . . . . 211-251 VI. INTERFERENCE CHARACTERISTICS OF VISUAL PROCESSES, A. FIORENTINI . . . 253-288 VII. DYNAMIC DEVICES,A. c. s. VAN HEEL . . . . . . . . . . 289-329 VIII. MODERNALIGNMENT 11.

C O N T E N T S O F V O L U M E I1 (1963) I. 11.

111. Iv. v. V1.

I. 11. 111.

I. 11.

III. Iv. V.

RULING,TESTINGAND USE OF OPTICAL GRATINGS FOR HIGH-RESOLUTION G. W. STROKE . . . . . . . . . . . . . . . . . . . 1-72 SPECTROSCOPY, THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS,J. M. BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-108 DIFFUSION THROUGHNON-UNIFORM MEDIA,R. G. GIOVANELLI. . . . 109-129 CORRECTION OP OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING, J. TSUJIUCHI . . . . . . . . 131-180 FLUC~UATIONS OF LIGHTBEAMS,L. MANDEL . . . . . . . . . . . . 181-248 METHODSFOR DETERMINING OPTICALPARAMETERS OF THIN FILMS,F. ABELBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249-288

C O N T E N T S O F V O L U M E I11 (1964) THE ELEMENTS OF RADIATIVE TRANSFER, F. KOTTLER . . . . . . . . APODISATION, P. JACQUINOT, B. ROIZEN-DOSSIER. . . . . . . . . . MATRIXTREATMENT~FPARTIALCOHERENCE,H. GAMO . . . . . . .

1-28 29-186 187-332

C O N T E N T S O F V O L U M E I V (1965) HIGHERORDERABERRATION THEORY,J. FOCKE . . . . . . . . . . . APPLICATIONS OF SHEARING INTERFEROMETRY, 0. BRYNGDAHL. . . . SURFACE DETERIORATION OF OPTICAL GLASSES,K. KINOSITA. . . . . OPTICAL CONSTANTS OF THINFILMS,P. ROUARD,P. BOUSQUET. . . . THEMIYAMOTO-WOLF DIFFRACTION WAVE,A. RUBINOWICZ . . . . . .

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

VI.

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

241-280 281-314

C O N T E N T S O F V O L U M E V (1966) I. 11. 111.

OPTICALPUMPING,C. COHEN-TANNOUDJI. %. KASTLER . . . . . . . . NON-LINEAR OPTICS,P. S. PERSHAN . . . . . . . . . . . . . . . . TWO-BEAM INTERFEROMETRY, W. H. STEEL . . . . . . . . . . . . . V

1-81 83-144 145-197

VI

INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFER FUNCTIONS. K. MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE V. LIGHTREFLECTION INDEX.R . JACOBSSON. . . . . . . . . . . . . . . . . . . . . . . DETERMINATION AS A BRANCH OF PHYSICAL VI . X-RAYCRYSTAL-STRUCTURE OPTICS.H . LIPSON.C. A . TAYLOR. . . . . . . . . . . . . . . . . . VII . THEWAVE OF A MOVINGCLASSICAL ELECTRON. J . PICHT . . . . . . . IV .

C O N T E N T S O F VOLUME V I (1967) RECENTADVANCES IN HOLOGRAPHY. E. N . LEITH.J. UPATNIEKS. . . . SCATTERING OF LIGHTBY ROUGHSURFACES. P . BECKMANN . . . . . .

I. I1. OF THE SECONDORDERDEGREEOF COHERENCE. M. I11. MEASUREMENT FRANCON. S. MALLICK . . . . . . . . . . . . . . . . . . . . . . IV. DESIGNOF ZOOMLENSES.K . YAMAJI. . . . . . . . . . . . . . . . OF LASERSTO INTERFEROMETRY. D . R . HERRIOTT. V . SOMEAPPLICATIONS STUDIESOF INTENSITY FLUCTUATIONS IN LASERS.J . A. VI . EXPERIMENTAL ARMSTRONG. A. W. SMITH. . . . . . . . . . . . . . . . . . . . . VII. FOURIERSPECTROSCOPY. G . A. VANASSE. H . SAKAI. . . . . . . . . . AT A BLACKSCREEN. PART11: ELECTROMAGNETIC THEORY. VIII. DIFFRACTION 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

C O N T E N T S OF VOLUME V I I (1969) I.

MULTIPLE-BEAMINTERFERENCEAND NATURAL MODES IN OPEN RESONATORS. G . KOPPELMAN. . . . . . . . . . . . . . . . . . . FILTERS.E. I1. METHODSOF SYNTHESISFOR DIELECTRICMULTILAYER DELANO.R . J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . I11. ECHOESAND OPTICAL FREQUENCIES. I . D . ABELLA. . . . . . . . . . WITH PARTIALLY COHERENT LIGHT. B. J . THOMPSON IV . IMAGEFORMATION THEORYOF LASERRADIATION.A. L. MIKAELIAN. M . L. V . QUASI-CLASSICAL TER-MIKAELIAN. . . . . . . . . . . . . . . . . . . . . . . . . VI . THEPHOTOGRAPHIC IMAGE.S. OOUE . . . . . . . . . . . . . . . . J.H. VII . INTERACTIONOF VERY INTENSELIGHT WITH FREEELECTRONS. EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1-66 67-137 139-168 169-230 231-297 299-358 359-415

C O N T E N T S O F VOLUME VIII (1970) SYNTHETIC-APERTURE OPTICS. J . W. GOODMAN. . . . . . . . . . . I. 1-50 51-131 OF THE HUMAN EYE.G . A. FRY . . . . . I1. THE OPTICALPERFORMANCE H . 2 . CUMMINS. H . L. SWINNEY. . . . 133-200 I11. LIGHTBEATINGSPECTROSCOPY. ANTIREFLECTION COATINGS. A. MUSSET.A. THELEN . . . 20 1-237 IV. MULTILAYER STATISTICAL PROPERTIES OF LASERLIGHT.H . RISKEN . . . . . . . . 239-294 V. THEORYOF SOURCE-SIZE COMPENSATION IN INTERFERENCE VI . COHERENCE 295-341 . . . . . . . . . . . . . . . . . . . . MICROSCOPY. T . YAMAMOTO VII . VISION IN COMMUNICATION. 343-372 H . LEV] . . . . . . . . . . . . . . . . 373-440 VIII. THEORYOF PHOTOELECTRON COUNTING. C. L. MEHTA . . . . . . .

C O N T E N T S OF VOLUME I X (1971) 1.

GAS LASERSAND THEIR APPLICATION TO PRECISE LENGTHMEASUREMENTS. A . L. BLOOM . . . . . . . . . . . . . . . . . . . . . . .

1-30

VII

PICOSECOND LASERPULSES,A. J. DEMARIA . . . . . . . . . . . . . OPTICAL PROPAGATION THROUGHTHE TURBULENT ATMOSPHERE, J. W. STROHBEHN. . . . . . . . . . . . . . . . . . .. . . . . . . IV. SYNTHESIS OF OPTICALBIREFRINGENT NETWORKS, E. 0. AMMANN. . . V. MODELOCKINGIN GAS LASERS,L. ALLEN,D. G. C. JONES . . . . . . v. M. AGRANOVICH, V. L. VI. CRYSTAL OPTICS WITH SPATIAL DISPERSION, GINZBURG. . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. APPLICATIONSOF OPTICAL METHODSIN THE DIFFRACTION THEORYOF ELASTICWAVES,K. GNIADEK, J. PETYKIEWICZ . . . . . . . . . . . VIII. EVALUATION, DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL B. R. FRIEDEN. SIGNALS,BASEDON USE OF THE PROLATEFUNCTIONS, 11.

111.

.

.

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

C O N T E N T S O F VOLUME X (1972) BANDWIDTHCOMPRESSION OF OPTICALIMAGES, T. S. HUANG . . . . . THEUSE OF IMAGETUBESAS SHUTTERS, R.w . SMITH . . . . . . . . TOOLSOF THEORETICAL QUANTUM OPTICS,M. 0. SCULLY, K. G. WHITNEY FIELD CORRECTORS FOR ASTRONOMICAL TELESCOPES, c.G. WY"E . . OPTICALABSORPTION STRENGTH OF DEFECTSIN INSULATORS, D. Y. SMITH,D. L. DEXTER . . . . . . . . . . . . . . . . . . . . . . . LIGHTMODULATION AND DEFLECTION, E. K. SITTIC . . . VI. ELASTOOPTIC DETECTION THEORY, C. W. HELSTROM . . . . . . . . . . VII. QUANTUM I. 11. 111. IV. V.

1-44 45-87 89-135 137- 164

165-228 229-288 289-369

C O N T E N T S O F VOLUME XI (1973) MASTEREQUATION METHODSIN QUANTUM OPTICS,G. S. AGARWAL. . RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . OF LIGHTAND ACOUSTIC SURFACE WAVES,E. G. LEAN. . 111. INTERACTION WAVES IN OPTICAL IMAGING, 0.BRYNGDAHL. . . . . . IV. EVANESCENT PRODUCTION OF ELECTRON PROBESUSINGA FIELDEMISSIONSOURCE, V. A.V. CREWE. . . . . . . . . . . . . . . . . . . . . . . . . . . THEORYOF BEAM MODEPROPAGATION, J. A. ARNAUD . VI. HAMILTONIAN INDEXLENSES,E. W. MARCHAND.. . . . . . . . . . . . VII. GRADIENT I. 11.

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

CONTENTS O F VOLUME XI1 (1974) I. 11. 111.

IV. V. VI.

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

1-51 53-100 101- I62

163-232 233-286 287-344

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

ON THE VALIDITY OF KIRCHHOFF'S LAWOF HEATRADIATION FOR A BODY I N A NONEQUILIBRIUM ENVIRONMENT, H. P. BALTES . . . . . . . . .

1-25

VlII

THE CASE FORAND AGAINSTSEMICLASSICAL RADIATION THEORY,L. MANDEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . MEASUREMENTS OF 111. OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION THE HUMANEYE,W. M. ROSENBLUM, J. L. CHRISTENSEN. . . . . . . IV. INTERFEROMETRI~ TESTING OF SMOOTH SURFACES,G. SCHULZ, J. SCHWIDER. . . . . . . . . . . . . . . . . . . . . . . . . . . . v. SELF FOCUSINGOF LASERBEAMSIN PLASMAS AND SEMICONDUCTORS, M. S. SODHA,A. K. GHATAK,V. K. TRIPATHI. . . . . . . . . . . . AND ISOPLANATISM, W. T. WELFORD . . . . . . . . . . VI. APLANATISM 11.

27-68 69-91 93-167 169-265 267-292

C O N T E N T S O F V O L U M E XIV (1977) OF SPECKLE PAII'ERNS, J. C. DAINTY. . . . . . . . . THE STATISTICS IN OPTICALASTRONOMY, A. LABEYRIE. HIGH-RESOLUTION TECHNIQUES IN RARE-EARTH LUMINESCENCE, L. A. RISE RELAXATION PHENOMENA BERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . OPTICALKERRSHUTTER,M. A. DUGUAY. . . . . . . IV. THEULTRAFAST HOLOGRAPHIC DIFFRACTION GRATINGS,G. SCHMAHL, D. RUDOLPH . . V. P. J. VERNIER . . . . . . . . . . . . . . . . . . . VI. PHOTOEMISSION, VII. OPTICALFIBRE WAVEGUIDES-A REVIEW,P. J. B. CLARRICOATS. . . .

I. 11. 111.

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

C O N T E N T S O F V O L U M E XV ( 1 9 7 7 ) I. 11. 111. IV. V.

THEORYOF OPTICAL PARAMETRIC AMPLIFICATION AND OSCILLATION, w . H. PAUL . . . . . . . . . . . . . . . . . . . . . . . . BRUNNER, OF THINMETALFILMS,P. ROUARD, A. MEESSEN. OPTICALPROPERTIES PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI . . . . . . . . . . . . . QUASI-OPTICAL TECHNIQUES OF RADIO ASTRONOMY, T. W. COLE . . . FOUNDATIONS OF THE MACROSCOPIC ELECTROMAGNETIC THEORYOF D~ELECTRIC MEDIA,J. VAN KRANENDONK, J. E. SIPE . . . . . . . . .

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

C O N T E N T S O F V O L U M E XVI (1978) LASERSELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY, V. S. LETOKHOV 1-69 RECENTADVANCES IN PHASEPROFILESGENERATION, J. J. CLAIR,C. I. . . . . . . 71- 1 17 ABITBOL. . . . . . . . . . . . . . . . . . . . . . HOLOGRAMS: TECHNIQUES AND APPLICATIONS, 111. COMPUTER-GENERATED W.-H. LEE . . . . . . . . . . . . . . . . . , . . . . . . . . . . 119-232 A. E. ENNOS . . . . . . . . . . . . . . 233-288 IV. SPECKLEINTERFEROMETRY, DEFORMATION INVARIANT, SPACE-VARIANT OPTICALRECOGNITION, D. V. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . . . . . 289-356 SURFACE-SPARK DISCHARGES, VI. LIGHT EMISSIONFROMHIGH-CURRENT R. E. BEVERLY 111 . . . . . . . . . . . . . . . . . . . . . . . . . 357-411 RADIATIONTHEORYWITHINA QUANTUM-MECHANICAL VII. SEMICLASSICAL FRAMEWORK, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . . 413-448 I. 11.

.

IX

CONTENTS O F VOLUME XVII (1980) I. 11. 111.

IV. V.

HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. DANDLIKER . . . . DOPPLER-FREE MULTIPHOTON SPECTROSCOPY, E. GIACOBINO, B. CAGNAC THEMUTUALDEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR OPTICAL PROCESSES, M. SCHUBERT, B. WILHELMI . . MICHIELSONSTELLAR INTERFEROMETRY, W. J. TANGO,R. Q.TWISS . . SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION,A. L. MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . .

.

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

CONTENTS O F VOLUME XVIII (1980) GRADEDINDEXOPTICALWAVEGUIDES:A REVIEW, A. GHATAK,K. THYAGARAJAN . . . . . . . . . . . . . . . . . . . . . . . . . . 1-126 11. PHOTOCOUNTSTATISTICSOF RADIATIONPROPAGATINGTHROUGH RANDOMAND NONLINEAR MEDIA,J. PERINA . . . . . . . . . . . . 127-203 111. STRONG FLUCTUATIONS IN LIGHTPROPAGATION IN A RANDOMLY INHOMOGENEOUS MEDIUM,V. I. TATARSKII, V. U. Z A V O R O ~ Y . I. . . . . . . 204-256 OPTICS:MORPHOLOGIESOF CAUSTICSAND THEIR DIFIV. CATASTROPHE FRACTION PAlTERNS, M. v. BERRY, c. UPSTILL . . . . . . . . . . . . 257-346 I.

CONTENTS O F VOLUME XIX (1981) I. 11.

111.

IV. V.

THEORY OF INTENSITY DEPENDENT RESONANCE LIGHTSCATTERING AND RESONANCEFLUORESCENCE, B. R. MOLLOW . . . . . . . . . . . . . 1-43 SURFACE AND SIZE EFFECTSON THE LIGHT SCAlTERING SPECTRA OF SOLIDS,D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . . . 45-137 LIGHT SCATTERING SPECTROSCOPY OF SURFACE ELECTROMAGNETIC WAVESIN SOLIDS,S. USHIODA. . . . . . . . . . . . . . . , . . . 139-210 PRINCIPLES OF OPTICAL DATA-PROCESSING, H. J. BUTTERWECK . . . . 21 1-280 THEEFFECTSOF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY, F. RODDIER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1-376

CONTENTS OF VOLUME XX (1983) I.

ULTRA-VIOLET BIDIMENSIONAL DETECASTRONOMICAL OBJECTS, G. COURTBS, P. CRWELLIER, M. DETAILLE, M. SA~SSE. . . . . . . . . . . . . . . . . . . . . . . 1-62 SHAPINGAND ANALYSIS OF PICOSECOND LIGHTPULSES, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . . . . . . . . . . . , 63-154 MULTI-PHOTON SCATTERING MOLECULAR SPECTROSCOPY, S. KIELICH . 155-262 COLOURHOLOGRAPHY, P. HARIHARAN. . . . . . . . . . . . . . 263-324 GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION, W. JAMROZ.B. P. STOICHEFF. . . . . . . . . . . . . . . . . . . 325-380 SOME NEWOPTICAL DESIGNS FOR

TION OF

11. 111.

IV. V.

. .

X

C O N T E N T S O F V O L U M E X X I ( 1984) I. 11. 111.

IV. V.

RIGOROUS VECTOR THEORIES OF DIFFRACTION GRATINGS, D. MAYSTRE. 1-68 69-216 THEORYOF OPTICALBISTABILITY, L. A. LUGIATO. . . . . . . . . . . AND ITS APPLICATIONS, H. H. BARRETT . . . 217-286 THERADONTRANSFORM ZONEPLATECODEDIMAGING: THEORY AND APPLICATIONS, N. M. CEGLIO, D. W. SWEENEY . . . . . . . . . . . . . . . . . . . . . . . . . . 287-354 FLUCTUATIONS, INSTABILITIES AND CHAOSIN THE LASER-DRIVEN NONLINEAR RINGCAVITY, J. c. ENGLUND, R. R. SNAPP, c. SCHIEVE . . . 355-428

w.

C O N T E N T S O F VOLUME X X I I (1985) OPTICAL AND ELECTRONIC PROCESSINGOF MEDICAL IMAGES, D. MALACARA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-76 QUANTUM FLUCTUATIONS IN VISION,M. A. BOWMAN, W. A. VAN DE GRIND, 11. P. ZUIDEMA. . . . . . . . . . . . . . . . . . . . . . . . . . . 77-144 OF BROAD-BANDLASER 111. SPECTRAL AND TEMPORALFLUCTUATIONS RADIATION, A. V. MASALOV. . . . . . . . . . . . . . . . . . . . 145-196 G. V. OSTROVSKAYA, METHODSOF PLASMADIAGNOSTICS, IV. HOLOGRAPHIC Yu. I. OSTROVSKY . . . . . . . . . . . . . . . . . . . . . . . . . 197-270 FRINGEFORMATIONS IN DEFORMATION AND VIBRATIONMEASUREMENTS V. USING LASERLIGHT,I. YAMAGUCHI . . . . . . . . . . . . . . . . 271-340 IN RANDOMMEDIA:A SYSTEMS APPROACH,R. L. VI. WAVEPROPAGATION 341-398 FANTE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.

C O N T E N T S O F VOLUME X X I I I (1986) I. 11.

111.

IV. V.

ANALYTICAL TECHNIQUESFOR MULTIPLESCATTERING FROM ROUGH SURFACES, J. A. DESANTO,G. S. BROWN. . . . . . . . . . . . . . . 1-62 PARAXIAL THEORY IN OPTICAL DESIGNIN TERMSOF GAUSSIAN BRACKETS, K. TANAKA . . . . . . . . . . . . . . . . . . . . . . . . . . . 63-1 12 OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES, P. J. MARTIN,R. P. NETTERFIELD . . . . . . . . . . . . . . . . . . . . . . . . . . . 113-182 A. TONOMURA . . . . . . . . . . . . . . 183-220 ELECTRON HOLOGRAPHY, PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT, F. T. S. Y U . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221-276

PREFACE In this volume reviews of current developments in a number of areas of modem optical physics and optical engineering are presented. In the first article the general theory of Fresnel lenses is summarized and techniques for the fabrication of micro-Fresnel lenses are described. Micro-Fresnel lenses are finding useful applications, for example, in laser-disk players and in optical communications systems. In the second article relatively recent developments regarding the generation of coherence by dephasing mechanisms in nonlinear mixing are described. This phenomenon is essentially quantum-mechanical and its elucidation has provided new insights into nonlinear processes in the presence of damping. In the article that follows a number of new interferometric techniques that utilize laser light are discussed. The high intensity and the high degree of spatial and temporal coherence of laser light has made it possible to overcome many limitations of traditional interferometry with thermal sources. The new developments have resulted in a remarkable increase in the precision, range and speed with which interferometric measurements can be performed. In the fourth article a thorough review is presented of investigations concerning the diffractive formation of unstable resonator modes. This subject is of importance in connection with a number of applications that utilize laser light. The article deals mainly with theoretical aspects, although some experimental results are quoted. The concluding article describes information processing with spatially incoherent light. It shows how complicated mathematical operations can be rapidly performed using simple optical systems. The underlying physical principles and some applications are discussed.

EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, NY 14627, USA May 1987

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CONTENTS I. MICRO FRESNEL LENSES by H . NISHIHARA and T . SUHARA(OSAKA.JAPAN)

1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. PRINCIPLES OF FRESNEL LENSES . . . . . . . . . . . . . . . . . . . . . 2.1 Phase shift function . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Diffraction efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. FABRICATION TECHNIQUES OF MICROFRESNEL LENSES . . . . . . . . . . 3.1 Photoreduction method . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Interference method . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Optical duplication . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Electron-beam writing . . . . . . . . . . . . . . . . . . . . . . . . . 4. FRESNEL LENSESFABRICATED BY ELECTRON-BEAM LITHOGRAPHY . . . . . . 4.1 Electron-beam writing system . . . . . . . . . . . . . . . . . . . . . 4.2 Blazing technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Focusing spot . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Focal length . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Diffraction efficiency . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Wavefront aberration . . . . . . . . . . . . . . . . . . . . . . 4.5 Lens array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Elliptical lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. WAVEGUIDE FRESNELLENSES. . . . . . . . . . . . . . . . . . . . . . . 5.1 Waveguide lenses: requirements and problems . . . . . . . . . . . . . 5.2 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fabrications and results . . . . . . . . . . . . . . . . . . . . . . . . 6. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I1. DEPHASING-INDUCED COHERENT PHENOMENA by L. ROTHBERG (MURRAYHILL. NJ. USA) 1. INTRODUCTION TO DEPHASING-INDUCED COHERENCE 1.1 Description and importance of the phenomenon .

1.2 Historical evolution of the problem 1.3 Overview of the article . . . . . 2. THEORETICAL TREATMENT. . . . .

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2.1 Conventional description of coherent nonlinear optical mixing . . . . . . . 2.1.1 Framework of the traditional theory . . . . . . . . . . . . . . . . 2.1.2 Calculation of nonlinear susceptibilities with damping . . . . . . . . . 2.1.3 Other features of multiresonant nonlinear optical mixing . . . . . . . 2.2 Diagrammatic picture of nonlinear optical processes . . . . . . . . . . . . 2.2.1 Double sided diagrams . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Dephasing-induced coherent behavior . . . . . . . . . . . . . . . 2.3 Relationship of pressure-induced coherent four-wave mixing to collisional redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dressed atom model of collision-induced coherence . . . . . . . . . 2.4 Field-induced resonances . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Fluctuation-induced extra resonances . . . . . . . . . . . . . . . . 2.4.2 Higher-order power and dephasing-induced resonances . . . . . . . . 3 . EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Classification of observed dephasing-induced coherent processes . . . . . . 3.2 Extra resonances between "unpopulated" excited states . . . . . . . . . . 3.2.1 Collision-induced resonances between 2P fine-structure components in Na vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Thermally induced excited state coherent Raman spectroscopy of molecular crystals . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Pressure-induced Hanle resonances between Zeeman sublevels of an excited state . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Collision-induced population grating resonances . . . . . . . . . . . . . . 3.3.1 Grating picture of four-wave mixing and the role of dephasing . . . . . 3.3.2 Characterization of pressure-induced population grating resonances in Na vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Collision-induced coherent Raman resonances between equally populated states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Hyperfine and Zeeman coherences in the ground state of sodium vapor 3.4.2 Collision-induced Hanle resonances in the ground state of sodium vapor 3.4.3 Collision-induced four-wave mixing lineshapes and velocity changing collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. SUMMARY AND FUTUREPROSPECTS. . . . . . . . . . . . . . . . . . . . 4.1 Effects of damping on coherent nonlinear optics . . . . . . . . . . . . . 4.2 Studies of dephasing mechanisms . . . . . . . . . . . . . . . . . . . . 4.3 Novel spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 47 52 54 54 56 58 59 63 64 65 67 67 67 68 70 77 80 80 82 86 87 90

92 94 94 96 97 98 99

111. INTERFEROMETRY WITH LASERS

by P. HARIHARAN (SYDNEY.AUSTRALIA) 1. INTRODUCTION.

1.1 Laser sources 1.2 Laser modes . 1.3 Laser linewidth

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1.4 Frequency stabilization . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Problems with laser sources . . . . . . . . . . . . . . . . . . . . . . 2. MEASUREMENTS OF LENGTH . . . . . . . . . . . . . . . . . . . . . . . 2.1 Definition of the metre . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Measurements of length . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Fringe counting . . . . . . . . . . . . . . . . . . . . . . . . . . . OF CHANGES IN OPTICAL PATH LENGTH . . . . . . . . . . 3. MEASUREMENTS 3.1 Closed-loop feedback systems . . . . . . . . . . . . . . . . . . . . . 3.2 Heterodyne methods . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Thermal expansion . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Michelson-Morley experiment . . . . . . . . . . . . . . . . . 3.3 Techniques using tunable lasers . . . . . . . . . . . . . . . . . . . . 3.3.1 Two-wavelength interferometry . . . . . . . . . . . . . . . . . . 3.3.2 Frequency-modulation interferometry . . . . . . . . . . . . . . . . 4. DETECTION OF GRAVITATIONAL WAVES . . . . . . . . . . . . . . . . . . . 4.1 Prototype interferometric detectors . . . . . . . . . . . . . . . . . . . 4.2 Methods of obtaining increased sensitivity . . . . . . . . . . . . . . . . 5. LASERDOPPLERINTERFEROMETRY . . . . . . . . . . . . . . . . . . . . . 5.1 Measurement of surface velocities . . . . . . . . . . . . . . . . . . . . 5.2 Measurements of vibrations . . . . . . . . . . . . . . . . . . . . . . 6. LASER-FEEDBACK INTERFEROMETERS. . . . . . . . . . . . . . . . . . . 7. OPTICALTESTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Unequal-path interferometers . . . . . . . . . . . . . . . . . . . . . . 7.2 Tests on ground surfaces . . . . . . . . . . . . . . . . . . . . . . . 7.3 Electronic measurements of optical path differences . . . . . . . . . . . . 7.3.1 Heterodyne techniques . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Quasi-heterodyne techniques . . . . . . . . . . . . . . . . . . . 7.3.3 Phase-stepping methods . . . . . . . . . . . . . . . . . . . . . 7.3.4 Residual errors . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SPATIALINTERFEROMETRY . . . . . . . . . . . . . . . . . . 8. HETERODYNE 8.1 Infrared heterodyne detection . . . . . . . . . . . . . . . . . . . . . 8.2 Infrared heterodyne stellar interferometry . . . . . . . . . . . . . . . . 8.3 Large infrared heterodyne stellar interferometer . . . . . . . . . . . . . . 9. INTERFEROMETRIC SENSORS . . . . . . . . . . . . . . . . . . . . . . . 9.1 Interferometric rotation sensors . . . . . . . . . . . . . . . . . . . . 9.1.1 Ring-laser rotation sensors . . . . . . . . . . . . . . . . . . . . 9.1.2 Passive interferometric rotation sensors . . . . . . . . . . . . . . . 9.1.3 Limits of sensitivity . . . . . . . . . . . . . . . . . . . . . . . 9.2 Fibre-optic interferometric sensors . . . . . . . . . . . . . . . . . . . 9.2.1 Rotation sensing . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Generalized fibre-interferometric sensors . . . . . . . . . . . . . . 9.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 10. PULSED-LASER AND NONLINEAR INTERFEROMETERS . . . . . . . . . . . . 10.1 Interferometry with pulsed lasers . . . . . . . . . . . . . . . . . . . 10.2 Two-wavelength interferometry . . . . . . . . . . . . . . . . . . . . 10.3 Second-harmonic interferometers . . . . . . . . . . . . . . . . . . . 10.3.1 Second-harmonic interferometers using critical phase-matching . . . . 10.4 Phase-conjugate interferometers . . . . . . . . . . . . . . . . . . . .

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114 114 115 116 117 117 117 118 118 119 120 121 123 125 127 127 129 129 129 130 130 131 131 132 133 134 136 138 138 138 139 140 140 140 142 143 144 144 144 144 147 148

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10.5 Interferometers with phase-conjugating mirrors . . . . . . . . . . . . . 10.6 Photorefractive oscillators . . . . . . . . . . . . . . . . . . . . . . 11. INTERFEROMETRIC MEASUREMENTS ON LASERS . . . . . . . . . . . . . . 11.1 Analysis of spatial coherence and wavefront aberrations . . . . . . . . . 11.2 Measurements of spectral linewidth . . . . . . . . . . . . . . . . . . 11.3 Heterodyne methods of frequency measurement . . . . . . . . . . . . . 11.4 Laser wavelength meters . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Dynamic wavelength meters . . . . . . . . . . . . . . . . . . . 11.4.2 Static wavelength meters . . . . . . . . . . . . . . . . . . . . 12. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

150 151 152 152 153 155 156 156 158 158 159 159

IV. UNSTABLE RESONATOR MODES by K . E. OUGHSTUN(MADISON.WI. USA)

167 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 2. GENERAL FORMULATION OF THE TRANSVERSE MODESTRUCTURE PROPERTIES . 170 2.1 Paraxial scalar wave propagation phenomena in open optical cavities . . . . 2.2 Canonical formulation of unstable cavity modes . . . . . . . . . . . . . . 180 181 2.2.1 Geometrical properties . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Diffractive properties . . . . . . . . . . . . . . . . . . . . . . . 184 2.2.3 Collimated and equivalent Fresnel numbers . . . . . . . . . . . . . 188 2.3 Transverse mode orthogonality in open cavities . . . . . . . . . . . . . . 190 2.3.1 Transverse mode orthogonality in optical cavities with a single diffracting 191 aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Transverse mode orthogonality and reciprocity in multi-aperture optical 196 cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Questions of existence and completeness of the transverse modes in open optical 201 cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 On the existence of the eigenvalues . . . . . . . . . . . . . . . . 202 2.4.2 Schmidt expansion of the cavity modes . . . . . . . . . . . . . . . 206 2.4.3 Nonstationary modes and the question of completeness . . . . . . . . 213 2.5 Polarization eigenstates and the vector modes of an optical cavity . . . . . . 221 2.5.1 Jones calculus and the polarization eigenstates . . . . . . . . . . . . 222 2.5.2 Vector modes of an optical cavity . . . . . . . . . . . . . . . . . 225 2.6 Standing-wave interference and the resonance condition . . . . . . . . . . 228 2.7 Spatial coherence of the transverse mode structure . . . . . . . . . . . . 230 2.7.1 Coherent mode representation . . . . . . . . . . . . . . . . . . . 231 233 2.7.2 Second-order coherence of the stationary field in an optical cavity . . . 240 3. PASSIVECAVITYMODESTRUCTURE BEHAVIOR . . . . . . . . . . . . . . . 3.1 Asymptotic behavior and the geometrical approximation . . . . . . . . . . 243 250 3.1.1 Asymptotic approximation of the rectangular cavity eigenvalue equation . 3.1.2 Asymptotic approximation of the cylindrical cavity eigenvalue equation . 257 260 3.2 Eigenvalue behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 269 3.3 Transverse mode structure . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Equivalent Fresnel number and magnification dependence . . . . . . . 270 3.3.2 Transverse mode hierarchy supported by an unstable cavity . . . . . . 276

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3.4 Aperture apodization and intracavity spatial filtering . . . . . . . . . . . 3.4.1 Aperture apodization in rectangular unstable cavities . . . . . . . . . 3.4.2 Intracavity spatial filtering in unstable ring resonators . . . . . . . . 3.5 Geometry-dependent properties . . . . . . . . . . . . . . . . . . . . . 3.5.1 Off-axis cavity geometry . . . . . . . . . . . . . . . . . . . . . 3.5.2 Elliptical aperture cavity . . . . . . . . . . . . . . . . . . . . . 3.5.3 Outcoupling mirror effects . . . . . . . . . . . . . . . . . . . . 3.5.4 Exotic cavity geometries . . . . . . . . . . . . . . . . . . . . . BEHAVIOR. . . . . . . . . . . . . . . . 4. ACTIVECAVITYMODE STRUCTURE 4.1 Equation of state of the active cavity mode structure . . . . . . . . . . . 4.1.1 Longitudinal mode expansion . . . . . . . . . . . . . . . . . . . 4.1.2 Outcoupled power and the cavity Q factor . . . . . . . . . . . . . 4.2 Saturable gain medium effects . . . . . . . . . . . . . . . . . . . . . 4.2.1 Laser amplifier gain and saturation . . . . . . . . . . . . . . . . . 4.2.2 Transverse mode structure behavior . . . . . . . . . . . . . . . . 5 . CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX . NUMERICAL TECHNIQUES AND SAMPLING CRITERIA . . . . . . . . . A.l Scalar wave propagation methods . . . . . . . . . . . . . . . . . . . A.l.l Cartesian Goordinate solution . . . . . . . . . . . . . . . . . . . A.1.2 Polar cylindrical coordinate solution . . . . . . . . . . . . . . . . A.2 Spherical wave coordinate transformation . . . . . . . . . . . . . . . . A.3 Numerical sampling criteria . . . . . . . . . . . . . . . . . . . . . . A.3.1 Guard band requirement . . . . . . . . . . . . . . . . . . . . A.3.2 Sampling interval requirement in Cartesian coordinates . . . . . . . . A.3.3 Fresnel zone requirement in Cartesian coordinates . . . . . . . . . . A.3.4 Sampling interval and Fresnel zone requirements in polar cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Thin-sheet gain-phase approximation . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

287 288 300 320 321 325 326 328 330 330 333 334 336 337 343 354 355 355 356 356 358 364 366 366 367 369 371 313 378

V . INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT by I . GLASER (REHOVOTH.ISRAEL) 1. INTRODUCTION.

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1.1 Coherent versus incoherent light for information processing . . . . . . . 1.1.1 Advantages and disadvantages of incoherent optical processing . . . 1.2 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Shift invariant linear systems . . . . . . . . . . . . . . . . . . 1.2.2 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Radon space . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING . . . . . . 2.1 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Correlations and convolutions . . . . . . . . . . . . . . . . . . 2.1.2 Time integrating scanners . . . . . . . . . . . . . . . . . . . . . 2.1.3 Radon space processors . . . . . . . . . . . . . . . . . . . . . 2.1.4 Sequential processors for general linear transformations . . . . . . .

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XVIll

CONTENTS

2.2 Shadow casting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Performance of shadow casting processors . . . . . . . . . . . . . 2.3 The lenslet array processor . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Special configurations . . . . . . . . . . . . . . . . . . . . . . 2.4 Spectral dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Wavelength spectrum correlators . . . . . . . . . . . . . . . . . 2.4.2 Dispersive processing of spatial information . . . . . . . . . . . . . 2.5 OTF synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Geometrical-optics OTF synthesis . . . . . . . . . . . . . . . . . 2.5.2 Diffractive OTF synthesis . . . . . . . . . . . . . . . . . . . . . 2.5.3 Holographic incoherent OTF synthesis . . . . . . . . . . . . . . . 2.6 Interferometric methods . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Direct, parallel, vector-matrix multiplication . . . . . . . . . . . . . . . 3. BIPOLARAND COMPLEX-VALUED SPATIALSIGNALS . . . . . . . . . . . . . 3.1 Multiple channel systems . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Optimal choice of the component spatial signals . . . . . . . . . . . 3.2 Temporal encoding . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Spatial encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Space segmentation . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Spatial carrier . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Polychromatic encoding . . . . . . . . . . . . . . . . . . . . . . . . OF INCOHERENT PROCESSING SYSTEMS . . . . . . . . . . . . 4. APPLICATIONS 4.1 Image pattern recognition . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Statistical pattern recognition . . . . . . . . . . . . . . . . . . . 4.1.2 Geometrical correlation methods . . . . . . . . . . . . . . . . . . 4.1.3 Non-correlation methods . . . . . . . . . . . . . . . . . . . . . 4.1.4 Associative memories via neural networks . . . . . . . . . . . . . . 4.2 Coded aperture imaging . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Optical tomographic reconstruction . . . . . . . . . . . . . . . . . . . 4.4 Incoherent Fourier transforms . . . . . . . . . . . . . . . . . . . . . 4.4.1 Computation with a vector-matrix multiplier . . . . . . . . . . . . 4.4.2 The Chirp-Z algorithm . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Fourier transform via the Radon transform . . . . . . . . . . . . . 4.4.4 Interferometric Fourier transform . . . . . . . . . . . . . . . . . 4.5 Digital optical processing . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Binary multiplication via analog convolution . . . . . . . . . . . . . 4.5.2 Logical operations using linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

420 422 423 427 432 434 435 436 438 439 441 445 458 461 463 463 465 468 470 471 473 477 477 479 479 481 488 490 491 494 494 494 495 496 497 497 498 500 502 503 503

AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511 523 527

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX.VOLUMES I-XXIV . . . . . . . . . . . . . . . . .

. .

E. WOLF, PROGRESS IN OPTICS XXIV 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1987

I

MICRO FRESNEL LENSES BY

H. NISHIHARAand T. SUHARA Dpt. of Electronics, Faculty of Engineering, Osaka University 2-1 Yamada-Oka, Suita, Osaka,565 Japan

CONTENTS PAGE

Q 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . .

3

Q 2 . PRINCIPLES OF FRESNEL LENSES

5

. . . . . . . . . . .

Q 3. FABRICATION TECHNIQUES OF MICRO FRESNEL LENSES . . . . . . . . . . . . . . . . . . . . . . . .

12

Q 4 . FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY

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

17

. . . . . . . . . . . . 26 Q 6 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . 35 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . 36 Q 5 . WAVEGUIDE FRESNEL LENSES

0 1. Introduction In general a glass lens (singlet) becomes thicker and heavier when one wants to increase the diameter (aperture) with the focal length fured. However, this is not the case in a Fresnel lens, which is constructed with stepped setback of many divided annular zones, as shown in Fig. 1. The optical properties are essentially the same as those of the singlet. Fresnel lenses have been used in lighthouses to collimate light beams for more than one hundred years, and recently have come widely into use in viewfinders of reflex-type cameras or as focusing lenses of overhead projectors. These Fresnel lenses mostly have a relatively large aperture but small numerical aperture (NA). Recently, microlenses with small apertures and large NA are required components in various optical systems, such as pickup lenses in laser-disk

/

\

0 -

\

0

\

0

I

l

l

I

I

I

l

I

l

l

l

l

l

l

I I

I

I

I I

I

l

l

Fig. 1. Illustration of a Fresnel lens 3

l

4

MICRO FRESNEL LENSES

[I, § 1

players and coupler lenses in optical components of optical communication systems. The two types of microlenses that have been used so far are the refraction type, such as ordinary convex lenses and graded-index lenses (OIKAWA and IGA[ 1982]), and the diffraction type, such as holographic lenses and zone plates (JORDAN, HIRSCH,LESEMand VAN ROOY [ 19701, FIRESTER [ 19731). Micro Fresnel lenses are intermediate between both types. The center portion is more like the refraction type, since the zone periods are much larger than the wavelength, and diffraction hardly takes place. The outer portion is more like the diffraction type, since the zone periods are close to the wavelength. Because of the quasi-periodical structure, Fresnel lenses have large wavelength dependency. Consequently, at a Wavelength other than the predetermined wavelength, the chromatic aberration and spherical aberration of Fresnel lenses are much larger than those of ordinary convex lenses. Fresnel lenses are closely related to Fresnel zone plates (FZP) (MYERS [ 19511, SUSSMAN [ 19601). Both zone patterns are the same. There are two types of FZP, that is, the absorption type and the phase type. In either case the absorption or the phase is constant over the half of each zone; that is, they are in binary distribution. The absorption type FZP will not be examined here, since the principle and the characteristics have been discussed by BORNand WOLF[ 19701and also the low diffraction efficienciesare not suitable for lenses. The phase-type FZP, or binary-type Fresnel lens, has a theoretical efliciency only up to 40% approximately. We are more interested in blazed type Fresnel lenses, that is, phase type FZP whose zone has a sawtooth profile, because higher efficiency can be expected. In section 2 the principle of the micro Fresnel lens is described with some design formulas. Various fabrication methods are reviewed in section 3. FUJITA,NISHIHARA and KOYAMA[ 19811 have developed a fabrication technique by computer-controlled electron-beam writing and have made micro Fresnel lenses successfully. By this technique one can write the zone pattern of an ideal Fresnel lens that is aberration-free at any predetermined optical wavelength. The new fabrication technique is explained in section4. The technique is also applied to fabricate a waveguide type Fresnel lens, which is described in section 5.

[I, 8 2

5

PRINCIPLES OF FRESNEL LENSES

6 2. Principles of Fresnel Lenses 2.1. PHASE SHIFT FUNCTION

Let us consider a lens that has a focal length f at a wavelength I, as shown in Fig. 2. A plane wave incident along the optical axis has a constant phase over the aperture, and the converging wave has the following phase retardation distribution $(r):

$(r) = k,(f - J7-3,

(1)

where r is the radius from the optical axis and k,, is equal to 2 4 I . Therefore an incident plane wave experiences a phase retardation $(r) by the lens and is converted into a converging wave, whose phase term is given by eJIwt -

+(+I.

When the focal length f is much larger than the lens aperture size, that is f %- r, then eq. (1) is approximated by t,ha(r) = - n r 2 / I f .

(2)

$(r) is the phase shift function, which is aberration-free, while $a(r) includes spherical aberration in the converging wave. The phase shift error A $ is given bY

A$=

$a(l)

- $(r).

Plane wave

(3)

i

Converging

Fig. 2. An incident plane wave is converted into a converging wave by the Fresnel lens.

6


Rayleigh’s criterion for negligible phase error requires

The boundary radius rb which satisfies eq. (4) is given by r,

=

4m.

When &(r/f 2r,

=

)6

(5)

rb

>

rmax

( b ) Fig. 4. (a) Phase shift function of the Fresnel lens; (b) thickness distribution.

under a given Anmax. In actual cases Anmaxis the deviation of the refractive index of the zone material from that of the surrounding medium (air). The total number of zones it4 for a lens of radius R and f-number F is derived from eq. (8):

For large F, M becomes

R M=41F ' Using the relation between F and the numerical

PRINCIPLES O F FRESNEL LENSES

[I, § 2

eq. (13a) can be rewritten as R (1 M=-

Jrm)

I

9

NA

and for small NA it becomes

R M = - NA.

2A

For example, when I = 0.633 pm, R = 0.5 mm and NA = 0.3 (F = 1.6), the total number of zones is 120. The minimum period of zones Amin appears at the outermost part and is given by Amin = r, - r,-,

Using eq. (9a),

(

Amin=R 1 -

J

1-

2If + (2M - 1 ) P 2MIf + (MI)2

and since M D 1, it is written as

Using eqs. (13a) and (14), one obtains Amin =

A . NA *

It is interesting to see that &in depends only on NA and not on the radius R , and /Imin is inversely proportional to NA. Therefore, it will be more difficult to fabricate a micro Fresnel lens of larger NA. In the case of NA = 0.3 and I = 0.633 pm, Amin becomes 2.11 pm.

2.3. DlFFRACTlON EFFICIENCY

The linear and uniform-period “thin” grating of a rectangular zone profile has a maximum theoretical diffraction efficiency of 40.5 %, a fact that is well known

10


11, § 2

(MAGNUSSONand GAYLORD[1978]). On the other hand, the linear and uniform-period “thin” grating of a sawtooth zone profile has a maximum theoretical efficiency of 100% under the condition given by eq. (12) (MAGNUSSONand GAYLORD [ 19781). By extending and applying this fact to a Fresnel lens that has circular zones with a sawtooth profile, we can expect the efficiency to be close to 100% if the lens is “thin” enough. When the thickness deviates from the optimum condition (12), the diffraction eficiency should be reduced. Theoretically the thickness dependence cannot be easily treated because of circular gratings. However, the trend should be similar to the curve shown in Fig. 21, which is calculated for a straight-line grating. However, if the lens is not thin enough, one has to take a completely different approach to efficiency consideration. For the criterion of the lens thickness, the parameter Q is used and defined by 2ndT nAz

Q=-,

where A is a zone period (KOGELNIK [ 19691). Q is smaller at the inner portion of a Fresnel lens and is maximum at the outermost portion because of the A variation. The maximum value Q,,, is expressed by NA as follows:

Q,,,

=

2n

nT (NA)’.

-

a

If Qm,, < 1 (i.e. NA is small), the lens is “thin” and therefore the Fresnel lens can have 100% efficiency. If 1, however, the assumption of “thin” Fresnel lenses does not hold any longer and the efficiency should be reduced. In the case of Q,,, $- 1, one has to treat the lens as “deep relief” grating (MAGNUSSON and GAYLORD[ 19771). For example, when T = 1 pm, I = 0.633 pm, n = 1.5, R = 1.5 mm and F/5 (NA = 0.3), the radius for Q = 1 becomes 1.3 mm. This means that the lens with the preceding specifications is not “thin” all over the aperture.

em,,$-

2.4. FOCUSING

When a plane wave is incident, the intensity distribution at the focal plane is expressed by the Airy function:

[I, § 2

PRINCIPLES OF FRESNEL LENSES

11

wherep = (2nR/Af )r, as shown in Fig. 5. The diffraction-limited spot diameter of 2wlIe2of l/e2 intensity is given by 2wlle2= 1.64 AF.

(22)

A lens of smaller F gives a smaller focal spot. The minimum attainable f-number Fminis determined by the minimum realizable outermost zone period Amln. From eq. (18) one obtains

The realizable Amin depends on the grade of the fabrication technique. The focal length is determined by the size of the zone pattern, which is expressed by eqs. (10a) or (lob). In the case of large F lenses the focal length, as given by eq. (lob), is inversely proportional to the wavelength under the same zone size, that is ;If

=

constant.

(24)

This fact means that generally the Fresnel lens has large chromatic aberration under a white light source. Even under a monochromatic light source, the lens has some spherical aberration. The aberration, however, is negligible in ordinary cases, and the lens exhibits diffraction-limited focusing performance regardless of the wavelength. On the other hand, in the case of smaller F lenses whose zone pattern should be expressed by eq. (9a) instead of eq. (9b), the lens again has a spherical aberration even under a monochromatic light source if the wavelength is different from the predetermined one. This feature is very important from the standpoint of fabrication techniques, as described in section 3.

I

m

Fig. 5. Intensity distribution on the focal plane and the l/e2 spot size.

12


[I, 8 3

g 3. Fabrication Techniques of Micro Fresnel Lenses 3.1. PHOTOREDUCTION METHOD

An important method for making Fresnel lenses (or zone plates) is to draw

an enlarged pattern and reduce it to the designed dimension by using a photo camera. This photoreduction technique is common in the production of semiconductor integrated circuits (IC). To obtain lenses of the phase type, the photoreduced pattern (absorption type) is used as a photomask in etching the substrate or duplicating in a photorefractive material. The pattern generation can be simplified by taking advantage of the circular symmetry. CAMUS,GIRARDand CLARK[I9671 drew a narrow sectorial portion of FZP pattern and rotated it to generate the circular FZP (7 cm diameter, f = 180 cm). KORONKEVITCH, REMESNIK, FATEEBand TSUKERMAN [ 19761 produced a Fresnel lens of the gradient-index type in photorefractive As$, film by using a binary photomask, which has an azimuthally modulated pattern, and rotating it so that the pattern is azimuthally averaged and results in the radially gradient exposure distribution. Whereas the conventional pattern drawing involved time-consuming hand work and lacked accuracy, the use of computer-controlled plotters has solved these problems and enhanced the fabrication flexibility. JORDAN,HIRSCH, LESEMand VAN ROOY[ 19701 demonstrated the fabrication of a Fresnel lens (1 x I cm2, F/15)by plotting a gradient-density pattern and subsequent photoreduction and etching. The computer technique was applied by ENGEL and HERZIGER[ 19731 to fabricate modulated zone plates that had functions of super resolution and beam-profile converter. Another interesting application, proposed by LOHMANN and PARIS[ 19671, is the production of variable Fresnel zone patterns to obtain a zoom lens effect. These examples show that the photoreduction method which uses computer plotting provides a high degree of flexibilityand reasonable accuracy. However, the technique requires rather complicated software, and in some cases, accuracy and resolution are inadequate and the pattern digitizing may cause deterioration of the optical quality.

3.2. INTERFERENCE METHOD

It is apparent from the discussion in section 2 that the Fresnel zones exhibit the same periodicity as the fringe produced by the interference between a plane

[I, § 3

FABRICATION TECHNIQUES OF MICRO FRESNEL LENSES

13

wave and a spherical (converging or diverging) wave. Therefore the zone pattern can be generated by holographically recording the interference fringe. Figure 6 illustrates a typical setup for this technique. Coherent light from a laser is split into two beams, one of which is converted into a spherical wave. They are then combined on axis to record the fringe in a high-resolution photographic plate. The fringe has a quasi-sinusoidal intensity distribution (Fig. 7 (a)); the linear recording results in a Gabor zone plate, which is characterized by the associated principal ( & 1st order only) foci and a relatively low efficiency. By making use of the nonlinear characteristic of the recording medium, a binary zone pattern (Fig. 7 (b)) can be obtained, and with appropriate choice of the recording condition, the FZP pattern (Fig. 7 (c)) can also be obtained. The pattern, being an absorption type, can be readily transibrmed into a phase type, as described in section 3.1. The interference method provides an effective and accurate means for making FZPs with a large number of zones, which is less likely to result from the photoreduction method. CHAMPAGNE [ 19681 reported the fabrication of a FZP having 960 zones in 4.3 cm diameter. CHAU[ 19691 fabricated zone plates of 2.5 cm diameter and 20 cm focal length and discussed the effect of the nonlinear recording on the zone plate optical characteristics. Fabrication of FZP for X-ray wavelength by interferenceof second harmonics of argon laser light has also been reported by NIEMAN,RUDOLPH and SCHMAHL [ 19831. A major drawback of Fresnel lenses fabricated by interference is that, as pointed out in section 2.4, they suffer from aberrations when used at a wavelength different from that of fabrication. This limits their application considerably. The problem is serious especially with lenses for laser diode light in the IR region, where no appropriate high-resolution recording material is '

COLLIMATING LENS

SPLITTERS

(ZONE PATTERN) Fig. 6. Optical arrangement for making Fresnel zone lenses by the interference method.

14


0 ' 0

I

Pl

I

[I, § 3

I

r2 213 RADIUS

D

Fig. 7. Transmittance as a function of the radial coordinate. (a) Gabor zone plate (intensity distribution of the interference fringe); (b) quasi-binary zone plate; (c) Fresnel zone plate.

available. BUINOV,KIT, MUSTAFINand SAVRASOVA [ 19751 and BUINOVand MUSTAFIN[1976] showed that the spherical aberration can be compensated by inserting a plane-parallel glass plate in the path of the focusing wave in the fabrication step. Another drawback of the FZP for use as a lens is the low efficiency resulting from the binary modulation. Higher efficiency can be obtained if the zone pattern is recorded as a volume (not plane) hologram of the phase type in a thick recording medium. In the inner zones, however, the volume effect does not work well because of the low spatial frequency, making it difficult to achieve high overall efficiency. The situation can be improved by making a volume hologram of the off-axis type with an inclined reference wave, but then the resultant high periodicity and associated Bragg effect cause considerable reduction in efficiency when the lens is used at a wavelength different from that for the recording. The efficiency of on-axis, volume-type lenses has been discussed in detail by NISHIHARA [ 19821. An aberration-corrected, off-axis lens was fabricated by KUWAYAMA, NAKAMURA, TANIGUCHIand SUDA [ 19841. Although no further specific examples will be described here, much interest and research (NISHIHARA, INOHARA, SUHARA and KOYAMA [ 19751, LATTAand POLE[ 19791, SOARES[ 198 11) have emerged regarding holographic optical elements (lenses) of both on-axis and off-axis types. The improvement in efficiency of thin Fresnel lenses requires a blazing technique. KOSUGE, SUGAMA,ONO and NISHIDA[1984] employed a computer-controlled, ion-beam etching technique with a mask zone pattern that was holographicdy recorded to blaze an off-axis lens of 5 x 5 cm2 area and 100 mm focal length, and they obtained efficiencies higher than 50%. Pure optical methods for blazing gratings have been developed, as reviewed by

[I, 8 3

FABRICATION TECHNIQUES OF MICRO FRESNEL LENSES

15

SCHMAHL and RUDOLPH[ 19761; the application to Fresnel lens fabrication, however, has resulted in difficultydue to the deep chirp in period. Recently the space harmonics multiple-interference method to synthesize a blazing exposure was adapted using a Fabry-Perot interferometer by FERRIERE,ILLUECA and GOEDGEBUER [ 19841, who demonstrated an efficiency of 65 % in a Fresnel lens o f f = 300 mm and F/15.

3.3. OPTICAL DUPLICATION

The standard optical photomask contact printing is an important technique, not only as a step for the conversion from absorption type to phase type, but also as an effective means for mass production. Fabrication of phase zone plates of F/2.4 by deep UV duplication was reported by KODATE,TAKENAKA and KAMIYA[ 19841, and zone plates of F/1 by UV duplication by TATSUMI, SAHEKIand TAKEI[ 19831. Efficiencies of approximately 40% have been obtained.

3.4. ELECTRON-BEAM WRITING

An important modem technique for micro Fresnel lens fabrication is computer-controlled electron-beam (EB) writing. The pattern generation by computer enables the fabrication of Fresnel lenses that are aberration free at arbitrarily prescribed wavelengths, and the direct writing of the pattern in the operational dimension by EB (without the photoreduction process) assures enough resolution. The technique also exhibits high flexibility in changing the lens specifications and in modulating or modifying the lens patterns. The technique has become more practical as a result of the recent progress on EB systems and EB resists. An ordinary EB writing system developed for IC photomask production can be used for Fresnel lens fabrication. However, since such systems are based upon digital X-Y scanning of EB, the writing involves complicated software, long computation time, and an inefficiently large number of data. In addition, the smoothness of the curved lines may not be sufficient. To eliminate these problems, a special EB writing system has been developed that incorporates analog circular scanning by applying sine and cosine waves to the X and Y axis and digital control of the diameter (Fig. 8). The fabrication of phase Fresnel zone lenses (FZLs) by EB was first

16

[I, § 3


SCANNING ELECTRON MICROSCOPE (HITACHI-AKASHI MSM-102)

1T I I

I

SEM CONTROL/ DISPLAY CONSOLE

..

I

Fig. 8. Block diagram of an electron-beam writing system designed for optical components fabrication.

demonstrated by FUJITA,NISHIHARA and KOYAMA[ 19811, who employed an of EB resist polymethyl analog/digital scanning technique to obtain FZLs (F/5) methacrylate (PMMA). Fabrications by EB of photomasks for duplication have been reported by TATSUMI,SAHEKIand TAKEI[ 19831, and KODATE, TAKENAKA and KAMIYA[ 19841. By making use of the high flexibility of the EB writing, a special FZL for beam-profile conversion (FUJITA,NISHIHARA and KOYAMA[ 19811) and off-axis FZLs and astigmatic FZLs (HATAKOSHI and GOTO[ 19841) are fabricated. In these examples nearly diffraction-limited focusing characteristics were obtained, whereas the efficiency was limited to approximately 40%.A technique for fabricating more efficient blazed Fresnel and KOYAMA lenses has been developed recently by FUJITA,NISHIHARA [ 19821; the technique will be described in section 4. The advantages of the EB writing technique have been also used to fabricate

[I, § 4

FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY

17

FZPs for X-ray wavelength. The fabrication of these FZPs requires high resolution, and a coherent X-ray light source that is suitable for the interference method is not available. Absorption-type FZPs of gold, which are selfsupporting (without substrate), have been fabricated by combinations of EB writing and X-ray lithography (SHAVER,FLANDERS, CEGLIOand SMITH [ 19791); EB writing and ion-beam etching (KERN,HOUZEGO,COANEand CHANG[ 19831);and EB writing and reactive ion etching (ARITOME, AOKIand NAMBA[ 19841).

5 4.

Fresnel Lenses Fabricated by Electron-beam Lithography

4.1. ELECTRON-BEAM WRITING SYSTEM

Ordinary electron-beam (EB) lithography systems write curved lines stepwise by line segments and cannot write smooth curves. To realize a smooth circular scanning of an electron beam for fabrication of a Fresnel zone pattern, FUJITA,NISHIHARA and KOYAMA [ 19811 have developed an EB lithography system with a specially designed deflection controlled by a minicomputer. They used a conventional scanning electron microscope (Hitachi-Akashi MSM-102) and a Melcom 70/10 minicomputer. The block diagram of the system is shown in Fig. 8, and the specifications are described in table 1.

4.2. BLAZING TECHNIQUE

A new method to fabricate blazed (sawtooth-profile) gratings has been proposed by FUJITA,NISHIHARA and KOYAMA[ 19821. A sawtooth profile can be formed by a suitably chosen electron-dose distribution because the etching rate of the resist depends on the electron dose. This process is depicted in Fig. 9. First, an electron-beam resist is spin-coated onto a glass substrate, which is deposited with a In,O, layer for preventing charging-up during exposure. The resists used are PMMA (positive type) or CMS (negative type), where CMS stands for chloromethylated polystyrene. The initial resist layer should be slightly thick, so that the final thickness after development T becomes

where T = 1.06 pm in the case of 1 = 0.633 pm and n

=

1.49 (PMMA).

[I, § 4


18

TABLE1 Specifications of the electron-beam writing system. Parameter

Value Linear scanning

Circular scanning

15,30 kV 0.1-1.0 nA 0.1-1.0 pn

15,30 kV 0.1-1.0 nA 0.1-1.0pm

3 x 3mmz

3 x 3mm2

X:analog scanning by ramp

X Y : analog scanning by sinusoidal signal (100 Hz)

Electron beam voltage current diameter Scanning area ~~

Direction

signal Y: digital scanning by 16 bit D/A converter Resolution

216 points

214 points

(radius)

100 ps- 10 s/line

10 ms/circle

-~~ ~

Scanning time

Electron Beam (30kV)

-CMS

1.5 pm

7-

Thin F i h lneOJFilm

-Glass

a)

Electron Beam Exposure

-Finished

b)

Substrate

Device

Development (Bubble Development Method)

Fig. 9. Cross-sectional view of (a) fabrication process of a blazed grating using electron-beam lithography; (b) result. The dot density corresponds with the electron dose.

[I. 8 4


19

Second, a grating pattern is drawn upon the resist by an electron beam of diameter 0.2 pm. The electron-dose distribution along the X direction, which has been determined on the basis of calibration curves (relationship between electron dose and etch depth), is given by changing the number of scanning times on the same position. The electron-dose versus etch-depth relationship is obtained from a cross-sectional scanning-electron-microscope (SEM) photograph of a blazed-profile grating that has been drawn by changing the dose from 0.3 x l o p 4to 1.7 x C/cmZlinearly. After exposure the sample is carefully developed, in a 1 : 1 solution of isopropyl alcohol and methyl isobutyl ketone. Figure 10 shows cross-sections of fabricated blazed linear gratings with periods of (a) 5 pm and (b) 10 pm. The diffraction efficiency of the grating at normal incidence of a He-Ne laser beam (A = 0.633 pm) has been measured, and 60% to 70% efficiency is obtained, where the efficiency is defined as the ratio of the diffracted to the incident flux.

~~

(a) A = 5pm

(b) A = lOpm Fig. 10. SEM cross-sectional photographs of the blazed gratings: grating period: (a) 5 pm,and (b) 1C pm.

20

[I, fi 4


4.3. EXAMPLES

Figure 11 shows a schematic view of the EB circular scanning on a PMMA thin film, which is spin coated onto a glass substrate covered with a film of In@,. The beam is circularly scanned a certain number of times at the innermost radius until the specified dose is given, and then the radius is increased by an amount (approximately 0.1 pm) that is smaller than the beam diameter of 0.2pm. This process is repeated until the outermost radius is attained. To obtain surfaces on which curvatures vary slightly for all grooves, several calibration curves (dose versus etch-depth) are obtained experimentally for different periods by repeating the procedure. Figure 12 shows a micrograph of the fabricated Fresnel lens, which has a focal length of 5 mm at 0.633 pm wavelength and a diameter of 1mm. The measurement result of the surface profile by means of the Talystep is shown in Fig. 13, where the desired curved profiles of each groove are observed. 4.4. CHARACTERISTICS

4.4.1. Focusing spot

The intensity profile on the focal plane is measured by an optical system, where the focal spot is magnified and imaged on a television camera by means

Electron Beam

-1

+ E l e c t r o n Beam R e s i s t (PMMA o r CMS)

Conductive In203 Film

Glass S u b s t r a t e

Fig. 11. Schematic view of the fabrication of a micro Fresnel lens using electron-beam lithography.

22


[I, § 4

Fig. 14. Intensity profile on the focal plane of the lens in Fig. 12 at a normal plane-waveincidence ( A = 0.633 pm). (a) Image of the spot; (b) video signal trace of a television camera.

conventional single glass lens of the same specifications and with Abbe number 50. It can be seen that the Fresnel lens has a chromatic aberration that is ten times greater than the single glass lens. Figure 16 shows a pattern that was photographed at the focal plane of blue light, when the lens was illuminated with a white light. The colored ring pattern results from the large chromatic aberration.

4.4.3.Diffraction efficiency The diffraction efficiency, which is defined as the ratio of the converging wave power to the incident plane wave power, is measured at approximately

[I, § 4


21

Fig. 12. Microphotograph of the fabricated Fresnel lens (diameter 1 mm; focal length 5 mm at 0.633 pm wavelength).

of an object lens of x 60 magnification. An example of the intensity profile measured at a normal plane-wave incidence is shown in Fig. 14. The focal spot size at half of the maximum intensity was about 4.2 pm, which is a nearly diffraction-limited size (1.031ZF = 3.3 pm). 4.4.2. Focal length

The Fresnel lens has a feature that enables the focal length to be predicted precisely by the size of the zone pattern. The measured focal length of a fabricated lens is 5 0.1 mm, whereas the design value is 5 mm. The dependency of the focal length on wavelength is also examined experimentally. In Fig. 15 the experimental values are compared with the theoretical values, and satisfactory agreement can be obtained. The dotted line depicts the case of a

Fig. 13. Surface relief profile of a fabricated cylindrical Fresnel lens.

[I, § 4


23

F r e s n e l Lens ( f = S m m , X=0.633um) 0 ; Measured Value

7 E E Y

c

+,

m c

:5

--+------

rl

rd

Ling,, Glass Lens (f=5mm,v=50)

u

0 E

0.5

0.6

0.7

Wavelength h ( w 1 Fig. 15. Dependence of the focal length on the wavelength.

60%. Optical microscope observation reveals that all grooves are not necessarily well fabricated and probably the chemical etching conditions are not optimized. Theoretically one can expect 100% efficiency in the center portion where the “thin” grating condition should hold. However, the efficiency should be reduced in the outer portion where the condition does not hold any more, since the period is shorter.

4.4.4.Wavefront aberration The Fresnel lens should have little wavefront aberration. From the measured results of wavefront aberration by means of a laser interferometer ZYGO ZAPP, it is found that the wavefront aberration is 0.266 1 at maximum and 0.054 I on average. This small value corresponds to the diffraction-limited focusing characteristics of the Fresnel lens. The imaging characteristics of the Fresnel lens are also examined. The optical source is a quasi-monochromatic light that is filtered from a mercury lamp, and the mesh pattern is imaged on a screen. Little distortion can be observed.

4.5. LENS ARRAY

Using the EB writing technique, one can fabricate a lens array, as shown in Fig. 17a (NISHIHARA and ENOMOTO [ 19841). Each lens is scanned where the

24


[I, t 4

Fig. 16. Colored ring pattern photographed at the focal plane of blue light, when the lens is illuminated with a white light.

[I, § 4


25

Fig. 17. (a) Example of a Fresnel lens array (diameter 0.5 mm; focal length 5 mm); (b)multiple images.

26


[I, § 5

stage is displaced successively at a predetermined interval of 0.7 mm. The diameter of each lens is 0.5 mm, and the scanning process takes 5 minutes. Figure 17b shows multiple images observed by this lens array under a white light source; chromatically aberrated images can be seen.

4.6. ELLIPTICAL LENS

By adjusting the sensitivity of X and Y deflections, an elliptical Fresnel lens can also be fabricated, as shown in Fig. 18. The ellipticity of the lens is 0.95. When the ellipticity is close to 1, the blazing technique can be used as well. The flexibility of the EB technique is useful when a lens with astigmatic aberration is required, for example, in the case of a distance-measuring optical head.

8 5. Waveguide Fresnel Lenses 5.1. WAVEGUIDE LENSES: REQUIREMENTS AND PROBLEMS

Although the foregoing sections have discussed Fresnel lenses for microoptics, lenses also are one of the most important components of integrated optics. In integrated optics (TAMIR[ 19751, HUNSPERGER [ 19821) devices (optical ICs) for optical communications, signal processing, and sensors are implemented by using a thin film waveguide in which the optical wave is confmed as a guided mode. Therefore waveguide lenses (ANDERSON,DAVIS, BOYDand AUGUST[1977], HATAKOSHI, INOUE,NAITO, UMEGAKIand TANAKA [1979]), that is, lenses constructed in wave-guiding structure, are required to perform imaging, collimating, focusing, and Fourier-transforming on a guided wave. For many applications the waveguide lenses must exhibit excellent performance, such as diffraction-limitedand aberration-free focusing characteristics, high efficiency, and large angle of view. The requirements are stringent, especially in constructing optical ICs for signal processing, such as integrated-optic radiofrequency (RF) spectrum analyzers. Because of their important functions and the difficulty in satisfying the requirements, waveguide

lenses have been considered to be a key component for integrated optics. In earlier works, lenses were fabricated in waveguides by changing the effective index of refraction in a lens area by overlay cladding or diffusion process. The resultant mode-index lenses (correspondingto singlets) exhibited large aberrations and offered few practical applications. Good performances

[I, § 5

WAVEGUIDE FRESNEL LENSES

Fig. 18. An elliptical Fresnel lens (average diameter 1 mm; focal length 5 mm).

21

28


[I, 3 5

have been achieved in Luneburg lenses and geodesic lenses, but these waveguide lenses involve difficulty of fabrication. Luneburg lenses require a highprecision, gradient-thickness deposition process, and geodesic lenses require an expensive and time-consuming precision mechanical grinding process. The difficulties of waveguide lenses arise from the necessary integration in a waveguide with other components. The lens material cannot be selected independently;the lens must be fabricated in a “given” waveguide, the material of which has been selected on the basis of good electro-optic, acousto-optic, waveguiding, and other characteristics. In contrast to the micro-optic counterpart, the medium surrounding the lens is not air but waveguide; the index difference available for the lens function is much smaller. The possible deviation in focal length from the designed value, which can be tolerated in many conventional and micro-optics applications, is fatal in integrated optics because the positions of all integrated components are rigidly fured. There has been much interest in diffraction-type waveguide lenses, that is, grating lenses and Fresnel lenses, since they eliminate many of the preceding problems. The major advantage of diffraction lenses is that they can be fabricated by the inexpensive and well-established planar microfabrication technique, which is compatible with fabrication of other components to be integrated. The focusing characteristics (focal length) of diffraction lenses are determined by the planar lens pattern (in particular the periodicity) and are not sensitive to the fabrication process variations. Therefore focal length control is easy and highly reproducible. Fresnel waveguide lenses are attractive for several reasons, including their inherent aberration-fiee focusing characteristics, the easy and reproducible fabrication, their potentially high efficiency, and their relatively large angle of view. 5.2. THEORETICAL CONSIDERATIONS

Figure 19 shows schematically the Fresnel waveguide lens configurations. Since the guided wave is confined in a planar waveguide, the lens function can be obtained by fabricating a structure that corresponds to a sliced (along the diameter) cross-section of a micro-optic Fresnel lens. The discussions in section 2 apply only if the radial coordinate is replaced by that along the lens aperture and the wavelength in the waveguide is used instead of that in free space. The required lens phase modulation @(x)and the phase modulation for a Fresnel lens GF(x)can be written as follows:

29


CONVERGING

/-

WAVE

/

/I== I

L

~'CRIN

\

\

/

I/

FRESNEL LENS

I

WAVEGUIDE LAYER SUBSTRATE

(a) G R I N F R E S N E L L E N S

CONVERGING G U I D E D WAVE

FRESNEL LENS

WAVEGUIDE LAYER SUBSTRATE

(b) GRTH F R E S N E L L E N S

Fig. 19. Fresnel waveguide lens configurations,

d j F ( x ) = d j ( x ) + 2 r n n , for

x,I V

z W

x

50-

w

-

I+

E h

rn 2

w

I 0.5

I

I

1.0

I

I

1.5

II ( = L / L F ) NORMALIZED PHASE MODULATION AMPLITUDE Fig. 21. Dependence of the efficiency of the Fresnel waveguide lenses on the phase modulation amplitude (lens thickness or refractive-index change).

32


optimum modulation amplitude can be written as

and the condition for the “thin” lens Q ,

< 1 is given by

An IT 1 - > - -. n, 2 F2 Equation (32), for example, shows that at least 6.3% mode index change is required for a lens of F/5. It is important to note that a lens which does not satisfy eq. (32) is not “thin” and is less efficient.

5.3. FABRICATIONS AND RESULTS

The first Fresnel waveguide lens was proposed and demonstrated by ASHLEYand CHANG [1978], and CHANG and ASHLEY [1980]. They fabricated SI Fresnel zone lenses (F/2.5,5) in BaO waveguide on a glass substrate with CeO overlay cladding, which was patterned by photolithography using a EB written photomask, and obtained nearly diffraction-limited focusing and an efficiency of 23 % . The work was an important step for research and development of integrated optics, since it demonstrated the feasibility of waveguide lens by lithography. MOTTIERand V A L E ~ [E19811 fabricated the same kind of lens (f= 10.2 mm, F/8.5, efficiency 19%) by patterned cladding of SiO, in a Si,N,/SiO,/Si waveguide, which is more suitable for integration. Research on Bragg-type grating lenses, to obtain higher efficiency with the SI structure, has also been reported (HATAKOSHI and TANAKA[ 19781). High-efficiency GRIN and GRTH Fresnel lenses have been demonstrated by SUHARA,KOBAYASHI, NISHIHARA and KOYAMA[1982] and SUHARA, NISHIHARA and KOYAMA [ 19831 in As,& waveguides on Si02/Si substrate. Amorphous As2S3, which exhibits low transmission losses in the near IR region, is a suitable waveguide material, and the refractive index can be changed by an EB irradiation. The index increment (up to appruximately 5%) can be controlled continuously by giving an appropriate EB dose. By the EB direct writing technique making use of this effect, GRIN and GRTH lenses of 1 mm aperture, F/3 and F/5, were fabricated. To write the lens, the EB was scanned along the optical-axis direction with small scanning line displacements (0.1 pm) so that the beam traces overlapped. For GRIN lenses the scanning speed or

11, § 5


33

(b) GRTH FRESNEL LENSES Fig. 22. Interference microphotographs of Fresnel waveguide lenses in As,S,/SiOJSi waveguide (f= 5 mm,F/5).

the number of scanning repetitions on a line was varied to give the gradient dose distribution, and for GRTH lenses the scanning width was varied to write the GRTH pattern. Figure 22 shows the interference microphotographs of the fabricated lenses. Figure 23 shows the typical light-intensityprofile on the focal line of the fabricated lens. Nearly diffraction-limited focusing characteristics (3 dB width of 3.5 pm for F / 5 , I = 0.83 pm) and efficiency of up to 61 % have been obtained.

Fig. 23. Measured intensity profiles of the focused light spot. (a) GRIN Fresnel lens F/3 1 = 1.06pm; (b) GRTH Fresnel lens F/5 1 = 0.83 pm.

34

[I, § 5


GRTH Fresnel lenses were fabricated also by VALE'TTE, MORQUEand MO'TTIER[ 19821 in Si,N,/SiOJSi waveguides with SiO, overlay cladding patterned by standard photolithography. In their structure a mode index change up to approximately 0.032 was available, and nearly diffraction-limited focusing characteristics and efficiencies of 60% to 70% were reported with lenses of F numbers of approximately 6. Realization of high-performance waveguide lenses in LiNbO, has been strongly desired, since the waveguide exhibits good waveguiding, electro-optic,

I .TITANIUM

INDIFFUSION

3,RESIST COATING

\i/ 4. E-BEAM WRITING AND DEVEMPING

ul S.Si-N

ETCHING

0

6.RESIST REMOVING

U

6.Si-N

REMOVING

Fig. 24. Proton-exchanged Fresnel lenses in Ti : LiNbO, waveguide: fabrication process and microphotograph.

[I, § 6

CONCLUSION

35

and acousto-optic characteristics, and therefore it is considered to be one of the most suitable materials for optical ICs. Fabrication of Fresnel lenses, however, has not been reported until recently because of the diflkulty in obtaining the required large index change. Most recently, SUHARA, FUJIWARA and NISHIHARA [ 19851 applied the proton-exchange technique, which gives an index change as large as 0.11, to fabricate a GRTH Fresnel lens in a Ti-indiffused LiNbO, waveguide. The fabrication process is shown in Fig. 24. The waveguide was coated with a thin Si-N mask layer, and the lens pattern written by EB was transferred to the mask layer by reactive ion etching. The waveguide was then immersed in molten benzoic acid for the patterned proton exchanging. Nearly diffraction-limited focusing properties and efficiencies as high as 70% have been obtained in the fabricated lenses of F/5. To conclude this section, it should be mentioned that the GRIN and GRTH Fresnel waveguide lenses have been used to construct actual prototypes of integrated optics devices for signal processing, such as RF spectrum analyzers, and their operation has been demonstrated by SUHARA,SHIONO,NISHIHARA and KOYAMA[ 19831, VALETTE, LIZET, MOTTIER, JADOT, RENARD, FOURNIER, GROUILLET, GODONand DENIS[ 19831 and SUHARA, FUJIWARA and NISHIHARA[ 19851. Although further improvements in performances are required, the importance of waveguide Fresnel lenses is increasing in the technology of integrated optics.

8 6. Conclusion We have reviewed the principles, characteristics, and fabrication techniques of micro Fresnel lenses. If the zone pattern is precisely fabricated, lens characteristics such as the focal length are obtained as designed. It is stressed that the electron-beam writing technique is important for obtaining a precise pattern and blazing zone profiles, and also for redesigning and fabricating the lenses of the different specifications. The problem that needs to be investigated is the efficiency; in particular the optimum relief profile for maximum efficiency requires further theoretical examination. Because of the relief structure, replicas can be obtained by the stamping method, and therefore the lenses are suitable for mass production. Micro Fresnel lenses will be used more widely in various optical systems in the future.

36


References ANDERSON, D. B., R. L. DAVIS,J. T. BOYDand R. R. AUGUST,1977, IEEE J. Quantum Electron. QE13, 275. ARITOME, H., H. AOKIand S. NAMBA,1984, Jpn. J. Appl. Phys. 23, L406. ASHLEY, P. R., and W. S. C. CHANG,1978, Appl. Phys. Lett. 33,490. BORN,M., and E. WOLF,1970, Principles of Optics, 4th Ed. (Pergamon Press, Oxford) p. 370. BUINOV, G. N., and K. S. MUSTAFIN,1976, Opt. Spectrosc. 41, 90. BUINOV, G. N., I. E. KIT, K. S. MUSTAFINand M. I. SAVRASOVA, 1975, Opt. Spectrosc. 38,88. CAMUS,J., F. GIRARDand R. CLARK,1967, Appl. Opt. 6, 1433. CHAMPAGNE, E., 1968, Appl. Opt. 7, 381. CHANG,W. S. C., and P. R. ASHLEY,1980, IEEE J. Quantum Electron. QE-16, 744. CHAU,H. H. M., 1969, Appl. Opt. 8, 1209. ENGEL,A., and G. HERZIGER,1973, Appl. Opt. 12,471. FERRIERE, R., C. ILLUECA and J. P. GOEDGEBUER, 1984, Multiple beam interferometry applied to the realization of phase Fresnel lenses and gratings, in: Congr. Int. Commission Opt. (KO-13), August 20-24, 1984, Sapporo, C6-2. FIRESTER, A. H., 1973, Appl. Opt. 12, 1698. FUJITA,T., H. NISHIHARA and J. KOYAMA, 1981, Opt. Lett. 6, 613. FUJITA,T., H. NISHIHARA and J. KOYAMA,1982, Opt. Lett. 7, 578. HATAKOSHI, G., and K. GOTO,1984, Grating lenses for optical components, in: Topical Meeting on Gradient-Index Optical Imaging Systems (GIOS), April 19-20, 1984, Monterey, ThE-El. 1978, Opt. Lett. 2, 142. HATAKOSHI, G., and S. TANAKA, HATAKOSHI, G., H. INOUE,K. NAITO,S. UMEGAKIand S. TANAKA,1979, Opt. Acta 26, 961. HUNSPERGER, R. G., 1982, Integrated Optics: Theory and Technology (Springer-Verlag,Berlin). JORDAN Jr, J. A., P. M. HIRSCH,L. B. LESEMand D. L. VAN ROOY,1970, Appl. Opt. 9, 1883. KERN,D. P., P. J. HOUZEGO, P. J. COANEand T. H. P. CHANG,1983, J. Vac. Sci. Tech. B1,1096. KODATE,K., H. TAKENAKA and T. KAMIYA,1984, Appl. Opt. 23, 504. KOGELNIK, H., 1969, Bell Syst. Tech. J. 48, 2909. KORONKEBITCH, V. P., V. G. REMESNIK, V. A. FATEEBand V. G. TSUKERMAN, 1976, Avtometrija 5, 3. KOSUGE,K., S. SUGAMA, Y.ONOand N. NISHIDA,1984, Ion-etched blazed holographic zone plates, in: Congr. Int. Commission Opt. (IC0-13), August 20-24, 1984, Sapporo, C6-9. KUWAYAMA, T., Y. NAKAMURA, N. TANIGUCHI and S. SUDA,1984, Aberration corrected off-axis holographic lens, in: Congr. Int. Commission Opt. (ICO-13), August 20-24, 1984, Sapporo, C6-6. LATTA,M. R., and R. V. POLE,1979, Appl. Opt. 18, 2418. LOHMANN, A. W., and D. P. PARIS,1967, Appl. Opt. 6, 1567. MAGNUSSON, R., and T. K. GAYLORD, 1977, J. Opt. SOC.Am. 67, 1165. MAGNUSSON, R., and T. K. GAYLORD, 1978, J. Opt. SOC.Am. 68, 806. MOTTIER,P., and S. VALEITE,1981, Appl. Opt. 20, 1630. MYERS,0.E., 1951, Am. J. Phys. 19, 359. NIEMAN,B., D. RUDOLPHand G. SCHMAHL,1983, Nucl. Instrum. Methods 208, 367. NISHIHARA, H., 1982, Appl. Opt. 21, 1995. NISHIHARA, H., and S. ENOMOTO,1984, Electron-beam direct fabrication of micro Fresnel lenses, in: Congr. Int. Commission Opt. (IC0-13), August. 20-24, 1984, Sapporo, B8-6. NISHIHARA, H., S. INOHARA, T. SUHARA and J. KOYAMA, 1975, IEEE J. Quantum Electron. QE-11, 794. M., and K. IGA, 1982, Appl. Opt. 21, 1052. OIKAWA, SCHMAHL, G., and D. RUDOLPH, 1976, Holographic diffraction gratings, in: Progress in Optics, Vol. 14, ed. E. Wolf (North-Holland, Amsterdam), p. 195.

I1

REFERENCES

31

SHAVER, D. C., D. C. FLANDERS, N. M. CEGLIOand H. I. SMITH,1979, J. Vac. Sci. Tech. 16, 1626. SOARES,O., 1981, Opt. Eng. 20,740. SUHARA, T., K. KOBAYASHI, H. NISHIHARA and J. KOYAMA,1982, Appl. Opt. 21, 1966. 1983, High-efficiency diffraction-type waveguide SUHARA, T., H. NISHIHARA and J. KOYAMA, lenses fabricated by electron-beam writing, in: Topical Meeting on Gradient-Index Optical Imaging Systems (GIOS), July 4-5, 1983, Kobe, F6. and J. KOYAMA,1983, IEEE J. Lightwave Tech. LT-1, SUHARA, T., T. SHIONO,H. NISHIHARA 624. SUHARA,T., S. FUJIWARAand H. NISHIHARA,1985, Proton-exchanged Fresnel lenses in Ti : LiNbO, waveguide, in: Topical Meeting on Gradient-Index Optical Imaging Systems (GIOS), September 26-27, 1985, Palermo, A3. M., 1960, Am. J. Phys. 28, 394. SUSSMAN, TAMIR,T., 1975, Integrated Optics (Springer-Verlag. Berlin). TATSUMI,K., T. SAHEKIand T. TAKEI,1983, High performance micro Fresnel lens fabricated by U. V. lithography, in: Topical Meeting on Gradient-Index Optical Imaging Systems (GIOS), July 4-5, 1983, Kobe, G5. VALEITE,S., A. MORQUEand P. MOITIER, 1982, Electron. Lett. 18, 13. S., J. LIZET,P. MOITIER,J. P. JADOT,S. RENARD,A. FOURNIER, A. M. GROUILLET, VALETTE, P. GODONand H. DENIS, 1983, Electron. Lett. 19, 883.



I1

DEPHASING-INDUCED COHERENT PHENOMENA BY

L. ROTHBERG AT& T Bell Laboratories Murray Hill, NJ 079074,U.S.A.

CONTENTS PAGE

1 . INTRODUCTION RENCE . . . . .

TO

DEPHASING-INDUCED

COHE-

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

41

3 2. THEORETICAL TREATMENT . . . . . . . . . . . . . . 45

5 3 . EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . § 4 . SUMMARY AND FUTURE PROSPECTS

. . . . . . . . . 94

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

67

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

98 99

8 1.

Introduction to Dephasing-Induced Coherence

1.1. DESCRIPTION AND IMPORTANCE OF THE PHENOMENON

The observation of dephasing-induced nonlinear optical mixing has verified the counterintuitive prediction (BLOEMBERGEN,LOTEM and LYNCHJR. [ 19781) that coherent processes can be caused by incoherent perturbations. The present review deals with coherent nonlinear optical phenomena that occur only in the presence of environmental perturbations usually associated with the decay of coherence between material quantum states. An understanding of damping in nonlinear quantum mechanical processes is important when working with near resonant nonlinear optical phenomena in condensed matter or in gas phase collisional environments. A perturbative view of nonlinear optical processes involves successive interactions of a material system with radiation fields. Each possible time ordering of the interactions can be associated with a quantum mechanical probability amplitude that contributes to the overall process (YEE and GUSTAFSON [ 19781). Interactions of the fields with either the material wavefunction I $) or its complex conjugate ( $1 modify the density matrix and create coherences, which can result in lightwave mixing. In the absence of damping the evolutions of I $) and ( $1 are independent, and the time sequence of interactions with I $) with respect to those with ( $1 is irrelevant. In this case the time reversal symmetry of the Hamiltonian prescribes that different time orderings of the field interactions contributing to coherent optical processes have probability amplitudes which are constrained to interfere destructively. Dephasing-induced coherent phenomena can be understood as the result of the removal of this destructive interference of amplitudes by incoherent damping processes. Clearly, these effects are intrinsically quantum mechanical and have no classical analog. The bulk of this article focuses on near resonant optical four-wave mixing, since this is the only case in which dephasing-induced resonances have as yet been observed. The term “near resonant” refers to the case where some or all of the applied fields are tuned close to intermediate one-photon allowed transitions. Near resonant four-wave mixing has been of practical importance 41

42

DEPHASING-INDUCED COHERENT PHENOMENA

[II, § 1

in wavefront conjugation (FISHER [ 1983]), Doppler-free spectroscopy (LEVENSON[ 1982I), dynamical measurements in the frequency domain (YAJIMA,SOUMAand ISHIDA [ 1978]), and resonant enhancement of coherent Raman spectra (DRUETand TARAN [ 19811). Examples of damping that can come into play are collisions in gaseous environments, phonons in crystals, and even spontaneous population decay under some circumstances. These can create electronic coherences whose contributions to nonlinear mixing can be substantial even quite far (hundreds of cm-’) from resonance (LYNCHJR. [ 19771). The study of dephasing-induced coherences has established the correct theoretical description of nonlinear phenomena in the presence of damping. This is important in the analysis of resonantly enhanced nonlinear optical lineshapes. Moreover, the “extra” resonances only observable as a result of dephasing mechanisms have been utiltzed to study the dephasing that causes them.

1.2. HISTORICAL EVOLUTION OF THE PROBLEM

Perturbative treatments of nonlinear interactions of light with matter were fist performed in the 1920s (DIRAC[1927]). The advent of lasers made possible the experimental study of nonlinear optical effects, beginning with second-harmonic generation in 1961 (FRANKEN,HILL, PETERS and WEINREICH[ 19611). A theory of the nonlinear susceptibilities governing second-order (I(’),“three-wave”) and third-order (xc3),“four-wave”) mixing in the absence of damping was published soon after by Bloembergen and co[ 19621). workers (ARMSTRONG, BLOEMBERGEN, DUCUINGand PERSHAN These authors used successive applications of time-dependent perturbation theory with coherent perturbation Hamiltonians due to applied light fields to write x(’) and xC3)explicitly. They obtained a result displaying 24 amplitudes (i.e. terms and sets of resonant denominators) contributing to a single tensor component of x ( ~ )It. would be difficult to account for damping within the context of the preceding formalism because that would require explicit expressions for perturbation Hamiltonians describing interactions with essentially random fields. To incorporate the effects of damping, an alternative approach was adopted by BLOEMBERGEN and SHEN [1964]. They used an iterative solution of the quantum-mechanical Liouville equation in the frequency domain where damping can be included phenomenologically to evaluate the steadystate density matrix to arbitrary order. The outcome of such a calculation for

11, § 11

INTRODUCTION

43

x(3) is a total of 48 terms (BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781) when damping is added phenomenologically at each level of the computation. This result was at variance with other approaches (BUTCHER[ 1965]), where phenomenological damping widths were added directly to the 24 sets of resonant denominators of the original theory. In the meantime the first four-wave mixing experiments, third-harmonic generation in LiF and coherent Raman resonances in various materials, were reported by MAKERand TERHUNE[1965]). The power of nonlinear spectroscopy quickly became apparent, and its development has been reviewed in detail (BLOEMBERGEN [ 19821). It was recognized that visible light could be used in transparent media to measure material dispersion and absorption in the infrared (LEVENSON and BLOEMBERGEN [ 19741) and ultraviolet (KRAMER and BLOEMBERGEN [ 19761) spectral regions. With the invention of the tunable dye laser, coherent Raman spectroscopy became an important analytical tool (LEVENSON and SONG[ 19801) because of its high sensitivity compared with spontaneous Raman scattering. In dilute systems the utility of four-wave mixing was found to be limited by nonresonant background from the solvent or host crystal, and the tunability of dye lasers was used to exploit resonance enhancement when tuned near allowed electronic transitions of the species of interest (HUDSON,HETHERINGTON, CRAMER,CHABAYand KLAUMINZER [ 19761, NESTOR,SPIROand KLAUMINZER [ 19761, CARREIRA, Goss and MALLOY [ 19771). The interpretation of these resonant coherent Raman lineshapes (CARREIRA, Goss and MALLOY[ 19771) led to some controversy (LYNCHJR., [ 19771) about the correct signs of damping terms LOTEMand BLOEMBERGEN in the resonant denominators of the third-order susceptibility. The aforementioned controversy was in part responsible for precipitating the seminal paper where the complete 48-term expression for x(’) governing coherent Raman scattering in the presence of damping was written out in detail (BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781). The genesis of dephasinginduced coherent phenomena is traceable to this paper, which contains an explicit prediction of “extra” resonances in four-wave mixing caused by incoherent perturbations. It should be pointed out, however, that the importance of the cancellation of amplitudes in the absence of damping was noted by Bloembergen in his farsighted 1965 monograph on nonlinear optics (BLOEMBERGEN [ 19651, pp. 29-30). The “extra” x(3) resonances arise from the additional 24 terms absent from the theory without damping included. It was shown that these terms can be grouped into 12 pairs, each of which is proportional to a difference of damping factors (rnn, - rnn,, - rn,n,,) that vanishes in a closed system (i.e. only composed of In), In’), In”)), when the

44


PI, s 1

only source of damping is spontaneous emission. Physically, the introduction of pure dephasing removes the destructive interference between these paired amplitudes, leading to four-wave mixing resonances whose characteristic feature is that their intensities are governed by the amount of pure dephasing. The first documented observationsof dephasing-induced coherent processes were “extra” Raman-type resonances between unpopulated excited states seen independently by PRIOR,BOGDAN,DAGENAIS and BLOEMBERGEN[ 19811 and by ANDREWS and HOCHSTRASSER [ 19811 using pulsed dye lasers. The former group demonstrated buffer gas collision-induced four-wave mixing resonances at different frequencies characteristic of the fine structure splitting of the sodium D-lines. The latter group studied similar phonon-mediated coherent Stokes resonances between vibrational levels of electronicallyexcited pentacene in a benzoic acid host crystal. Shortly after, collision-induced population grating resonances (BOGDAN,PRIORand BLOEMBERGEN[ 198I], BOGDAN,DOWNERand BLOEMBERGEN [ 1981aJ) and collision-induced Raman resonances between equally populated ground state levels (BOGDAN, DOWNERand BLOEMBERGEN [ 1981al) were observed.

1.3. OVERVIEW OF THE ARTICLE

The purpose of this review paper is to provide a pedagogical introduction to both theoretical and experimental studies to characterize and apply dephasinginduced coherence. In $ 2 a brief synopsis of several different pictures of dephasing-induced nonlinear optical mixing is presented. The evolution of alternative and more general formulations of collision-induced resonances is an excellent example of the symbiotic relationship between theory and experiment. Much of the theoretical work motivated by the observation of dephasinginduced four-wavemixinghas not only clarified the nature of these resonances, but it has also stimulated further experimental work. Following a brief review of the foundations of nonlinear optical theory, the correct procedure for iterative solution of the density matrix equations in the frequency domain is discussed and applied to dephasing-induced resonances ($ 2.1). Using a timedomain perturbative expansion of the density operator instead leads to a diagrammatic view of nonlinear processes. Double-sided diagrams of the density matrix evolution lead to a more physical understanding of the role of damping and can be adapted more easily to microscopic models of the dephasing mechanisms at issue ($ 2.2). Dressed atom models of pressureinduced four-wave mixing ($2.3) clarify its relationship to the collisional

11, § 21

THEORETICAL TREATMENT

45

redistribution of radiation. Moreover, this theoretical framework permits one to relax the impact approximation for collisions and make predictions for arbitrary field strengths and detunings from resonance. Finally, we review several nonperturbative approaches to nonlinear phenomena and conclude with notes on the effects of laser power and laser field fluctuations on coherent optical mixing as they relate to dephasing-induced resonances. Extensive experimentalefforts, both to characterize these resonances and to apply them as a unique tool, are reviewed in $ 3. The types of dephasing-induced four-wave mixing observed to date are classified into three categories as follows: (1) resonances between unpopulated excited states ($ 3.2), (2) population grating resonances ( 5 3.3), and (3) resonances between equally populated ground states (0 3.4). The predicted properties of each type and a summary of efforts to characterize them are interspersed with a selection of applications, including the study of dephasing mechanisms, velocity changing collisions, population dynamics, and novel atomic and molecular spectroscopy. In conclusion, $ 4 takes a view of the future in terms of areas in which theoretical work is needed and experimental anomalies persist. Several examples of as yet undiscovered dephasing-induced processes of interest are discussed. Some speculation on further applications of dephasing-induced coherence is included.

4 2. Theoretical Treatment 2.1. CONVENTIONAL DESCRIPTION OF COHERENT NONLINEAR OPTICAL MIXING

The purpose of this section is to illustratehow dephasing-induced resonances fit into the larger scheme of nonlinear optics. The traditional picture and theory of coherent processes are briefly reviewed to establish the appropriate language

and notation to understand dephasing-induced four-wave mixing. Several excellent reviews and texts with more detail are available (SHEN [1984], [ 19821, BLOEMBERGEN [ 19651). LEVENSON

2.1.1. Framework of the traditional theory In sufficiently strong electromagnetic fields the electronic response of materials can no longer be adequately described by linear response theory. Naively, one can think of electrons being shaken by the electric fields of incident

46

[It 8 2


radiation. With the aid of material nonlinearities, they can acquire Fourier components at all possible combinations of the applied frequencies describing their spatial oscillation. Ensembles of these accelerating charges tied in phase to the fields behave like antenna arrays and can radiate cooperatively at these new frequencies. In the language of quantum mechanics, incident fields create coherent superpositions of quantum states, which can reradiate coherently leading to nonlinear frequency generation. By analogy with linear electromagnetic theory, it is convenient to describe the electronic response by a material polarization P related to the applied fields E by susceptibilities. Conventionally

P = x")E

+ x('):EE + x'~':EEE+

*

,

(2.1)

where fl) is the usual linear susceptibility and 2'") are nonlinear analogs. The expansion of the polarization in powers of field makes sense when perturbation theory is valid. Far from material resonances the importance of the nth order correction to P scales approximately as (E/EB)%,where EB( ez/rgOh,)is a field magnitude characteristic of the electronic binding energy. Therefore, successive terms generally become smaller sufficientlyfast to solve for perturbatively when none of the incident fields is near material resonances. Naturally, near resonance the validity criteria for perturbation theory are more complex, having to do with the relative size of Rabi precession frequencies, material relaxation parameters, and field detunings from resonance. More precise conditions for the suitability of perturbation theory in the near resonant case have been discussed by many authors (e.g., DELONE and KRAINOV[ 19851, DICKand HOCHSTRASSER [ 1983a1). For the purposes of most of this paper we assume it to hold, although exceptions are considered in 5 2.4 and § 3.4. The susceptibility tensors x describe how efficiently a material couples incident fields E of arbitrary frequency and polarization,

-

E(w, r, t ) = hE,(r, t) exp(i(k * r - wt)). E,(r, t) is a complex field envelope function of space (r) and time (t), the subscript m refers to a Cartesian direction rii describing the field polarization. The field has frequency w and wavevector k. The case of four-wave mixing is discussed here for the sake of concreteness. Four-wave mixing results from third-order corrections to the polarizability in eq. (2. l), which for the example w, = w1 - o, + w3 would be written explicitly as q 3 T w 4 , r, t ) =

x$L(-

w4,01

-

x exp(i[(k, - k,

+ C.C. ,

02,

w3)Ek(w1)E:(o,)Em(o3)

+ k3) r - w4t]) *

(2.2)

11, § 21


47

where C.C. denotes the complex conjugate of the entire expression. x$L is one element of an 81 ( = 34) element fourth-rank tensor that governs the coupling E(w2),E(m3),and “output” E(m4)).The oscillating of four plane waves (E(o,), polarization P acts as a source of radiation E in Maxwell’s wave equation

where the definition PNL= P - x(’)Eis invoked and would be equal to Pj(3)(m4,r, t ) for our case. Given the fairly unrestrictive approximation that the envelope functions Em(r,t ) are slowly varying on the scale of a wavelength and oscillation period, a simple equation for the growth of “output” E(m4) can be derived from eq. (2.3). We can write aE.

2=

az

.2m2 j PNL(m, z, t ) exp( - i(kz - at)) kc2

1

~

(2.4)

for the growth of a wave propagating along z, where the backward wave is neglected (SHEN[ 19841). Rewritten for the four-wave mixing example,

(2.5)

where Ak = (k, - k, + k , - k4) * 2 is the wavevector mismatch and energy conservation w1- w2 + u3- m4 = 0 has been assumed. The intensity of the output wave varies as the square of its field and is therefore proportional to the intensities of each input wave and Jx(~))’.Photon momentum need not be strictly conserved in an extended medium, but naturally when it is conserved (Ak = 0) coupling is the strongest. Because of material dispersion, this “phase matching” condition can be difficult to satisfy and may limit the useful interaction lengths for nonlinear frequency conversion. Phase matching prescribes the direction of the generated wave precisely, leading to the characteristic directionality of coherent nonlinear optical mixing. 2.1.2. Calculation of nonlinear susceptibilities with damping

As pointed out in the introduction, it is difficult to include damping rigorously in standard time-dependent perturbation theory. Without damping only 24 terms are calculated for x(3) (ARMSTRONG, BLOEMBERGEN, DUCUINGand PERSHAN [ 19621). Since the density matrix describes a statistical ensemble

48


W,8 2

average, it is possible to describe how stochastic damping affects its evolution without precise knowledge of the damping interaction. From the density matrix it is easy to calculate the material polarization

P

=

NTrbp),

(2.6)

where N is the density of dipoles, p contains the dipole matrix elements, and p is the density operator. Diagonal elements of p represent state populations,

and off-diagonal elements describe coherences between states. The evolution of the density matrix is given by the quantum-mechanical Liouville equation

where the Hamiltonian is H = Ha + Hcoh+ Ha is the material Hamiltonian in the absence of fields, and the commutator [Ha, p ] vanishes when eigenstates of H, are chosen as a basis. Hcohincludes electric dipole interactions of the fields with the material, and Hrandomdescribes the source of damping. This latter term is incorporated phenomenologically in (2.7), a procedure which is valid when Hrandomobeys the relationship

where z, is the correlation time of Hrandom(BLOEMBERGEN [ 19651). Specscally, it is the time for correlation ( Hrandom(t)Hrandom(t- z)> to fall to zero. Physically, the inequality (2.8) guarantees that the perturbation strength, described by the precession rate (IHrandomI/fi),is sufficiently weak not to disturb the system appreciably before the perturbation loses memory and attempts to alter system in another direction. To treat the random perturbation phenomenologically also requires that the timescale z, on which it acts be well separated from the timescale on which the applied field acts. This criterion is expressed by the condition

I:[ 0, -G

,

(2.8a)

where QR is the precession rate of the applied field's Bloch vector. In the case of collisions the correlation time of the perturbation can be considered to be the collision duration z, . Relation (2.8a),forms the basis of the impact approximation, which allows collisional dephasing to be approximated phe-

11, § 21

49


nomenologically (BERMANand LAMB[ 19691). It can be rigorously demonstrated that these conditions lead to an exponential destruction of coherence as implicit in eq. (2.7) (SARGENT,SCULLYand LAMBJR. [ 19741). To determine the steady-state response of the density matrix, it is convenient to solve eq. (2.7) iteratively in the frequency domain. Corrections to p in nth order, p'"), with Fourier component w, = on- + w,can be written in terms of p'" - 1) (0, - ,). The appropriate equations are (BLOEMBERGEN and SHEN [ 19641) - iconp,$)(wn) = - iwklp,$)(on) -

-

'k/&)(%)

[Hcoh(W),

P'"-

1)1k/

(2.9)

and - icon&)(wn) =

R-pzA m

Rmk&'

m

-i [Hcoh(4, P'" ?I

- 1)(%

-

Alkk

9

(2.10)

where ok,and rk,are frequency differences and phenomenological damping rates between lk) and 11). The R, are population transfer rates from l j ) to li) . The unperturbed density matrix p ( O ) is typically given by the populations p$" in the absence of fields. Each iteration of (2.9) and (2.10) introduces an additional field factor in each subsequent correction to p. This justifies the expansion assumed in eq. (2.1) and means that x'") is simply related to p'") by (2.1) and (2.6). General expressions for x(3)resulting from repeated application of (2.9) and (2.10) are given by FLYTZANIS [ 19751. All 48 terms from such a treatment to obtain x ( ~ ) (- w,, o,- o,,w3) are written explicitly by BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781, including 24 terms not in the expression derived without damping (ARMSTRONG,BLOEMBERGEN, DUCUINGand PERSHAN[ 19621). These 24 terms can be arranged into 12 pairs, which are proportional to expressions of the form (rkl - rk, - rm1) which vanish in a system of only states Ik), [ I ) , and Im) when there is no proper dephasing (SARGENT,SCULLYand LAMBJR. [ 19741). In the special case of a two-level system this means that these extra terms cancel when the familiar relationship T2 = 2T, holds between population and coherence decay times, Tl and T 2 . We now derive expressions for x(3) specific to the case of Fig. la, which depicts an excitation diagram appropriate to dephasing-induced four-wave mixing resonances between unpopulated excited states le) and le') . Only terms in f 3 ) with three near resonant denominators are retained. The physics

50


-. le>-

-

W1

W1

Ig>

-

le>

le>

lg>

(b)

Fig. 1. Energy level diagrams and tuning schemes for several dephasing-induced four-wave mixing resonances. (a) Raman resonance between unpopulated excited states; (b) population grating resonance; (c) Raman resonance between equally populated ground states.

of these resonances is contained in the second order corrections to the density matrix, p$?(o, - m2). These are calculated by successive application of Hcoh(Ol) and ffcoh( - 0 2 )in eq. (2.9), where

Hcoh(0)

h - ‘pE 2

= -exp(iot)

+ C.C..

It is important to note that distinct resonant contributions to p222(w, - w 2 ) arise from both possible orderings of these perturbations. The coherence between excited states le) and le’) is then given by

(2.11)


11, § 21

51

The coherence p2(!J(20, - 0,)strictly responsible for coherent emission at 0,= 20, - o, is calculated from (2.11) by an additional application of Hcoh(ol) in (2.9). It is then simple to extract xC3)via eq. (2.6) to obtain

(2.12) where

pi:)

1 is assumed. Expression (2.12) shows the usual resonances at 0, = oeg when incident frequencies coincide with material resonances. The effect of damping on these resonances is merely to broaden the resonance width and reduce the peak amplitude. The right-hand term inside the brackets, however, describes a resonance where the difference between incident frequencies equals the difference between excited state frequencies. This resonance is unusual in several respects. First, the states involved, le) and le’), are both unpopulated in zeroth order. Second, the relaxation rate of coherence responsible is re,eand does not depend on electronic dephasing between ground and excited states. Most notably, the strength of the coherence between le) and le’) is proportional to proper dephasing - reg- re,g), and thus these resonances are “dephasing induced”. We defer a more detailed discussion of the quantitative predictions of eq. (2.12) and their experimental verification to 5 3.2. As can be seen by a comparison with the full 48-term formula for f 3 ) ( - w,, o1- cu,, 0,) (PRIOR,BOGDAN, DAGENAIS and BLOEMBERGEN [ 1981]), expression (2.12) represents 3 of the 48 terms. The 48 terms are reduced to 24, since we have assumed 0,= w,, and then reduced to only 3 terms when contributions nearly resonant in three denominators are assumed to dominate. (These triply resonant terms are selected by applying Hcoh in an order so that each is nearly resonant with a material transition.) One of the three terms, that corresponding to the term “1” in brackets, is not resonant in w1 - w,. The other two terms have been combined and belong to the pair of quantum-mechanical probability amplitudes that destructively interfere in the absence of pure dephasing. These two terms correspond to the damping ‘‘correction factors” (BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781) K2 of PRIOR,BOGDAN, DAGENAH and BLOEMBERGEN[ 19811. The resonances they represent are concrete manifestations of damping that cannot be described by perturbative theories of x(3) unless damping is added at each level of the calculation. =

o1= oePg, 0, = weg, and

52


111, s 2

It is instructive to relate eq. (2.12) to the general form for xC3) in eq. (1) of PRIOR,BOGDAN,DAGENAISand BLOEMBERGEN[ 19811. The notation becomes identical for It) = Ig), Ik) = l e ) , l j ) = le’) with o, = o, = ol, o, = o,,and w, = 0,.Next, we have omitted the “ordinary” resonance described by the left hand term in their parentheses and the local field corrections L = (n’ + 2)/3, where n are the refractive indices (SHEN[ 19841). [ 1981J also have a factor of PRIOR,BOGDAN, DAGENAIS and BLOEMBERGEN from 3 ! different possible permutations of applied fields (LYNCHJR. [ 19771) neglected here, whereas eq. (2.12) has an extra factor of f ( = 2 - 3, resulting Finally, the identity from a factor of two difference in the definition of Hcoh. (re,g - I‘, - I‘ge,)= 0 must be used to make the expressions equivalent so that ground state 18) has been assumed to have infinite lifetime in obtaining (2.12). 2.1.3. Other features of multiresonant nonlinear optical mixing We have already cited the importance of intermediateone photon resonances as a practical aid in enhancing four-wave mixing. Moreover, the introduction of damping under these conditions causes new resonances that, as will be shown in 3, make possible new types of spectroscopy and studies of damping mechanisms. It is the purpose of the present subsection to point out a number of additional considerations that can be important near material resonances. These are of help in interpreting the results of the dephasing-induced coherence experiments presented in 93, and they give a feeling for the richness of information in, and complexity of, resonant nonlinear mixing. 2.1.3.a. Znhomogeneous width One of the important contributions of xC3) processes to modern spectroscopy is reduction of inhomogeneouswidth as in Doppler-free two-photon absorption and four-wavemixing. In these cases, all velocity groups contribute equally, and the lineshapes can be analyzed in terms of more fundamental broadening mechanisms. With intermediate resonances, however, some velocity groups may be preferentially selected for excitation bandwidths less than the inhomogeneous width. The general case of electronically resonant coherent antistokes Raman scattering (CARS) in a Doppler-broadened medium is treated by DRUET,TARANand BORDB [ 19791, who classify contributions to xC3)according to the extent to which they are Doppler free. Similarly, other authors [ 1983b1) have dis(OUDARand SHEN[ 19801, DICK and HOCHSTRASSER cussed the case of inhomogeneous crystal field broadening and the line narrowing that can be realized in resonant nonlinear spectroscopy.

I L § 21


53

2.1.3.b. Saturation Moderately intense fields may be sufficient to make effects of higher order than f 3 ) significant, and the treatment according to eqs. (2.9) and (2.10) becomes impractical. On resonance in a two-level system, saturation begins when the Rabi precession frequency o,= fi - ' p E becomes comparable to the damping width r of the transition. The usual two-level result for power broadening (YARIV[ 19751) is equivalent to an expansion of p to all orders. Saturation of four-wave mixing has been treated by many authors (e.g., OLIVIERA, DE ARA~JJO and RIOS LEITE[ 19821, AGRAWAL [ 19831, GRYNBERG, PINARD and VERKERK[ 19841). Other papers that refer explicitly to the case of dephasing-induced resonances are reviewed in detail in § 2.4. 2.1.3.c. Eflects of intermediate state relaxation times When tuned far from resonance (detuning A %- reg), the intermediate state can be thought of as having virtual population with an uncertainty lifetime t 4 A . Near a one-photon resonance the actual relaxation rates re,,rgg and re,become important (YAJIMAand SOUMA[1978], OUDAR and SHEN [ 19801). For example, a coherently modulated real population p22)(oI - 02) can coherently scatter a field E ( q ) in a four-wave mixing process, and the population relaxation time TI = (ree + rgg)/2 therefore determines the linewidth of the corresponding resonance. In fact, the beautiful analysis and experiments of Yajima (YAJIMA[ 19751,YAJIMAand SOUMA[ 19781,YAJIMA, SOUMAand ISHIDA [ 19781) have demonstrated that one can determine both population and coherence relaxation times from resonant four-wave mixing lineshapes. It is, however, necessary to model the population dynamics to obtain such information. This type of four-wave mixing resonance is discussed in more detail in § 3.3 in the context of collision-induced population grating resonances. N

2.1.3.d. Discrimination against nonresonant terms Frequently, nonresonant contributions to nonlinear susceptibilitiescan interfere with the terms of interest and alter resonance lineshapes. Exploiting intermediate resonance selection rules by judicious choice of field polarizations can sometimes be used to suppress nonresonant background (SONG,EESLEY and LEVENSON[ 19761, OUDARand SHEN[ 19801). 2.1.3.e. Absolption and dispersion Near one-photon resonances, absorption of incident and emitted waves can be of practical importance in observing four-wave mixing signals. Changes in

54


s

[II, 2

phase matching efficiency can also be critical because of dispersion of the refractive index in the vicinity of material transitions (OUDARand SHEN [ 19801). This can determine the preferable geometry for a nonlinear mixing experiment (e.g., PRIOR[ 19801).

2.2. DIAGRAMMATIC PICTURE OF NONLINEAR OPTICAL PROCESSES

The perturbative expansion of the density matrix eq. (2.7) in the time domain (SLICHTER [ 19631,WARD[ 19651) leads to an alternative picture of dephasinginduced coherent phenomena that is physically clearer. Moreover, it has the advantages of being more easily applied to coherent transient phenomena (YEE and GUSTAFSON [ 19781) and to microscopic modeling of damping processes (OMONT,SMITHand COOPER[ 19721). In this section we follow closely the work of FUJIMOTO and YEE [ 19831 to show how the time domain formalism , which the leads to a diagrammatic representation of contributions to x ( ~ ) in circumstances leading to dephasing-induced coherence can be identified in a general way. With several simple rules, terms of f 3 ) can be identified and DRUETand TARAN[ 19771). calculated by inspection (YEE, GUSTAFSON,

2.2.1. Double-sided diagrams Equation (2.7) for the evolution of the density matrix can be transformed to a form which can be solved by direct integration (FUJIMOTO and YEE [ 19831). that Given an initial state at time to described by p(to) and a perturbation Hcoh occurs at time t - z, we can write &, at any subsequent time t as

where 4,s w,, - irk, and z is the t h e since the perturbation has occurred. Thus the first term represents the field-free evolution of the system, whereas the second term takes into account the effects of perturbations at all possible times t - z. Equation (2.13) can be solved iteratively,given an appropriate succession of perturbations Hcoh. One can find the (n + 1)st order correction to the density matrix, p(" I), due to a specific element of the nth order correction, PI;), via +

11, § 21


55

The A J z ) equal exp( - is2,z) and propagate the density matrix between interactions. At the time t - z of the initial field perturbation, the density matrix has been taken to be p p ) ( t - z) = A,(t - z - to)pii(to), where p,(ro) is the initial condition. Equation (2.14) expresses the physical situation where Hcohinteracts with the ket component of p'"), changing the state in which the density matrix evolves fram 11) ( m I to I k ) ( m1. Alternatively, Hcohcan interact with the bra component as in (2.15) to modify I Z) ( mI to I Z) ( n 1. Each of these can be responsible for creation of coherences and populations in (n + 1)st order. It is convenient to formulate a diagrammatic representation of the physics in eqs. (2.14) and (2.15). Since bra and ket can interact separately with the applied fields, each is represented by a vertical line where higher points denote later times. Field interactions are depicted by vertices with wavy lines as photons entering or leaving the vertex, as in Fig. 2 (cf. YEE, GUSTAFSON, DRUETand TARAN[ 19771). Using these double-sided Feynman-like diagrams any sequence of interactions contributing to p(") that does not include incoherent population feeding terms can be represented*. Naturally, there is a one-to-one correspondence between possible diagrams and terms in the density matrix expansion for x ( ~ ) , and this is worked out explicitly by PRIOR [ 19841. The 48 terms in x ( ~ ) (- o,,w1 - o,,w 3 ) arise from 3! possible time orderings of the field interactions, each of which can occur with bra or ket (23 variations). Steadystate terms in xC3)derived from the frequency domain expansion detailed in the previous section can be calculated from the diagrams by inspection using a few simple rules (YEE, GUSTAFSON, DRUETand TARAN [ 19771).

* Incoherent population feeding terms, like those involving pmm in eq. (2.10), can still be depicted in a more general diagrammatic scheme (MUKAMEL[1982], BOYD and MUKAMEL [ 19841).

56


(b)

(a) If>

<jl

If>

<jl

li>

<ji

li>

<jl

li>

<jl

TIME

I- li li>

<jI

(C)

Fig. 2. Possible field interactions with a system in state ti) ( j ( using double-sided diagrams. (a) Absorption by bra; (b) emission by bra; (c) emission by ket; (d) absorption by ket.

2.2.2. Dephasing-induced coherent behavior The 12 diagrams having all interactions with only bra or only ket have been referred to as “parametric”. In the case of four-wave mixing the 36 “nonparametric” diagrams can be associated with the sets of three sharing common resonant denominators. As discussed in 0 2.1 and identified in eq. (2.12), two of these three led to an extra resonant denominator, whose amplitude is proportional to ‘‘correction factors” which vanish in the absence of damping. Physically, the amplitudes corresponding to these diagrams are related because they derive from different time orderings of the same field interactions with bra

11, § 21


57

and ket. In the absence of stochastic damping processes, the evolution of bra and ket can be described independently without respect to the relative ordering of field perturbations. When damping is involved, the evolutions of bra and ket are coupled and the diagrams at issue can no longer be expected to cancel. YEE and FUJ~MOTO [ 19841 have identified pairs of diagrams that lead to dephasing-induced behavior by isolating diagram sets whose denominators have the correct algebraic structure for interference which is removed by damping. Their results serve to put pressure-induced coherent processes in a more general context, and so we describe them here in detail. They distinguish between two types of dephasing-induced behavior, both of which can be understood in terms of removal of destructive interferences between quantummechanical amplitudes, and therefore cannot be described by single-sided density matrix diagrams. The first type involves a “local” interference of contributions to p@), meaning a cancellation only for a specific choice of incident field frequencies. This interference is removed by damping. The second type involve terms that cancel for any set of incident frequencies but manifest “extra” resonant denominators in the presence of pure dephasing. These, of course, are the ones that we have focused on to this point. The algebraic forms for pairs of contributions to p(”) that can lead to these behaviors can be used to identify all possible diagram pairs for dephasinginduced coherence. The first type of behavior results from contributions like pl‘“’ + d”’= M[p2(zm1 - O a b + irab) + p1(zm2 - mcd + ircd)l x (Em1 - ~ , , + i ~ , ~ ) - ’ ( C ~ ~ - m ~ ~ + i(2.16) ~ ~ ~ ) - ~ ,

where pi are dipole matrix elements and M contains all remaining denomi[ 19841). If M has no nators, fields, and matrix elements (YEEand FUJIMOTO nearly resonant factors when the term in square brackets vanishes, then p(”)has a minimum when the condition pZ(cwl

-

+ p1(z02 - O c d )

=

(2.17)

is met. The corresponding coherent process has amplitude proportional to p 2 r a b + p l T c d , and the interference implicit in (2.17) is relaxed by increased dephasing. Strictly, such behavior initiated by collisions should be termed pressure enhanced, since the interference is not complete even in the absence of pure dephasing. An example of such a local interference would be in the two-photon absorption from 3s to 4D in sodium vapor (BJORKHOLM and LIAO[ 19741).

58

s

[II, 2


Both fine-structure components of 3P can serve as intermediate levels, and amplitudes involving each of these contribute to coherent two-photon absorption x ( ~ ) . When w1 is tuned between 3P,,, and 3P3,, and 0,+ W2 = 03s 4D9 a particular value of o,results in an interference of the contributions from 3P,,2 and 3P3,, that could be removed by collisions. One can think of the collisions as broadening the intermediate states to encompass frequency values where the condition (2.17) will no longer be met. Arbitrary numbers of diagrams could interfere in this way to permit dephasing-induced behavior of this kind. The second type of behavior takes the algebraic form

py) + p p ) = hf(,%o, + ,%02 - me,- + ire,-)x [(xu, - o a b - irab)- + (zm2 - mcd + ired)- ‘1

9

(2.18)

where now the terms contain a common resonance factor, which cancels when me,- = mob + mcd irrespective of the field frequencies. Equation (2.18) can be rearranged to be similar to (2.1 l), and it also predicts the appearance of an extra resonance at Zol + Co2 = we,-, whose peak amplitude goes as (rob

+

rc,

- ref)/Cy

Figure 3 displays the forms of pairs of subdiagrams that can exhibit these types of algebraic resonance structures and therefore pressure-induced behavior. As can be seen, the dephasing-induced “extra” resonances (involving Figs. 3f and 3g) on which we have concentrated are special cases of the first type of behavior. Experimental demonstrations of the types of resonance covered by Fig. 3f will be discussed in $ 3.2 and those characteristic of Fig. 3g in $3.3. A third type of pressure-enhanced coherent resonance has been observed that cannot be characterized by these diagram pairs, since more than one participating state must be initially populated; this is reviewed in § 3.4. To my knowledge no dephasing-induced coherent phenomena represented by Figs. 3a to 3e have been reported. 2.3. RELATIONSHIP OF PRESSURE-INDUCED COHERENT FOUR-WAVE MIXING TO COLLISIONAL REDISTRIBUTION

The spectral redistribution of near resonance light by collisions has been studied extensively (HUBER[ 19691, MOLLOW[ 19691, CARLSTEN, SZBKEand RAYMER [ 19771). The analogy with collision-induced four-wave mixing has been made by several authors (PRIOR, BOGDAN, DAGENAISand BLOEMBERGEN [ 19811, GRYNBERG [ 1981a,b]) and, since it is both qualitatively and quantitatively useful, we elaborate on it here.

c

11, 21

59


TYPE I

i

i

i

i

(b)

(C)

TYPE II i

i

i

i

(g) Fig. 3. Schematic representation of all possible subdiagrams exhibiting dephasing-induced behavior. Type-I and -11 behaviors are described in text. (Horizontal wavy lines can represent either absorption or emissions.) (After YEE and FUJIMOTO [1984].)

2.3.1. Dressed atom model of collision-induced coherence The interpretation of pressure-induced resonances in four-wave mixing using [ 1981a,b], MIZRAHI,PRIOR and a dressed state picture (GRYNBERG MUKAMEL [ 19831) has provided a great deal of insight into the nature of these resonances and clarified their similarities to, and differences from, collisional

60


PI, § 2

redistribution. Moreover, the dressed state formalism is not restricted in validity to the impact approximation (§ 2.1.2), and quantitative analysis of the resonance outside the impact regime can be used to predict the far wing detuning dependence of collision-induced four-wave mixing. Conversely, measurements of the detuning dependence of pressure-induced coherences can be used to extract collision potentials (see $3.2.3). An additional feature of the dressed atom model is that, since the laser-atom interaction is already diaPINARD gonalized, it facilitates the treatment of strong fields (GRYNBERG, and VERKERK[ 19841). Using a dressed atom model, GRYNBERG [ 1981a,b] has reproduced the dephasing-induced coherence of (2.1 1) responsible for four-wave mixing resonances between unpopulated excited states. His derivation is summarized below both because the theoretical approach is generally useful in nonlinear optics and because the way that collisional dephasing enters is instructive. Physically, collisional dephasing of material levels corresponds to population transfer in the dressed level picture, which can create a coherence between dressed states. The dressed state analog to Fig. l a is illustrated in Fig. 4, where the dressed states la), Ib), and Ic) are associated with Ig), le), and le’) in zeroth order. A and A’ are the detunings of fields E ( u 2 )and E ( o J from oegand o,.,,and n,n’ are field occupation quantum numbers for E(o,)and E(w,), respectively. 4 A ) are given The dressed eigenstates to first order of perturbation theory (o, by

t

Fig. 4. Dressed state energy level diagram corresponding to Fig. la. l a ) , jb), and Ic) are meant to correspond in zeroth order to ( g , n , n ’ ) , ( e , n - l , n ’ ) , and (e‘,n,n’- l ) , respectively.

11, I21


61

where w, and &c are the resonant Rabi frequencies h-’pegE(w2) and h - ‘pe.gE(w,).In the impact approximation (w,,A 4 l / ~ = )collisions , can be considered to dephase material eigenstates ie) and lg) by different fixed phases $e and qg due to the different scattering potentials for le) and 18) during the collision (BERMAN[ 19781). The difference $e - qg is given by (2.20) where Ueg(t)is the energy level spacing between le) and ( g ) while in the field ofthe perturber. After the collision, ia,n,n’) has become (GRYNBERG [ 198 lb])

(2.21) The state 1 a,n,n‘) has a non-zero projection on I b ) and ic), signifying that the collision redistributes population amongst dressed states. From (2.21) Grynberg calculates the population transfer and coherence between dressed states caused by the collision to be (2.22)

- ORWR --

444’

[ 1 - exp(i&)

where (pee. = $eg damping parameters rub

=

+ 1 - exp(i$,.,)

- (1 - e~p(i$~,,)l ,

(2.23)

In the impact regime the usual phenomenological

r are defined by

( - exp(i$ub)>

9

(2.24)

where ( ) denotes an average over all possible collision trajectories. The and therefore pbc energy denominator appropriate to (2.23) is (A - A‘ + iree,), has a resonance when detunings A and A‘ are equal. Formula (2.23) then displays the interfering terms of eq. (2.11) that describe the dephasing-induced coherence. The corresponding collision-induced four-wave mixing resonance

62


PI, § 2

can be thought of as a level crossing resonance (A = A‘) of the dressed states, where collisional population transfer creates a coherence. Note that eq. (2.22) can be used to calculate the collisional redistribution of radiation (MOLLOW [ 19771). It was pointed out by MIZRAHI,PRIORand MUKAMEL [ 19831 that some care must be taken in using the foregoing interpretation for cooperative phenomena, since the coherence has been derived for a single dressed atom. These authors also used a dressed state picture to model the nonlinear susceptibility,solving the Liouville equation with a tetradic T matrix formalism (MUKAMEL[ 19821). They show that both pressure-induced single-atom resonances and coherent cooperative four-wave mixing resonances occur for w, - 0,= mee, (i.e. A = A’), each resulting from collisional dephasing which removes destructive interferences. These resonances differ, however, in several important respects: The coherent process is directional (must be phase matched), and emission occurs at w, = 2 0 , - 0,.The single-atom radiation is emitted isotropically and takes on the atom transition frequency and width. Moreover, the pressure and detuning dependences of these resonances are different. Both of these authors’ dressed state calculations can apply for large detunings where the impact approximation fails, and they can therefore be used to model the detuning dependence of pressure-induced four-wave mixing from a microscopic point of view. A detuning independent damping width r (eq. [ 2.241) can no longer be defined, and it is necessary to use eq. (2.20) and a model potential Ueg(t)to evaluate populations (2.22) and coherences (2.23). LISITSAand YAKOVLENKO [ 19741 evaluate (exp(i$,,)) with a stationary phase approximation, since cancellation is observed for all $eg not near zero [ 19791). This when averaged over collision trajectories (cf. YEHand BERMAN means that only times t o , where (2.25) contribute to population transfer and coherence. The times to during collisions therefore correspond to level crossings between dressed states, and the pressure-induced four-wave mixing can be understood in terms of “extra” level crossing resonances. In fact, the form of eq. (2.22) evaluated using the stationary phase approximation is equivalent to the Landau-Zener curve crossing formula (LANDAU and LIFSHITZ[ 19741). The interpretation of (2.25) in the bare-atom picture would be that the “molecule” formed by the atomperturber system at bond distance corresponding to to has an eigenstate

11, § 21


63

separation now in resonance with the laser frequency. Population transfer and creation of coherence therefore become possible during the collision. From this point of view it becomes clear that the details of the interaction potential in the nonimpact (“quasi-static”) collision limit must have a profound effect on pressure-induced four-wave mixing (see also $3.2.3). Grynberg’s calculations show that the coherence responsible for dephasinginduced four-wave mixing resonances between “unpopulated” excited states falls off as fast or faster with detuning than the collisional population of these states. Equal fall-off would be observed in the special case where le) and le’) experience the same collision potentials ($ 3.2.3). In general, where le) and Ie’) have very different electronic potentials, it would be difficult to observe dephasing-induced four-wave mixing outside the impact reghe. The tetradic T matrix formalism (MUKAMEL [ 19821) also permits calculation of the detuning dependence in the quasi-static regime. This approach is based on computing the 2n-time correlation function from a Hamiltonian describing the microscopic interaction potentials. Multiphoton process lineshapes are obtainable through a variety of approximation schemes. Among these is a “factorization approximation”,where the multiphoton lineshape can be decomposed into single photon lineshapes. This makes contact with the large amount of work done on linear absorption and fluorescence far from resonance (ALLARD and KIELKOPF[ 19831). Using this technique, BOYDand MUKAMEL [1984] have calculated the absorptive (imaginary) part of f 3 ) pertaining to an incoherent pump-probe experiment (HILLMAN,BOYD, KRASINSKI and STROUD JR. [ 19831). This experiment is analogous to the coherent resonances which are the subject of this review in that it cannot be described without the use of double-sided diagrams. We return to this point in $4.2.

2.4. FIELD-INDUCED RESONANCES

So far we have reviewed a number of theoretical treatments of nonlinear optical processes and illustrated how external stochastic perturbations can affect near resonant coherent processes. The most dramatic manifestations of damping are dephasing-induced extra resonances, which can be understood qualitatively and quantitatively in a variety of ways. The effect of disturbances such as collisions is to destroy the phase coherence of oscillators so that cancellations of amplitudes for nonlinear mixing are no longer exact. It is natural to ask whether stochastic fluctuations of the fields intrinsic to the

64


[II, $ 2

generation of coherences can also remove this interference and cause “extra” resonances. This question and other issues related to laser fluctuations are addressed in 5 2.4.1. In 5 2.4.2, we discuss strong-field effects, which are pertinent here for several reasons. First, intensity effects can modify dephasing-induced coherent phenomena. Second, collisionally assisted higher-order nonlinear phenomena can simulate dephasing-induced four-wave mixing and in some cases can be difficult to distinguish experimentally. In addition, several new types of dephasing-inducedcoherent phenomena have been predicted that are observable in the presence of strong fields. We briefly touch on nonperturbative theoretical treatments of coherent processes that are important near resonance. 2.4.1. Fluctuation-induced extra resonances Several authors have treated the effects of laser linewidth on coherent Raman spectra using phenomenological convolution procedures (e.g., YURATICH [1979], TEETS [1984]). EBERLY[1979] has reviewed the effect of laser fluctuations on single-atom phenomena, but the influence of field phases on coherent four-wave mixing has only recently received attention. A more complete accounting of the effect of fluctuations on coherent Raman scattering shows that laser linewidth effects are not simple (DUTTA [ 19801, AGARWAL and SINGH[ 19821) and that resonance intensity enhancements by field fluctu[ 19761) are also ations as in multiphoton absorption (LAMBROPOULOS expected. A two-level model of the pressure-induced “extra” resonances in four-wave mixing (PIER4) has been proposed by AGARWALand COOPER[1982]. Although these resonances require the participation of three states by definition, it is possible to project the dynamics onto a two-level space. This model facilitates the descriptionof the effects of laser fluctuations on pressure-induced resonances. These authors use phase diffusion models for the incident laser fields, which are assumed to be uncorrelated. There, laser fluctuations alone cannot cause resonances at o1- w, = we,e,and collisions are still required. The laser bandwidths are simply predicted to add to the PIER4 width, although the resonance intensity should depend on the laser fluctuations. AGARWAL and KUNASZ[1983] appear to be the first to have predicted coherent resonances induced solely by laser fluctuations. They branded these by the acronym FIER, making analogy to PIER4. Their hypothesized fourwave mixing resonances differ greatly from the dephasing-induced resonances we have discussed so far. Although they occur at o,- o, = we,, they are

11,s 21


65

nonparametric, the emission being at atomic frequences weg and oeg,. The phase-matching criteria are also different from PIER4, and they are destroyed rather than enhanced by collisions. Such resonances confirm the qualitative speculation that the “relaxation associated with laser fluctuations” (AGARWAL and SINGH[ 19821) can be responsible for cooperative phenomena. More recently, the elegant work of PRIOR,SCHEKand JORTNER[ 19851 has shown that stochastic phase fluctuations of the pump fields are equivalent to dephasing processes such as collisions. Specifically, the laser bandwidths from a phase diffusion model of the laser field cause exponential relaxation of the off-diagonal elements of the density matrix. The conventional formalism of 0 2.1 can then be used to calculate x(’). Since the phase diffusion widths y1, y2, y3 for fields E(w,), E ( 0 2 ) and E(w3)enter in a way similar to collisional widths r, analogous results are obtained. First, stochastic fluctuation-induced extra resonances (SFIER) occur at w1 - w, = we,, which have the same emission frzquency and phase matching criterion as PIER4. These resonances have amplitudes proportional to clustered stochastic width (yl + y3 - yI3), where y,3 is the appropriate convolved phase diffusion width. This expression is reminiscent of the correction factors (reg + rge, due to damping in eq. (2.12). For uncorrelated laser fields E(q) and E(w3), y13 = y1 + y3 and extra resonances like PIER4 do not occur, in agreement with previous predictions (AGARWALand COOPER[1982]). When the fields are correlated, yI3 # y1 + y3 and coherent SFIER occur. Physically, ifthe fields are correlated, then some time orderings of interactions are preferred and double-sided diagram pairs that otherwise interfere are no longer exactly cancelling. PRIOR, SCHEKand JORTNER[ 19851 also predict that in this limit of correlated fields, the laser linewidths do not simply add to the resonance width, an observation recently confirmed for coherent two-photon absorption (ELLIOIT,HAMILTON, ARNETTand SMITH[ 19851). 2.4.2. Higher-order power and dephasing-induced resonances Near one-photon resonances even modest intensity lasers can cause saturation and the breakdown of perturbation theory. It is therefore necessary to develop nonperturbative theories of nonlinear optical mixing. The traditional approach for strong fields interacting with a two-level system has been to describe the dynamics in a rotating frame, where the physical response is nearly time independent in the electric dipole and rotating wave approximations (SLICHTER[ 19631). Several authors have transformed the Liouville equation to obtain such a description (DICK and HOCHSTRASSER[1983a],

66


[I4 § 2

WEITEKAMP, DUPPENand WIERSMA[ 19831). DICKand HOCHSTRASSER [ 1983al develop a method to treat some fields perturbatively while allowing others to be strong. Since they assume phenomenological damping rates, their results are valid until o, 1/rc, which is typically not restrictive. They have applied the formalism to calculate power effects on lineshapes in dephasinginduced coherent three-wave and four-wave mixing. They also predict powerinduced resonances for o,- o,= oege in the absence of damping when one field is strong (a xc5)effect). Scanning the strong field through resonance tends to provide information about the light source while scanning the weak field yields traditional spectroscopic data. Some of the results of their calculations are presented in 8 3.2.2, where their experimental work is detailed. Another nonperturbative theory was used by WEITEKAMP, DUPPEN and WIERSMA [ 19831 to calculate picosecond coherent transient nonlinear mixing. There, a careful accounting of damping is not necessary, since the time ordering of field interactions is prescribed and dephasing is not required to see the resonances analogous to those which are the subject of this chapter. Other authors have proposed schemes to trigger four-wave mixing by radiative relaxation (FRIEDMANNand WILSON-GORDON[ 1982, 19831, WILSON-GORDON and FRIEDMANN [ 1983, 19841). They propose generation of spontaneous scattering by a three-photon process or by pumping a level le” ) with a strong nonresonant field to initiate the damping to induce four-wave mixing. Strictly, these are higher-order processes that should be described by x ( ~ ) x, ( ~ ) etc. , ACARWAL and NAYAK[ 19841 have made analogous predictions of radiative relaxation-induced nonlinear mixing using a nonperturbative theoretical model. A different sort of damping-induced coherence involving high powers has been proposed by PEGGand SCHULZ[ 19831. There, a two-level atom is probed by a strong amplitude-modulated field with o,% T,A. For a particular value of modulation frequency an interference between balanced pathways results in no coherence between levels, and a coherence can be induced by spontaneous emission or collisions.

-

11,

I 31

EXPERIMENTAL RESULTS

67

6 3. Experimental results 3.1. CLASSIFICATION OF OBSERVED DEPHASING-INDUCED COHERENT

PROCESSES

Nonlinear optical techniques have become increasingly useful with the rapid technological improvements in lasers. We have seen in $ 2 that for a quantitative understanding of resonant effects, damping must be properly taken into account. In the present section we review only the most extreme demonstrations of this point, experimental observations of coherent phenomena that do not even occur without damping. The resonances at issue are classified into three types, schematically depicted in Figs. la-c. The first observed and characterized were four-wave mixing resonances between initially unpopulated excited states ($ 3.2). These have been coined PIER4 (pressure-induced extra resonances in four-wave mixing) or DICE (dephasing-induced coherent emission) and are described by the diagram pairs of Fig. 3f. Next, we discuss completely degenerate frequency resonances due to collision-induced population gratings ($ 3.3). Resonances of this kind are commonly used in phase conjugation and measurement of relaxation parameters. The full role of damping has only recently been appreciated, and double-sided diagrams of the form in Fig. 3g are required to describe these resonances. Finally, we document a third type of resonance that would not occur without damping: collisioninduced coherent Raman resonances between equally populated ground states ($3.4). These cannot be reduced to any of the diagram pairs of Fig. 3, since initial population in two states is required. These three classifications subsume all of the dephasing-induced coherent nonlinear optical resonances so far reported experimentally. Many other types are possible, and several of these are discussed in $ 2 and $ 4.

3.2. EXTRA RESONANCES BETWEEN “UNPOPULATED EXCITED STATES

The theory of extra resonances between “unpopulated” excited states was presented in $ 2 . They have been labeled “extra” because of the additional resonant denominators in the correction factors K , and K , of BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781. The extra denominator is the one in curly brackets in eq. (2.12) leading to dephasing-induced resonances at Oefe= w1 - 0,.States le) and le’) are initially unpopulated, although they

68


[II,8 3

must be populated by the fields to some extent in order to produce a coherence between them (ANDREWSand HOCHSTRASSER [ 19811, DAGENAIS [ 19821). Although coherence cannot be created without population, merely to “create the possibility of energetically accessing excited states with a detuned laser” (DAGENAIS [ 19821)by damping is insufficient to create the coherences integral to dephasing-induced coherent mixing. A simple example illustrates that collisionally induced populations do not always contain the information necessary to describe extra resonances in four-wave mixing. The coherence p,‘.’,’ of (2.1 1) has a resonance for w1 - w, = a,.,, even though populations pi:) and pi.’,! are essentially frequency independent for A, A‘ % reg, There are nonlinear mixing experiments that require damping-induced population transfer to prepare the state from which they occur, and these should not be confused with dephasing-induced coherent processes. Examples of these are cases where collisions help to populate states from which coherent mixing processes can subsequently take place (DAGENAIS [ 19811, EWARTand OLEARY[ 1982, 1984a,b]). These do not involve removal of interferences by damping and are not reviewed here. 3.2.1. Collision-inducedresonances between ’Pfie-structure components in Na vapor The level diagram appropriate to coherent Raman resonances between unpopulated excited states is that of Fig. la. Here, we take (8)to represent the 32S ground states of sodium and l e ) , le’) to denote the 32P1,2, 3,P3,, states, respectively. For the purposes of this discussion we ignore the additional level degeneracy, since it does not affect the basic physics. The relevant susceptibility is given by eq. (2.12), the resonance of interest being that where w1 - w, = we,e.The amplitude is proportional to (reg + fggt - re,,), which vanishes when all of the damping is caused by spontaneous emission, &, = f,y. The idea behind the experiments to test the theory is to add inert buffer gas pressure p so that 4,becomes (3.1)

and the interference of amplitudes is no longer complete. The pressurebroadening parameters have been measured by trilevel echo (MOSSBERG, WHITTAKER, KACHRUand HARTMANN[ 19801) to be yeg

x ygef z

=

y = 5.5 MHz/ToIT.

11, J 31


69

The PIER4 experiments were done under the following conditions : A G rrlz,,

(3.2)

q z , ok 4 reg, refg 9

A, A’ P

reg,re’g,

A, A’ P k , . U, k , . u

.

(3.3) (3.4) (3.5)

Conditions (3.2) through (3.4) guarantee that the impact approximation (2.8a) used to derive eq. (2.12) is valid, provided buffer gas pressures are sufficiently low that three-body collisions are negligible. The criteria of (3.3) also preclude saturation phenomena. Conditions (3.4) and (3.5) mean that the lasers are tuned “far” from one-photon resonances at wI = we,gand 0,= weg.This avoids confusion amongst the resonances implicit in (2.12), minimizes problems resulting from resonant absorption, and simplifies interpretation, since far outside the Doppler width k . u no velocity group is preferentially selected. Using (3.1) through (3.5) with (2.12), the output intensity 1(04)can be expressed as (BOGDAN,DOWNERand BLOEMBERGEN [ 1981bl)

The first experiments demonstrating pressure-induced resonances at w1 - w2 = we,e(PRIOR, BOGDAN,DAGENAISand BLOEMBERGEN [ 19811) were done with pulsed dye lasers separated by the 17 cm- line structure splitting in sodium. Because of high-intensity x(5)effects (see also DAGENAIS [ 19813) and the large laser linewidths, the quantitative predictions of (3.6) were not verified. The experiment was therefore repeated with high-resolution, single mode continuous dye lasers. The apparatus used is approximately as shown in Fig. 5 , where one of the lasers has 100 MHz resolution and a monochromator is used before the photomultiplier tube, since o, differs from w , by 17 cm- I . A nearly forward scattering geometry is used (PRIOR[ 1980]), where spatial discrimination of the output is possible even for completely degenerate mixing. Since beams at w, ,w; and w2 are nearly copropagating, the experiment is nearly Doppler free so that the omission of Doppler shifts in (2.12) is valid. Residual Doppler width is discussed in detail in 5 3.3 and 5 3.4. Phase-sensitive detection is used along with polarization discrimination

70


[II, § 3

LOCK- IN AMPLIFIER

cw

DYE

INPUT AND OUTPUT

MHZ VIEWS INTO OVEN

Fig. 5. Experimental apparatus for high-resolution, pressure-induced four-wave mixing in sodium vapor. (After ROTHBERG and BLOEMBERGEN[ 1984al.)

(E(w4),E ( q ) IE(o;), E(o,))to eliminate background scattering. Typical conditions were several mTorr of sodium with 10-1000 Torr of helium and laser detunings of 15 GHz. The continuous-wave laser experiments verify (BOGDAN,DOWNER and BLOEMBERGEN[ 1981bl) the Z :, intensity dependence of four-wave mixing signal predicted by eq. (3.6). Fig. 6 illustrates the measured behavior of PIER4 with pressure. The ratio of the peak signal on resonance to the nonresonant signal saturates with helium pressure at the value of 4 as prescribed by (3.6). The full width increases linearly at the predicted rate of 2 y = 11 MHz/Torr, once yp becomes large compared with the instrumental width. The integrated intensity increases linearly with pressure, also in accord with (3.6). These experiments served to c o n h the theory of damping for nonlinear quantum-mechanical processes and to resolve the controversy over signs of damping terms in x ( ~ ) as , discussed in § 1.2. The results also demonstrated the feasibility of measuring excited state splittings and line broadening of excited state transitions using four-wave mixing. 3.2.2. Thermally induced excited state coherent Raman spectroscopy of molecular crystals Pure dephasing-induced four-wave mixing in solids was demonstrated by ANDREWSand HOCHSTRASSER [ 1981J and labeled “DICE” (dephasing-

11, § 31

71


P la LL

0

25

50

75

10

He PRESSURE ( t O r r )

Fig. 6 . PIER4 signal characteristics as a function of buffer gas pressure. (a) Ratio of peak height to nonresonant signal; (b) resonance full width; (c) integrated intensity of resonant signal. (After BOGDAN,DOWNER and BLOEMBERGEN [1981b].)

induced coherent emission). The coherent Stokes Raman resonances (CSRS) between excited states of pentacene in a benzoic acid crystal they observed are the condensed-phase analog of PIER4. In this case le) and le’) represent differentvibrational levels of pentacene in its first excited singlet state. The role of collisions is played by phonons, which can be turned “on” and “off’ by varying the crystal temperature. These resonances are spectroscopically useful for several reasons. First, they can be used to alleviate spectral congestion and to resolve vibrational structure in complex systems where molecular beams and polarization spectroscopy are

12


[II, 8 3

not appropriate. Since they are not subject to broadening by electronic dephasing reg and re.,,it is possible to resolve vibrational bands of electronically excited states and obtain structural information that would be difficult to get in any other way. Second, one can measure pure dephasing rates between excited state pairs without measuring both longitudinal and coherence relaxation times T , and T, independently. Third, one can study the dephasing mechanism (phonons) that gives birth to the resonances. Examples of all three have been demonstrated by Hochstrasser and co-workers and are reviewed below. In addition, it is also possible to see line narrowing in inhomogeneously broadened systems as has been worked out by several authors (DICKand HOCHSTRASSER [ 1983b], OUDARand SHEN[ 19801). For the case of dilute pentacene in a benzoic acid host crystal, le) represents the vibrationally unexcited ( u = 0) level of the first excited singlet state S , , and le') has in addition one quantum ( u = 1) of a 747 cm- vibration. A level Ig') corresponding to the same vibrational mode in the ground state (at 755 cm- I ) has been incorporated into the theory (ANDREWSand HOCHSTRASSER [ 1981]), but the essential physics of the DICE resonances is contained in the nonlinear mixing susceptibility of eq. (2.12). Two pulsed dye lasers with o,- o,in the vicinity of 750 cm- are used to observe coherent Stokes (CSRS) and antistokes (CARS) resonances. At 4.5 K a CSRS resonance at w, - o,= o,, = 747 cm- is observed when tuned directly to the ground state to S , transition (0, = oeg). Fig. 7 plots four-wave mixing intensity versus o,- o,and shows no such resonance at 4.5 K when detuned by 16.8 cm- I . On resonance, S, becomes populated and CSRS subsequently occurs from S , as with the collisionally initiated f 5 ) processes (DAGENAIS [ 19811). A ground = 755 cm- ' is observed for any state CSRS resonance at w, - w, = ogeg detuning as would be expected. When the temperature is raised at 16.8 cm- ' detuning, a thermally induced excited state CSRS resonance appears in Fig. 7 at 747 cm- This resonance can be regarded as the removal of a destructive interference of contributions to x(3) by crystal phonon perturbations. Characteristically, the resonance amplitude is given by a difference of damping factors (ANDREWSand HOCHSTRASSER [ 1981]),

T(T)= re,+ re,g - rere, ,

(3.7)

cj ;(c.i

where = + r;i) + G, with r representing the pure dephasing contribution and T denoting temperature. If re,z re,,and the ground state is stable (r,, = 0), then (3.7) can be rewritten T ( T )=

r;g+ r;,,,

(3.8)

11,

I 31


73

A-16.8crn-'

761 755

747

>

k U J

Z

w I-

s UJ

Y l UJ V

X

12.5. 24.7K

A

'.

Fig. 7. CSRS spectrum as a function of temperature for A = 16.8 cm- Note that the band at 747 cm- I grows relative to that at 755 cm- with increasing temperature. (After ANDREWS and HOCHSTRASSER [ 198 I].)

'

demonstrating that the resonance is dephasing induced. The ground state CSRS resonance can be used for a normalization to correct for laser intensity effects on the dephasing-induced resonances. The ratio R ' of dephasinginduced resonance peak intensity to that for the ordinary CSRS can be calculated from the respective third-order susceptibilities, leading to the result (ANDREWS and HOCHSTRASSER [ 19811

74


[II, § 3

20

10

T(K)

Fig. 8. Growth of pure dephasing r ( T ) as a function of temperature, as derived from experimental measurements of R’ and eq. (3.9). The solid line is the fit to Arrhenius form (3.10). The point X represents twice the observed pure-dephasing contribution to the 18) -+ le’) transition [1981].) and is an independent estimate of T(T).(After ANDREWSand HOCHSTRASSER

In Fig. 8, expression (3.9) is plotted versus temperature for the known values ogk- oefe = 8.4 cm- I , re,e = 0.22 f 0.02 cm(HESof A = 16.8 cmSELINK and WIERSMA[1980]) and the measured values R ‘ at different temperatures. The solid curve in Fig. 8 is a fit to the Arrhenius form

’,

T(T)

=

9.7cm-’exp(- 13.8cm-’/kBT),

(3.10)

where k, is Boltzmann’s constant. The value of 13.8 cm-’ is consistent with the dephasing being associated with a known phonon of that frequency in benzoic acid (HESSELINK and WIERSMA[ 19801). The exponential activation of dephasing shown in (3.10) has been observed in coherent transient experiments (AARTSMA and WIERSMA[ 1976]), and the theory of optical dephasing has been stimulated by such data. For example, microscopic calculations of the dephasing using Redfield relaxation theory and a model Hamiltonian have been done by DE BREEand WIERSMA[ 19791). Other dephasing-induced Raman studies of molecular excited states were done in ferrocytochrome-C in solution by ANDREWS,HOCHSTRASSER and

11, § 31


Fig. 9. CA and CSRS spectra of ferrocytochrome-C in solution at pump (0,) wavc..ngth 4200A. (a) Concentration 9.6 x lo-' M; (b) concentration 18 x M. (After ANDREWS, HOCHSTRASSER and TROMMSDORFF [1981].)

TROMMSDORFF [ 19811. These studies probe the n-+ Izr electronic excitation known as the Soret band. The vibrational structure in the absorption spectrum is not resolved and the nature of the broadening mechanism was not known. The CSRS spectra are shown in Fig. 9. The band at 1362 cm- is known to be a ground state vibration of ferrocytochrome-C, which indicates the oxidation state of the iron atom. A new band at 1339.5 cm- ' is observed and assigned to the corresponding excited state vibration. On the basis of the third-order power dependence, it was concluded that this resonance between initially unpopulated excited levels must be dephasing induced. In general, however, it is not easy to identify dephasing-induced nonlinear mixing without varying the dephasing mechanism experimentally (e.g., APANASEVICH [ 19841). The linewidths are useful in analyzing the relaxation rate of the Soret state. The ground state coherent Raman resonance has half width rgtg = 5 cmwhereas the excited state dephasing-induced analog has re,e = 8.5 cmThe latter number puts a lower limit on the Soret state lifetime at 300 fs. If the ground-state and excited-state vibrations are dephased at the same rate

76


(rLfg = &,J and the measured 5 cm-

width

[II, I 3

is due to pure dephasing

(r',,= I",,),then the Soret state does not decay for at least 700 fs. These

times were much longer than previously thought, and it is clear that the Soret band is not lifetime broadened. TROMMSDORFF, ANDREWS, DECOLAand HOCHSTRASSER [ 19811 have compared these results with those in femcytochrome-C. Since the DICE resonances are typically measured with high-intensitypulsed dye lasers, it is important to understand the intensity behavior of these resonances to interpret them quantitatively. DICK and HOCHSTRASSER [ 1983al have modeled the experimental case of pentacene in benzoic acid (see earlier), using the nonperturbative theory described in 0 2.4.2. One of the fields is chosen to violate the conditions (3.3) and (3.4), which preclude saturation behavior by taking it to have much smaller detuning than in the actual experiment. A sample result from their work is illustrated in Fig. 10, showing a Stark splittingin both the DICE and ordinary material resonances. Dick and Hochstrasser also point out that power-induced resonances can occur at frequencies typically associated with DICE resonances even without dephasing.

a w-0 I

-5

0

5

Fig. 10. Strong-fieldeffect on CSRS resonance calculated by DICKand HOCHSTRASSER [ 1983al. Damping parameter values are chosen to correspond to those measured for pentacene. In the notation of this review, a,,, = and W = f t y , (in cm- '). The strong field is denoted by the double-lined arrow and is detuned from the ig) --t Ie) transition by 1 cm-'.

11, § 31

I7


3.2.3. Pressure-inducled Hanle resonances between Zeeman sublevels of an excited state The effects of level degeneracy on PIER4 were treated theoretically by GRYNBERG [ 1981~1,who calculated amplitude and polarization properties of the coherent emission as a function of incident light polarization and external magnetic field. His prediction of collision-induced resonances between exactly degenerate excited state Zeeman sublevels in four-wave mixing was verified by the experiments of SCHOLZ,MLYNEK,GIERULSKI and LANGE [1982], and SCHOLZ, MLYNEKand LANGE[ 19831. The transitions studied were simple J = 0 ground state to J = 1 excited state transitions in ytterbium and barium. A single laser frequency from a pulsed dye laser is used in a phase conjugate geometry, and the Zeeman splitting of excited state sublevels ( l e ) , l e ' ) ) is swept in energy with a magnetic field. The apparatus and polarization scheme are depicted in Fig. 11. Argon collision-induced Hanle type (HANLE[ 19241)

M

LASER

Fig. 1 1 . (a) Experimental layout for collision-induced Hanle resonances. P: analyzer, PD: photodetector, F: attenuating filter. (b) Polarization geometry with respect to magnetic field axis z. e, (i = 1,2,3) are beam polarizations and ep is the analyzer polarization setting. (c) The J = 0 + J' = 1 atomic transition excited by near resonant radiation. (After SCHOLZ, MLYNEK and LANGE[1983].)

78


PI, § 3

four-wave mixing resonances described by xC3) of eq. (2.12) are observed at zero magnetic field when we, = o1- o,= 0. The detuning dependence of these resonances is used to study collision potentials between 'P barium and ground state argon. The general theory behind this relationship outside the impact regime has been discussed in detail in 0 2.3.1. As pointed out there, the collision potentials of le) and le' ) must be similar for coherence to be created with reasonable efficiency at large detunings (A > 1/rc). Zeeman sublevels of the same excited state are therefore ideal for this purpose. It is easy to derive an appropriate expression for the Hanle resonance intensity in the impact regime. Here, of course, o1= w, = w,, and we,eis the Larmor precession frequency OO

=

gpBB

7

(3.11)

where g is the Landt factor for J = 1, p B is the Bohr magneton, and B is the applied magnetic field. We assume that pressure broadening rates yeg and yerg for the optical transitions are identical and that the linewidth re,e is much greater than the Zeeman splitting 0,.Then, from equations (2.12) and (3.1), (3.12) where p is the argon pressure. The factor K ( A ) is added by SCHOLZ,MLYNEK and LANGE [1983, eq. 11 and assumed to describe detuning dependence outside the impact regime. Inside the impact regime K ( d ) = 1. SCHOLZ, MLYNEKand LANGE[ 19831 measured the quantity (3.13) at detunings where the impact approximation is no longer valid as shown in Fig. 12. They modeled the ratio K(A)/K(- A), using Van der Wads potentials for both 'P, and IS, barium collisions with argon, and standard pressure broadening theory (SZUDY and BAYLIS[ 19751). The resulting theoretical values of R ( d ) are also shown in Fig. 12 as the solid line fit to the data. Qualitatively, the asymmetry of four-wave mixing efficiency with A is easy to understand in the molecular picture introduced in 52.3.1. Since the ground state interaction with the buffer gas is more repulsive, for most values of intermolecular separation the "molecular" electronic transition will be redshifted and detuning below resonance should be preferred (cf. YEH and BERMAN[ 19793).

11, § 31

19


0

-0 2 p:

o-o

4

0

-0

-0.6

-0 8 -

0

04

0.8

DET U N I N G

12

/

16

rad.sec-'

Fig. 12. Ratio of magnetic field-dependent contributions to four-wave mixing for detuning above and below resonance (eq. [3.13]). The experimental points are averages of several measurements; error bars are chosen to include all individual measurements. The dashed line is the prediction of optical Bloch equations. The solid line is a fit for a model Van der Waals interaction potential between barium and argon. (After SCHOLZ, MLYNEKand LANGE[1983].)

The collision-induced Hanle resonances are an excellent illustration of the relationship of PIER4 to collisional redistribution discussed in 0 2.3.1. SCHOLZ,MLYNEKand LANGE [ 19831 have also demonstrated that information about collisions can be obtained. We should, however, add acautionary note with regard to extracting collision potentials. It has been contended (MUKAMEL [ 19821) that pressure-induced four-wave mixing is an excellent tool to study the breakdown of Lorentzian behavior at large detuning because the amplitude (rather than lineshape) of the effect depends on collisional dephasing. This is misleading, since it is still necessary to map out a lineshape by performing the mixing experiments at many detunings. The intrinsic limitations of four-wave mixing due to nonresonant background make it unlikely that it will be competitive with linear fluorescence excitation (YORK, SCHEPS and GALLAGHER [ 19751, ALLARDand KIELKOPF [ 19831) for studies of collision pot entials .

80

111, § 3


3.3. COLLISION-INDUCED POPULATION GRATING RESONANCES

Population grating resonances in four-wave mixing have been used, for example, to measure dye relaxation times in solution (YAJIMA,SOUMAand ISHIDA[ 1978I), to study velocity changing collisions in atomic vapors (LAM, STEELand MCFARLANE [ 1982]), and to monitor excitation transport in solids (SALCEDO, SIEGMAN, DLOTT and FAYER[ 19781). The role of dephasing in these nearly degenerate four-wave mixing processes has not been fully appreciated and documented until recently. This section focuses on collisioninduced population grating resonances which, in a two-level system, do not occur in the absence of pure dephasing. A qualitative understanding of these resonances is emphasized, and several examples of quantitative applications are included. Often incoherent feeding terms play an important role in the population dynamics, and multiple level models using sets of equations like (2.10) are necessary to interpret experimental data. Theoretical work on the secular terms that describe these resonances (YAJIMAand SOUMA[1978], OUDARand SHEN[ 19801) has provided the necessary framework to extract both population and coherence relaxation times from frequency domain measurements. A generalized diagrammatic model such as that used by BOYD and MUKAMEL [ 19841 also incorporates incoherent population flow. 3.3.1. Grating picture of four-wave mixing and the role of dephasing

The sequential application of perturbing laser fields in solving the Liouville eq. (2.7) suggests a picture of four-wave mixing as the coherent scattering of the third incident wave from a grating established by the first two. This grating is a periodic spatial and/or temporal modulation of coherences pj, and/or populations pji. In this section we concentrate on the population gratings, and the importance of dephasing in their occurrence. Physically, the origin of coherently modulated population can be understood in terms of interference between incident light waves. The total intensity in the interaction region resulting from incident fields E(w,) and E(w2)is given by I , + 2 cc $E2(w,) + 4E2(w2)

- J q w ,1E(w2) cos[(w,

- 02)t

+ (k,- k , ) . PI

9

(3.14)

where E ( w , ) and E(w2)are the field amplitudes and we have taken the field phases to be zero. The last term of (3.14) is modulated in space when the beams are not parallel and in time when the frequencies are not identical.

11,

J 31


81

The intensity modulation can be translated by absorption into a spatial and temporal modulation of excited- and ground-state populations. These excitedstate population “excesses” and ground-state population “holes” appear as an index modulation, and a third beam can scatter coherently from this grating. With nonresonant radiation in a vapor this four-wave mixing process can only occur when collisions (or fluctuations) permit absorption to couple the material levels. It is important to keep in mind that the collisional dephasing inherent in collisional absorption is essential to observing the difference frequency resonance in four-wave mixing. Resonant pumping of a two-level system without collisions would not be sufficient to induce a population modulation resonant at o,- o,= 0 (see eq. [3.15] below). There will,of course, be single-photon resonances in the four-wave mixing when o,= oegand o, = oeg. Applying eqs. (2.9) and (2.10) to the two-level system of Fig. lb, the population modulation responsible for dephasing-induced four-wave mixing is (ROTHBERG and BLOEMBERGEN[ 1984a1)

(3.15) where Ak = k, - k, and u is the absorber velocity. We have incorporated the Doppler shifts dk . v into the terms in brackets but omitted shifts k * v in the single-photon resonances, since conditions (3.2)-(3.5) are assumed to hold. The appropriate susceptibility xC3)is proportional to the population modulation (3.15), which contains all of the essential physics. Note that the resonance in brackets does not occur in the absence of pure dephasing when 2(xT,)- = reg= r;; = (xTl)- As before, random perturbations induce coherent resonances by dephasing interfering probability amplitudes for nonlinear processes. In the case of population grating resonances in four-wave mixing, the relevant diagrams are those of Fig. 3g. The grating picture provides a sound qualitative basis for understanding the collision-induced population grating resonances implicit in eq. (3.15). The nonlinear mixing is resonant for o,- 0,- Ak u = 0 because the population grating scatters most efficiently when it is stationary in time and space. Spatial washout of the grating modulation can occur if the nonlinear medium is free to move. This statement is equivalent to the statement that the resonance is susceptible to residual Doppler broadening Ak . u. The homogeneous spectral

’.

82

PI, 8 3


width of the resonance predicted by (3.15) is r:; = (nT,)- and reflects the rate at which ground state “holes” are Wed by excited state “excesses”. Put another way, the grating can only follow an intensity modulation that is slow compared with the material system’s longitudinal relaxation rate. Note that the resonance is not pressure broadened, since pure collisional dephasing without quenching will not affect the population relaxation rate. 3.3.2. Characterization of pressure-induced population grating resonances in Na vapor The first population grating resonances observed where the role of dephasing was demonstrated explicitly were reported by BOGDAN, PRIOR and BLOEMBERGEN [ 19811). Using pulsed dye lasers tuned 24 cm- from the 2S1/2+2P3/2transition of sodium in the buffer gas, the verified the p h e dependence of Z(o,) derived from eqs. (3.1) and (3.15). Higher resolution studies using the apparatus depicted in Fig. 5 revealed collision-induced Raman resonances between ground state hyperfine levels (BOGDAN,DOWNER and BLOEMBERGEN [ 1981a1, 0 3.4.1). The quantitative spectral predictions of [ 1984al at eq. (3.15) were investigated by ROTHBERGand BLOEMBERGEN resolution adequate to measure The experimental conditions satisfied criteria (3.2)-(3.5) with lasers detuned from 2S1,2+ 2P1,2by 30 GHz. All of the field polarizations were parallel so that interference as in (3.14) is possible. Several mTorr of sodium with several hundred Torr of inert buffer gases were probed using the nearly Doppler-free geometry of Fig. 5 (PRIOR[ 19801). A helium collision-induced population grating resonance in sodium is shown in Fig. 13. The width contribution from Doppler broadening is negligible, since at sufficiently hagh b&er gas pressures, the population grating is held in place. This collisional narrowing of the Doppler width was first observed by WITTKE and DICKE[ 19561 in microwave spectroscopy and is discussed in detail in $3.4.3. The discrepancy between the observed 34 MHz non-Lorentzian line and the predicted homogeneous width of 20 MHz from (3.15) was attributed to a breakdown of the simple two-level model. This was ascribed to optical pumping of the ground state hyperfine levels of sodium (ROTHBERGand BLOEMBERGEN [ 1983a]), We return to this point later, since it is clarified by the data of Fig. 14. There, nitrogen gas is added to sodium-helium mixtures and quenches the excited states of sodium. The quenching broadens and eventually eliminates the 2P excited state grating &) but leaves a sharp resonance due to a residual long-lived grating. This occurs when a 2S population modulation pi:) persists because the “excesses” do not refill the ground

‘

c:.

11, § 31

83


34 MHZ

Fig. 13. Intensity I(w4)of dephasing-induced population grating resonance in four-wave mixing as a function of w1 - w,. A = 30 GHz below ie)(zP,,2) and p = 700 Torr helium. (After ROTHBERG and BLOEMBERGEN [1984a].)

state “holes” from which they were formed. Instead, population can pool in a “sink” state lg’), which does not scatter as efficiently, and the homogeneous line width of the remaining grating depends on the recovery rate R,. In the case of sodium the sink levels 1 g’ ) would be other hyperfine and Zeeman levels of 2S, and the ground state would have very long relaxation times determined by spin exchange. Adding cesium vapor to enhance spin exchange was shown to broaden these sharp resonances, corroborating this picture. A simple quantitative model based on adding a nonresonant level Ig’) in eqs. (2.10) was formulated to explain the data of Figs. 13 and 14 (ROTHBERG and BLOEMBERGEN [ 1983a1).These authors calculate an effective optical pumping rate r,, to a nonresonant level lg’) and show that the four-wave mixing intensity becomes

+ (RgB+ RggT rop)2}, (w1 - %I2 + (RgTg+ RggJ2

a1- a2)’

+

(3.16)

84


111, § 3

I

0

v, - Y Fig. 14. Collision-induced population grating resonance behavior in the presence of excited state quenching by N,. A = 30 GHz below le)(2P,,2). (After ROTHBERGand BLOEMBERGEN [1984aI.)

where r E Rge + RgZe.Equation (3.16) is nearly identical to eq. (46) of YAJIMA and SOUMA[ 19781, which describes a somewhat different three-level system. For weak optical pumping (0 < - r,, < R,,g + Rgg,)lineshapes like those of Fig. 13 are predicted. Physically, the dip at line center results from an interference between the scattering from ground-state and excited-state gratings over the bandwidth where both populations can follow the intensity modulation. This results in the apparent broadening of the line over the expected width of (nT,)-'= 20 MHz. The origin of the optical pumping was ascribed to the slightly different detunings from the S to P resonance of the F = 1 and F = 2 hyperfine components (presumably ( g ) and ( g ' ) ) .The data of Fig. 14

11, § 31


85

can also be rationalized in terms of eq. (3.16). There, quenching of the excited state by nitrogen degrades the grating lifetime, broadening the resonance. A residual grating remains because nitrogen quenching does not properly refill the ground state “holes” (rap # 0), and a sharp resonance of half width Rg,g+ R,. is observed. LAM, STEELand MCFARLANE[1982] have observed similar sharp resonances due to collision-induced spectral cross relaxation while exciting population gratings within the inhomogeneous bandwidth of sodium vapor. ROTHBERG and BLOEMBERGEN [ 1984al have also observed that the excited state gratings in sodium are quenched by xenon. The bottom trace in Fig. 15 illustrates a collisionally induced population grating in xenon buffer gas, which has a width of less than (nT,)-= 20 MHz. These authors attribute this to

E l E; E 2 €out

I1 II I1 II

PXe=100 torr PHe=200

Pxe = 100 torr PHe = 0

V, - V2 (MHZ)

Fig. 15. Collision-induced population grating resonances in Xe/He buffer gas mixtures. d = 20 GHz below le)(ZP,,2).(After ROTHBERGand BLOEMBERGEN [1984a].)

86

[II, § 3


excimer formation between Na(2P) and Xe, which leaves only a ground-state grating. Such two-body “sticking collisions” have been observed in alkali rare gas systems by many groups (BOUCHIAT, BROSSELand POTTIER[1967], EWARTand O’LEARY [ 19821,TAM, MOE,PARKand HAPPER[ 19751). In the top trace of Fig. 15 these metastable complexes are shown to be collisionally dissociated by added helium (ROTHBERGand BLOEMBERGEN [ 1984a]), with the spectrum reverting to that of Fig. 13 taken in pure helium. These processes can be modeled quantitatively using eqs. (2.10), and their observation shows that collision-induced population gratings can be useful in studying chemical reaction dynamics. These dephasing-induced four-wave mixing resonances have lineshapes sensitive to population relaxation and can be instructive if it is possible to model the system under study appropriately.

3.4. COLLISION-INDUCED COHERENT RAMAN RESONANCES BETWEEN

EQUALLY POPULATED STATES

It has long been recognized that coherent Raman susceptibilities vanish when the Raman levels are equally populated. This section is concerned with collision-induced coherences between such equally populated ground state levels as in Fig. lc. The associated four-wave mixing resonances cannot be described by single diagram pairs as in Fig. 4, since population in two initial levels is important. Collisions, however, play the same role in destroying an interference between contributions to x ( ~ ) . From eq. (2.9) the coherence between states (8) and ( g ’ ) in second order is calculated to be

where conditions (3.2)-(3.5) are assumed to hold. At the Raman resonance o,- o2= ogSg this can be rewritten

(3.18)

11, § 31


87

where re, = reg, = r. The term (pi?$ - pi:)) is zero when ig) and Ig’) begin with equal populations. The adjacent term has amplitude proportional to damping so that the coherent Raman susceptibility no longer vanishes when collisions are introduced. Strictly, these resonances should be termed “pressure enhanced”, since r does not completely vanish at zero pressure due to spontaneous emission. Just as in the case of population modulation, the coherence modulation of (3.18) can be thought of in terms of coherence gratings. The coherence is resonant when the modulation frequency equals the level spacing and cancels without damping. The grating decays temporally with coherence decay rate n- ‘rga and washes out spatially at rate n- ‘dk . u corresponding to the residual Doppler width. These properties are discussed in light of experimental results in sodium vapor. 3.4.1. H y p e f i e and Zeeman coherences in the ground state of sodium vapor The experimental apparatus used to study four-wave mixing resonances associated with the coherences of (3.18) is that of Fig. 5, where conditions (3.2) through (3.5) hold. Collisionally generated coherent Raman resonances between F = 1 and F = 2 hyperfine components of Na 3,s states were first discovered in studies of population grating resonances (BOGDAN,DOWNER and BLOEMBERGEN [ 1981al). Later it was demonstrated that similar Raman resonances between Zeeman components contributed to the degenerate frequency resonance at o,- o2= 0 under some polarization conditions (BLOEMBERGEN, DOWNERand ROTHBERG[ 19831). Population grating and coherence grating contributions to four-wave mixing in sodium can be distinguished by exploiting angular momentum selection rules. For linearly polarized fields in zero magnetic field, creation of coherence between lg) and Ig’) in the S state of sodium requires the polarizations of E(o,)and E ( q ) to be orthogonal (AmF = & 1). Parallel polarized beams create only population gratings (AmF = 0). A more detailed discussion of the physics and other combinations of polarizations can be found elsewhere (ROTHBERGand BLOEMBERGEN[ 1984b1). A spectrum of collision-induced coherent Raman resonances between Zeeman levels of ground state sodium in an external magnetic field is shown the peak substructure in Fig. 16. The resonances occur for w1- w, = resulting from incipient breakdown of the weak-field (linear) Zeeman effect, causing different level pairs 18) and ig’) to have slightly different splittings.

88


I

-100

0 100 u1-u2 (MHZ)

Fig. 16. Collision-induced coherent Raman resonances between equally populated Zeeman sublevels in the ground state manifold of sodium. Polarization configuration noted is with reference to the direction of external magnetic field B. A = 30 GHz below le)(*Pli2) and p = 700 Tom helium. (After ROTHBERG and BLOEMBERGEN [1984b].)

~

Zeeman x H y p e r f i n e 1x21

0

500 -

z 200r C

a

4a

100-

I I

v

I

I

I I

I

II I I

? X

I

10 X

0 6

s

11, 31


89

At low buffer gas pressure p the resonance peak intensities (Fig. 17) increase as p z as would be predicted by eqs. (3.1) and (3.18). A p 3 dependence takes over at higher pressure, concomitant with a decrease in the line width because of a collisional narrowing of the Doppler width, as in Fig. 18. In the language of the grating picture, velocity changing collisions prevent washout of the spatial modulation of coherence. This begins to narrow the line width (i.e., increase the grating lifetime) when the mean free path becomes less than the grating constant II Ik, - k,l - l . The mean free path then varies inversely with pressure, as does the resonance line width, and the resonance peak intensity therefore increases as p 3 as in Fig. 17. The line width and line-shape variation with pressure have been used to study velocity changing collisions, and this is discussed in 5 3.4.3. An instrumental width of 5 MHz persists after the collisional elimination of Doppler width. In order to demonstrate the homogeneous width due to coherence decay rate rg,g, cesium was added to increase rg,gthrough spin

70.

Helium data Strong

1 0

50-

-

30.

N

5

20-

v

s3 LL

10

CT

-

5

I 30 100

300

1000

log PRESSURE (Torr) Fig. 18. Plot of resonance full width (FWHM) versus pressure exhibiting collisional narrowing of the residual Doppler width of pressure-induced Zeeman resonances. Boxes and open circles are the expected forms of narrowing for strong and weak velocity changing collision models, respectively (see 3.4.3).The asymptotic l/p line width dependenceis explained in the text. (After ROTHBERGand BLOEMBERGEN [1984b].)

90


[II, § 3

exchange collisions, and a line broadening was observed (ROTHBERGand BLOEMBERGEN [ 1984b1). The experimental studies bear out the predictions of eq. (3.18), demonstrating collisionally enhanced coherences between equally populated ground states. Collisions make it possible to study resonances in the ground state manifold using only optical spectroscopy. 3.4.2. Collision-induced Hanle resonances in the ground state of sodium vapor The resolution in the experiments cited in the previous section was ultimately limited by relative frequency jitter between lasers. This can be eliminated by using only a single CW laser and varying the level spacing wgrginstead of tuning the lasers. This is accomplished by sweeping an external magnetic field, and Hanle-type level crossing resonances (cf. 3 2.3 and 0 3.2.4) are observed. The coherence of (3.18) remains appropriate to these circumstances, where now o1- w2 is always zero and ugfg is given by the Larmor frequency of eq. (3.1 1). The Hanle resonances therefore have properties very similar to the Zeeman resonances of 3.4.1. Experiments of the aforementioned type have been done in a phase conjugate geometry (as in Fig. 11) and have demonstrated collision-induced

I -100

-50 0 50 8, (milligauss)

100

Fig. 19. Collision-induced Hanle resonance in four-wave mixing. The experimental points ( x ) are compared with a Lorentzian fit (solid curve). Note that the full width corresponds to about 24 kHz. A = 50 GHz below le>('P,,,) andp = 3050 Torr argon. (After BLOEMBERGEN, ZOUand ROTHBERG[1985].)

11, I 31


91

four-wave mixing resonances (Fig. 19) sharper than 30 mGauss (20 KHz) (BLOEMBERGEN, Z o u and ROTHBERG[ 19851). These widths are limited by magnetic field inhomogeneities and power broadening as well as fundamental contributions to Again, collisional narrowing eliminates the residual Doppler width at high buffer gas pressures. The high resolution permits the Na-Na spin exchange contribution to rgfg of several kHz (HAPPER[ 19721, HAPPERand TANG[ 19731) to be observed directly. These sharp resonances also enable relaxation of Zeeman coherences due to phase interrupting collisions with xenon to be measured (BLOEMBERGEN [ 19851). A modification of the preceding experiments using a single laser at w2 and generating frequencies o,with a phase modulator both eliminates relative laser frequency jitter and allows Hanle resonances and Zeeman resonances to be studied independently (BLOEMBERGEN and ZOU [ 19851). The Hanle resonances are unique in the sense that, in the absence of a magnetic field, the choice of quantization axis is arbitrary. The most natural choice is the light-propagation direction so that the selection rule for two-photon 3% 4 3’s transitions would be Am, = 0 corresponding to population modulation. To this point we have assumed a quantization axis along one of the light-polarization directions and associated the resulting Am, = & 1transitions with the creation of Zeeman coherence. The resolution of this apparent anomaly provides a great deal of insight into the nature of these collision-induced coherences. It follows from eq. (3.18) that the strength of the collision-induced part of Zeeman coherence pit,’ (w, - w 2 ) is proportional to (3.19) R can beclearlyidentified as the spatialmodulationof3S(lg), 1g’))to 3P(le)) pumping rate when w , = o2. When the pumping rate is faster than the ground state equilibration rate ( R > rgTg), the lasers alter steady-state populations and pip,’.via collisionally assisted optical pumping with rate R . It is clear that collision-induced coherences between Zeeman levels are equivalent to collisionally assisted modulated transverse optical pumping. This is demonstrated in the beautiful experiments of BLOEMBERGEN and Z o u [ 19851, where the dependences of collision-induced Hanle resonance intensities on detuning from the D, and D, lines of sodium are measured. At large detuning, R < rgrg and the four-wave mixing intensities are equal. At small detuning, R % r a n d strong optical pumping occurs. The saturated transverse polarization is expected to be twice as large for D, pumping as for D, pumping, and the corresponding

pg)

92


[II, 8 3

four-wave mixing signal is indeed four times as large, confirming this relationship. In the limit of large optical pumping ( R > r)perturbationtheory fails and Stark splitting of the Hanle resonances is observed (BLOEMBERGEN [ 19851).

34.3. Collision-induced four-wave mixing lineshapes and velocity changing collisions Traditionally, the effect of collisions on optical spectra is a pressure broadening of the resonances due to collisional dephasing of the resonant levels. Concomitant narrowing of the Doppler widths is too slight to observe. Exceptions occur in the case of vibrational and rotational absorption or Raman resonances in small molecules where the dephasing is small because the levels experience similar collision potentials. PINE [ 19801 has reviewed experiments that observe collisional narrowing in small molecule Raman and infrared spectroscopy. Unfortunately, dephasing is not always negligible and molecular collision potentials are difficult to model. The collision-induced coherent Raman resonances between Zeeman levels permit collisional narrowing in an atomic system using optical spectroscopy to be observed for the first time. These resonances exhibit a Doppler narrowing observable over several orders of magnitude. These occur in simple atomic systems, where the Raman levels have nearly identical scattering potentials and are therefore ideal to study velocity changing collisions. In fact, to assign a single velocity to a superposition state after a scattering event is only strictly valid in this limit (BERMAN[ 19781). The four-wave mixing susceptibilities must be averaged over velocity distributions P(u) to obtain correct lineshapes as discussed by DRUET, TARAN and BORDB (1979). The collision-induced part of coherence (3.18) is properly evaluated by the convolution

in the assumed limit of detuning much larger than Doppler widths given by relation (3.5). Typically, the expression (3.20) leads to CARS lineshapes that are the convolution of a Gaussian velocity distribution and a complex Lorentzian. As explained in Q 3.4.1, however, when velocity changing collisions begin to help preserve the spatial modulation implicit in the E ( w , )E*(w2)term of the coherence, the residual Doppler width decreases. In this limit the velocity of import is the diffusive velocity 5,which tends to wash out the grating, and

11, § 31


93

not the instantaneous thermal velocity u. Thus P(u) and u of (3.20) should be replaced by P(5) and 5when velocity changing collisions occur on the timescale of the measurement n/ldkl Jvl. Dicke (DICKE [ 19531, WITTKE and DICKE [ 19561) has proved that, in the high-pressure limit where Doppler width goes as mean free path (i.e., lip), the distributions P(%) are Lorentzian regardless of the microscopic collision model. Equation (3.20) then predicts the complex “off-resonance CARS” lineshapes of DRUET,TARANand B O R D[~19791 at low buffer gas pressure and Lorentzian four-wave mixing lines in the highpressure limit, in good agreement with experimental results for the collisioninduced resonances (ROTHBERG and BLOEMBERGEN [ 1983b, 1984b1). Using the full susceptibility as obtained from (3.20), the lineshapes of Fig. 16 can be reproduced by ab-initio calculation from the sodium wavefunctions using the experimentally measured Doppler width ] A k (5 (ROTHBERG[ 19831). We note in passing that the collision-induced Hanle resonance shape of Fig. 19 deviates from Lorentzian behavior. This occurs because of the very small angle between k , and k, used in those experiments. The grating spacing becomes nearly as large as the interaction region, and residence time of the atoms in the field becomes an important factor. Atoms with high diffusive velocities 5 are more likely to leave the interaction region, leading to a suppression of the line wings (BLOEMBERGEN, ZOU and ROTHBERG[ 19851). The Doppler narrowing in collision-induced Zeeman resonances has been used to measure velocity changing collision cross-sections and diffusion coefficients for sodium in rare gases (ROTHBERGand BLOEMBERGEN [ 1983b1). In the intermediate narrowing regime before l/p behavior is obtained, the four-wave mixing lineshapes are sensitive to the details of velocity changing collisions. In principle, one can extract P(5) from the resonance profiles by fitting (3.20), but the entire collision kernel (BERMAN[1978]) cannot be obtained. Since the scattering potentials of the Zeeman levels are nearly equal, classical models of velocity changing collisions are valid and the collisioninduced resonances have been used to test them. Figure 18 compares the narrowing of the experimental linewidth in helium with that predicted by the strong and weak velocity changing collision models of RAUTIANand SOBEL’MAN [ 19671. The models are constrained by the low pressure Doppler width and the asymptotic l/p high-pressure behavior so that there are no adjustable parameters. Similar analysis has been done for argon and xenon and was used to evaluate the assumptions of these models (ROTHBERGand BLOEMBERGEN [ 1983b1).

94


[IL § 4

6 4. Summary and Future Prospects The scientfic contributions of dephasing-induced coherent phenomena can be roughly divided into the following three general but not orthogonal categories : (1) Demonstrating the correct approach to perturbation theory for nonlinear quantum mechanical processes with damping; (2) Studying dephasing mechanisms; (3) Making unique types of spectroscopic measurements. This section is partitioned accordingly. The main results of the preceding sections are reviewed, and areas where problems remain are pointed out. Finally, we speculate briefly on future work involving dephasing-induced coherences.

4.1. EFFECTS OF DAMPING ON COHERENT NONLINEAR OPTICS

The observation of dephasing-induced resonances in four-wave mixing has veritied the perturbative approach to damping in nonlinear processes put forth by BLOEMBERGEN and SHEN[ 19641. These extra resonances can be understood to arise from the removal of a cancellation between quantum-mechanical amplitudes by incoherent perturbations. Phenomenological damping must be considered at each level of the calculation in order to retain the amplitudes describing these dephasing-induced coherences. Aside from additional resonances, the interfering terms that vanish in the absence of dampingcan produce substantial corrections to resonant four-wave mixing when incoherent perturbations are present. Rigorous treatment of damping is also required to obtain the correct signs of phenomenological damping terms in resonant coherent Raman scattering, and hence for a correct analysis of four-wave mixing lineshapes. Three types of coherent optical mixing resonances that do not occur without damping have been studied experimentally. These are all four-wave mixing resonances and can be classified as Raman resonances between initially unpopulated excited states (0 3.2), population grating resonances ( 5 3.3), and Raman resonances between equally populated states ( 5 3.4). All three types have been demonstrated in vapors where collisions are the incoherent perturbation, and the first type has also been observed in the condensed phase where phonons are the source of dephasing. The experimental data recorded so far are for the most part in agreement with model perturbative calculations for steady state x ( ~ ) .

s

11, 41

95

SUMMARY AND FUTURE PROSPECTS

An exception where anomalies remain are the studies of interference between population grating and Zeeman resonances in four-wave mixing using polarization configurations where neither is forbidden by selection rules (ROTHBERG and BLOEMBERGEN [ 1984a1). Since the population and Zeeman coherence modulations are created by components of the same incident fields, it was expected that the radiating polarizations PC3)corresponding to each would be phase locked, either exactly in or 180" out of phase. Interference lineshapes should then be symmetrical which is not what was observed. ROTHBERG [ 19831 speculated that this was a result of a collisional dephasing of different state-ordered pathways contributing to x(3) due to the large degeneracy of sodium. The resolution of this puzzle should contain interesting physics. Within the context of steady-state perturbation theory many other types of damping-induced resonances have been predicted and appear worthy of further investigation. The collision-induced resonances discussed in Q 2.2.2 corresponding to diagram pairs of Figs. 3a-e (YEEand FUJIMOTO [ 19841) have not been observed. Several authors have worked out the theory for dephasinginduced extra resonances in xC2) (BLOEMBERGEN, LOTEMand LYNCHJR. [1978], DICKand HOCHSTRASSER [1983b]). These should be possible to observe in noncentrosymmetrical media, where xC2) does not vanish by symmetry. As touched upon in Q 2.3.1, the incoherent analog to the dephasing-induced nonlinear mixing resonances has been reported (HILLMAN, BOYD,KRASINSKI and STROUDJR. [ 19831). Using a pump-probe experiment, they observe absorption dips of width (nT,)- in a xC3)'' profile of width 2(5cT2)- ', which could not occur in the absence of pure dephasing where 2T,- = T; '. The physical reason for these dips is the same as that for the dephasing-induced coherences, involving removal of destructive interference between diagram pairs by incoherent perturbations (BOYDand MUKAMEL[ 19841). Dip widths of 37 Hz were obtained, showing great promise for application to ultrahighresolution spectroscopy, dynamical measurements, and characterization of laser sources. Another area of recent activity has been the interference of different-order corrections to the electric polarization as, for example, in the multiphoton ionization of xenon (JACKSON,WYNNEand KES [1983]). There, resonant enhancement of ionization on the 5p6 5p56s transition is absent because of cancellation of linear and nonlinear polarizations. In the particular case of xenon this interference would not be removed by collisions because both ~ ( " ( 3 0 ) and ~ ( ~ ' ( 3 0resonant ) denominators depend on the same phenomenological damping width. If, however, a similar system with a near

'

'

96


PI, J 4

intermediate resonance for incident frequency w (or one near 2w) were at issue, collisional dephasing should restore the missing ionization resonance. Little work has been done outside the context of the assumptions of steady-state perturbation theory. The breakdown of the impact approximation at large detuning has received some attention (§ 2.3.1 and J 3.2.3), but further theory and experiments are required. In particular, the regime where laser pulses are shorter than collision times, where three-body collisions become important, and where incident fields are strong (failure of condition 3.3) promise to be interesting. Power effects on dephasing-induced resonances as discussed in 9 2.4.2 and § 3.2.2 are especially important to understand if these resonances are to be used as general spectroscopic tools with pulsed lasers. The development of a valid formalism for damping in nonlinear optics when field precession cannot be ignored on the incoherent perturbation timescale has progressed with the use of dressed state models and nonperturbative techniques. Saturation measurements on dephasing-induced resonances should provide appropriate data to test these theories. The pioneering work of DICK and HOCHSTRASSER [ 1983al in this area needs to be tested systematically and extended to the case where microscopic modeling of the damping process is possible. Coherent transient analogs to pressure-induced extra resonances in fourwave mixing have been proposed by BERMANand GIACOBINO [ 19831. The electronic state coherences created by collisionally assisted absorption could WHITTAKER,KACHRUand be detected using trilevel echoes (MOSSBERG, HARTMANN [ 19801).

4.2. STUDIES OF DEPHASING MECHANISMS

Since the coherences at issue are both created and destroyed by dephasing, their resonances contain a great deal of information about the associated perturbations. For example, fitting of interatomic collision potentials has been done by studying collision-induced four-wave mixing at large detuning from intermediate resonances (§ 3.2.3). Experiments on coherences induced by phonons have produced quantitative data concerning which phonons are responsible and Arrhenius prefactors for rate coherence formation (8 3.2.2). These kinds of measurements should be possible for discrete line excitations in many types of condensed matter. Perhaps analogous time-resolved experiments could be done where phonon-induced four-wave mixing would be used as a probe to produce information about phonon generation and decay caused by a separate excitation.

11, I 41

SUMMARY A N D FUTURE PROSPECTS

91

Naturally the coherences responsible for the four-wave mixing detailed here are also destroyed by incoherent perturbations and population decay. Dephasingrates between excited state or ground state pairs by atomic collisions have been measured directly in the frequency domain without the need to measure electronic dephasing independently. Similarly, the phonon dephasing and lifetime decay contributions to line broadening have been deduced for vibrational states of electronically excited molecules in condensed phase. The broadening of the dephasing-induced population grating resonances reflects the population dynamics of the system under study. Examples in $ 3 include measurement of quenching collision rates, excimer formation and dissociation rates, and effective optical pumping rates. Frequency domain studies of population relaxation are not plagued by the experimental difficulties of making ultrafast measurements in the time domain but must be modeled carefully. Reduction of Doppler dephasing in the collision-induced coherent Raman resonances between Zeeman levels has been analyzed to learn about velocity changing collisions and diffusion coefficients ($ 3.4.3). These resonances are ideal to study velocity changing collisions, since the phase change in collisions with superpositions of Zeeman sublevels is genuinely negligible. Thus even study of Doppler narrowing in a regime where three-body collisions are frequent may be possible. With the excellent signal-to-noise ratios obtainable, quantitative measurements can be made on collisions where difFractive scattering also plays a role (BERMAN[ 19781). The discovery of fluctuation-induced extra resonances, as discussed in 0 2.4.1, might provide new spectroscopic possibilities but will almost certainly be a valuable test for models of laser fluctuations. This will enable a proper treatment of source fields in nonlinear spectroscopic applications. Perhaps more importantly, the ability to characterize and understand laser fluctuations may be a valuable design aid in the construction of technologically important lasers where bandwidth is critical, such as semiconductor lasers and stabilized dye lasers for frequency standards. Experiments to observe fluctuation-induced coherent emission and additional theory to guide and interpret these investigations are exciting avenues to be pursued.

4.3. NOVEL SPECTROSCOPY

The information available from line shapes and line strengths of dephasinginduced resonances has been summarized in the previous section. Here, we

98


[II, § 4

concentrate on the possibilities for investigating new energy levels through dephasing-induced coherence. The demonstration of Raman resonances between essentially unpopulated excited states that occur in the presence of dephasing makes vibrational spectroscopy of electronically excited states feasible (0 3.2.2). It is difficult to obtain this structural information in any other way, since large excited state populations would be required. Moreover, the dephasing-induced resonances have been demonstrated to be appropriate even in molecules where spectral congestion obscures any structure in the electronic absorption band. Application to gas-phase species where molecular beams do not suffice to remove spectral congestion should be fruitful. Coherences between equally populated ground states have also been produced by damping. In essence it is possible to do all optical spectroscopy between ground-state Zeeman levels with resolution comparable with radiofrequency techniques. This may be useful for frequency standards, since better than one part in 10" resolution has been demonstrated in a simple collisional environment. The linewidths of these collision-induced Hanle resonances remain largely instrumentally limited, but not by the laser source, and several orders of magnitude improvement is plausible. Ground-state population gratings in nondegenerate systems that persist indefinitely are not difficult to envision. Collision-inducedresonant coherent Raman scattering between ground-state rotational levels of large molecules with lines spaced by less than the thermal energy k,T may be detectable when collisional broadening is less than the level spacing. Currently, this type of spectroscopy is impossible, since the coherent Raman susceptibilityvanishes for equally populated Raman levels. Moreover, spontaneous Raman scattering is exceedingly difficult, since the Raman shifts may be only a few cm- I . Much progress has been made in the understanding of damping in nonlinear optics. The manifestations of damping are proving to be useful tools for spectroscopy and for the study of dephasing processes. It is hoped that this review will be of use in the continuing evolution of this exciting field.

Acknowledgements I am indebted to Professor Nicolaas Bloembergen who has contributed much to this field and to my own education in it. Many of the results and perspectivesin this review reflect his mark on nonlinear optics. I thank him also for his communication of recent results prior to publication. Special thanks go

111

REFERENCES

99

to Professors Yehiam Prior and Michael Downer for valuable discussions. I am also grateful to those authors who permitted use of their figures in this review.

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I11

INTERFEROMETRY WITH LASERS BY

P. HARIHARAN CSIRO Division of Applied Physics Sydney, Australia 2070

CONTENTS PAGE

$ 1. INTRODUCTION

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

105

0 2. MEASUREMENTS O F LENGTH . . . . . . . . . . . . 110 $ 3. MEASUREMENTS O F CHANGES IN OPTICAL PATH 113 LENGTH . . . . . . . . . . . . . . . . . . . . . . . .

5 4. 0 5.

DETECTION O F GRAVITATIONAL WAVES . . . . . . . 118 LASER DOPPLER INTERFEROMETRY . . . . . . . . . 120

. . . . . . . . 125 $ 7 . OPTICAL TESTING . . . . . . . . . . . . . . . . . . 127 $ 6 . LASER-FEEDBACK INTERFEROMETERS

$ 8. HETERODYNE SPATIALINTERFEROMETRY . . . . . . 132

5 9. INTERFEROMETRIC SENSORS . . . . . . . . . . . . 5 10.PULSED-LASER AND NONLINEAR INTERFEROMETERS 0 11.INTERFEROMETRIC MEASUREMENTS ON LASERS . .

152

$ 12.CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

158

138 144

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 159 REFERENCES

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

159

$1. Introduction Until the early 1960s, the only light sources available were thermal sources. The most commonly used source for interferometry was a pinhole illuminated by a mercury vapour lamp through a filter that isolated the green line. Such a source provided at best only a crude approximation to coherent illumination and, in addition, gave only a very limited amount of light. The development of the laser made available for the first time an intense source of light with a remarkably high degree of spatial and temporal coherence. As a result, lasers have now largely replaced thermal sources such as the classical mercury arc for interferometry (INGELSTAM [ 19761). The use of lasers has removed most of the limitations of interferometers imposed by thermal sources and has led to the development of several new and interesting techniques in interferometry. Some of these will be discussed in this review.

1.1. LASER SOURCES

Many different types of lasers have been used for interferometry. Gas lasers, such as the helium-neon (He-Ne) laser, are widely used, since they are inexpensive and provide a continuous output in the visible region. Another very useful gas laser is the carbon dioxide laser, which can be operated in the infrared on any one of a number of transitions in the 10.6 pm and 9 pm bands. Dye lasers can also provide a continuous output. In addition, with any given dye the output wavelength can be varied continuously over a fairly wide range, typically about 70 nm, by incorporating a tunable wavelength-selective element in the resonant cavity. Since a large number of laser dyes are available, laser operation can be obtained over the entire visible spectrum. Heterostructures fabricated with the GaAs-GaAlAs system have made possible semiconductor lasers, which operate continuously at room temperature at a wavelength between 840 nm and 910 nm, depending on the temperature. Semiconductor lasers have the advantages of small size, simplicity of operation, and high efficiency, and are now being used to an increasing extent in interferometry. 105

106

INTERFEROMETRY WITH LASERS

[III, § 1

In addition to these lasers, pulsed lasers such as the ruby laser have also been used in optical interferometry. With the ruby laser short pulses of light (A = 694 nm, typical duration a few nanoseconds) can be produced by using a Q-switch in the laser cavity. An alternative to the ruby laser is the neodymium-doped yttrium aluminium garnet (Nd :YAG) laser, the output from which can be converted to visible light (A = 530 nm) by a frequency-doubling crystal. 1.2. LASER MODES

The ouput from a laser operating in the TEM,, mode exhibits complete spatial coherence. However, the laser cavity has a number of resonant frequencies given by the expression v = m(c/2L),

(1.1)

where c is the speed of light, L is the separation of the mirrors, and m is an integer. If more than one of these frequencies lies within the Doppler-broadened gain profile, the laser may oscillate in two or more longitudinal modes, resulting in a severe reduction in the temporal coherence of the output (ERICKSON and BROWN[ 19671). Single-frequency operation can be obtained with a He-Ne laser by using a short cavity, whose length is chosen so that only one longitudinal mode falls within the gain profile, although a disadvantage then is that the gain available is limited and the power output is low. Single-frequency operation can also be obtained fairly readily with semiconductor lasers, since the number of longitudinal modes decreases rapidly as the current is increased. With proper design the threshold for single-frequency operation can be brought within the normal operating range. However, with other lasers some type of mode selector is necessary to obtain single-frequency operation. One technique described by SMITH[ 19651 involves the use of a beam splitter and an extra mirror, forming an auxiliary resonant cavity. Oscillation is then possible only on a mode common to both cavities. A simpler method of ensuring single-frequency operation is to use a Fabry-Perot etalon (usually a plane-parallel plate of fused silica) in the laser cavity (HERCHER[ 19691). 1.3. LASER LINEWIDTH

Because of resonance narrowing, the spectral linewidth of a laser is much smaller than both the natural linewidth of the laser transition and that of the

111, § 11

INTRODUCTION

107

resonator. The fundamental lower limit on linewidth is set by phase noise due to spontaneous emission; this leads to a Lorentzian line shape with a linewidth given by the modified Schawlow-Townes relation AvF

=

(~ZV/P)(AV,)’,

(1.2)

where h is Planck‘s constant, P is the power in the laser mode, and Avc is the linewidth of the passive cavity resonator. Spontaneous emission leads to a fundamental linewidth in a typical He-Ne laser of a few hundredths of a hertz. Accordingly, under most conditions the observed linewidth is determined essentially by mechanical noise. Variations in the separation of the mirrors because of vibrations or thermal fluctuations result in a dither of the output frequency about its nominal value, and considerable care is necessary to reduce such disturbances to the point where the observed linewidth approaches the fundamental linewidth. With semiconductor lasers an additional source of line broadening is fluctuations in the refractive index of the active medium resulting from variations in the electron population density induced by spontaneous emission events. The fundamental linewidth is then given by the expression Av;

=

AvF(l

+ a’),

(1.3)

where tl is the ratio of the changes in the real and imaginary parts of the refractive index (HENRY[ 19821). For GaAlAs lasers the factor (1 + a’) is of the order of 30, and Avk is typically about 40 MHz. However, the fundamental linewidth can be reduced by several orders of magnitude by using an external resonator to increase the cavity Q. The spectral linewidths of lasers are sufficientlynarrow to enable observation of beats produced by superimposing the beams from two lasers operating on the same transition (JAVAN,BALLIKand BOND [1962]). Beats can also be observed with a single laser oscillating in more than one longitudinal mode (HERRIOTT[ 1962]), the beat frequency corresponding to the separation of these modes. These beats can be interpreted as being due to a set of moving interference fringes, the number of fringes passing any point on the detector in unit time being equal to the beat frequency. Accordingly, for the beats to be detected the dimensions of the detector must be small compared with the spacing of the fringes. SIEGMAN[ 19661 has shown that there is a trade-off between the angular field of view of the detector and its effective area, their product being equal to 1’. The phenomenon of beats is easily explained in classical terms on the basis

108


m,§ 1

that the amplitude and phase of each beam do not vary appreciably over the coherence time which is equal to l/Av, where Av is the laser linewidth. A more detailed analysis in terms of quantum mechanics made by MANDEL[ 19641 shows that even with two light beams derived from completely independent sources, the correlation between the intensities at two space-time points is a periodic function of their separation, indicating the presence of transient interference effects. These effects cannot be observed normally with light from thermal sources, for which the average number of photons in the same spin state falling on a coherence area in the coherence time (the degeneracy parameter 6) is less than However, they become observable with laser beams for which 6 is greater than unity.

1.4. FREQUENCY STABILIZATION

The frequency of the output from a free-running laser, even when it is oscillating in a single longitudinal mode, is not stable, since it depends on the optical path between the two mirrors. This can vary over time because of thermal and mechanical effects. As a result, the output frequency may shift its position within the gain profile. Typically, in the case of a He-Ne laser, the output frequency may vary in the long term by as much as 1 part in lo6. Because of this, some method of stabilizing the output frequency is necessary for precise measurements over long optical paths. Frequency stabilization also leads in most cases to a substantial reduction in the observed linewidth. One of the earliest methods of frequency stabilization applied to the He-Ne laser was stabilization on the Lamb dip (ROWLEYand WILSON[ 19721). More recently, methods of stabilization based on the use of adjacent, orthogonally [ 19721, BENNETT, polarized modes (BALHORN, KUNZMANN and LEBOWSKY WARDand WILSON[ 19731, GORDON and JACOBS[ 19741) or on Zeeman splitting induced by the application of a transverse magnetic field to the laser (MORRIS,FERGUSON and WARNIAK [ 19751, FERGUSON and MORRIS[ 19781, UMEDA,TSUKIJIand TAKASAKI[ 19801) have been used. Frequency stability to 1 part in 10' over long periods can be obtained by these techniques (BROWN 119811, CIDDORand DUFFY[ 1983]), provided care is taken to avoid optical feedback and stray magnetic fields. The highest degree of frequency stability (a few parts in l O I 3 over a few minutes) is obtained by techniques based on saturated absorption. This method of frequency stabilization was first used with a methane cell to stabilize a He-Ne laser at a wavelength of 3.39 pm (HALL[ 19681) and was subsequently

111, § 11

INTRODUCXION

109

extended to the visible region, where both I2’I2 and 1291,have absorption lines that lie within the gain profile of a He-Ne laser operating at a wavelength of 633 nm (HANESand DAHLSTROM[ 19691, KNOX and PAO [ 19701). This technique of frequency stabilization now provides a number of wavelengths that have replaced the 86Kr and 198Hglamps as standards for interferometric measurements of length (BRILLETand CEREZ[ 19811). With the carbon dioxide laser the most common method of frequency stabilization is based on saturated fluorescence (FREEDand JAVAN[ 19701). This permits stabilization on any one of a number of transitions in the 10.6 pm band to 1 part in 10l2. Frequency stabilization of dye lasers is normally based on the use of an external, temperature-controlled Fabry-Perot cavity as a reference. With semiconductor lasers a short length of a single-mode optical fibre can be used as the reference element, permitting a very compact package (WOLFELSCHNEIDER and KIST [ 19841).

1.5. PROBLEMS WITH LASER SOURCES

Several practical problems can be encountered with laser sources because of the very high spatial and temporal coherence of laser light. One such problem is speckle, which is particularly noticeable when a diffuser is used to obtain an extended source. Its effects can be minimized when observing or photographing the fringes by using a moving diffuser or, better still, a combination of two moving diffusers (LOWENTHAL and ARSENAULT [ 19701). Another problem is light reflected or scattered from the surfaces of the various elements in the optical paths. Because this stray light is coherent with the main beam but has traversed an additional optical path d, its amplitude a, adds vectorially to the amplitude a of the main beam, as shown in Fig. 1.1, resulting in a phase shift A$

= &/a)

sin @ ,

( 1.4)

where $ = (274A)d.Spatial or temporal variations in either the amplitude of the stray light or its phase relative to that of the main beam will give rise to noise in the interferogram. Finally, care must be taken to avoid optical feedback due to light reflected back into the laser from the interferometer, since changes in either the amplitude or phase of the reflected light can cause changes in the output power or, in some cases, even the frequency of the laser (BROWN [ 19811).

110


Fig. 1.1. Phase shift of a beam produced by stray light that is coherent with it but has traversed an additional optical path.

Q 2. Measurements of Length The very narrow linewidth of a laser makes it possible to obtain interference fringes with good visibility even with optical path differences of several hundred metres and has literally revolutionized length interferometry.

2.1. DEFINITION OF THE METRE

In 1960 the metre was defined in terms of the wavelength of the orange line from a 86Kr discharge lamp. The wavelength of the 3.39 pm line from the He-Ne laser stabilized by saturated absorption in methane was the first to be measured with high accuracy relative to this standard. Repeated measurements on this laser line, as well as on other stabilized laser lines, showed that the accuracy of such measurements was limited to a few parts in lo9, largely because of the uncertainty associated with the 86Kr standard itself. Simultaneously, experiments were initiated at the major national laboratories to compare the frequencies of the 3.39 pm He-Ne laser line, as well as other laser lines in the visible region, with the frequency of the 133Csclock, which is the primary standard of time (see EVENSON,DAY,WELLS and MULLEN [ 19721, WHITFORD [ 19791, and BAIRD[ 1983a1). Such comparisons could be made with an accuracy of the order of a few parts in lo'', and they gave a value for the speed of light, whose accuracy was limited essentially by the uncertainties related to the primary standard of length, the wavelength of the 8% lamp. The realization that the *%r standard was no longer adequate for

111, § 21

MEASUREMENTS OF LENGTH

111

measurements of the wavelengths of stabilized lasers with the degree of accuracy justified by their characteristics led to the decision (see PETLEY [ 19831) to freeze the value for the speed of light and redefine the metre in terms of this value. Practical methods for realizing the metre include the vacuum wavelengths of a number of stabilized lasers. These have a reproducibility of a few parts in 1O’O.

2.2. MEASUREMENTS OF LENGTH

A major problem in metrology is the limited coherence length of light from thermal sources such as discharge lamps. This limitation is virtually eliminated with laser sources, which have made possible measurements of even quite large lengths in a single step. In addition, because of the high intensity and good visibility of the fringes with a laser source, the limiting precision attainable in length measurements, even with simple photoelectric setting methods, is high, theoretically of the order of 1 part in 10l2 (CIDDOR[ 1973)). The range of wavelengths availablewth lasers also makes it possible to apply the method of exact fractions very effectively to absolute measurements of distances. A highly suitable source for such measurements is the carbon dioxide laser, which can be used to produce a number of lines whose wavelengths are known to a high degree of accuracy and which can also be tuned rapidly under computer control from line to line (BOURDET and ORSZAG[ 19791, GILLARD and BUHOLZ[ 19831, WALSHand BROWN[ 19851). Lasers have opened up new applicationsof length interferometry, such as the measurement of earth strains, for which interferometers with a length of more than a kilometre have been used (VALIand BOSTROM[1968], BERGERand LOVBERG [ 19691). These interferometers provide a strain sensitivity of a few parts in 10” for strain rates up to a maximum value of s - ’ with a long-term stability of the same order.

2.3. FRINGE COUNTING

Lasers have made electronic fringe counting a very practical technique for length interferometry.Typically, an optical system is used, giving two uniform interference fields in one of which an additional phase difference of 4 2 is introduced between the interfering beams. Two detectors viewing these fields provide signals in quadrature, which can be used to drive a bidirectional

112


IIII, § 2

counter. Fringe-counting interferometers of this type have been described by GILLILAND, COOK,MIELENZand STEPHENS [ 19661 and by MATSUMOTO, SEINOand SAKURAI [ 19801. Fringe counting systems of this type have also been used in free-fall instruments for absolute measurements of gravitational acceleration (ZUMBERGE,RINKER and FALLER[ 19821, ARNAUTOV, BOULANGER, KALISH, KORONKEVITCH, STUS and TARASYUK [ 19831). In anDther method of fringe counting (DYSON,FLUDE,MIDDLETON and PALMER [ 19721) the two beams emerging from the interferometer are linearly polarized at right angles and traverse a 4 4 plate oriented at 45 O , which converts them into right-handed and left-handed circularly polarized light, respectively. The two beams therefore combine to produce a linearly polarized beam whose plane of polarization rotates through 360" for a change in the optical path difference of two wavelengths. The optical path difference can then be monitored continuously by a polarizer controlled by a servo system (ROBERTS [ 19751). HOPKINSON [ 19781 has made a detailed analysis of the possible errors in such a system, which has been called an 'optical screw'. A better fringe-counting system, which has the advantage that its operation is not affected by variations in the intensity of the beams due to low-frequency laser noise, uses two optical frequencies (DAHLQUIST,PETERSONand CULSHAW [1966], DUKESand GORDON[1970]); these are generated by a He-Ne laser which is forced to oscillate simultaneously at two frequencies v1 and v, ,separated by a constant differenceof about 2 MHz, by applying an axial magnetic field. As shown in Fig. 2.1, these two waves, which are circularly polarized in opposite senses, go through a A/4plate which converts them to orthogonal linear polarizations.A portion of the beam is then split off; this goes through a polarizer, which mixes the two frequencies and is incident on a detector D,. The output from this detector at the beat frequency (v, - vl) is used as a referencefrequency.The main beam goes to a polarizing beam-splitter at which one frequency (say v l ) is reflected to a fixed cube corner C, ,while the other frequency v,, is transmitted to a movable cube comer C,. Both frequencies leave the interferometer along a common axis and, after being made to interfere by a polarizer, are incident on another detector D, . The outputs from the two photodetectors D, and D, are taken to a differential counter. If the cube corner C, is stationary,the frequenciesof the two outputs are the same, and no net count accumulates. However, if C, is moved, the net count gives the change in optical path in wavelengths. This interferometer is now used widely for industrial measurements over distances up to 60 m. LIU and KLINGER[ 19791 have shown how it can also be used for measurements of angles, straightness, flatness and squareness.

111, § 31

113

MEASUREMENTS OF CHANGES IN OFTlCAL PATH LENGTH

Beam Solenoid

expander

Display

Fig. 2.1. Fringe-countinginterferometer using a two-frequencylaser. (AAer DUKES and GORDON [ 19701, 0 Copyright 1970 Hewlett-Packard Company. Reproduced with permission.)

Fringe counting has also been used for the measurement of small vibration amplitudes (STETSON [ 19821). In this method an additional, linearly increasing phase difference is introduced between the two beams in a Michelson interferometer by means of a heterodyne phase-shifter, and the number of zero crossings in the output for each vibration cycle is counted. If the frequency of the signal generated by the heterodyne phase-shifter is much lower than, and incommensurate with, the vibration frequency, the number of zero crossings per vibration cycle converges, in the limit, to the peak-to-peak vibration amplitude in units of a quarter wavelength. If measurements are made over 1000 cycles, a peak-to-peak amplitude of ;1/2 can be measured with an accuracy of ;1/2000. The minimum detectable vibration amplitude is set by the ratio of the frequency generated by the phase shifter to the vibration frequency and is typically a few nanometres.

Q 3. Measurements of Changes in Optical Path Length A number of techniques have been developed that can be used with laser interferometers for measurements of small changes in optical path length.

114


[IIL I 3

3.1. CLOSED-LOOP FEEDBACK SYSTEMS

One group of methods is based on phase compensation. Changes in the output intensity from the interferometer are detected and fed back to a phase modulator in the measurement path so as to hold the output constant (CATHEY, HAYES,DAVISand PIZZURO[1970], SPRAGUEand THOMPSON [1972], JOHNSON and MOORE[ 19771, FISHER and WARDE[ 19791). The drive signal to the modulator is then a measure of the changes in the optical path. 3.2. HETERODYNE METHODS

Another group of methods is based on optical heterodyning.In one technique a frequency difference is introduced between the two beams in the interferometer (CRANE[ 19691, LAVAN,CADWALLENDER and DEYOUNG [ 19751). Systems that have been used for this purpose include a rotating 114 plate in series with a fixed 1214 plate, a rotating grating, and two acousto-optic modulators operated at slightly different frequencies. The electric fields resulting from the two beams emerging from the interferometer can then be represented by the relations El@)= a, cos(2nv1t +

$1),

(3.1)

E2(t) = a2 cos(2nv2t + $J,

(3.2)

where a, and a2 are the amplitudes, v1 and v2 are the frequencies, and $, and $2 are the phases of the two wavefronts. The output from an ideal square-law detector would be, accordingly, I ( t ) = lE,(t) + E2(012, = la2 + lu2 2

1

2

2

+ ; [ a ; cos(4lrv,t + $1) + a; cos(4kv*t + $2)] + ala2 cos[24v,

+ V2)t + ($1 + $211

+ a 1 4 cosPn(v1 - v2)t + ($,

- (p2)I.

(3.3)

The second and third terms on the right-hand side of eq. (3.3) correspond to components at frequencies of 2 vl, 2 v2 ,and (v, + vz), which are too high to be followed by practical detectors. Accordingly, eq. (3.3) reduces to

I ( $ )= I, + l2+ 2(I,12)”2c0s[27t(v1- v2)t + (+I - $211 , where I ,

= iu;

1 2 and I z = 5a2.

(3.4)

111, I 31

MEASUREMENTS OF CHANGES IN OPTICAL PATH LENGTH

115

The output from the detector consists of a steady current on which is superposed an oscillatory component at the difference frequency (v1 - v2). The phase of this modulation, which can be measured accurately with respect to an electronic reference signal, then directly gives the phase difference between the interfering wavefronts. In another technique two mirrors are attached to the two points between which measurements are to be made, fonning a Fabry-Perot interferometer, and the frequency of a slave laser is locked to a transmission peak of the interferometer, so that the wavelength of the slave laser is an integral submultiple of the optical path difference in the interferometer. Any change in the separation of the mirrors then results in a corresponding change in the wavelength of the slave laser and hence in its frequency. These changes can be measured by mixing the beam from the slave laser at a fast photodiode with the beam from a reference laser, whose frequency has been stabilized and measuring the beat frequency. This technique replaces measurements of the fringe order by measurements of frequency. Since measurements can be made in the frequency domain with very high precision, it is an extremely powerful technique which has found many applications.

3.2.1. Thermal expansion One such application has been in precise measurements of coefficients of thermal expansion (WHITE 119671). Figure 3.1 is a schematic of the experimental arrangement used for such measurements on low-expansion and SHOUGH[ 19811). A problem with materials such as fused silica (JACOBS such a setup is that measurements normally can be made only over a very limited range, since it is possible, as the sample length changes, for the resonant frequency of the cavity to move outside the gain bandwidth of the laser. Accordingly, after a preassigned frequency excursion the frequency of the slave laser is automatically unlocked from the original cavity resonance and locked to an adjacent cavity resonance that lies well within the laser gain bandwidth. Measurements have been made by this method, using a 100 mm sample, with a precision better than 1 part in lo8. A modified version of this system has also been described for comparing the coefficients of thermal expansion of samples taken from different parts of a large mirror blank of low-expansion material (JACOBS,SHOUGHand CONNORS [ 19841).

116


[III, § 3

Temperature Controlled Enclosure 1 Tunable Laser Detector -

t Spectrum AnalYSer

I

I r

-j I

I

pn Convertf

I

Amplifier High Voltage OP-AMP

Fig. 3.1. Apparatus for measurements of thermal expansion. (JACOBS and SHOUGH[1981].)

3.2.2. The Michelson-Movley experiment

Laser heterodyne techniques have made it possible to verify the null result of the Michelson-Morley experiment to a very high degree of accuracy [ 19641, BRILLETand HALL[ 19791). (JASEJA,JAVAN,MURRAY and TOWNES

The latter experiment used a He-Ne laser operating at a wavelength of 3.39 pm, whose output frequency was locked to a resonance of a very stable, thermally isolated confocal Fabry-Perot cavity mounted along with it on a rotating horizontal granite slab. Any variations in the delay within this Fabry-Perot interferometer due to effects such as those predicted by classical theory would manifest themselves as frequency shifts of the laser, which could be detected with a very high degree of sensitivity by miXing the beam from this laser with the beam from a stationary laser that was stabilized by saturated absorption in methane. However, the experiment revealed no effect greater than of that predicted by classical theory. 5x

111, § 31

MEASUREMENTS OF CHANGES IN OPTICAL PATH LENGTH

117

3.3. TECHNIQUES USING TUNABLE LASERS

The availability of electrically tunable lasers has made possible a number of other techniques for measuring optical path lengths as well as changes in optical path length. Two of these techniques described by OLSSONand TANG[ 19811 are outlined below. 3.3.1. Two-wavelength intellferometry

In two-wavelength interferometry the interferometer is alternately illuminated with two wavelengths A1 and A,, which differ by such an amount that the phase differences between the beams emerging from the interferometer at these two wavelengths differ by 90". If, then, AI1 and M2are the changes in the output intensity at these two wavelengths because of a small change A$ in the phase difference q5 between the interfering beams, we have

AZ, = ( A sin @)A$

(3.5)

A12 = (A cos $ ) A $ ,

(3.6)

and

where A is a constant, so that A$

=

(AZ? + A I y A

(3.7)

This technique can be implemented conveniently with a setup using a GaAlAs semiconductor laser, whose output can be switched at a frequency of 10 kHz between two wavelengths separated by about 0.3 nm and has been used to detect a phase modulation of 0.0003 radian at a frequency of 500 Hz. 3.3.2. Frequency-modulation interferometry

The basic principle of frequency-modulation interferometry is to measure the change of the optical wavelength AA, at a mean wavelength A, required to change the phase difference 4 between the two interfering beams by an integral multiple N of 2n. N, A, AA, and the optical path difference d are then related by the expression N

=

dAA/A2.

(3.8)

118

[IIL § 4


If the laser wavelength is linearly swept at a tuning rate

d

=

p, we then have

A2/T0B,

(3.9)

where To is the time required for the phase difference between the beams to change by 27c. Hence, by measuring the period of the intensity variations of the output from the interferometer,the optical path differencecan be obtained. This technique can be used with an electronically tuned dye laser to measure optical thicknesses ranging from a few micrometres to several centimetres.

8 4. Detection of Gravitational Waves Gravitational waves can be thought of as an alternating strain that propagates through space, affecting the dimensions and spacing of material objects embedded in it. Possible sources of such gravitational waves include collapsing supernovas, pulsars, and binary stars. Early experiments to detect gravitational waves used short resonant detectors, but it was soon realized that a long wide-band antenna would have many advantages. The high sensitivity obtainable by laser interferometry also made this approach seem promising.

4.1. PROTOTYPE INTERFEROMETRIC DETECTORS

Early prototypes of such a wide-band detector (Moss, MILLER and FORWARD [ 19711, FORWARD [ 19781) consisted of a Michelson interferometer with the beam splitter and the end reflectors attached to separate, freely suspended masses. As shown in Fig. 4.1, the arm lying along the direction of propagation of a gravitational wave does not experience a strain, so that it acts as a reference, whereas the strain produced by the gravitational wave acts on the other arm of the interferometer, causing a differential motion of the beam splitter and end reflector at the frequency of the gravitational wave. When the direction of the gravitational wave is at right angles to the plane of the interferometer, the changes in length of the two arms have opposite signs, resulting in a doubling of the output. A detailed analysis of the response of such an antenna has been made by RUDENKOand SAZHIN[ 19801. The ultimate limit to the sensitivity of such an interferometric detector is set by photon noise and is approximately (see DREVER,HOGGAN,HOUGH, MEERS,MUNLEY,NEWTON,WARD,ANDERSON, GURSEL, HERELD,SPERO

111, § 41

119

DETECTION OF GRAVITATIONAL WAVES

Direction of propagation of gravitational wave I

Fig. 4.1. Schematic of a laser interferometer for detecting gravitational waves.

and WHITCOMB [ 19831) E =

(Ih~/16nZL~Zz)'/~,

where E is the strain due to the gravitational wave pulse and z is its duration, h is Planck's constant, L is the length of each arm,and Z is the laser power. Theoretical estimates of the intensity of bursts of gravitational radiation which reach the earth from outer space suggest that their detection requires a sensitivity to strains of the order of a few parts in lo2'. With typical laser powers this would lead to unrealistically long arms (> 100 km).

4.2. METHODS OF OBTAINING INCREASED SENSITIVITY

One way of obtaining increased sensitivity is by using multiple reflections between two mirrors in each arm to increase the effective paths (BILLING, MAISCHBERGER, RUDIGER,SCHILLING,SCHNUPPand WINKLER[ 19791). However, a problem then is scattered light from the mirrors that has traversed a shorter optical path and can have its phase modulated by residual fluctuations

120


[IIL § 5

in the laser frequency (SCHILLING,SCHNUPP, WINKLER, BILLING, MAISCHBERGER and RUDIGER[ 19811). An alternative approach that is not affected by this problem and has a number of other advantages involves the use of two identical Fabry-Perot interferometers, about 1OOOm long, at right angles to each other (DREVER, HOUGH, MUNLEY,LEE, SPERO, WHITCOMB,WARD, FORD,HERELD, ROBERTSON, KERR,PUGH,NEWTON, MEERS,BROOKSand GURSEL[ 19811). The mirrors of these interferometers are mounted on four nearly free test masses, and their separations are continuously compared by locking the output frequency of a laser to the transmission peak of one interferometer, while the optical path in the other is continuously adjusted by a second servo system so that its transmission peak coincides with the laser frequency. The corrections applied to the second interferometer then provide a signal that can be used to detect any changes in the length of one arm with respect to the other, over a wide frequency range, with extremely high sensitivity. Even with such a system, very high laser powers would be needed to reach the required sensitivity. Two promising ways of obtaining the necessary sensitivity with limited laser power have been discussed by DREVER, HOGGAN, HOUGH, MEERS, MUNLEY, NEWTON, WARD, ANDERSON, GURSEL, HERELD,SPEROand WHITCOMB [ 19831. Since most of the light incident on the two arms is reflected back to the source when the transmittance of the interferometer is a minimum, an additional mirror can be introduced in front of the laser so as to return part of this light to the interferometer. Another method, which can be applied to gravitational wave signals of known frequency, is to use a modified system in which two waves circulate through the two arms in opposite senses. If the number of reflections in each arm and its length are arranged to give a light storage time equal to half the period of the gravitational wave, the two circulating beams will accumulate opposite phase shifts, giving a gain in sensitivity by a factor equal to the ratio of the overall storage time to the period of the gravitational wave.

4 5. Laser Doppler Interferometry Laser Doppler velocimetry (YEH and CUMMINS[ 19641) makes use of the fact that light scattered from a moving particle has its frequency shifted by an amount proportional to the component of its velocity in a direction determined by the directions of illumination and viewing. This frequency shift can be detected by the beats produced either by the scattered light and a reference

111, B 51

121

LASER DOPPLER INTERFEROMETRY

beam or by the scattered light from two illuminating beams incident at different angles. To distinguish between positive and negative flow directions, an initial frequency offset can be used between the two interfering beams (STEVENSON [1970]). This technique is now used widely to measure flow velocities (see DURST,MELLINGand WHITELAW[ 19761).

5.1. MEASUREMENT OF SURFACE VELOCITIES

Laser Doppler interferometry is a similar technique that has been used for the measurement of surface velocities. In applications such as the propagation of shock waves in solids, the polished surface of the moving specimen is used as one of the end mirrors in a Michelson interferometer (BARKERand HOLLENBACH[1965]). The output beam from the instrument is then amplitude-modulated at a frequency corresponding to the Doppler shift. However, such a simple setup is limited in its velocity capability, since for a wavelength of 633 nm, a velocity of 0.1 m/s would give a fiinge frequency of 316 MHz. The fringe frequency can be reduced and the interferometer made direct reading in velocity by using the modified optical arrangement shown in Fig. 5.1, in which the laser beam is reflected off the specimen before entering the interferometer, where it is split into two beams, one of which is subjected to a delay before it is recombined with the other (BARKER[1971]). With this

Target

specimen

Laser

Detectors

Fig. 5.1. Laser interferometer for direct measurement of surface velocities. (BARKER[ 19711.)

122

[III, f 5


Polarizer

Frequency shifter

Beam

Laser

Wollaston

Polarizing beam-splitter

I Fig. 5.2. Optical system of an industrial laser Doppler interferometer used to measure the and SOMMARGREN [1984].) velocity and length of moving surfaces. (TRUAX,DEMAREST

configuration the total fringe count at any time is proportional to the average velocity over the delay interval, whereas the fringe frequency is proportional to the acceleration (KAMEGAI [ 19741). The constant of proportionality can be conveniently chosen by varying the delay time. In addition, since the beam is focused to a small spot on the moving surface, measurements can be made on diffusely reflecting surfaces. The measurement range can be extended to cover even higher velocities as well as rapid variations in velocity by using interference between two beams that are incident upon the surface at different angles (MARON [ 1977, 19781). An industrial application of laser Doppler interferometry is to measure the velocity of moving material such as, for example, hot rolled steel, in order to cut it to given lengths. Figure 5.2 is a schematic of a velocimeter developed for DEMAREST and SOMMARGREN [ 19841). As shown, two this purpose (TRUAX, orthogonally polarized beams produced by a Wollaston prism are focused on the surface by a lens at angles of incidence equal to 8. The scattered light collected by the same lens goes to a polarizing beam splitter oriented at 45" so that half of each polarization is directed to the two detectors. It can then be shown that the frequency of the beat signal at the two detectors

111.8 51

LASER DOPPLER INTERFEROMETRY

123

does not depend on the viewing direction and is given by the relation V, =

(2/A) I upl sin 8 ,

where up is the component of the velocity of the material parallel to its surface. In addition, the signals from the two detectors are 180" out of phase, so that by subtracting one signal from the other, the Doppler signal can be doubled while cancelling out variations caused by surface structure (BOSSEL,HILLER and MEIER[ 19721).

5.2. MEASUREMENTS OF VIBRATIONS

Laser heterodyne techniques can also be used to analyse surface vibrations. In the simplest situation, involving only sinusoidal motion, one of the beams in a Michelson interferometer is reflected from a point on the vibrating specimen, whereas the other is reflected from a fixed reference mirror (DEFERRARI, DARBYand ANDREWS[ 19671). Measurements are facilitated by using an arrangement in which a known frequency offset is introduced between the beams by diffraction at a B r a g cell (EBERHARDT and ANDREWS[ 19701). The output from a detector then consists of a component at the offset frequency (the carrier) and two sidebands. The amplitude of the vibration can be determined by a comparison of the amplitudes of the carrier and the sidebands, whereas the phase of the vibration can be obtained by comparison of the carrier with a reference signal (PUSCHERT[ 19741). A modification of this method, which can be used to study nonsinusoidal vibrations (OHTSUKAand SASAKI[ 1974]), uses two successive ultrasonic modulators to generate two frequency-shifted beams, one of which is reflected from the vibrating surface and the other from a reference mirror. These beams are incident on a photodetector that generates a beat signal whose phase can be detected by means of a phase demodulator circuit using an electronic reference signal. The output of the demodulator, which is a direct measure of the displacement of the specimen, can then be displayed in real time on an oscilloscope. These techniques make it possible to measure nonsinusoidal vibration amplitudes of the order of a few nanometres as well as sinusoidal vibration amplitudes down to a few thousandths of a nanometre at frequencies ranging from about 50 kHz upwards. Measurements at lower frequencies are hampered by the fact that the signal is buried in low-frequency (l/f) noise. One way of minimizing the effects of

124

[IIL 3 5


tM*

Ultrasonic light modulator

generator Signal

-1

Lcl Bias

fotfc+fm

-\

He-Ne laser

i f,ff,

I.

Ah

Balanced modulator

fc

11

Detector

/

/

M2

11 Detector

YT

Spectrum analyzer

Oscillator

Tr Oscillator

,1

Recorder Fig. 5.3. Laser interferometer for measurements of slowly varying displacements. (OHTSUKA and

ITOH [1979].)

such noise down to about 500 Hz is by using bispectral analysis (SASAKI,SATO and ODA[ 19801). However, the most widely used method is to shift the desired signal to a higher frequency range. One technique that has been used to detect low-frequency vibrations involves phase modulating the reference beam (BASSANand CILIBERTO[ 19801). Another technique, which has been applied to measurements of very slowly varying displacements (OHTSUKAand SASAKI [ 19771, OHTSUKA and ITOH [ 1979]), uses an ultrasonic light modulator to produce a two-frequency laser beam (frequenciesf, + f, + f, and f, + f, - f,) which, as shown in Fig. 5.3, is split in a Michelson interferometer into two components: one of them is the signal beam which is modulated in phase by reflection at the vibrating mirror M,, whereas the other is a reference beam which is reflected at the fixed mirror M,. When these reflected beams are superposed at a detector, it can be shown that the output is given by an expression of the form

+ 1s + 2(IRzs)1’2cos[$p. - @s(t)]} + +{zR+ I , + 2(zRzS)ll2COS[& - & ( t ) ] }

r(t) = i ( Z R

cos4nfmt,

(5.2)

111, 8 61

LASER-FEEDBACK INTERFEROMETERS

125

where ZR and Z, are the intensities of the reference beam and the signal beam, respectively, & is the constant part of the phase difference between the reference beam and the signal beam, and $ ~ ~ ‘ is s ( tthe ) time-varying phase shift of the signal beam. As can be seen, information on the low-frequency displacements of the object contained in the second term of eq. (5.2) is converted into amplitude modulation of a signal at the difference frequency 2f, and can be separated from low-frequency noise quite efficiently. Experiments with such a system have shown that measurements are possible on square waves with a frequency of 0.2 Hz down to amplitudes of 0.3 pm (OHTSUKAand ITOH [ 19791, OHTSUKA and TSUBOKAWA [ 19841.)

6 6. Laser-Feedback Interferometers The laser-feedback interferometer (ASHBYand JEPHCOTT [ 19631) makes use of the fact that the intensity of the beam from a laser can be influenced by feeding even a very small portion of the output back into the laser cavity. This feedback is provided, as shown in Fig. 6.1, by an external cavity consisting of M,, the output mirror of the laser, and an external mirror M, . The laser output then varies cyclically with the separation of M, and M, ,a change in the optical path length of the external cavity of 1/2 resulting in one cycle of modulation. The operation of this interferometer can be analysed very simply by considering M, and M, as a Fabry-Perot interferometer that effectively replaces M, as the output mirror of the laser (CLUNIEand ROCK[ 19641). The reflectance of this Fabry-Perot interferometer is

R=

rt

+ rz(1 - A,), - 2r2r3(1- A,) cos$ 1 + (r,r3), - 2r2r3COS$

9

where r, and r, are the reflectances for amplitude of M, and M, ,A is the loss due to scattering and absorption in M, , and $ is the phase difference between

Detector

Laser cavity

External cavity

Fig. 6.1. The laser-feedback interferometer.

126


WI,§ 6

successive interfering beams. Typically, with a laser mirror having a transmittance of 0.008, the output can be made to vary by a factor of four by using an external mirror with a reflectance of 0.1. The response of such a system decreases at high frequencies because of the finite time required for the amplitude of the laser oscillation to build up within the cavity. Measurements with a He-Ne laser and a spinning reflector have shown that the depth of the modulation decreases by 50percent at a modulation frequency of 100 kHz. A detailed analysis of this interferometer, including the factors contributing to the finite response time, has been made by HOOPERand BEKEFI[ 19661. An interesting observation with such an interferometer is that if the laser is oscillating simultaneously on two transitions that share a common upper or lower level, and if M, reflects only one of these wavelengths, the output power at the other wavelength varies in antiphase to the power at the first wavelength. This effect can be seen with a He-Ne laser and has been used to measure plasma densities at 3.39 pm with a detector sensitive to radiation at 633 nm (ASHBY,JEPHCOTT,MALEINand RAYNOR[ 19651). An n-fold increase in the sensitivity of this interferometer can be obtained by passing the beam back and forth n times within the external cavity, which contains the medium under study. A spherical external mirror has been used for this purpose (GERARDO and VERDEYEN[ 1963]), but a system that is easier to align and allows the isolation of a selected higher-order beam consists of a focusing lens used in conjunction with a plane laser output mirror and a plane return mirror (HECKENBERG and SMITH[ 19711). Another method of obtaining increased sensitivity for measurements on plasmas is by using a longer wavelength, typically the 10.6 pm line from a carbon dioxide laser, to take advantage of the large dispersion of the electrons (HEROLDand JAHODA [ 19691). Laser-feedback interferometershave several advantages, including simplicity and high sensitivity, since a small change i n the gain results in a large change in the output. However, a problem is that when the change in optical path exceeds 4 2 , ambiguitiescan arise. A way around this is to use a laser oscillating on two orthogonally polarized longitudinal modes at slightly different frequencies. If the ratio of the lengths of the laser cavity and the external cavity is properly chosen, the signals corresponding to the two polarizations can be separated and used to obtain phase data in quadrature (TIMMERMANS, SCHELLEKENS and SCHRAM[ 19781).

111, t 71

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OPTICAL TESTING

8 7. Optical Testing The development of lasers has led to many new advances in the application of interferometry to tests on optical components and optical instruments.

7.1. UNEQUAL-PATH INTERFEROMETERS

Much effort has gone over the years into the design of interferometers for optical testing mainly because of the limitations imposed by the poor spatial and temporal coherence of light from thermal sources. These limitations are virtually eliminated with laser sources, so that very simple optical arrangements become possible. A typical example is the compact laser unequal-path interferometer shown in Fig. 7.1, which can be used for testing large concave surfaces (SHACKand HOPKINS[ 19791). This interferometer contains only one precision optical component, a beam-splitter cube with a plano-convex lens cemented to one surface. The image of the centre of curvature of the convex surface of this lens formed in the beam splitter lies just outside the input face of the cube. A pinhole spatial filter is placed at this point in the beam from the laser, which is brought to a focus by a microscope objective. Interference fringes are formed by the beams reflected from the surface under test and the spherical surface of the lens cemented to the beam-splitter cube, and these fringes can be viewed through an eyepiece. An even simpler arrangement that can be used as a lateral shearing interferometer with a nominally plane wavefront consists, as shown in

-I I

Mirror under test

4 cube

Plano-convex lens

Fig. 7.1. Laser unequal-path interferometer for testing large concave surfaces. (SHACK and HOPKINS[1979].)

128


est wavefront

Radially sheared wavefronts Fig. 7.2. Laser shearing interferometers: (a) lateral shear (MURTY[1964]); (b) radial shear (ZHOUWANZHI[1985]).

Fig. 7.2a, of a plane-parallel plate (MURTY[ 19641). A modification of this instrument (HARIHARAN [ 1975a1) uses two separate plates with an air gap; this has the advantage that a tilt can be introduced between the two sheared wavefronts to make the interpretation of the fringes easier. A number of very simple optical systems have also been described for radial-shearing interferometers, consisting essentially of the two spherical surfaces of a thick lens. Interference takes place either between the directly transmitted wavefront and the wavefront that has undergone one reflection at each surface (STEEL[ 19751) or, as shown in Fig. 7.2b, between the wavefronts reflected from two spherical surfaces (ZHOUWANZHI[ 1984, 19851). In addition a number of simple methods are available for testing the parallelism of the surfaces of a glass plate, or the angles and pyramidal error of a right angle prism, with the component itself acting as an interferometer (MCLEOD[ 19741).

s

111, 71

4

OPTICAL TESTING

129

7.2. TESTS ON GROUND SURFACES

Laser interferometers also permit tests on he-ground surfaces before they are polished. Nominally flat surfaces can be tested with a He-Ne laser by using an interferometer in which light is incident obliquely on the surface. Since it is not necessary to equalize the paths, a simple optical setup is possible, using two gratings to divide and recombine the beams (BIRCH[ 19731, HARIHARAN [ 1975bl). Such a system also has the advantages that it is easy to align and can compensate for the low specular reflectivity of the surface. The need to test a ground surface is greatest during the production of aspheric optics. In this case one can make use of the fact that a specular reflection is obtained at longer wavelengths. MUNNERLYN and LATTA [ 19681 were the first to show that useful measurements could be made with a carbon dioxide laser at a wavelength of 10.6 pm. Further progress was made possible by the development of the pyroelectricvidicon. KWON,WYANTand HAYSLETT [ 19801 have built an infrared Twyman-Green interferometer with a carbon dioxide laser source and a pyroelectric vidicon camera to view the fringes and have used it to test a large off-axis mirror for a collimator. Infrared laser interferometry is also useful for testing optical components made of infrared transmitting materials. Three lateral shearing interferometers for this purpose have been described by KWON[ 19801. 7.3. ELECTRONIC MEASUREMENTS OF OPTICAL PATH DIFFERENCES

Lasers have also made feasible new techniques using electronics for directly measuring the optical path difference between the test and reference wavefronts at an array of points covering the interference pattern. These techniques are capable of high precision and are not affected by variations in the level of illumination across the field.

7.3.1. Heterodyne techniques In one goup of techniques, which is directly related to heterodyne interferometry, a frequency difference is introduced between the two beams (CRANE [ 19691). As a result, the irradiance at any point in the interferencepattern varies sinusoidally at the difference frequency (see $3.2). The phase difference between the two interfering wavefronts at any selected point can then be determined by comparing the phase of the electrical signal from a movable detector, which can sample the pattern at different points, with that from a

130


[III, § 7

stationary reference detector (SOMMARGREN and THOMPSON[ 19731, MASSIE,NELSONand HOLLY[ 19791). Measurements can be speeded up by using an image dissector camera to scan the pattern (MOTTIER[ 19791, MASSIE [ 19801).

7.3.2. Quasi-heterodyne techniques In quasi-heterodyne techniques the optical path difference between the interfering beams is made to vary linearly with time, either by introducing a frequency offset between the beams or by means of a suitable linear phase modulator, and the output current from a detector located at any point P ( x , y ) on the fringe pattern is integrated over a number of equal segments (typically four) covering one period of the sinusoidal output signal (WYANT[1975], STUMPF[ 19791, SCHAHAM [ 19821). If these outputs are I,, I,, 13,and 14, respectively, the phase difference between the interfering wavefronts at this point is given by the relation tan q%Y)

= (1, -

13)/(12 - 1.4).

(7.1)

Quasi-heterodyne methods have the advantage that a CCD array can be used as the detector to make measurements simultaneously at a very large number of points covering the interference pattern.

7.3.3. Phase-stepping methods In one method (BRUNING,HERRIOTT, GALLAGHER, ROSENFELD, WHITE and BRANGACCIO [ 19741) the optical path difference between the interfering wavefronts is changed in a number of equal steps, and the correspondingvalues of the irradiance at each data point in the interference pattern are measured and stored. The values of the intensity at each point can then be represented by a Fourier series, whose coefficients can be evaluated to obtain the original phase difference between the interfering wavefronts at this point. Typically, an eight-step staircase modulation of the optical path difference is used (GROSSO and CRANE[ 19791). A simpler version of this method originally described by CARRE[ 19661 for length measurements involves only four measurements at each point, corresponding to four equal phase steps. These measurements provide enough data to calculate the original phase difference between the wavefronts as well as the phase step. If the phase step is known, only three measurements of the intensity are required. Typically, one value of the phase shift can be zero, whereas the

111, § 71

OPTICAL TESTING

131

other two values are 90" and 180" (FRANTZ,SAWCHUKand VON DER OHE [ 19791, DORBAND[ 1982]), in which case tan

9 = [2Z(90) - I(0) - I(l8O)]/[Z(O) - I(l8O)l.

Alternatively, a phase step of 120" can be used (HARIHARAN, OREB and LEISTNER [ 19841). Because of the relatively small memory requirements as well as the simplicity of the algorithm for calculating the phase difference, an inexpensive microcomputer can be used with these methods. Measurements can be made over a 100 x 100 array of points with a precision of k 2" in a few seconds. A typical system for such measurements is shown in Fig. 7.3.

7.3.4. Residual errors Since it is possible to make measurements rapidly and store the data, the effects of vibration and air currents can be minimized by averaging a number of observations. Similarly, errors due to the interferometer optics can be eliminated by subtracting readings made without the test piece, or with a standard, from readings made with the test piece (BRUNING [1978], HARIHARAN, OREBand LEISTNER[ 19841). SCHWIDER, BUROW,ELSSNER,GRZANNAand SPOLACZYK [ 19831 have made a detailed analysis of other sources of systematic errors such as deviations of the phase steps from their nominal values. Perhaps the most serious problem, when making measurements to All00 or better, is unwanted reflections and scattered light within the interferometer, which can add coherently to the wavefront from the test piece and result in significant systematic errors.

7.4. APPLICATIONS

Laser interferometers with digital phase measurement systems are now used extensively in the production of high-precision optical components (YODER, GROSSOand CRANE[ 19821). A particularly interesting area of application is [ 19791, HARIHARAN, OREBand ZHOU in testing aspheric surfaces (DOHERTY WANZHI[ 19841). Another area is in evaluating the residual surface roughness of polished surfaces, where rms surface rmghness down to 0.01 nm can be [ 19811, HUANG [ 19841, BHUSHAN,WYANTand measured (SOMMARGREN KOLIOPOULOS [ 19851).

132


ALTERNATIVE POSITIONS OF TEST PIECE FOR MEASUREMENTS

CONTROLLER

J A b

11

LOGIC

Fig. 7.3. Schematic of a typical system for phase-stepping interferometry. (HARIHARAN, OREB and LEISTNER[1984].)

8 8. Heterodyne Spatial Interferometry One of the earliest applications of interferometrywas in measurements of the angular diameters of stars. A star can be modelled as a small incoherent source

111, 5 81

HETERODYNE SPATIAL INTERFEROMETRY

133

over which the intensity distribution follows some simple law. The angular diameter of the star can then be obtained from interferometric measurements of the complex degree of coherence of the radiation reaching the surface of the earth at points separated by different distances. The possibility of using a laser as a local oscillator for heterodyne detection of light from a star was first studied experimentally by NIEUWENHUIJZEN [ 19701. In trials with a 2 m telescope, light from a star was combined with light from a He-Ne laser at a photocell, and radiofrequency signals arising from interference between the light from the laser and components of the light from the star at very nearly the same frequency were observed.

8.1. INFRARED HETERODYNE DETECTION

Heterodyne detection is based on the fact that a photodetector has a square-law response to the incident radiation field. The total electric field at the surface of the photodetector can be described by the relation (see Q 3.2) E ( t ) = EL cos q t

+ E , cos o,t,

(8.1)

where, in this case, the first term on the right-hand side represents the field due to the laser, which consists of a fixed frequency, whereas the second term, which represents the field due to the star, contains power over a range of frequencies. The response of an ideal square-law detector would then be

+ E$ C O S ~o,t + ELEs COS(O,+ w,)t + ELEs COS(W,- o,)t.

I ( t ) = E t COS' q t

(8.2)

However, since the detector has a limited frequency response, its output actually is I'(t) = iEZ

+ i E $ + ELE, COS(W,- w , ) t .

(8.3)

The second term on the right-hand side, which contains frequencies ranging from zero up to the maximum frequency to which the detector responds, represents the heterodyne signal. Typically, this covers a frequency bandwidth of about 1 GHz. The actual optical frequency bandwidth is twice this because of detection of radiation in this bandwidth both above and below the laser frequency. A major attraction of the heterodyne technique is that the spectral resolution obtained is extremely high, since it is defined by the bandwidth of the detector

134


WI,I 8

and its associated electronics. In addition, much higher sensitivity is obtained over this limited bandwidth than is possible with direct detection, because the output is proportional to the product of the intensities of the laser and the star. The properties of optical heterodyne detectors have been analyzed in detail by SIEGMAN[1966], who has shown that they offer no advantages over conventional detectors with weak thermal sources in the visible region. However, their relative sensitivity improves directly with increasing wavelength. Their use at a wavelength of 10.6 pm appears particularly attractive because of the 8 to 14 pm atmospheric transmittance window as well as the availability of a powerful source of coherent radiation in the carbon dioxide laser. The strong directivity and frequency selectivity of the heterodyne detection process also gives good discrimination against noise sources such as black-body radiation from objects at room temperature, which can be appreciable at this wavelength (TEICH[ 19681).

8.2. INFRARED HETERODYNE STELLAR INTERFEROMETRY

Heterodyne measurements of the angular diameter of the sun were first made at 10.6 pm by GAYand JOURNET [ 19731, using a carbon dioxide laser and two HgCdTe photodiodes with a bandwidth of about 400 MHz. Correlation of the amplified currents from the detectors gave a sinusoidal output, the equivalent of fringes in an interferometer, because of the diurnal motion of the sun. The fringe amplitude was found to be a function of the separation of the detectors. The angular diameter of the sun, calculated from the separation of the detectors corresponding to the first zero of the fringe amplitude, was 33 minutes of arc. Subsequently, JOHNSON,BETZ and TOWNES [ 19741 constructed a two-element heterodyne spatial interferometer operating at 10.6 pm with a baseline of 5.5 m and successfully tested it on a number of astronomical sources. This instrument used two independent telescopes with an effective aperture of about 80 cm, each consisting of a steerable heliostat followed by a fixed off-axis focusing mirror and flat mirrors that directed the beams to a fixed focus. The two heliostats were situated on an East-West baseline with a centre-to-centre separation of 5.5 m. At an operating wavelength of 10.6 pm, this separation would give an angular resolution of about 0.5 second of arc. As shown in Fig. 8.1, a high-speed germanium-copper photoconducting detector located at the focus of each of the telescopes mixed the light from the star with a beam from a stabilized 1 W carbon dioxide laser, which acted as the local oscillator. The amplified signals from the two detectors went to a

135


1

Telescope

Teles

5MHz offset

Laser

2

\

I

Detector

Detector

Delay lines

Computer

b

b

Processor

+

Fringe amplitude Fig. 8.1. Schematic of an infrared heterodyne stellar interferometer. (JOHNSON,

TOWNES[1974].)

BETZ and

136


[I14 § 8

correlator where they were multiplied together. The amplitude of the sinusoidal output signal from the correlator (which was the equivalent of fringes in a conventional stellar interferometer) was proportional to the degree of coherence between the wavefields at the two telescopes. The natural frequency of this signal was determined by the motion of the star across the field of view, and for a horizontal East-West baseline of length D was given by the relation

f, = [ 520 cos 6 cos H ] / A ,

(8.4)

where B is the rotation rate of the earth, 6 the declination, and H is the hour angle of the source. To produce a signal at a convenient frequency for further processing and also to avoid interaction between the two lasers, they were phase locked with a frequency difference of 5MHz. This 5 M H z carrier frequency was finally removed in a single-sideband demodulator to give the natural fringe signal. T o observe interference of two beams over a frequency bandwidth of Av, the difference in the path lengths must be small compared with c/Av. The signals from the photodetectors, after amplification, were therefore passed through adjustable radio frequency delay lines, which compensated for the changes in the two optical paths as the star was tracked across the sky. In this case the 1500 MHz bandwidth of the radio frequency signals required the path lengths to be equalized only to within a few centimetres. Observations have been made with this interferometer on a number of infrared sources including M-type supergiants and Mira variables to obtain information on the temperature and spatial distribution of circumstellar dust shells (SUITON, STOREY,BETZ, TOWNESand SPEARS[1977], SUTTON [ 19791).

8.3. LARGE INFRARED HETERODYNE STELLAR INTERFEROMETER

Since the sensitivity of such a system is proportional to the available collecting area, it is advantageous to use larger telescopes. This is possible in the infrared, since for a given degree of atmospheric turbulence the diameter of a telescope that is diffraction limited increases as A6I5. Thus, at a wavelength of 10 pm with reasonably good seeing conditions, the useful diameter of each telescope could be as much as 3.8 m, making it possible to study faint objects. Another major improvement would be an interferometer whose baseline could be extended up to, say, 100 m and could also be changed in orientation. This would give, in addition to better angular resolution, information on any

111, I 81


137

departure from circular symmetry of the source. Finally, measurements of the phase of the signal would also permit making accurate positional measurements as well as mapping regions that are not symmetrical with respect to inversion. TOWNES [ 19841 has made a detailed analysis of the factors involved in the design of an optimum system and has given a description of a large infrared heterodyne stellar interferometer that is under construction. This instrument is basically made up of two telescope units, each of which consists of a 2 m flat mirror rotating about two axes and a 1.65 m paraboloid with its optic axis horizontal. This arrangement gives a compact system which is sturdy enough to be mounted on a trailer so that it can be moved, when required, to a new site to change the baseline. Once in position, each mirror rests on kinematic mounts on a concrete pad set into the ground so that it is no longer supported by the trailer. During observations, He-Ne laser interferometers are used to monitor the positions of the telescopes with respect to invar posts set in bedrock, as well as the optical path lengths within the telescope. In addition, the large flat mirrors are pointed using interferometric measurements at four positions around the edge of each mirror to determine its angular position relative to the horizontal optic axis of the parabola. The local oscillators on each telescope are carbon dioxide lasers that are locked in phase. Phase locking is done by sending a beam from one laser to the other telescope, where it is compared with the output from the second laser whose phase is controlled with a fast feedback circuit. To eliminate the effects of any changes in the optical path between the two telescopes, part of the first laser beam is retuned along its original path and its relative phase on return is automatically adjusted by a variable element in the path. Very fast HgCdTe detectors are used with a bandwidth of 4 GHz. Corresponding to this bandwidth, the path lengths can be equalized to better than 1 cm with a variable-length radio frequency cable. This interferometer is expected to provide angular resolution down to 0.001 second of arc as well as an astrometric precision of about 0.010 second of arc. Since it will measure fringe phase as well, it should be possible to map complex infrared objects with high resolution. Estimates of its sensitivity indicate that, with integration times up to an hour, it could be used to study a few thousand stellar objects.

138


[IK§9

9. Interferometric Sensors The development of lasers has opened up a completely new field of applications for interferometers, namely their use as sensors for various physical quantities.

9.1. INTERFEROMETRIC ROTATION SENSORS

One of the earliest applications of an interferometer as a sensor was Sagnac's experiment for detecting rotation in an inertial frame, using an interferometer with two beams travelling around the same circuit in opposite directions. When such an interferometer rotates with an angular velocity 61 about an axis making an angle 8 with the normal to the plane of the beams, a fringe shift is observed corresponding to the introduction of an optical path difference

where A is the area enclosed by the light path. This formula is valid for light propagating in an accelerating frame of reference as well as in media other than a vacuum (POST [ 19671, LEEB,SCHIFFNER and SCHEITERER [ 19791). 9.1.1. Ring-laser rotation sensors The ring laser (ROSENTHAL [ 19621, MACEKand DAVIS[ 19631)was the first practical method for detecting rotation in an inertial frame by purely optical means. Rotation of a ring laser (see Fig. 9.1) shifts the frequencies of the clockwise-propagating and anticlockwise-propagating modes by equal amounts in opposite senses, giving rise to an optical beat whose frequency can be measured. A very stable beat can be obtained because the optical cavity is common to both the modes and they are affected equally by temperature changes and any mechanical disturbances. However, at low rotation rates problems arise because of mode locking, mainly because of back scattering from the mirrors. ARONOWITZ [1971] has made a detailed analysis of the physics of such a ring-laser rotation sensor, whereas ROLANDand AGRAWAL [ 19811have described some techniques for eliminating mode locking. The most widely used method of avoiding mode locking is the introduction of dither (KILLPATRICK [ 19671); an alternative is the use of a four-frequency ring laser

139

INTERFEROMETRIC SENSORS

pherical mirror

Glass ceramic block

Output mirror

Detector

prism Cathode

Fig. 9.1. Ring laser for rotation sensing. (ROLANDand AGRAWAL [1981].)

HAMBENNE, HUTCHINGS, SANDERS,SARGENT and SCULLY [ 19801, STATZ,DORSCHNER, HOLTZand SMITH[ 19851) in which the cavity supports two pairs of counter-propagating waves with opposite circular polarizations. Ring lasers are now used widely in inertial navigation systems because of their many advantages over mechanical gyroscopes; these include fast warm-up, rapid response, large dynamic range, insensitivity to linear motion, and freedom from cross-coupling when used for multi-axis sensing. (CHOW,

9.1.2. Passive interfrometric rotation sensors Mode locking can be avoided by using a passive ring interferometer as a rotation-sensing element with an external laser and measuring the difference in the delays for the two directions of propagation (EZEKIEL and BALSAMO [ 19771). One way is to use a Faraday cell within the cavity to cancel out the difference in the optical path lengths for the counter-propagating beams. Another scheme uses a single laser and two acousto-optic modulators to generate two independently controlled optical frequencies that are locked to the two resonant frequencies of the ring (SANDERS,PRENTISSand EZEKIEL [ 19811).

140


m,§ 9

9.1.3. Limits of sensitivity The theoretical limit of sensitivity for the ring laser is set by spontaneous emission in the gain medium and is given by the relation (DORSCHNER, HAUS, HOLTZ,SMITHand STATZ[ 19801)

A51 = A0P~,/4A(n,,z)'l2 ,

(9.2)

where A, is the vacuum wavelength, P is the optical perimeter and r, is the linewidth of the cavity, nph is the photon flux in the laser beam, and z is the averaging time. A similar expression is also obtained for a passive resonator, the limit in this case being determined by photon noise. Practical problems that limit the sensitivity of passive resonators are misalignment and back scatter at the mirrors. However, the effects of back scatter can be virtually eliminated by a phase modulator placed in one of the beams before it enters the resonator (SANDERS,PRENTISSand EZEKIEL [ 19811). Both ring lasers and passive rotation sensors can be made to deliver performance close to the theoretical limit.

9.2. FIBRE-OPTIC INTERFEROMETRIC SENSORS

With the development of lasers it became possible to build analogues of conventional two-beam interferometers with single-mode optical fibres. Early fibre interferometers used gas lasers, but semiconductor lasers have now replaced them almost completely, because apart from their small size and high efficiencyGaAlAs lasers operate in the near infrared at a wavelength at which the losses in silica fibres are much lower than in the visible region. High sensitivity can be obtained with fibre interferometers because it is possible to have very long paths in a small space. In addition, because of the extremely low noise level, sophisticated detection techniques can be used. 9.2.1. Rotation sensing The first application of fibre interferometers was in rotation sensing, where a closed multi-turn loop made of a single fibre was used to replace a conventionalring cavity with mirrors in order to increase its effective area (VALI and SHORTHILL [1976]). Very small phase shifts can be measured and the sense of rotation determined with such an interferometer by introducing a nonreciprocal phase modulation and using a phase-sensitive detector (ULRICH


Polarization controller

141 Polarization controller

Polarizer

Semiconductor laser Coupler Detector I Modulated signal

1 -

e

a

de

n

d

AC generator

Fig. 9.2. All-fibre rotation sensor. (BERGH,LEFEVREand SHAW[1981].)

[ 19801). BERGH,LEFEVRE and SHAW[ 19811 have described a system based entirely on optical fibres, in which, as shown in Fig. 9.2, normal beam splitters are replaced by optical couplers and the phase modulator consists of a few turns of the fibre wound around a piezoelectric cylinder. A wider measurement range can be obtained with a closed-loop system, in which the phase difference caused by rotation is compensated by a nonreciprocal phase shift generated within the ring by suitably positioned acousto-optic frequency shifters (CAHILL and UDD [1979], DAVISand EZEKIEL[1981]). Fibre-interferometric rotation sensors have the advantages of very small overall size and relatively low cost. A detailed analysis made by LIN and [ 19791 shows that they are an attractive alternative to ring-laser GIALLORENZI rotation sensors. Practical limitations to their sensitivity are set by noise from a number of causes. The effects of back-scattering can be avoided by using a broad-band source such as a superluminescent diode, whereas nonlinear effects can be minimized by equalizing the intensities of the two beams (EZEKIEL, DAVISand HELLWARTH [ 19821). The effects of fibre birefringence can be eliminated either by using an input polarizer in conjunction with a polarization-preserving fibre (BURNS, MOELLER,VILLARUEL and ABEBE [1984]) or by using a single-polarization fibre. If proper care is taken to

142


“11, § 9

minimize temperature variations, vibration, and external magnetic fields, performance close to the limit set by shot noise can be obtained. 9.2.2. Generalized fibre-interj‘ierometric sensors

Other applications of fibre-interferometric sensors are based on the fact that the optical path length in a fibre is affected by its temperature and also changes when the pressure changes or the fibre is stretched. Figure 9.3 is a schematic of a generalized fibre interferometer showing the principal components of such a sensor. An optical layout analogous to the Mach-Zehnder interferometer is common, since this avoids optical feedback to the laser, although optical arrangements based on the Michelson and the Fabry-Perot interferometers have also been used (IMAI, OHASHIand OHTSUKA[1981], YOSHINO, KUROSAWA, ITOHand OSE [ 19821). Instead of beam splitters, optical-fibre couplers are used, permitting an all-fibre arrangement with a considerable reduction in noise. Measurements are made with either a heterodyne system or a phase-tracking system. Phase modulation is introduced either by an integrated-optic phase shifter or by a fibre stretcher in the reference beam, and the resulting output signal is picked up by a photodetector followed by a demodulator (JACKSON,PRIEST, DANDRIDGEand TVETEN [ 19801, DANDRIDGE and TVETEN[ 19811). Other detection schemes have also been

Optical fibre sensing element

Detector system Fig. 9.3. Schematic of a typical fibre-optic interferometric sensor. (GIALLORENZI, BUCARO, DANDRIDGE, SIGEL,COLE,RASHLEIGH and PRIEST[1982].)

111, § 91


143

used, involving either a modulated laser source (GILLES,UTTAM,CULSHAW and DAVIES [ 19831) or laser-frequency switching (KERSEY,JACKSON and CORKE[ 1983I). Slow drifts can be eliminated by a phase-tracking system using an electronic feedback loop or by tuning the emission frequency of the semiconductor laser source (DANDRIDGE and TVETEN[ 19821). Optical phase shifts as small as radian can be detected with such sensors. A very compact interferometric sensor can be set up with a single-mode GaAlAs laser and an external mirror coupled by a single-modefibre to the laser to form a laser-feedback interferometer (DANDRIDGE, MILES and GIALLORENZI [ 19801). In one mode of operation the laser current is held constant and the output power is monitored by a photodetector; alternatively, the drive current to the laser required to maintain a constant output is measured. An increased measurement range can be obtained by mounting the mirror on a piezoelectric translator and using an active feedback loop to hold the optical path from the laser to the mirror constant at a suitable operating point on the response curve.

9.2.3. Applications Fibre interferometers have been used as sensors for mechanical strains and changes in pressure and temperature (BUCARO,DARDY and CAROME [ 19771, BUTTERand HOCKER[ 19781, HOCKER[ 19791, JACKSON, DANDRIDGE and SHEEM[ 19801, LACROIX, BURES,PARENTand LAPIERRE [ 19841). They can be used for measurements of magnetic fields either by using a magnetostrictive jacket on the fibre or by bonding the fibre to a magnetostrictive element (YARIV and WINSOR[ 19801, DANDRIDGE, TVETEN,SIGEL,WESTand GIALLORENZI [ 19801, RASHLEIGH[ 19811, WILLSONand JONES [ 19831, Koo and SIGEL [ 19841). Electric fields can also be measured by using a single-mode optical fibre bonded to a piezoelectric film or jacketed with a piezoelectric polymer as a detector (Koo and SIGEL[ 19821, DE SOUZAand MERMELSTEIN [ 19821). Yet another application has been to monitor variations in the output [ 19801). Much of wavelength of a semiconductor laser (SHEEMand MOELLER the work in these areas has been reviewed by GIALLORENZI, BUCARO, DANDRIDGE, SIGEL,COLE,RASHLEIGH and PRIEST[ 19821 and by KYUMA, [ 19821. TAI and NUNOSHITA


144

5 10.

[III,$ 10

Pulsed-Laser and Nonlinear Interferometers

Lasers can be used to produce light pulses of very short duration ( 1.

3.3 TRANSVERSE MODE STRUCTURE

The transverse field structure of the modes supported by an unstable resonator with a sharply defined feedback aperture has been investigated with both numerical and analytical techniques. Early numerical results were published by SIEGMANand ARRATHOON [ 19671, SANDERSON and STREIFER[ 1969a,b], SHERSTOBITOV and VINOKUROV[ 19721, RENSCH and CHESTER[ 19731, CHESTER [ 1973b], SIEGMAN [ 19741,RENSCH[ 19741, KARAMZIN and KONEV [ 19751, and STEER and MCALLISTER [ 19751. A rigorous exposition of the required sampling criteria, based on the collimated Fresnel zone structure of an unstable cavity, was first given by SZIKLASand SIEGMAN[ 1974, 19751. These sampling criteria are described in Appendix A of the present chapter. SIEGMANand SZIKLAS[1974] have also presented a numerical procedure based on a Hermite-Gaussian beam expansion of the cavity field; this set of basis functions is naturally suited to a wide variety of beam propagation problems, since they are eigenmodes for free-space propagation (SIEGMAN [ 19731). Other numerical propagation techniques and refinements thereof have been described by PERKINS and SHATAS [1976], SIEGMAN [1977], SOUTHWELL [ 1978, 19811, LAX,AGRAWAL and LOUISELL [ 19791, LATHAM and SALVI[ 19801, OUGHSTUN[ 19801, AGRAWAL and LAX[ 1981a1, and FEIT and FLECK[ 19811. Analytical approaches to describe the transverse mode structure have also been developed by CHENand FELSEN[ 19731, HORWITZ [ 1973,19761,MOOREand MCCARTHY [ 1977b],B m s and AVIZONIS [ 19781, CHO, SHINand FELSEN[1979], NAGEL,ROGOVIN,AVIZONISand B u n s [ 19791, CHOand FELSEN[ 19791, NAGEL and ROGOVIN[ 19801, and LUCHINI and SOLIMENO [1982]. The theoretical predictions of both the cavity field

270

UNSTABLE RESONATOR MODES

[IV, § 3

mode structure and the divergence of the outcoupled field have been found to be in good agreement with the experimental measurements of KRUPKEand SOOY[ 19691, ANAN’EV,CHERNOVand SHERSTOBITOV [ 19721, FREIBERG, CHENAUSKY and BUCZEK[1972, 19741, WISNER,FOSTERand BLASZUK [ 19731, CHODZKO,MIRELS,ROEHRSand PEDERSEN[ 19731, GRANEKand MORENCY[ 19741, ZEMSKOV,ISAEV,KAZARYAN,PETRASHand RAUTIAN [ 19741, RENSCH[ 19741, ANAN’EV,BELOUSOVA, DANILOV, SPIRIDONOV and TROFIMOV [ 19741, ANAN’EV,GRISHMANOVA, PETROVAand SVENTSITSKAYA [ 19751, PHILLIPS,REILLY and NORTHAM [ 19761, CHODZKO, MASONand CROSS[ 19761, FREIBERG, FRADINand CHENAUSKY [ 19771, [ 19781, MUMOLA,ROBERTSON, STEINBERG, KREUZERand MCCULLOUGH CHODZKO,MASON,TURNERand PLUMMER[ 19801, SPINHIRNE,ANAFI, and GARCIA[ 19811, ANAFI,SPINHIRNE, FREEMANand OUGHFREEMAN STUN [ 19811, SPINHIRNE, ANAFIand FREEMAN [ 19821, OUGHSTUN, SPINHIRNE and ANAFI[ 19841, and DAN’SHCHIKOV, DYMSHAKOV, LEBEDEV and RYAZANOV [ 19821. The present section reviews the salient features of the transverse mode structure of an unstable cavity, as primarily described in the preceding list of references. The analysis begins with the fundamental Fresnel number and magnification properties of the cavity mode structure and concludes with a description of the transverse mode hierarchy that is capable of being supported by an unstable cavity. 3.3.1. Equivalent Fresnel number and magnification dependence The physically important parameters for describing the diffractive mode structureproperties of an unstable resonator with a single, sharp-edgefeedback aperture are the cavity magnification and the equivalent Fresnel number. The cavity magnification describes the dominant first-order, geometrical properties of the transverse mode structure [see eqs. (3.13) and (3.24)]. Superimposed on this geometrical contribution are the edge-scattered waves from the feedback aperture, and their constructive (or destructive) interference is quasiperiodic with respect to the equivalent Fresnel number of the cavity (see the discussion following eq. [2.55]). These results are completely borne out by the results of extensive numerical and asymptotic analyses. Figures 16 and 17 depict the intensity and phase distributions of the dominant mode structures that are incident on the feedback aperture of an M = 2.5 unstable caVity with Neq= 0.5 and Neq= 1.5, respectively. These results were obtained by SIEGMAN and SZIKLAS[ 19741 with the Hermite-Gaussian beam

IV, § 31

PASSIVE CAVITY MODE STRUCTURE BEHAVIOR

27 1

Fig. 16. Relative intensity and phase distributions of the dominant mode structure incident on the feedback aperture ofanM = 2.5 unstable cavity withN,, = 0.5. (After SIEGMANand SZIKLAS [ 19741.)

212


IIV, § 3

Fig. 17. Relative intensity and phase distributions of the dominant mode structure incident on the feedback aperture of an M = 2.5 unstable cavity with Nes = 1.5. (After SIEGMAN and SZIKLAS [ 19741.)

IV, s 31


213

expansion technique. Similar behavior with some additional fine structure, particularly at the higher Fresnel number, was obtained with a fast Fourier transform (FFT) method (SZIKLAS and SIEGMAN[ 19741); it is believed that this fine structure would appear in the Hermite-Gaussian (HG) beam expansion calculations if more terms were retained. The dominant mode eigenvalue magnitude was found to be y = 0.632 (HG calculation) and y = 0.640 (FFT calculation) for the N,, = 0.5 cavity, whereas for the Neq = 1.5 cavity the results were y = 0.510 (HG calculation) and y = 0.557 (FFT calculation). The smoothing effect of the overtruncated Hermite-Gaussian beam expansion calculations is then seen to yield a more lossy mode structure, particularly at the larger Fresnel number. The transverse mode structure formation in a small Neq unstable cavity (Neq 5 1) is almost completely dominated by edge diffraction effects, even to the extent that the diffractive properties overpower the geometrical properties for sufficiently small values of Neq (OUGHSTUN[ 1981bl). This phenomenon is evident in Fig. 16. In effect, the feedback aperture acts much like a spatial filter in the diffractive formation of the cavity mode structure. As such, it strongly discriminates against the higher-order azimuthal structure in favor of the 1 = 0 mode of a cylindrical cavity, as it would also discriminate against the antisymmetrical mode structure in favor of the symmetrical mode structure of a rectangular cavity. Figures 8 and 10 clearly show that this behavior continues at large Fresnel numbers. This discriminatory effect becomes more predominant with decreasing Fresnel numbers because of the decrease in the relative size of the feedback aperture with respect to the transverse extent of the central intensity core region of the cavity mode (OUGHSTUN[ 1983a1). A large Fresnel number cavity does not possess the excellent transverse mode discrimination found in a small Fresnel number cavity because of the large Fresnel zone structure retained in the cavity by the feedback aperture. The Fresnel zone structure over the feedback aperture of an unstable cavity may be defined in the following manner: Consider the sagittal distance between the exiting geometrical mode phase front and the feedback mirror surface of a cylindrical, confocal unstable cavity. This is readily found to be

(3.86) for r < a , , where a , is the transverse radial extent of the feedback aperture, R , is the radius of curvature of the feedback mirror, zT is the cavity length, A4 is the magnification, and r is the radial distance from the cavity optical axis. A

214


[IV, § 3

radially dependent equivalent Fresnel number function may then be defined as r2 2 Neq(r)= -A(r) z NeqI a:

(3.87)

for r < a l . At the feedback mirror edge (r = a , ) this function is equal to the equivalent Fresnel number of the cavity. The associated cavity Fresnel zones over the circular feedback aperture are then concentric circles whose radii satisfy Neq(r) = n + f

where n

=

Neq

(3.88)

0, 1,2,. .., and

(3.89)

O, 0 and N:q (3.130) becomes

(3.132b) =

Im Neq 2 0. With these substitutions eq.

(3.133) and an exponentially damped term appears, whose damping coefficient Nlq depends only on the parameters a and a = As a consequence, as the equivalent Fresnel number of the cavity increases, the eigenvalues 7;') of the apodized aperture cavity approach the geometrical values 7;') more rapidly than in the case of the cavity with a sharp-edged feedback aperture. For the case (3.94~)of an exponentially smoothed feedback aperture edge with "edge smoothness parameter" E satisfying E 6 a, the first order approximation of the eigenvalue spectrum is

4s.

(3.134) As in the previous case, as the equivalent Fresnel number of the cavity increases, the eigenvalue spectrum 6''approaches the geometrical spectrum 6') faster than in the case of a sharp-edged aperture. When the edge smoothness parameter satisfies the inequality

(3.135) the nature of the oscillations of the eigenvalues 7;') is approximately that for the sharp-edged aperture case. On the other hand, if

1

(3.136)

IV, § 31

299


then the nature of the oscillations of the eigenvalues 7;’) depends strongly on the value of the edge smoothness parameter. Finally, for the case (3.94d) of a feedback aperture with a partial smoothness of degree k (ke a), the eigenvalue spectrum is found to be given by

(3.137)

T -

0 -

-aJ ,

,

0

1

,

,

2

,

, 3

,

,

,

4

I

5

6

7

8

9

Neq

Fig. 22. Dominant (n = 0) eigenvalue magnitude and phase as a function of the equivalent Fresnel number Neq of an M = 2 cavity with an exponentially smoothed feedback aperture for several values of the edge smoothness parameter E.

300


[IV, § 3

As in the previous cases, the eigenvalue spectrum 7;') approaches the geometrical spectrum 72)more rapidly than for the sharp-edged aperture cavity, but now the approach is not exponential but goes as a - ( k +l ) . The behavior of the dominant (n = 0) eigenvalue magnitude and phase as a function of the cavity equivalent Fresnel number Neqfor the case of an exponentially smoothed feedback aperture is depicted in Fig. 22 for several values of the edge smoothness parameter E with M = 2, Each value of E depicted here satisfies the inequality (3.135), at least in the weak sense, where [M/(M2- 1)]'/*/27t = 0.13 for the present example. For larger values of E the oscillations in both the eigenvalue magnitude and phase are almost completely damped out.

3 A.2. Intracavity spatial filtering in unstable ring resonators Another viable approach to enhance the mode discrimination properties and reduce the equivalent Fresnel number dependence of unstable cavity modes is to employ a spatial filtering process within the cavity. A properly designed spatial filter would significantly reduce the high spatial-frequency components associated with the edge diffraction phenomena at the outcoupling-feedback aperture of the cavity, thereby directly influencing the diffractive formation of the cavity mode. The low-loss, low-order spatial modes of an optical cavity possess a higher relative degree of focusable irradiance than do any of the higher-loss, higher-order spatial modes capable of being supported by the cavity; it follows that the introduction of a suitably sized spatial filter aperture at an intracavity real focus will significantly increase the relative losses of all the higher-order modes while only slightly increasing the loss of the lowestorder mode, thereby increasing the transverse mode discrimination of the cavity. In addition, this approach would serve to minimize the deleterious effects of high-order intracavity phase aberrations and high spatial-frequency amplitude distortions that naturally occur in large laser systems. The basic ring cavity geometry considered by OUGHSTUN,SLAYMAKER and BUSH [ 19831 is depicted in Fig. 23. In the positive or forward direction of propagation around the cavity, a plane wave geometrical mode originating at the feedback aperture dl will be magnified by the factor M after a single round-trip iteration, whereas in the negative or reverse direction it will be demagnified. The entirety of the forward wave mode magnification occurs in propagation through the confocal mirror pair, at the real focus of which the spatial filter aperture d2is introduced. In the remainder of the cavity the forward mode is collimated parallel to the optical axis of the resonator (this

IV, § 31

30 1


-____-_________--___-_

f-----

FEEDEACK-

I

I

I

I

I

I

I I

I

I I

I

-+-\

APERTURE

/ - -

-- --_--

I

-

\

CONFOCAL MIRROR PAIR

(b)

Fig. 23. Ring resonator geometry with an intracavity (real) focus and spatial filter aperture. In (a) the cavity comprises an odd number of mirrors and so is an effectively positive-branch cavity in the plane of the figure, whereas in (b) the cavity comprises an even number of mirrors and is a negative-branch cavity in that plane.

302


[IV, § 3

collimated beam condition is not necessary for the spatial filtering approach to be applicable). The only differencebetween the two cavity configurationsdepicted in Fig. 23 is the inclusion of an additional turning flat mirror in configuration (b). This results in a fundamental difference in the symmetry properties of those two configurations (AL’TSHULER, ISYANOVA, KARASEV,LEVIT,OVCHINNIKOV and SHARLAI[ 19771). Because of the presence of the internal focus, the orientation of the cavity mode in the vertical direction (out of the plane of the figure) is inverted in both cases after a single round-trip propagation through the cavity. Hence, in either case the cavity is negative branch in the vertical dimension. For a ring resonator with an internal focus and an odd number of mirrors (Fig. 23a), the horizontal orientation of the cavity mode (in the plane of the figure) is left unchanged in a single round-trip propagation, whereas for an even number of mirrors (Fig. 23b), the horizontal orientation is inverted in a single round-trip propagation. As a consequence, for a ring resonator with an internal focus and an even number of mirrors the cavity is negative branch in both the vertical and horizontal directions. In that case the transverse cavity mode is inverted about the optical axis in a single propagation through the cavity (in both the forward and reverse directions). However, for a ring resonator with an internal focus and an odd number of mirrors, the cavity is negative branch in the vertical direction but is positive branch in the horizontal direction. The cavity mode is then inverted about the horizontal meridional plane in a single round-trip propagation. The mode inversion properties of the particular cavity configuration directly influence the odd-order aberration sensitivity of the cavity mode (AL’TSHULER, ISYANOVA, KARASEV,LEVIT,OVCHINNIKOV and SHARLAI [ 19771, OUGHSTUN[ 1985a1). For the present analysis only the ideal unaberrated mode properties of the passive cavity are considered. In that case the branch type has no influence on the transverse mode properties, provided that the cavity is symmetrical with respect to inversion about the vertical and horizontal meridional planes. It is assumed here that the paraxial optical system of the ring cavity is rotationally symmetrical about the cavity optical axis. For the present the transverse geometries of the feedback aperture d,and spatial filter aperture d2are left unspecified with only the requirement that they are symmetrical with respect to inversion about the vertical and horizontal meridional planes. The integral equation for the unfiltered passive cavity mode structure in the

IV,

s 31


303

forward direction of propagation around the cavity is

where B

=

z2 (M + l)fl - M z ~- . M

(3.139)

Here M = f 2 / f l > 1 is the geometrical magnification of the cavity (in the forward direction), where fz and f, are the magnitudes of the focal lengths of the confocal mirror pair (see Fig. 23). The total collimated Fresnel number of the cavity (when the transverse geometry of the feedback aperture dlis circular with radius a, and no spatial filter is present) is given by N,=

- MU: ~

,

(3.140)

a,B and the total equivalent Fresnel number is

(3.141)

A case of special interest occurs when z2 =

M(M

+ l)f,

- MZz,

,

(3.142)

in which case B = 0 and the effective propagation distance around the cavity is zero. In that case the feedback aperture is imaged back upon itself with magnification M, and a self-imaging condition is present in the ring cavity (PAXTON and SALVI[1978]). When this condition is satisfied, both of the cavity Fresnel numbers are infinite. That special case is then equivalent to the geometrical optics limit as the wavelength of the cavity field goes to zero. However, it is important to note the fundamental difference between these two cases. In the geometrical optics limit as A + 0, all wave phenomena disappear. When the self-imaging condition is satisfied, however, only the diffraction phenomena due to the feedback aperture are eliminated in the paraxial approximation. In the latter case the integral equation (3.138) for the transverse mode structure reduces exactly to that in the geometrical mode theory of SIEGMAN

304


",

§3

and ARRATHOON[ 19671, whereas in the former it reduces asymptotically. The results of a numerical calculation of the eigenvalue spectrum of such a selfimaging resonator are given in Fig. 15 ($3.2). The integral equation for the unfiltered passive cavity mode structure in the negative direction of propagation around the cavity is

(3.143) At the outcoupling-feedback aperture of the cavity the counterpropagating wave fields are related by (see eq. [2.67]) u,t(x, y ) = u,(x, y ) exp

1

(3.144)

with 7; = y,, . Here R is the radius of curvature of the geometrical mode phase front in the negative direction incident upon the outcoupling aperture of the cavity, given by (3.145) The location of the intracavity focus for this spherically diverging geometrical mode is given by Az=

MZf? R+z2-Mfl'

(3.146)

where Az denotes the distance of the reverse mode focus C from the focus F of the confocal mirror pair, as depicted in Fig. 24a. Notice that the self-imaging condition (3.142) also applies to this reverse propagating field. When the spatial filter aperture is introduced at the intracavity focus F, the focus C of the reverse propagating mode will shift over to F so as to minimize its loss at that aperture in the strong spatial filtering limit, as indicated in Fig. 24b. From eq. (2.80) the eigenvalues of the counterpropagating modes will always be identical so that one need only consider the forward propagating mode properties. From eq. (2.71) the passive cavity mode propertles of the

IV, S 31

\

\

4 \ \ \ \

\

\

I

\ \

\

\

\ \ \

:5 \ 9g ............\..

g'

I

't

$5 1 0 cc ...(... ........ z a 8% \

I

I

I

I

I

I

I

I

I

I

I I

7 f

I

I I

I

I

I

I

I

I I

I

I I I I I

tI I

I

I

I I

I I

I

I I

I

;

I

1

\

\

\ I

'

............4 ......./ ............

I

I

I I I

'I

I

1


I

..j ............I

\ I \I

k-4

h

4

'Fj OD

0

.5

305

306


[IV,8 3

forward travelling wave are found to satisfy the coupled pair of equations

(3.147a)

Here u1 is the cavity field incident upon the feedback aperture dl and uz is that incident upon the spatial filter aperture dz. Notice that the cavity mode field distribution incident upon the ideal spatial filter aperture plane is proportional to the Fourier transform of the cavity mode over the feedback aperture dl,and that the cavity mode field distribution incident upon the outcoupling-feedback aperture plane is (to a good approximation) proportional to the Fourier transform of the cavity mode over the spatial filter aperture dzwhen atzis sufficiently small so that the Fraunhofer approximation is applicable. This approximate Fourier transform relationship between the cavity mode field at the outcoupling and spatial filter aperture planes then suggests that the dominant transverse cavity mode is very nearly a Gaussian in the presence of strong spatial filtering. Note, however, that this is not true if the spatial filter aperture size is made too small so that a majority of the irradiance in the focal plane is apertured. In such an "overfiltered" case, even though the Fourier transform relationship between the cavity mode field at the outcoupling and spatial filter aperture planes is still valid, the severe edge dfiaction effects incurred at the focal plane aperture will dominate the cavity mode formation process and the resultant mode structure will no longer be Gaussian. 3.4.2.1. Single-stage spatial filtering; the focal point aperture resonator In polar cylindrical coordinates the integral equation (3.147) for the passive forward cavity mode properties (when both apertures dl and dzare circular

IV,

s 31


307

and centered on the cavity optical axis) may be written as

x exp

['ifl&)rz] r, i-

(1 -

dr, , (3.148a)

for the dominant azimuthally symmetrical ( I = 0) cavity mode. Here a is the radius of the spatial filter aperture and a, is the radius of the feedback aperture at the outcoupling plane. Let the spatial filter radius a be sufficiently small so that the cavity mode is approximately a Gaussian, which may be written as ul(rl) = e-Er: ,

(3.149)

where the parameter a remains to be determined in terms of the cavity parameters. If the inequality a 0 is is the radius of the spatial filter aperture d,, the spatial filter radius index. If the inequality

A4 m o ,

g,

0 , otherwise,

0,

if

g>o,

- g , otherwise

,

(77)

466

INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT

[V,§ 3

(and similarly for go, g,, and g, for complex-valued g) is not the optimum choice, even though it does keep the component spatial signals (and, hence, system noise) minimal. The proper choice of the component signals is by no means simple. We shall concentrate on the bipolar case and treat the complex-valued case only briefly. Following LOHMANNand RHODES[ 19771 and MAIT [ 1986b], we can write g+ (x, Y ) = M

X , 39

+ $(A Y)l

9

v) = fr - g(x9 r) + J/(x, A1

g- (4

*

(78) Equation (77) would imply that $(x, y) = I g(x, y ) I. Since the spatial bandwidth of Ig(x, y) 1 for bipolar g(x, y ) can be much higher (possibly infinite) than that of g(x. y ) itself, this is usually not an acceptable choice. A usable $ must satisfy the following requirements: $(x, y) must be real. $(x, y) must be integrable. $(x, y ) must be band limited (with spatial bandwidth not substantially higher than that of g(x, y)). $(x,y) must be bounded in space, not taking much more space than g(x, y ) itself. $(x, Y ) 3 I g(x9 Y )I. In the general case not all possible bipolar spatial signals have $functions that satisfy these requirements. In practice, however, one can find a $ that satisfies all conditions at least approximately(for example, its very high spatial frequency terms are not zero, yet are neghgible). A method of obtaining a suitable $(x, y) is through the use of analytic function theory (MAIT [ 1986b1). Although this method is mathematically elegant, it is difficult to apply to practical, often complicated spatial signals. An alternative, also demonstrated by MAIT[ 1986a1, is iteration. We start with an unsatisfactory solution, namely that of eq. (77). This solution will be called g,, (x, y) and gmin,- ( x , y). The corresponding $ is +

$min(x, Y ) =

I

Y )I.

(79)

We now iterate $dx, Y> =

+min(X,

B [ $(x, y)]

=

Y)

3

$(x, y) * [4B2sinc(2Bx, 2By)l:

iteration cycle:

$n(x,y) = P { B [+n - l(x, y)]}

.

v, § 31

BIPOLAR AND COMPLEX-VALUED SPATIAL SIGNALS

467

Here the operator P keeps $ sufficiently large so that both g, and g- are non-negative, whereas the operator B makes $ band limited within a spatial frequency limit of B. MAIT[ 19851 discusses the convergence of this iteration. He shows that for discrete signals (as is always the case when we use a digital computer) the iteration always converges quite rapidly. To get a similar procedure for the complex-valuedg(x, y), we rewrite eq. (73) as

c g,sin 2

Zm{g} =

I-0

2 -Re{g} - -Im{g}

fi

)

+$ ,

where g , g, and $ are all functions of (x,y). The iteration process is similar to that of eqs. (80):

468


[V,§ 3

3.2. TEMPORAL ENCODING

The use of several independent non-negative real channels for processing bipolar or complex-valued spatial signal calls for multiple optical subsystems (at least part of the optical system must be replicated) and multiple image detectors. In addition to the extra complexity and bulk, the issue of channel balancing must also be taken care of. Many optical information processing schemes offer response times that are significantly shorter than necessary for most applications. If we “time share” a single optical system between several logical channels, we gain in overall system compactness and circumvent the balancing problem. One possibility is using temporal modulation, similar to that used in radio broadcasting. If g(x, y), our complex-valued (or bipolar-real) spatial signal, varies slowly enough in time, we can mathematically define g’(x,y , t ) so that g’(x, y, 2) = Re { g k y ) exp2 njvt}

+ b(x, y ) ,

(84)

where v is the temporal frequency of the carrier, and b ( x , y ) is a (real) bias, satisfyingb(x, y ) 2 Jg(x,y ) 1. The temporally encoded spatial signal then can go through the processing channel. There are two distinct cases. (1) One only of the two signals is complex valued or bipolar; the other spatial signal is non-negative real. For addition both input signals must be temporal carrier modulated by the same temporal frequency following eq. (84). The output would come out carrier modulated by the same temporal frequency. For multiplication, convolution, or correlation the non-negative real spatial signal can be unmodulated. The output would be modulated. (2) Both input signals are complex valued and/or bipolar. For addition both spatial signals must be modulated by the same temporal frequency. The output is also modulated by the same frequency. For multiplication, convolution, or correlation each spatial signal may be modulated by a different temporal frequency. The frequency of the modulation of the output signal is the sum of the two frequencies. Although the idea of temporal encoding, as presented above, is simple and elegant, its practical implementation is often difficult. The encodingprocess can be implemented (for bipolar-real spatial signals) by switching between two component spatial signals using, for example, polarized light and some electronically controlled polarizer or polarization rotator such as a PLZT device or liquid crystal. One such example is the work of INDEBETOUW and POON


469

[ 19841. Alternatively, the modulator can be integrated in the system itself as in RHODES[ 1977al. The RHODES[ 1977al system used a cube beam splitter, so that two pupil masks are presented in an OTF synthesis system in the same effective location (Fig. 32). One pupil has the pupil function p1 and the other p,. We note that light passes twice through each mask, so pi is the square of the complexamplitude transmittance functions of the actual masks. If the optical path through both masks is the same, the effective pupil of the system is PI + la2. However, we may shift one mirror (M2 in Fig. 32, for example) by 1/4,changing its optical phase by n. For this case the effectivepupil of the system would come out to be 8, - p2. We can now quickly switch between the mirror positions (using, for example, a piezo-driven mirror mount) and get sequentially the two pupil functions and,hence, two PSFs. Rhodes shows how to design the masks from desired bipolar PSFs. Demodulation is easy for scanner systems where the output is essentially temporal. For parallel processors such as OTF synthesis systems, we need an “area temporal demodulator” or an array of phase-locked amplifiers. BARRETT, GMITRO and CHIU[ 19811 showed how a certain type of TV camera, based on the image orthicon, can be modfied to do precisely this desired functioning. They flipped the polarity (the sign of the y ) of the TV camera by applying a square wave temporal signal, at a frequency much higher than the frame rate (they used frequencies up to over 100 kHz), to the grid of the image-orthicon tube. This frequency is matched to the expected temporal frequency of the output of the optical system. As the tube integrates in time during each frame period - the accumulated signal - the camera acts as an array of lock-in

Fig. 32. Phase switching incoherent OTF synthesis. From RHODES [1977a,b].

470


[V, 8 3

amplifiers. Unfortunately, this technique cannot be applied to most contemporary video tubes; image-orthicon based cameras are now considered obsolete and their production was discontinued several years ago.

3.3. SPATIAL ENCODING

An alternative to temporal encoding, with its implementation difficulties, is to use more of the spatial dimensions. Here we assume, as it is often the case, that the space-bandwidthproduct of our optical system is greater than that of the input and the expected output (bipolar or complex) spatial signals. There are two related approaches to space representation of complex-valued or bipolar-real spatial signals: (1) area segmentation, in which the area [inx = (x, y ) space coordinates] is segmented; parts are allocated to different non-negative real componentssignals of the signal ( g - and g , , or g l ) . (2) spatialfrequency modulation,which is analogous to the temporal modulation scheme discussed earlier. Here the signals are shifted in spatial frequency [ f =(j”,,&) coordinated], so that the mathematical phase of the complexvalued (or bipolar) signals is converted to the geometrical phase of the modulation frequency. We first recall how some operations affect the amount of spatial and spatial frequency terrains occupied by spatial signals. Let us assume two input signals, g(x) and h(x), each taking non-zero values in space domain only for 1 x I < r and in spatial frequency domain for If\ < p. If we now select a shift xs, the shifted signalg(x - x,) will be different from zero only inside a circle of a radius r about x,. Similarly, if we frequency-shift the signal by a spatial frequency s,, the shifted signal, g(x) exp (2njx ef,), would have non-zero values in Fourier space only inside a circle of a radius p centered at f,.We say that the signal has a space bounds of 2r (the diameter of the circle) and frequency band of radius

2 P. Now let us look at the results of different operations on the following shifted signals = g(x - x,)

9

(85a)

h(x - x,)

(85b)

g,(4 = g(4 exp(2.njx.L)

(85c)

h&)

(854

h,(x)

=

=

h(x)exp(Znjx.S,).

v, § 31

BIPOLAR A N D COMPLEX-VALUED SPATIAL SIGNALS

471

All four spatial signals have space bounds of 2r m d frequency band of 2p, as do g(x) and h(x); g, and h, are shifted in space by x,, whereas g, and h, are shifted in spatial frequency by f , . Now g, + h, = ( g + h), has the same bounds and band as g and h and is centered in space at x,. g, + h, = ( g + h), is also of the same bounds and band as g and h and is shifted in frequency to f , . g, x h, = (g x h), also has the same bounds and shifts as the individual spatial signals. The frequency band, however, is doubled. g, * h, = ( g * h), (by applying the Fourier transform to both sides and using the former relation). Bands and shifts are not changed, but the space bound is doubled. [ g(x) exp (2 njx *&I 1 x [ h(4exp (2 njx =

[gW x h W l

fh

exp”2njx*(fg + f h ) l .

11 (864

Thus the product of two frequency-shifted signals is frequency shifted by the sum of the shifts of the multiplicands

Here the convolution of two space-shifted signals yield an output signal which is shifted by the sum of the individual shifts.

3.3.1. Space segmentation If we have a complex-valued, or bipolar-real, spatial signal that is bounded in space, we can derive its non-negative real components (as shown previously in 0 3. l), then shift each component signal in space so they would not overlap. Since for convolution/correlation, the space bounds of the output are larger than those of each of the input spatial signals, sufficient “empty” space must be reserved. For the cases where only one of the two input signals is not non-negative real, this simple solution is adequate. However, as shown in 5 3.1 earlier, when both input signals are bipolar (or complex-valued),a single complex-valued or bipolar convolution turns into nine (six ifwe use the GOODMAN and WOODY[ 1977al shortcut) non-negative real convolutions for complex input (four for bipolarreal input). A way of doing all the necessary operations (as well as the additions, although not the subtractions) together is shown by GLASER[1981a]. In

412


[V,I 3

I

h

IBm= 1 , 1

f

h

9

It B

u

Fig. 33. Convolution of bipolar-real spatial signals by area segmentation. From GLASER [ 198 1a].

Fig. 33A we see how simple shifts of the g , and the g - components in one segmented input plane, and the h , and h - in the other, yield all four convolution components at once at the output plane. Now, as shown in Fig. 33B, arranging and replicating the components of two input bipolar signals will get us the components of the bipolar result directly in the output plane. A similar technique, shown in Fig. 34, gives us directly all the three components of the complex-valued output signal from those of the input signal in one physical convolution step.

Fig. 34. Convolution of complex-valued spatial signals by area segmentation. From GLASER [ 198 1 a].

BIPOLAR A N D COMPLEX-VALUED SPATIAL SIGNALS

473

Fig. 35. Multiplication of bipolar-real spatial signals by area segmentation

The situation for multiplications and additions is simpler in the sense that no extra area must be reserved for the results. For additions we simply have to shift each component in the same way in both input planes. For multiplication of bipolar signals we may use the arrangement shown in Fig. 35 and, similarly, for complex-valued signals. In both cases we have to do the additions and subtractions necessary to obtain the components of the output by other means after the multiplication. 3.3.2. Spatial carrier

LOHMANN[ 1977aJ suggested the use of spatial modulation as a method for encoding bipolar-real or complex-valued spatial signals for incoherent processing. The method is related to the concept of off-set reference holography of LEITHand UPATNIEKS [1962] and to the single-side-band holography concept of LOHMANN[ 19651; both earlier works, however, were concerned with the problem of recording and reconstructing the complex-amplitude of coherent wavefronts using irradiance-sensitive photographic emulsions. We take eq. (84)and modify it - instead of a temporal carrier at v we f,,,). introduce a spatial carrier that has a spatial (2-D) frequency f, = (f,,,, The space-modulated spatial signal g is

gW = Re {gW exp(24x.L)) + b ( 4 ,

(87)

Since g(x) must be non-negative real, the bias term b(x), which is non-negative real, &st be sufficiently large. There are (like the $bias of the multiple channel representation) two conflicting requirements : ( I ) b ( x ) should be as small as possible because excess signal introduces flare, noise, and error. (2) B(f)= F { b ( x ) } should take as little frequency space as possible, so that efficient use of the system space-bandwidth product is possible.

414


[V,8 3

LOHMANN[ 1977al discussed three types of bias (1) Constant bias, in which max [ J g ( x j)3 , if x is inside the bounds of g(x) , (0 otherwise (a smooth slope should connect the two regions).

ba(x) =

9

(88a)

In Fourier space this bias is ideal because it takes almost no area at all. If g(x) is bounded by r, the area B , ( f ) in frequency domain is about ( l / r ) 2 ,which is the same as the smallest resolvable spatial-frequency element. For most of the space, however, b,(x) is much larger than necessary. ( 2 ) Maximum contrast bias, in which bb(x)

=

IdX) I

(88b)

Clearly, this bias is precisely as high as necessary to make g(x) non-negative. However, the frequency domain spread of B b ( f )can be intolerably large. For example, although sin (k x) has a modest frequency band, that of I sin (k x) I is infinite. Maximum constant bias must be used with care. ( 3 ) Holographic bias, in which

u-4= 1 + Ig(x)12 .

(88c)

At first, it looks like hybrid between the constant bias of eq. (88a) and the maximum constant bias of eq. (88b). However, we note that 1 g(x)1' = g(x) f(x). Its Fourier bandwidth is thus always exactly twice that of the original signal g(x) and can never become infinite. The three Lohmann bias types are clearly not the only possible choices. Indeed, one may try to adapt an optimum bias that depends on the particular g(x) and on the system spatial bandwidth in its form. For example, we can extend the iterative method discussed earlier in 0 3.1 as follows:

B { b ( x ) = 9' ( - i b ( x ) },]circ )J'( Pmax

and each iteration cycle is given by

v, I 31


415

where pmaxis the frequency band allowance for b(x). Having obtained g(x) and, possibly, h(x) from the original g(x) and h(x), we now look at how to use them. Convolution (and correlation) are most conveniently done with spatial carriers. In Fourier space

G(f)= q g < x > > =

p{$[g(x)exp(2njnjf,*x)+ g * ( x ) e x p ( - 2 n j ~ . x ) l

=

G ( f - f c ) + G * ( L -f)+ B ( f ) .

+ b(x)) (89)

Applying the same notation to h, we get

q g * h ) = G(f)H(x) =

$ [ G ( f - f,)H(f - f,) + G * ( f

-f)H * ( f , - f)l

+ [ B , ( f ) B h ( f ) ]+ [other terns] .

(9 0 4

The first term in square brackets contains the signals of interest, and the second term is the Fourier transform of the convolution of the two bias terms. The third, “other terms”, contain cross products. Since G and H are shifted by f,, G*( - ) and H*( - ) are shifted by -fc, and the B terms are unshifted, all cross products are zero provided 1 f,1 is large enough. Thus if both g and h have no frequency terms higher than p and the bias terms have no frequency terms higher than p,,,, the condition Ifcl

( P + Pmax)

(90b)

would ensure that all cross terms would be zero. Since each of the preceding terms occupies its own region in frequency space, it is easy to separate (demodulate) them. For example, we can detect the output optical spatial signal with a vidicon (or a 2-D CCD) camera, where the scan direction is set parallel to f,.This would convert the optical spatial modulation into an electronic temporal modulation ; an abundance of electronic demodulation techniques exist (see, for example, KATZIR, YOUNG and GLASER [ 1984,19851). Multiplication of carrier-encoded spatial signals is more complex, since spatial frequencies (includingf,)are not invariant under multiplication. If both g(x) and h(x) are encoded using the same spatial frequency fc, F{g(x) h(x)} = G ( f ) * H(f)would come out shifted by 2fc. G(f)* H(f) has and as other terms will twice as much frequency bandwidth as G ( f ) or H(f), appear modulated at f,;we must have ,fI 1 3 4p (for p,, = p). In that case the total bandwidth requirement from the optical system is lop. On the other hand,

416


[V,I 3

there is no need to use the same carrier frequency for both input signals. GLASER [ 1985al analyzes the case when the two spatial carriers have the same absolute value but a different direction. If = Ifc,hI = 4p and the angle between the two frequencies is 2n/3(120"),we get a total system requirement

a

\

b

-

Fig. 36. Multiplication of carrier-encoded signals Fourier space diagram. Here W is p in the (a) both input spatial signals are modulated by the same spatial carrier; text and p = p,,. (b) spatial carriers are at an angle of cos- '(0.4) = 66"25'18". Note the increased utilization of the optical spatial frequency bandwidth for b. From GLASER[1985a].

of 6p. However, the demodulation process is difficult because one cannot define a square in frequency space where only terms of interest are non-zero; the demodulation process is not separableinf, andf,. A better choice (Fig. 36) is If,/= 5pandanangleof2cos~'(2/5)(132"50'37").Thesametotalsystem frequency bandwidth of 6pis sufficient and, because demodulation is separable, it can be carried out sequentially- fmt in one dimension and then in the other.

v, s 41

APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS

411

3.4. POLYCHROMATIC ENCODING

Several types of incoherent optical processors are achromatic. Such processors, like the shadow casting correlator, the lenslet array processor, the geometrical optics OTF synthesis system, and others, can process several data streams simultaneously by using a different wavelength for each. Furthermore, some of these can actually offer controlled different behavior in each wavelength band. For example, s t h e mask for the lenslet array processor or the shadow casting system is made with conventional subtractive color photography technology, it has three independent dye layers transparent to cyan, magenta, and yellow light. They would independently modulate red, green, and blue light, respectively. Thus we can introduce a single white light input and obtain, in parallel (using, for example, a color video camera for image pickup) three output spatial signals. We can also introduce three different inputs in parallel (using dichroic beam combiner mirrors or an RGB color CRT) and have a single optical system replace three. The availability of three channels is exactly what is needed for the three non-negative components of a complex-valued spatial signal. WIERSMA[ 19791 and GORLITZ and LANZL[ 19791 describe dsractive OTF synthesis systems where the pupil mask is made of three apertures, each covered with a color filter. Although dsractive OTF synthesis is wavelength dependent, each aperture sees only one wavelength band and can be scaled to that wavelength. KELLY[ 19611 described a similar system with a shadow casting processor.

8 4.

Applications of Incoherent Processing Systems

As we saw from the previous sections, information processing with spatially incoherent light offers a combination of simplicity and performance that is attractive in many fields. Some fields where high speed information processing is desirable include the following:

Industry

- automated inspection - “machine vision” for flexible automation (robotics) - massive processing for industrial research and development

478

[V,8 4


- nuclear medicine - imaging with y-rays - transaxial tomography - automated interpretation or screening of medical imagery,

Medicine

including X-ray, tomography, y-ray, and microbiological specimens - radiation treatment planning (on-site tomography) - picture archiving and communication (PACS) Communication - cost-effective video links - “picture-phone’’ - multiplexing and demultiplexing for lightwave communication - efficient facsimile for continuous-tone imagery

- image transmission and analysis for C31

Military

- navigation and terminal guidance systems

- radar/sonar analysis - control of phased arrays In table 2 we see how different processing functions relate to these application fields. The remainder of this section will review several of these processing functions and the work done on their implementation. TABLE2 “Matrix” for some processing functions and potential fields of applications. Processing functions

Application fields Industrial

~~~~

~

Pattern recognition Image compression Coded aperture imaging Tomographic reconstruction Radar/ultrasound imaging Digital optical processing Image restoration a

+ : Direct

Medical

~

+ a

?b

+ + -

+ + + +

?

?

?

-

significance for this field.

?: Likely to be useful in this field in the future.

- : No known potential use in this field.

?

Communication

Military

v, I 41


479

4.1. IMAGE PATTERN RECOGNITION

One definition of image pattern recognition systems is “systems that tell us if a certain object, or an object from a given list, was found in a scene.” Usually we would like this system to tell us more; for example, the location of the object in the scene. Image pattern recognition applications include 2-D problems, where 2-D objects or 3-D objects viewed from a fixed orientation are to be recognized, and 3-D object recognition where the relative orientation of the object is not known beforehand. There is little work on optical 3-D recognition; discussion here will be limited to the 2-D problem. Two major approaches to pattern recognition theory are syntactic pattern recognition, dealing with the topological relations of substructures in the image, and statistical recognition, which sees an input scene as a whole and applies statistical, or probabilistic, mathematical tools to assert its similarity to the object of interest. Syntactic algorithms are heavily oriented to “if-then-else” computing and to nonlinear operations. Little work was done on their optical implementation. Statistical methods usually contain computational-intensive linear transforms. These are ideally suited to optical implementation. A new approach is the use of neural models. These algorithms can provide a contents addressable (as opposed to location addressable) storage mechanism. When presented with an imperfect image of a stored object, contents addressable storage retrieves its perfect image. At present most of the interest in this approach stems from academic research on the algorithmic structure of biological memory systems. However, successful neural-model systems may become useful in their own right. Some early works on implementing neuralmodel associative memories in optics have been reported. 4.1.1. Statistical pattern recognition

Statistical pattern recognition, also called decision theoretical pattern recognition, is based on the notion that both the input scene, and any available image of the searched object, are samples, or instances, of some random processes. In other words, when we look at one possible input scene, which is defmed as the 2-D irradiance (for example) distribution, gin,the value of the irradiance at any given ( x , y ) is unknown a priori. Its probability density function for each (x, y), and the joint probability functions for gin(x,y ) and gi,(x’, y’), however, are assumed to be (at least approximately) available. Similar knowledge is available about the random process that represents all possible images of the object to be recognized.

480


tv, I4

Statisticalpattern recognition is concerned with trying to evaluate how likely is one given input scene image to contain an image of the required object. TO make a decision about this likelihood, we usually have to define some cost functions - how much damage would be incurred if we wrongly “found” the object, when it was not there (the false alarm situation), and what is the cost ofnot finding the object when it was, actually,in the scene. These cost functions vary greatly from one practical application to another. We would rather have a high false alarm rate for a cancer detection system but would prefer not to have the wrong parts picked by a robot on an assembly line. Because of this wide spectrum of problems, several different recognition criteria have evolved. We can adequately define the statistics of our problem in terms of the statistical correlation matrices. If, for simplicity, we describe the input scene as a finite discrete l-D sequence gi(with i = 1, . ..,N ) , the statistical correlation matrix J C ] = (&) is

where E { - * > is the expectation operator, and giis assumed to be real. It is important to differentiate between statistical correlation, defined here, and geometrical correlation, as used throughout this chapter (for some special random processes the statistical correlation can be calculated with the aid of the geometrical correlation; this is hardly the general case). In this chapter unqualified “correlation” refers to geometrical correlations. The general procedure of statistical pattern recognition involves two steps, as follows: (1) A linear transformation which should bring the statistical correlation matrix of the object image random process into a simpler (diagonal or nearly diagonal) form. (2) A decision step where the values of selected elements (called features) of the transformed scene are evaluatedusing some nonlinear (for example, threshold) criteria. For a general review of statistical pattern recognition theory the reader is referred, for example, to DEVIJVER [ 19821. If the problem is known to be shift invariant, it makes sense to use a shift-invariant transform above. Shift-invariant linear transformations can always be expressed as convolutions or geometrical correlations. This case is ofparticular interest to us because there are many optical methods for obtaining geometrical correlations and convolutions. The more general, nonshift invariant case is also of interest, and other optical methods can be used there.

v, I 41


48 1

4.1.2. Geometrical correlation methods

A simplistic way to look at spatial correlation is template matching. The correlation formula g Q h = j j g(t, q) h(5 + x, q + y ) d t d r] (all functions here are real) can be considered as putting h over g and looking how closely they fit, then shifting h and trying again. From this naive description one may think that h should be an image of the object we are looking for. For some data this is often the case, and early work on (coherent) optical pattern recognition used correlation with a direct image of the searched object (the best known example is VANDERLUGT [ 19641). Later work (for example VANDERLUGT[ 19701) showed that often this is not the best solution. ARCESE,MENGERTand TROMBINI [ 19701 came with probabilistic analysis of correlation-based image pattern recognition. For simplicity we shall deal here mostly with the one-dimensional discrete and finite case where we have an input scene g,, an object “image” oi(with i = - v, . , .,0,. .., + v) where both are random processes; we wish to find a reference, or model, function hi where the correlation g Q h has the best chance of separating scenes containing an image of our object from scenes which do not. Furthermore, we want the correlation with h to give us the precise locution of the object in the scene. Thus if the object is not present in g,, we want lg 0 h I to be as small as possible. When the scene contains o,, we want one term of the correlation to have a relatively large value, indicating the location of the object. Mathematically, the desired h should satisfy

... ,

E{oQ

... where

gv-

1

and similarly for h and

0.

E{gQh}=O, when g does not contain the searched object,

482


[V,§ 4

From this, Arcese and colleagues showed that the equation for h is

JC]h = E{o}.

(92b)

The desired h thus is

h

=

J C ]- ‘ E { o } .

(92c)

The relation between o and h thus depends on the form of , [ C ] .In the general case one has to obtain ,[ C ] from statistical data. A useful, and often adequate, approximation is the exponential process

if Ci,k

= {ili-kl

i=k,

, otherwise,

(93a)

where p G 1 is a constant. We get

hi =

-@{oi-

I}

+ (1 + p 2 ) E { o i } - pE{o,+ I } .

(93b)

For an uncorrelated scene ( p = 0) this gives hi = E{oi}, whereas for a highly correlated ( p z 1) case we get hi = -E{oi+ 2E{oi} - E{oi+,}, which is the discrete version of the second spatial derivative of o. ARCESE,MENGERTand TROMBINI [ 19701 also analyzed the continuous 2-D case. Here the statistical correlation “matrix“ is a four-variable continuous function, C ( x , y ; x ’ , y’). However, when the problem is shift invariant, we can rewrite it in the form C ( x - x ’ , y - y’). The equation for h(x, y ) is

11

C(X - 5,Y - rl)h(t, v ) d t d r l = E { o ( x , y ) } .

(94)

If our spatial signal is highly uncorrelated, C(x,y) = b ( x , y ) and we get h ( x , y ) = E { o ( x ,y ) } ; for a highly correlated exponential process the optimum h(x, y ) is approximately h(x, y ) = V 2 E { o ( x y)}, , where V 2 = @/ax2 + a2/dy2is the Laplacian operator. Ultimately, for more general cases we must also account for the statistics of “false” objects and for statistical correlation matrices that are not exponential. This may be done by analyzing statistically many input scene samples and object images. 4.1.2.1. Optical correlators for pattern recognition

One major problem in incoherent cptical implementation of correlators that correlate the incoming scene with optimized model functions such as those of the form V 2 E { o ( x , y ) }is that these h ( x , y ) models always come out to be

v, fi 41


483

bipolar. We have two possible ways of dealing with this problem: (1) Using one of the schemes for indirect representation of bipolar spatial signals. Since only the h ( x , y ) function is bipolar [ g ( x , y ) is non-negative because it is an incoherent image], this is relatively easy. (2) Changing both the input sceneg(x, y ) and the object image o(x, y ) so each becomes statistically uncorrelated and h(x, y ) = E { o ( x ,y ) } can be used. Both

Fig. 37. Carrier-modulated spatial signals for pattern recognition correlator: (a) the carriermodulated model spatial signal h_(x,y);(b) a carrier-modulated input scene g ( x , y ) (not to the same scale). From KATZIR, YOUNG and GLASER[1985].

484 484

INFORMATION PROCESSING PROCESSING WITH WITH SPATIALLY SPATIALLYINCOHERENT INCOHERENT LIGHT LIGHT INFORMATION

P, P, II44

spatial signals signals must must be be changed changed because because only only with with aa nonlinear nonlinear operation operation the the spatial will become both non-negative real and nearly statistimodified spatial signals modified spatial signals will become both non-negative real and nearly statistically uncorrelated. uncorrelated. cally GAMBLE and VERBER VERBER Sherman and and co-workers co-workers (SHERMAN, (SHERMAN, GRIESER,GAMBLE Sherman GRIESER, and 19831, SHERMAN, SHERMAN, GRIESER, GAMBLE, VERBER and and DOLASH DOLASH[[19831, 19831, and and [[19831, GRIESER, GAMBLE, VERBER VERBER [ 19841) explored the second possibility. SHERMAN, GAMBLE and SHERMAN, GAMBLEand VERBER[ 19841) explored the second possibility. They used used the the same sametransformation transformation on on the the input input scene sceneand and the the model. model.Their Their They transformation can can be be written written approximately approximately as as transformation

Clearly, this is a shift-invariant but nonlinear transformation. The Sherman system was a holographic incoherent OTF synthesis system where the hologram was prepared from a processed image of the object (using an approximation of eq. (95)), and the input signal was presented on a CRT monitor (with a narrow spectral bandwidth phosphor) using electronic processing of the video signal from a vidicon camera obtaining L{ g(x, y)}. Note that ha3x3 %

00

0.25 --0.25

00

0.25 --0.25

11

0.25 --0.25

00

0.25 --0.25

00 ame time

me of the :omplexity into electronics. However, it is possible to use indirect representation of bipolar signals so that the optical system would not require on-line [ 1984,19851 and electronic pre-processing. KATZIR,YOUNGand GLASER FURMAN and CASASENT [1979] used spatial carrier encoding for the preprocessed image of the object, from which they produced a hologram for their incoherent holographic OTF synthesis system. The input scene had to be spatial-carrier encoded too, but since the input scene is non-negative, carrier encoding was done by optically multiplying with a grating of the proper spatial frequency. This could be achieved by imaging the input scene on a Ronchi ruling in the input plane of the OTF synthesis system. Figure 37 shows the carrier-encoded model and one input scene, whereas Fig. 38 contains an output


485

Fig. 38. Output spatial signal from a carrier-encoded pattern recognition correlator. This output is for the input signals of Fig. 37. (a) The raw output spatial signal; (b) the same, after AM demodulation of the video signal, as seen on a video monitor; (c)a video frame of the demodulated output spatial signal, recorded from an oscilloscope screen; (d) one video line from the same image, also recorded from an oscilloscope screen. From KATZIR, YOUNGand GLASER [1985].

example. To demodulate the output, Katzir and co-workers used a simple analog electronic AM demodulator on the video signal from a vidicon; for pattern recognition only the absolute value of the output is of interest. OTF synthesis is hardly the only way to do pattern recognition via correla[ 19841 tion. Some alternativesinclude the scanning correlator of INDEBET~UW and the shadow casting bipolar correlator described by ARCESE,MENGERT [ 19701. Indeed, much of the research on correlators of all types and TROMBINI is motivated by pattern recognition applications. 4.1.2.2. Rotation invariant correlation

Another problem with the use of correlation for object recognition is the fact that correlation is generally not rotation invariant. There are several attempts

486


[V, 8 4

to overcome this problem, some of which are correlation based and some use other linear transformations. GLASER and KATZIR[ 19821noted that, for the industrial environment, one can often put registration marks of one’s choice on each workpiece. These marks or fiducials can be optimized for detection by optical correlation by making them circularly symmetrical (hence rotation invariant) and with a sharp, well-defined correlation peak. It also has to be easy to print, so it cannot contain a grey scale -only black and white. As an example, Glaser and Katzir used a Fresnel zone pattern for the fiducial and a bipolar Fresnel pattern (one with values of 1 and - 1 instead of 1 and 0) for the correlation model. An example of their results, using two channel holographic incoherent OTF synthesis, is shown in Fig. 39; where Fig. 39A shows a typical input scene and

Fig. 39. Recognition of a fiducial mark with bipolar holographic incoherent OTF synthesis: (a) typical input scene; (b) output. GLASER and KATZIR [1982].


487

Fig. 39B is the output. By encoding workpieces with three fiducial marks each, where the centers of the fiducials are arranged in a non-isosceles triangle, the location, orientation, and even the identity of the workpiece can be easily derived by computing (digitally) the distances between the three observed correlation peaks. For applications in fields other than manufacturing, and even for several problems in the industry, putting fiducials on objects is not possible. For these we must be able to deal with an image of the unmodified object. ARSENAULT, Hsu and CHALASINSKA-MACUKOW [ 19481 analyzed the use of geometrical correlation model functions h(x, y ) that can detect rotated objects. They analyzed two types of rotation invariant model functions, or “filters”: (1) Optimum circular jilters (OCF), in which the output of the correlation system does not vary when the object is rotated 0

8h,cF

= Oo@

(964

hoCF

Here o is an image of the object and 0, is o rotated by an angle 8. For this case

(2) Circular harmonicjilter (CHF), in which there is no attempt to keep the output of the correlator rotation invariant, but only to keep the absolute value of the peak of the correlation invariant. It may change phase, and the rest of the correlation pattern may rotate with the object

I LO Q ~ C H IF[O,0) I

=

I

1

~ C H F(030)

I.

(97)

Arsenault and co-workers show that the C H F filter must be a term in some linear orthogonal expansion of o(x, y). They show that, using polar coordinates r = (x’ + y2)1/2and d = tan-’ (y/x), hCHF(r,8) = u(r) eJ@(r) eJMe= h,(r)

eJMs,

(9 8 4

where u(r), $(r) and A4 are real, and M is an integer multiple of 271, and where h,(r) depends on A4 so it satisfies the orthogonal circular harmonic expansion W

C

o(r, d ) =

M=-aa

where

271

h,(r)ejMe,

488


By substituting A4 = 0 above, we see that h,,,

[V, I 4

of eq. (96b) is a special case of

hCHF *

Naturally, since either hOCFor the more general h,,, are “less matched” to any particular orientation than the optimal nonrotation-invariant model function, they should give lower S/N value. Arsenault and co-workers report a factor or about 10 between the h,,, for A4 = 1 and the nonrotation-invariant model functions with their experimental objects. This would make the approach useful for some objects (where the initial S/N is high) but not for all objects. A useful feature of the Arsenault h,,, lilter is that the phase of the output peak indicates the rotational orientation of the object. Possibly, CHF detection can be followed by physically rotated, nonrotation-invariant correlation for verification. A significant improvement in signal/noise performance is possible if one uses linear combinations of several harmonic terms (A4 values in eq. 9%) with proper selection of complex-valued weighting factors. This was demonstrated by SCHILSand SWEENEY[1986]. 4.1.3. Noncorrelation methods

We noted before that the use of correlation for pattern recognition offers automatic shift invariance. We also saw that obtaining rotation invariance with correlation carries a substantial S/N penalty. Alternative methods of achieving rotation invariant correlators included the introduction of rotation to the input scene using an image rotating prism or electronic image rotation (on a CRT). These approaches require added complexity and semi-sequential,hence slower operation. For several real-life applications the situation is even more complicated; some require not only shift and rotation invariance but also scale (magnification) invariance and perspective invariance. One attempt to cope with scale and rotation was to use a geometrical transformation of the form h(x, y ) + h’(r’, 0) and g(x, y ) 4g’(r’, 0), where B = tan - ( y / x ) and Y‘ = log[(x2 + y 2 ) 1 ’ 2 ]Rotations . and scale changes in the ( x , y ) coordinates become shifts in the ( r ’ , 0) ones, so the shift invariance of correlation is transformed into scale + magnification invariance. Unfortunately, shifts in the ( x , y ) system do not transform into any simple motion in the (r’ , 0) system, so ( x , y ) shift invariance is lost. We thus see that there is a significant class of image pattern recognition problems where correlation-based techniques are inadequate. One possibility is to use one of the statistical pattern recognition algorithms and carry out the transformation part on an optical matrix-vector multiplier

v, c 41

APPLlCArIONS OF INCOHERENT PROCESSING SYSTEMS

489

discussed in Q 4.1.1. However, even without complete statistical analysis it is easy to come up with simple, workable solutions. For example, one typical pattern recognition application is conveyer belt workpiece identification. Here we have some assembly or inspection system to which workpieces are moving on a conveyer belt. A “vision” module observes the moving belt and tells a robot arm whenever a workpiece of a specific type is on the belt; it also gives the location and orientation of that workpiece. Since the belt is in motion, only one-dimensional shift invariance is required. Because the height of the vision module above the belt is fixed, no scale invariance is necessary. What is needed, thus, is rotation, l-D shift invariance, or x-8 invariance. We can define a modified x-0 correlation: cx-e(x,O)=

S_I,.

g(x-t,q)he(t,q)dtdq,

(99)

-02

where h, is h rotated by an angle 8. KATZIR,GLASER and MENAKER[ 19851 and GLASER [ 1985bl report on the use of a lenslet array processor (LAP), using the base function set of Fig. 40, to demonstrate x-0 correlation. Although the number of discrete 8 and x

Fig. 40. Base function set for x-0 correlation on the lenslet array processor. From GLASER [ 1985b].

490


[V, § 4

positions they used (9 x 9) would be inadequate for most practical objects, and although they used a nonoptimized h(x, y), this demonstration does suggest a potential solution to the problem. Note, however, that the space-bandwidth product of the LAP (less than about 100’) and other matrix-vector multipliers is substantially lower than that of optical correlators, particularly holographic OTF synthesis. 4.1.4. Associative memories via neural networks GABOR[ 19641 noted that, like biological memory, holography can recall a nearly perfect image of an object from some partial input and stored information. He named this capability associative memory. Later, HOPFIELD[ 19821 noted a possible similarity between the mathematical behavior of associative memories and some “emergent collective properties” in solid state physics. He noted that the structure of a network of a large number of switching elements (neurons) has more influenceon the system behavior than the internal operation of the switching elements themselves. Using these ideas, Hopfield described his neural network model for associative memory. Let us assume a system for which its status can be described by a state vector x, which has N elements, where each can have only one of two possible states (1 or 0). As the system evolves with time, x may converge into any of several stable limitpoints x(’), id’), . .. , dS); if we start with x(’) + A, where I/ A 11 4 1, it would converge to d’). A specific example can be described by

Hopfield suggests, for obtaining a desired set of stored patterns (stable limit points), d’), . . ., dS), to use of the form

2 ( 2 ~ s “-’ 1)(2~‘,“’- 1 ,

if

1# k ,

( 1OOb)

otherwise.

Other neutral network models use multiple stages similar to eq. (100a) and/or variations on the derivation of of eq. (100b). FARHATand PSALTIS[1984] and FARHAT,PSALTIS,PRATAand PAEK [ 19851 investigated the use of incoherent optical processing for the implemen-

v, B 41


49 1

tation of neural network models. They used the GOODMAN, DIAS and WOODY [ 19781 vector-matrix multiplier (discussed in 5 2.7 and Fig. 31) to do T l , k ~ l , iand n , a network of the linear transformation part of eq. (lOOa), photodiodes with LEDs to carry out the threshold operation. Note that Tl,kcan be redefined so all U, will be equal. The Hopfield model and its optical implementation by Psaltis, Farhat, and their co-workers receives a partial, noisy binary (1’s and 0’s only) pattern and retrieves the complete pattern from “memory”. In this sense it can be considered pattern recognition. It now serves mostly as a rather powerful tool for research on the algorithmic aspects of biological memory. It seems too early to predict the influence of this approach on practical applications of pattern recognition, although its mathematical elegance and the possibility of simple optical implementations are likely to attract more work and results.

4.2. CODED APERTURE IMAGING

Coded aperture imaging is a technique that was developed for medical diagnosis, specifically the imagmg of the spatial distribution of y-ray emitting isotopes inside a patient’s body. Similar techniques were adopted also for industrial applications, such as nuclear energy plant inspection, and also as a general research tool. A major conflict in the use of radioactive materials in medical diagnosis is that nuclear radiation is harmful. If some source of this radiation needs to be introduced into the human body, as little of it as possible must be used. On the other hand, medical diagnosis depends on the quality of available imagery. Images obtained with very little radiation are ridden with spatial noise, particularly (for short-wave radiation such as y rays) photon noise. There are two conflicting needs - having as little radiation exposure to the patient as possible, and getting more radiation for image detection. With radiation of longer wavelength, such as visible and IR light, “faster” optics, that is, lenses with larger aperture, can be used. Unfortunately, the only practical kind of optics that work with yrays is shadow casting, and the only imaging devices available, the pinhole and the y-ray collimator, are notoriously inefficient. The resolution of a pinhole camera (with yrays where the wavelength is much smaller than the pinhole diameters) is inversely proportional to the diameter of the pinhole, and its radiometric efficiency is proportional to the area of the pinhole. An alternative to the pinhole camera, the pray collimator, is a block of thick lead with many small long holes. For a given resolution its radiometric efficiency is comparable with that of the pinhole camera.

492


[V, § 4

So far we have assumed that the pray camera must directly produce a usable image. A complete departure from this line of thought is coded aperture imaging. Here the camera gives a transformed image, which is later back-transformed to obtain a usable picture. Such systems permit better radiometric efficiency (more photons) and reasonable resolution in the final, post-processed, image. We recall that the small number of available photons is the major source of spatial noise in pray imagery. A side benefit of some of the coded aperture systems is their ability to get some limited three-dimensional imagery. There are many variations on coded aperture imaging; clearly we can cover only few of these here. The reader may refer, for example, to BARRETIand SWINDELL [ 19811 for a more complete review of this field. Since no refractive optics is available for pray imagery, most coded aperture imaging systems are based on the simple shadow casting correlator of Fig. 14A. We convolve (or correlate) the irradiance distribution of the pray emitting object with the transmittance function of the lead mask. As shown in Q 2.2, the scale of the convolution (or correlation) depends on the distance between the object (input) to the mask (siin Fig. 14A) and the distance between the mark to the output (detection) plane (so). From eq. (33) we recall that the object is magnified by a factor of ( - s,/s,) and the mask (convolution PSF) by (so + si)/si.The major problem now is to find a PSF that can be easily de-convolved. One class of such PSFs is random arrays of pinholes. If we choose the locations of the pinholes in the array so that we have about 50% open area, good autocorrelation peak, and very low autocorrelation side lobes, we have a unifomly redundant array. To recover the object from the coded image, we correlate (usually on a computer) the coded image with a similar pattern, except that now, instead of having z = 1 for the holes and z = 0 elsewhere, we use values of 1 and - 1. Unfortunately, uniformly redundant arrays are scale sensitive. Since medical objects (human beings) are three dimensional, siis variable. With uniformly redundant arrays, scale variation causes the appearance of artifacts in the reconstructed image. BARRETT[ 19721 suggested the use of the Fresnel zone pattern for the mask. The Fresnel pattern satisfies all the previous requirements and offers some clear advantages over uniformly redundant arrays - the scale sensitivity is eased and a simple method for optical reconstruction exists. One way to look at a Fresnel zone pattern coded imagery is to consider it as approximate incoherent simulation of the GABOR[ 19481 on-axis hologram. Each point of the object is represented by a Fresnel zone pattern, much like the convolution-withquadratic-phase used in propagating coherent wavefronts. Furthermore, the scale of the Fresnel zone pattern PSF in the coded image varies with si as it

v, 3 41


493

should in holography. Indeed, BARRETT’S[ 19721 Fresnel coded images were reconstructed like holograms. A small copy was prepared and was illuminated by laser light. A diffusing screen (ground glass) was used to view one plane of the reconstructed object at a time. As the screen was moved, a different plane would come into focus. Unfortunately, this version of Fresnel coded imaging shared another property with the Gabor hologram - when reconstructed, several diffraction orders would mix together, introducing artifacts in the reconstructed image. To overcome these limitations of Fresnel coated imagery, BARRETT,WILSON and DEMEESTER [ 19721 borrowed another concept from holography, namely the spatial carrier concept (off-axis holography) of LEITH and UPATNIEKS [ 19621.They used a linear grating adjacent to the object and an off-axis Fresnel zone pattern for the mask. The transmittance of their off-axis Fresnel zone pattern can be written as

i + f sign[Re(exp{2njc[(x - x,,)’ + y’]})] = i + i sign (Re { exp (2 njcxg) exp [2 nj(4cx0)x]

z(x,y) =

x exp(2nJc(x2 + y2)1}),

where sign(a)

= -

1 if a > O , 1 otherwise.

This expression has the form of a quadratic phase (with some constant phase) carrier encoded on a spatial frequency fcarrier = ~ c x , , c; and x,, are constants. If the spatial frequency of the grating that was placed next to the object is approximately equal to fcarrier, we have a carrier-encoded convolution with quadratic phase - an incoherent emulation of off-axis holography. Later, with developments in digital electronics and their adoption into medical diagnosis, there was increased interest in apertures that can be easily de-convolved with a computer and give some three-dimensional information. One example is the rotating slit aperture described by KUJOORY,MILLER, BARRE=, GRINDIand TAMURA[1980]. Here the aperture is a simple one-dimensional slit. It gives a single dimension projection of the 3-D or 2-D object. As we rotate the slit and acquire many such projections, we get a set of all one-dimensional projections-Radon transform of the object. Since inverse Radon transforms are widely in use (for tomography) in medicine, decoding is simple.

494


[V,s 4

4.3. OPTICAL TOMOGRAPHIC RECONSTRUCTION

The introduction of computer aided tomography (CAT) was a great step forward in medical diagnosis. Unfortunately, many medical centers, particularly in developing countries, found that for them modern medicine is not affordable. Several projects were initiated with the objective of using simple optical technology instead of expensive digital electronics and provide “cheap” tomography machines. Some of this work is described in 0 2.1.

4.4. INCOHERENT FOURIER TRANSFORMS

A widely cited advantage of coherent optical processing is the ease with which it permits Fourier transformation of complicated imagery (although for real-time applications this ease depends on the availability of high quality 2-D spatial light modulators). Although no direct method for Fourier transforming with spatially incoherent light exists, the basic tools with which one can build a 2-D Fourier transform system are available. 4.4.1. Computation with a vector-matrix multiplier The discrete Fourier transform (DFT), the Fourier transform of finite discrete (sampled and space bound) data into finite discrete frequency space, is well known. One common form is

c

K - 1

G,

=

2n gkexp( - j K m k ) ,

m

= 0,

..., K -

(102a)

k=O

for the one-dimensional discrete Fourier transform, and K-1 L - I

Dm,n

C 1C= 0 dk,lexp[ - j ( $ m k + ? n l L) ] ,

= k=O

m = O,..., K - 1 ,

n = O ,..., L - 1

(102b)

for the two-dimensional one. These two relations are special cases of the discrete linear transformation. There are several incoherent methods for obtaining arbitrary linear transformations, or vector-matrix multiplication, using spatially incoherent light, as described in 5 2. Not surprisingly, many of these methods for vector-matrix multiplication were tested using the Fourier transform.

v, I 41


495

4.4.2. The Chirp-Z algorithm Let us have a second look at how coherent light gives us a Fourier transformation. In coherent illumination lenses multiply the complex amplitude of the incoming wavefront by a quadratic phase of the form exp [2njc(x2 + y 2 ) ](c is a real constant that depends on the focal length of the lens and the wavelength; it may be positive or negative). The propagation of the complex amplitude can be given by the Fresnel formula, which (as shown, for example, in GOODMAN [1968]), can be written as convolution with another quadratic phase exp[2njc’(xZ+ y’)] (c’ is another real constant, depending on the wavelength and the distance along the z axis of propagation; it is positive). Figure 41 shows two coherent optical Fourier transform configurations with their block-diagram equivalents. Since multiplication with incoherent light is simple and several incoherent methods for incoherent convolution exist, we have the necessary building blocks. Incoherent convolutions are more complicated than multiplications; the “multiply-convolve-multiply”variant (Fig. 41b) is preferable to the “convolve-multiply-convolve” one of Fig. 4 1a. Furthermore, for most applications we want only the modulus, or the absolilte value, of the Fourier

X

a

?k

X

b

Fig. 41. Chirp-2 equivalents of some coherent optical Fourier system: (a), (b) two Chirp-Z KATZIR algorithms; (c) and (d) their coherent optics realizations, respectively. From GLASER, and TOSCHI[1984].

496


[V,8 4

transformed image. The second multiplication, which atrects only its phase, is not necessary for these applications. We are still left with one problem - quadratic phases are complex valued. This must be taken care of by using one of the schemes described in $3. Lastly, for the Chirp-2 algorithm we should have c = - c ’ in the quadratic phase expressions given earlier. Thus the “mu1tip1y-convo1ve-mu1tip1y” Chirp-2 transform can be written as

G‘(X’,Y’)= { [ g ( x , Y ) x h(x,y)l * h * ( x , y ) ) x W , Y ) = M X l Y ) x h(x9 Y)l@ h(x9 Y ) ) x h ( x , v ) ?

(103a)

where

h(x, y)

=

exp [ - 2njc(x2 + y’)] , c > 0 .

As shown, for example, by RABINER, SCHAFERand RADER [ 19691 and by BLUESTEIN and LEO [1970], G ’ ( x ’ , y ’ )is related to the Fourier transform G ( L ,4)= F ; ( g ( x Y)> , by (103b) There are many combinations of complex-value representations and incoherent convolution methods ; several were tested. For example, STEPHENS and ROGERS[ 19441used simple shadow casting with biased real and imaginary [ 19721 used a collimated shadow casting convolver channels; RICHARDSON with bias (he was interested only in the real part of the Fourier transform); WILSON,DEMEESTER,BARRETTand BARSACK[1973] used y-ray shadow casting with spatial carrier encoding; and GLASER,KATZIRand TOSCHI [ 19841 used holographic OTF synthesis with spatial carrier encoding. 4.4.3. Fourier transform via the Radon transform The central slice theorem (eq. 22 in 2.1.3) says that the one-dimensional Fourier transform of any one-dimensional projection of a 2-D spatial signal equals the value of the two-dimensional Fourier transform along a line that goes through the origin of the Fourier plane and is perpendicular to the direction of the projection. Since simple analog electronic methods (based, for example, on surface acoustic wave devices) of computing the I-D Fourier transform are available, TICKNOR,EASTONand BARRETI [1985] suggested and demonstrated the following system for 2-D Fourier:


497

(1) A set of 1-D projections of the 2-D image are produced with a scanning system. (2) Each projection undergoes a 1-D Fourier transform using an acoustoelectric device-based analog electronic processor. (3) The resulting Fourier data are placed in their correct Fourier space location using a storage CRT display or some other scan converter. Despite the use of scanning and electronic processing, the method is fast. It can provide the complex Fourier transform with fairly high space-bandwidthproduct. 4.4.4. Interferometric Fourier transform

In Q 2.6 we discussed the use of spatially incoherent shear interferometers for two-dimensional Fourier transforms. The interferometer can produce either a biased red part or a biased imaginary part of the transform. It operates in quasi-monochromatic, spatially incoherent, light and is fast (parallel) and provides reasonable space-bandwidth-product. It is easy to change the scale of the Fourier output. Like all interferometers it requires careful alignment and is quite sensitive to vibrations.

4.5. DIGITAL OPTICAL PROCESSING

Electronic information processing has become almost synonymous with digital processing. Although analog electronic processing techniques have not been completely replaced by digital approaches yet, the trend in that direction is clear. There are several advantages to digital processing, as follows: Precision. In an analog system the precision is usually dictated by the quality and the noise characteristics of its worst component or subsystem. With digital systems, once the components are sufficiently precise to qualify (and tolerances of & 30% are typical), any arbitrary precision is attainable by either adding enough such components or by accepting slower system response. Programmability. Many (although not all) digital systems can perform multiple arbitrary tasks with minor or no change to their “hardware”; only the software has to be modified. Only rudimentary programming is possible on some analog systems; many analog processors allow no programming at all. Reusability. Many digital systems can be easily reconfigured for different tasks, alleviating the need to redesign a different system for each task. These advantages of digital computing are paid for in system bandwidth. For

498


[V,§ 4

example, an analog system of 0.4% precision and a bandwidth of 12 MHz is equivalent to a digital processor with about 100 MHz bandwidth. The major disadvantage of digital systems, thus, is as follows: Performance. The performance limitations of digital systems (in speed versus cost and bulk) relative to some analog alternatives results from the following constraints: bandwidth is used by digital systems to obtain high system precision with low accuracy components; nonparallel operation results from the impossible complexity that is necessary if we wish to implement a massively parallel architecture with the overhead of digital technology. The preceding arguments seem to apply also to optical processing. However, many researchers now believe that eventually optical digital processing can combine the advantages of digital electronics with those of analog optics. Indeed, considerable research into optical digital processing is now in progress, as exemplified in a recent survey by ARRATHOON [ 19861. The research can be divided into two major activities. (1) Optical switching research tries to capitalize on ultra-fast, nonlinear optical phenomena where optical switching in less than 10- l 3 second seems feasible. (2) Optical interconnections and architectures where the high parallelism of optics is coupled with some switching (possibly electronic) mechanism to obtain very high throughput through massive parallelism. In this chapter we shall limit our discussion to the second approach. Some of the work on digital optical architectures is oriented toward pure, fixed-function,signal processing systems where no programmability is required. Another part of the work is oriented toward logical processing systems where flexibility, similar to that of electronic digital computers, is the goal. Work related to incoherent methods for implementing both types of processors will be briefly summarized. 4.5.1. Binary multiplication via analog convolution

In standard binary notation we have N digits (“bits”), bi, i = 0, . . . ,N - 1 ; we can write any number n, where 0 < n < (2N - 1) in the form N- 1

n

=

2

bi2‘

i=O

Standard, normalized, binary notation has also the requirement that each bit, bi, must be either 0 or 1. Adding two binary numbers or multiplying them must therefore be done sequentially from right (b,) to left (bN-,) to allow manipu-

v, § 41


lation of the carry digits. For example, to add ,,5 do

=

499

,0101 to 107 = ,0111 we

0101 0111 ___ 0100

1

(carry)

0110 0100 1 0100 0100

(carry)

1

(carry)

~

0000 -

1100 (final result) ,

which gives 21100 = 1012.Alternatively, we can tolerate non-normalizedbinary notation and write 0101 0111 0212 (excess binary result) .

~

We can later take out the carry and normalize the result. This mode of operation seems to offer little advantage for electronics, where the binary electronic components cannot handle the digit “2” anyhow, and for addition, where the number of steps saved is nearly the same as the number added in post-normalization. For optical binary multiplication, however, we have a different situation. Let us consider ,7 x ,,5 = ,0111 x ,0101, as an example 0111 x 0101 0111 0000 0111 011211 ~

(0111 x 1 x 1) (0111 x 0 x 2) (0111 x 1 x 4 ) (result = ,011211 = 1035)

Now, looking carefully at the preceding example, we see that ifwe consider each of the two multidigital numbers to be a spatiul signal [ g ( x ) = binfeger(x,A)],then the preceding operation is simply a convolution between the spatial represenand SPEISER[ 19771 suggested this tations of the two numbers. WHITEHOUSE

500


[V, § 4

algorithm for optical digital multipliers. A digital vector-matrix multiplier that used this algorithm was designed, for example, by GUILFOYLE [ 19841.

4.5.2. Logical operations using linear transformations Commutative binary logical functions can be implemented optically using a point nonlinear function. The general techniques used for each logic gate can be written as follows: output = P

(iNCI

1

input,

,

where

P(x) is some point nonlinear function that produces either 1or 0 as output and ouiput and input,(where I = 1, .. .,N ) are binary digits that can take values of 0 or 1 .

For example, for the NOR(a,b)

=

NOT(a OR 6 ) operation we select:

1 , i f x

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