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
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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
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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 . . . . . . . . . . . . . . . . . . . .
xv 108 109 110 110 111 111 113
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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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 . . . . . . . . . . . . . . . . .
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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
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Q 3. FABRICATION TECHNIQUES OF MICRO FRESNEL LENSES . . . . . . . . . . . . . . . . . . . . . . . .
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Q 4 . FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY
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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
MICRO FRESNEL LENSES
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)
<< 1, eq. (5) is rewritten by
32AF3,
using the f-number of the lens expressed by F = f / 2 r b . In the case of 2 = 0.633 pm, the diafneter 2rb, within which the phase shift error may be neghgible, is calculated to be 2.5 mm for F/5 and 0.6 mm for F / 3 . When F is small, therefore, one must use $(r) given by eq. (1) instead of $a(r).
2.2. ZONES
If the phase shift function $(r) is modified to have a modulo II phase shift structure, as shown in Fig. 3, one obtains the phase shift function of a phase-type FZP or binary-type Fresnel lens, denoted by $ z ( ~ )and , the phase distribution on a zone becomes a rectangular profile. If $(r) is modified to have a modulo 2 n phase structure, as shown in Fig. 4, one obtains the phase shift function of a blazed-type Fresnel lens, denoted by i,hF, and the phase distribution on a zone becomes a sawtooth profile. $F is expressed by $F(r) = $(r) + 2 m 72, r, < r < r,
+ 1,
(7)
where r,,, is the inner radius of the mth zone and is determined by
Then, one has
If $a(r) instead of $(r) is taken as t/&(r), that is, the lens has a large f-number, eq. (9a) can be replaced by r,
=
Jmf.
7
PRINCIPLES OF FRESNEL LENSES
I
rm r m + l
( b ) Fig. 3. (a) Phase shift function of the phase-type Fresnel zone plate; (b) thickness distribution.
From eqs. (9a) and (9b) the focal length f is given by
f = r;
(mA)’ 2mA -
, for small F,
rf for large F. (lob) 2m1’ The phase shift results from the product of the change in refractive index, An, and the relative thickness TI1 of the zone material, that is AnT/1. Fresnel lenses can be produced by modulating the index change An or the thickness T. Since large maximum values Anmaxcannot be expected in optical materials, the thickness modulation type should be considered. Therefore, the gradientthickness (relief) structure is expressed by f=-
This is schematically shown in Fig. 4b. To have the optimum phase shift 211, therefore, the optimum thickness T is given by To,,
=
AIAn
(12)
MICRO FRESNEL LENSES
8
( a )
> 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
MICRO FRESNEL LENSES
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
MICRO FRESNEL LENSES
[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
MICRO FRESNEL LENSES
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
MICRO FRESNEL LENSES
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
MICRO FRESNEL LENSES
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
FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY
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
MICRO FRESNEL LENSES
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
MICRO FRESNEL LENSES
[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
FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY
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
FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY
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
MICRO FRESNEL LENSES
[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
FRESNEL LENSES FABRICATED BY ELECTRON-BEAM LITHOGRAPHY
25
Fig. 17. (a) Example of a Fresnel lens array (diameter 0.5 mm; focal length 5 mm); (b)multiple images.
26
MICRO FRESNEL LENSES
[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
MICRO FRESNEL LENSES
[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
WAVEGUIDE FRESNEL LENSES
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,
(27)
where n, is the effective index of refraction of the guided wave (mode index), I the wavelength in free space, and x, denotes the zone edge. Under the assumption of a "thin" lens, the phase shift is given by k,AnL, with mode index modulation An and lens thickness L. Therefore Fresnel lenses can be constructed by modulating An or L; the gradient-index (GRIN) Fresnel lens
30
[I, § 5
MICRO FRESNEL LENSES
(Fig. 19(a)) is described by
[F+
An(x) = Anmax
I] , L
const.,
=
and the gradient-thickness (GRTH) Fresnel lens (Fig. 19(b)) is described by L ( x ) = L,,,
[$I + 11,
An = const.
To have the modulo-2a phase modulation, in either case, the modulation amplitude must be optimized so that koAnm,L = 2 n or koAnL = 271 is satisfied. The binary approximation of the phase modulation results in the step-index (SI) Fresnel zone lens. Theoretical analysis of “thin” GRIN and GRTH Fresnel waveguide lenses has been given by SUHARA,KOBAYASHI, NISHIHARA and KOYAMA[ 19821. Figure 20 shows the calculated intensity profile on the focal line. The parameter I denotes the ratio of the modulation amplitude to the previously given optimal value. The diffraction-limited 3 dB spot width of the main lobe on the focal line is given by 2a=2
(neLxM) *
1.38 = 0.88 FA ,
~
ne ~~
GRIN/GRTH
- 400 -
~~
FRESNEL LENS
H c,
w
E ffl z
gw 2 0 0 H
w
S I FRESNEL LENS
2B
1 O
-3n
-2Tl
-n
0
n
2n
3n
NORMALIZED D I S T A N C E FROM O P T I C A L AXIS ( k X M / f ) X
Fig. 20. Theoretical curves of the light intensity distribution on the focal line of the Fresnel waveguide lenses.
[I, § 5
31
WAVEGUIDE FRESNEL LENSES
where F = f / 2 x M is the F number. As seen in Fig. 20, the spot width is approximately independent of 1, that is, the diffraction-limited width is obtainable even in lenses of I # 1 and the SI lens. Figure21 shows the calculated dependence of the lens efficiency, defined in this section as the ratio of the power in the focus main lobe to the total input power, on the parameter 1. A maximum efficiency of 90.2 %, which is limited only by diffraction, can be obtained in GRIN and GRTH lenses, whereas the maximum efficiency for a SI Fresnel zone lens is 30.5%. There are some methods to produce the mode index change (modulation) required for the lens fabrication. The most effective method to create a large change would be to remove the waveguide material within the lens area and fill the vacancy with different material; the current technology, however, gives unsatisfactory results. In some waveguide materials the refractive index can be changed by a diffusion process, such as ion exchanging, or by electron (light) beam irradiation. Since the mode index is a function of the refractive indexes and the thicknesses of the layers forming the waveguiding structure, it can be changed by a modification of the structure, such as by changing the guide thickness or overlay cladding the guide. The amount of the available mode index change, however, is much smaller than that for micro-optic lenses. By using eqs. (19) and (26)-(29), the maximum of the Q parameter for the
-* Y
r
I 100 -
2M=100 GRIN/GRTH FRE,SNEL LENS
-
>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
MICRO FRESNEL LENSES
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
WAVEGUIDE FRESNEL LENSES
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
MICRO FRESNEL LENSES
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
MICRO FRESNEL LENSES
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.
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E. WOLF, PROGRESS IN OPTICS XXIV 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1987
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
DEPHASING-INDUCED COHERENT PHENOMENA
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$L(-
w4,01
-
x exp(i[(k, - k,
+ C.C. ,
02,
w3)Ek(w1)E:(o,)Em(o3)
+ k3) r - w4t]) *
(2.2)
11, § 21
THEORETICAL TREATMENT
47
where C.C. denotes the complex conjugate of the entire expression. x$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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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$" in the absence of fields. Each iteration of (2.9) and (2.10) introduces an additional field factor in each subsequent correction to p. This justifies the expansion assumed in eq. (2.1) and means that x'") is simply related to p'") by (2.1) and (2.6). General expressions for x(3)resulting from repeated application of (2.9) and (2.10) are given by FLYTZANIS [ 19751. All 48 terms from such a treatment to obtain x ( ~ ) (- w,, o,- o,,w3) are written explicitly by BLOEMBERGEN, LOTEMand LYNCHJR. [ 19781, including 24 terms not in the expression derived without damping (ARMSTRONG,BLOEMBERGEN, DUCUINGand PERSHAN[ 19621). These 24 terms can be arranged into 12 pairs, which are proportional to expressions of the form (rkl - rk, - rm1) which vanish in a system of only states Ik), [ I ) , and Im) when there is no proper dephasing (SARGENT,SCULLYand LAMBJR. [ 19741). In the special case of a two-level system this means that these extra terms cancel when the familiar relationship T2 = 2T, holds between population and coherence decay times, Tl and T 2 . We now derive expressions for x(3) specific to the case of Fig. la, which depicts an excitation diagram appropriate to dephasing-induced four-wave mixing resonances between unpopulated excited states le) and le') . Only terms in f 3 ) with three near resonant denominators are retained. The physics
50
DEPHASING-INDUCED COHERENT PHENOMENA
-. le>-
-
W1
W1
Ig>
-
le>
le>
lg>
(b)
Fig. 1. Energy level diagrams and tuning schemes for several dephasing-induced four-wave mixing resonances. (a) Raman resonance between unpopulated excited states; (b) population grating resonance; (c) Raman resonance between equally populated ground states.
of these resonances is contained in the second order corrections to the density matrix, p$?(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)
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
(b)
(a) If>
<jl
If>
<jl
li>
<ji
li>
<jl
li>
li>
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
THEORETICAL TREATMENT
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
(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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
EXPERIMENTAL RESULTS
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
DEPHASING-INDUCED COHERENT PHENOMENA
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
DEPHASING-INDUCED COHERENT PHENOMENA
[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
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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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E. WOLF, PROGRESS IN OPTICS XXIV 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1987
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 . . . . . . . . . . . . . . . . . . . .
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ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 159 REFERENCES
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$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
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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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
[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
INTERFEROMETRY WITH LASERS
[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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
[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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
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
127
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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
[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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRY WITH LASERS
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
HETERODYNE SPATIAL INTERFEROMETRY
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
INTERFEROMETRY WITH LASERS
[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
HETERODYNE SPATIAL INTERFEROMETRY
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
INTERFEROMETRY WITH LASERS
[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
INTERFEROMETRY WITH LASERS
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
INTERFEROMETRIC SENSORS
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
INTERFEROMETRY WITH LASERS
“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
INTERFEROMETRIC SENSORS
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
INTERFEROMETRY WITH LASERS
144
5 10.
[III,$ 10
Pulsed-Laser and Nonlinear Interferometers
Lasers can be used to produce light pulses of very short duration (<0.1 ps) and extremely high intensity. The availability of sources of coherent light with such characteristics has led to a number of new interferometric techniques, many of which are based on the use of nonlinear optical elements.
10.1. INTERFEROMETRY WITH PULSED LASERS
Q-switched ruby lasers have been used very effectively in interferometric studies of gas flow. Several types of interferometers for such work have been described by TANNER[ 1965,1966,19671,who has shown that, because of the very high temporal and spatial coherence of light from such lasers, a simple optical setup can be used. In addition, SCHMIDT,SALZMANNand STROWALD [ 19751have used a TEA nitrogen laser producing pulses with a duration of less than 500 picoseconds for interferometric studies of short-lived plasmas.
10.2. TWO-WAVELENGTH INTERFEROMETRY
The first application of nonlinear optics in interferometry was in studies of plasmas to make simultaneous measurements at the fundamental laser wavelength (A = 694 nm) and its second harmonic (A = 347 nm) generated by passing the beam through a frequency doubler (ALCOCKand RAMSDEN [ 19661). Because of the large dispersion of the electrons, such measurements permit separate evaluation of the densities of the electrons and the other components of the plasma. 10.3. SECOND-HARMONIC INTERFEROMETERS
In second-harmonic interferometers a fringe pattern is produced corresponding to the phase difference between two second-harmonic waves generated from the original wave at the fundamental frequency. A nonlinear Twyman-Green interferometer is shown in Fig. 10.la (HOPF, TOMITA and AL-JUMAILY [ 1980I). The beam from a Q-switched Nd :YAG laser (A, = 1.06 pm) is incident on a frequency-doublingcrystal, and the green (A, = 0.53 pm) and infrared beams emerging from this crystal are separated by a dichroic beam splitter (BS,) so that each beam traverses one arm of the
111, I 101
PULSED-LASER A N D NONLINEAR INTERFEROMETERS
Green
---
4L
Green
Test piece "
Ir
. \
h I
145
I
V Do u bIe r
1 - --
t-4
' BS,
IR
MI
(a)
Green
\ Doubler
\ Test piece
'
t
Doubler
IR laser
-
(Cl Fig. 10.1. Second-harmonic interferometers; analogues of (a) the Twyman-Green, (b) the Mach-Zehnder, and (c) the point-diffraction interferometer. (HOPF,TOMITA and AL-JUMAILY [1980], LIEPMANN and HOPF[1985].)
interferometer. The green beam serves as the reference, and the test piece is placed in the infrared beam. When the two beams return to the crystal, the infrared beam undergoes frequency doubling to produce a green beam. Another dichroic beam splitter (BS,) brings the two green beams out to the observation plane.
146
INTERFEROMETRY WITH LASERS
[111, 10
The optical path difference between the beams at any point ( x , y) due to the test specimen is then
P@,
v) = m, - 1) &, 34
(10.1)
9
where d ( x , y ) is the thickness of the test specimen at that point and n, is its refractive index in the infrared, whereas the resulting change in the interference order is
N x , v) = 2(n, - 1)
4 x 7
YYA2 .
(10.2)
Since the same small fraction of the power in the infrared beam is converted into green light at each pass, fringes with good visibility are obtained. If the beam splitter BS2 and the mirror M2 in Fig. 10.la are removed, a nonlinear Fizeau interferometer is obtained. In this setup both the infrared and green beams go through the object, and the difference in their optical paths is A x , v) = a n 2 - a,)
4 x 9
34 .
(10.3)
where n2 is the refractive index of the test specimen for green light. When the two beams return to the crystal, the infrared beam undergoes frequency doubling and fringes are obtained. The interference order at any point (x, y) is then “x,
A = 2(n2 - n,)
4 x 9
YYA2
.
(10.4)
On the other hand, with a conventional Fizeau interferometer using green light the interference order would have been ” ( x , Y ) = 2(n, - 1) 4 x 3 YYA2 .
(10.5)
A comparison of eq. (10.4) with eq. (10.5) shows that a desensitized interferogram is obtained in both these cases, corresponding to an equivalent wavelength
A,,
= A2@2
- 1)/(n2 - n,)
9
(10.6)
which is typically around 50 pm. An interferometer using two frequency-doubling crystals that functions in a similar manner and can be considered an analogue of the Mach-Zehnder interferometer is shown in Fig. 10.Ib. A detailed study of this interferometer has been made by HOPFand CERVANTES [ 19821, who have shown that it is well suited to studies of phase objects with large changes in optical thickness. [ 19841 have analysed some other ALUM,KOVAL’CHUK and OSTROVSKAYA
111, § lo]
PULSED-LASER A N D NONLINEAR INTERFEROMETERS
147
optical configurations and pointed out the advantages of interferometers of this type for measurements of the electron density in plasmas. A third type of interferometer belonging to this group (LIEPMANN and HOPF [ 1985]), which has some similarities to the point-diffraction interferometer, is shown in Fig. 10. 1~In . this interferometer the object 0 is placed in the front focal plane of the lens L, so that the complex amplitude in the back focal plane of L, corresponds to the two-dimensional Fourier transform of the complex transmittance of 0. A frequency-doubling crystal X, placed in this plane converts some of the infrared light to green light. The amplitude of the second harmonic, which is proportional to the square of the amplitude at the fundamental frequency, therefore corresponds to the square of the Fourier transform of the original object wave. In the image plane this second harmonic wave (which constitutes the reference wave) should have a complex amplitude proportional to the self convolution of the object, but it is found experimentally that it is very nearly a plane wave. The other second-harmonic wave that is generated by the frequency-doubling crystal X, is a faithful reproduction of the transmitted wave at the fundamental frequency. However, since its wavelength is half that of the fundamental wave, the phase shifts due to the object are doubled. Accordingly, an interferogram is obtained in the image plane that is a direct representation of the shape of the object wavefront, but with twice the sensitivity of a conventional interferometer. 10.3.1. Second-harmonic inte$erometers using critical phase-matching Another group of second-harmonic interferometers makes use of the distortions of amplitude and phase produced by phase mismatching in a frequency-doubling crystal to produce an interference pattern containing information on the test wavefront (HOPF,TOMITA,AL-JUMAILY, CERVANTES and LIEPMANN [ 19813). The complex amplitude of the second harmonic from a frequency-doubling crystal operated at low efficiency is given by the relation (ZERNIKEand MIDWINTER [ 19731) a2 =
[
]
8n2L sin (AkL/2) exp (iAkL/2) , d,, a: 21% AkL/2
~
(10.7)
where a , is the complex amplitude of the incident fundamental wave and I , its wavelength, L is the length of the crystal and n, its refractive index at the
148
INTERFEROMETRY WITH LASERS
[111, § 10
fundamental wavelength, d,, is the nonlinear coefficient, and the factor Ak is given for a type-I doubler by the expression Ak
=
2n(n, - n,)/2,
(10.8)
where n, is the refractive index for the second harmonic and 1, is its wavelength. For critical phase matching the angle 8 between the normal to the wavefront and the optic axis is set so that Ak = 0. Normally, any change in the angle 8 will then result in a phase mismatch. However, this problem can be minimized if the matching angle 8, is made equal to 90°, a condition known as noncritical phase matching, which can be obtained in some frequencydoubling crystals by adjusting the temperature. The phase mismatch resulting from a tilt of the object wavefront in a critically phase-matched frequency doubler can be used to produce a fringe pattern in two ways. One method is based on the effect of the wavefront aberration on the term [ sin(AkL/2)/(AkL/2)] in eq. (10.7), which modifies the amplitude of the second-harmonic wave (MAKER,TERHUNE,NISENOFF and SAVAGE [1962]). The value of this term and, hence, the amplitude of the second harmonic drops to zero in the directions for which AkL/2 = m,where m ( f 0) is an integer, giving rise to dark fringes. This type of interferometer is well suited to measurements of small wavefront distortions as, for example, in studies of self-focusing. Another type of interferometer can be constructed with two frequencydoubling crystals. If the first crystal is noncritically phase matched, it generates a green wavefront that is a faithful replica of the original infrared wavefront. However, the green wavefront generated by the second crystal that is critically phase matched has a phase distortion superimposed on it because of the factor exp(iAkL/2) on the right-hand side of eq. (10.7). The interference pattern observed will therefore correspond to this phase distortion, which is a measure of the local changes in slope of the wavefront. Because of its sensitivity to phase gradients, this type of interferometer is most useful for studying localized wavefront errors where, even though the gradients are appreciable, the actual phase deviations are too small to be measured by a conventional interferometer.
10.4. PHASE-CONJUGATE INTERFEROMETERS
In phase-conjugate interferometry the wavefront under study is compared with its complex conjugate. This technique, which was first demonstrated using
111, 5 lo]
PULSED-LASER A N D NONLINEAR INTERFEROMETERS
149
holography (BRYNGDAHL[ 1969]), has the advantage that a reference wavefront is not necessary. In addition, the sensitivity is double that of conventional interferometry, and the fringe contrast is always unity, regardless of the intensity distribution over the test wavefront. Two types of interferometers based on the generation of a conjugate wave in real time were proposed in a theoretical study by HOPF[ 19801. The first, shown schematically in Fig. 10.2a, uses a conjugate wave generated by four-wave mixing (see YARIV[ 19781). In this setup the crystal located in the signal beam in the upper arm of the interferometer is also illuminated by a coherent pump beam derived from the same laser. The conjugate wave generated by the crystal travels down the lower arm of the interferometer to the second beam splitter, where it is combined with the signal beam to produce an interference pattern. A schematic of another phase-conjugate interferometer based on degenerate three-wave mixing is shown in Fig. 10.2b. In this arrangement the signal wave is imaged by the lens L, inside a noncritically phase-matched crystal, where it
L1 Signal
-
(a)
/ Crystal Ll
Ir
Second-harmonic
L2
150
[III,
INTERFEROMETRY WITH LASERS
8
10
is mixed with an orthogonally polarized pump beam generated from the same laser in a frequency doubler. The desired interference pattern is obtained directly in the output from the crystal and is imaged by the lens L, into the observation plane. The operation of Mach-Zehnder and Michelson phase-conjugate interferometers has been demonstrated experimentally by BAR-JOSEPH,HARDY, KATZIRand SILBERBERG [ 19811. Phase conjugation was achieved in thin films of eosin in gelatin, using light from an argon-ion laser at a wavelength of 5 14 nm.
10.5. INTERFEROMETERS WITH PHASE-CONJUGATING MIRRORS
Replacement of the mirrors in a conventional interferometer with phaseconjugating mirrors results in systems with new physical properties. In such an interferometer any nonuniform phase changes such as might be caused by air turbulence or distortions in the optics are cancelled out. In addition, the interference pattern is unaffected by angular misalignment of the mirrors. A theoretical analysis of such an interferometer has been made by BASOV, ZUBAROV, MIRONOV,MIKHAILOVand OKULOV[1980], who have also presented some experimental results with an interferometer of the form shown schematically in Fig. 10.3, which used a single phase-conjugating mirror to reflect back both the beams. This interferometer was shown to exhibit a very interesting effect resulting from the use of stimulated Brillouin scattering to generate the phase-conjugate waves.
Movable mirror
Phase-distorting plate Phase-coniugating
(Lightguide containing CS,)
0
Detector
0
Detector
Fig. 10.3. Interferometer with a single phase-conjugating mirror using stimulated Brillouin and OKULOV [1980].) scattering. (BASOV, ZUBAROV, MIRONOV, MIKHAILOV
111,s lo]
PULSED-LASER A N D NONLINEAR INTERFEROMETERS
151
With a phase-conjugating mirror a beam that has traversed a shorter path A1 before reaching the phase-conjugating mirror is reflected back with a phase lag equal to (2n/A)Al, so that if the frequency of the light remains unchanged, the reflected beam returns to the beam splitter with unchanged phase for any value of Al. However, Brillouin scattering introduces a frequency shift AvB for the Stokes component, so that the reflected waves reach the beam splitter with a phase difference A@ = (2n/c)AvBAl.
(10.9)
As a result, the output from the interferometer was found to vary cyclically with the optical path difference, with a spatial period of a few centimetres. A Michelson interferometer using a self-pumped phase conjugator in one arm has also been demonstrated by FEINBERG [ 19831. The phase conjugator consisted, in this case, of a single crystal of BaTiO,, the pump beams being generated from the incident beam by internal reflection at the crystal faces (FEINBERG [ 19821). In this interferometer, when a sheet of semifrosted plastic was placed in front of the phase conjugator, the fringe pattern vanished momentarily but reappeared after a few seconds.
10.6. PHOTOREFRACTIVE OSCILLATORS
The phenomenon of two-beam coupling in a photorefractive crystal has been used to obtain oscillation in a ring cavity (WHITE,CRONIN-GOLOMB, FISCHER and YARIV[19821). As shown by FEINBERG and BACHER[1984] and by RAJBENBACH and HUIGNARD [ 19851, the frequency of the oscillation is, in general, slightly different from that of the pump beam. A detailed theory has been developed by YEH [ 19841 and by YARIVand KWONG [1985] to explain these detuning effects. This theory, as well as experiments by KWONG,YARIV,CRONIN-GOLOMB and URY[ 19851 using the arrangement shown in Fig. 10.4, shows that a linear relationship exists between the frequency of oscillation and the displacement of one of the mirrors (say M,) over each period of a wavelength of the change in the optical path. This phenomenon provides an interesting new class of interferometers that can convert changes in optical path length directly into changes in frequency.
152
INTERFEROMETRY WITH LASERS
[III,
c 11
Detector
Fig. 10.4. Experimental arrangement used to measure frequency detuning of a photorefractively and URY [1985].) pumped, unidirectional ring oscillator. (KWONG,YARIV,CRONIN-GOLOMB
8 11. Interferometric Measurements on Lasers Interferometers have been used widely to study the characteristics of the output beams from lasers. This application has become particularly important with the development of semiconductor diode lasers and tunable dye lasers.
11.1. ANALYSIS OF SPATIAL COHERENCE AND WAVEFRONT ABERRATIONS
Proper design of the structure of a semiconductorlaser is necessary to ensure operation in a single spatial mode. A modified version of a wavefront-reversing and MONTALTI [ 19791to examine the interferometerwas used by BERTOLOTTI spatial coherence of the output of semiconductor lasers and to study the variations of modal structure with pulse length and aging. Operation in a single longitudinal mode can also be obtained by proper design. As the current is increased above threshold, one mode is preferentially amplified while the others progressively disappear. However, even when single-mode operation has been obtained, the angular divergence of the beam is greater in a direction perpendicular to the plane of the junction. This is because index guiding is the predominant effect in this plane, whereas only gain guiding operates in the plane of the junction (COOKand NASH[ 1975I). The result of this asymmetry is an elliptical focal spot, which is undesirable in applications such as optical data recording or readout.
111, I 1 1 1
INTERFEROMETRIC MEASUREMENTS O N LASERS
153
Beam expander Collimator
3-
Semiconductor laser
Observation plane
Fig. 1 1 . 1 . Interferometer used for measurements of the shape of the output wavefront from a semiconductor laser. (TATSUNO and ARIMOTO [ 19811.)
.
Early measurements of the astigmatism of the output wavefront were made by studying the intensity distribution in the image plane, but this method is inadequate when the aberrations are small. Direct measurements of the shape of the output wavefront from different types of GaAlAs double-heterostructure lasers were first made by TATSUNO and ARIMOTO [ 1981] using the radial-shear interferometer shown in Fig. 11.1. If the diameter of the reference wavefront leaving the beam-expandhg telescope is large compared with that of the test wavefront in the lower arm, the resulting interferogram directly gives the aberrations of the test wavefront. Other workers such as ARQUIE[ 19821 and CREATH[ 19851 have used a point-diffraction interferometer for such tests. Maximum sensitivity, as well as rapid analysis of the results, is possible with a system using heterodyne techniques for phase measurement (HAYESand LANGE[ 19831, LEUNGand LANGE [ 19831).
11.2. MEASUREMENTS OF SPECTRAL LINEWIDTHS
The usual method for evaluating the spectral linewidth of a source from measurements of the visibility of the fringes in a Michelson interferometer breaks down with lasers because of the very large delays that would be required to give useful results.
[HI, J 11
INTERFEROMETRY WITH LASERS
154
In such cases a solution is the use of heterodyne methods, in which the signal to be studied is mixed at a square-law detector with a reference signal that has a slightly different frequency. The output from this detector then contains a component at the beat frequency (see $3.2), from whose spectral characteristics it is possible to determine the spectral characteristics of the original signal. The need for a reference signal that is spectrally pure and free from drift can be avoided by taking afraction of the test signal, shifting its frequency by a suitable amount, and then mixing it with the original signal. Since the two signals are statistically identical, it can be shown that the self-coherence function of the original signal can be obtained from measurements on the beat (HALMOS and SHAMIR [ 19821, ABITBOL, GALLION, NAKAJIMA and CHABRAN [ 19841). Figure 11.2 is a schematic of an experimental setup for such measurements. If the field at the detector due to the beam reflected at the beam splitter B, and transmitted by the beam splitter B, is written as E , ( t ) = E, cos[ot
+ $(t)],
(11.1)
the field due to the other beam, which has its circular frequency shifted by an amount a by the acousto-optic modulator and has a delay z introduced by the single-mode fibre, is given by the expression E*(t) = E, cos[(o
+ o)t + +(I - z)] .
-
--
r
Laser
Optical isolator
(11.2)
-
-
Acousto-
11
Spectrum analyser
*
Detector
Fig. 11.2. Experimental arrangement used for measurements of laser linewidth by heterodyne GALLION, NAKAJIMA and CHABRAN[ 19841.) interferometry. (ABITBOL,
111, § 11)
INTERFEROMETRIC MEASUREMENTS ON LASERS
155
The beat-frequency signal is then proportional to cos[Qt +
$(l - 7) -
$(t)].
(11.3)
If z >> (l/Av) where A v is the laser linewidth, $(t - z) and @(t)are statistically independent and the variance of the phase fluctuations of the photocurrent is twice the variance of the phase fluctuations in the output of the laser. Accordingly, the spectral width of the beat signal, which can be measured by a radio frequency spectrum analyser, is twice the linewidth of the laser. An alternative method, which has been used with semiconductor lasers by ABITBOL,GALLION,NAKAJIMA and CHABRAN [ 19841, makes use of the fact that a change in the injection current results in a change in the refractive index of the cavity and, hence, in the output frequency. This phenomenon can be used to modulate the output frequency at rates in excess of 1 MHz. The optical arrangement used is similar to that shown in Fig. 11.2, except that the acousto-optic modulator is no longer needed, and a square-wave modulation is superposed on the base current with a period
T
=
T/(m + +),
(11.4)
where m is an integer. Measurements of the beat frequency for different values of m and for different base currents can be used to evaluate the modulation characteristics of the laser. In addition, measurements of the spectral width of the heterodyne signal as a function of the base current, at constant modulation current, make it possible to plot the variation of the laser linewidth with the output power. These measurements confirm the expected 1/P dependence of the laser linewidth (see eq. 1.2). 11.3. HETERODYNE METHODS OF FREQUENCY MEASUREMENT
Heterodyne methods can also be used for precise measurements of the frequency of a laser. One method (BAY,LUTHERand WHITE[ 19721) that can give the frequency of a laser to a few parts in 10' involves modulating the laser beam (frequency vL) at a microwave frequency ( vM) to produce two frequencies vL ? ,v . If, then, the length of a Fabry-Perot interferometer and the modulation frequency are adjusted so that both these frequencies are transmitted with maximum intensity, we have (11.5)
156
INTERFEROMETRY WITH LASERS
[HI,
I 11
where N is the interference order for the frequency v, ,and is the difference in the interference orders for the two frequencies. BAIRD[1983b] has described another method that can be used for very accurate comparisons of the frequencies of two lasers, whose frequencies are too far apart to give a measurable beat frequency. This method uses two slave lasers, which can be tuned over a small range around the frequencies of the two lasers to be compared. These slave lasers are locked to two transmission peaks of a very stable Fabry-Perot interferometer so that their wavelengths are integral submultiples of the optical path in it, and their frequency offsets with respect to the lasers to be compared are measured.
11.4. LASER WAVELENGTH METERS
Interferometric wavemeters meet the need for instruments that can measure the output wavelength of tunable lasers with an accuracy commensurate with their linewidth. Dynamic wavemeters, in which the measurement involves the movement of some element, usually have greater accuracy but can be used only with cw lasers; static wavemeters can also be used with pulsed lasers. 11.4.1. Dynamic wavelength meters The most common form of dynamic wavelength meter consists of a two-beam interferometer in which the number of fringes is counted as the optical path is changed by a known amount. One form (HALLand LEE[ 19761) uses a folded Michelson interferometer with the two end reflectors mounted back-to-back on a single carriage. In another, shown in Fig. 11.3, two beams, one from the laser whose frequency is to be determined (typically, a dye laser) and another from a reference laser (a frequency-stabilized He-Ne laser) traverse the same two paths in opposite directions (KOWALSKI, HAWKINS and SCHAWLOW [ 19761). The fringe systems formed by these two wavelengths are imaged on two detectors D, and D, . If, then, the end reflector is moved through a distance d, we have 11
=
(N,n,/N,n2)1, >
(11.6)
where N , and N, are the numbers of fringes seen by D , and D,, and n, and n, are the refractive indices of air at 1,and A,, respectively. If fractions are not counted, the precision is approximately 1/N,or about 6 parts in lo7, for a change in the optical path difference of 1 m.
111, I 1 I ]
INTERFEROMETRIC MEASUREMENTS ON LASERS
$1 Detector
D,(A
157
11
B 1
Reference laser [ h p )
Dye laser ( h l )
c------)
Detector
D,(
hp)
Fig. 11.3. Optical system of a dynamic wavelength meter SCHAWLOW [1976].)
Movable cube-corner
(KOWALSKI,
HAWKINSand
Higher precision can be obtained by measurements of the fractional order number for the reference laser. One way of doing this is by phase-locking an oscillator to an exact multiple of the frequency of the ac signal from the reference channel (HALLand LEE [ 19761, KOWALSKI, TEETS,DEMTRODER and SCHAWLOW [ 1978I); another is by digital averaging of the two signal frequencies, and a third is a vernier method in which the counting cycle starts and stops when the phases of the two signals coincide (KAHANE,OSULLIVAN, SANFORDand STOICHEFF[ 19831). MILAN, PULLICINO,ROUSSEL and MORET-BAILLY [ 19841 have given a detailed description of a high-precision dynamic wavemeter, and MONCHALIN, KELLY,THOMAS, KURNIT,SZOKE, ZERNIKE,LEE and JAVAN [ 19811 have made a detailed analysis of possible sources of error. Under favourable conditions a precision of 1 part in lo9 is possible. Another type of dynamic wavemeter uses a scanning spherical Fabry-Perot and POLE[ 19801). If the separation of the mirrors interferometer (SALIMBENI in such an interferometer is changed slowly, coincidences will be observed between the transmission peaks corresponding to the two wavelengths at intervals corresponding to the condition in,,?, =
m2A2 = p ,
(11.7)
where m, and m2 are the changes in the integer orders and p is the change in the optical path difference. The precision is enhanced in this case by a factor equal to the finesse of the Fabry-Perot interferometer, but defocusing effects limit the range of movement of the mirrors. With a range of movement of 25 mm, a precision of 1 part in lo7 can be attained.
158
INTERFEROMETRY WITH LASERS
[III, 3 12
11.4.2. Static wavelength meters
Static wavemeters can be classified according to the type of interferometer used. The sigma-meter (JUNCARand PINARD[ 1975, 19821) uses four Michelson interferometers with optical path differences of 0.5 mm, 5 mm, 50 mm, and 500 mm sharing a common beam splitter and reference mirror. An approximate value of the wavelength of the dye laser is used to calculate the integer order of the first interferometer. The wavelength determined from the fractional fringe order in this interferometer is then used to calculate the integer order for the next interferometer, and so on. The actual wavelength is obtained with an accuracy of 1 part in 10' using the fractional order in the last interferometer. A second type of static wavemeter uses multiple Fabry-Perot interferometers (typical spectral ranges 3 GHz, 30 GHz and 300 GHz, respectively) (BYER,PAULand DUNCAN [ 19771, FISCHER,KULLMERand DEMTRODER [ 19811). The intensity distributions across the fringes in these interferometers are read by photodiode arrays, and the fractional fringe orders are calculated by a minicomputer with an accuracy of about 1%. In this case also the wavelength is found by a series of approximations, as described for the sigmameter, with an accuracy of about 5 parts in lo9. A very simple static wavemeter developed by SNYDER[ 19771 is based on the Fizeau interferometer. A collimated beam from the dye laser is incident on two uncoated fused-silica flats separated by about 1 mm and making an angle of about 3 minutes of arc with each other. A linear photodiode array located at a point where the shear between the reflected beams is zero is used to record the intensity distribution in the fringe pattern (SNYDER[ 19811, GARDNER [1983]). The integral interference order is then calculated from the spatial period of the fringe pattern, following which the wavelength is calculated from the positions of the maxima and minima. Up to 15 measurements a second can be made with an accuracy of 1 part in lo7. This wavemeter has been used for and SNYDER[ 19841); measurements on pulsed lasers (MORRIS,MCILRAITH another instrument for the same purpose using five coated wedge interferometers operated in transmission has been described by LEEand SCHAWLOW [ 19811.
6 12. Conclusions It is now just over 25 years since the advent of the laser. Within this period lasers have revolutionized optical interferometry. The main reasons for this
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dramatic change have been the obvious advantages of lasers over conventional thermal sources, namely a high-intensity output with almost perfect spatial and temporal coherence. Two other factors that have contributed to this revolution have been the progressive replacement of the human eye by photodetectors and the increasing use of sophisticated techniques of signal processing made possible by digital computers. These developments, as well as the availability of tunable lasers, have led to new methods of measurement, some based on heterodyne techniques and others involving direct evaluation of the interference phase. They have resulted in unprecedented increases in the range, speed, and precision of measurements. Finally, interesting possibilities have been opened up by the use of singlemode optical fibres and nonlinear crystals to build new types of interferometers with entirely new applications. There is little doubt that we can expect more new and important developments in the next few years.
Acknowledgements
I would like to thank Dr. W.H. Steel and P.E. Ciddor, with whom I have had many helpful discussions, and C.M. Chidley who was responsible for the drawings.
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E. WOLF, PROGRESS IN OPTICS XXIV 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1987
IV
UNSTABLE RESONATOR MODES BY
KURTEDMUND OUGHSTUN* Department of Electrical and Computer Engineering University of Wisconsin Madison, WI, U.S.A.
*Current address: Department of Computer Science and Electrical Engineering University of Vermont Burlington, VT, USA
CONTENTS PAGE
$ 1. INTRODUCTION .
. . . . . . . . . . . . . . . . . . .
167
$ 2. GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES . . . . . . . . . . . . . . . 168
. . . $ 4. ACTIVE CAVITY MODE STRUCTURE BEHAVIOR . . . . $ 5. CONCLUDING REMARKS . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . $ 3 . PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
240 330 354 355
APPENDIX. NUMERICAL TECHNIQUES AND SAMPLING CRITERIA . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 REFERENCES
. . . . . . . . . . . .
,
. . . . . . . . . .
378
8 1. Introduction Theoreticalresearch into the diffractive mode structure properties associated with an optical resonator was launched in 1961 by Fox and Li with the publication of their well-known paper on the resonant modes in a maser interferometer. Subsequent analysis by BOYD and KOGELNIK [ 19621 presented a classification of open resonators, composed of a pair of arbitrary spherical mirrors, based upon their associated diffraction losses, as was first suggested by Fox and LI [ 19611. This classification was in the form of a two-dimensional stability diagram in tenns of the cavity mirror radii R , and R , (with the radius of curvature of a convex mirror taken there as a negative quantity) and the mirror separation (or cavity length) d. They showed that stable, low-loss resonator operation occurred when the inequality 0 Q (1 - d / R , ) ( l - d / R , ) < 1 was satisfied. However, when the product (1 - d/R 1 - d/R,) was either negative or greater than unity, stable operation was shown to be impossible since the associated diffraction losses increased dramatically*. Resonator geometries in this high-loss or unstable region were accordingly called unstable and were subsequently avoided by most researchers until 1965. Interest in unstable resonators was initiated in 1965 with the publication of SIEGMAN’S foundational paper on unstable optical resonators for laser applications. In this paper Siegman postulated an approximate geometrical analysis of the dominant, lowest loss transverse mode that was remarkably accurate in predicting the outcoupling losses associated with an unstable cavity. This geometrical mode analysis was further developed by KAHN[1966] and by [ 19671 and extended to the higher-order, higherSIEGMAN and ARRATHOON loss transverse modes of such resonators. At this point it became clear that unstable resonators possess several significant advantages over their stable counterparts for many laser applications, particularly in the high energy regime * Subsequent numerical calculations by Fox and LI [I9631 showed that the transition between the stable and unstable regions of resonator operation is characterized by a continuous, gradual change in the associated diffraction losses. A detailed review of the multiple-beam interference [1969]. See also the and mode properties of stable open resonators is given by KOPPELMAN [1963, 19641. papers by KOGELNIKand LI [1966a,b] and VAINSHTEIN 167
168
UNSTABLE RESONATOR MODES
[IV, 8 2
(SIEGMAN [ 19711). First, unstable cavities provide a large mode volume even when the overall cavity length is short. The transverse mode structure, being geometrically dominated, yields a daractive field distribution that, to a good approximation,uniformlyfills this large mode volume. Second, the “d8ractive” output coupling is also geometrically dominated and, to a good approximation, is independent of the mirror sizes, depending only on the overall cavity magnification. Third, and importantly, unstable cavities can be designed to possess significant transverse mode discrimination in favor of a single, wellbehaved cavity mode (SIEGMAN [ 19651, BARONE [ 19671, BERGSTEIN [ 19681, SIEGMAN and MILLER[ 1970]), whose output field is highly directional (LA TOURETTE, JACOBS and RABINOWITZ[ 19641, SINCLAIRand COTTRELL [ 19671, ANAN’EV, SVENTSITSKAYA and SHERSTOBITOV [ 1968,1970), TREACY [ 19691). On the basis of this foundational work, research into the transverse mode structure properties of unstable resonators has been actively pursued in both the theoretical and experimental arenas. Although the geometrical behavior is relatively simple, the daractive behavior is deceptively so, and several fundamental questionsremain unanswered to date. This article presents an overview of the present understanding of the diffractive formation of unstable resonator modes. It is the author’s intention to provide a consistent formulation of the fundamental mathematical theory and to bring together several viewpoints of the underlying physical processes involved. This review of the published research to date is by no means complete and partially reflects the author’s personal research interests. Experimental results are quoted; however, the experimental aspect of the subject is not considered explicitly.
8 2. General Formulation of the Transverse Mode Structure Properties The basic mathematical formulation of the transverse mode structure properties of unstable optical cavities is presented in this section for the purpose of establishing a uniform notation. The analysis begins with the general scalar wave diffraction formulation of the transverse eigenstructure supported by an idealized optical cavity. The electromagnetic nature of the optical cavity field is then accounted for by the polarization eigenstate of the resonator. The ultimate startingpoint in describingthe optical field supported by a given cavity geometry is the electromagneticfield equations of Maxwell. These basic equations,together with both the boundary conditions at all the surfaces of the cavity, the shape and size of the cavity, and the material properties of the cavity
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GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
169
volume completely specify the electromagnetic eigenmodes of oscillation within the cavity. For time-harmonic (steady-state) phenomena in vacuum in the absence of charge and current sources, each Cartesian component of the two electromagnetic field vectors E and H is found to satisfy the scalar Helmholtz equation V*u(r)+ k2u(r) = 0.
(2.1)
Here
is the vacuum wavenumber of the monochromatic field of angular frequency o with wavelength 1,where c is the speed of light in vacuum. A complete description of the electromagnetic field requires that both the magnitudes of the (real-valued) field vectors and the polarization state of the field are everywhere specified as functions of both position and time. However, because of the very high frequencies of oscillation associated with optical fields, the instantaneous values of any of these quantities cannot be measured so that only certain long-time averages taken over intervals that are large compared with the optical periods are of any physical use. In this manner a very natural connection may be made between the magnitude of the time-average value of the Poynting vector of the electromagnetic field, given by
and the intensity distribution associated with a single complex scalar wave function u(r), given by
z = u*u ,
(2.4)
where u(r) satisfies the scalar Helmholtz equation and, in addition, is constrained by the polarization state of the field. A rigorous development of such a scalar represeatation of the electromagnetic field, given by GREEN and WOLF [ 19531 and WOLF[ 19591, justifies this association (see also LAX, LOUISELL and MCKNIGHT[1975]). This interpretation of the scalar optical field is adopted here.
170
[IV, I 2
UNSTABLE RESONATOR MODES
2.1. PARAXIAL SCALAR WAVE PROPAGATION PHENOMENA IN OPEN OPTICAL CAVITIES
Following the analysis of SIEGMAN[ 19791, consider a generalized passive open cavity that consists simply of an output coupling device M and an optical system that may contain any number of generalized complex on-axis paraxial elements that completely contain the transverse extent of the cavity field. A single round-trip propagation through the cavity may then be represented by the unfolded cavity cell depicted in Fig. 1, where the field begins at some convenient reference plane (usually chosen at the output coupler of the cavity) and, via the basic process of propagation, is transferred by the optical system comprising the remainder* of the cavity back to that same reference plane. This simple model encompasses both standing-wave linear and travelling-wavering cavities. It is assumed here that the output coupler in a standing-wave cavity is described by an arbitrary complex reflection coefficient p(x, y ) in the dimension transverse to the direction of propagation of the cavity mode field. This translates to an arbitrary transmission coefficient in the case of a travellingwave ring cavity. In either case the function p(x, y ) contains arbitrary amplitude and phase variations which are applied to the incident field at that plane. The transverse geometrical extent of the feedback portion of the outcoupling
OUTPUT C 0 UP L E R pcx, Y )
-
OUTPUT COUPLER p (x, Y)
PAR AXIAL OPTICAL SYSTEM
i
l
;
1%
r----------l
I I
I
I
L
REFERENCE PLANE
(3
1 I
I
_ _ _ _ _ _ _ _ _ _ _JI
-
POSITIVE DIRECTION NEGATIVE DIRECTION
REFERENCE PLANE
Fig. 1. Generalized model of an open optical cavity characterized by a single paraxial optical system with at most a single diffracting aperture.
* By remainder it is meant here that the output coupler of the cavity is considered as a distinct element of the cavity separate from the paraxial optical system.
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171
aperture, denoted by d,is defined as the minimum, simply connected closed curve (or set of closed curves) such that I p ( x , y)l is nonzero when (x, y) E d and 1 p(x, y)l = 0 whenever (x, y) $ = A Finally, it should be noticed that the generalized open cavity model considered here may be either of the stable or of the unstable type. There are two possible directions of propagation through the generalized open cavity, as indicated in Fig. 1. For a strictly monochromatic scalar wave field propagating along the positive z direction through the unfolded cavity, in which case the reference plane is situated just in front of the output coupler, the complex field may be written as
where the wave function Q satisfies the homogeneous scalar wave equation. Within the paraxial approximation the complex scalar wave disturbance u(r) satisfies the paraxial Helmholtz equation V:u(r)
+ 2ik-M r ) = 0 , az
where: V denotes the transverse Laplacian operator. The mathematical relation between the complex source and image scalar wave disturbances at the cavity reference plane is then given by the following form of the Fresnel-Kirchhoff diffraction integral (see BORN and WOLF [ 19751):
where the integration is taken over the transverse extent of the cavity feedback aperture. Here c(x, xo;y , yo) is the distance of the total optical propagation path in a single round-trip iteration in the positive direction through the cavity from the source point located at (xo, yo) in the reference plane to the image point located at (x,y) in the same plane. For a strictly monochromatic scalar wave propagating along the negative z direction through the unfolded cavity, in which case the reference plane is situated just after the output coupler-feedback aperture, the complex field must be written as
172
UNSTABLE RESONATOR MODES
[IV, 8 2
where ut(r) satisfies the paraxial Helmholtz equation aut(r) V$ut(r) - 2ik= 0
az
(2.9)
The dagger is used here to distinguish this field structure from that propagating in the positive z direction through the cavity. In this case the Fresnel-Kirchhoff diffraction integral becomes
Again, the integration is taken only over that portion d of the transverse plane of the feedback aperture where p(x, y) is nonzero, but now because the field ut itself is zero exterior to d at this plane. If the reference plane is moved just to the right of the output coupler in Fig. 1, the diffraction integral for the reverse propagating field becomes
(2.11) By uniqueness it then follows that
This expression may be considered as providing the analytical continuation of d.Finally, the quantity Ct(x, x,; y, yo) is the total optical propagation path in a single round-trip iteration in the negative direction through the cavity from the source point (xo, yo) in the reference plane to the image point (x,y) in the same plane. The propagation distances C(x, xo, y, yo)and Ct(x, xo ;y, y o )appearing in the Fresnel-Kirchhoff diffraction integrals (2.7) and (2. lo), respectively, describe the optical path length of an arbitrary ray originating from a source point at (xOryo)and propagating through the optical system (in the appropriate direction) to an image point at (x,y). A general expression for the propagation distance between any two transverse reference planes in a stigmatic, paraxial optical system can be obtained in terms of the paraxial ray transfer matrix between those two planes (GERRARDand BURCH[1975]). The resultant expression for the distances, when substituted in eqs. (2.7) and (2. lo), will then yield a general scalar wave diffraction formulation for the propagated disturbut(x,y) outside of the domain of the aperture
Iv, 5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
173
ance that is expressed in terms of the paraxial ray-transfer matrices that are characteristic of the optical elements of the open cavity. The optical propagation path of an arbitrary paraxial geometrical ray in an optical system is completely specified by the canonical variables ( x , y, u, u ) of the ray at a given transverse plane of the optical system and the ray transfer matrix of that system. Here ( x , ~ )specifies the location of the ray in that transverse plane and (u, u ) are the direction tangents of the ray at that particular point. It is important to notice that the expressions of interest here account for both meridional and skew rays. It is reasonable to assume that each of them is separable in x and y in the manner 5(x, x o ; Y , Yo) = h ( x , xo) + CJY, Yo) - L
9
(2.13)
where L represents the total optical path length along the optical axis common to both Cx(x, xo) and 5,,(y,yo). The separated expressions 5, and 5, are then restricted to meridional rays. The total propagation distance as given by (2.13), however, has allowance for skew rays. Such a separation of the propagation distance will be assumed in the following analysis so that attention will be focused on the expression of each meridional distance. Within the approximation of paraxial Gaussian optics, an output meridional ray vector (yz,n2uz) due to some input meridional ray vector (yl, nlul) is related by a ray transfer matrix that is characteristic of the paraxial optical system separating the input and output planes (GERRARD and BURCH[ 19751, where (2.14)
Here n denotes the (real-valued)index of refraction of the medium in which the ray lies, and the product nu is the optical direction tangent of that ray. An important property of the paraxial ray transfer matrix is that it is unimodular, viz . (2.15)
which is an invariant of any stigmatic paraxial optical system. The inverse ray transfer matrix (for propagation in the reverse direction through the optical system) is then given by (2.16)
174
UNSTABLE RESONATOR MODES
IIV, I 2
The ray transfer matrix for translation between two reference planes separated by the axial distance t in a medium of refractive index n is given by
T=
(A
*:).
(2.17)
For reflection from a spherical surface ofradius R, the paraxial reflection matrix is found to be
R = ( 2nlR O), 1
(2.18)
where n is the refractive index of the medium in which the geometrical ray lies. The radius of curvature of the reflecting surface is taken here to be positive if the surface is convex to the incident ray and is negative ifthe surface is concave. Finally, the paraxial ray transfer matrix for refraction across a spherical interface separating two dielectric media of refractive indices n, and n2 is given bY
R=(
-9
O), 1
(2.19)
where
(2.20) is the refracting power of the spherical surface with the incident ray in the medium of index n, . The same sign convention for the radius R still applies. The paraxial optical characteristics of any given symmetrical optical system may then be constructed from these three basic ray transfer matrices. If the matrices M I,M,, ...,M, represent, in order, the various translation, reflection, and refraction operations that occur in propagation from reference plane 1 to reference plane n through a given optical system, the overall ray transfer matrix characteristicof the optical system between these two reference planes is given by M
=
M,M, -
,
* *
M,M2Ml
.
(2.21)
An extension of this matrix formalism to take into account translational and angular misalignment of the optical elements may be obtained through the and BURCH introduction of an augmented ray transfer matrix (GERRARD
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GENERAL FORMULATION O F THE TRANSVERSE MODE STRUCTURE PROPERTIES
175
[1975], OUGHSTUN[1981a, 1982~1).A more general extension to nonorthogonal optical systems is given by ARNAUD[ 19701. As a simple application of the paraxial ray transfer matrix formalism, consider a spherical wave diverging from an axial point source 0. Associated with the diverging spherical wave phase front is a diverging family of paraxial rays (a ray pencil). At a transverse plane a distance R from 0 along the axis, a typical paraxial ray of that family has canonical variables (y, nu) with u = y/R. Consequently, for all members of this paraxial family of rays, the radius of curvature of the spherical phase front associated with them at a particular transverse reference plane is given by the ratio (2.22)
A positive value of R indicates a diverging phase front, whereas a negative value indicates a converging phase front. It is convenient to specify the radius of curvature of a spherical phase front by its reduced value (2.23) where nu is the optical direction tangent of the associated ray. The advantage of using this reduced value is that whenever a ray crosses a planar interface separating two optical media, there is no change in the optical direction tangent of the ray, since by the paraxial form of the law of refraction, nlul = n2u2. Hence, there is no change in the reduced radius of the spherical phase front. Consider now the propagation of a spherical wave through a stigmatic optical system that is characterized by the paraxial ray transfer matrix ($ :). The canonical variables of the family of rays associated with the wavefront emerging from the optical system are then given by Y , = AY, + B f i I U l
nzuz = Cy,
7
+ Dn,u, .
(2.24)
On dividing the first equation by the second and using eq. (2.23), there results r,
=
-1 B ____
Ar,
Cr,
+D ’
(2.25)
where r , is the reduced radius of curvature of the spherical wavefront incident upon the optical system and r, is that exiting from the optical system. This simple result is commonly referred to in the literature as the ABCD rule.
176
UNSTABLE RESONATOR MODES
. .
. ...*
PARAXIAL OPTICAL SYSTEM
l--i,--+---
L
_I_
r2
----I
Fig. 2. Propagation of a spherical wave through a stigmatic optical system.
An expression for the meridional propagation distance C,,b2, y l ) describing the total optical propagation length of a paraxial meridional ray through a stigmatic optical system may now be obtained. Let the paraxial ray transfer matrix for the optical system between an input transverse reference plane Pl and an output transverse reference plane P2be given, as shown in Fig. 2. The canonical variables for the input and output meridional rays are then related by eq. (2.24), which may be solved for the optical direction tangents nlul and n2u2 independent of each other, giving
nlul
=
Y2 - AYI ----, B
(2.26) The axial intersection point 0, of the input ray is then located at the reduced distance r1=---
y1 -
BY1 Y2
nlul
(2.27a)
- AYI
from the input reference plane PI,and the axial intersection point 0, of the associated output ray is located at the reduced distance r 2 = - y2 n2u2
-
BY2 DY2 - Y1
(2.27b)
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GENERAL FORMULATIONOF THE TRANSVERSE MODE STRUCTURE PROPERTIES
177
from the output reference plane 9. These input and output rays may be considered members of a family of meridional paraxial rays diverging from a source point 0, in object space and converging to an image point 0, in image space, as indicated in Fig. 2. By the theorem of Malus and Dupin (see BORN and WOLF [1975], GERRARD and BURCH [1975]) the optical path length measured along all of the possible paraxial rays from an object to an image point in any stigmatic imaging system is the same. Accordingly, a spherical phase front emanating from 0, generates another spherical phase front centered on O2 after propagation through the optical system. Let L denote the total optical path length along the optical axis from reference plane 9,to reference plane P2,where L
=
92
n(z) dz ,
(2.28)
91
with n(z) denoting the index of refraction distribution of the optical system along the optical axis. From Fig. 2 the meridional propagation distance is seen to be given by L plus the sum of the sagittae of the input and output phase fronts at the reference planes 9, and P2,so that
Y2
Yf
1,(Y2,Y,) = L - -L + 2r, 2r2
= L + AY: + DY2 - 2YlY2 2B
9
(2.29)
where eqs. (2.27a) and (2.27b) have been employed. The expression (2.13) for the total propagation distance in the positive direction through the paraxial optical system from a point (xo,yo) in the to a point ( x , y ) in the reference plane 9?!is then gven by reference plane 9,
(2.30) where allowance has been made for the possibility that the meridional ray transfer matrices in the x and directions may differ. In a similar manner the total distance of propagation in the reverse (or negative) direction through the to a point ( x l ,y,) optical system from a point ( x o , y o )in the reference plane 9??
178
UNSTABLE RESONATOR MODES
in the reference plane 9, is found to be 5+(x,X O , Y , Y O )
=
-L -
D,x;
+ A,x’ - 2 x 0 ~ 2BX (2.3 1)
With these results the Fresnel-Kirchhoff diffraction integrals for propagation in the positive and negative directions through the paraxial optical system depicted in Fig. 1, as given in eqs. (2.7) and ( 2 .lo), are now completely specified in terms of the ray transfer matrix of that system. In the usual manner (see BORN and WOLF [ 19751) the entire expression for the propagation distance must be employed in the exponent of the integrand of the diffraction integral, since the quantity exp(ik((x, x,, ;y , y o ) )will oscillate rapidly over the transverse domain ofthe aperture A However, some effective value ce,may be substituted in the denominator of the integrand, since the variation of [ itself over the transverse domain of d is very small compared with the axial propagation distance for the cavity geometries of interest here. With these substitutions eq. (2.7) becomes 1
u(x, y ) = - __ eikL 15eR
The appropriate value of Ceff to be employed in this expression is determined by the requirement that the boundary value
must be reobtained from this diffraction integral in the limit as B, + 0 and By + 0 together such that L --+ 0 and A x = D , -+ 1 and A , = Dy 4 1 consistent with the Lagrange invariant of the paraxial optical system. In this limit the stationary phase asymptotic approximation of the preceding integral is found to be
IV,8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
179
so that
id= J B P y
*
With this result the Fresnel-Kirchhoff diffraction integral for propagation in the positive direction through the optical system becomes
K
x exp {in
[
A , x ~+ D,x2 - 2 ~ x 0 Bx (2.32)
In a similar manner the diffraction integral for propagation in the negative direction through the optical system is found to be
a
(2.33)
where the fact that if = has been employed in obtaining this result. These particular forms of the Fresnel-Kirchhoff diffraction integral for scalar wave propagation phenomena in the positive (or forward) and negative (or reverse) directions through a given optical system provide the basis upon which the remaining analysis of the transverse mode structure properties of unstable optical cavities is drawn. In this formalism the nondiffracting stigmatic imaging system of the cavity is completely accounted for in the diffraction kernel. A similar analysis has been given by BAUES [ 19691, COLLINS[ 19701, and ARNAUD[ 1971, 19731.
180
UNSTABLE RESONATOR MODES
[IV, § 2
2.2. CANONICAL FORMULATION OF UNSTABLE CAVITY MODES
For an open optical cavity, eqs. (2.32) and (2.33) can be viewed as expressing the integral relationships between the initial transverse scalar wave field disturbances uo(xo,y o ) and u$(xo,yo) at the feedback aperture reference plane of Fig. 1 and the corresponding resultant transverse scalar wave field disturbances u(x, y ) and ut(x, y) incident back upon that same reference plane after one complete round-trip propagation through the cavity in the positive and negative directions, respectively. The only edge-diffraction phenomenon occurring in that single iteration is that resulting from the feedback aperture d of the cavity. For a self-reproducing transverse mode to be set up in the cavity, each of the transverse field distributions incident back upon the reference plane must be linearly related to each corresponding initial field disturbance at that plane (Fox and LI [ 19611). Hence, in the positive direction
4%Y ) = 7u&,
Y)
(2.34a)
9
whereas in the negative direction (2.34b)
J(x, Y ) = 7tut;(x, Y ) *
With eq. (2.34a), eq. (2.32) becomes
xexp{ii[71 A x X'0 + D X x z- 2 ~ x 0 BX
(2.35) whereas with eq. (2.34b), eq. (2.33) becomes
(2.36)
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GENERAL FORMULATION O F THE TRANSVERSE MODE STRUCTURE PROPERTIES
181
where the superfluous 0 subscript on the complex field amplitudes has been replaced by a mode index n that labels both the complex field eigenfunction and its associated eigenvalue 7, (or 7;). Equations (2.35) and (2.36) constitute a canonical formulation of the transverse eigenstructure supported by an open optical cavity in the positive and negative directions, respectively, when there is at most a single diffracting aperture within the cavity. These equations are equally applicable to both stable and unstable cavity geometries. 2.2.1. Geometricalproperties
The distinction between stable and unstable cavity geometries is most readily obtained from a geometrical analysis of the cavity mode field (KAHN [ 19661, SIEGMAN [ 19761). With attention restricted to a single meridional plane of the cavity, a given input spherical phase front of reduced radius r, = R,/n results in a spherical phase front of reduced radius r = R/n given by [cf. eq. (2.25)] 1 --
Dlr,
+C
r
Blr,
+A
(2.37)
after a single round-trip propagation through the cavity in the meridional plane described by the paraxial ray transfer matrix :( If the ray transfer matrix in the orthogonal meridional plane is different, the resultant reduced radius in that plane will then differ from that given in eq. (2.37). In that case the resultant wave front will, after a single iteration, be astigmatic when the initial wave front is spherical, and the cavity mode properties in these two meridional planes will be different. The geometrical eigenmode phase front produced by an optical cavity in a particular meridional plane is that wave front whose curvature is unchanged after a single round-trip propagation through the cavity, so that r, = r in that particular meridional plane. From eq. (2.37) this reduced curvature eigensolution is found to be given by
i).
(2.38)
where the optical invariant relation AD - BC = 1 has been employed. This geometrical solution for the phase front of the cavity mode describes the global behavior of the daractive mode phase structure incident upon the reference plane in Fig. I in both the positive and negative directions of propagation through the cavity.
182
UNSTABLE RESONATOR MODES
[IV, J 2
The branch type of the cavity is characterized by the stability of the iterated paraxial ray transfer matrix of the cavity optical system (KOGELNIKand LI [ 1966a1, SIEGMAN [ 19761).The n-fold iterated ray transfer matrix of the open cavity is given by the nth power of the paraxial ray transfer matrix of the cavity optical system, which may be evaluated by means of Sylvester’s theorem (see GERRARD and BURCH[ 19751) to give
where cose=;(A + D ) = g
(2.40)
is called the g-parameter of the cavity. Notice that the g-parameter is the same in both the positive and negative directions of propagation through the cavity. Such iterated sequences are classified as either stable or unstable according to the value of the trace ( A + D) = 2g of the system matrix. A stable system occurs when the cavity g-parameter satisfies the inequality -l
(2.41a)
In that case 8 is real and ray eigenvector solutions exist which are trapped inside the cavity. From eq. (2.38) it is seen that such stable solutions have complex phase curvatures. An unstable system satisfies the inequality (2.41b) in which case the phase curvature given by eq. (2.38) is real. For / g /> 1 the trigonometric functions appearing in eq. (2.39) become hyperbolic functions, and ray eigenvector solutions do not exist that are trapped inside the cavity. The case I gj zz1 corresponds to a marginally stable system for which the cavity mode structure properties are in a transition between a stable and an unstable system. For lgl = 1, 8 is real, which is characteristic of a stable open cavity, whereas g satisfies the inequality (2.41b) for unstable operation. The geometrical magnification M for the self-consistent wave-front solution of an unstable open cavity is specified by the pair of equations y = Ay,
+ Bnu,
=
My,,
(2.42a)
IV, 21
nu
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
= Cy,
+ Dnv,
=
Mn, .
183
(2.42b)
That is, both the displacement and slope of each member of the family of rays associated with the self-consistent wavefront solution of the cavity are magnified by the same factor M after a single round-trip propagation through the unstable cavity. The solution of the equation pair (2.42) for M in terms of the ray transfer matrix elements alone is found to be (2.43a) where the relation AD - BC = 1 has been employed. The two different magnilications specified by (2.43a) are Mp = g
M,,
=
+ Jg2
g-
-
1,
d m ,
(2.43b) (2.43~)
and are related by MpMn= 1 .
(2.43d)
Dependent on the sign of the cavity g-parameter, one or the other of the two solutions M , and M,, corresponds to the geometrically expanding unstable cavity mode solution with lMl > 1. The other solution then corresponds with a demagnifyingreversed form of the same geometrical wave, with a geometrical demagnification of Il/MI on each iteration through the optical cavity. One is then left with two possible choices for the cavity mode magnification in both the positive and negative directions of propagation through the unstable cavity. It is a natural choice to specify that IMl > 1 in the positive direction through the unstable cavity. Depending on which of the two magnification factors M , or Mn corresponds to the geometrical magnification M, one has a generalized equivalent of either a positive-branch or a negative-branch unstable cavity (KRUPKEand SOOY[ 19691). For a positive-branch unstable cavity a given meridional ray associated with the self-consistent wave-front solution of the cavity emerges with a magnified displacement and slope on the same side of the optical axis after a single round-trip iteration. This implies that the cavity mode has either none or an even number of internal foci in a single round-trip propagation through the cavity. The appropriate geometrical solution is then given by (2.44a)
184
UNSTABLE RESONATOR MODES
A4 = Mp = g + Jp >1,
[IV, 8 2
(2.44b) (2.44~)
For a negative-branch unstable cavity, on the other hand, a given meridional ray associated with the self-consistent wave-front solution of the cavity emerges with a magnified displacement and slope on the opposite side of the optical axis after a single round-trip iteration. This implies that the cavity mode has an odd number of internal foci in a single round-trip propagation through the cavity. The appropriate geometrical solution is then given by g=-
M
A + D
=
2 M,
=
(2.45a)
<-1, g-
,/-
< - 1,
(2.45b) (2.4%)
Similar remarks apply in the negative direction of propagation through the optical cavity. 2.2.2. Diflractive properties For an unstable optical cavity with a rotationally symmetrical optical system the mode structure in the positive direction is given by (cf. eq. [2.35])
[A(x$ + y $ ) + D(x2 + y’) - 2(x0x + yOy)] (2.46) where the finite extent of the feedback aperture is accounted for by the complex reflection coefficient p(xo, yo). The transverse field distribution appearing in this integral equation possesses a globally spherical phase front with radius given either by eq. (2.44~)for a positive-branch cavity or by eq. (2.45~)for a negative-branch cavity. In order to reduce this integral equation to its simplest form, one first extracts this spherical phase variation from the cavity field
Iv, 8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
185
(SIEGMAN[ 19761). Hence, let (2.47)
u,(x, Y ) = un(x,y ) exp
with a similar expression for un(x0,yo). Substitution into eq. (2.46) then yields 1
7,un(x, y ) = - -eikL
P(Xo 9 Y o ) un(xo ,Yo)
AB
+ Since (A
+ B/r)= M
(
3
D - - (x'
+ y')
- 2(xox + yO y) ]}dx, dy,
and ( D - B / r ) = 1 / M , one then obtains
{
x exp i
7T ~
MBR
[ ( x - Nx,)'
+ 0,- My,)']
This form of the integral equation for the modified transverse mode structure vn(x,y) involves only the cavity magnification M, the equivalent propagation distance B that is specified by the paraxial ray transfer matrix of the cavity optical system, and the output coupler-feedback aperture reflection coefficient p(xo, yo). The constant phase shift kL may be absorbed into the complex eigenvalue yn. Under the change of variables x; = MXo ,y; = M y , , where M is negative for a negative-branch cavity, the preceding integral equation becomes
{
x exp i
7T ~
MBI
[ ( x - x ; ) ~+
0,- y;)']
dxh dy; ,
(2.49)
where the upper sign choice corresponds to a positive-branch cavity (M > 1) and the lower sign choice corresponds to a negative-branch cavity (M c - 1). This form of the cavity diffraction integral embodies Siegman's canonical
186
[IV, 8 2
UNSTABLE RESONATOR MODES
model for a single round-trip propagation through an arbitrary unstable cavity with a single diffractingaperture (SIEGMAN [ 19761). According to this model a single round-trip propagation of the cavity field through the generalized resonator system in the positive direction may be synthesized in the following manner, as depicted schematically in Fig. 3. First, the modified cavity field u(x, y ) incident upon the reference plane just in front of the cavity feedback aperture is multiplied by the feedback reflection coefficient p(x, y ) . This field is then magnified outward to produce the field IMI-'p(x/M,y/M) u(x/M,y/M), where M is the cavity magnification. The resulting field is then propagated over the equivalent free-space distance MB back to the same reference plane. This free space propagation may be numerically computed [ 19681, SHERMAN using the plane wave spectrum representation (GOODMAN [ 19821)with afast Fourier transform algorithm (SZIKLAS and SIEGMAN [ 1974, 19751). A description of this numerical approach and its associated sampling requirements in both Cartesian and polar cylindrical coordinates is given in the Appendix. A repeated Fox-Li type of numerical application of this canonical propagation model will then yield the cavity mode structure if the process converges. This solution will then satisfy the integral equation given in (2.49). The field distribution outcoupled from the cavity is then obtained by multiplying the
OUTPUT COUPLERFEEDBACK APERTURE pcx, Y )
1
: .
REFERENCE
pcx,
Y)
:/ 1
, '
Fig. 3. Equivalent collimated propagation geometry through a generalized unstable cavity system in the positive direction.
Iv, f 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
187
modified cavity field u(x,y) incident upon the reference plane by the transmission function of the output coupler and applying eq. (2.47) to reintroduce the proper geometrical phase curvature. A similar analysis may also be applied to the mode structure that propagates in the negative direction through the cavity. In general, a hierarchy of transverse modes un(x,y) is found to satisfy the linear homogeneous Fredholm integral equation (2.49), each with its own associated complex eigenvalue Tn (SIEGMANand MILLER[ 19701, O’NEIL, HEDINand FORGHAM[ 19771). The magnitude yn = I Y n / of the series of eigenvalues of the set of transverse eigenmodes supported by an optical cavity may always be ordered such that y1>
y2
2
y3
2 *.
. .
Each eigenvalue is physically related to the fractional power loss of its corresponding cavity eigenmode. The total flux of the cavity mode u,(x, y ) fed back into the cavity at the feedback aperture is given by
and the total flux of that same cavity mode field fed back into the cavity after a single round-trip propagation through the cavity is given by
pi’’=
Jim 1s
IP(x,y)~i2’(x,y)I2dxdy
-00
00
=
IP(X,Y) ~i0’(X9Y)IZ dxdy,
--oo
where the relationship (2.34a) has been applied. The fractional power loss associated with that cavity mode is then seen to be
Hence, in the ordering given above, yl is the dominant (or first) eigenvalue associated with the lowest loss spatial cavity mode, y2 is the second eigenvalue associated with the first higher-loss, higher-order cavity mode, and so on.
188
UNSTABLE RESONATOR MODES
[IV, § 2
2.2.3. Collimated and equivalent Fresnel numbers The specific cavity eigenstructure described by eq. (2.49) depends on the functional form of the feedback reflection coefficient p(x,y ) . Of particular interest is the case of a circular feedback aperture of diameter 2a, with unity reflectivity within the aperture and zero outside, so that (2.5 1) Although such a step discontinuity is impossible to realize fully in experiment, a vast majority of resonator analyses assumes such a Kirchhoff boundary condition. Under the change of variables
5 = xlMa,,
rl = y/Ma, 3
to = xb/Ma,
rlo = yb/Ma, ,
(2.52)
the integral equation (2.49) becomes, with (2.5 l),
where the integration is taken over the unit circle C. The parameter N, appearing in this equation is defined as (2.54) and is known as the collimated Fresnel number of the cavity. Notice that it carries the same sign as does the cavity magnification M . This particular form of the integral equation for the modified mode structure then involves only the geometrical magnification and the collimated Fresnel number of the unstable cavity. As a consequence, it is well suited for a variety of parametric studies regarding the modified eigenstructure and eigenvalue behavior of passive unstable cavities. The collimated Fresnel number N, physically represents the number of Fresnel half-period zones of a plane wave filling the magnified feedback aperture as viewed from the center of the reference plane one iteration removed (and, hence, separated by the equivalent collimated distance MB). The col-
IV, Q 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
189
h a t e d Fresnel number of an unstable cavity then describes the diffractive phenomena that occur in a single iteration through the optical cavity. As a consequence, it is the parameter upon which the sampling criteria for the numerical evaluation of the transverse mode structure are based (see the Appendix). However, the collimated Fresnel number does not describe the Fresnel zone structure of the cavity mode field distribution, since this is the result of many iterations through the optical cavity. A parameter that does describe the Fresnel zone structure of the cavity mode field needs to be defined in terms of the number of Fresnel half-period zones present in the geometrical mode phase front at the feedback aperture as viewed from the center of the reference plane one iteration away (SIEGMANand ARRATHOON [ 19671). To that end the equivalent Fresnel number Neq of an unstable cavity is defined such that the quantity Neq1/2 is equal to the sagittal distance between the geometrical mode phase front and the edge of the cavity feedback mirror (ANAN’EV,SVENTSITSKAYA and SHERSTOBITOV [ 19691, ANAN’EV [ 1971,19721).For the present canonical model of an unstable cavity, the quantity N e q l is then seen to be equal to the sagittal distance between the expanding geometrical mode phase front and the corresponding converging wave phase front [whose radius is given by the other sign choice in eq. (2.38)] at the edge of the cavity feedback aperture (ANAN’EV [ 19721). Hence, within the paraxial approximation
where r , is given by eq. (2.38). With these substitutions and eqs. (2.44) and (2.45) one obtains the following relation due to SIEGMAN(1976): M2 - 1
Neq= ___ Nc . 2M2
(2.55)
The physical importance of the equivalent Fresnel number in describing the diffractive properties of unstable cavity modes may be understood through a consideration of the edge diffraction effects at the cavity feedback aperture (ANAN’EV, VINOKUROV, KOVAL’CHUK, SVENTSITSKAYA and SHERSTOBITOV [ 19701, ANAN’EV and SHERSTOBITOV [ 19711, ANAN’EV [ 19721). Diffraction at the cavity feedback aperture produces not only a reflected feedback field but also an edge scattered field whose virtual 5ource is the edge of the feedback aperture. A portion of this scattered radiation gives rise to a converging wave whose self-consistent radius of curvature satisfies eq. (2.38). The radiation
190
UNSTABLE RESONATOR MODES
[IV, $ 2
scattered along other directions rapidly escapes from the unstable open cavity and is of little further concern here. For a positive-branch cavity the reduced geometrical radius of curvature of the magnifying cavity field is r + and that of the associated converging wave is r - ,whereas for a negative-branch cavity the reduced geometrical radius of curvature of the magnifying cavity field is r - and that of the associated converging wave is r + . The radiation in the converging wave is nearly completely retained within the cavity even after several roundtrip propagations, whereas the flux of the geometrically magnifying field decreases by the factor 1 - y: after each roundtrip propagation. Consequently, the intensity of the converging wave, which is negligible near the edge of the cavity feedback aperture, is geometrically amplified as it propagates inward toward the cavity optical axis to such an extent that it significantly influences the entire structure of the cavity field. It is this mechanism of dsractive feedback, along with its interaction with the magnifying cavity field, that determines the diffracthe mode properties of an unstable cavity. All of this is contained quite naturally in the integral eq. (2.53). The radiation in the diverging cavity field that is incident upon the edge of the feedback aperture and then scatters into the converging wave travels a distance NeqAbetween these two wavefronts. Hence, when the equivalent Fresnel number changes by unity, the phase shift between the diverging and converging waves changes by 2n and the interaction between these two waves is essentially unchanged. As a consequence, the diffracthe properties of unstable resonators with sharply defined feedback apertures will be quasiperiodic with respect to the equivalent Fresnel number of the cavity. Furthermore, the dsractive properties of unstable resonators will be critically dependent on the shape of the feedback aperture edge, which gives rise to these edge-scattering effects. Finally, notice that the equivalent Fresnel number of an unstable cavity does not approach the collimated Fresnel number as the magnitude of the cavity magnification approaches unity from above and the unstable geometry approaches the boundary of the stable region. As a consequence, the equivalent Fresnel number is a useful concept only in the unstable cavity regime.
2.3. TRANSVERSE MODE ORTHOGONALITY IN OPEN CAVITIES
Fundamental to the eigenstructure properties are the orthogonality relations satisfied by the set of transverse eigenmodes capable of being supported by an optical cavity. A mathematical development of the general orthogonality properties for open optical cavities is presented in this section. This analysis
Iv, 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
191
is partly based upon work of SIEGMAN[ 19791, WRIGHTand FIRTH[ 19821, and OUGHSTUN[ 1982bl.
2.3.1. Transverse mode orthogonality in optical cavities with a single diflracting aperture Following the analysis of SIEGMAN [ 19791, consider the generalized passive open cavity depicted in Fig. 1, which may be either of the stable or unstable type. Let there be at most a single diffracting aperture d within the cavity, which in the present model, is located in the plane of the output coupler. For an unstable cavity this aperture corresponds to the diffractive output couplerfeedback aperture of the resonator. For a stable cavity, however, the output is usually derived through a partially transmitting reflector, which may be of sufficient size so as to completely contain the transverse extent of the cavity mode field. The complex reflection coefficient of this element is denoted by P(X, Y ) . From eq. (2.35) the integral equation describing the transverse eigenstructure of this generalized optical cavity at the specified reference plane for propagation in the positive direction through the cavity is y,pn(x,y) = --
i
AB
eikL
[A(xz
+ y ; ) + D ( x 2 + y’) - ~ ( x x ,+ yy,)] (2.56)
where it has been tacitly assumed that the paraxial optical system of the cavity is rotationally symmetrical about the optical axis. The shape of the aperture d, however, is left arbitrary. The integration here extends over only that portion d of the transverse plane of the output coupler where the feedback reflection coefficientp(x, y ) is nonzero. The integral equation describing the transverse eigenstructure at the same reference plane for propagation in the negative direction is, from eq. (2.36),
[D(xi
+ y g ) + A(x’ + y’) - ~ ( x x ,+ yy,)] (2.57)
192
[IV, I 2
UNSTABLE RESONATOR MODES
Again the integration is taken over only that portion d of the transverse plane of the output coupler where p(x, y) is nonzero but now because the field u: itself is zero outside of d at this particular reference plane. The analytical continuation outside the domain of d of the field u i incident upon the output coupler is given in eq. (2.12). Notice that the transpose of the kernel in eq. (2.56) is simply that in eq. (2.57), so that these integral equations for the counterpropagating cavity modes form a conjugate pair. The integral operator of eq. (2.57) is then the adjoint of that in eq. (2.56), as are their kernels. However, these kernels are not Hermitian, as can easily be seen (e.g., STREIFERand GAMO[ 19741). Consider now defining the pair of modified eigenfunctions
Un(x, y ) = a,’/’p’/’(x, y) u,(x, y ) exp U,?(x,y ) = a;
‘l’p-
”’(x, y) uL(x, y) exp
1
- D)(x’ + y 2 ) , (2.58a) - D)(x’
J
+ y’) , (2.58b)
valid over the domain (x, y) E d,where an is an arbitrary scaling factor. This definition is, in essence, a shift in the reference plane to an imaginary midplane of the output coupler together with the extraction (or application) of a spherical phase front with reduced curvature 1/r given by the average of the curvatures of the diverging and converging geometrical phase fronts in eq. (2.38). Substitution of eq. (2.58a) into (2.56) then yields
{
x exp i -[(A 2n”B
1
+ D)(x’ + xg + y’ + yg) - 4(xx0 + yyO)] dx,
dy, ,
and substitution of eq. (2.58b) into (2.57) yields -
jji ~ ; ( x ,y ) = 2e i a AB
{
x exp i
~
2B ;
[(A
+ D)(x’ + xg + y’ + yg) - 4(xx0 + yy,)]
1
dx, dy, .
These two integral equations are of the same identical form, which may be
IV, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE S T R U ~ R E PROPERTIES
193
written as 7,Un(x,Y) = ~ j d u n ( x o , Y 0 ) K ( x Jx0,Yo)dxo ; dY0
(2.59)
with or without the superscript +, where
[ ( A + D ) ( x 2 + y 2 + xz + y;) - 4(xx0 + yyO)l
(2.60) is the symmetrized kernel of the transverse mode integral equation. As a consequence of uniqueness, the modified transverse mode eigensolutions in the positive and negative directions of propagation through the optical cavity are identical, that is, U,(X,Y)
7,
=
=
U,(X,Y)
(2.61)
9
yn.
(2.62)
With this identification attention is now turned to the orthogonality properties satisfied by these modified eigenfunctions. Upon forming the overlap integral of U,(x, y ) with U,(x, y) over the feedback aperture d of the output coupler, one obtains [with eq. (2.59)]
jjd
u n (x, y ) u r n(x,
y ) dx dy
194
UNSTABLE RESONATOR MODES
[IV, § 2
With this result and eq. (2.61), and with a suitable choice of the scaling constants for these modified modes, the two sets of functions { U,} and { Ui} are found to satisfy the following orthogonality relations (assuming that these are nondegenerate modes): (2.63a) n
m
(2.63b)
(2.63~) From these results it then follows that no further distinction need be made between the elements of the two sets { U,(x, y ) } and { V i ( x ,y)} of the modified eigenfunctions of the cavity. From eqs. (2.58) and (2.63) the resonator eigenmodes u, and ui are found to be bi-orthonormal in the sense
(2.64) where the second relation holds, since the field u L ( x , y ) identically vanishes outside of the feedback aperture d at this particular reference plane. This is the basic orthogonality relationship satisfied by the eigenmodes of the cavity, first obtained by SIEGMAN [ 19791. Each separate set of transverse eigenmodes {u,} and {u:} is then found to satisfy the following modified orthogonahty relations :
(2.65)
Iv, § 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
=
c(,6,
.
195
(2.66)
In a travelling-wave ring resonator the two sets of transverse modes {u,} and {unf} represent the counterpropagating travelling wave fields supported by the cavity that are, in the absence of any generalized scattering sources, uncoupled from each other. These two sets of oppositely directed mode fields do, however, form a bi-orthonormal set in the sense of eq. (2.64). Furthermore, from eq. (2.58) they are simply related by
(2.67) In addition, the associated losses of these counterpropagating fields are identical. The experimental measurements of the counterpropagating mode losses in an unstable ring resonator by Pozzo, POLLONI,SVELTOand ZARAGA [ 19731 and FREIBERG, CHENAUSKY and BUCZEK[ 19741 confirm this important result. On the other hand, in a standing-wave cavity the two sets of transverse modes {u,} and {ui} represent the counterpropagating travelling wave components whose superposition yields the standing-wave cavity field. These two sets of eigenmodes represent the same set of transverse eigenmodes of the cavity in the sense that if u, is the cavity mode field incident upon the feedback aperture of the cavity, then uL is derived from that same cavity mode field just after it has been fed back into the cavity, and is given by eq. (2.67). Finally, the transverse overlap integral appearing in the fundamental bi-orthogonality relation
(2.68) between the oppositely directed cavity mode fields is invariant along the cavity
196
UNSTABLE RESONATOR MODES
[IV, § 2
optical axis (ARNAUD [ 19761). Hence, this bi-orthogonality relation is satisfied at all transverse reference planes within the cavity (WRIGHT,O’BRIENand FIRTH[ 19841).
2.3.2. Transverse mode orthogonality and reciprocity in multi-aperture optical cavities It is intuitively obvious that the introduction of an additional diffracting aperture within an unstable cavity can have a significant influence upon the transverse mode structure behavior. In particular, it was thought that a judicious placement of an additional sharp-edged diffracting aperture in a negative-branch travelling-wave ring resonator would serve to increase the diffraction losses dramatically in one direction while only slightly altering that in the opposite, preferred direction, thereby yielding a unidirectional ring laser (ANAN’EV,SVENTSITSKAYA and SHERSTOBITOV [ 19691, ANAN’EV[ 19721, BUCZEK,FREIBERG and CHENAUSKY [ 19731, Pozzo, POLLONI, SVELTO and ZARAGA[ 1973,19741, SIEGMAN [ 19761, SHENG[ 19791, OUGHSTUN [ 1982b], OUGHSTUN, SLAYMAKER and BUSH[ 19831, RIGROD[ 19831). However, the experimental work of Pozzo, POLLONI,SVELTOand ZARAGA[ 1973, 19741 and the later numerical analysis of OUGHSTUN, SLAYMAKER and BUSH[ 19831 showed that the resulting difference in the losses between the counterpropagating modes was insufficient by itself to produce unidirectional oscillation, as was first proposed. Nearly perfect unidirectional oscillation in a ring resonator has been achieved by using an external retro-reflection mirror to feed the outcoupled reverse wave back into the intracavity forward wave field (see BUCZEK, FREIBERGand CHENAUSKY[1973], and MAK, USTYUGOV, FROMZEL’ and KHALEEV [ 19741). The recent analysis of WRIGHTand FIRTH [ 19821 has proved that such directional discrimination between the counterpropagating modes in a multi-aperture ring resonator is not possible (see also WRIGHT,O’BRIENand FIRTH[ 19841, and OUGHSTUN[ 19841). This important proof is now considered in detail; a portion of this analysis is based on correspondence between SIEGMAN[ 19821, HARDY[ 19831, WRIGHTand FIRTH[ 19831, and OUGHSTUN[ 1983bl. Consider, then, a generalized passive optical cavity that consists simply of a diflractive output coupler with feedback aperture d,, an optical system that may contain any number of on-axis paraxid elements that completely contain the transverse extent of the cavity mode field distribution, and an additional diffracting aperture d2distinct from d l .A single round-trip propagation through the cavity may then be represented by the unfolded iterative cavity cell
Iv, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
DIFFRACTIVE OUTPUT COUPLER
DlFFRACTlVE OUTPUT COUPLER
APERTURE
i'
1 ; c: ::) -t+ f ------___ r----------
L
i' 1
Al
r---------
I
I I
I
I
I
+ I
J
L
1
A2 0 2 (C,
D2)
---_____
PAR AXIAL OPTICAL SYSTEM
I+ .1
PAR AXIAL OPTICAL SYSTEM POSITIVE DIRECTION
~
197
.
NEGATIVE DIRECTION
Fig. 4. Generalized model of an open optical cavity containing two distinct diffracting apertures separated by two paraxial optical systems.
depicted in Fig. 4. Here, the cavity field begins at some convenient reference plane (chosen at the plane of the output coupler) and is propagated through the paraxial optical system comprising the cavity first to the internal diffracting aperture d2and then back to that same reference plane. This simple model then encompasses both standing-wave* and travelling-wave resonators. For purposes of convenience it is assumed here that the paraxial optical system of the cavity is rotationally symmetrical about the cavity optical axis. The transverse geometries of the apertures dland d2are, however, left arbitrary. Unless otherwise stated, these two apertures are assumed to have unit transmission (or reflection) coefficients with finite support (i.e., they are hard apertures). In the positive direction of propagation through the open cavity, the optical system is described by the two paraxial ray transfer matrices
as indicated in Fig. 4, where the internal diffracting aperture d2lies between these two imaging systems. If the aperture d2is allowed to go to infinity, the
* Due to the symmetry implied by a standing-wave cavity and the fact that only a single additional diffracting aperture is considered in the present model, it follows that this aperture must be located at the so-called back mirror of the cavity so that it is encountered only once in a single pass through the cavity.
198
UNSTABLE RESONATOR MODES
overall ray transfer matrix for the cavity is then given by
(2.69) In the negative direction of propagation through the cavity cell, the optical system is described by the two paraxial ray transfer matrices
')I:-
A2 B2
Bl (C, D , ) - ' = ( - Z : (C, D I P ' = ( - : The overall ray transfer matrix in this direction is then given by
("c
*): -
A1
A1 Bl
;)-'=(Cl
D ] - ' C
C2B, + D2D1 -A2B1 - B2D1 (2.70) - C d 1 - D2Cl A d , + B2Cl The linear homogeneous integral equation describing the passive cavity mode structure properties in the positive direction may simply be written as (OUGHSTUN[ 1982bl; see also KUSHNIR[ 19801)
=(
.
7nu,(x,Y)
r r
J J4 + A X ~ , Y ~ ) IB2 1
= --
[A2(x:
r i1
+ y:) + D2(x2 + y 2 ) - 2(xx2 + yy,)]
I
dx, dy, ,
I
(2.71a)
x exp i -[Adx: + Y:) + Dl(x: + A )- 2(X,X, + Y2Yl)l dx, dYl9 (2.7 1b)
where $,(x2, y 2 ) is the transverse mode field distribution incident upon the aperture plane of d2,and u,(x, y) is that transverse mode field distribution incident upon the outcoupling aperture plane. Substitution of eq. (2.71b) into (2.7 la) then yields
Iv, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
199
where the kernel function
-2[(5+*)x,+ B , Bz
(".'>yz]})dxzdy2 B , Bz
(2.73)
is proportional to the impulse response of the diffracting aperture d2in the positive direction. The integral appearing in eq. (2.72) is then seen to be the cross-correlation of the feedback aperture field with the impulse response of the intracavity aperture. In the limit as the diffracting aperture dzgoes to infinity, the impulse response function (2.73) becomes
(2.74) where the identifications given in eq. (2.69) have been employed. Substitution of this result into eq. (2.72) results in the usual expression for the cavity mode structure when no other additional diffracting apertures are present in the cavity [cf. eq. (2.56)]. The integral equation describing the passive cavity mode structure properties in the negative direction may be written
+ AAx: +
A ) - 2(x,xi
1
+ Y Z V ~ dxi )~ d ~ i 9
(2.75b)
where t,bL(x,, y z ) is the transverse mode field distribution incident upon the aperture plane of dzin the negative direction, and uL(x, y ) is that transverse mode field distribution incident upon the outcoupling aperture plane. Substitu-
200
[IV, 8 2
UNSTABLE RESONATOR MODES
tion of eq. (2.75b) into (2.75a) then yields
.
r r
where the kernel function
-2
-+ [(n: + ;> + (i2 Ir,> -
-
x2
-
11)
y2
dx2dy2 (2.77)
is proportional to the impulse response of the diffracting aperture d2in the negative direction. In the limit as the diffracting aperture Sa, goes to infinity, eq. (2.77) reduces to the appropriate form such that eq. (2.76) goes over the the usual expression for the cavity mode structure incident upon the outcoupling aperture plane when no other additional diffracting apertures are present in the cavity [this is the analytical continuation of eq. (2.57)]. Comparison of the kernel function (2.77) in the negative direction with its counterpart (2.73) in the positive direction shows that they are related by
Furthermore, the transpose of the kernel function (2.73) in the positive direction is easily found to be
(2.79) so that the integral equations given in eqs. (2.72) and (2.76) for the counterpropagating cavity modes form a conjugate pair; that is, (2.76) is the conjugate equation of (2.72). Hence, in any discrete (i.e., numerical) determination of the eigenvalues associated with the pair of integral equations (2.72) and (2.76), one
Iv, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
201
must obtain as a result that
7;
=
Yn
(2.80)
9
irrespective of the aperture placement within the cavity. Furthermore, the fundamental bi-orthogonality relation (2.81) JJ-cc
between the counterpropagating mode fields is seen to be satisfied in the discrete sense (i.e., in any numerical evaluation of the preceding overlap integral). Since the kernel functions of the integral equations (2.72) and (2.76) are non-Hermitian, a rigorous proof (i.e., one not based on any finitedimensional matrix representation) of these important relations in a hard-edge, multi-aperture optical cavity has yet to be given. Nevertheless, the physical simplicity of these two relations makes it desirable to claim their general validity. The extension of this analysis to cavities with several discrete, hardedge internal diffracting apertures is straightforward. 2.4. QUESTIONS OF EXISTENCE AND COMPLETENESS OF THE TRANSVERSE
MODES IN OPEN OPTICAL CAVITIES
The major difficulty of the general mathematical formulation of the properties characterizing the transverse mode structure behavior in open optical cavities is simply due to the non-Hermitian, complex character of the cavity eigenmode equation. In general, the pair of integral equations describing the transverse eigenstructure in the positive and negative directions of propagation through an open cavity may be written 7,%(X,Y)
=
JJdu.(xl9Y,)H(x.Y;
7n&,Y)
=
[[du2(xl,Yl)HYx,Y;
xl9YI)dxl dY,
(2.82a)
>
x,,’Y,)dx, dY1
9
(2.82b)
where the integration is taken over the transverse extent of the cavity feedback aperture d,and where the two kernel functions are related by HT(x,y; X,,Y,)
=
H(x,,y,; X,Y)
=
HYx, Y ; X I ,Y , ) ,
(2.83)
so that H t is the adjoint of the kernel H . For an open cavity with a single
202
[IV, 8 2
UNSTABLE RESONATOR MODES
hard-edge diffracting aperture 4 the modified eigenfunctions U, and @ defined by eq. (2.58) are both found to satisfy the linear homogeneous Fredholm integral equation
where
-
--1 AB
[ ( A + D ) ( x 2+ y 2 + x:
+ y:)
- 4(xx1 + yy,)]
(2.85)
Clearly, this kernel function is complex symmetrical but not Hermitian.* The usual, extensively studied theory for either real symmetrical or complex Hermitian kernels (e.g., COURANTand HILBERT[1953], AKHIEZERand GLAZMAN [ 19661) is not applicable in the case of central interest here. At the present time one is left either with specialized arguments or with broad assumptions (or a combination of both) regarding the most fundamental properties of the transverse eigenmode equations arising in the theory of open optical cavities. These concern the fully unresolved questions on the existence and the completeness of these transverse eigenmodes. 2.4.1. On the existence of the eigenvalues
A proof of the existence of at least one nonzero eigenvalue has been given by NEWMANand MORGAN[1964], COCHRAN[1965], and HOCHSTADT [1966] for the special class of open optical cavities in which the integral equation (2.84) is separable in the x and y coordinate directions (the feedback aperture is then rectangular). The two-dimensional transverse eigenmode equation then reduces to a pair of one-dimensional eigenmode equations of the
* The terminology used here is that of MORSEand FESHBACH [I9531 for the adjoint and Hermitian conjugate (or Hermitian adjoint) of an operator or kernel function. That is, the kernel function K ( 5 ; tl) has as its adjoint the kernel function Kt(T; = K ( 5 , ; t),which is the transpose, whereas its Hermitian conjugate is Kc*’(<;cl) = K * ( t , ; t),which is the complex conjugate of the transpose. This terminology is somewhat different from that used by mathematicians and quantum-mechanicists, who take the adjoint of an operator to mean its Hermitian conjugate (e.g., AKHIEZER and GLAZMAN [I9661 and JORDAN [1969]). Unfortunately, both sets of notation are used freely in the open literature.
cl)
IV, J 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
203
form
(2.86) where K(5, 5,)
=
J i $eikLexp -
[ ( A + D)(5’
+ <:)
- 4<<,]
Here 2a is the aperture width in the considered direction, where the aperture is assumed to be centered on the cavity optical axis. The existence of at least one nonzero eigenvalue ?in of the integral equation (2.86) is guaranteed by the following
Theorem (NEWMAN and MORGAN[ 19641):Let G( <)and H( <)be any bounded functions on the closed interval - 1 < 5 < 1, and let F( 5) be any integral function of Jinite order such that (2.88) Then the integral equation
(2.89) has at least one nonzero eigenvalue.
A similar theorem has been proved by COCHRAN [ 19651 for Holder continuous kernels with exponent greater than one half and with nonvanishing trace. For the integral equation given in (2.86) with the kernel function (2.87), the condition (2.88) becomes
(2.90) provided also that a Z / M# 0. Here g = (A + 0 ) / 2 is the usual cavity g-parameter. Furthermore, since the argument of the exponential in eq. (2.90) is an even function of t l , the integral equation (2.86) has at least two
204
[IV, § 2
UNSTABLE RESONATOR MODES
eigenvalues for all but certain exceptional values of the quantity a2(g + 1)/AB, one such exceptional value being zero (see NEWMANand MORGAN[ 19641). This existence proof has also been extended to the case when the feedback aperture is circular (HOCHSTADT [ 19661). It is also shown there that an infinity of nonvanishing eigenvalues of the integral equation (2.84) exists for both rectangular and circular apertures, except possibly for a countable set of values of the quantity a2(g + l ) / D . The existence of the eigenvalues of eq. (2.84) with an arbitrary feedback aperture shape dl remains to be proved. However, the preceding existence proof is sufficient for the majority of optical cavity geometries of practical interest. A formal expression for the complex-valued eigenvalues may be obtained from eq. (2.84) by integrating both sides of this equation over all space, Tn
O0
un(x, y ) d x dy
- w
=
JJddxl
dy1 un(x1 ,yl)
JJ:
W , y ; x1,Yl) dx dy.
From eq. (2.85) and the definition of the cavity g-parameter, one has that
and one then obtains
1
.
jjn = -
g
JJ
O0
(2.91a)
un(x,y ) dx dy
- w
In terms of the original (unmodified) cavity eigenfunctions, this expression becomes
jjn
=
1 D
l[dUn(x,y)exp
.
-
JJ
O0
(2.91b)
un(x,y)dxdy
- m
Hence, given a modified eigenfunction Un(x, y ) that satisfies eq. (2.84) [or the unmodified eigenfunction un(x, y ) ] , its associated eigenvalue y,, is uniquely determined by the appropriate form of eq. (2.91), provided that the associated
IV, 8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
205
field integrals exist and that the denominator in these two expressions does not vanish. Another expression of interest is obtained from the integral
jj
02
vn
7%
un(x, Y ) u;(x, Y ) d x dY
-a2
and one then obtains the rather simple expression
(2.92)
JJ
un(x, Y ) ~ ( xY ),d x d~ -a2
In particular, for m
=
n there then results
(2.93a) J J-co
(2.93b)
This result associates the eigenvalue magnitude with the cavity mode loss Tn of the nth eigenmode as given in eq. (2.50). These expressions are useful for
206
[IV, 8 2
UNSTABLE RESONATOR MODES
determining the complex eigenvalue yn from a numerical Fox-Li type of evaluation of the cavity mode field u,(x, y ) that is incident on the cavity feedback aperture d.
2.4.2. Schmidt expansion of the cavity modes Attention is now turned to a description of the eigenmodes of a standingwave optical cavity as provided by the Schmidt expansion theorem (SCHMIDT [ 19071). By employing the Schmidt theory, STREIFER and GAMO[ 19641 were able to expand the kernel of the cavity integral equation in an appropriate set of Schmidt functions that are orthogonal and complete over the transverse domain of the cavity. Their analysis is extended here to the canonical formulation of the cavity mode structure for both stable and unstable cavities. Once again, attention will be restricted to the class of open optical cavities that are separable in the x and y coordinate directions. Consider then the generalized standing-wave cavity model depicted in Fig. 5. With the reference plane at the feedback aperture and the reference plane at the back element of the cavity taken at the midplanes of those elements, the paraxial ray transfer matrices in the positive direction between these two reference planes are then as they are given in Fig. 5. The overall ray transfer
FEEDBACK APERTURE REFERENCE PLANE
BACK ELEMENT REFERENCE PLANE
FEEDBACK APERTURE REFERENCE PLANE
Al
A2
A,
POSITIVE DIRECTION
. )
Fig. 5. Generalized standing-wave cavity model with a diffracting aperture at both the feedback aperture and back element reference planes.
IV, 5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
207
matrix from the feedback aperture reference plane back to that same reference plane in the positive direction is then given by
(2.94)
Notice that A = D for this symmetrical placement of the cavity reference plane. The integral equation for the self-consistent cavity field incident upon the chosen feedback aperture reference plane is then given b y
( D , x $ + A l x Z- 2 x x , ) ) dx, ,
(2.95a) (A,x:
+ D,x$ - 2x,x,)
-- a 1
(2.95b) where un is the associated field incident upon the back element reference plane. Here the hard-edge feedback aperture dl is taken to extend from - a, to a, in the x direction, whereas the hard-edge aperture d,of the back element of the cavity is taken to extend from - a, to a, in the x direction. Substitution of eq. (2.95b) into (2.95a) then yields the general integral equation
[D,xZ - x2(x + x,)]) dx,
.
(2.96)
In the limit as a, goes to infinity (i.e., in the limit as the aperture d,of the back
208
[IV, B 2
UNSTABLE RESONATOR MODES
cavity element becomes unbounded), the inner integral becomes
which, when substituted into eq. (2.96), yields (Ax'
+ Dx:
-~xx,)
(2.97) where the identifications given in eq. (2.94) have been employed here. Equation (2.97) is simply the integral equation for a single aperture optical cavity. The general integral equation (2.96) for the transverse cavity modes may be
(2.98a) with the symmetrical, complex but non-Hermitian kernel
L(x,, x)
=
L(x, x,)
=
(2I:[
,i : ~
[D,x: - x2(x + x,)]) dx,
(2.98b)
Following the analysis of STREIFERand GAMO[ 19641, let (2.99a)
(2.99b) in terms of which the kernel L(x,, x) may be written (2.100)
J
-a2
IV,5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
209
Notice that
so that eq. (2.100) may also be written as (2.102) J
-a2
Schmidt then defines two positive semidefinite Hermitian kernels LAX2
,4 )= =
J
-al
3sinc J-Bl
(2
(xi - xz)
(2.103b) where the superscript (*) is used here to denote the Hermitian conjugate. The two sets of eigenfunctions { $ n } and { $ n } of these two kernels, the solutions of the integral equations (2.104a) (2.104b)
(2.105a)
($n
9
$m)d
=
j
a1
e(x,)
$m(xl)
hnm *
(2.105b)
- 0 1
Furthermore, the eigenfunctions of these two sets have the same eigenvalues
210
UNSTABLE RESONATOR MODES
[IV, § 2
(2.106a) (2.106b) The eigenfunctions $n(x2) are defined by eq. (2.104a) over the back element reference plane with aperture d 2 ,whereas the eigenfunctions are defined by eq. (2.104b) over the feedback aperture reference plane with aperture d,. They are related by the kernel L,,(x, ,x,) which describes the Fresnel-Kirchhoff propagation phenomena between these two reference planes (see. eq. [2.95]). Equation (2.106) may be considered as giving the analytical continuation of these eigenfunctions outside the domain of their respective apertures. A special case of interest occurs when the aperture d,becomes infinite in extent and the cavity model represents that of a single aperture resonator. The kernel L,, then becomes (2.107) and the integral equation (2.104b) simplifies to (2.108) which admits a continuum of “eigenvalues” given by p,
=
eie,
(2.109)
where 8 is real. The eigenfunctions $, are then determined by m
(2.110a) -m
whereas the eigenfunctions $, are given by m
$,(XI)
=
~ i ~ ) ( xx2) , ,$n(X2)dx2.
e1eS
(2.1 lob)
-m
Attention is now turned to the ke’mel expansion in terms of the Schmidt functions defined earlier. Following the analysis of STREIFERand GAMO [ 19641 (see also STREIFER [ 19651, HEURTLEY and STREIFER[ 1965]), the
Iv, 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUmURE PROPERTIES
21 1
kernel L,,(x,, x,) may be expanded in terms of the two sets of Schmidt functions { &} and { $n} as L,,(x,, x2) =
c P,$n(X2)
(2.11 1)
e(XJ*
n
Since the kernel L,, is square-integrable, eq. (2.11 1) is then an equality in the mean-square sense. Furthermore, if the convergence of this expansion is uniform with respect to either variable, then it is uniformly convergent with respect to both. With this expansion the original kernel L given in eq. (2.102) becomes
S-
a2
UX,, x)
=
L,2(x, x2) L,2(x,9 x2) dx2 a2
C
= m
pmpndCz(x1)
C ( X )($J:,
+m)&
9
(2.112)
n
where (2.113)
(2.1 14)
(2.115)
(2.117)
(2.118)
L
=
Yp@p
(2.119)
212
[IV, § 2
UNSTABLE RESONATOR MODES
with element Lpmgiven by eq. (2.118). Here p is a diagonal matrix with elements Prim
=
~
n
L
9
(2.120)
n
whereas the elements of the symmetrical matrices Y and
Km = (k,$C)4=
J
a1
&xx*) VMXl) dx1
Q,
are given by
9
(2.12 1a)
3
(2.12 1b)
-a1
and @nm
=
(4:
9
$m)A
=
J
a2
$n
- a2
(xz 1$m (x2 1dx2
respectively. Since each of the infinite matrices Y and represents an isomorphic transformation in a Hilbert space, they are then both unitary (for example, HALMOS[ 19571). Such unitary transformations correspond to lossless transmission (ONEILL[ 19631, GAMO[ 19641) so that the associated loss information is contained in the p matrices appearing in eq. (2.119). However, since the eigenfunctions @, and $, are defined only over the respective domains of the apertures at2and dl,this information represents only the loss over these two cavity apertures and does not account for the loss outside of these two aperture domains. The transverse mode function u(x) that satisfies eq. (2.98a) is now expanded in the set of Schmidt functions { %} defined over the feedback aperture dl (2.122) By the Schmidt expansion theorem, if u(x) is continuous, the preceding expansion converges absolutely and uniformly. By operating on eq. (2.122) with the form (2.117) of L(x, ,x), one obtains an infinite number of linear simultaneous equations in place of the integral equation (2.98a); these may be written in matrix form as -1LT
=
Ty,
(2.123)
where y is a diagonal matrix with elements R, the eigenvalues of the integral equation (2.98a). The elements Tpnof the nth column of T are the expansion coefficients for the nth transverse mode pattern on the feedback aperture atl. Nontrivial solutions of (2.123) exist for those values y,, which satisfy the characteristic equation det IL - iyl
= 0,
(2.124)
where the left-hand side of this equation is the Fredholm determinant of (2.123).
IV, 5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
213
The major advantage of employing the Schmidt expansion of the cavity modes in place of directly solving the integral equation (2.98a) is due to the Schmidt expansion theorem, which states that if the p, are ordered such that IPOI 2
IPlI
2 IPZI 2 * . .
9
the best finite N-term approximation of the kernel L,, in the mean square sense is given by the first N terms of the Schmidt kernel expansion. That is, the quantity =
J;l1 J
N- 1
az
I~ lZ(X1 x2) ~ -
c fn(X2) gn(x1)I2dx, dx2
n=O
- a2
is a minimum if and only if N- 1
c fn(x2)gn(xl) c Pn$n(X2) N- 1
=
n=O
@(X1).
n=O
Hence, the sense of the approximation that results when the infinite matrices appearing in the eigenvalue equation (2.123) are replaced by ones of finite rank (for computational purposes) is known. This kernel expansion method has been employed by STREIFERand GAMO[1964] and STREIFER[1965] to determine the transverse mode structure in both low- and high-loss optical and STREIFER [ 19651for cavities with rectangular apertures and by HEURTLEY optical cavities with circular apertures. Their results are in excellent agreement with the previously published numerical results obtained with a Fox-Li type of approach. 2.4.3. Nonstationary modes and the question of completeness Up to now the attention of this article has been focused upon those field distributions u, that reproduce themselves, to within a complex factor yn, on a single propagation pass through an open optical cavity. These selfreproducing field distributions are the eigenfunctions of the equation PlPn
=
(2.125)
%'nun
and are called the stationary modes of the cavity. Here PI is the integral operator describing a single propagation pass through the cavity. With attention again restricted to the single transverse dimensional case, t h s propagation operator is given by n
Pl
=
-i J
dx, L(X, x,) , -4
(2.126)
214
[IV, $ 2
UNSTABLE RESONATOR MODES
where the domain of integration is specified by the cavity feedback aperture dl at which position the cavity reference plane is situated. If a field distribution f ( x ) at this plane is given by the series (2.127)
then M propagation passes through the cavity results in the field distribution
Let the eigenvalues be ordered such that yo > y1 > y2 The preceding expression may then be rewritten as
> . ,where y,, * *
=
1 7, I .
(2.128)
Hence, if yo > yl, the effect of several propagation passes through the cavity is simply to attenuate the higher-order mode components (n > 0) in the series decomposition (2.127) relative to the first mode component. As the number of passes M increases indefinitely, the field structure f,(x) then converges to the stationary mode uo(x) that has the least power loss. This process is referred to as the natural selection of modes and is the basis of the Fox-Li type of approach for the numerical determination of the stationary modes supported by an optical cavity. The efficacy of this process is clearly dependent on the ratio yo/yl, which determines the rate at which the higher-order modes attenuate. For example, if yo = y1 > y2, the field structures f M ( x ) will then converge towards a linear combination of the first two stationary modes, given by f&> = ao7oMU,W + a , 7 ? 4 ( x ) . The validity of expression (2.127) for a given function f ( x ) depends on both the class of functions f ( x ) considered and the question of the completeness of the stationary modes u,(x) of the open cavity. The difficulty here arises from the non-Hermitian character of the propagation kernel for the integral operator describing these modes. To address this problem, BALASHOVand BERENBERG [ 19751have employed a Schmidt expansion of continuously variable operators to determine a complete orthonormalized system of functions, which they call the nonstationary modes, for a given open cavity. Following their analysis, consider the set of linear operators {P,}, where P , is the integral operator
Iv, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
215
Here L is the symmetrical complex but non-Hermitian kernel given by eq. (2.98b). Consider now the self-adjoint nonnegative operators defined by
EN = PF,*'P,,
(2.130a)
P,Pp) ,
(2.130b)
and
K,
=
where P p ) is the Hermitian conjugate of the operator PN.The eigenfunctions of these two operators are defined by
- -
KNvN,,
=
K N 'Nn
=
A;,,
Gn 9
(2.13 la)
'Nn
9
(2.13 1b)
and
where A,,, > 0. Furthermore, the corresponding eigenfunctions of these two equations are related by the equation 'N'Nn
=
' N n 'Nn
.
(2.132)
Since the kernel L(x, x l) of the cavity integral equation is a symmetrical function, one can always define the solutions of eqs. (2.13 I) in such a way that the corresponding nonstationary modes form a complex conjugate pair, that is, V,, = V;,,. BALASHOV and BERENBERG[ 19751 call these functions the nonstationary modes of the open cavity in contrast to the stationary modes given by eq. (2.125). These nonstationary modes are also called quasimodes, since they may be interpreted as representing certain aspects of a radiation field (KARBOWIAK [ 19641). The operator PN is a Fredholm operator acting in the Hilbert space L, of square-integrable functions (defined over the cavity feedback aperture). The solutions of eq. (2.131) then have a simple physical interpretation from the theorem on the minimum-maximum properties of the eigenvalues of a nonnegative continuously variable operator (AKHIEZERand GLAZMAN [ 19661). From this theorem it is seen that the system of nonstationary modes { pNn} I
216
UNSTABLE RESONATOR MODES
[IV, § 2
represents the fundamental field configurations travelling across (as opposed to along the axis of) the open cavity with the optimal (i.e., least-loss) energy transfer (BALASHOVand BERENBERG[ 19751). From the Schmidt theory it follows that if the set of operators { R N }and, consequently, the set { K N } are continuously variable, then for each N the two and { VNn}form complete orthonormalized systems in sets of functions { the appropriate Hilbert space (BALASHOVand BERENBERG[1975]). Any function f is then transformed by the operator PN into a distribution that has the expansion
rNn}
(2.133) where the scalar product (VNn,f)is defined over the cavity feedback aperture d,as (2.134) These nonstationary modes then allow one to properly consider the formation of a laser radiation field from an arbitrary initial field distribution. In general, the system of nonstationary modes { VNn}is not identical with the system of stationary modes { u n } of an open cavity for any value of N. From eq. (2.133) the general relationship between these two sets of modes is given by (2.135) However, these two mode systems will be identical if the propagation operator P, is normal (BALASHOVand BERENBERG[ 1975]), viz. P 1P(*) 1 = P[*)P, .
(2.136)
From eqs. (2.98b) and (2.126) it is seen that this very special property will be satisfied either when B , = 0 or when A, = D , = 0. Only the confocal cavity with an intracavity (real) focus has this property, and the set of stationary modes of this resonator form a complete orthogonal system (BOYD and GORDON [ 19611, BALASHOVand BERENBERG [ 19751). In that special case the transverse field distribution incident on each of the cavity end mirrors is a finite Fourier transform of the field at the opposite mirror [see eqs. (2.95a) and (2.95b)l.
Iv, 5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
217
Consider now obtaining a criterion for the validity of the expansion of an arbitrary function f ( x ) in terms of the stationary modes of an open cavity. Any function f is transformed by the propagation operator P , into a distribution that has the expansion (BALASHOVand BERENBERG[ 19751)
n=O
where m > 0 is an arbitrary but fixed integer. When N increases, the ratio (2.138)
tends to zero, where If,/ is the norm of the residual term f , .In almost all cavity geometries, with the exception of those close to the afore-mentioned confocal configuration, the value of N at which the residual term f, is negligible is identical with the number of propagation passes necessary to establish a regime with a small number of stationary cavity modes. Since unstable cavities typically have a high degree of transverse mode selectivity, the value of N at which f , is negligible is close to unity. In that case one may write m
(2.139) to a good degree of approximation. This is just the completeness relation for the stationary modes of an optical cavity proposed by OUGHSTUN[ 1982al. This relationship is seen to be sufficient when one is considering only the stationary field behavior in an unstable cavity. In that case the residual term f l represents a transient component of the field structure that does not excite any of the stationary modes of the cavity; however, it does couple to the nonstationary mode structure. To develop this point more fully, consider the following form of eq. (2.137): (2.140) where f l is the residual term after a single propagation pass through the cavity. From eqs. (2.134) and (2.135) one obtains (2.141)
218
[IV, § 2
UNSTABLE RESONATOR MODES
Substitution of this result into eq. (2.140) and using (2.135) then yields the expansion r n r n r n
C C C 1=0 m=O
PIf(x1) =
( V l l , un)(un,
Vlm)(Vlrn,P1f)V~Ax1)
n=O
(2.142)
+ fl(Xl).
With eq. (2.133) one then sees that c o r n
=
(~11,fI)
C (VlI, un)(un, VIrn)('lrn, (vl,, P 1 f ) - mC =O n = O
'If)
*
(2.143)
Hence, the scalar product of the residual term f l with the members of the system of nonstationary mode functions does not, in general, vanish unless the system of stationary modes of the cavity is complete. Furthermore, from eq. (2.140) one also sees that (U,,fI)
=
0
(2.144)
9
so that the residual term does not excite any of the stationary mode functions. It is in this sense that the expansion (2.139) is sufficient when considering the stationary field behavior in an optical cavity. However, it clearly is insufficient when considering the process of formation of the laser radiation field (as in a Q-switched laser), and one must then introduce the system of nonstationary modes. The expansion (2.139) may be extended (without proof) to the twotransverse-dimensional generalized optical cavity depicted in Fig. 1 with an arbitrary feedback aperture A Let { UJx, y ) } and { Ui(x, y)} be the two sets of modified stationary eigenmodes of the cavity that satisfy eq. (2.59) with the symmetrical kernel K(x, y ; xo, yo). Let f(x, y ) be a continuous function that is an integral transform with the symmetrical kernel K ( x , y ; x o , yo) of a piecewise continuous function g(xo, yo), viz. f(X, v) =
JJ
g(x0 ,yo) ~ ( xy ;, xo ,yo) dxo dyo .
(2.145)
d
This propagated function may then be expanded in the set of modified eigenfunctions { Un(x,y ) } of this kernel as (2.146)
(2.147)
IV, 8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
219
due to the orthonormality relations (2.63). The expansion (2.146) holds in the same approximate sense as does eq. (2.139). A similar expansion in terms of the counterpropagating set of modified eigenfunctions { U i ( x , y ) } also holds. These expansions are sufficient when considering only the stationary field behavior in an unstable cavity. In terms of the original, unmodified set of transverse eigenmodes {u,(x, y ) } defined in the positive direction of propagation through the cavity, given by eq. (2.56), the expansion (2.146) becomes [using eq. (2.58a)l f ( x , y ) = pl/’(x, y ) exp
( A - D)(x’
+ y’) (2.148)
(A- D)(x2+y2)
(2.149)
where f ( x , y ) is given by eq. (2.145). To cast these two expressions into a form appropriate for the set of stationary eigenmodes { u n } ,one can define
F(x,y ) = p-
l’’(x,
y ) f ( x ,y ) exp
( A - D ) ( x 2 + y’)
G(x, y ) = p - ‘/’(x, y ) g(x, y ) exp
With this substitution eq. (2.145) becomes
[ A ( x ~+,y’,)
+ D(x’ + y 2 ) - 2(xx0 + yy,)] (2.151)
where the function G(xo,y o ) is piecewise continuous and the function F(x, y ) is continuous. Furthermore, eqs. (2.148) and (2.149) become (2.152)
(ll
x exp i -(A - D ) ( x 2 + y 2 ) ) dx dy .
)
(2.153)
Equations (2.151) through (2.153) express the expansion of the continuous function F ( x , y ) that is derived from a given piecewise continuous function G ( x , y ) after a single propagation pass through the cavity in the positive direction in terms of the set of stationary eigenmodes {un}. In a similar manner, if one defines the modified functions Ft and G t by
+ y’)
Ft(x, y ) = pl/’(x, y ) f ( x , y ) exp
(A - D)(x’
G t ( x , y ) 3 p’/’(x, y ) g(x, y ) exp
(A - D ) ( x 2 + y 2 )
then eqs. (2.145) through (2.147) become
(2.155) (2.156)
x exp( -i--(A-D)(x’+y’) IT
(2.157)
Equations (2.155) through (2.157) express the expansion of the continuous function F t ( x , y ) that is derived from a given piecewise continuous function G t ( x , y ) after a single propagation pass through the cavity in the negative direction in terms of the set of stationary eigenmodes { u ; } . Finally, since ~ t ( xy ), = p(x, y ) ~ ( xy ), exp
(2.158)
I v , $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
221
and
(2.159) it is then seen that the expansion coefficients given in eqs. (2.153) and (2.157) are identical, provided that one starts with the same function f ( x , y). Furthermore, it is seen that (2.160a) (2.160b)
2.5. POLARIZATION EIGENSTATES AND THE VECTOR MODES OF AN OPTICAL CAVITY
A complete description of the optical field in an open cavity requires that the polarization state be known everywhere within the cavity. For many simple cavity geometries the cavity field is either circularly polarized or completely linearly polarized along some preferred direction, and the preceding scalar analysis is entirely adequate. However, if anisotropic elements are placed within a cavity, they not only alter the polarization state of the outcoupled field but also may dictate the transverse mode structure the cavity field may assume (DENTE [ 19791, FINK[ 19791, LATHAM[ 19801, CHODZKO, MASON,TURNER and PLUMMER[1980]), thereby altering the mode losses and the resonant frequencies of the cavity field structure. In addition, the polarization state of the cavity field will, in general, be different in the various cavity sections that are separated by anisotropic elements, and this will modify the standing-wave interference phenomena between the counterpropagating cavity fields. A description of the polarization eigenstates of an open optical cavity based on the Jones matrix method (JONES [ 19411, HURWITZ and JONES [ 19411, JONES [ 19421) and the synthesis of the cavity vector modes (TYLER and SOUTHWELL [ 19801, FERGUSON [ 19821) is presented in this section. This treatment is based and SKROTSKII [ 19721, in part on the excellent review paper by MOLCHANOV of which many authors are unaware. The analysis here is restricted to the stationary field properties of an optical cavity. A description of the dynamics
222
UNSTABLE RESONATOR M O D E S
[IV, § 2
of the polarization eigenstates of a quasi-isotropic laser may be found in the paper by LE FLOCH,ROPARS,LENORMANDand LE NAOUR[ 19841. 2.5.1. Jones calculus and polarization eigenstates The electric field vector E of a transverse, plane, coherent light field propagating along the z axis is uniquely defined simply by specifying the vector components Ex and Ey along the x and y axes, respectively, of an arbitrarily oriented Cartesian coordinate system. Let these components be given by the real parts of the expressions
E ox ei(d'-w)
(2.16 1a)
EY = EOY ei(4+-w)
(2.16 1b)
Ex
=
where Eo, and Eoy are the real-valued amplitudes and $, and $, are the phases. With the harmonic time dependence suppressed, one obtains the following equation for the polarization ellipse (BORNand WOLF[ 19751): (2.162) where 6 = $y - $, is the phase difference between the two vector components. The principal axis of the polarization ellipse is at an angle $ (0 < $ < R) with the x axis, given by tan 2$
=
tan 2a cos 6 ,
(2.163)
where (2.164) and the principal semiaxes u and b of the ellipse are in the ratio b -=tax,
(2.165)
U
where - i n < x < i n a n d sin 2x
=
sin 2a sin 6 .
(2.166)
The direction of rotation of the electric field vector is determined by the sign of the quantity sin 6. In the traditional terminology the polarization is said to
IV, 8 21
GENERAL FORMULATIONOF THE TRANSVERSE MODE STRUCTURE PROPERTIES
223
be righthanded (a+)if the electric field vector, when viewed in the direction from which the light is coming, rotates clockwise about the polarization ellipse; in that case sin 6 > 0 and 0 < x d $x. For left-handed polarization (a-) the rotation is counterclockwise so that sin 6 < 0 and - f n d x < 0. The electric field vector of the light wave may be represented by the column matrix (2.167)
where (2.168)
is known as the Jones vector. The various forms that the Jones vector can assume then correspond to the various polarization states of the associated optical field. The state of polarization is altered when the light field interacts with anisotropic or polarizing optical elements. Under the assumption that these elements do not alter the wave front of the field, any such interaction can be considered as a linear transformation of the polarization state according to the relation (2.169)
which may be written in matrix form as E2= ME,.
(2.170)
If the polarizing elements are lossless, the matrix M is unimodular. The elements aij of the matrix M depend on the orientation of the x and y axes of the selected Cartesian coordinate system. The rotation of this system through an angle 0 about the z axis (the propagation direction) transforms a matrix M’ into a matrix M given by
M
=
S( - 0)M’S(O),
(2.17 1)
where
s(e)=
cos 0 sin B -sin 0 cos 0
(2.172)
is the rotation matrix. The matrix M that describes a given polarizing element
224
UNSTABLE RESONATOR MODES
[IV, § 2
takes its simplest form in the coordinate system that is linked to the principal axes of the matrix of this element. Each matrix denoted by a superscript prime will be assumed here to be in its principal form. A system of N polarizing elements A ,A,, . ..,A, may be described by a singe matrix M that is uniquely defined by the matrices M k= s( - 6,) MLS(6,) of all the elements A, along with the orientations of their principal axes and the direction of propagation of the light field. Here ML is the principal form of the matrix describing the element A,. If the field propagating in the positive z direction encounters, in order, the elements A A2,. .., A,, then
,
M
=M ,M,-
1.
*
*M,M,.
(2.173)
For propagation in the opposite direction (the negative z direction) one obtains
Mt
=
MlM2 . MN- ] M N , * *
(2.174)
where Mtis the adjoint of the matrix M.In general, Mt # M. When an optical field interacts with a polarizing or anisotropic element, its polarization state usually is altered. However, there are certain polarization states, known as eigenpolarizations,which are not affected by a given element. These polarization states then satisfy the equation
Me = l e ,
(2.175)
where l is generally a complex factor. The Jones vector e that determines the polarization eigenstate is then the eigenvector of the matrix M,which corresponds to the eigenvalue A. The eigenvalues are then determined by the characteristic equation
.$-
l ' ; ;
=
0
7
(2.176)
whose solutions are the two eigenvalues A, an( l2 given by
In general, these eigenvalues are complex, and it is convenient to write them in the form
l 1= alei+l,
(2.178a)
A,
(2.178b)
=
~,e-~&,
where a, and u2 are real and positive and where and +1 are real. These eigenvalues correspond to the two polarization eigenstates to be
IV, $21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
225
denoted as El
=
E2 =
(;;)
5
(t) .
(2.179a)
(2.179b)
The complex numbers ,Il and ,I, are then seen to describe the changes in the amplitude and phase of each of these two polarization eigenstates resulting from the interaction with the anisotropic element. The moduli a , and a, give the amplitude transmission of the element for these polarizations, and the sum $, + $2 gives the phase shift between the polarization eigenstates. Two elliptically polarized plane wave fields propagating along the same direction are orthogonal if the azimuths of the principal axes of the corresponding polarization ellipses are at right angles, the directions of rotation are opposite, and the ellipticities are equal. In particular, the two polarization eigenstates e, and e2 are orthogonal if and only if 5,5:+?1@=0.
(2.180)
However, in general, the polarization eigenstates are associated with different losses and are not orthogonal. Consider now an optical system that is represented by the two polarization eigenstates E , and E, corresponding to different eigenvalues I, and I,. The matrix representation of this system is then described by the expression (JONES (2.181) # 0, since the eigenvectors E , and E, are linearly where d = (,ti2 independent. With this representation one can, in principle, design an optical system so as to possess a specified pair of polarization eigenstates. Further discussions of the properties of this matrix for several forms of the system anisotropy are given by JONES[1942] and MOLCHANOV and SKROTSKII [ 19721.
2.5.2. Vector modes of an optical cavity In a resonator the electric field or Jones vector E of the stationary cavity field must be invariant in a single round-trip propagation pass through the cavity.
226
UNSTABLE RESONATOR MODES
[IV, § 2
That is, both the transverse field distribution and polarization state of the cavity field must be self-reproducing. With reference to Fig. 1, let K represent the integral operator that propagates the field in the positive direction from the chosen reference plane through the cavity back to that plane in a single pass, and let M be the resultant polarization matrix of the optical system encountered in that pass. The eigenvalue equation for the cavity field may then be symbolically written as
MKE = Y'E,
(2.182)
where 7' is the complex eigenvalue that accounts for both the propagationdependent and the polarization-dependent loss and phase changes. In the absence of any polarizing elements (i.e., for an isotropic cavity) the polarization matrix is simply the identity matrix, and the eigenvalue equation for the vector field becomes
KE= YE,
(2.183)
which separates into two identical integral equations for the scalar components of the electric field vector. The solutions of these two scalar equations are degenerate and the polarization state of the cavity field is unspecified, although it must be spatially uniform across the transverse dimension of the field. If the anisotropy is spatially uniform across the transverse dimension of the cavity field so that MKE = KME and
Me = h
(2.184)
independent of the transverse coordinate position, then the eigenvalue equation given in (2.182) becomes
(Ku -
~ U ) E=
0,
(2.185)
where
E
= ue
(2.186)
and ;i, = y'/L. Since e # 0, the scalar quantity in eq. (2.185) must vanish, so that Ku
=
Yu,
(2.187)
and this is just the eigenvalue equation for the scalar transverse mode structure of the cavity. Equation (2.184) is simply that given in eq. (2.175). Hence, the of the two polarization eigenstates are given by eq. (2.177) and eigenvalues may be expressed in the form given in eq. (2.178). The moduli a, and u2 determine the losses experienced by the vector modes due to the anisotropic
Iv, 8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
227
elements, and the quantity ($1 + $J determines the difference Av between the oscillation frequencies of the modes associated with these two polarization eigenstates, where
(2.188) and L denotes the optical path length traversed in a single round-trip propagation through the cavity. If a , > u2, then the polarization state of the cavity field is whereas i f u , = u2, the polarization state is any linear combination of the two polarization eigenstates E , and E ~ Specific . examples of the polarization eigenstates for several forms of the spatially uniform anisotropy are given by MOLCHANOV and SKROTSKII [ 19721, VETKINand KHROMYKH [ 19721, and BLOKand DE KO0 [1975]. For propagation in the negative direction through the same cavity one obtains in place of eqs. (2.184), (2.186), and (2.187) the relations
M t ~ =t
Et
=
,
utet,
Ktut = yfuf.
(2.189) (2.190) (2.191)
From eq. (2.62) one has that yt = 7 and, in addition, the transverse scalar field distributions ut and u are simply related by eq. (2.67). For a standing-wave cavity one must have Mt = M, so that I = IT and 8 = et and the polarization eigenvectors for the counterpropagating cavity fields are the same. However, for a travelling-wave resonator in general Mt # M [cf. eqs. (2.173) and (2.174)], so that the polarization eigenstates for the counterpropagating cavity fields are usually different. This occurs, for example, in a ring resonator with a Faraday rotator, a circular phase plate, and an arbitrary phase anisotropy (MOLCHANOV and SKROTSKII [ 19721). Quite exotic behavior can result when the cavity anisotropy is spatially nonuniform. In that case one must determine the field at the plane of each anisotropic element present that is spatially nonuniform, since each such element will locally mix the field components between the polarization eigenstates. The appropriate propagation operators must then be applied between all such elements. When such local mixing occurs, the spatial distribution of the transverse mode structure at that plane is modified in a way dictated by the element anisotropy (DENTE[ 19791, FINK[ 19791, LATHAM[ 19801, TYLERand SOUTHWELL [ 19801, FERGUSON [ 19821). One then cannot construct a single equivalent polarization matrix M at any given reference plane with the exception
228
UNSTABLE RESONATOR MODES
[IV, 8 2
of the plane of a single nonuniform anisotropic element within the cavity. It is clear that a simple scalar description of the cavity field is completely inadequate in this situation, since the vector eigenvalue equation cannot be separated in the manner of eqs. (2.184) and (2.187).
2.6. STANDING-WAVE INTERFERENCE AND THE RESONANCE CONDITION
The daractive mode loss Ynand phase shift q$, per round-trip propagation in an open cavity are related to the complex eigenvalue y,, of the integral equation (2.46) for the transverse mode structure by the equation [cf. eq. (2.50)]
yn
=
(1 - yn)1/2e-i%,
(2.192)
where the overall propagation phase shift kL appearing in eq. (2.46) has been absorbed into the term @,. Here $ J ~ is defined in the principal range - n < 4b, ,< n. For a standing-wave cavity, L is given by twice the axial end-mirror separation ( L = 2zT), whereas in a travelling-wave ring cavity L is the axial distance about the cavity. The mode-dependent phase term qn represents the additional phase shift experienced by the nth cavity mode relative to a uniform plane wave field in a single round-trip propagation through the cavity. Resonance occurs in a standing-wavecavity when the end-mirror separation zT and the cavity field wavelength 1 are such that the quantity yneikzT is real, so that (2.193) where q is an integer. When this condition is satisfied, there exists an integer number of axial nodes in the field between the cavity end mirrors with a node at each end mirrur. The cavity will support resonances only for those values of A and z, that satisfy this condition. The frequency spacing between the longitudinal modes satisfying (2.193) is then (2.194) where p is the refractive index of the cavity medium. The relation between this cavity resonance condition and the standing-wave interference effects in a Fabry-Perot cavity as described by the Airy distribution is given by
IV, 8 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
229
KOPPELMAN[ 19691. The classical multiple-beam interference effects in a Fabry-Perot interferometer arise from the partial wave superposition only in the case when the effects of edge diffraction caused by the cavity mirrors are negligible. The effects of diffraction may be included by expanding the input plane wave field as a superposition over the set of stationary cavity modes in the sense of eq. (2.152). To a certain degree of approximation, each mode is enhanced by the factor (1 - p exp( - 2BmzT))- because of the multiple reflections between the cavity mirrors (KOPPELMAN [ 1969]), where pis the reflection coefficient of the cavity feedback mirror and ,8 is the propagation constant of the nth mode. The resultant field distribution incident upon the cavity feedback mirror is then given by the superposition of each of the initially excited, resonance-enhanced modes. For a travelling-wave ring cavity resonance occurs when the cavity length L = 22, and the field wavelength A are such that the quantity yneiw is real. The resonance condition is then (2.195) and the longitudinal mode separation frequency is C
Av=--, 4pzT
(2.196)
which is one half that of the standing-wave cavity with the same diffractive mode properties. It is interesting to notice that the dsractive formulation of the transverse, scalar mode structure of an optical cavity, as embodied in eq. (2.46) for example, is fundamentally a travelling-wave formulation. The only distinguishing features described so far between these two cavity types appear in the transverse mode polarization eigenstates (under nonreciprocal conditions of the cavity anisotropy) and the longitudinal mode resonance conditions and consequent mode separation frequencies. These features can be used to advantage in obtaining unidirectional oscillation in a ring cavity (HUTCHINGS, WINOCUR,DURRET,JACOBSand ZINGERY[ 19661, CHERNEN'KII[ 19721, CIRKELand SCHAFFER[1972], VETKIN and KHROMYKH [1972], TOMOV, FEDOSEJEVS and RICHARDSON [ 19771). Other methods for achieving near unidirectional oscillation in a ring cavity include tailoring the transverse profile of the cavity gain medium with respect to the forward-to-reverse wave mode volume ratio and employing external reverse wave suppression 'mirrors or
230
UNSTABLE RESONATOR MODES
[IV,§ 2
externally injected fields (BUCZEK, FREIBERGand SKOLNICK[ 19711, FREIBERG, CHENAUSKY and BUCZEK[ 1973, 19741).
2.7. SPATIAL COHERENCE OF THE TRANSVERSE MODE STRUCTURE
The final fundamental property connected with the transverse mode structure of an optical cavity is its degree of spatial coherence. The early analysis of WOLF[ 19631 and its extension by FAN[ 19641, STREIFER[ 19661, ALLEN, GATEHOUSE and JONES [1971], and GORI[1980], have shown that the multiple diffraction of an initially incoherent, quasi-monochromatic optical beam propagating through an iterated sequence of equidistant apertures can result in a field that is completely spatially coherent after a sufficient number of iterations. As a consequence, the spatial coherence of the optical field in a laser should not be attributed solely to the stimulated emission process of the laser medium. Up to this point in the analysis of the transverse mode structure properties of an optical cavity it has been assumed that the optical field is strictly monochromatic [cf. eqs. (2.5) and (2.8)J. This idealization results in a formalism that is incapable of providing any information about the coherence properties of the cavity field. This unnecessary restriction has only recently been removed by WOLF and AGARWAL [ 19841, who have developed a new theory of open cavity modes based on an integral equation fur the cross-spectral density of the steady-statefield of arbitrary spectral composition. The solutions of this equation are shown to be expressible as quadratic forms involving the stationary modes of the strictly monochromatic theory. If there is no degeneracy, each of these modes is found to be completely spatially coherent (within the framework of second-order coherence theory) at each temporal frequency; in that case the conventional, strictly monochromatic theory is completely adequate in its description of the stationary field properties of the cavity. However, if several transverse modes are excited and persist simultaneously, the cavity field cannot be completely spatially coherent and the conventional theory is inadequate. These important results are borne out by the earlier experiments of BERTOLOTTI,DAINO,GORIand SETTE [1965]. The analysis of the present section closely follows that of WOLF and AGARWAL [ 19841.
GENERAL FORMULATION OF THE TRANSVERSE MODE S T R U ~ U R E PROPERTIES
IV, f 21
23 1
2.7.1. Coherent mode representation The description of the second-order spatial coherence properties of the cavity field is based on the coherent mode representation of fields of any state of coherence resulting from the work of WOLF [ 1981a, 1982, 19861, which is briefly reviewed here (see also WOLFand AGARWAL[ 19841). Let r denote the position vector of a typical point P in some finite closed domain D of free space, and let V(r, t) be the complex analytical signal representation of a stationary field in D (e.g., MANDELand WOLF[ 19651, BORNand WOLF[ 1975]), where t denotes the time. The mutual coherence function of the field is then given by the ensemble average m1,r2,4=
(V*(r,,t)W,,t+
3)
9
(2.197)
which represents the correlation between the field at r2 and the complex conjugate of the field at rl at the times t + z and t, respectively. The temporal Fourier transform of the mutual coherence function then yields the crossspectral density
(2.198) For a large class of stationary fields the cross-spectral density function is a continuous, Hermitian, nonnegative definite, Hilbert- Schmidt kernel that admits a Mercer expansion (WOLF [ 19821) W I ,
r29 0)=
c U 0 )eb.1,
0)* k 2 , 0)9
(2.199)
n
which is absolutely and uniformly convergent, where n
is the Fredholm integral equation defining the eigenfunctions $, and their associated eigenvalues An . The eigenfunctions of (2.200) are orthonormalized over the domain D, so that
(2.201) Furthermore, it is well known that the integral equation given in (2.200) has at least one nonzero eigenvalue (AKHIEZER and GLAZMAN [ 19661). Finally, it is
232
[IV, § 2
UNSTABLE RESONATOR MODES
understood that the index n stands for an ordered triplet of indices (4, n,, n 3 ) when the domain D is three-dimensional. The Mercer expansion (2.199) expresses the cross-spectral density W ( r , ,r,, w ) of a statistically stationary field of any state of coherence as a linear combination of cross-spectral densities
W(n)(rl,r29 0) =
$ 3 1 9
0) $ ? I @ , ¶
(2.202)
0)
that factorize with respect to the two spatial variables (WOLF[ 19821). They then represent fields that are completely spatially coherent in the space-frequency domain, since their associated complex degree of spectral coherence at the frequency w is unimodular (MANDELand WOLF[ 1976]), viz.
Each term W(")(rl,r,, w ) in the expansion (2.199) then may be regarded as being associated with a natural mode of oscillation of the field (WOLF[ 1982]), satisfying the Helmholtz equations (WOLF[ 1981al)
V'W(")(rl, r,, w ) + k2W(")(r,,r,, w ) = 0, j
=
1,2,
(2.204)
where V ' is the Laplacian operator acting on the variable 9 , and k = w/c is the free-space wavenumber. Clearly the cross-spectral density W ( r ,,r,, w ) also satisfies the Helmholtz equations (2.204). One may also construct an ensemble of monochromatic wave functions U(r, o)eciwrwith the same angular frequency w such that the cross-spatial density of the field is their cross-correlation function
W(r,
9
'2
Y
w ) = ( U*(r,, 0 )
w,
3
0 ) )w
,
(2.205)
where the suffix o indicates that the average is taken over this ensemble (WOLF [ 1981bl). Each function U(r, w ) may be expanded in a series of the eigenfunctions IC/&, w ) of the integral equation (2.200) as
U(r, 0) =
c
%(O) k
o - 9
0) 9
(2.206)
"
where the factors a,(o) are random coefficients satisfying the condition (a?Xw)a,(w)>,
= U w )
k, .
(2.207)
In addition, each random function U(r, o)satisfies the Helmholtz equation
V2U(r,o)+ k2U(r,w ) = 0 throughout the finite closed domain D.
(2.208)
IV, 5 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE
PROPERTIES
233
Finally, if one substitutes the Mercer expansion (2.199) for the cross-spectral density function W(rl, r,, o)in the Fourier inverse of eq. (2.198), the mutual coherence function (2.197) of the field is found to be given by (2.209)
is the mutual coherence function associated with the mode labelled by the index n. Unlike the degree of spectral coherence p(")(rl, r, , o),the complex degree of coherence (2.211)
of a mode in the space-time domain is, in general, not unimodular. As a consequence, r(")(rl, r, ,z) is a mutual coherence function of a field that is not completely coherent in the space-time domain, whereas it is necessarily completely coherent in the space-frequency domain. 2.7.2. Second-order coherence of the stationary Jield in an optical cavity Following the analysis of WOLFand AGARWAL[ 19841, consider a scalar optical field distribution across the transverse reference plane situated just before the output coupler (in the positive direction) of the generalized optical cavity depicted in Fig. 1. This distribution is assumed to be characterized by a statistical ensemble that is stationary, at least in the wide sense. Let Wo(rl, r,, a)be the cross-spectral density of the field at two points on the reference plane that are specified by the two-dimensional position vectors rl and r,. Under a single round-trip propagation through the cavity this field distribution produces a new field distribution that is incident back upon the same reference plane. Let Wl(rl, r,, o)be the cross-spectral density of this new field distribution. Continuing this process, let W,(rl, r,, o)be the crossspectral density of the field distribution incident upon the reference plane after j complete round-trip propagations through the cavity. From the two-dimensional version of eq. (2.205), the cross-spectral density Wj is given by y - ( r l ,r,, w ) =
< U;(rl,
o)q(r2, w ) >
(2.212)
234
UNSTABLE RESONATOR MODES
[IV,8 2
where the average is taken over an appropriate statistical ensemble of random functions V, . Each function q.(r, w ) is a boundary value of the cavity field that satisfies the Helmholtz equation (2.208). Hence, the fields q, and q are related by the linear transformation
q,l(r, w ) =
L
K(r, r’, w ) q ( r ’ , w ) d2r‘ ,
(2.213)
where j = 0, 1,2, . .. . Here K(r, r’, w ) is the propagation kernel (independent of j ) for a single round-trip propagation through the open cavity of monochromatic light of angular frequency w. Substitution of eq. (2.213) into (2.212) yields the following relation between the cross-spectral densities W, and W, : +
?.+ ,(r1, r2, w ) = S , S , K * ( r l , r ; , 0)m,,r;, w ) x
y.(r.(r;, r;, w ) d2r; dzr; .
(2.214)
After a sufficient number of round-trip propagations through the cavity, a steady-state condition is reached such that
q+
r29 0) =
a(w) q r 1 , r,,
0)
(2.215)
for each frequency w. The proportionality factor 40)represents the loss that occurs in a single round-trip transit. Since the “diagonal values” W,+ ,(r, r, w ) = o(w) W,(r,r, w ) represent spectral densities, they are necessarily real and positive so that o(w) > 0 .
(2.216)
Substitution of W,+ from the self-consistency relation (2.215) into eq. (2.214) then yields
X
K(r2, r;, w ) d2r; d2r; ,
(2.217)
where the superfluous index j has been dropped. This integral equation is the basic equation for the second-order spatial coherence of the stationary transverse mode structure of an optical cavity due to WOLFand AGARWAL [ 19841. Its solutions W ( r l ,r,, w ) are the boundary values of the cross-spectral densities on the cavity reference plane of the transverse modes capable of being supported by the open cavity. The cavity fields U(r, w ) appearing in the foregoing cross-spectral density
IV, § 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
235
functions W(rl, r,, o)are taken as the modified eigenfunctions of the cavity [if not, they may always be redefined as such through eq. (2.58)], satisfying the integral equation (2.59). The integral equation (2.217) then is seen to apply equally to the conjugate sets of stationary modes { U,} and { Un}, where U, is taken in the positive direction and Un in the negative direction of propagation through the cavity (see Fig. 1). ,Hence, the second-order coherence properties of the counterpropagatingmodes in an open, passive optical cavity are identical at the chosen cavity reference plane. The same arguments also apply for a multi-aperture optical cavity. The properties of the solutions of the integral equation (2.217) were investigated by WOLF and AGARWAL[1984] through the introduction of a bi-orthogonal expansion (MORSE and FESHBACH [1953]) of the nonHermitian kernel K(r, r’, o).Let {y,(o)} and {u,(r, o)}be the sets of eigenvalues and eigenfunctions, respectively, of the Fredholm integral equation* (2.218) Wr1, r2, 0) u,(r2, 0) d21; y,,(o) u,+,, w ) =
s,
and let {F,(o)}and {u,(r, o)}be the sets of eigenvalues and eigenfunctions, respectively, of the corresponding integral equation with the Hermitian conjugate kernel K*(r,, rl, o),viz. F,(w) u,(rl, o)=
K*(r2,rl, o)u,(r2, 0) d2r2 .
(2.219)
For open cavities the kernel K(r, ,r,, o)is defined on a finite domain d and is a continuous function in both the spatial variables rl and r, for every o.Such kernels belong to a class of square-integrable kernels for which the following two results hold: (1) To each eigenvalue 7, of eq. (2.218) there corresponds an eigenvalue of eq. (2.219), where
B,
A = 7:
(2.220)
r,
Furthermore, the degrees of degeneracy of 7, and are the same. (2) The corresponding eigenfunctions of eqs. (2.218) and (2.219) are (with suitable normalization) orthonormal over the domain d,viz.
(2.221) * Clearly, this equation defines the stationary transverse mode structure of the passive optical cavity.
236
[IV,§ 2
UNSTABLE RESONATOR MODES
Consider now the bi-orthogonal expansion (WOLFand AGARWAL [ 19841) (2.222) Upon substitution of this expansion into eq. (2.217) one obtains the relation
do)W(r, r2 9
9
0)=
C C7%~)
ym(o) wnm(w)
n
uXr1, 0)u m ( r z
9
(2.223)
m
where W ( r i ,r;, w ) un(ri,o)o*,(r;, o)d2r; d2r;
wnm(o)=
.
(2.224)
Multiplication of both sides of eq. (2.223) by uN(rl,o)t&(r2, o)and integration with respect to rl and r, over the domain d yields, with eqs. (2.221) and (2.224),
C C 730) n
Ym(w)*wnm(o) hndm,
=
do)w
d o )
7
(2.225)
m
which then implies that (with no summation) (2.226)
[ 4 w ) - %(a)7M(0)1 W N M ( W ) = 0 *
Hence, either wNM(o) = 0 or a(w) = 7$(o)yM(w).The 6rst case is of no interest, since the corresponding term does not contribute to the summation in eq. (2.223). The other case yields the important result that =
7Xo) 7do)
(2.227)
9
thereby relating the eigenvalues of the integral equations (2.217) and (2.218). Two separate cases need now be considered. 2.7.2.1. Case I: Nondegenerate eigenvalues Let the eigenvalue oNM(o)be nondegenerate in the sense that there are no other pairs yw(o),yMr(o) of eigenvalues of the integral equation (2.218) for ) = 7$.(0) yM(w). Equation (2.226) implies that with a which 7 % ~ yM(o) particular choice
d o )= 7E(m) 7 / ( 0 )
(2.228)
of a nondegenerate eigenvalue of (2.217), wNM(o) = 0 unless N = k and I = M, and the expansion (2.223) then reduces to the single term W(r1 r, w ) = 9
9
Wkl(0)
uE(r1,
0) %(‘2
9
0)*
(2.229)
Since the cross-spectral density function W ( r , ,r,, w) is necessarily Hermitian
IV, § 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE
PROPERTIES
231
[i.e., W(r,, rl, w ) = W*(r,,r,, w ) ] ,then, applied to eq. (2.229), this condition demands that w d w ) u2(rz , w ) 4 r 1 ,w ) = @[‘I(w)uk(rl,w ) uT(rZ, w ) ,
and hence (2.230) At fixed w, the left-hand side of this equation is seen to be a function of rl alone, whereas the right-hand side is a function of r, alone, where rl and r2 are independent variables. This is possible only if each side is independent of the spatial variables, so that one may write u,(rr
0) =
Ukb.9
(2.23 1)
9
where the factor akl(w)is independent of r. Substitution of this result into eq. (2.229) then yields
W(rl ,r2
=
wkl(w)akl(w)@(rl , 0) uk(r2,a).
(2.232)
If one now substitutes eq. (2.232) into (2.224) there results %,(a)
=
%/(W)
x Jduz(r;, w ) u,(r;, w ) d2r; =
wk/(w)% / ( w )~k,~,,
s,
uk(r;, w ) y*,(r;, w ) d’r;
9
(2.234)
where the bi-orthonormality relation (2.22 1) has been employed. Hence, w,,,(w)
=
0 unless n
=
m
=
k,
Wkk(O) = Wk/(W)
(2.234a) (2.234b)
On substitution of (2.234b) into (2.232) one sees that the admissible solutions of the integral equation (2.217), which are now denoted by* Wk(rl, r,, w), are given by Wk(r19r,, w ) = wkk(w)u,*(rl, 0) uk(r2,w ) .
(2.235)
From eq. (2.228) the corresponding eigenvalues, which are now denoted by
* It is important to notice here that the subscript k that now labels W has a different meaning from the subscript used in eqs. (2.212) to (2.215) and subsequently dropped.
238
UNSTABLE RESONATOR MODES
[IV, 3 2
uk(w), are then given by 40) =
R(o)MO).
(2.236)
Since 1 Ykl2 = 1 - Yk,where Ykis the single round-trip relative power loss of the kth stationary cavity mode (cf. eq. [2.192]), the eigenvalue uk(w)is simply the relative amount of power retained in the cavity mode per round-trip propagation. The factor wkk(w)appearing in eq. (2.235) depends on the normalization. Let
\uk(r,w)l' d2r = 1 .
(2.237)
The right-hand side of eq. (2.235) can then be identified with the twodimensionalversion of the Mercer expansion given in eq. (2.199). In the present case this expansion consists only of a single term and uk(r7
O)=
$k(',
(2.238)
O)9
(2.239)
U r n )*
Wkk(W) =
Hence, each solution (2.235) of the integral equation (2.217), which may now be written as wk(rl
9
r2
7
0)= &(W)
uk*(rl9 0) uk(rZ
3
0) 9
(2.240)
is also a mode in the sense of the general theory of the coherent-mode representation of fields of any state of coherence. Furthermore, eqs. (2.235), (2.237), and (2.239) yield
(2.241) Since Wk(r,r, o)represents the spectral density at the frequency o at the point r (see MANDELand WOLF [ 19761, WOLF [ 1982]), eq. (2.241) implies that &(a)is a measure of the steady-state rate at which energy at the frequency w is fed back into the cavity from the reference plane at the cavity feedback aperture. Since each solution (2.240) factorizeswith respect to the two spatial variables r , and r2,its degree of spectral coherence at the frequency w is unimodular [cf. eqs. (2.202) and (2.203)]. Hence, eq. (2.240) is the cross-spectral density of a field distribution that is spatially completely coherent at the frequency w over the feedback aperture d of the cavity. That is, if there is no degeneracy of the eigenvalue uNM(u),the integral equation (2.217) only admits solutions that represent the cavity optical field that is spatially completely coherent within the
IV, 21
GENERAL FORMULATION OF THE TRANSVERSE MODE STRUCTURE PROPERTIES
239
framework of second-order correlation theory at each frequency o (WOLFand AGARWAL [ 19841). When a laser operates in the steady state, the integral equation (2.217) will apply to each frequency component that is present in the spectrum of the laser field [the modes uk(r, o)will now be those of the active cavity]. Substitution of eq. (2.240) into the Fourier inverse of (2.198) then yields the following expression for the mutual-coherence function of the kth laser mode:
rk(r,,r,,
z)
=
I0*
&(o)uE(rl, w ) uk(r2,o)e - i w r d o .
(2.242)
Finally, the spectrum of the kth laser mode at a point r on the feedback aperture
d is given by Sk(r, 0)= Wk(r,r, o)= &(a)Iuk(r,o)I2 .
(2.243)
2.7.2.2. Case II: Degenerate eigenvalues The factorization of the cross-spectral density into a product of a function of rl and a function of r,, where each function is independent of the other spatial variable, is known to be both a necessary and sufficient condition for complete second-order spatial coherence (MANDELand WOLF [ 198 I]). Hence, if the cavity operates on more than one transverse mode, the cavity field cannot then be fully spatially coherent across the feedback aperture reference plane. When there is a degeneracy, one obtains in place of eq. (2.229) an expression of the form (WOLFand AGARWAL [ 19841) (2.244) Since the cross-spectral density is necessarily Hermitian, the expansion coefficients satisfy the constraint cIk(w) = c&(w). If one diagonalizes the matrix C with elements c k / ( o ) so that C = TtAT, where T is a unitary matrix with elements tkI(w),and also normalizes the cavity eigenmodes uk(r, o)as in eq. (2.237), the degenerate solution (2.244) takes the form of its Mercer expansion
(2.245)
(2.246) The field across the cavity feedback aperture d is then seen to be no longer spatially completely coherent.
240
UNSTABLE RESONATOR MODES
[IV, 8 3
Substitution of eq. (2.245) into the Fourier inverse of eq. (2.198) then yields the following expression for the mutual-coherence function of a degenerate mode:
r(r,,r2, 4 = Ck JOm
4(w)fk*(rl,
01
fk(r2, w ) e-io7dw.
(2.247)
Finally, the spectrum of this mode at a point r on the feedback aperture d is given by (2.248)
These results are in complete agreement with the experimental results of BERTOLOTTI, DAINO,GORIand SEITE [ 19651. Using a two-pinhole Young’s interference arrangement, they measured the visibility of the interferencefringes due to the outcoupled field from a helium-neon laser, where one pinhole was fixed along the optical axis defined by the laser cavity and the other pinhole was used to sample the field at several transverse distances from it. They found that, in the nondegenerate case when there is only a single transverse mode supported by the cavity (the TEM,, mode), the degree of coherence 1pI2I = I y(rl, r,, 0) I is unity, even when several longitudinalmodes are present. However, in the degenerate case when there are several transverse modes simultaneously supported by the cavity (the TEM,, and TEM,, modes in their experiments),they found that the degree of coherence may take any value from zero to one according to the cross-sectional sample of the outcoupled field.
8 3. Passive Cavity Mode Structure Behavior The particular transverse mode structure properties that are characteristic of passive unstable cavity geometries are now considered. This analysis is based on the general formulation of the transverse mode structure properties developed in the previous section. Of particular interest here is the eigenvalue behavior and the stationary cavity field structure as described by the canonical integral equation [cf. eq. (2.46)] 7nm%,(X,
Y ) = --
P(Xo9Yo) %n(XmYo)
[A(xg + yg) + D ( x 2 + y’)
- 2(~,,x+ y,y)]
IV,
s 31
24 1
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
where the overall propagation phase shift kL has been absorbed into the phase of yn. A simple illustration of the unstable cavity geometries encompassed by this model is given in Fig. 6 for both positive and negative branch cavities. The field u,, appearing in eq. (3.1) is that incident on the feedback aperture reference plane indicated in the figure. Two special geometries are of interest here, depending on the geometry of the cavity feedback aperture [as described by the reflection coefficient p(x, y ) ] .
FEEDBACK APERTURE REFERENCE PLANE
GEOMETRIC MODE PHASE FRONT
/ :
END MIRROR
L ZT+ POSITIVE BRANCH UNSTABLE CAVITY (M>1) FEEDBACK APERTURE REFERENCE GEOMETRIC MODE PLANE \: ./ PHASE FRONT
..
END MIRROR
L Z
T
d
NEGATIVE BRANCH UNSTABLE CAVITY (Mc-1)
Fig. 6 . Elementary positive- and negative-branch unstable cavity geometries. Notice the real focus at F in the negative-branch cavity.
242
UNSTABLE RESONATOR MODES
[IV, 8 3
The 6rst is the rectangular geometry for which p is separable as
The transverse field structure is then also separable in the x and y coordinate directions, where unm ( x , Y ) =
u!?’(~)u 2 ’ Q
*
(3.3)
On substitution of these two expressions into eq. (3.1) it is found that u?)(x) and u!$(y) both satisfy the linear homogeneous Fredholm integral equation
where the complex eigenvalue ynm appearing in eq. (3.1) is given by Ynm
= Yn Ym *
(3.5)
If one now extracts the (geometric) spherical phase variation from the field u, (see eq. [2.471) and performs the change of variable
(3.6) where the upper sign choice corresponds to a positive-branch cavity (M > 1) and the lower sign choice to a negative-branch cavity ( M <- 1). For the idealized case of a hard feedback aperture of linear dimension 2a1, centered on the cavity optical axis with unity reflectivity within the aperture and zero outside, the integral equation (3.6) becomes, under the change of variable 5 = 5/al, 5, = tA/Mal to normalized coordinates,
where N, = Ma:/B1 is the collimated Fresnel number of the cavity [cf. eq. (2.254)]. Notice that both 5 and 5, are measured on the same coordinate scale.
IV,
I 31
243
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
The other case of interest is that of a circular feedback aperture of diameter 2al with unity reflectivity within the aperture domain and zero outside. In that case one obtains [cf. eq. (2.53)]:
where the integration is taken over the unit circle C ((g change of variable to polar cylindrical coordinates
5,
=
ro cos 0,
qo = r, sin 0,
+ ~2 < 1). Under the
5 = r cos $, q
=
r sin $,
(3.9)
and with the azimuthal decomposition of the cavity field CO
4 5 , d = u(r, $1 = I=
C- w
(3.10a)
uI(r) e"+,
c 00
v ( G 9
rlo) = u(ro,@ = I=
4(ro) eile
9
(3.10b)
-CO
it is found that each azimuthal component radial mode satisfies the integral equation
(3.1 1)
Here the subscript lis the azimuthal mode index and the subscript n is the radial mode index. Both r and ro are measured on the same radial coordinate scale.
3.1. ASYMPTOTIC BEHAVIOR AND THE GEOMETRICAL APPROXIMATION
Consider first the asymptotic behavior of the integral equation (3.7) for the transverse rectangular cavity modes in the limit as the collimated Fresnel number Nc approaches infinity. The quadratic phase factor $(lo)= (5, - c/M)2 appearing in the integral of this equation possesses an interior stationary phase = i / M , at which $'(Q")) = 0 for 151 < IMI and $"(5$)) = 2. The point at i$)
244
UNSTABLE RESONATOR MODES
IIV, § 3
asymptotic behavior of the integral appearing in eq. (3.7) then may be determined from the stationary phase approximation (ERDBLYI[ 19561, COPSON t 19651)
f(B) e'
+ __ iV$'(B)
I"@(@)
-_ f (a) _ eiV@(a) i v+'( a)
(3.12)
for a < 5, < p, as v + oc). The first term in this asymptotic expansion is due to the interior stationary phase point, whereas the last two terms are due to the end point contributions. With this expansion the asymptotic approximation of eq. (3.7) is easily found to be
(3.13) for 151 < /MI as N,-,CO. Inclusion ofjust the 6rst term in the preceding expansion yields the geometrical mode equation of SIEGMAN and ARRATHOON [ 19671,which may be written as
(3.14) It represents the purely geometrical optics form of the integral equation (3.7) and expresses, in essence, the conservation of energy in a simple geometrical magnification of the cavity field. A physically reasonable set of eigenfunctions of (3.14) is given by the family of polynomials*
p(5)
=
5'
(3.15)
* Any function of the set vjG)( 5) = 5'" satisfies eq. (3.14) with gG)= M - ( W '+I 2 ) , where w can assume any value. Hence, eq. (3.14) actually admits a continuum of eigenvalues and is consequently insufficient to determine the transverse mode structure of the cavity. The requirement that I,$() be square integrable over [ - IMI,lMl]demands that Re w 3 0, but w is otherwise unspecified.
IV, § 31
245
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
with associated (real-valued) eigenvalues (3.16) where j = 0, 1, 2, . . . . The corresponding geometrical mode losses are then given by (3.17)
The higher-order (j> 1) transverse modes of this set are then seen to have significantly higher losses, since their intensity distributions are increasingly concentrated near the outer edges of the cavity feedback aperture and are thus more rapidly outcoupled from the cavity. One would then expect significant transverse mode discrimination in an unstable resonator if the effects of diffraction at the feedback aperture edge can be made weak. The geometrical modes of the two-transverse-dimensional rectangular cavity with the same magnification in each coordinate direction are then given by
with associated eigenvalues (3.19)
f o r m , n = 0 , 1 , 2,... . For the subset of even geometrical modes of eq. (3.15) ujG)( - 5) and with this result eq. (3.13) becomes
=
u>~)([),
(3.20)
for j
=
0, 2, 4,
. .. .
For the subset of odd geometrical modes
246
UNSTABLE RESONATOR MODES
[IV,§ 3
dG)(- 4) = - ujG)(C), and olne then obtains
(3.21) for j = 1,3,5, . .. .The second term in each ofthese two expressions represents the superposition of the cylindrical secondary waves diffracted from the opposite edges of the cavity feedback aperture. For large Fresnel number cavities the transverse mode structure is then seen to be dominated by its geometrical solution, which is completely determined by the cavity magnification, plus a secondary edge diffracted wave that is characteristic of both the cavity magnification and collimated Fresnel number. On the other hand, for small Fresnel number cavities one would expect the transverse mode structure to be dominated by the edge diffracted wave contributions. Consider now the asymptotic behavior of the integral equation (3.11) for the transverse cylindrical cavity modes in the limit as the collimated Fresnel number approaches infinity. If one first replaces the Bessel function appearing in the integrand of this expression by its large argument asymptotic approximation, given by* (see ABRAMoWrTZ and STEGUN [ 19641)
J1(0
-
1 ~
(2K t ) " 2
[exp i(5 - $Zn - in) + exp - i ( l - $ 1 -~in)],
(3.22)
the integral equation (3.11) then assumes the approximate form
(3.23)
* Notice that this expansion is not valid at the lower limit of integration r, = 0; however, the factor r, appearing in the integrand of eq. (3.11) helps to neutralize this singularity.
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
241
Consider first a positive-branch cavity (M > 1). In that case the first integral appearing in the preceding expression does not possess a stationary phase point interior to the integration domain [0,1], so that its asymptotic behavior is simply due to the end-point contributions. The second integral possesses an interior stationary phase point at rp) = r/M for 0 < r < M. Application of eq. (3.12) then yields the asymptotic approximation
(3.24) for 0 < r < M as N, -+ m . Inclusion of just the first term in this expansion yields the geometrical mode equation for the cylindrical cavity with solutions
(3.25) which are independent of the azimuthal index 1. For continuity at the origin one must require that n > 0 for I # 0. Notice that the discrimination against the higher-order transverse modes in favor of the lowest-order geometrical mode is the same in both the rectangular and cylindrical cases [cf. eqs. (3.19) and (3.25)]. The second term on the right in eq. (3.24) represents the secondary wave diffracted from the cavity feedback aperture edge. For large Fresnel number cavities the transverse mode structure is again seen to be dominated by its geometrical solution, which is determined by the cavity magnification, plus a secondary edge diffracted wave that is characteristic of both the cavity magnification and collimated Fresnel number. For a rectangular cavity this diffractive contribution was seen to be dependent on the rectangular symmetry (or parity) of the mode; for a cylindrical cavity it is seen to be dependent on the azimuthal symmetry of the mode through the index 1. For a negative-branch cavity ( M c - 1) the fkst integral appearing in eq. (3.23) now possesses an interior stationary phase point at rJ0)= - r/M for 0 < r < 1 MI, whereas the second integral does not possess a stationary phase point interior to the integration domain [0, 11. Application of eq. (3.12) then
248
UNSTABLE RESONATOR MODES
yields the asymptotic approximation
(3.26) for 0 < r < IM( as INJ -+ co. Inclusion of just the first term in this expansion yields the geometrical mode equation for the negative-branch cavity with solutions (3.27) An illustration of the feedback aperture edge diffraction phenomena appearing in the transverse field structure of a cylindrical, confocal unstable cavity is given in Fig. 7.* The cavity magnification is M = 1.44 and the collimated and equivalent Fresnel numbers are N, = 25.0 and Neq= 6.5, respectively, The solid curves in the figure depict the radial behavior of the azimuthally symmetrical (1 = 0) Mractive field structure (intensity and phase) incident upon the feedback aperture reference plane after a single round-trip propagation through the cavity, which results when a uniform plane wave field is initially incident upon that aperture. This diffracted field structure is readily seen to be characteristic of the collimated Fresnel number of the cavity. The dotted curves in the figure depict the relative intensity and phase structure of the dominant, azimuthally symmetrical radial mode supported by the cavity. This self-reproducing transverse field structure is seen to be characteristic of the equivalent Fresnel number of the cavity. Notice the pronounced central intensity core in the cavity mode about the origin and the depressed intensity level in the remainder of the mode field structure relative to that of the single iteration dBractive field. This accounts for the reduced power loss of the cavity * These cylindrically symmetrical diffractive field structures were numerically determined in a Fox-Li type of iteration procedure employing an angular spectrum representationof the single iteration propagation phenomena using a quasifast Hankel transform algorithm (see Appendix A) with 4096 radial sample points.
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
249
1.0 1
0.8 >
c
5 w
0.6: ::: h h i i ; ;; ;. ... ..: ::. i:i
-c 5
a
-I W
K
0 . 0
r
-0.4d 0
I
a1
r
Mat
Fig. 7. Transverse field structure incident on the outcoupling aperture-feedback reference plane of a confocal, cylindrical unstable cavity of magnification M = 144, collimated Fresnel number N , = 25.0, and equivalent Fresnel number Neq = 6.5. The solid curves depict the intensity and phase of the diffractive field structure after a single round-trip propagation through the cavity that results when a uniform plane wave field is incident upon the feedback aperture. The dotted curves depict the relative intensity and phase structure of the dominant, azimuthally symmetrical radial mode supported by the cavity.
mode (Yoo = 0.366) relative to that of the geometrical mode plane wave field (Yo$,G) = 0.518). Notice also the relative planarity of the phase structure over the feedback aperture domain 0 < r < a, : exterior to this feedback region the outcoupled field phase structure is seen to be conically divergent because of the diffracted field from the circular feedback aperture edge at r = a,. Other approximate techniques for determining the losses in unstable resonators have been described by KAHN[1966] and BARONE[1967] with
250
UNSTABLE RESONATOR MODES
[IV, § 3
similar results. BERGSTEIN[ 19681 and STREIFER [ 19681 have extended this analysis to include all orders in the expansion, but only in the limit as the diffractingaperture becomes infinite in extent. A more rigorous extension of the asymptotic approach described here that yields the large Fresnel number diffractive behavior of the transverse mode structure and eigenvalues characteristic of unstable cavity geometries with a sharply defined feedback aperture of finite extent has been given by HORWITZ [ 1973, 19761 for unstable resonators with a rectangular outcoupling aperture and by BUTTS and AVIZONIS [ 19781 for a circular outcoupling aperture (see also the criticism by LANDAU[ 19761). An entirely different approach to this problem, based on the [ 19691, has been coupled-mode theory of waveguides pioneered by WEINSTEIN taken by CHENand FELSEN[ 19731, with further development by SANTANA and FELSEN[ 19761, CHOand FELSEN[ 19791 and CHO,SHINand FELSEN [ 19791. In this alternative, hybrid approach the open cavity configuration is considered as a metallic waveguide structure in the direction transverse to the cavity axis, with the waveguide boundaries defined by the cavity end mirrors. The effects of finite mirror size are accounted for in a ray-optical treatment of the reflection phenomena at an open-ended waveguide. The later analysis of CHO, SHIN and FELSEN[1979] and CHO and FELSEN[1979] leads to expressions for the transverse mode eigenstructure that are equivalent to the asymptotic forms of HoRW1Tz [ 19731 and BUTTSand AVIZONIS [ 19781. This agreement between the asymptotic and coupled-mode approaches would then seem to dispel the critique by LANDAU[ 19761. The full asymptotic approach is now briefly considered. 3.1.1. Asymptotic approximation of the rectangular cavity eigenvalue equation
With the definitions t
=
(3.28)
xNc
the integral equation (3.7) becomes &I(()
=
( - i ~ ) ” ’ ~-’ ,u(Co)exp[it(to -‘>Idto, M
(3.30)
which is the same form as suggested by HORWITZ [ 19731, who considers the
IV,
B 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
25 1
asymptotic behavior of the integral (3.31)
as t -+ a.The first four terms of the stationary phase approximation of this integral are (HORWITZ[ 19731)
+ o(t-5/21,
(3.32) where the primes denote differentiationwith respect to the coordinate variable 5. Substitution of the first two terms of this expansion into eq. (3.30) yields the expression [cf. eq. (3.13)] +&u(-l)G+((),
(3.33)
where
(3.35)
corresponds to the choice of field parity u(5) = f u( - 5). where the The eigenvalue equation (3.33) may be readily solved by means of an iterative [ 19731). If one lets 5 -+ 5/M in eq. (3.33) and substitutes procedure (HORWITZ
252
IIV, 8 3
UNSTABLE RESONATOR MODES
the resulting expression back into eq. (3.32), there results
On repeated application of this procedure one obtains the expression
P u ( ~= u)
(in) + -
ED(-
[
1) G , (Mnl-1) -
+
XG*
(A) (3.36)
For n sufficiently large one may take
v(L) x v(O), M"
(3.37)
provided that u ( 5 ) is continuous at
5 = 0. Furthermore, from eq. (3.33) (3.38)
A - 1
Substitution of eqs (3.37) and (3.38) into (3.36) then yields ,i"u([)/u(
- 1) =
&[= + G, (L) I-1 w -'
Evaluation of this expression at 5
= -
1 then yields a polynomial equation for
I= M'/2p. However, ifone substitutes eq. (3.36) back into eq. (3.32), one finds that, because of the derivatives of u( 5) at the two endpoints, the higher-order terms in the asymptotic approximation of I(5) are at least comparable with the terms considered. In order to circumvent this difficulty HORWITZ [ 19731 introduces the set of functions
(3.40)
IV, 8 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
253
with n
M - 2 m , n = 0 , 1 , 2,...,
Mn=
(3.41)
m=O
in terms of which the inhomogeneous term in eq. (3.33) may be written as (3.42) The set of functions En(5) maps onto itself under the integral operator of eq. (3.30). This then suggests an expansion of the eigenfunctions u( 5) of the form (3.43) where the functions fn(5) are assumed to be slowly varying over the domain [ - l,l]. The integral of concern is now of the form (3.44) which possesses the asymptotic approximation (HORWITZ [ 19731)
(3.45) as t + CQ, where a,, =-,
Mn
(3.46)
M"-l
(3.47)
(3.48)
and E is as given in eq. (3.34). If one substitutes the expansion (3.43) into eq. (3.30) and equates the coefficients of&( 5) on either side, one then finds that, with eq. (3.45), the functions fn(5) will be slowly varying over the domain [- 1311.
IIV, § 3
UNSTABLE RESONATOR MODES
254
For practical purposes one uses in place of the infinite series (3.43) the finite sum N
o(5)
=
h(5) +
c [ W O f , ( Ok a-Of,(-0 1
9
(3.49)
n= 1
where h( 5) is taken to have the same parity as u( 5 ) and N is chosen to be sufficiently large to ensure that h( 5) is slowly varying over the interval [ - l,l]. Substitution of this expansion into eq. (3.30) with (3.45) and equating coefficientsof En(& 5 ) results in the finite set of coupled functional equations (HORWITZ[ 1973]),
(3.5oc) The set of equations (3.50~)may be iterated to yield, with the aid of eq. (3.46), 1
&L+l ( 0 = ML'2
n
--fl(.fn(O)9
=
192,. .. 3
(3.51)
where Sn(0
= r1(r2(. *
.(rn(0)))
f
(3.52)
Substitution of eq. (3.51) into eqs. (3.50a) and (3.50b) then gives
(3.5 3a)
(3.5 3b)
c
IV, 31
PASSIVE CAVITY MODE STRUCTURE BEHAVlOR
255
From eq. (3.48) with n % 1 one finds that d,’ N 1; one may then take d,‘ as an adequate approximation in the final analysis. From eq. (3.53b) the functional form of fl([) is found to be
=
1
(3.54) where
(3.55) with
(3.56) From eq. (3.53a) one has, using h(1) = & h( - l),
(3.57) where
(3.58) By analogy with the iterative procedure applied to eqs. (3.33) and (3.39, one finally obtains (cf. eq. [3.39])
(3.59) where N’ is chosen to be sufficiently large to ensure that h([/MN’)z h(0). By taking N large enough, H , ([) may be made as slowly varying over [ - 1,1] as ill $so be slowly varying over this interval. Equations desired, so that h(z) w (3.53b) and (3.51) then ensure that the set of functions f,( 5), n = 1 , 2 , 3 , . ..,
256
UNSTABLE RESONATOR MODES
[IV,8 3
will also be slowly varying over [ - 1,1]. Hence, by retaining only the first two terms in the asymptotic expansion of the integral I,( c), one is indeed neglecting only terms that are O(t-3/2),as was desired. There were two major steps in this derivation: First, N integrations were’ performed over the set of functions En( 5 ) until the inhomogeneous term in the functional equation was reduced to EN(5), which can be made slowly varying by choosing N to be sufficientlylarge. Second, eq. (3.57) was iterated N times to derive h( C ) with the requirement that EN(l)/M” - m const. These two steps may be combined with the single requirement that E N ( [ )= const. (HORWITZ [ 19731). Equation (3.59) then requires that h( 5) becomes aconstant. For the symmetrical case, eq. (3.57) then yields
( I - l)PS(J) - &A+;
1
=
(3.60)
0,
where (3.61) (3.62)
Equation (3.60) then yields the polynomial equation
IN+ + (&A; - 1 ) P + &(A; - A ; ) x N (3.63)
+..*+&(A;+l-A;)=O.
For the antisymmetrical case, H - ( C ) becomes vanishingly small in the asymptotic limit as N - r c g , so that h ( l ) vanishes and eq. (3.53b) yields
Pk(2) = 0 ,
(3.64)
which yields the polynomial equation x N + & ~ ; x N - l
+
... + & A , = o .
(3.65)
The roots of eqs. (3.63) and (3.65) then yield the asymptotically approximate eigenvalue spectrum for the symmetrical and antisymmetrical modes, respectively, of the integral equation (3.30). In the geometrical optics limit as t + w , in which limit 8 - r 0, eq. (3.60) for the even-parity case yields the nontrivial solution 1, so that p = 1/M1/2, which is the geometrical mode eigenvalue that corresponds to a uniform amplitude spherical wave field distribution over the feedback aperture domain [ - 1,1]. The higher-order geometrical modes postulated by SIEGMAN[ 19651,
x=
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
251
however, are not recovered by this full asymptotic treatment. Of the set of polynomial modes obtained by neglecting the endpoint contributions to the diffraction integral (BERGSTEIN[ 19681, STREIFER[ 1968]), only the zerothorder mode approximates a solution of the full equation. This is reflected in the fact that only for this geometricalmode does the diffraction integral exist when the limits are extended to infinity (SANDERSONand STREIFER[ 1969a1). 3.1.2. Asymptotic approximation of the cylindrical cavity eigenvalue equation With the definitions fi=Mjj,
(3.66)
t=wc,
(3.67)
the integral equation (3.11) for each azimuthal component radial mode of a cylindrical unstable cavity assumes the form
)2"o(r)= 2t( -iY+'
1,'
:( ) [*(ri + I-3 1
rou(ro)J, -rro
exp it
dr,, (3.68)
which is the same form as that suggested by BUTTSand AVIZONIS [ 19781, who consider the asymptotic behavior of the integral I ( r ) = ( - iy+ ' 2 t
1,'
;( ) [ (rg + -
rou(ro)J, -rr,
exp it
;)Id?,
(3.69)
as t + m . Following the analysis from eqs. (3.22) to (3.24), the asymptotic approximation of Z(r) is found to be
(2
=0 -
+ u(l)E,(r)
(3.70)
258
[IV,8 3
UNSTABLE RESONATOR MODES
as t -,00. The singularity at the origin appearing in the function E,(r) may be removed in an ad hoc fashion by assuming that the radical times the two trigonometric functions, which is recognized as the sum of the asymptoticforms of two Bessel functions, should be replaced by those Bessel functions (BUTTS and AVIZONIS[ 19781). This substitution then yields -
E,(r) = -
( $)]
( - i>' exp [it 1 + 1 - (r/M)2
(g)-
i
JI+ I
(31 (3.71)
This technique of "fixing up" the singularity is similar to that used by KELLER [ 19621 to remove the caustic on the optical axis generated by the diffracted edge rays from a circular aperture. Beginning with the function E1(r),one can obtain the set of functions EJr) by propagating E,(r) through the unstable cavity n - 1 times using the stationary phase approximation. The result is
(3.72)
(3.73) with
(3.74)
The integral of concern is now of the form [cf. eq. (3.44)] I&) = ( - iy+ 2t
roEn(r,)Jl($ ro) exp[ir(ri 0
'>1
+ M2
dr, , (3.75)
which possesses the asymptotic approximation (BUTTSand AVIZONIS [ 19781)
IV, 8 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
259
as t -,co for r < M . The first term in this expression arises from the interior stationaryphase point contribution at r, = r/M, whereas the second term arises from the endpoint contribution. For the azimuthally symmetrical case (I = 0), one assumes the solution N n= 1
where Nis chosen to be sufficientlylarge so that EN+l(r) is nearly constant over the interval [0,1]. Substitution of this expansion into eq. (3.68) and using the preceding relations yield the expression
(3.78) On equating the constant terms on each side and equating coefficients of E,(r), one obtains
-
a,+
, = a,lA
;I-
1 = a,EN+, ,
-
*
= a,/A",
(3.79a) (3.79b)
so that a, =
(d - 1)P-
(3.80)
7
EN* 1
and
&, = 1 +
c a,E,(l). N
,=
(3.81)
1
Using eq. (3.80) in (3.81) then gives N
XN(X - 1 ) = EN+ + (X- 1) C
E,(l)
IN-",
(3.82)
n= 1
which is a polynomial equation in Xof order N + 1. Given mode is determined from eqs. (3.77) and (3.80). For I # 0 one assumes the solution
the corresponding
(3.83)
260
UNSTABLE RESONATOR MODES
tIV, $ 3
where N is again chosen to be saciently large so that E N + l ( r )is nearly constant over the interval [0,1]. The resulting equations are then 1 a,==,
(3.84)
1"
and N
IN= C
E,,(1)XNTn
(3.85)
n=l
Equation (3.85) is a polynomial equation in fi of order N. Given corresponding mode is determined from eqs. (3.83) and (3.84).
the
3.2. EIGENVALUE BEHAVIOR
The eigenvalue spectrum characteristic of passive unstable cavities with a sharply defined feedback aperture has been studied with both numerical and analytical techniques. The first extensive studies were carried out by SIEGMAN and MILLER[ 19701, who employed a Prony routine to extract the eigenvalue spectrum numerically from the Fox-Li type of convergencehistory of the cavity mode structure. Modifications of this numerical approach have been developed by MURPHYand BERNABE [ 19781, GERCK [ 19791, GERCK and CRUZ[ 19791 and LATHAM and DENTE[ 19801. A spectral approach based on the discrete Fourier analysis of the cavity field iteration history has also been described by FEITand FLECK[ 19811. These purely numerical studies have yielded results that are in excellent agreement with the results of both the asymptotic analyses of HORWITZ[ 19731 and B u n s and AVIZONIS [ 19781 and the coupled-mode waveguide analysis of CHOand FELSEN [ 19791 and CHO, SHINand FELSEN [ 19791. The present section provides an overview of these results. Consider first the eigenvalue behavior correspondingto the transverse mode structure in a single transverse dimension of a sharp-edged rectangular aperture cavity. The eigenvalue magnitude as a function of the equivalent Fresnel number for an M = 2.9 unstable cavity is depicted in Fig. 8 for both the symmetrical (even parity) and antisymmetrical (odd parity) modes; these results were determined from the asymptotic expressions (3.63) and (3.65), respectively (see HORWITZ [ 19731). Notice that the lowest-loss mode (i.e., the mode with the largest value of y = 171) is always symmetrical and that its eigenvalue magnitude oscillates about the geometrical value yG = l/M1/' This . oscillation is periodic in the equivalent Fresnel number of the cavity with
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
26 1
0.7
- I/MI /2
0.6
0.5 0.4
Y 0.3 0.2 0. I
a 0
6 0.6
0.5
8
10
1
12
14
16
18
20
*=I
0.4 Y 0.3 0.2
0.1
{ b
0 6
8
10
12
14
16
18
20
Neq Fig. 8. Eigenvalue magnitude behavior as a function of the equivalent Fresnel number for (a) the symmetrical and (b) the antisymmetrical mode of a single transverse-dimensional M = 2.9 unstable cavity. (Afer HORWITZ[1973].)
maxima occurring at (or very near to) integer plus one half values of Neqand minima occurring at (or very near to) integer values of Neq[see the discussion following eq. (2.55)]. Mode crossing points for the dominant mode structure appear only at integer values of Neqbut not at every integer value; at each of these crossing points multimode operation will occur. The amplitude of oscillation of the dominant eigenvalue magnitude about the geometrical value is found to decrease with increasing equivalent Fresnel number and vanishes
262
[IV, 8 3
UNSTABLE RESONATOR MODES
in the asymptotic limit as Neq +co, as described by eq. (3.63). The detailed study by HORWITZ 119731 has also shown that the only eigenvalue which separates from the remaining eigenvalue spectrum is the dominant symmetrical eigenvalue, as illustrated in Fig. 8. Finally, the next-to-lowest-loss mode is found to be either symmetrical or antisymmetrical but at crossing or cusping points it is always that of a symmetrical mode. The eigenvalue behavior corresponding to the transverse mode structure of a cylindrical unstable cavity with a sharp-edged circular feedback aperture is depicted in Figs. 9 to 13. These results are a mixture of purely numerical calculations (as may be found in SIEGMAN and MILLER[ 19701) and the results of the asymptotic expressions (3.82) and (3.85) resulting from the work of BUTTSand AVIZONIS[ 19781. The numerical and asymptotic approaches have yielded results that are in excellent agreement with each other; surprisingly,the asymptotic results (obtained in the limit as Neq +a)are found to be reasonably accurate down to Neq = 1. The eigenvalue magnitude behavior as a function of the equivalent Fresnel number of an M = 2 cylindrical unstable cavity is depicted in Fig. 9 for the azimuthally symmetrical (I = 0) mode structure and in Fig. 10 for both the I = 0 and the 1 = 1 mode structure. Notice that an I = 0 mode is always the lowestloss mode and that its eigenvalue magnitude oscillates about the geometrical
0.8
0.6
Y" 0 0.4
0.2
0
0
2
4
6
10
8
12
14
16
eq
Fig. 9. Eigenvalue magnitude behavior as a function of the equivalent Fresnel number for the azimuthally symmetrical ( I = 0) modes of a cylindrical, M = 2 unstable cavity. (Atter SIEGMAN and MILLER[1970] and Bums and AVIZONIS[1978].)
18
IV, § 31
263
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
1
0.6
1=0
- -I M
0.4
yno
0.2
y 1
0
20
22
24
26
28
30
Neq
0.6
1
I= I
0.4
7"I
- -I M2
0.2
0
20
22
24
26
28
30
Neq
Fig. 10. Eigenvalue magnitude behavior as a function of the equivalent Fresnel number for (a) the I = 0 and (b) the I = 1 mode of a cylindrical, M = 2 unstable cavity. (After BUTTS and AVIZONIS [1978].)
value yG = 1/M. As in the rectangular cavity case, the eigenvalue magnitude of the lowest-loss mode oscillates periodically about the geometrical value with maxima occurring at (or very near to) integer plus one half values of Neqand minima occurring at (or very near to) integer values of Neq, at which points multimode operation will occur. The amplitude of oscillation of the dominant eigenvalue magnitude about the geometrical value is found to decrease with increasing Fresnel number and vanishes in the asymptotic limit as Neq+ 00, as
UNSTABLE RESONATOR MODES
264
[IV, 8 3
described by eq. (3.82). The only major difference in eigenvalue behavior between the cylindrical and the rectangular cavity is that the dominant eigenvalue has never been found to separate from the remaining eigenvalue spectrum for the cylindrical cavity geometry as it does for the rectangular case (Bums
0.3
0.2
Y"0
0 .I
0
01
Fig. 1 1. Eigenvalue magnitude and phase behavior as a functionofthe equivalentFresnel number for the azimuthally symmetrical (I = 0) modes of a Cylindrical, M = 5 unstable cavity. (After SIEGMANand MILLER[1970].)
IV, 5 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
265
and AVIZONIS[ 19781). Finally, the next-to-lowest-loss mode is found to be either symmetrical (I = 0) or nonsymmetrical (I # 0), but at crossing points in the dominant eigenvalue magnitude it is always that of a symmetrical mode. The complex eigenvalue behavior is depicted in Figs. 11 and 12 for the I = 0 mode structure of an A4 = 5 unstable cavity (see SIEGMANand MILLER [ 19701). The associated periodicity of both the eigenvaluemagnitude and phase in terms of the equivalent Fresnel number of the cavity is readily evident in Fig. 11. Notice that when Neq= integer + $ and the dominant eigenvalue is at (or very near to) its relative maximum, its phase is always nearly zero. The eigenvalue trajectories in the complex 7-plane as a function of Neqare shown in Fig. 12, where the Fresnel number values are indicated in parentheses. These trajectories circle periodically about the origin in a clockwise direction and decrease inwards toward the geometrical value yG = l/M on the real axis as Neqincreases. Similar behavior is observed in the rectangular aperture case with the additional phenomena that when mode separation occurs there, the
Fig. 12. Complex eigenvalue trajectoriesin the complex 7-plane as a function of the equivalent Fresnel number (whose values are indicated in parentheses) for the azimuthally symmetrical ( I = 0) modes of a cylindrical, M = 5 unstable cavity.
266
[IV, § 3
UNSTABLE RESONATOR MODES
dominant eigenvalue ceases to circle about the origin and begins to circulate [ 19731). about the geometrical value on the real axis (HORWITZ The validity of these theoretical results has been demonstrated expenmentally by FREIBERG, CHENAUSKY and BUCZEK[ 19721, who measured the loss associated with the dominant mode structure of an unstable confocal COz laser system as a function of the equivalent Fresnel number of the cavity. The results of their relative power loss measurements for an M = 2 cavity system have been translated into a dominant mode eigenvalue magnitude and plotted in Fig. 13 along with the theoretical eigenvalue curves. Their results clearly exhibit a discontinuous change in the slope of the dominant eigenvalue behavior at Neq E 1.1, as predicted by theory. This may then be taken as an indirect verification of a change in transverse mode structure at this point. The dominant eigenvalue magnitude behavior of the I = 0 mode structure as a function of the cavity magnification M of a cylindrical unstable cavity is illustrated in Fig. 14. Each curve is at a fixed value of the cavity equivalent Fresnel number, as indicated in the figure. The upper set of curves is for integer
0. 75
0.70
yn0 0.65
0.60
0.55
0
.2
.4 .6
.8
1.0
1.2
1.4 1.6
1.8
2.0 2.2
Neq
Fig. 13. Experimental measurementsof the dominant mode eigenvalue magnitude for an M = 2 unstable resonator as a function of the equivalent Fresnel number of the cavity. The theoretical curves for the eigenvalue spectrum are included for purposes of comparison.
IV, 8 31
267
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
= 0.5 = 1.5 = 6.5
M
M
Fig. 14. Eigenvalue magnitude behavior as a function of the cavity magnification M at several fixed values of the equivalent Fresnel number for the azimuthally symmetrical ( I = 0) modes of a cylindrical unstable cavity.
plus one half values of Neq for which the dominant eigenvalue is at (or near to) a local maximum,"whereas the lower set is for integer values of Neqfor which the dominant eigenv$ue is at (or near to) a local minimum, and a mode crossing occurs. It is readily evident from these curves that the geometrical eigenvalue yG = 1/M is obtained in the asymptotic limit as Neq-+00.
268
[IV,§ 3
UNSTABLE RESONATOR MODES
As was mentioned earlier, the higher-order geometricalmodes as well as their associated eigenvalues are not fully recovered in the asymptotic theory described in the previous section. For example, from eq. (3.25)one would expect that the dominant eigenvaluemagnitude of the 1 = 1 mode structure of a cylindrical cavity would, in some fashion, approach the limiting value l/M2 as Neq 400; inspection of Fig. 10 shows that this most likely is not the case. On the other hand, from eq. (3.16)one expects that the dominant eigenvalue magnitude of the antisymmetrical mode structure in a single transverse dimension of a rectangular aperture cavity would approach the limiting value l/M3/’; inspection of Fig. 8 shows this to be the case. Numerical calculations of the transverse
0
I
2
3
4
5
6
7
j Fig. 15. Eigenvalue magnitudes as a function of the mode index j for both the symmetrical and the antisymmetrical mode of a self-imaging unstable cavity with (MI= 2. The solid curve describes the geometrical behavior given by y, = IM(- u + ‘I2?
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
269
mode structure and associated eigenvalue spectrum for a one-dimensional negative -branch unstable ring cavity with a self-imaging aperture, for which the effective cavity length vanishes and the Fresnel number is infinite, confirm this behavior for both the dominant eigenvalue magnitude of the antisymmetrical mode structure and the first higher-order eigenvalue of the symmetrical mode structure (PAXTONand SALVI [1978]). The eigenvalue magnitudes as a function of the mode index j for both the symmetrical ( j = 0, 1,2, ...) and antisymmetrical( j= 1,2,3, .. .) modes of this self-imagingcavity (as calculated by PAXTONand SALVI[1978]) are illustrated in Fig. 15 along with the - ( J + ' I 2 ) for JMJ = 2. The geometrical mode approgeometrical curve % = JMJ ximation is then seen to fail for j > 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
UNSTABLE RESONATOR MODES
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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
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
UNSTABLE RESONATOR MODES
[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,
The radii of the cavity Fresnel zones at the feedback aperture are then given bY n
+f
'I2 9
(3.90)
where n = 0, 1,2,. .., subject to the inequality n + f < Neq. These radial Fresnel zones are indicated in Fig. 18, which depicts the transverse field structure of the dominant mode incident on the feedback aperture for two Neq = 6.5 (so that f = $) confocal unstable cavities, one with M = 1.44 and the other with M = 2. The large central intensity cores of the intracavity mode structures illustrated here axe seen to occupy the central Fresnel zone (n = 0) of the cavity at the feedback aperture. It is this central core region that the intracavity field propagates out from and constructs the remainder of the cavity field. The resulting edge-scattered wave component from the feedback aperture edge that propagates back into the cavity provides the diffractive feedback to this central core region and accounts for the Fresnel number dependence of the cavity mode structure (ANAN'EV[ 1972, 19751, SHERSTOBITOV and VINOKUROV [ 19721). ANAN'EV[ 19711 has pointed out that a laser with an unstable cavity corresponds to a system composed of a driving generator and an amplifier with a matching telescope between them. The role of the generator is played by the central intensity core that occupies the central Fresnel zone of the cavity and the role of the amplifier by the remaining peripheral zone of the cavity, with the edge diffracted field at the feedback aperture edge providing the controlling feedback to the central intensity core. It is this mechanism of diffractive
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
IV, § 31
215
1.0
7r-
0.8 >
0
-7-
7-
1.0
0.8 >
0.2
-7-
0
Fig. 18. Radial behavior ofthe relative intensity and phase structure ofthe dominant, azimuthally symmetrical mode incident on the feedback aperture plane for two confocal unstable cavities with Neq = 6.5, with magnifications M = 1.44 for the upper figure and M = 2.0 for the lower figure. The solid curves depict the intensity distributions and the dashed curves depict the phase distribution in each case. The dotted lines indicate the boundaries of the cavity Fresnel zone structure over the feedback aperture in each case.
feedback into a converging wave field and its interaction with the magnifying or diverging cavity field that determine the diffractive properties of the cavity mode structure. It is important to recognize that the converging and diverging wave fields inside the cavity are intimately related to each other, as is evident
276
UNSTABLE RESONATOR MODES
[IV, § 3
in both the geometrical theory described in 0 2.2.1 and the diffractive theory described in 0 2.3.1. Fundamentally, the converging wave field is transformed by diffraction into the diverging wave solution (ISAEV,KAZARYAN, PETRASH and RAUTIAN[ 19741). As stated by ANAN’EV[ 19751: “Both converging and diverging waves form two different complete systems of functions which can be used equally satisfactorily for expanding an arbitrary signal as a series. Expansions produced by these two methods naturally give identical final results.” These expansions are just those given in eqs. (2.151) to (2.153) and eqs. (2.155) to (2.157) (OUGHSTUN[ 1982a1). This interrelationship between the converging and diverging wave fields in an unstable resonator can be used to alter the transverse mode structure properties through the introduction of internal reflecting surfaces within the cavity (as is presented by the gain tube windows). Such internal reflecting surfaces can produce additional converging wave fields, which tend to increase the divergence of the outcoupled field distribution (ANAN’EV, GRISHMANOVA, and SVENTSITSKAYA [ 19761) and, hence, should be avoided. A PETROVA beneficial method of altering this interrelationship is to apodize the feedback aperture; the effects of apodization are considered in § 3.4. Figure 18 also illustrates the magnification dependence of the dominant mode structure at a fixed value of the cavity equivalent Fresnel number. The diffraction structure of the mode is clearly not a simple linear mapping in the cavity magnification. However, the relative mode loss is found to depend simply upon M, at least to a very good approximation, as seen in Fig. 14.
3.3.2. Transverse mode hierarchy supported by an unstable cavity The final point of interest regarding the fundamental transverse field structure of an unstable resonator is the hierarchy of modes that are capable of being supported by a given cavity. The most extensive study to date on this matter [ 1977 1, who have computed has been given by O’NEIL,HEDINand FORGHAM the first 36 transverse modes of an unstable cavity using a Fox-Li iteration procedure together with a Gram-Schmidt orthogonalization procedure (e.g., see COURANT and HILBERT [ 19531). The resonator they considered was an M = 2.5 nonconfocal, cylindrical unstable cavity with equivalent Fresnel number Neq = 3.17 and collimated Fresnel number N, = 7.53. In addition to a circular feedback aperture of diameter d , = 3.6 cm, the cavity had two additional circular apertures centered upon the cavity optical axis: The first was situated in the feedback aperture plane with a diameter equal to the geometrical mode diameter of the exiting field at that plane, and the second was situated
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
277
in front of the cavity end mirror plane with a diameter equal to 1.09 times the geometrical mode diameter at that plane. ONEIL,HEDINand FORGHAM [ 19771 employed a cylindrical coordinate diffraction code to determine the set u,,(r) of radial modes, labeled by the index m, at several individual values of the azimuthal index n, where the angular variations e-in@are cosinusoidal in form. The composite mode structure (both radial and azimuthal behavior) is denoted there by TEM,, . To determine the azimuthally symmetrical (n = 0) set of radial modes, the lowest order radial mode uoo(r) was first determined numerically. The next higher-order radial mode uol(r)was then determined using the Gram-Schmidt orthogonalization procedure, which was continued in order to generate an orthogonal set of radial modes uoo(r), uol(r), uO2(r), . . . . The same procedure was used for each value of the azimuthal index n and was continued in each case until all the radial modes at a given azimuthal index n had been determined. A total of six radial modes was found for each of the first six values of n. A sampling of the results of this analysis is given in Fig. 19. A pair of TEM,, modes is shown in each figure: The upper plot is the intensity distribution, and the middle plot is the phase structure of the transverse field distribution incident upon the cavity feedback aperture, whereas the lower plot is the far-field intensity distribution due to the corresponding outcoupled field. The calculated eigenvalue magnitude y,, of each of these modes is given in Fig. 20, both as a function of the radial index m at fixed values of n and as a function of the azimuthal index n at fixed values of m. The following ordering for these 36 modes is then obtained: 1/00
> Yo1 > 1/10 > 1 / 1 1
= 1/41
> 751 > 742
’
1/20
’
723
> Yo2
> 1/04 > Y52 > 1/14 > 1/53 > 1/43 > ’YO5
’
> 734 > 715 > Y33 > 1/44 > 1/35 > Y54 725 > y45 > Y55 * As is the general rule, the first two lowest-loss modes are azimuthally symmetrical (n = 0), followed by the first two radial modes with azimuthal index n = 1 (cos 8 dependence). At a fixed value of the azimuthal index n, the eigenvalue magnitude y,,, typically decreases with increasing radial order m,as seen in the upper graph of Fig. 20. An exception to this typical behavior occurs for n = 3. Furthermore, at a fixed value of the radial index m,the eigenvalue magnitude y,,,,, typically decreases with increasing szi.nuthal order n, as seen in the lower graph of Fig. 20, although this dependence is not as pronounced as the dependence on the radial index. = y24
“V, § 3
UNSTABLE RESONATOR MODES
278
A
\
Fig. 19 (pp. 278-285). Intracavity intensity, phase and far-field intensity patterns for several of the lower order modes of an M = 2.5, Neq = 3.17 nonconfocal cylindrical unstable cavity. The near- and far-fieldintensity patterns are normalized to those ofthe TEM,, mode. (Atter ONEIL, HEDINand FORGHAM [1977].)
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
219
EAK INT = .25
%
Fig. 19 (continued)
PEAK IN1= .68
280
UNSTABLE RESONATOR MODES
I
Fig. 19 (continued)
PASSIVE CAVITY MODE STRUCTURE :BEHAVIOR
M A X PHASE = 1 9 nrod
V r
MAX PHASE = 4 mod
w
Fig. 19 (continued).
28 1
282
UNSTABLE RESONATOR MODES
30
N - 18
Fig. 19 (continued).
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
Fig. 19 (continued).
283
284
UNSTABLE RESONATOR MODES
Fig. 19 (continued).
[IV, § 3
PASSIVE CAVITY MODE S T R U C T U R E BEHAVIOR
PEAK INT = 31
PEAK IN1 = .08 QUAL1W = 15 9%
Fig. 19 (continued).
285
286
UNSTABLE RESONATOR MODES
[IV, § 3
0.6
0.4 Gm
0.2
0 0
0.6
I
2
4
3
5
RADIAL INDEX
rn
..... ...............................
4 ..............*-----------*m= ....................... em.5
1
0.4 Ynm
0.2
0 0
I
2
3
4
5
AZIMUTHAL INDEX n
Fig. 20. Eigenvalue magnitude behavior for the TEM,, modes of an M nonconfocal cylindrical unstable cavity.
=
2.5, Nes
=
3.17
The peak intensity and beam quality* of the far-field intensity distribution due to the outcoupled field structure of these modes are depicted in Fig. 2 1 as a function of the azimuthal index n at fixed values of the radial mode number m. Although these far-field performance measures typically decrease with increasing azimuthal order, the behavior for the m = 3 radial mode is a striking exception. In general, it cannot be assumed that a lower-order mode possesses a better peak far-field intensity and beam quality than a higher-order mode. * The beam quality is taken here as the fractional integrated far-field intensity contained within the Airy disc that is defined by the far-field diffraction pattern due to the outcoupled field of the TEM, mode.
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
287
>
$
1.0
z W
+
z 9
0.8 0.6
W
rI U
Y
i
0.4 0.2
o 0
I
2
3
4
AZIMUTHAL INDEX n 0.6
k
0.5
13
z
0.4
{
0.3
3 w
0.2
dc
0.1
U
i
I
2
3
4
AZIMUTHAL INDEX n
Fig. 21. Far-field peak intensity and beam quality behavior as a function of the azimuthal mode index n for the outcoupled TEM,, modes of an M = 2.5, Neq = 3.17 nonconfocal cylindrical unstable cavity.
3.4. APERTURE APODIZATION AND INTRACAVITY SPATIAL FILTERING
The dzractive mode structure properties of a passive unstable cavity may be modified either through an alteration of the dzractive feedback from the feedback aperture edge of the cavity (aperture apodization) or through a modification of the spatial spectrum of the intracavity field (spatial filtering). The effects of apodizing the feedback aperture have been considered by ZUCKER[ 19701, ANAN’EV and SHERSTOBITOV [ 19711, SHERSTOBITOV and VINOKUROV[ 19721, and most extensively by BEL’DYUGIN, ZEMSKOV, MAMYAN and SEMINOGOV [ 19741. A coupled-mode waveguide treatment has
288
UNSTABLE RESONATOR MODES
[IV, § 3
also been given by SANTANA[ 198 11. The special case of an unstable cavity with a Gaussian reflectivity feedback mirror has been treated by CASPERSON and LUNNAM[1975], GANIELand HARDY[1976], MCCARTHYand LAVIGNE [ 1984, 19851, and LAVIGNE,MCCARTHY and DEMERS [ 19851; this analysis has recently found direct application in the transverse mode analysis of optical cavities with phase conjugate mirrors (see SIEGMAN, BBLANGERand HARDY [ 19831). The effects of intracavity spatial filteringin an unstable ring cavity have been considered by KORZHENEVICH, RATNERand SOLOV'EV [ 19721, POZZO, [ 19791, SHENG[ 19791, POLLONI, SVELTOand ZARAGA[ 19731, SOUTHWELL and most extensively by OUGHSTUN,SLAYMAKER and BUSH[ 19831 (see also OUGHSTUN[ 19841). A variation of this technique for standing wave cavities has been given by ANAN'EV,GRISHMANOVA, PETROVA and SVENTSITSKAYA [ 19751. Both of these approaches can signficantly reduce the Fresnel number dependence of the cavity mode structure, thereby increasing the transverse mode discrimination properties of the cavity. The analysis here begins with a review of the detailed analysis of the apodization effects given by BEL'DYUGIN, ZEMSKOV, MAMYANand SEMINOGOV [ 19741 and concludes with a description of the analysis of intracavity spatial filtering given by OUGiISTUN, SLAYMAKER and BUSH[ 19831.
3.4.1. Aperture apodization in rectangular unstable cavities The integral equation describing the transverse mode structure in a single transverse dimension of a rectangular aperture unstable cavity with a separable reflectivity coefficient p(x, y ) is given by eq. (3.6). Under the change of variable ( = (A13)'/2x, (A = M(A13)'/2ythis equation becomes
which may also be expressed as
(3.92) where
(
u(x) = ij(x) exp in M; ~
lX2).
(3.93)
IV, J 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
289
This is the form of the equations used by BEL’DYUGIN,ZEMSKOV,MAMYAN and SEMINOGOV [ 19741. They consider reflection coefficients p,b) that belong to the following classes : (1) Mirrors with sharply defined edges (3.94a)
fl
and N = a:/BL = NJM in the transverse coordinate system of where a = eqs. (3.91) and (3.92). In unnormalized coordinates a = a, the radius of the feedback aperture. (2) Mirrors with a truncated Gaussian reflectivity
where u is, in general, a function of a, and where p2Cy) is given by eq. (3.94a). (3) Mirrors with an exponentially smoothed edge (3.94c) where p 2 ~is) given by eq. (3.94a)’ E is the “edge smoothness parameter” and is generally a function of a (usually E 4 a), and where the symbol * denotes the convolution operation. (4) Mirrors with a partially smoothed edge of degree k (3.94d) where p2(y) is again given by eq. (3.94a), and where p,(y) is a function that is ~ where E is a function concentrated in the interval ( - E, E ) and equals 1 / 2 there, of a with ke 4 a. Each reflection coefficient p u b ) is a function that can be differentiated k - 1 times at the mirror edges. If one introduces the complex g-parameter [see eq. (2.44a)I
g = g + $u2,
(3.95)
where c( # 0 only for the case of the reflection coefficient of the type (2), then the complex cavity magnification may be expressed as [see eq. (2.44b)I
M
=g+
(g’ - 1)1/*.
(3.96)
290
UNSTABLE RESONATOR MODES
[IV, § 3
When a = 0, the parameter M is real and is simply the usual cavity magnification. The integral equations (3.91) and (3.92) clearly hold in this case. They also apply to the case (3.94b) where the function p,(y) in the integrands of these equations is simply p2(y) and M is given by eq. (3.96) with (3.95). 3.4.1.1. Infinite extent, uniform reflectivity feedback aperture Consider fist the solutions of eq. (3.92) for the case a = co. Assume that the Fourier transformation V ( 0= 28
Sm
(3.97)
u(x) eicxdx
-a
of the function u(x) exists. Application of this transformation to eq. (3.92) then yields the functional equation (3.98) whose analytical solutions are given by (3.99a)
(3.99b) for n
=
0, 1,2, . . . . The inverse Fourier transformationof eq. (3.99a) then yields
for n = 0, 1,2, .. . . This result was also obtained in a different approach by BERGSTEIN[ 19681 and STREIFER[ 19681. 3.4.1.2. Finite, shalp-edged feedback aperture For the case (1) of a sharp-edged feedback aperture with a fjnite value of a, the Fourier transformation of eqs. (3.91) and (3.92) yields the equivalent integral equations in the spatial frequency domain
a
y V ( t ) = L e x p ( - i4nM L )
-a
V ( i - v ) w8 V) d v ,
(3.101)
IV, J 31
29 1
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
YP(<)
=
a e x p (
-I-). M t 2
471
1
*
P(M<- -)v sin (av)
dv.
(3.102)
71V
--Q)
Notice that
in which limit the integral equation (3.101) reduces to the functional equation (3.98). With eqs. (3.93) and (3.100) one finds that the solution of eq. (3.102) for a = co in the space domain is x2)
n(M2 - 1) d" exp (i dx" M
=2nHn[(i n(M2 - 1))lt2x],
(3.103)
where A is a normalization coefficient and d" Hn(5) = ( - 1)"erz-exp(-52) d 5"
(3.104)
are the Hermite polynomials. The Fourier transform of eq. (3.103) is (3.105) where Ak = S(")(<) is the kth derivative of the delta function S(5). It follows from eqs. (3.91) and (3.92) that the functions v(x) are analytical, approach zero at infinity, and belong to the class L, . In essence eqs. (3.91) and (3.92) are defined only in the interval ( - a, a) for the sharply defined aperture case. However, because of the analyticity of the kernel, the functions specified in the interval ( - a , a) are defined along the entire axis ( - 00, co). These equations may then be viewed as an analytical continuation of the functions onto the entire real axis. Furthermore, the eigenvalue 7 appears solely as a result of the reproducibility condition imposed on the cavity field incident on and defined over the cavity feedback mirror, that is, in the interval ( - a, a). Indeed,
292
UNSTABLE RESONATOR MODES
[IV, § 3
the eigenvalues p are formally given by
p=
I”,
u*(x) Ru(x) d x (3.106)
7
u*(x) u(x) dx where R is the integral operator for a single round-trip propagation through the cavity. Thus, in order to obtain the characteristic behavior of the cavity mode structure, it is sufficient to consider the field behavior only in the region of the feedback mirror, that is in the interval ( - a, a);this clearly remains true for the more general reflectivity functions given in eqs. (3.94). The norm of a function defined in the interval ( - a, a) is given by
11 u 1;
=
Sr,
u*(x) U(X)dx
.
(3.107)
If u(x) is the field distribution incident on the cavity feedback mirror, its norm is then proportional to the energy of the field over that aperture. BEL’DYUGIN, ZEMSKOV, MAMYAN and SEMINOGOV [ 19741 assume that two functions fl(x) and f2(x) are related by fi % f2 if
II f d x ) II 4 II f d x ) II . 0
0
(3.108)
This concept of the smallness of one function compared with another is convenient for the subsequent analysis in the following manner: If one assumes that f,(x) is the field on the cavity feedback mirror and fi(x) is the correction to the field, the inequality (3.108) means that the correction to the energy off, is small compared with the energy as a whole. As a zeroth-order approximation, let G0’(X) = fi,,(X)PM0(X)
9
(3.109a) (3.109b)
where pMu is given by eq. (3.94a) and 8,,(x) is given by eq. (3.103). This is a natural choice from the viewpoint of geometrical optics, since there the solutions of eq. (3.102) are concentrated in the interval (-Ma, Mu) and are approximately zero outside of this interval. The zeroth-order approximation ?Ao) is simply the geometrical approximation of the eigenvalue spectrum [cf. eq. (3.16)]. From eq. (3.106) the first order approximation of yn can be found
IV,
s 31
293
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
immediately from
(3.110)
where
(.
u ( x ) = O(x) exp in- M;
(3.111)
12)
Consider now the correction to the zeroth-order approximation (3.109a). The Fourier transformation of this function is
(3.1 12) where (3.113) As the first-order approximation, let
fp(5 ) = ip'(4 ) + p (5) .
(3.114)
Substitution of this expression into eq. (3.102) then yields
+ M1/2 exp(
-i
52)
J -a,
p ( M < - v)
sin (a v) ~
"V
dv.
(3.115)
294
[IV, 8 3
UNSTABLE RESONATOR MODES
Let the correction Pi1)(<)be represented by the sum (3.1 16) where C,, is the coefficient in the Hermite polynomial Hn( 5) associated with Ck. Equation (3.115) then yields the following set of equations for &)((): 00
For large values of a, one can assume that y,, = 7;') = M-(" '/') and that the preceding integral equation can be replaced by the functional equation (by letting a go to infinity in the integrand) +
=
-[
dk sin(Mut) nt ] k - * e x p ( -iGt2) M dtk
-
11.
(3.117)
-
The assumption that 7, yio) is valid if the Fourier transform of the correction function f'il)(t) is small in the sense of eq. (3.108). The solution of the functional equation (3.117) can be written as
where for j O0(j, M ) =
=
0,
for j 2 1 .
This procedure may be continued by using the expression I p ( x ) = f i p ( x )pMa(x)
+L(Z)(X)
(3.119)
IV, § 31
295
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
as the second approximation and employing the eigenvalues 7:') from eq. (3.110) to obtain the second approximation functions, followed by the determination of $'), and so on. Consider now the expression (3.110) for the first order approximation 7:') of the eigenvalues. In terms of the functions uio)(x), this expression becomes
[
a
uio)*(x)guio)(x) d x (3.120)
J
uio)*(x)uLo)(x)d x -a
where k is the integral operator appearing in eq. (3.92). The zeroth-order approximation functions are ui0)(x) = %(X) PMU(X)
(3.121)
9
where the functions un(x) are defined by eq. (3.100). In order to evaluate the numerator in eq. (3.120) it is necessary first to determine the functions WiO)(X) = R u y ( x ) .
(3.122)
The Fourier transformation of eq. (3.122) yields W,'O'(5) =
EAr>+ FA51
(3.123)
9
where
x
jm
exp[ - i
-w
sin (uv) 4n(M2 - 1)
dv. (3.124)
7CV
The functions pn(()differ from the functions V,(<) given in eq. (3.99a) only by an integral factor, which depends on 5. In the limit as u + co this factor tends to unity, whereas for large but finite values of u it depends weakly on 5 and is approximately unity. To a good approximation one may then write
(3.125)
296
UNSTABLE RESONATOR MODES
[IV, 3 3
The integral factor in this expression is simply the Fresnel integral. Finally, the first three of the functions F,({) are Fo(8
=
0
(3.126a)
Y
x exp i n(M2 ( M
u 2 ) exp
( - i 2) sin (i <), 47cM
x exp i n ( M 2 - l)u2)exp( - i Z ) o c o s ( ; { ) . ( M 4 M
(3.126b)
(3.126~)
The general form of F,(<) may be found in the paper by BEL’DYUGIN, ZEMSKOV, MAMYANand SEMINOGOV [1974]. In real space eqs. (3.126) become
fo(x) = 0
(3.127a)
9
e i ~ ( a+*x 2 )
L(x)
=
sin(2mx),
(3.127b)
M2 - 1 312 - 8 7 1 ? i ( ~ ) ei&(az + x z ) u cos(2nax)
U
x sin(2mx) - i - cos(2mx M
(3. 27c)
The functions fo(x), fi(x),f 2 ( x )are small compared with the functions u,!joO’(x), v , ( O ) ( x ) , U $ ~ ) ( X ) ,respectively, in the sense of eq. (3.108) for sufficiently large values of a.
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
291
Equation (3.120) with (3.127) then yields the eigenvalues
(3.128) for large values of a = ,/-. The first term in this expression is the purely geometrical contribution, the second term is the first term of the asymptotic expansion of the Fresnel integral, and the quantity rnis the contribution from the wave f i ( x ) , given by
ro=o,
(3.129a)
( "%
x exp in-
'a')
(3.129b)
In terms of the equivalent Fresnel number Neq= (M' - l)a2/2Mof the cavity, these expressions become
(3.130)
(3.131) and where ro= 0. Notice the similarities of these expressions with the asymptotic expressions obtained in § 3.1.
3.4.1.3. Apodized feedback apertures Consider first the case (3.9413) of a feedback mirror with a truncated are Gaussian reflectivity. In this case both M, as given by eq. (3.96), and Neq
298
UNSTABLE RESONATOR MODES
[IV, § 3
complex valued, and they can be represented in the form
M
= M'
+ iM" ,
(3.132a)
N eq = N + W eq eq'
where M" = Im M > 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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
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
UNSTABLE RESONATOR MODES
[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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
-____-_________--___-_
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
UNSTABLE RESONATOR MODES
[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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
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
UNSTABLE RESONATOR MODES
",
§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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
I
..j ............I
\ I \I
k-4
h
4
'Fj OD
0
.5
305
306
UNSTABLE RESONATOR MODES
[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
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
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<-
1
(3.150)
a:
is satisfied, then over the feedback aperture the Gaussian field (3.149) may be approximated by the first two terms in its Maclaurin series expansion. With this approximation eq. (3.148b) may be directly evaluated to yield
which is to be substituted into the integral equation (3.148a). The maximum value of the argument of the Bessel function appearing in that equation is given by (3.152)
308
UNSTABLE RESONATOR MODES
where
a = m -u - 1 2%
(3.153)
where the real number m > 0 is is the radius of the spatial filter aperture d,, the spatial filter radius index. If the inequality
A4 m<-
(3.154)
7c
is satisfied, then the Bessel function appearing in the integrand of eq. (3.148a) may be approximated as
Jo(G rlrz)
a
exp( - ___ A2Mzf :
for 0 f r, f a. The inequality (3.154) constitutes what is called the strong spatial filtering limit. In that limit the integral equation (3.148a) yields the algebraic equation
ye-&=.
-
_ _2 M
[J,(mn) + J,(mn)] eikZT
x exp{
--[
m2t 8MZa: Jz(mn) + J,(mn)
x exp{ -i"[ The Gaussian parameter a=
J,(m$ 8Nc J,(mn) + J4(m7c) CI
2
ag]]. 2
is then given approximately by
2m2$J,(mn) a:(16M2 + mZnZ)[J,(m7c)+ J,(mz)]
(3.155)
With this result the magnitude and phase of the complex eigenvalue of the
IV, s 31
309
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
dominant cavity mode in the strong spatial filtering limit are found to be 2 M
y r - [J,(m.n) + J4(mx)]
x exp
m2n?J2(mn) J4(m79 1+ {-(16M2 + m2n?)(J2(mn)+ J4(mn))[ 2(J2(ma) + J4(m7c))
(3.156a) arg(7) E kz,
+ A -(3.156b)
These equations also describe the eigenvalue behavior for the dominant reverse cavity mode. The behavior of the dominant eigenvalue magnitude y as a function of the spatial filter radius index m is depicted by the family of curves in Fig. 25 for
I .o
0.8 M = 1.5
0.6
Y
0.4
0.2
0 0
0.2
0.4
0.6
0.8
1.0
1.2
I .4
1.6
rn
Fig. 25. Behavior of the dominant eigenvalue magnitude as a function of the spatial filter index m for several values of the cavity magnification M in the strong spatial filtering regime.
3 10
UNSTABLE RESONATOR MODES
[IV, § 3
several values of the cavity magnification M. For small values of m the eigenvalue magnitude is given by m2n2
y z -exp(
4M
-=)4M m2n2
z
m2n2
; m61,
which increases quadratically with the spatial filter index. For larger values of m within the strong spatial filtering domain (3.154), the full expression (3.156a) is required to give a reasonably accurate estimate of the dominant eigenvalue magnitude. For sufficiently large values of the cavity magnification M, the exponential factor weakly influences the eigenvalue behavior about m = 1. In that case the first maximum of the eigenvalue magnitude occurs at the first value of m’ > 0, satisfying the equation J,(m’n) = 2J,(m’n),
(3.157)
which is satisfied when m‘ 2 1.2. The first maximum in the eigenvalue magnitude is then given approximately by (3.158) when
Af .
a i 1.2- 1 2a1
(3.159)
This value of the spatial filter aperture radius at which the maximum in the dominant eigenvalue magnitude occurs closely corresponds to the f is t minimum in the Any pattern generated by an unobscured circular aperture with uniform plane wave illumination. The l/e half-width of the intensity profile of the dominant forward cavity mode incident on the outcoupling aperture of the cavity is, from eqs. (3.149) and (3.155), given by
which, to first order, varies inversely with the spatial filter index m. From these results it is seen that in the strong spatial filtering limit, the eigenvalue magnitude and intensity distribution of the dominant cavity mode are independent of the cavity Fresnel number.
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
311
Numerical calculations of the forward propagating transverse mode structure and eigenvalue spectrum as a function of the spatial filter index m yield results that verify these analytical approximations (OUGHSTUN,SLAYMAKER and BUSH [1983]). The ring resonator considered there is that depicted in Fig. 23a with a cavity magnification M = 1.50, collimated Fresnel number N , = 10.80 and equivalent Fresnel number Nes = 3.00 (in the absence of a spatial filter). The strong spatial filtering regime of this cavity is then m < 0.477, where m is the spatial filter index defined in eq. (3.153). The eigenvalue spectrum (magnitude and phase) as a function of the spatial filter radius index m is depicted in Fig. 26. The solid curves in this figure represent the numerical results and the dotted curves represent the analytical predictions of the dominant mode eigenvalue behavior given by eq. (3.156). It is seen that the analytical approximation is in excellent agreement with the numerical results for the dominant eigenvalue behavior for m 5 0.8. Notice that the first 1 = 0 mode with eigenvalue magnitude yP is the dominant mode of the cavity for all values of m. Another important parameter in assessing the performance of the intracavity spatial filter aperture is the relative cavity mode power loss incurred at the spatial filter aperture plane, given by (3.161) where i(r, $) is the relative cavity mode irradiance distribution incident on the spatial filter aperture plane, a = mAfl/D, where D = 2a, is the spatial filter aperture radius, and where
IomIo2=
i(r, $)r d r d$
=
1
(3.162)
for normalization purposes. The numerically determined behavior of the relative spatial filter aperture loss for the diffractive cavity mode in the forward direction as a function of the spatial filter aperture index rn is depicted by the solid curve in Fig. 27. The dashed curve in that figure represents the theoretical behavior of the relative spatial filter aperture loss for the dominant azimuthally symmetrical geometrical cavity mode. Unlike the behavior for the geometrical cavity mode, which monotonically decreases with increasing spatial filter aperture radius, the relative spatial filter aperture loss for the daractive cavity mode oscillates with very nearly the same periodicity as does the dominant eigenvalue magnitude as depicted in Fig. 26a. In addition, the relative minima in the relative spatial filter aperture loss conveniently occur at (or very near to) the same points as do the relative maxima in the dominant eigenvalue magni-
3 12 UNSTABLE RESONATOR MODES
[IV,§ 3
I
<
ra, W
4 r
Y
3-
2 Lo
m
U
m
I
1-
W
cn
ARG
Q
I
a W
3
3z
ElO )
.".".'"'.. ........._...,,, ,,,,,,,,,_........ ...''. \
0
ANALYTIC .APPROXIMATION
-1
3
'
0
W
c7 W
-2
-3
-4
I-
1
0
I
0.2
I 0.4
1 0.6
1 0.8
1
1
1.0
12
I
I
1
1
1.4
1.6
1.8
2.0
I
I
I
1
2.2
2.4
2.6
2.8
1
3.0
m
Fig. 26b. Fig. 26. Behavior of (a) the magnitude and (b) the phase of the eigenvalue spectrum as a function of the spatial filter aperture radius index for the intracavity focal point aperture ring resonator with M = 1.5. The dotted curves represent the analytical approximation of the dominant eigenvalue behavior.
W W
314
3
UNSTABLE RESONATOR MODES
\
0141
\
\ GEOMETRIC CAVITY MODE
\
I
a Q a: w
010'
5 LL
2
9 F
008.
a a
006.
2
+
4 g
004-
t
Ooa 01
1
06
I 08
I
10
I 12
I
14
I 16
I
I
I
I
I
I
20
22
I
18
24
26
28
30
m
Fig. 27. Behavior of the relative spatial filter aperture loss for the dominant cavity mode as a function of the spatial filter aperture radius index m for the intracavity focal point aperture resonator with M = 1.5.
tude, whereas the relative maxima in the spatial filter aperture loss occur very near to the points at which the relative minima in the dominant eigenvalue magnitude occur. The oscillatory behavior of the dominant eigenvalue magnitude and relative spatial filter aperture loss as a function of the spatial filter aperture radius is found to correspond closely to the intensity distribution of the cavity mode structure incident upon the outcoupling aperture plane of the resonator (OUGHSTUN, SLAYMAKER and BUSH [ 19831). In the first cycle of oscillation of yl0), which lies within the domain 0 < m < 1.8, the cavity mode structure is essentially Gaussian in shape, in agreement with the representation (3.149), particularly for values of m in the neighborhood of m = 1.3. This fundamental domain then corresponds to the strong spatial filtering regime of the cavity mode structure. Within this domain yf attains the maximum value (Yf)max =
0.814
at m = 1.3, which is almost identical to its unfiltered value yf = 0.816. The analytical approximation predicts a maximum value of 0.723 at m = 1.1, in fair
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
315
agreement with the exact numerical result. As the spatial filter aperture size is increased beyond the strong spatial filtering domain, higher spatial frequency information is retained in the cavity field and the Gaussian field approximation is no longer valid. The accuracy of the approximate analysis of the eigenvalue spectrum is seen to increase as the spatial filter aperture size is allowed to go to zero. In that limit, however, the validity of the assumption that the transverse mode structure is Gaussian breaks down. This aspect is reminiscent of a variational technique. The approach does represent an analytical approach to the numerical Fox-Li iteration procedure to the transverse eigenmode problem. Its success is in part due to the excellent transverse mode discrimination that exists in the strong spatial filtering regime. 3A.2.2. Two-stage spatial filtering; the focal line aperture resonator One practical difficulty in the successful application of an intracavity spatial filter aperture to obtain enhanced transverse mode discrimination along with a decrease in the aberration sensitivity in an unstable laser is the large incident irradiance (or incidence) that is apertured by the spatial filter aperture. In order to reduce this flux loading, the single focal point aperture may be replaced by a pair of separate, mutually orthogonal, rectangular spatial filter apertures. In that case the spatial filtering of the cavity mode is performed in two stages, which significantly reduces the flux loading on each of the apertures. In addition, the intracavity field focal point is replaced by a pair of separate intracavity field focal lines; this significantly reduces the local flux level of the irradiance distribution incident on each aperture. The geometry of the general ring resonator configuration with a pair of mutually orthogonal, intracavity focal line aperture spatial filters is depicted in Fig. 28. With the x Cartesian coordinate being in the plane formed by the ring geometry and perpendicular to the cavity optical axis, and the y Cartesian coordinate being perpendicular to that plane, the fist spatial lilter (in the forward direction) is seen to aperture the field in they direction. In general, the transverse mode properties of this cavity are different in the x and y coordinate directions if either the magnifications M, and My are different or if the outcoupling-feedback aperture has an elliptical cross-section. If a, and a,, represent the principal transverse radii of the cross-sectional geometry of the feedback aperture in the x and y coordinate directions, respectively, then the outcoupled geometrical mode has an elliptical cross-section with inner aspect ratio Ei = 5 (3.163a) ax
316
[IV, § 3
UNSTABLE RESONATOR MODES
I
I
CYLINDRICAL MIRROR
MIRROR
_ _ _ _ _ _ _ _ _ _ _ _ -- -ELLIPTICAL OUTCOUPLING APERTURE
x FOCAL LINE
SPHERICAL MIRROR
Fig. 28. Ring resonator geometry with two mutually orthogonal intracavity focal line aperture spatial filters. The geometrical mode profile of the forward propagating wave is indicated by the dashed lines.
and outer aspect ratio (3.163b) Furthermore, according to eqs. (3.139) and (3.140) and the comparison between Figs. 23 and 28, the collimated Fresnel numbers in the x and y directions of the cavity (in the absence of the focal line apertures) are given by
IV,I 31
M:a: Ncy =
317
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
A[z2 + M;zl
+ M;(1 + M x l f l x
,
(3.164b)
- My(l+ Mylflyl
respectively. In order to have an isotropic geometrical beam cross-section, it is then required that the feedback aperture of the cavity be circular and that the outer geometrical boundary of the outcoupled mode is circular, so that a, = ay = a ,
(3.165a)
,
Mx = My = M
.
(3.165b)
Moreover, in order to have an isotropic transverse dsractive mode structure in addition to an isotropic cross-section, it is required that both the focal line aperturestransmit the same amount of spectral content in the x and y directions (specified by the filter aperture index m = lD/Af, where l is the spatial filter aperture half-width) and that the collimated Fresnel numbers of the cavity in the x and y directions be equal, viz. A’, = N c y . From eqs. (3.164) one then requires that (3.166) f l y = Mflx. For such an isotropic cavity design the dominant Mractive mode properties will be, to second order*, rotationally symmetrical in the absence of the focal line apertures. With the focal line apertures present the transverse mode structure will no longer be rotationally symmetrical since the effective spatial filter aperture of the cavity possesses a square cross-section in the isotropic case. It is self-evidentthat for equal cavity magnification and Fresnel numbers, the focal point aperture and isotropic focal line aperture cavities will have equivalent transverse mode properties when the areas of the transverse limiting aperture of the spatial filters are equal. In that case the half-width of each of the line filters is given by l = s1n 112 a , (3.167) where a is the radius of the equivalent focal point aperture. The numerically determined passive cavity mode structure properties of an isotropic ring resonator with an ideally situated pair of orthogonal focal line apertures are found to be, in many respects, similar to those of the equivalent focal point aperture resonator (OUGHSTUN, SLAYMAKER and BUSH [ 19831).
* The cylindrical mirrors of the isotropic cavity are equivalent to the spherical mirrors of the cavity geometry in Fig. 23 only in the parabolic (or second-order) approximation.
318
UNSTABLE RESONATOR MODES
[IV, 8 3
In order to obtain a close comparison with the forward wave mode properties of the focal point aperture cavity considered previously, an isotropic focal line aperture cavity is considered there with the same magnification and collimated and equivalent Fresnel numbers. The eigenvalue magnitude spectrum as a function of the spatial line filter aperture half-width index m is depicted in Fig. 29a. The oscillatory behavior of the dominant eigenvalue magnitude here is similar to that exhibited for the focal point aperture resonator, given in Fig. 26a, except that in the present case mode crossings occur in such a way that a different transverse mode is the dominant field structure supported by the cavity in each cycle of oscillation. In the first cycle of oscillation, which lies within the domain 0 < m 1.735, the dominant eigenvalue is j j 3 , which is of the 1 = 0 type for m = 00 .* In that cycle the cavity mode structure is essentially Gaussian along the two coordinate axes, particularly in the neighborhood around m = 1.1. This fundamental domain then corresponds to the strong spatial filtering regime of the cavity mode structure. Within this domain y3 attains the maximum value
-=
at m = 1.1, which is nearly identical to the unfiltered dominant eigenvalue magnitude y1 = 0.807. The direct equivalence through eq. (3.167) of the eigenvalue spectrum of the focal line aperture cavity with that of the focal point aperture cavity (Fig. 26a) is readily evident. The most relevant parameter in assessing the performance of the intracavity spatial line filter pair is the relative cavity mode power loss incurred at each spatial filter aperture plane, given by =
1-
s s s‘ Dy’2 -DJ2
‘ &(x, y ) d x dy
(3.168a)
-6
for the x spatial line filter, and
3 = 1-
-D,/2
Ij(x,y)dxdy
(3.168b)
--I
for the y spatial line filter. Here I&, y ) is the relative cavity mode irradiance distribution incident upon the x spatial line filter aperture plane, fJx, y ) is that incident upon the y spatial line filter aperture plane, c = mlf / D is the spatial line filter half-width, Di (i = x, y ) is the length of each spatial line filter (taken
* The introduction of the spatial line filter pair into the cavity removes the second-order circular symmetry of the dominant cavity eigenmode supported by the resonator.
319
PASSIVE CAVITY MODE STRUCIWRE BEHAVIOR
rn
0 18-
0 16-
In
9
014-
U W
3
& W
0 12-
a Q
W U
010-
5
U
0 08-
006-
0 04
-
002-
01
1
06
1
08
I
10
I 12
I
14
I
16
I
18
I
20
I 22
I
24
I 26
I
28
I 30
rn
Fig. 29. Behavior of(a) the eigenvalue magnitudes and (b) the individual and total relative spatial filter aperture losses for the dominant forward wave mode as a function of the spatial line filter aperture half-width for the isotropic intracavity focal line aperture ring resonator with M = 1.5.
320
UNSTABLE RESONATOR MODES
[IV, § 3
to be infinite), and where
[(
m
ij(x,y)dxdy=l, j = x , y
(3.169)
J J - m
for normalization purposes. The total relative cavity mode power loss due to the spatial line filter pair is then given by
g.= Yx+ Yy- YxYy.
(3.170)
The numerically determined behavior of each of the relative spatial line filter aperture losses Yxand Yyand the total loss % for the diffractive cavity mode in the forward direction as a function of the spatial filter aperture half-width index m are depicted by the solid curves in Fig. 29b. The dashed curve in that figure represents the behavior of the relative spatial filter aperture loss for the equivalent focal point aperture cavity [obtained by way of the relation (3.167)], which is in close proximity to the behavior of L?& for the focal line aperture cavity. As was the stated purpose for employing a pair of spatial line filter apertures in lieu of a single focal point aperture, the relative loss incurred at each spatial line filter is significantly reduced from that at the single focal point aperture. In addition, the relative minima in each of the individual spatial line filter aperture losses very conveniently occur at (or very near to) the same points as do the relative maxima in the dominant eigenvalue magnitude, whereas the relative maxima in the spatial line filter aperture losses (with the exception of the maximum at m = 0) occur near to the predicted mode instability points of the cavity. A further reduction in the relative intracavity power loss due to the spatial filter arrangement may be obtained through a proper apodization of the cavity feedback aperture. Indeed, the apodization and spatial filtering techniques are mutually beneficial in the sense that, although both serve to enhance the transverse mode discrimination, the first w ill minimize the intracavity losses and the second will reduce the aberration sensitivity of the dominant cavity mode structure (OUGHSTUN [ 1986aJ). 3.5. GEOMETRY-DEPENDENTPROPERTIES
Some final properties of interest in the difFractive mode structure behavior of unstable cavities, which arise because of the particular geometrical arrangement of the cavity, are now briefly considered. These include the effects of an off-axis cavity geometry, the dependence of the mode structure on the feedback aperture geometry, and outcoupling mirror effects.
IV, 8 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
321
3.5.1. Off-axis cavity geometry
In an off-axis cavity geometry the center of the feedback aperture is offset from the cavity optical axis. This may be achieved in a standing-wave cavity simply by tilting the feedback mirror so that more radiation is outcoupled from one side of the cavity than from the other side; when the surface at one edge of the feedback mirror is parallel to that of the cavity end mirror, the optical axis is at that mirror edge and, in the geometrical approximation, all of the radiation is outcoupled past the opposite edge in that meridional plane (ANAN’EV, CHERNOV and SHERSTOBITOV [1972]). For small tilt angles the dfiractive mode structure is found to degenerate as the cavity feedback symmetry is destroyed (KRUPKEand SOOY [1969], CHESTER[1972, 1973a1,
2
OUTCOUPLED GEOMETRIC MODE PROFILE
t ttf
Fig. 30. Off-axis ring resonator geometry. The dotted lines indicate the on-axis mirror position and optical axis location. The geometrical mode profde of the forward propagating wave is indicated by the dashed lines.
322
UNSTABLE RESONATOR MODES
[IV, 3 3
HORWITZ [ 19761, PERKINSand CASON[ 19771, CASON,JONES and PERKINS [ 19781, SANTANAand FELSEN [ 19781, OUGHSTUN[ 1981b, 1983a1);however, as the tilt angle is increased further, several points are reached at which the diffractive feedback symmetry is partially restored and a well-behaved off-axis state appears (OUGHSTUN[ 1983a1). The practical aspects of such an off-axis cavity arrangement have been described by PHILLIPS,REILLYand NORTHAM [ 19761, SUTTON,WEINERand MANI [ 19761, and WEINER[ 1977, 19791. An off-axis cavity geometry may be achieved in a travelling-wave ring resonator simply by decentering the optical axis at the feedback aperture through a translation of one of the cavity turning flats, as shown in Fig. 30. The dotted lines in the figure indicate the mirror position and optical axis location in the on-axis or centered cavity arrangement. The additional path length introduced by the displacement to a decentered system may be compensated in one of the remaining collimated cavity sections in such a way that the cavity magnification and Fresnel number remain unchanged; this is assumed to be the case here. The dimensionless parameter A denotes the degree of decentration of the cavity, where the centerline of the feedback aperture is decentered with respect to the centerline of the outcoupled geometrical mode boundary by the amount d a , , as indicated in Fig. 30. The decentered optical axis location with respect to the centerline of the cavity feedback aperture, of width 2a,, is then given by A (3.171) xg = a1 M+1
GEOMETRIC M O D E PHASE FRONT
Fig. 31. Converging wave phase front at the feedback aperture of the decentered ring resonator.
IV,8 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
323
for a negative-branch cavity (in the plane of the decentration), as indicated in Figs. 30 and 31. A simple explanation of the transverse mode structure behavior of a decentered cavity may be readily obtained through a consideration of the difference in equivalent Fresnel numbers at the opposite sides of the feedback aperture in the meridional plane of the decentration (SHERSTOBITOV and VINOKUROV [ 19721, OUGHSTUN, BUSHand SLAYMAKER [ 19811). As illustrated in Fig. 31, the edge-scattered waves from the feedback aperture feed into the converging wave of radius R, which for the off-axis cavity is centered on the decentered optical axis. The equivalent Fresnel numbers N& and Ne;, defined by the sagittal distances from the respective feedback aperture edges to the converging wave geometrical phase front (for a collimated field incident upon the feedback aperture), are then different, as illustrated. From the geometry of Fig. 3 1 these [ 19811) Fresnel numbers are given by (OUGHSTUN,BUSHand SLAYMAKER
(3.172) where Neqdenotes the equivalent Fresnel number of the on-axis cavity arrangement. The Fresnel number difference between the opposite edges of the feedback aperture is then
6Neq N.& - Nei
=
A 4Neq-. M+1
(3.173)
For an on-axis system with integer plus one half equivalent Fresnel number, the converging waves from opposite sides of the feedback aperture will be in phase in the central core region near the optical axis, and a relative maximum will occur in the transverse mode discrimination [see the discussion following eq. (2.55)]. This condition is maintained in the off-axis configuration when SNeq is an integer, so that relative maxima in the off-axis mode discrimination will occur when
M+ 1
Am,, = -n
(3.174)
4Neq
for n = 0, 1, 2, 3, . .. . On the other hand, destructive interference results when 6Neq is an integer plus one half, so that relative minima in the off-axis mode discrimination will occur when
(3.175)
324
[IV,§ 3
UNSTABLE RESONATOR MODES
for n = 0, 1,2,3, . .. . For an on-axis system with an integer-equivalent Fresnel number, the preceding relations are interchanged and relative maxima in the off-axis mode discrimination occur when A is given by eq. (3.175), whereas relative minima occur when A is given by eq. (3.174).
1
01
0
0.2
0.4
0.6
DEGREE OF DECENTRATION
0
-
I4
I
0.8
I .o
t
1.2
A
(b)
max
a
z 1.6 -
0 I
4
-
-
a ON-AXIS SEPARATION
min
m in DEGREE OF DECENTRATION
A
Fig. 32. Behavior of (a) the eigenvalue magnitudes and (b) the transverse mode separationratio as a function of the (dimensionless) degree of decentration parameter A for the off-axis ring resonator with M = 2.5 and Nes = 2.5. The predominant azimuthal structure of the dominant transverse mode is indicated in parentheses in (a).
IV, 8 31
PASSIVE CAVITY MODE STRUClWRE BEHAVIOR
325
Numerical calculations of the eigenvalue spectrum and mode separation ratio as a function of the degree of decentration A in an M = 2.25, Neq= 2.5 unstable, negative-branch ring cavity c o n h these predictions, as illustrated in Fig. 32 (OUGHSTUN, BUSHand SLAYMAKER [ 19811). The mode separation ratio is a measure of the transverse mode discrimination of the cavity, being the ratio of the dominant eigenvalue magnitude to the next-lowest-loss eigenvalue magnitude. The predicted values of and Lin, given by eqs. (3.174) and (3.175), respectively, are indicated in Fig. 32b. As can be seen, an optimum design range for off-axis mode operation occurs for values of A between 0.6 and 1.0. Similar off-axis behavior is obtained for a standing-wave cavity when considered as a function of the angular misalignment of one of the cavity mirrors (OUGHSTUN [ 1983a1). 3.5.2. Elliptical aperture cavity The majority of unstable cavity geometries considered in practice possess either rectangular or cylindrical symmetry for obvious reasons. However, the analysis of the previous subsection has clearly shown that, even when this geometrical symmetry is removed, a certain “dsractive symmetry” may still be obtained under certain definite conditions at which there exists a well-behaved mode structure with exceptional transverse mode discrimination. The question then arises as to whether or not unstable cavities with more general feedback aperture shapes will also possess good mode discrimination properties. A first step toward answering this question is to consider a cavity whose optical elements are cylindrically symmetrical but whose feedback aperture is elliptical with major axis along the x coordinate axis and with aspect ratio (3.176) The transverse mode structure is then most conveniently expressed in elliptical-cylindrical coordinates x = $acoshpcos
J/,
y = $a sinhp sin J/,
z = z,
(3.177)
where the surfaces p = const. describe elliptical cylinders, whereas the surfaces J/ = const. describe hyperbolic cylinders. The equivalent Fresnel number of the cavity is then found to be J/ dependent, given by Neq(J/) = N$)(cos2 $ + ez sinZJ/) ,
where
E’
=
(3.178)
N$)/N$), with N$) being the equivalent Fresnel number in the
326
UNSTABLE RESONATOR MODES
[IVY§ 3
x-meridional plane and N,C) being that in the y-meridional plane. The cavity Fresnel number will then be constant along any one of the coordinate curves $ = const., as depicted in Fig. 33 for an elliptical aperture cavity with N g ) = 5.5, N g ) = 2.5 and E = 0.674. It is then evident that a single transverse mode cannot be supported by such a cavity geometry unless the aspect ratio is such that the equivalent Fresnel number range does not contain an integer value. This difficulty may be removed either by apodizing the feedback aperture (thereby removing the Fresnel number dependence of the cavity mode structure) or by employing cavity elements whose optical power is also $ dependent (thereby removing the $ dependence of the Fresnel number). Unfortunately, such elements are difficult to fabricate, particularly for small aspect ratios. 3.5.3. Outcoupling mirror effects In most practical applications of unstable cavities to laser systems, the outcoupled field is not taken past the feedback mirror but, rather, is obtained
Fig. 33. Equivalent Fresnel number contours over the feedback aperture of an elliptical aperture cavity with aspect ratio E = 0.674.
IV, § 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
321
Fig. 34. Outcoupling mirror arrangement in an unstable cavity.
by reflection from an outcoupling mirror, as illustrated in Fig. 34. Not only does this arrangement result in a slight alteration of the geometry of the feedback and outcoupled fields, but it also results in an additional diffracting aperture within the cavity. The typical analysis of the cavity system with the outcoupling aperture taken at the feedback mirror plane is an accurate approximation of the actual system, provided that the outcoupling mirror is kept sufficientlyclose to the feedback mirror and that the inclination angle 8 is kept greater than 40 degrees, wherein the relative geometrical changes are smallest (OUGHSTUN [ 19801). The influence of the additional edge diffraction source at the back side of the outcoupling aperture is found to enhance the peaking of the cavity mode intensity structure, which, in turn, tends to increase the associated eigenvalue magnitude (LATHAMand SMITHERS [ 19821). For this additional diffractive feedback to be negligible, the displacement d of the outcoupling mirror from the feedback mirror should satisfy the inequality
(3.179) Not only is this a requirement on the model accuracy for comparison with experimental results, but it is also a requirement on the experimental design so that a more smoothly varying mode structure can be obtained.
328
UNSTABLE RESONATOR MODES
[IV, s 3
3.5.4.Exotic cavity geometries An important area of practical interest is the design of an unstable cavity that is matched to a given lasing medium. Because of the peculiarities of the gain medium, such as a very short transverse gain span, and because of the limitations imposed by high-power mirror loading effects and cavity size, as well as the requirements of excellent transverse mode discrimination, several exotic cavity geometries have been developed. The simplest is a cavity with an asymmetrical magnification, as described by HOFFMAN[ 19811. One very important class of cavity design for high-power chemical laser applications is the annular cavity geometry. In these devices the gain medium is contained in a thin annular region so that a large mode volume is obtained. However, such an annular propagation geometry provides little radial mode discrimination and hardly any azimuthal mode discrimination so that it must be coupled to a compacted cavity section which must itself provide this discrimination. This coupling between the annular and compacted cavity sections may be obtained by means of a variety of axicon-type elements that, in essence, provide a conformal mapping between a circular and an annular mode region and vice versa. Unfortunately, such optical elements are very dscult to fabricate, since the optical surface quality tolerances become exceedingly severe in the region of the optical axis (HAYES,UNDERWOOD, LOOMIS,PARKSand WYANT[ 1981]), which is geometrically mapped into a cylinder in the annular cavity section. The behavior of such annular cavity devices has been described by MUMOLA, ROBERTSON, STEINBERG, KREUZER and MCCULLOGH [ 19781, MURPHYand BERNABE [ 19781, OGLAND[ 19781, DENTE[ 19791, CHODZKO,MASON, TURNERand PLUMMER[ 19801, SPINHIRNE, ANAFIand FREEMAN [ 19821, and STEPANOV and SHCHEGLOV [ 19821. Other approaches to this problem that do not rely upon an axicon-type element for the field conversion between the annular and compacted cavity sections have been described by CHODZKO,MASONand CROSS[1976], FRADINand CHENAUSKY PAXTONand ERKILLA[1977], and FREIBERG, [ 19771. Because of the relatively poor transverse mode discrimination afforded by most annular cavity geometries, it may be desirable to introduce an adaptive optical element within the cavity for the purpose of transverse mode control. Such an approach is necessary in order to compensate for any uncontrollable phase aberrations that may originate in the annular gain medium (PAXTON [ 19841) or that may arise from any slight angular or translational misalignment of the axicon-type element (MURPHYand BERNABE[1978], SHELLAN,
IV,
s 31
PASSIVE CAVITY MODE STRUCTURE BEHAVIOR
329
HOLMES, BERNABE and SIMONOFF [ 19801, SWANTNER [ 1982]), which is the most critical optical element of the cavity. The experimental results of SPINHIRNE, ANAFI,FREEMAN and GARCIA [ 19811, ANAFI,SPINHIRNE, FREEMAN and OUGHSTUN[1981], SPINHIRNE, ANAFIand FREEMAN[1982], and OUGHSTUN, SPINHIRNE and ANAFI[ 19841have shown the promise of such an intracavity adaptive optical system, but have also demonstrated that a multidither system is poorly suited to the problem of intracavity mode control. A completely deterministic approach to intracavity adaptive optical mode control, based on the well-defined aberration sensitivity of an unstable cavity (ANAN’EV[ 19721, OUGHSTUN[ 1981a,b, 1982c, 1983% 1986a,b]), has been shown to yield the optimum correction for several low-order aberration sources (OUGHSTUN [ 1986b1). This new approach does not rely on any multidither technique, which is itself an aberration source (albeit controlled) within the cavity (OUGHSTUN[ 1982c, 1983a1, OUGHSTUN,SPINHIRNE and ANAFI [ 19841). A rather complete overview of the geometrical and diffractive theories of the aberration sensitivity measure of an unstable cavity may be found in the review paper by OUGHSTUN [ 1986bl. One approach of current interest that circumvents some of the problems associated with high-power cavity designs is the coupled cavity arrangement. Such a configuration has previously been employed for the enhancement of the laser intracavity power (SEE,GUHAand FALK[ 19811). The present interest centers around the mutual coupling of several lasers with (nearly) identical unstable cavities so as to form a coherent phased-array output (PALMAand FADER[ 19851, FADER[ 19851, MARCUSE [ 19851). To date, the analysis has focused on a one-dimensional semiclassicaldescription of the locking phenomena between the individual cavities, as described by the SPENCERand LAMB [ 19721 model. Under the assumption that each of the individual cavity fields is completelyphase locked to theothers, forming a so-called supermode, the transverse mode structure of such an arrangement would be described by a set of coupled integral equations; the equation for each cavity of the array then contains the self-feedback for that particular cavity plus the injected field from each of its directly coupled neighbors. If all the cavities are identical, the set of equations is degenerate and one is left with a single integral equation for the transverse mode structure of each cavity, where the injected field may now be considered as a part of the self-feedback. However, if the cavities are not all identical the transverse mode structure of each cavity of the configuration will, in general, be different from the others and the entire set of coupled integral equations must be solved in a self-consistentfashion. Finally, one should notice that the polarization properties of the cavity fields must be carefully considered
330
UNSTABLE RESONATOR MODES
[IV, 8 4
to ensure that the injected fields are of the same polarization state as the cavity field.
0 4. Active Cavity Mode Structure Behavior The final general topic of interest in this review is the analysis of the active cavity mode structure. By active cavity it is meant here that the full effect of a saturable gain medium is included in the analysis of the cavity mode field structure. Of particular interest is the equation of state of the active cavity mode structure and the connection of this transverse mode structure with the longitudinal modes of the resonator and the passive cavity modes. Notice that the passive cavity analysis provides a fundamental understanding of what may be called the purely optical properties of the cavity, whereas the active cavity analysis provides an understanding of the dynamical properties of the laser field. When the results of the passive and active cavity analyses are compared, the influenceof the saturablegain medium on the mode structure properties can be distilled.
4.1. EQUATION OF STATE OF THE ACTIVE CAVITY MODE STRUCTURE
Consider again the generalized model of an optical cavity depicted in Fig. 1. As in the passive cavity analysis, this model represents a single propagation pass in either a standing-wave cavity in which the optical axis (taken here as the z-axis)originates at the output coupler-feedback reference plane at z = 0 and, in the positive direction, progresses to z = Z, at the back element of the cavity and then back to the output coupler-feedback reference plane at z = 2zT, or a travelling-wave ring cavity of axial perimeter L = 2zT. The only diffraction loss in the cavity is assumed to occur at the output coupler that is described by a complex amplitude reflection coefficientp(rT), where r, denotes the transverse coordinate position of a point in the feedback-outcoupling aperture reference plane. The complex nature of p allows one to account for the cavity optical system at this single plane (assuming that no additional edge diffraction sources appear in the cavity). Following the analysis of SIEGMAN[ 19801, the vector wave equation for the and dynamic cavity field 8(rT, z,t ) is given by (see also SARGENT,SCULLY LAMB[1974]) +
[
v2
a
- potJ- az
&&()
“1
a2
7 8 = po - 9. at
at2
IV, I 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
331
The electric field vector 8 and polarization B are real-valued vector functions of both space and time. The conductivity B here represents a linear distribution of the losses inside the cavity, including the outcoupling loss at the feedback aperture. The electric polarization vector B(rT, z, t) represents the effects of the atomic transition of the laser medium, including the effects of spontaneous emission; its quasi-linear form is 9’ = g ( b ) 8 , where g ( 8 ) is the gain coefficient of the saturable laser medium. For a strictly monochromatic field propagating along the positive z direction through the unfolded cavity, the electric field vector may be written as [cf. eq. (2.511 8(rT, Z, t ) = 8(rT, Z, t) ei(kz-mf),
(4.2)
where k = o/c. In addition, the electric polarization vector likewise may be written as
9’(rT, z, t ) = 9’(rT, z, t) ei(kz- mr) .
(4.3)
The angular frequency w may assume any arbitrary value near to the atomic line center; however, it is convenient to choose w so as to satisfy the axial mode equation 2oL/c = 2 q 0 , where qo is an integer chosen so that w is near the atomic line center (see 02.6). Under both the paraxial and slowly varying envelope approximations for the complex fields E and P,the wave equation (4.1) becomes
. boundary where V% is the transverse Laplacian operator and cr, = a / 2 ~ ,The condition on the complex field amplitude I at the feedback aperture reference plane is 8(rT, z
=
o + , t ) = p(rT) 8(rT, z = 2L-9 t ) ,
(4.5)
where z = 0 indicates that the field is just leaving the reference plane, whereas z = 2L- denotes the field amplitude just incident upon that plane. The steady-state, dflractive transverse eigenmodes of the passive cavity are solutions of the paraxial Helmholtz equation [cf. eq. (2.6)] +
pi:
+2
4 U n m ( r T , z)
=
0
(4.6)
These passive cavity eigenmodes satisfy all of the boundary conditions in the
332
[IV,5 41
UNSTABLE RESONATOR MODES
cavity except that at the feedback aperture reference plane, where they satisfy the reduced boundary condition ynmunm(rT,
=
O’)
= d r T ) unm(rT,
z
=
2L-)
3
(4.7)
where ,y is the eigenvalue of the passive cavity eigenmode u,,, . These modes and their associated losses are characteristic of the purely optical properties of the cavity; the vector nature of these modes describes the polarization state imposed on the cavity field by the passive cavity (see Q 2.5). An adjoint set of passive cavity eigenmodes uL,(rT, z) also exists with identical eigenvalues ynm = ynm (see 5 2.3) and, provided there are no nonreciprocal elements in the -t cavity, identical polarization states. The adjoint set of modes {u;,] form a bi-orthonormal set to { unm}that is useful in determining mode expansions of both the cavity field and the electric polarization vector (see 82.4.3). The functional relation between this conjugate pair of modes is given by eq. (2.67) for a single aperture cavity. SIEGMAN[ 19801 has introduced a set of “oscillation eigenmodes” for the resonator given by
Each of these modes then satisfies the boundary condition (4.5). They also satisfy the extended wave equation
p+.
+ 2ik(g +2
) ] E n m=0.
(4.9)
Each of these oscillation eigenmodes then represents a continuous wave field that could just be sustained by the cavity if the volume was filled with a homogeneousgain-phase medium with a net gain that is just sufficientto permit the En, mode to oscillate in the steady state. Because of the boundary condition (4.9, each such mode has y,”, less power at z = 0’ Gust after reflection from the feedback mirror) than at z = 2 L - (just incident upon the outcouplingfeedback aperture reference plane), thereby accounting for the diffractive output loss of the cavity at that plane. If one then sets (4.10a) so that a,,,
c 1 + ip,, = In ; , 2L
Ynm
(4. lob)
IV, 3 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
333
where an, and fin, are both real, then the oscillation eigenmodes (4.8) for the resonator become (4.11) The quantity an, = Re[(c/2L) ln(l/Tnm)]is then seen to be the temporal decay rate, whereas fin, = Im[(~/2L)ln(l/~~,)] is the frequency shift associated with each oscillation eigenmode. These oscillation eigenmodes of the cavity provide a convenient basis set for the resonator field E and complex polarization vector P ; hence, let (4.12) (4.13)
This is the basic cavity equation of motion for each transverse mode amplitude subject to the boundary condition A,,(O
+9
t ) = Anm(2L- t ) Y
9
(4.15)
which states that each mode amplitude is continuous across the output coupler-feedback aperture plane. Equations (4.14) and (4.15) are the basic equations of state of the active cavity mode structure due to SIEGMAN [ 19801. 4.1.1. Longitudinal mode expansion Because of the finite (round-trip) cavity length 2L and the boundary condition (4.15), each transverse mode amplitude A,,(z, t ) may be expanded as a Fourier series of longitudinal modes (4.16) where the integer q labels each individual axial mode. Substitution of this expansion into eq. (4.14) then yields the reduced equation of motion for each axial mode amplitude
334
UNSTABLE RESONATOR MODES
where P,,,(t)
3
1 Jo2L
P,,(z,
t) exp( -
F)
dz ,
2L
[IV, 8 41
(4.18)
In the absence of gain and dispersion effects the resonance frequency of each longitudinal mode is given by
The longitudinal mode spacing is then seen to be m/L. For a standing-wave cavity L = z,, where z, is the end mirror separation and Acu = m/zT [see eq. (2.194) with p = 11; for the equivalent travelling-wave cavity L = 2zT and Aw = m/2zT [see eq. (2.196)]. The additional term fin, appearing in eq. (4.19) is a small frequency shift related to the phase of the eigenvalue associated with each transverse eigenmode [see eq. (4. lo)]. Finally, it is convenient to number the individual longitudinal modes q with respect to the central value qo so that (qo + q ) is the actual longitudinal mode number. 4.1.2. Outcoupledpower and the cavity Q factor Consider the idealized situation for which the resonator is oscillating at steady state in a single transverse mode that is defined by the passive cavity*. From eqs. (4.11) and (4.12) the time-average Poynting vector along the positive z direction of the active cavity field is found to be (in MKS units)
(4.20b) where y,, = I ynm I. This quantity is the net power propagating in the positive z direction through the resonator. The power just incident upon the output * With a saturable gain medium the resonator in steady state may oscillate in a single transverse mode which is usually different from that defined by the passive cavity alone. In general, a single transverse mode of the active cavity is a superposition of the passive cavity modes [see eq. (4.12)].
IV, 5 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
335
coupler-feedback aperture of the cavity is then P(z
=
2L-) =
$(:)
112
IAnm(z= 2L-)12,
(4.21)
whereas the power fed back into the cavity at that plane is
P(z = 0 ' )
=
(2)
112
IAnm(z= O+)12y:m = y,',P(z = 2 L - ) ,
(4.22)
by virtue of the boundary condition (4.15). The outcoupled power is then given by (4.23) and the fractional power loss is simply (4.24) This is precisely the result obtained in the passive cavity case [see eq. (2.50)] and is a direct consequence of the assumption that the resonator is oscillating in a single transverse mode of the passive cavity. Furthermore, the time-average stored electromagnetic energy in the cavity is given by
(4.25) where it has been assumed that only a single longitudinal mode is present [see eq. (4.16)]. The ratio of the time-average electromagnetic energy stored in the cavity to the outcoupled power is then related to the external Q factor of the cavity by (COLLIN [ 19661) (4.26) where ynm = 1 vnm 1 < 1. As before, this result is strictly valid only under the approximation that the resonator is oscillating in a single transverse mode that is defined solely by the passive cavity.
336
UNSTABLE RESONATOR MODES
4.2. SATURABLE GAIN MEDIUM EFFECTS
In the presence of a saturable gain medium the transverse mode structure of the cavity is altered from its passive cavity form. That is, both the cavity and the gain medium now determine the transverse mode structure of the resonator. Since the support of a given mode now also depends on the availability of gain, the higher-order transverse mode losses may be decreased relative to that of the lowest-loss mode of the passive cavity. In the central core region of the cavity all of the transverse modes are competing for atoms yielding gain. However, since the higher-order transverse modes occupy a larger relative volume of the gain medium, they are able to extract energy from atoms that are not available to the lower-order transverse modes of the cavity and, as a consequence, may even become dominant. In general, the active cavity field may be expressed as a linear combination of the passive cavity modes, as expressed by eq. (4.12) with (4.11). The gain medium is then seen to couple the various passive cavity modes together to form the active cavity mode, where each mode amplitude coefficient satisfies the equation of motion (4.14). This coupling becomes nonlinear for any significant degree of saturation in the active medium. In light of these results the importance of excellent transverse mode discrimination is now evident. For a cavity that possesses this property the higher-order transverse modes are rapidly rejected by the cavity in favor of the single lowest-order mode, so that in the presence of a saturable gain medium the cavity itself will still predominantly determine the transverse field structure. In this case a moderate fluctuation in the gain medium (resulting from a perturbation in the flow conditions, for example) will be manifested in a minor fluctuation of the cavity field structure. However, for a cavity with poor transverse mode discrimination the medium properties strongly influence the field structure through the nonlinear coupling described earlier. Any mild fluctuation in the gain medium will then be manifested in a strong fluctuation of the cavity field, and well-behaved cw operation will be difficult to maintain. The purpose of the present section is to describe the effects of a well-behaved saturable gain medium on the transverse field structure of an unstable cavity. The details of the cavity medium are beyond the scope of this article (see, for example, MAITLANDand DUNN[ 19691). However, the general behavior may be described using a fairly simple model of the laser amplifier gain due to RIGROD[ 1963, 19651 that was later extended by AGRAWAL and LAX[ 19791 to include the effects of interference between the counterpropagating cavity fields. This analysis is restricted to the case of homogeneous line broadening in which the line shape does not change during gain saturation.
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
331
4.2.1. Laser amplfir gain and saturation Following the analysis of RIGROD[ 19631, consider a collimated beam of uniform intensity propagating in the z direction through an active medium with homogeneous line broadening. The differential change in intensity with distance is then given by dZ dz
-=
(A+
- a)z,
1 Z/Zo
(4.27)
where go is the small-signal gain, Zo is the saturation intensity parameter of the medium*, and a is the average loss coefficient (caused, for example, by absorption and scattering). Define the normalized radiation intensity by P = Z/Zo so that dZ = Z0dP. Equation (4.27) may then be rewritten as (4.28) When the loss parameter a of the medium vanishes, one obtains (4.29) which, when integrated over the length L of the amplifying medium, gives (4.30) where
and P2 are the normalized input and output intensities, respectively.
4.2.1.1. Incoherent approximation In a laser the radiation propagates back and forth between the cavity end mirrors (for a standing-wave cavity) so that it is amplified during each transit through the gain medium of length L and suffers losses at the end mirrors. As a first-order approximation of the field distribution in the cavity, one may take the dominant geometrical mode structure of the unstable cavity. In addition, the losses occurring in the homogeneous gain medium may be neglected in eq. (4.27); all losses are then assumed to occur upon reflection at the cavity end mirrors, including the outcoupling losses. Let the optical axis of the resonator * The saturation intensity of an active medium with homogeneous line broadening is that value of the local field intensity which reduces the gain to half the small signal gain.
338
UNSTABLE RESONATOR MODES
[IV, 8 41
be along the z direction and let the normalized radiation intensities in the positive and negative z directions be denoted by 8, and p-, respectively. Neglecting the standing-wave interference between these two counterpropagating fields, one may take the local intensity at any transverse position z in the cavity as the (incoherent) sum I(z) = I, + I - . With this substitution in the denominator of the gain term of eq. (4.27), one obtains (SCHULZDUBOIS[ 19641) (4.31) Hence, at any plane z within the active medium one sees that
B+B-
=
c,
(4.32)
where C is a constant. Let the active medium completely fill the cavity volume between the cavity end mirrors with effective reflectances rI and r,, where r l = l - a l - z l , at z = O , r,=l-a2-t2,
at z = L .
(4.33)
Here a, and a, are the dissipative losses and t , and t, the respective mirror transmittances. Let b1denote the normalized intensity reflected back into the cavity at mirror 1 and propagating in the + z direction. After passage through the active gain medium the normalized intensity incident upon mirror 2 in the + z direction is 8,. The outcoupled normalized intensity at mirror 2 is then t2& and the normalized intensity reflected back into the cavity and propagating in the - z direction is (4.34a)
8 3 = 1282
After passage through the active gain medium the normalized intensity incident on mirror 1 in the - z direction is f14, where 81 = r184
7
(4.34b)
and the outcoupled normalized intensity at mirror 1 is t1#?,,,.From eq. (4.32) one finds that PIP4 = B2B3 so that, with eqs. (4.34), (4.35)
IV, I 41
ACTIVE CAVITY MODE STRUCTURE BEHAVlOR
339
From eqs. (4.31) and (4.32) the power flow in the + z direction in the resonator is given by
- 1_ _dB+ _ -
B,
dz
go
(4.36)
1 + B + + CIP,’
which, when integrated over the length L of the resonator, yields (4.37)
In the same manner, for gain in the
- z direction in the
resonator, one obtains (4.38)
Upon adding these two equations together and making use of eqs. (4.34) and (4.39, one fiaally obtains (RIGROD[ 19651)
- 112 (4.39) In addition, from eq. (4.35) there results
(4.40) The total output radiation intensity from both mirrors is then proportional to tlB4 + t2P2. When the dissipative losses at both mirrors are equal, a , = a2 = a and t , = 1 - a - rl , t, = 1 - a - r,. The total output radiation intensity from both mirrors is then tlZ4 + t2z2 = zo
1 - a - (r1r2)1/2 [g& 1 - (r1r2)1/2
+
ln@1r2)1/21.
For a symmetrical resonator defined by rlr2 = r2, r radiation intensity at each mirror is given by ZOUT
=
1 1-a-r 210(g&
1-r
+ In r) = $z0
t
~
a+t
(g&
=
(4.41)
1 - a - t, the output
+ ~nr) .
(4.42)
Hence, when a , = a 2 ,one can always replace an asymmetrical resonator by an equivalent symmetrical resonator that has the same relation between the total
340
UNSTABLE RESONATOR MODES
outcoupled power and the unsaturated gain g&. Finally, when mirror 2 is perfectly reflecting ( I , = 1, tz = a2 = 0), the output radiation intensity is given by (4.43) which corresponds to the outcoupled radiation intensity at each of the two identical mirrors of a symmetrical resonator with twice as much unsaturated gain. The optimum output coupling (transmittance) of mirror 1 is that value which yields the maximum output intensity. Upon differentiatingthe right-hand side of eq. (4.43) with respect to t , and remembering that r , = 1 - a, - t , , one obtains the expression t,
1 - a, - t ,
a,
a, + t ,
-=2
[g& +
(4.44)
I.:/’]
for the optimum output coupling. For an unstable cavity with geometrical magnification M, one would let I , = l/M2 in the preceding expression to determine the optimum cavity magnification for a given unsaturated gain product g,& and dissipative loss parameter a, . Further analysis of lasers with unstable resonators in the geometrical optics approximation has been given by ANAN’EV,SHERSTOBITOV and SHOROKHOV [1971] and by MOOREand MCCARTHY[ 1977al. 4.2.1.2. Coherent interference eflects The near-resonant interaction of an electromagnetic field with an active medium results in a saturable gain function g = g(Z), where Z(r) represents the total local intensity of the cavity field. In the previous analysis the total local intensity Z was conveniently replaced by the incoherent sum of the intensities associated with the positive and negative propagating cavity fields, so that Z = Z, + I - . In that case the counterpropagating waves of the cavity field experience identical gain. This approximation clearly precludes the coherent interference effects that occur between the counterpropagating cavity fields. Under the opposite assumption of complete coherence the total local intensity is given by the coherent sum of local field intensities, so that Z = I u + u - I 2, where u and u - are the counterpropagating cavity fields. This idealization, taken by AGRAWAL and LAX [ 1979, 1981bl and LOUISELL, LAX, AGRAWAL and GATZKE[1979], results in the complete interference of the counterpropagating cavity fields which, in turn, produces a periodic spatial variation +
+
IV, 8 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
341
in the gain saturation along the cavity axis. As a result of this, it is found that the counterpropagating cavity fields do not experience identical gain as is generally assumed. In the paraxial approximation the counterpropagating cavity fields are found to satisfy the coupled pair of equations (AGRAWAL and LAX [ 19791) (v2
+2 d ) u +
(..+
2iki)u-
=
ikg+u+ ,
(4.45a)
=
ikg-u- ,
(4.45b)
where the saturated gain function is given by (4.46) with
Here 61 = (w - wub)/rab is the detuning parameter of the cavity field frequency w from the atomic frequency difference wab = w, between the upper and lower atomic states, where rubis the associated homogeneous broadening linewidth. The cavity fields u , and u - have been scaled in units of the saturation field amplitude uo = I;’’, where I , is the saturation intensity of the medium. Equation (4.46) represents the modified gain equation due to AGRAWAL and LAX (1979). Since g + # g-, the counterpropagating cavity fields experience different gain. The incoherent result of identical gain in the opposing directions is obtained by setting b = 0 in eq. (4.46). In the geometrical optics approximation the counterpropagating normalized field intensities 8, = 1 u I 2 and 8- = I u - I are found to satisfy the coupled set of ordinary differential equations (AGRAWAL and LAX [ 19791) +
(4.49a) dPdz
2(W - 1 )
+
L
+ (W - 1)(L - Z)B-
=
- Re(g-18-
9
(4.49b)
342
[IV, s 41
UNSTABLE RESONATOR MODES
where M' and M" are the partial (or one-way) magnifications for the + z and - z propagating cavity fields, respectively (the overall cavity magnification is then M = M'M"), and where L is the cavity length. Since the product fi+/?is not invariant as it is in the incoherent case [cf. eq. (4.32)], eq. (4.49) cannot be solved in closed form by Rigrod's method. Numerical solutions of these equations, indicated by the solid curves in Fig. 35, show that the output
;
8 1 Incoherent t
t
In 2
w I-
f b W
N
-I
4:
9 K
0
z
o
i
2
3
4
5
6
j
A X I A L DISTANCE z
sZ=I >
8-
-
E In z
t -I-n-c o-h-e -r e-n --
i
W
I-
f
2-
0 0
1
2
3
4
A X I A L DISTANCE
5
6
7
2
Fig. 35. Normalized intensity variation of the geometrical mode along the resonator axis of a positive-branch, confocal unstable cavity with partial magnifications M' = 2.25 and M" = 1.0. The cavity length is L = 7.3 m, and the region between z = 3.3 m and z = 5.3 m consist of the homogeneously broadened gain medium with small signal gain go = 4/m. The coherent interaction of the counterpropagating cavity fields is indicated by the solid curves, and the incoherent and LAX [1979].) interaction is given by the dashed curves. (Afier AGRAWAL
IV, 8 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
343
intensity level is decreased from that obtained in the incoherent case (indicated by the dashed curves), the amount of reduction increasing with detuning. In addition, the intensity variation along the resonator axis is decreased from its incoherent behavior (AGRAWAL and LAX [ 19791). In many unstable gas laser systems the active gain medium is flowing either parallel or transverse to the cavity optical axis. Because of this flow condition the spatial hole burning resulting from the periodic saturation of the extended gain medium along the optical axis (with spatial period equal to one half the wavelength of the cavity field) is washed out to a degree dependent on the flow velocity. The actual behavior in that case would then lie somewhere between the solid and dashed curves of Fig. 35. 4.2.2. Transverse mode structure behavior The influence of a saturable gain medium on the transverse mode structure of an unstable cavity is now considered for the case of a homogeneously broadened transition. In the examples considered here the active gain medium is flowing either along the direction of the cavity axis or transverse to it. The coherent interference effects between the counterpropagating mode fields upon the local gain coefficient are then inconsequential and (to a good approximation) the incoherent expression (4.3 1) applies. Consider first the idealized situation in which the medium is optically homogeneous so that there are no gain-phase variations. The only alteration of the cavity mode structure from its passive cavity state is then solely caused by saturation effects in the gain medium. For connection to be made with the passive cavity results of § 3.3, a cylindrical, confocal, positive branch unstable cavity with magnification M = 1.44 and collimated and equivalent Fresnel numbers N, = 25.0 and Neq= 6.5, respectively, is considered here. The associated passive cavity mode structure and Fresnel zone behavior is depicted in Figs. 7 and 18. The extended gain medium is represented by a single gain-phase sheet situated approximately midway through the cavity (see OUGHSTUN[ 1983a1). The gain coefficient at that plane is given by (4.50) where go = 0.01 cm- ’ is the longitudinally averaged small-signal gain coefficient of the extended medium, I, = 3.404 WJcm’ is the average saturation intensity, and I is the local two-way intensity of the cavity field, given by the
UNSTABLE RESONATOR MODES
RELATIVE PHASE
I
\. .. ..
CAVITY
... I
I 02
I
I 04
.......... . . . . ..... .. .. . . I
I
06
.....
I,
1IM
...... I 08
10
ria2
Fig. 36. Radial behavior ofthe relative intensity and phase structure ofthe dominant,azimuthally symmetrical mode incident upon the feedback aperture plane of a positive-branch confocal unstable cavity of magnification M = 1.44 and equivalent Fresnel number Neq = 6.5. The solid curves depict the active cavity mode structure and the dotted curves depict that of the passive cavity.
incoherent sum Z = I , + I-. The dominant azimuthally symmetrical transverse mode structure incident on the feedback mirror of the active cavity is depicted by the solid curves in Fig. 36. A comparison of this irradiance structure with its passive cavity counterpart (indicated by the dotted curve in Fig. 36) shows that the idealized saturable gain medium decreases the height of the central intensity peak relative to the outer diffraction rings of the cavity mode. It does not, however, alter the dzractive structure of the mode phase profile (OUGHSTUN[ 1983a1). Similar results have also been obtained by LAX, AGRAWAL, BELIC,COFFEYand LOUISELL[ 19851 with complete allowance made for the coherent interference effects between the oppositely directed cavity fields. The intracavity power of the dominant mode incident upon the cavity outcoupling aperture is 75.77 W, and the outcoupled power is 35.91 W. The flux eigenvalue of the active cavity mode [defined as the square root of the
IV, § 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
345
ratio of the feedback power to the total power incident upon the outcoupling aperture, as given in eq. (2.50)] is then yF = 0.725,
which is approximately 9% less than that of the ideal passive cavity mode. The majority of this decrease is simply due to the inclusion here of all the reflection and transmission loss factors present in the cavity (roughly 11.6%). The problem of the mode deformation resulting from gain saturation in a wellbehaved homogeneously broadened gain medium in an unstable cavity is then seen to be more severe than that which occurs in stable cavity geometries (see STATZand TANG[ 1965]), particularly for the intensity distribution. The average degree of saturation of the active gain medium for this mode is (Z) /Z, = 0.271, where ( I ) is the spatially averaged two-way intensity of the cavity field across the gain-phase sheet. This then corresponds to a weakly saturated system in the following sense: In the limit as ( I ) /Io co,then g + 0 and the passive cavity mode results. In this limit the gain is fully saturated and the passive cavity mode is obtained. On the other hand, as ( I ) / I , + 0, then g -+ g, and, although the cavity power increases exponentially with time, the power-normalized cavity mode is identical with that of the passive cavity. In this limit the gain is weakly saturated and the passive cavity mode is again obtained. Hence, if the average degree of saturation of the active gain medium were increased for the resonator considered here (by decreasing the saturation intensity Zo), one would find that the central intensity peak would initially decrease relative to the outer dsractive ring structure of the mode, but would then increase back toward its passive cavity form as the gain medium became more fully saturated. The location of the mode diffraction peaks and valleys would remain essentially unaltered (as would the phase structure) throughout this variation in the degree of saturation, since these properties are solely due to the edge diffraction at the feedback aperture (in a single aperture unstable cavity) when the medium is optically homogeneous. The saturation of the gain medium does influence the aberration sensitivity of an unstable cavity, tending to drive this measure towards its geometrical value (OUGHSTUN[ 1983aJ). In a real laser system with flowing gain the medium will not be optically homogeneous, and the transverse mode structure of the cavity will consequently be aberrated. As an illustration, consider the analysis by OUGHSTUN[ 1983al of the experimental adaptive resonator system reported by SPINHIRNE, ANAFI, FREEMANand GARCIA[1981] and ANAFI, SPINHIRNE,FREEMANand OUGHSTUN [ 19811(see also OUGHSTUN, SPINHIRNE and ANAFI[ 19841). The ideal resonator system is just that described earlier with transverse mode
346
UNSTABLE RESONATOR MODES
[IV, s 41
structure depicted in Fig. 36. Because ofthe nonuniform transverse small-signal gain profile in the experiment, eq. (4.50) is modified to the azimuthally averaged function (4.51) where 1, is the saturation intensity employed previously and where the ad hoc small-signal gain function go(rT)= go(l - ur?
+ br;
- cr:)
, r < $D
(4.52)
accounts for the radial decrease in gain that is due to collisional deactivation with the cylindrical gain tube wall of diameter D = 10.2 cm, where the collimated geometrical mode diameter is 2Mu, = 8.812 cm. Here go is the smallsignal gain employed previously, and the empirically determined coefficients appearing in eq. (4.25) are a
=
1.5606 x 10-’/crn2,
b
=
8.664 x 10-4/cm4,
c = 2.249 x 10-’/cm6,
where rT is expressed in centimeters. With 0.0612 wavelength ( A = 10.6 pm) of astigmatism introduced into the cavity by the thermal bowing of the discharge tube windows, the cavity mode structure depicted in Fig. 37 is obtained. The majority of the mode deformation seen here results from the intracavity astigmatism. However, because of the decrease in the small signal gain near the outer region of the mode volume, the intracavity and outcoupled power levels are decreased from their previous values to Pi,= 67.7 W and Po,,= 30.7 W, respectively, with a resultant increase in the flux eigenvalue to yF = 0.739. Isointensity contour plots of the intracavity mode and the resultant far-field irradiance structure for this resonator are given in Fig. 38. The far-field power in the bucket* is Pb = 7.39 W with a relative beam quality of 1.165 and Strehl ratio i(0,O) = imaX= 0.640. The analysis by OUGHSTUN[1983a] of this particular resonator system continues on to consider the influence of a nonideal deformable mirror placed * The so-called “power in the bucket” is the far-field power contained within the central Airy disc of a reference field, which is taken here as the outcoupled geometrical mode of the ideal unaberrated cavity. The radial obscuration ratio of the outcoupled geometrical mode is = 1/M = 0.695, and the first zero in the resultant far-field reference diffraction pattern occurs at rgln = 0.913 ,lf/D, within which is contained 28% of the total far-field flux in that refer‘ence field. The relative beam quality is then defined as the ratio of this reference percentage to that [1981b].) for the diffractive cavity mode (see OUGHSTUN
IV, 8 41
341
ACTIVE CAVITY MODE S T R U n U R E BEHAVIOR
+7r
1.0
>
0.8
cv)
W v)
a
0.6
I
I-
n
f
w o
2
W
I-
0.4
a
a
A
W
A W K
K
0.2
0 0
1.0
2.0
3.0
4.0
5.0
X (cm)
-
+7r
1.0
0.8 W
m
t
2
s
w
z
0.6
20I
2
-
w =
41
R E L ATlV E
w >
F 0.4 4
w
K
0.2
-7r-
0
v1
I
0
1.0
2.0
3.0
4.0
5.0
Y (cm)
Fig. 37. Transverse active cavity mode structure (normalized intensity and phase) in the x and M = 1.44, Neq = 6.5 positive-branch confocal unstable cavity. The saturable gain medium has a radially decreasing small-signal gain, and 0.0612 wavelength of astigmatism is introduced into the cavity by the thermal bowing of the discharge tube windows.
y meridional planes that is incident on the feedback aperture plane of an
within the cavity (see also OUGHSTUN[ 1982d], OUGHSTUN,SPINHIRNE and ANAFI[1984]). With its dithers on in the open-loop state, the deformable mirror surface profile fluctuates in a somewhat random manner about its ambient profile, and this introduces a random phase aberration within the
348
UNSTABLE RESONATOR MODES
Y
X
Pin
=
67.7 watts
Pout
=
30.7 w a t t s
yF=
0.739
Y
im a x= 0.640 i(O.0)
=
0.640
a;,,
=
1.165
P b
=
7.39 w a t t s
Fig. 38. Isointensity contour plots of the intracavity mode structure incident on the outcoupling aperture plane (upper figure) and the resultant far-field irradiance structure due to the outcoupled field distribution from the cavity (lower figure). The active cavity is the same as that considered in Fig. 37.
X
iMAX = 0.352 i(0. 0)= 0.049
I
I
I
Y
Y
iMAX=0.318 i(0. 0) = 0.086
Fig. 39. lsointensity contour plots of the intracavity mode structure incident on the outcoupling aperture plane (upper figures) and the resultant far-field irradiance structure due to the outcoupled field distribution from the cavity (lower figures) for two different intracavity phase aberration structures. The active cavity is the same as that considered in Fig. 38.
I
Y
Y
*2
$
W
Y
P
M
350
UNSTABLE RESONATOR MODES
IW s 41
cavity. Such a random phase structure may be thought of as being due to turbulence in a rapidly flowing gain medium, which would result in a timevarying optical inhomogeneity. The computed intracavity mode irradiance structures for two different open-loop deformable mirror surface profiles are illustrated in the upper portion of Fig. 39, and the corresponding far-field irradiance profiles are given in the lower part of the figure. For each mirror profile considered here, no single aberration is greater than one tenth of a wavelength, and, excluding tilt, each is below 0.05 wavelength center to edge, as is the difference (see OUGHSTUN,SPINHIRNE and ANAFI [1984]). The localized nature of the aberration (because of a strong localized surface error in the deformable mirror profile located in the first quadrant of the transverse coordinate system depicted in Fig. 39) is nevertheless sufficiently severe to alter the intracavity mode structure completely, resulting in a confused far-field irradiance pattern. Of more critical consequence here, however, is the observation that slight changes in the introduced phase aberration structure only slightly alter the intracavity mode structure properties (a 4% change in intracavity power and a 3.4% change in outcoupled power with almost no change in the flux eigenvalue), whereas they significantly alter the resultant far-field properties (a corresponding 47% change in relative beam quality and a 44% change in power in the bucket taken about the ideal aligned optical axis). The performance of this laser system is then seen to be strongly driven by the internal aberration sources to the extent that the cavity has lost its control over the transverse mode structure. Similar results have also been obtained by SIEGMANand SZIKLAS[ 19741 and SZIKLAS and SIEGMAN[ 19751 for the case of a transversely flowing gain medium that is optically inhomogeneous. Their analysis considered realistic flow conditions in a high-power gas-dynamic laser, including the refractive index inhomogeneities caused by the aerodynamic shock waves emanating from the sidewalls of the flow device. With this same medium the transverse mode structure of the resonator device was numerically determined at two different cavity Fresnel numbers with the same geometrical magnification (A4= 2.5). First, a low Fresnel number cavity of Neq= 0.5 was considered for the purpose of maintaining excellent transverse mode control in the presence of the medium inhomogeneities. The passive cavity mode structure for this resonator is depicted in Fig. 16 and the resultant active cavity mode structure is given in Fig. 40. In spite of the presence of a gain distribution that favors the upstream edge of the cavity volume, the central intensity core of the active cavity mode retains the peak intensity characteristic of the passive cavity mode that is
IV,
I 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
35 1
Fig. 40. Relative intensity and phase distributions of the active cavity mode structure incident upon the feedback aperture of anM = 2.5, Nes = 0.5 unstable resonator with a transverse flowing saturable gain medium. (After SIEGMANand SZIKLAS[1974].)
352
UNSTABLE RESONATOR MODES
indicative of excellent mode discrimination. The phase-front structures of the passive and active cavity fields are also quite similar. The calculated intracavity power incident upon the outcoupling-feedback aperture plane of this resonator is Pi,= 52.0 kW with an outcoupled power of Po,,= 41.0 kW. The flux eigenvalue is then yF = 0.46, which is decreased by 27 % from its passive cavity value of y = 0.632. A higher Fresnel number cavity of Neq = 1.5 was then considered for the purpose of increasing the power extracted from the saturable gain medium. The passive cavity mode structure for this resonator is depicted in Fig. 17, and the resultant active cavity mode structure is given in Fig. 41. The intracavity intensity distribution in this case exhibits a more noticeable lean toward the upstream side of the gain flow and substantially more distortion that is correlated with the aerodynamic shock waves. In addition, the central intensity core of the active cavity mode has been significantly reduced from its passive cavity predominance, indicating a partial decrease in the transverse mode discrimination. The phase distribution, however, is not seriously distorted beyond that obtained in the Neq = 0.5 case. The calculated intracavity power incident upon the outcoupling-feedback aperture plane of this resonator is P,, = 104 kW with an outcoupled power of Pout= 88.5 kW, roughly twice that for the Neq = 0.5 case. The flux eigenvalue is then yF = 0.386, which is decreased by 24% from its passive cavity value of y = 0.510. As the Fresnel number of the cavity is increased further, the degradation of the cavity field becomes more pronounced (see SZIKLASand SIEGMAN[ 19751) as the cavity medium increasingly predominates the transverse mode structure formation process. Any fluctuation in the gain medium flow will then be directly translated into a fluctuation in the cavity field with minimal resistance from the cavity. Such is not the case for the Neq = 0.5 cavity at this level of aberration, since the cavity there still strongly rejects the higher-order mode structures. Similar numerical results have been obtained by KARAMZINand KONEV[ 19751, OUGHSTUN[1981b, 1983a1, and LAX, AGRAWAL,BELIC, COFFEY and LOUISELL[ 19851. The importance of maintaining transverse mode control is then clearly seen. At all times the cavity must predominantly determine a single transverse mode, all other modes being highly unfavored. This may readily be accomplished by keeping the cavity Fresnel number below unity (preferably at Neq = OS), but this does not allow one to extract all of the available power efficiently from the gain medium (unless the cavity is folded so as to increase the cavity field-medium interaction volume). One may also employ the artifacts of aperture apodization or spatial filtering or some other exotic cavity design in
IV, I 41
ACTIVE CAVITY MODE STRUCTURE BEHAVIOR
353
Fig. 41. Relative intensity and phase distributions of the active cavity mode structure incident upon the feedback aperture of an M = 2.5,Neq = 1.5 unstable resonator with the same transverse and SZIKLAS[1974].) flowing saturable gain medium as in Fig. 40. (After SIEGMAN
354
UNSTABLE RESONATOR MODES
[IV, 8 51
order to increase the cavity Fresnel number (and, in effect, the cavity mode volume for a fixed cavity length) without a corresponding decrease in transverse mode discrimination (usually through a decreasein the cavity mode dependence upon the Fresnel number). The application of a deterministic intracavity adaptive optical system for the sole purpose of transverse mode control in the presence of uncontrollable phase aberrations has been described by OUGHSTUN[ 1982d, 1983a, 1986al and KOVAL’CHUK, RODIONOVand SHERSTOBITOV [ 19831.
8 5. Concluding Remarks This article has attempted to present an in-depth overviewof the fundamental mode structure theory of unstable optical cavities for laser applications. Much of this analysis has focused on the passive cavity mode structure properties, since it is in this purely optical arena that the more fundamental mathematical properties reside, several of which yet remain unresolved (in the rigorous sense). Indeed, two of the most fundamental properties of transverse mode theory in open optical cavities have yet to be completely answered, these being the questions of existence and completeness. In spite of this, much has been accomplished since the publication of the foundational paper on the resonant modes of a maser interferometer by Fox and LI in 1961 and the subsequent invention of unstable resonators by SIEGMAN in 1965. Still yet, much remains to be done. Despite the length of this article, several topics of interest to the subject of unstable resonators have either been neglected or have only been slightly mentioned. These include the analysis of marginally stable resonators (NAGEL, ROGOVIN, AVIZONISand BUTTS [ 19791, NAGEL,ROGOVINand AVIZONIS [ 19801, NAGELand ROGOVIN [ 19801);the analysis ofthe aberration sensitivity and of unstable cavities (ANAN’EV[ 19721, CHESTER[ 1973a1, SANTANA FELSEN [ 19771, OUGHSTUN[ 1981a,b, 1982c, 1983a, 1986a,b,c]); and the virtual source theory of unstable resonator modes (SIEGMAN and ARRATHOON [ 19671, SOUTHWELL [ 19811). Experimental techniques have largely been ignored, although some (more fundamental) experimental results have been quoted. Finally, it is of some historical interest to notice the close relationship between the publications in the Soviet and the American and European journals (this would be readily evident if the reference list here was assembled in chronological order). In spite of many obstacles, a productive scientific debate has persisted for many years in this international forum.
IVI
APPENDIX
355
Acknowledgements
To my good friend and colleague Walter J. Fader, to Edward A. Sziklas, Anthony J. DeMaria, David C. Smith, James M. Spinhirne, Judith B. Addison, and to all my former colleagues and friends at United Technologies Research Center with whom I have had the pleasure of working for several formative years, to the faculty, staff and students of the Electrical and Computer Engineering Department at the University of Wisconsin at Madison, and to my wife Joyce and daughters Marcianna and Kristen who have put up with me through all of this. Without each of you this work would have not been possible.
Appendix. Numerical Techniques and Sampling Criteria An important area of past and recent research interest is the numerical determination of the transverse mode structure supported by a given optical cavity. Since this topic does not fall into the mainstream of transverse mode theory but, rather, is a tool of more practical interest with applicability to a wide variety of optical problems, it is best treated here as an appendix. The numerical determination of the transverse mode structure for a stable cavity was first described by Fox and LI [1961], based upon the Fresnel-Kirchhoff diffraction integral.* Similar approaches for unstable cavities have been described by RENSCHand CHESTER[ 19731, CHESTER[ 1973b], RENSCH[ 19741, STEIERand MCALLISTER [ 19751, and PERKINSand SHATAS [1976]. A more general approach based on the angular spectrum of plane waves method (see GOODMAN [ 19681, SHERMAN [ 19821) was described by SZIKLASand SIEGMAN[1974, 19751 with specific application to unstable cavities. Since this approach does not necessarily entail the use of the paraxial approximation found in the first approach, it is inherently a more precise mathematical formulation of the problem. As a consequence, only this approach will be considered in detail here. With the availability of the fast Fourier transform algorithm (FFT), the numerically efficient and accurate solution to a wide variety of optical wave propagation problems in Cartesian coordinates was made possible. However, that particular coordinate system does not take full advantage of the rotational symmetry present (to some degree) in most optical systems of interest. To take * The limit of applicabilityofthe Fresnel approximation for wave propagation in large aperture optical systems has been described by FEIOCK[1978].
356
UNSTABLE RESONATOR MODES
[IV
advantage of this inherent symmetry, a quasi-fast Hankel transform algorithm (QFHT) was developed by SIEGMAN [ 19771. Any asymmetries present in the field (such as that introduced by phase aberrations in the optical system) may then be taken into account with this approach by modeling the various azimuthal componentsintroduced into the field structure. As the optical system becomes more asymmetrical with increasing aberrations, the advantage of the QFHT over the FFT approach is lost.
A.l. SCALAR WAVE PROPAGATION METHODS
For monochromatic, scalar wave optical propagation phenomena in homogeneous, isotropic media, solutions to the scalar Helmholtz equation
(V' + k2)u(r) = 0
(A. 1)
are sought subject to given, well-prescribed boundary conditions. For the present analysis the boundary value problem of interest may be stated as follows: Given the input scalar wave function u(rT, 2,) that is propagating in the positive z direction at a transverse plane z = z, (where rTis the component of r transverse to the z axis), determine the resultant scalar wave field u(rT, z) at some later transverse plane z > 2,. The exact formal solution of this boundary value problem in both Cartesian and polar cylindrical coordinates is now briefly reviewed and put into a context suitable for numerical analysis. A. 1.1. Cartesian coordinate solution
At the transverse plane located at z = z,, the initial scalar wave function = u(x, y, z), satisfies the two-dimensional spatial frequency Fourier transform pair u(rT, z),
U(X,Y, 20) =
SS,
C(vx, v,,, zo) =
fi(vx, v,,, z), exp[i2n(vXx+ v,,y)] dv, dv,, , u(x, y, z), exp[ - i2n(vxx + v,,y)] dx dy ,
(A.2) (A.3)
where a( v, , vr ,z,) is the spatial frequency spectrum of the initial transverse disturbance. At a later transverse plane at z situated a distance Az = z - z, from the initial plane, the propagated scalar wave disturbance u(x, y, z) and its spatial frequency spectrum fi(v,, v,, z) also satisfy a similar two-dimensional
IVI
357
APPENDIX
Fourier transform pair relationship, viz.
NV,, vv, z)
=
srm
u ( x , y, z) exp[ - i2n(vxx + v,,y)] d x dy
.
(A.5)
Since the propagated field (A.4) must satisfy the Helmholtz equation (A. 1) with the boundary value (A.2), the propagated spectrum is found to be simply related to the initial field spectrum according to the equation ~( v, , v,,, z)
=
fi(v,, v,,, zo) exp[iB(v,, v,,) A21 ,
(A.6)
where Az = z - z, is the propagation distance between the two transverse planes, and P(v,, v,)
= [k2 - (242(v? + v,)l2 =
k [ 1 - P(V;
+ v,2)]1/2
112
(A.7)
is the propagation function with k = 2n/A. Consequently, given the initial transverse structure of the optical field u ( x , y , z,) at the plane z = z,, ,its spatial frequency spectrum a( v,, v,,, zo) is first determined from eq. (A.3). This initial spectrum function is then propagated the distance Az = z - z,, according to eq. (A.6), resulting in the spatial frequency spectrum ii( v,, vv, z). The transverse behavior of the propagated field u ( x , y , z) at the plane z is then given by the inverse Fourier transformation (A.4). The resultant field is then the exact solution to the prescribed boundary value problem for the scalar Helmholtz equation (the effects of apertures are typically approximated by the usual Kirchhoff approximation). For efficient numerical calculations the required Fourier transformations may be numerically determined by using the fast Fourier transform algorithm*. However, for the propagated field distribution to be computed accurately, proper consideration must be given to the calculational domain and sampling interval size for the required Fourier transformations. This is presented in $ A.3.
* The evaluation of the required Fourier integrals may also be accomplished using B-splines [1982]). This approach allows one (LAX,AGRAWAL and LOUISELL[1979], LAXand AGRAWAL to deal accurately with sharply cut-off fields at aperture edges using a continuous Fourier integration procedure.
358
[IV
UNSTABLE RESONATOR MODES
A. 1.2. Polar cylindrical coordinate solution
Across the input plane the initial disturbance u(r, 0, zo) may be expressed as a Fourier series expansion in the azimuthal parameter I and as a Hankel or Fourier-Bessel expansion in the radial transform parameter p. This expansion has the general form cn
u(r, 0, zo)
C
=
uI(r, zo) eile
(A.8)
/= - m
= I=
2
m
eire.27c
J0
E,(p, zo) JI(27cpr)p dp .
64.9)
-a2
Because of the following orthogonality relation for the Bessel functions
r.0
(A. 10)
it follows that each Ith-order azimuthal component u,(r, zo) of the scalar wave field and its radial transform P,(p, z,,) are related by the Ith-order symmetrical Hankel transform pair (A. 11)
(A. 12) At a later plane z a distance Az = z - zo from the input plane, the propagated azimuthal component disturbance uI(r,z) and its radial spatial frequency spectrum E,(p, z) satisfy a similar Ith-order symmetrical Hankel transform pair relationship, viz. U I k
z ) = 211
EI(p, z ) = 2 x
Jom JOm
%(p, z ) J , ( 2 ~ p9 dp ,
(A.13)
uI(r, z ) JI(2xpr)r d r ,
(A. 14)
where the propagated spectrum is given by (A.15)
z’j/(p, z ) = cI(p, zo) eiP(P)&,
with B(p) = [ k 2 - ( 2 7 ~ p ) ~ I=”k~[ 1 -
A2pZ]1’2.
(A.16)
IVI
APPENDIX
359
For efficient numerical calculations the preceding Fourier-Bessel transformations may be numerically computed using the quasi-fast Hankel transform algorithm developed by SIEGMAN [ 19771. Such a numerically efficient Hankel transform algorithm can be particularly useful for optical beam propagation problems that possess complete cylindrical symmetry or only have low-order azimuthal variations. The algorithm for such a numerical calculation is simply as follows: Since the radial spectrum of each azimuthal component of the scalar wave field propagates as exp[iP(p) 1123 independently of the azimuthal mode number 1, then given an initial disturbance U ( I , 0, z o ) comprising the individual azimuthal component fields uI(r,zo),as specified in eq. (A.8), each initial azimuthal component disturbance is propagated to the transverse plane at z by fis t calculating its radial spatial frequency spectrum EI(p, zo).Each propagated spectrum is then determined from eq. (A.15), and the propagated azimuthal component disturbance is determined from the Hankel transformation (A. 13). Finally, the total propagated scalar wave disturbance u(r, 0, z ) is obtained by summing over each propagated azimuthal component disturbance u,(r, z ) multiplied by its individual azimuthal variation, viz. u(r, e, z ) =
m
C
ul(r, z ) eitO
(A. 17)
/= - m
This approach then yields the exact solution to the planar boundary value problem for the scalar Helmholtz equation in cylindrical polar coordinates. A. 1.2.1, The Gardner transformation and the quasi-fast Hankel transform algorithm For wave propagation phenomena with cylindrical symmetry it then becomes necessary to evaluate a given Hankel transform pair numerically; the pair may be written (A.18) (A. 19)
with the same enhanced efficiency presently available in Cartesian coordinates with the FFT. However, there does not exist an exact discrete Hankel transform evaluated by the FFT algorithm. This being the case, the recent algorithm developed by SIEGMAN[ 19771 for numerically evaluating Hankel transforms proceeds by an indirect method that provides at least some of the
360
UNSTABLE RESONATOR MODES
[IV
improvement in numerical efficiency offered by the FFT. The new algorithm proceeds by employing an exponentially nonlinear change variable r = roe"",
p
= pOean,
(A.20)
called the Gardner transformation, with scale factor a, to convert each Hankel transform integral into a cross-correlation integral, which can then be evaluated on a discrete sampled basis by two successive FFTs. The resulting algorithm is not a true fast Hankel transform in that it does not contain the full Nlog,(N) computational efficiency of the FFT but, rather, is a quasi-fast Hankel transform in that some of this computational efficiency is retained (see SIEGMAN [ 19771). In order that the QFHT algorithm be properly implemented, special care must be taken in choosing the parameters related to the Gardner transformation and the sampling requirements for the particular propagation geometry being considered. Following Siegman's notation, let ro and po be the lower-end sample points in real and transform space, respectively, and let N denote the total number of radial sample points employed. In order to obtain criteria for selecting the parameters N, a, r,, and po, it is necessary first to assume the existence of maximum values of the coordinates r and p, denoted by (A.21)
above which the functions $(r) and pg(p), respectively, become sufficiently small so that the integrals appearing in eqs. (A. 18) and (A. 19) may be truncated at these values with only a small resultant error. The product fib represents the space-bandwidth product for the radial function and its transform. The smallest sample values in r and p are ro and po (when n = 0), respectively, whereas the largest values are determined by locating the ( N + 1)th sample points rN and pN (which are not actually retained for the numerical computation) at the upper limits, viz. b
=
roeaN,
/3
=
pOeaN.
The Hankel transform integrals (A. 18) and (A. 19) are then effectively truncated to the respective domains po5pSB,
ro5r5b.
(A.23)
,
The outermost sample points rN- and p N - are actually located just inside the upper limit in each domain. Finally, if K, and K , represent the sampling rates at the lower and upper ends of the domains (A.23), it then follows that
IVI
APPENDIX
36 1
all of these parameters are interrelated according to the following equations: (A.24)
(A.25)
(A.26) These three equations may be solved implicitly to determine the values of the corresponding to any chosen value of N and any parameters a, rope, and assumed values of the number of sample points per cycle K, and K2 at the upper and lower ends of the truncated integration ranges. Equation (A.25) can then be solved iteratively for a using the method of interval bisection. Alternatively, given a desired value of the space-bandwidth product pb, the required number of radial sample points N is then given by N
=
1 K,Bb ln(KJ3b) = - In(K,pb).
(A.27)
01
Notice that the required number of sample points is much more sensitive to the upper-end sample spacing K , than to the lower-end spacing K , . The scaling parameter a and the product ropo are then determined from eqs. (A.25) and (A.26). The required value of the space-bandwidth product /?b is determined by the geometry of the propagation problem through the value of the collimated Fresnel number of the propagation step. The remaining parameters appearing in the QFHT algorithm are then determined in a self-consistent manner from the required value of j?b and purely numerical considerations. A. 1.2.2. Lower-end correction to the quasi-fast Hankel transform algorithm Because of the discrete sampling of the Gardner transformation, the Hankel transform functions f(r) and g ( p ) are not described for positive values of r < ro and p < po , respectively. In order to account for this small argument behavior, a lower-end correction to the QFHT must be employed. To illustrate this, consider eq. (A.19), which may be written as
362
UNSTABLE RESONATOR MODES
[IV
where (A.29) (A.30)
(A.3 1)
where f ( r ) is assumed known for all r 2 0. The integral g ( p ) is just that evaluated by the QFHT algorithm. If f ( r ) is space limited and b is properly chosen so that rf(r) is negligible for r 2 b, then the third integral E,(p) is negligible and may be ignored. For 1 # 0, J,(2npr) = 0 at r = 0, so that for sufficiently small but nonvanishing values of r, the first integral go(p) is also negligible. The contribution of that integral, however, is not negligible for 1 = 0 iff(r) is nonvanishing near the origin. In that case the contribution of this lower-end integral must be incorporated into the calculation ofg(p). In a similar manner the inverse Hankel transform may be written as
(A.33)
(A.34) (A.35)
where g(p) is assumed known for all p 2 0. The integral T ( r ) is just that evaluated by the QFHT algorithm, and the integral %(r) is negligible if fl is properly chosen so that pg(p) is neghgible for I # 0 with sufficiently small values of po > 0, but it is not negligible for 1 = 0 unless g(p) is vanishingly small near the origin. The evaluation of the lower end integrals go(p) and fo(r) due to OUGHSTUN[ 19801 for 1 = 0 is now considered (see also AGRAWAL and LAX [ 1981al). For sufficiently small values of the lower end point r, ,the lower end integral go@)may be directly evaluated by expanding the function f ( r ) in a Maclaurin
IVI
363
APPENDIX
series as
f(r)=f(O)
+ if”(0)r2 + . . . ,
where the fact that, for I = 0, f(r) is symmetrical about the origin [and hence, f ’ ( 0 ) = 01 has been employed. Substitution of the first two terms of this expansion into (A.29) for the lower end integral go(p) yields
and, sirmlarly, for the inverse transform the lower end integral is given by fo(r) E d o )
r
J1(271Por)+ g”(0)
P;
[J1(271Por)- J&.nP,r)l .
(A.37)
Notice that the second term in both of these expressions vanishes at the origin and is very small in comparison with the corresponding first term over the domain of the calculation for sufliciently small values of po and r, ,respectively, and may then be ignored. Furthermore, notice that to compute these two lower-end integrals accurately, the behavior of the functions at the origin must be known at all times. These values may easily be determined from eqs. (A.18) and (A.19), as (A.38)
and rf(r) J,(O) dr E 2 4 ,
(A.39)
where a,, is the Kronecker delta function. The final Hankel transform algorithm then proceeds as follows (OUGHSTUN [ 19801): Given the function f(r) with space-bandwidth product p, calculate its QFHT pair function g ( p ) from po to /3 and determine its value g ( 0 ) at the origin from eq. (A.39). The total spectrum g(p) is then given by adding the lower-end integral go(p) to g(p), so that
(A.40)
364
UNSTABLE RESONATOR MODES
[IV
and similarly
(A.41)
These two numerically determined functions then represent an accurate discrete approximation to the Hankel transform pair (A.18) and (A.19) with space-bandwidth product fib, provided that proper sampling is maintained. A.2. SPHERICAL WAVE COORDINATE TRANSFORMATION
The spatial frequency transform approach to scalar wave propagation problems, combined with the FFT or QFHT algorithms, should provide a faster and more accurate computational procedure than, for example, a direct numerical evaluation of the equivalent Fresnel-Kirchhoff integral equation. However, both the computational efficiency and accuracy are reduced if the optical beam field under consideration is primarily a spherically diverging or converging beam. In that case, some comparatively slowly varying transverse amplitude and phase structure multiplies a spherical wavefront. The spherical wave factor in the converging or diverging beam increases the spread of the transform in spatial frequency space without a corresponding increase in real information. The increased spread in the frequency domain then requires more closely spaced samples in real space. In order to avoid these difficulties, SZIKLASand SIEGMAN[ 19741 introduced a coordinate transformation based on Gaussian beam theory that removes the known spherical wave factor. In the paraxial approximation the complex scalar wave disturbance satisfies the reduced Helmholtz equation azU
a2u
ax2
ay2
au = 0 , az
- + - + 2ik-
(A.42)
where the monochromatic scalar wave field is of the form u(x, y , z) exp [i(kz - at)]and the z variation of the complex amplitude u(x, y , z ) is assumed to be slow in comparison with the variation of the wavenumber k. The general coordinate transformation is begun by extracting an arbitrary Gaussian spherical wave [ l/q(z)] exp[ik(x2 + y2)/2q(z)] from the complex amplitude function by letting
(A.43)
IVI
APPENDIX
365
where q(z) is the complex curvature of the Gaussian beam, given by q(z) = z - ib when z is measured from the beam waist, where b = nw;/A is the and LI [ 1966a1). confocal beam parameter with beam waist w, (see KOGELNIK The function o(x’, y’, z’) is expressed in terms of the complex coordinates ux ux x’(x, z) = - = -, q(z) z - ib
(A.45) dz’(z) - u2 dz q2(z)’
(A.46)
(A.47) where u is an arbitrary complex constant. Under this transformation the function v(x’, y’, z’) is found to satisfy the paraxial Helmholtz equation (A.48) in the primed coordinate system. Because an underlying beam divergence (or convergence) has been removed, the variation of u(x‘, y’, z’) in the primed coordinate system can be made to resemble that of a collimated field. For propagation phenomena involving either a spherically diverging or a spherically converging field, one may choose the origin of the z coordinate to be located at the focus of the spherical wave. Furthermore, let the confocal parameter b be chosen much smaller than the magnitudes of any of the values of z at which the field u(x, y , z) is to be determined. The preceding coordinate transformations are then given by x’ =
ux
-
(A.49)
Z
aY y’ = -
(A.50)
Z
(A.51)
366
UNSTABLE RESONATOR MODES
[IV
and it is most convenient to choose a to be real valued. A fixed array of sample points in the primed coordinate system now corresponds to one in real space that is linearly expanding or contracting with the propagation distance z. A scalar wave beam field u(x, y, z) that is diverging linearly with the propagation distance in the unprimed (real) coordinate system will then possess a near constant width in the primed coordinate system, and the function D(x’,y’, z’) may be efficiently calculated using a fixed sample grid in the transverse direction. The same result clearly holds in polar cylindrical coordinates.
A.3. NUMERICAL SAMPLING CRITERIA
Beam field propagation calculations employing either the FFT or QFHT algorithms require proper attention to the questions of aliasing, sampling, and windowing of the discrete transforms (SZIKLASand SIEGMAN[ 19751). Since the coordinate transformation of the preceding section may be used to transform all diverging or converging beam problems into equivalent collimated beam problems, the analysis here need only address the problem for a collimated beam with an initial diameter 2a, that is propagated for a distance L in free space. The associated propagation phenomena are then characterized in the Fresnel approximation by the collimated Fresnel number N, = a:/;U, of the transformed propagation geometry.
A.3.1. Guard band requirement The discrete Fourier transform procedure with a uniform array of sample points actually solves the problem of an infinite array of identical, iterated cells located side by side with equal spacing to the full width of the cell domain. The information in each of these cells must then be separated from that in each adjoining cell by an adequate guard band in order to avoid having dif€racted energy from one cell spill over into the neighboring cells. As a simple guideline, SZIKLAS and SIEGMAN [ 19751 consider the diffracted field due to a uniform amplitude plane wave field normally incident upon a slit aperture of width 2a,. After a propagation distance L the fractional amount of energy that is diffracted outside a transverse domain of width 2Ga,, where G > 1, and thus spills over into the adjoining cells, is given approximately by <&I.
(A.52)
361
APPENDIX
IVI
The inequality expressed here indicates that this fractional amount of “energy spillover” into the adjoining cells is to be kept below some specified value E , . Upon solving this quadratic equation for the guard band ratio, one obtains the requirement G(El)> 1 +
1
[ 1 + (1 + 27?NC&,
~
7cE1)”2]
.
(A.53)
For sufficiently small values of c1 this expression reduces to that obtained by SZIKLASand SIEGMAN[1975]. The functional behavior of G given by the equality in eq. (A.53)as a function of the collimated Fresnel number N, for several values of the plane wave energy tolerance E , is illustrated in Fig. 42. Because of the inequality appearing in eq. (A.53),these curves represent the minimum allowable values of the guard band ratio G for a specified tolerance E , . For all nonvanishing values of E , the guard band ratio is seen to approach unity asymptoticallyfrom above as N, approaches infinity. However, numerical calculations have shown that the guard band ratio should not be decreased below approximately 1.1. A.3.2. Sampling interval requirement in Cartesian coordinates For a uniformly illuminated slit aperture of width 2a1 the spatial frequency spectrum of the apertured field is given by ii( v)
=
2a1 sinc(2mra, v) .
(A.54)
For a discrete Fourier transform based on a cell of width 2Ga,, this spectrum is sampled at discrete intervals in spatial frequency space whose width, as specified by the Whittaker-Shannon Sampling Theorem, is AV=-.
1
(A.55)
2Ga,
With Np sample points the highest spatial frequency sampled is then given by v,,,
NP . 4Ga1
N AV = =9
~
2
(A.56)
The fractional amount of residual energy present in the spatial frequency spectrum above this maximum sampled value is approximately 2G
E(V,,,)
.
E-
*NP
(A.57)
368
UNSTABLE RESONATOR MODES
12
11
10
9
a
a 0
2 7 a: 0 2
$
6
0
a: Q 3
u
5
4
3
2
1
0
I
I
10
20
I
I
I
I
60 COLLIMATED FRESEL NUMBER NC 30
40
50
I
I
I
70
80
90
Fig. 42. Parametric family of curves for the guard band ratio G as a function of the collimated Fresnel number of the propagation geometry for several values of the plane wave energy loss tolerance E , .
From a numerical point of view this quantity is the fractional amount of energy that is aliased from high spatial frequencies back into lower ones in the performance of the FFT. To keep this error below a fractional tolerance c, then
APPENDIX
IVI
369
requires that the inequality (A.58) be satisfied for the total number of sample points (in one transverse dimension) in the discrete Fourier transform. A.3.3. Fresnel zone requirement in Cartesian coordinates
Consider now the edge wave scattered from one edge of the same uniformly illuminated slit aperture of width 2 a , . The maximum spatial frequency present in the calculation cell of width 2Ga, at a propagation distance L from the aperture is (A.59) To sample this rapidly oscillating component adequately at the output plane then requires that the total number of sample points be given by (NJ2)Av = vb,,, or with Av given by eq. (A.55) one then obtains the requirement (SZIKLASand SIEGMAN[ 19751) N,, 3 4G(G
+ l)Nc.
(A.60)
The same result is obtained if one considers the minimum number of sample points necessary to have two points per cycle of oscillation in the finest grain Fresnel zone of the slit aperture as viewed from the outer edge of the calculational domain in the observation plane. With the guard band ratio given by the approximate expression G = 1 + l/(n?Nccl)and with c2 = E ~ the , number N i of sample points specified by eq. (A.58) is
whereas the number N,, specified by eq. (A.60) is
Since the Fresnel zone requirement for the total number of sample points is more stringent than the sampling interval requirement, the former is adopted.
370
[IV
UNSTABLE RESONATOR MODES
With the use of the equality in relation (A.53) for the guard band ratio G and one obtains the functional behavior of N p given by the equality in with % = relation (A.60), depicted in Fig. 43. Because of the inequalities present in these 500
400
300
-+
0
2
(3
I?II
a
z
200
100
0 0
I
I
I
I
I
10
20
30
40
50
I 60
I
70
I 80
1 90
COLLIMATED FRESNEL NilMBER N,
Fig. 43. Parametric family of curves for the required number of Cartesian sample points N,, in one transversedimension as a hnction ofthe collimated Fresnel number of the propagation geometry for several values of the plane wave energy loss tolerance E , .
IVI
APPENDIX
37 1
two expressions, these curves represent the minimum allowable values of the total number of Cartesian sample points in one transverse dimension as a function of the collimated Fresnel number N, for the indicated values of the tolerance E ] . Numerical studies indicate that the choice = 0.01 yields highly accurate results and may then be considered as a tight tolerance. On the other hand, the choice E , = 0.03 results in some energy loss resulting from dsractive spillover, which may be unacceptable for some cavity mode calculations. The compromise choice of = 0.02 yields accurate results without being overly severe on the required number of transverse sample points. A.3.4. Sampling interval and Fresnel zone requirements in polar cylindrical coordinates
Consider first the determination of the numbers of sample points per cycle K , and K2 at the lower and upper ends of the truncated integration domain employed in QFHT algorithm. First of all,for the Nyquist sampling rate to be met it is required that K , 2 2 and K 2 2 2. At the outer end of the calculation region both the field and spectrum amplitudes are small, and the particular choice of K , is arbitrary to a certain degree. It is numerically found that the choice
K,
=
4
(A.61)
is sufficient for typical beam propagation problems with an adequate guard band ratio. At the lower end of the calculation region the lower limit sample coordinates r, and po are chosen to be not larger than 1/K, of a cycle at the highest frequency present in each domain so that (A.62) (A.63)
Since the sampling requirements near the origin are most critical for the 1 = 0 azimuthal field structure, that case will be employed to determine the required lower end sampling. For the I = 0 field structure the Bessel function kernel at the highest radial spatial frequency present in the numerical calculation is Jo(2n/lr).The first zero of this function occurs at 27$r = 2.41, and it is desired to have r,, well below this value. As a consequence, let 27r/3r0 = 2 . 4 1 ~ , ,
312
UNSTABLE RESONATOR MODES
[IV
where E~ < 1 is some positive number. On substituting this result into eq. (A.62) one obtains
It is found numerically that decreasing E~ much below the value 0.26 does not gain very much in the accuracy of the code (with the lower-end correction). With this value a conservative estimate of the lower end sampling rate is found to be
K,
=
10.
(A.64)
Numerical accuracy is lost i f K , is dropped below this value, whereas not much additional accuracy is,gained if K , is increased above it. All that remains now to be determined is the Gardner transform parameter a, and that is specified by the Fresnel number of the propagation problem. In order to have two sample points per cycle for the finest grain Fresnel zone present in the aperture of radius a,, as seen from the outer edge of the calculation region, with guard band ratio G in the observation plane a distance L from the aperture, the radial sampling interval at the edge of the aperture plane must be given by ArN =
LA 4(G + l)a,
(A.65)
With ArN = ab and b = Gal one then obtains the expression a=
1
4G(G + 1)N,
(A.66)
From eq. (A.26) the physically required space-bandwidth product is then given by 1 j?b = -4G(G + 1)N,. K*
(A.67)
On substituting this result into eq. (A.27), one finally obtains N
=
4G(G + l)Nc In
(A.68)
[ 19801). as the physically required number ofradial sample points (OUGHSTUN
IVI
APPENDIX
313
In Cartesian coordinates the number of required sample points in one transverse dimension is given by N,, = 4G(G + l)Nc. The additional factor 1n((K,/K2)4G(G + l)Nc) appearing in eq. (A.68) is due to the nonlinear change of variable in the Gardner transformation. With the equality in relation (A.53) for the guard band ratio, the resultant functional behavior of the number N of radial sample points, given by the equality in relation (A.68) as a function of the collimated Fresnel number of the propagation geometry, is illustrated in Fig. 44 for several values of the tolerance factor tl. For the purpose of comparison the behavior of the total number of Cartesian sample points N: for an equivalent two-dimensional Cartesian coordinate calculation is also illustrated. As can be seen, the required number of radial sample points is about an order of magnitude less than the required total number of Cartesian sample points. The ratio of the total number of cartesianto-polar sample points, given by P = 4G(G N2
+ l)Nc
(A.69)
is illustrated in Fig. 45 for several values of the plane wave power loss tolerance 6 , . This ratio may be considered as an estimate of the maximum number of azimuthal modes that may be described in a polar coordinate calculation without exceeding the required number of points in the equivalent Cartesian coordinate calculation. For a smaller number of azimuthal modes required for a particular problem, the polar coordinate code will be computationally more efficient than the Cartesian coordinate code, whereas for a larger number of required azimuthal modes the Cartesian coordinate code is the optimum numerical approach.
A.4. THIN-SHEET GAIN-PHASE APPROXIMATION
The final problem of interest here concerns the incorporation of the extended gain medium into the calculation of the transverse mode structure of a given resonator. Typically one models the continuous distribution of the gain medium, along with its refractive index inhomogeneities, as a discrete series of longitudinally uniform gain-phase segments in which the properties of the medium are assumed constant within cach axial segment. The medium properties present in each axial segment are then concentrated on a single transverse plane situated within that hegment, with free-space propagation
314
UNSTABLE RESONATOR MODES
0
10
20
30
40
50
60
70
COLLIMATED FRESNEL NUMBEH NC
Fig. 44. Parametric family of curves for the required number of radial sample points N and the total number of Cartesian sample points N,' as a function of the collimated Fresnel number of the propagation geometry for several values of the plane wave energy loss tolerance E , .
315
APPENDIX
120
110
100
80
NP2 N
70
60
50
40
30
20
10
0
0
I
I
I
I
I
I
I
I
1
10
20
30
40
50
60
70
80
YO
COLLIMATED FRESNEL NUMBER N,
Fig. 45. Parametric family of curves for the ratio of the total number of Cartesian sample points to the number of radial sample points N as a function of the collimated Fresnel number of the propagation geometry for several values of the plane wave energy loss tolerance E , .
assumed between each neighboring gam-phase plane. This simplification is known as the thin-sheet gain-phase approximation. In the limit as the number of gain-phase segments becomes very large, this approximation accurately
316
[IV
UNSTABLE RESONATOR MODES
models the actual continuous medium distribution. However, the associated amount of numerical computation becomes unbearable in this limit so that one must accurately approximate the actual medium by only using a small number of gain-phase segments. Criteria for determining the axial length z of a given gain-phase segment such that the thin-sheet gain-phase approximation is valid have been derived by MILONNI[ 19771. Following his analysis, let the gain medium be described by the susceptibility x(r) such that the gain and refractive index are given by (A.70) 1 n(r) z 1 + -Re ~ ( r, ) 2%
(A.7 1)
respectively. The optical field satisfies the paraxial wave equation V$u(r)
&(r)
k’
+ 2ik __ = - - X(r) u(r) , az %
(A.72)
where V; is the transverse Laplacian operator and z is the distance along the propagation direction. The formal solution of this equation may be written as 4 x 9
y , 4 = udx, y , 4 +
4 x 9
y, 4
(A.73)
9
where
[ ( x - x‘)’
+ 0,- y’)’]
is the Fresnel-Kirchhoff solution of the homogeneous wave equation and describes the free-space propagation of the field from the plane z = 0 to the observation plane at z. The solution of the inhomogeneous equation is given by
[(x - x‘)’
+ 0,- y’)’]
dx‘ dy’ , (A.75)
311
APPENDIX
IVI
and describes the field at the observation plane that is due to the dipoles induced by the field throughout the propagation volume between the plane at z = 0 and the observation plane at z. With a Taylor series expansion of the product function ~ ( x y’, ’ , z’) u(x’,y‘, z’) about the point ( x , y, z’) and integration over (x’, y’) of each term in the integral of eq. (A.75), there results ~ J xY, , Z) = 2% ik
r
X(X,
Y , z’) u(x, Y , z’) dz‘
0
1
- -V$
4% i
joz -
(z z’) ~ ( xy , z’) u(x, y , z’) dz‘ 4
- -V: Jo ( z - z’)2 ~ ( xy,, z’) u(x, y , z‘) dz’ + . . , 16k%
(A.76)
where terms involving odd derivates of xu have integrated to zero. MILONNI[ 19771 then introduces the function
[ ( x - x‘)’
+ 0,- y‘)’]
:,3
(A.77) On comparing this expression with eq. (A.76), it is seen that the inhomogeneous solution u,(x, y, z ) may be accurately approximated by u(x, y, z) if the propagation distance z is sufficiently small that (A.78) Z
so that the second and higher order terms in the expansions (A.76) and (A.77) are negligible compared with the first term, and provided that (A.79)
378
UNSTABLE RESONATOR MODES
[IV
When these two conditions are satisfied, eq. (A.73) may be approximated as
[(x - x ‘ ) ~+ 0, - Y ’ ) ~ ] dx’ dy‘ . (A.80)
This result is the propagation equation for the optical field in the thin-sheet gain-phase approximation. This approximation is valid provided that the length z of the medium segment is sufficiently small that the conditions (A.78) and (A.79) are satisfied. The first condition imposes a limit on the magnitude of diffraction within a given segment and, in essence, requires a large Fresnel number propagation between adjacent gain-phase sheets (RENSCH[ 19743). The second condition imposes a limit on both the magnitude and the variation of the medium susceptibilitywithin the segment; a similar criterion was argued by SIEGMAN and SZIKLAS[ 19741. Clearly both conditions must be satisfied for an accurate representation of the medium-field interaction to be obtained.
References ABRAMOWITZ, M., and LA. STEGUN,eds, 1964, Handbook of Mathematical Functions (U.S. Gov. Printing Office, Washington, D.C.). AGRAWAL,G.P., and M. LAX, 1979, Effects of Interference on Gain Saturation in Laser Resonators, J. Opt. SOC.Am. 69, 1717. AGRAWAL, G.P., and M. LAX, 1981a, End Correction in the Quasi-Fast Hankel Transform for Optical-Propagation Problems, Opt. Lett. 6, 171. AGRAWAL, G.P., and M. LAX, 1981b, Analytic Evaluation of Interference Effects on Laser Output in a Fabry-Perot Resonator, J. Opt. SOC.Am. 71, 515. AKHIEZER, N.I., and I.M. GLAZMAN, 1966, Theory of Linear Operators in Hilbert Space, vol. I (Ungar, New York). and D.G.C. JONES,1971, Enhancement of Spatial Coherence during ALLEN,L., S. GATEHOUSE Light Propagation in Bounded Media, Opt. Commun. 4, 169. A.L. LEVIT,V.M. OVCHINNIKOV and S.F. AL’TSHULER, G.B., E.D. ISYANOVA, V.B. KARASEV, SHARLAI,1977, Analysis of Misalignment Sensitivity of Ring Laser Resonators, Sov. J. Quantum Electron. 7, 857 (Kvantovaya Elektron. 4, 15171. and K.E. OUGHSTUN,1981, Intracavity Adaptive ANAFI,D., J.M. SPINHIRNE,R.H. FREEMAN Optics. 2: Tilt Correction Performance, Appl. Opt. 20, 1926. ANAN’EV, Yu.A., 1971, Angular Divergence of Radiation of Solid-state Lasers, Sov. Phys.-Usp. 14, 197 [Usp. Fiz. Nauk 103, 705). ANAN’EV, Yu.A., 1972, Unstable Resonators and their Applications (Review), Sov. J. Quantum Electron. 1, 565 [1971, Kvantovaya Elektron. 1, 31. ANAN’EV, Yu.A., 1975, Establishment of Oscillations in Unstable Resonators, Sov. J. Quantum Electron, 5, 615 [Kvantovaya Elektron. 2, 11381.
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RIGROD,W.W., 1963, Gain Saturation and Output Power of Optical Masers, J. Appl. Phys. 34, 2602. RIGROD,W.W., 1965, Saturation Effects in High-Gain Lasers, J. Appl. Phys. 36, 2487. RIGROD,W.W., 1983, Diffraction Loss of Stable Optical Resonators with Internal Limiting Apertures, IEEE J. Quantum Electron. QE-19, 1679. SANDERSON, R.L., and W. STREIFER,1969a, Unstable Laser Resonator Modes, Appl. Opt. 8, 2129. SANDERSON, R.L., and W. STREIFER,1969b, Laser Resonators with Tilted Reflectors, Appl. Opt. 8, 2241. SANTANA, C., 1981, Ray-Optical Calculation of Eigenmode Behavior of Unstable Laser Resonators with Rounded Edges, Appl. Opt. 20, 2852. 1976, Unstable Open Resonators: Two-Dimensional and SANTANA, C., and L.B. FELSEN, Three-Dimensional Losses by a Waveguide Analysis, Appl. Opt. 15, 1470. SANTANA, C., and L.B. FELSEN,1977, Effects ofMedium and Gain Inhomogeneities in Unstable Optical Resonators, Appl. Opt. 16, 1058. 1978, Unstable Strip Resonators with Misaligned Circular SANTANA, C., and L.B. FELSEN, Mirrors, Appl. Opt. 17, 2352. SARGENT,M., M.O. SCULLYand W.E. LAMBJR., 1974, Laser Physics (Addison-Wesley, Reading, MA). SCHMIDT,E., 1907, Zur Theorie der Linearen und Nichtlinearen Integral Gleichungen, Math. Ann. 13, 433. E.O., 1964, Pulse Sharpening and Gain Saturation in Travelling-Wave SCHULZ-DUBOIS, Masers, Bell Syst. Tech. J. 43, 625. 1981, Coupled Optical Resonators for the Enhancement of SEE, Y.C., S. GUHAand J. FALK, Laser Intracavity Power, Appl. Opt. 20, 1211. J.B., D.A. HOLMES,M.L. BERNABEand A.M. SIMONOFF,1980, Adaptive Mirror SHELLAN, Effects on the Performance of Annular Resonators, Appl. Opt. 19, 610. SHENG,S.C., 1979, Diffraction-Biased Unstable Ring Resonators with Possible Applications in Laser Gyroscopes, IEEE J. Quantum Electron. QE-15, 922. SHERMAN, G.C., 1982, Introduction to the Angular-Spectrum Representation of Optical Fields, Proc. SOC.Photo-Opt. Instrum. Eng. 358, 31. SHERSTOBITOV, V.E., and G.N. VINOKUROV, 1972, Properties of Unstable Resonators with Large Equivalent Fresnel Numbers, Sov. J. Quantum Electron. 2,224 [Kvantovaya Elektron. 3, 361. SIEGMAN, A.E., 1965, Unstable Optical Resonators for Laser Applications, Proc. Inst. Electr. Electron. Eng. 53, 277. SIEGMAN, A.E., 1971, Stabilizing Output with Unstable Resonators, Laser Focus 7, 42. SIEGMAN, A.E., 1973, Hermite-Gaussian Functions of Complex Argument as Optical-Beam Eigenfunctions, J. Opt. SOC.Am. 63, 1093. SIEGMAN, A.E., 1974, Unstable Optical Resonators, Appl. Opt. 13, 353. SIEGMAN,A.E., 1976, A Canonical Formulation for Analyzing Multielement Unstable Resonators, IEEE J. Quantum Electron. QE12, 35. SIEGMAN, A.E., 1977, Quast Fast Hankel Transform, Opt. Lett. 1, 13. SIEGMAN, A.E., 1979, Orthogonality Properties of Optical Resonator Eigenmodes, Opt. Commun. 31, 369. SIEGMAN, A.E., 1980, Exact Cavity Equations for Lasers with Large Output Coupling, Appl. Phys. Lett. 36,412. SIEGMAN, A.E., 1982, private communication. SIEGMAN, A.E., and R. ARRATHOON, 1967, Modes in Unstable Optical Resonators and Lens Waveguides, IEEE J. Quantum Electron. QE-3, 156.
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SIEGMAN,A.E., and H.Y. MILLER,1970, Unstable Optical Resonator Loss Calculations using the Prony Method, Appl. Opt. 9, 2729. SIEGMAN,A.E., and E.A. SZIKLAS,1974, Mode Calculations in Unstable Resonators with Flowing Saturable Gain. 1: Hermite-Gaussian Expansion, Appl. Opt. 13, 2775. SIEGMAN, A.E., P.A. B~LANGER and A. HARDY,1983, Optical resonators using phase conjugate mirrors, in: Optical Phase Conjugation, ed. R.A. Fisher (Academic Press, New York). SINCLAIR, D.C., and T.H.E. COTTRELL,1967, Transverse Mode Structure in Unstable Optical Cavities, Appl. Opt. 6, 845. SOUTHWELL, W.H., 1978, Asymptotic Solution of the Huygens-Fresnel Integral in Circular Coordinates, Opt. Lett. 3, 100. SOUTHWELL, W.H., 1979, Mode Discrimination of Unstable Resonators with Spatial Filters and by Phase Modification, Opt. Lett. 4, 193. SOUTHWELL, W.H., 1981, Virtual-Source Theory of Unstable Resonator Modes, Opt. Lett. 6, 487. SPENCER, M.B., and W.E. LAMBJR., 1972, Laser with a Transmitting Window, Phys. Rev. A5, 884. and H.R. GARCIA,1981, Intracavity Adaptive SPINHIRNE, J.M., D. ANAFI,R.H. FREEMAN Optics. 1: Astigmatism Correction Performance, Appl. Opt. u),976. SPINHIRNE, J.M., D. ANAFIand R.H. FREEMAN, 1982, Intracavity Adaptive Optics, 3: HSURIA Performance, Appl. Opt. 21, 3969. STATZ,H., and C.L. TANG,1965, Problem of Mode Deformation in Optical Masers, J. Appl. Phys. 36, 1816. STEIER,W.H., and G.L. MCALLISTER,1975, A Simplified Method for Predicting Unstable Resonator Mode Profiles, IEEE J. Quantum Electron. Q E l l , 725. STEPANOV, A.A., and V.A. SHCHEGLOV, 1982, Use of Unstable Resonators in CW Chemical Lasers with Radial Flow of a Gas Mixture, Sov. J. Quantum Electron. 12, 1142 (Kvantovaya Elektron. 9, 17721. STREIFER, W., 1965, Optical Resonator Modes Rectangular Reflectors of Spherical Curvature, J. Opt. SOC.Am. 55, 868. Opt. SOC.Am. 56, 1481. STREIFER, W., 1966, Spatial Coherence in Periodic Systems, .I. STREIFER, W., 1968, Unstable Optical Resonators and Waveguides, IEEE J. Quantum Electron. QE4,229. STREIFER,W., and H. GAMO,1964, On the Schmidt expansion for optical resonator modes, in: Proc. Symp. on Quasi-Optics, ed. J. Fox (Wdey, New York) pp. 351-365. SUITON, G.W., M.M. WEINERand S.A. MANI, 1976, Fraunhofer Diffraction Patterns from Uniformly Illuminating Square Output Apertures with Noncentered Square Obscurations, Appl. Opt. 15, 2228. SWANTNER, W., 1982, Wavefronts of Axicon Systems, Opt. Eng. 21, 333. SZIKLAS,E.A., and A.E. SIEGMAN,1974, DifFraction Calculations using Fast Fourier Transform Methods, Proc. Inst. Electr. Electron. Eng. 62, 410. SZIKLAS,E.A., and A.E. SIEGMAN,1975, Mode Calculations in Unstable Resonators with Flowing Saturable Gain. 2: Fast Fourier Transform Method, Appl. Opt. 14, 1874. TOMOV,I.V., R. FEDOSEJEVS and M.C. RICHARDSON, 1977, Unidirectional Travelling Wave Operation of a Mode-Locking Nd:Glass Ring Laser, Opt. Commun. 21, 327. TREACY,E.B., 1969, Diffractive Coupling for a C02 Laser, Appl. Opt. 8, 1107. TYLER,G.A., and W.H. SOUTHWELL,1980, Synthesis of the Vector Resonator Modes from Scalar Results, Opt. Lett. 5, 42. VAINSHTEIN, L.A., 1963, Open Resonators for Lasers, Sov. Phys.JETP 17, 709 [Zh. Eksp. & Teor. Fiz. 44, 10501. VAINSHTEIN, L.A., 1964, Open Resonators with Spherical Mirrors, Sov. Phys.-JETP 18, 471 [1963, Zh. Eksp. & Teor. Fiz. 45, 6841.
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VETKIN,V.A., and A.M. KHROMYKH, 1972, Competition between Longitudinal Modes in a Ring Laser with an Anisotropic Resonator, Sov. J. Quantum Electron. 2,240 [Kvantovaya Elektron. 3, 591. WEINER,M.M., 1977, Far-Field Energy, in the Geometric Mode Limit, of Loaded Unstable Resonators with Centered or Comer Obscurations, Appl. Opt. 16, 1790. WEINER,M.M., 1979, Modes of Empty Off-Axis Unstable Resonators with Rectangular Mirrors, Appl. Opt. 18, 1828. WEINSTEIN, L.A., 1969, Open Resonators and Open Waveguides (Golem Press, Boulder, CO). and P.R. BLASZUK,1973, Unstable Resonator for CO, ElectricWISNER,G.R., M.C. FOSTER Discharge Convection Lasers, Appl. Phys. Lett. 22, 14. WOLF,E., 1959, A Scalar Representation of Electromagnetic Fields: 11, Proc. Phys. SOC.74,269. WOLF,E., 1963, Spatial Coherence of Resonant Modes in a Maser Interferometer, Phys. Lett. 3, 166. WOLF,E., 1981a, A new description of second-order coherence phenomena in the space-frequency domain, in: Optics in Four Dimensions - 1980, eds M.A. Machado and L.M. Narducci, AIP Conf. Proc. 65,42. WOLF,E., 1981b, New Spectral Representation of Random Sources and ofthe Partially Coherent Fields that they Generate, Opt. Commun. 38, 3. WOLF,E., 1982. New Theory of Partial Coherence in the Space-Frequency Domain. Part I: Spectra and Cross Spectra of Steady-State Sources, J. Opt. Am. 72, 343. WOLF,E., 1986, New Theory of Partial Coherence in the Space-Frequency Domain. Part 11: Steady-State Fields and Higher-Order Correlations, J. Opt. SOC.Amer. A3, 76. WOLF,E., and G.S. AGARWAL, 1984, Coherence Theory of Laser Resonator Modes, J. Opt. SOC. Am. Al, 541. WRIGHT,E.M. and W.J. FIRTH,1982, Orthogonality Properties of General Optical Resonator Eigenmodes, Opt. Commun. 40,410. WRIGHT,E.M. and W.J. FIRTH,1983, private communication. WRIGHT,E.M., D.P. O'BRIENand W.J. FIRTH,1984, Reciprocity and Orthogonality Relations for Ring Resonators, IEEE J. Quantum Electron. QE20, 1307. ZEMSKOV, K.I., A.A. ISAEV,M.A. KAZARYAN, G.G. PETRASHand S.G. RAUTIAN,1974, Use of Unstable Resonators in Achieving the Diffraction Divergence of the Radiation Emitted from High-Gain Pulsed Gas Lasers, Sov. J. Quantum Electron. 4, 474 [Kvantovaya Elektron. 1, 8631. ZUCKER,H., 1970, Optical Resonators with Variable Reflectivity Mirrors, Bell Syst. Tech. J. 49, 2349.
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E. WOLF, PROGRESS IN OPTICS XXIV 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1987
V
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT BY
I. GLASER The Weizmann Institute of Science, Department of Electronics Rehovoth IL-76100, Israel
CONTENTS PAGE
$ 1. INTRODUCTION .
. . . . . . . . . . . . . . . . . . .
391
$2. PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING . . . . . . . . . . . . . . . . . . . . . 400 $ 3. BIPOLAR AND COMPLEX-VALUED SPATIAL SIGNALS
.
463
$ 4. APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . 477
. . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . $ 5. CONCLUDING REMARKS
.
502 503 503
0 1. Introduction In this chapter methods and applications are described of information processing with spatially incoherent light. Although historically some methods for incoherent optical processing preceded both electronic and coherent-optical information processing, the overwhelming success of electronics and the mathematical elegance of coherent optics have placed incoherent optical processing somewhat in the shade. Nevertheless, interest in incoherent methods persisted and, lately, increased; there are many applications where incoherent optical processing offers the desirable mixture of practical simplicity and sufficient power. Because of the diversity of methods and approaches to incoherent optical processing, it is impractical to cover them all in one chapter. This chapter therefore addresses only one group of approaches, namely those which use spatially incoherent light. Some rather significant works on other methods for incoherent optical processing were therefore excluded. Next, in this introduction, we have a brief overview of the rationale for spatially incoherent optical processing, after which a short survey of linear systems theory is presented. It is brought here primarily to define notations and conventions that will be used throughout this chapter. Methods for information processing with incoherent light can, like any applied field, be classified either by their physicalprinciples or by their designated applications. The first classification has the advantage of clearly showing the relations and the relative capabilities of different approaches. However, for applied work, one must relate to the needs and methodology of their designated applications. We describe the physical theory of different approaches in 8 2; § 3 deals with one problem common to all incoherent optical processing schemes: how to process bipolar (negative-positive) or complex-valued data with light intensities that must be non-negative real. Several approaches are described. Section 4 shows how all those approaches and algorithms relate to some real problems.
39 1
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§ 1
Why optical information processing?
Today information processing has become almost synonymous with digital electronics. Digital electronics is a dominant and mature technology. However, as its performance is being improved, its limits are better understood. For most computing applications, electronic information processing is more than adequate. There are, however, some important tasks that are difficult (in some cases impossible) to execute with electronic computers. Some problems require electronic systems with disproportionate bulk, complexity and cost. Examples include real-time two-dimensional signal processing for certain image analysis applications in medicine and defense, real-time complex radar and sonar signal analysis, and efficient, low-error-rate, video bandwidth compression. Other problems may be beyond the capabilities of electronic digital processing for some time. Seismic analysis, global meteorological forecasting, and threedimensional aerodynamics are typical examples. There are several examples where optical systems can “compute” much faster than electronic computers. Calculating the two-dimensional Fourier transform on an electronic computer requires (using the so-called fast Fourier algorithm), for loo2 points input, about lo8 multiplications and a similar number of additions. Coherent optical systems can perform that in about 5x seconds. This is equivalent to a rate of better than 10I6 operations per record. Now, since the temporal period of a light wave is 10- l 5 second, how can light process information in less than 10 - I 6 of a second? The answer is that it cannot. The very high throughput of optical processing does not result from ultrafast response but from an extreme degree of inherent parallelism. An optical Fourier transform system does a lo8 operations in parallel. Electronic systems that do few tens of operations in parallel are called, in contrast, “massively” parallel. At this point we must recall that, unlike optical processing, digital electronics is a mature, well-developed, and widely accepted technology. It offers much greater flexibility than optical systems. Nevertheless, there are many cases where an optical system, usually aided by some electronics, can solve a problem and offer capabilities beyond those of electronics alone.
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v 7
§ 11
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1.1. COHERENT VERSUS INCOHERENT LIGHT FOR INFORMATION
PROCESSING
Let us consider two interferometers: The Michelson inteferometer (Fig. 1A) has a beam splitter that splits an incoming light flux into two parts, then recombines them after each part has been reflected from a mirror. A detector measures the energy flux density (irradiance) of the combined flux. When one of the mirrors is shifted, the measured irradiance may fluctuate. The Young inteferometer (Fig. 1B) has a screen which blocks all incoming light except at two small openings. At a second plane we observe the spatial distribution of the irradiance of the light that was combined from the two openings. We note that each of these interferometers has two light paths with a time difference z between them. In the Michelson interferometer we shift the “2nd mirror” by A 5 to obtain a time difference of z = 2 6 5n/c (n is the refractive index of the medium and c is the velocity of light in vacuum). With Young’s interferometer (assuming x 4 z and X e z ) this time difference is given by z = Xxn/(zc);(A(, z , x and X are defined in Fig. 1). So far, as we considered perfectly coherent light, both interferometers behaved similarly. Coherent light propagating in the x direction can be written as E = E, exp [2zj(kx - v t ) ] . (j = k = l/A, where A is the wavelength, v E nc/k is the temporal frequency of light, E is the electric field, and E,, is the amplitude; we follow the scalar approximation and the complex amplitude
J-1,
1st mirror
T
:en
area
-z
L I l
defector .3
Fig. 1. Two interferometers: (A) Michelson interferometer; ( B ) Young interferometer.
394
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§1
’.
convention.) For visible, or near visible, light v E 10’’ s - No detector can respond to such frequencies - all we can measure is the time average of the irradiance la ( lE12) ; (( * . ) stands for time averaging and the proportionality factor depends on the unit system in use). When the two beams in the interferometer are combined, the detector sees Idet= factor x ( IE, + Eb12) = I,[ 1 + cos(2nvz)l (I, depends on the unit system, the losses in the interferometer, and the amplitude of the incoming waves; for simplicity we assume that both beams are equal). Now, let us introduce non-monochromatic light into the interferometers : Ip0,,,= JZ,dv. Each spectral component results from an electric field E, = E,,, exp [2 nj(kx - v t - 441. For each E , we get irradiance output of the = I,,,[ 1 + cos(2nvz)], where I , , a I, - all cross-frequency terms form ZOut,, cancel out in the time averaging process; the total output irradiance is I,,,dv = {f (I,)dv} + { f [Z,cos(2nvz)] dv}. The second term is clearly the real part of a Fourier relation. Thus, if I, # 0 only for (v, - Av/2) < v < (v, + Av/2), we shall get a non-neghgible second term (“fringes”) only when l/Av;,,z is called the coherence time of the ( -),z, < z < ( + ,),z, where ,,z light, and lc0 = c/(nz,,) is its coherence length. Let us now put, in Young’s interferometer (Fig. 1B) just left to the screen, some extended source of light that has sufficient coherence time to give us visible fringes,,,z > Xxn/(zc). However, with this source each atom or molecule of this extended source emits light independently of any other atom or molecule, so that the phase of the light that travels through one of the screen openings is statistically independent of the phase of the light that went through the other opening. We can model such light as EL = E; exp{2nj[kx - v t + & ( t ) ] } (n = 1,2), where $,(t) and^ &(t) are two random functions (for each opening), statistically independent of each other. Because of this statistical independence, the interference term would average out giving + < l e x p { 2 ~ j [ - v ( t +z)+ +22(t+ d1}t2>; no fringes would be seen. As we move the openings closer to each other (making X smaller), light from the two openings may cease to be statistically independent and fringes would start to appear on the area detector. The maximal value of X for which we can observe fringes is termed the transverse coherence size of the light, measured at the plane of the screen. For quantitative analysis of temporal and spatial coherence the reader is referred to the published works on coherence theory, such as BORNand WOLF [ 19751. An optical system is temporally incoherent if the coherence length of the light
V , § 11
INTRODUCTION
395
it uses is substantially smaller than typical path differences in the system. It is spatially incoherent if the transverse coherence size of the illumination at the input (object) plane is such that, even if the system was dsraction limited, it would be unresolved. (Note that this definition relates the transverse coherence size to the angular size of the system aperture -not its actual resolving power.) An example of light that is coherent temporally but is spatially incoherent is light from a low pressure gas discharge spectral lamp which is spectrally filtered. An example of spatially coherent, temporally incoherent light is light coming from a distant star. 1.1.1. Advantages and disadvantages of incoherent optical processing Incoherence adds redundancy to optical systems. Temporally incoherent light adds redundancy because many spatial noise generating effects are wavelength dependent but, for a system with no chromatic aberrations, the optical signal itself is not. For example, a dirt speck on an ordinary lens would cause a Fresnel diffraction pattern in monochromatic light. This pattern would become much less pronounced in polychromatic light. When the same optical system is illuminated with spatially incoherent light, many Fresnel patterns would be created, each shifted in space. These patterns would practically average out each other, leaving no visible indication of the presence of dust other than (usually negligible) decrease in the overall contrast. Quantitative analysis on the effect of coherence on spatial noise was carried out by LOWENTHAL and CHAVEL[ 19771. Spatial incoherence contributes two-dimensional redundancy, whereas temporal incoherence contributes one-dimensional redundancy. For this and other reasons spatial incoherence is usually more effective in decreasing “noise” than temporal incoherence. The noise suppression capability of incoherent light does not come free, however. Coherent light has deterministic phase, so it can be used to represent complex-valued signals directly. To measure the phase of coherent light, we can use some interferometric scheme. With incoherent light the optical phase is immeasurable. We must get by with irradiance which is non-negative real. With coherent light rather simple optical systems give us direct physical access to the two-dimensional Fourier spectrum of a spatial signal. With temporally incoherent light this can be done only with special compensation optics, and with spatially incoherent light indirect methods must be used. Incoherent systems are simpler to interface (both to the outside world and to electronics), less sensitiveto dirt and other environmental problems, and can
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INFORMATION PROCESSING WITH SPATIALLY 1NCOHERENT LIGHT
[V, $ 1
be built from widely available optical and electro-optical components. Coherent systems, which do not rely on indirect methods for complex-value representation, can offer higher ultimate performance (usable space-bandwidthproduct) than incoherent ones. Practical coherent optical systems usually require the use of some transducer (spatial light modulator) to convert the incoming incoherent light, or electronic signal, to coherent light. At present the performance of two-dimensional spatial light modulators leaves much to be desired. Since no analogue problem exists with incoherent processors, they are much simpler to use for red-life applications. We also note that polychromatic spatially coherent light shares some of the properties of both coherent and spatially incoherent light. Such polychromatic systems often use an optical Fourier transform of some sort, but when they do so, they need spatial light modulators (for real-time operation) just like their coherent counterparts. They are better than coherent systems in noise suppression, but not quite as good as spatially incoherent systems.
1.2. LINEAR SYSTEMS
In this section we review some basic concepts, terminology, and symbols that are widely used to describe signal processing systems. We follow the conventions set by GOODMAN [ 1968]. We shall use the term system to describe a “black box” that takes some input function, gi,(x, y ) and processes it to deliver an output function go,,((, q). This may be written as
Since this system transforms the input function, often the word transformation or transform is used as its synonym. If a system satisfies
it is called a linear system. It can be shown (for example, MOORE[ 19681) that for transformations of continuous functions one can find a kernel function,
v > s11
391
INTRODUCTION
K(x,y; 4, q), such that
j j
10
y{gin(x, y>>=
-10
gin(x,y ) ~ ( xy ,; 4, q) dx dy
9
(3)
-03
whereas for finite, discrete, input gk,Ithe kernel takes the form Kk,&m,n and the transformation becomes .4p{gk,I> =
N-1
N-1
k=O
1=0
c
gk,IKk,l; m,n
(4)
‘
This is a two-dimensional extension of vector-matrix multiplication: Y { g } = [ K l g . It is also important to note that it is always possible to rearrange the elements of any n-dimensional finite discrete space into some other dimensionality as long as we do not change the total number of elements. We can then find an equivalent [ K ] in this new space. PRATT[ 19751 shows, for example, how a two-dimensional discrete function gk,I, where k, I = 0, . . . ,N - 1, can be unfolded into ,gi for which i = 0, .. . N 2 - 1. We use i = Nk + I and the matrix , [ K ] is extracted from the kernel Kk,&m , n , using , [ K ] = ( l K i ; p )where i = Nk + I and p = Nm + n. 1.2.1. Shift invariant linear systems
A special class of linear transformations is the class of shifrinvariant or isoplanatic transformations. A shift invariant system is one where for any gin(x, Y ) andfor any
then y{gin(x - ~
0Y ,- Y O ) >
(
~
0
Y9 O )
v
= gout(< - ~ 0 , - Y O )
*
(5)
The kernel has only two independent variables K(x, Y ; 5, ~ 1 ) = h(x - 4, Y - V ) .
(6)
The two-dimensional function h defines the output of the system. From an impulse gin(x,Y ) = HX - ~ 0Y , - Y O ) we get gout(& V ) = h(4 - ~ 0 q, - Y O ) . [Here 6(x, y ) is Dirac’s delta “function”.] The shape of the output does not depend on the location of the input. We shall use the notation gin
* h = gout(x9 Y ) =
J---
J m
gin(5,q)h(x-5,Y-ll)d5dq,
(7)
--m
and call goutthe result of the convolution of ginwith h. We note that convolution
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§1
is commutative: g * h = h * g. The degenerated kernel h is named the impulse response or the point spread function (PSF) of the system. Another way to look at shift invariant systems is through the concept of geometrical correlation (or template matching). Correlation is defined as the amount of overlap between two functions with some given relative shift g@h
= J--m-m
g(5, q)h*(5 - x, q --Y)dtd?,
Jm -m
(8)
where h* is the complex conjugate of h. Convolution and correlation are related: g(x, y) @ h(x, y ) = g(x, y) * h *( - x, - y). Correlation is generally not cumulative. Also, for (x, y ) = (0,O) we have [g 0 g] (0,O)= J J 1 g(t, q) 1 d 5 d q, which may be interpreted as the “total energy” contained in the spatial signalg. 1.2.2. The Fourier transform The Fourier transform is of special significance to signal processing in general and to optical systems in particular. Unfortunately, different textbooks use slightly different definitions of the Fourier transform. This chapter will use GOODMAN’S [ 19681 definition G(f.9 4 )= F{g(x, Y ) ) =
J
a2
g(x, y ) exp [ - 2 nj(fxx + s,~)Idx dy ,
(94
--m
from which it can be shown that the inverse Fourier transform is g(x, Y ) = 8-‘ { g K =
JP, J
9
A>>
m
~ ( . t . . , ~ , ) e x p ~ n j ( . +s,~)I t . X d ~ d ~ , . (9b) --oo
Using this convention, the spatial frequencies ( f x ,S,) are the inverse of the spatial period: l/period = (L2 + S,2)1/2.We use lower case letters for functions defined in the “real” ( x , y ) space coordinates and upper case letters for functions in “spatial frequency” (f,,S,) coordinates. One very useful property of the Fourier transform is the convolution/correlation theorem F i g * h} = G H
and F{g@c} = GC*.
(10a)
V , 8 11
INTRODUCTION
399
1.2.3. Radon space We define as the projection of a 2-D function g ( x , y ) on the x axis the integral
1g ( x , y ) dy. We also define a rotated coordinate system (x,, ye), where 8 is the rotation angle and both coordinate systems share the same origin
So the same g ( x , y ) gets another mathematical expression when written as a functional of the rotated coordinate system g,(x,, ye). The projection of our function on the new x , axis is
1
co
pge(xe) =
(12)
g(x,,y,)dy,,
-cc
This .g,(x,) is actually a function of two variables, 8 and x,. We shall sometimes use the notation .g,(x), where x stands for x , . Thus we have defined a transformation that takes some two-dimensional function and extracts a set of projections from it: a { g ( X ,Y>>= .g,(x) .
(13)
This transformation is known as the Radon transformation; for a detailed description see, for example, BARRETT[ 19841. 1.2.4. Sampling Lastly, we note that "practical" continuous functions, such as g ( x , y), can usually be expressed also as discrete sequences gk,[,where k = 0, . . .,N, - 1 and I = 0, . . . ,N, - 1 (N, and N, may be infinite). The conversion process, or sampling, is given by gk.1 = d
X O
+
YO
+ lA) .
(14)
The sampling theorem states that, for a band limited function g ( x , y), satisfying for all f, and f,such that %(x,
Y>>= 0
and if A<-
1 2fmax
9
,/m
>f,,, (154
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V, I 2
in eq. (14), complete precise reconstruction of g(x, y ) from gk,l is possible. 1/(2fmax)is called the Nyquist sampling rate. If, also, g(x,y) = 0 for all (x’ + y2)l/’ > x,,,, it is known as aspace-limitedor bounded spatial signal and its discrete representation is also fmite, since no more than (~X,,,/A)~ terms of gk,] are non-zero.
8 2. Physical Approaches to Noncoherent Optical Processing One thing that is particularly noticeable about incoherent optical processing is its plurality of approaches and methods. These range from a “simple” overlap of two transparencies to complex hybrid opto-electronic systems. This situation reflects the versatility of the laws of optics; it also partly reflects the fact that incoherent optical processing is an immature field: It is dynamic and evolving but still not calcified into a fixed, well-defined form.
2.1. SCANNING
In Fig. 2 we see some common image scanners. These devices are widely used to convert two-dimensional spatial distributions of irradiance, transmission, or reflectance into one-dimensional electronic signals, or alternatively,
Fig. 2. Some image scanners: (A) drum; (B) “laser”; (C) vidicon; and (D) charge coupled devicd (CCD).
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
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“write” an electronic signal into some image receptor (photographic film, xerographicdrum, or phosphor, for example). They are essential to applications such as image transmission, electronic image storage, digital graphics, and electronic image processing. All the above image scanners follow a similar principle: a spot of light (or an electron beam for the vidicon of Fig. 2C) moves relative to a twodimensional surface. Its motion is line-by-line: it is continuous in one dimension and discrete in the other. (In the charge coupled device (CCD) of Fig. 2D the scanning is done electronically after converting light into electric charge; it is discrete in both dimensions.) If the spot of light (or electron beam for the vidicon) is a mathematical point, the reflected or transmitted flux, as a function of time gScan(t)can be written as gscan(l) = g ( x , Y )
(16)
9
where x = v ( t mod T ) + x m i n ,
y
=
[-
mteger ( 3 l A Y 9
where v is the scan velocity in the x (continuous) direction, xminand xmaXare the scan limits, Tois the “retrace” (dead) time between scan lines, Ay is the scan step in the discrete y dimension, and (tmod T) and integer(t/T) are the fractional and integer parts of t/T; t = integer(t/T) + ( t mod T ) and 0 6 (tmod T ) 1.
-=
2.1.1. Correlations and convolutions 2.1.1.1. Reading Assuming that Ay satisfiesthe Nyquist condition of eqs. ( 15),gscm(t)contains all the information necessary to reconstruct g(x,y). It is a precise temporal representation of our spatial function. Now, let us assume we scan with a spot of light that has a spatial irradiance distribution c(x, y) of some finite extent. Instead of gscan(t)of eq. (16), we get gsc,,,
.O)
=
fS
g(5, 49 4
5 - x , v - Y) d5 d v
x and y are given as functions o f t in eq. (16).
Y
(17)
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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Since c(x, y ) is not complex valued, we can use eqs. (7) and (8) from 3 1.3 and rewrite eq. (17) as
Thus, if we reconstruct a spatial signal gc(x,y ) from g,,,,, c(t), we get
2.1.1.2. Writing In the foregoing equation we assumed a perfect reconstruction. From the sampling theorem the ideal spot shape for reconstructing g(x, y) from gscan(t) has the form rided(x,y ) = S(x) sinc (y/Ay),where sinc (a) = sin (.a)/( na). This reconstruction spot is not physically realizable: sinc(y) has areas where it gets negative, and no optical system can create a spot as narrow as 6(x). In practice, reconstruction spots of the shape r(x, y ) z exp [ - n(x/Ax)’] exp x x [ - n(y/Ay)’] are widely used. If, instead of the above r ( x , y ) , we use a spot of shape rh(x,y ) = r(x, y ) * h(x, y), we shall reconstruct gh(x,y ) 2 g(x, y ) * h(x, y). In practice the spatial frequency bandwidth of h(x, y), f,, max, , and A,max, are lower than those of r(x, y); we use rh(x,y ) h(x, y). 2.1.1.3. Implementations A modified drum scanner was used by SWINDELL [ 19701; both “read” and “write” heads were placed on two areas of the same drum. The write head projected the desired c(x, y ) pattern using slide-projector-like optics. The reflected light was detected and amplified electronically, and modulated the intensity of a light source in the write head. Both pattern recognition and image restoration were demonstrated with the Swindell system. Mechanical scanning correlators/convolvers are conceptually simple and effective, but have two drawbacks: They are relatively slow and are dficult to apply directly to a scene - one must prepare a transparency or a photograph first. One alternative is to use an electronic scanner. GELUK[1979] built a scanning convolver that is based on an imaging electronic image intensifier (Fig. 3). He placed a deflection yoke around the intensifier tube so he could shift the intensified image. By applying TV raster signals to the yoke, he got TV scan-like shift of the image at the output end of the intensifier. This was imaged onto a transparency, creating a fast scanner. Since convolution is commutative, Geluk could put the transfer function c(x, y ) on the transparency and the input
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Shifted light Intensity distribution flw-x: y-<)
Light intensity distribution f (x,y) on phothocathode
I
output
L47e
I
Lh\ I
ll
Ph o tom ult i DIier
Tr a n srn i t t a nc e distribution g ( x . Y )
V
Image intensifier in deflection yoke
Fig. 3. Geluk's electro-optical scanning convolver. After GELUK[ 19791.
function g(x, y) into the input end of the scanning intensifying tube. When the detected signal was fed into a standard TV monitor, the Geluk convolver could be used just like a video camera, giving real-time convolution of the ongoing external scene. Another work on a rapid scanning correlator by INDEBETOUW and POON [ 19841used mirrors to scan a projected c ( x , y) pattern on the area of an object. Although laser light was used, the system is mathematically incoherent because, being sequential, light from two scan positions cannot interfere. 2.1.2. Time integrating scanners There are many cases when we want to convolve, or correlate, a temporal (1-D) signal with some stored signal. One way of doing that is shown in Fig. 4. A temporal signal is used to modulate some light source, which illuminates a moving transparency. A detector at a static point behind the moving transparency measures the light flux that gets through. If the transparency has
404
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
r v 9
I2
moving tvunspuvency
Fig. 4. Basic time integrating scanner.
a (irradiance) transmittance distribution c(x), the modulated light source irradiance impinging on the transparency is g(t), the location of the detector is x = 5, the transparency is moving at constant velocity u, and the integrator sums the detector reading over the complete duration of the signal, we get the output
where r = -5 u
9
Sr(7)
=
~(5) 9
cr(t> = ~ ( 0 2 ) .
This is precisely the one-dimensional form of the correlation integral of eq. (8), and, hence, s are all real.) To carry out this concept, we need a modulated light source, a moving transparency, and an array of detectors/integrator. Depending on the temporal frequency bandwidth requirement, sources varying from incandescent lamps (typical response frequency of few Hz to few hundreds of Hz)to LEDs and lasers can be used. Alternatively, we may prefer using mechanical, acoustooptical or electro-optical modulators with a constant light source. We note that even though some of the sources emit coherent radiation, time integration is phase amnesiac, hence incoherent. There is similar variety in methods for obtaining “moving transparencies”. When we do not need a very high velocity u and we do not want to change c(x) too often, a physical moving transparency s = g @ c. (g, c
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
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would be just h e . For example, SKENDEROFF and SYMANIEC[1969] described a system that used a rotating disk or drum as the moving transparency. Acoustic light modulators offer higher velocities and “programmability”; KELLMAN[ 19801 describes schemes to use such modulators for time integrating scanning correlator. Finally, one can keep the transparency static and move the detector instead. This is possible with a modified CCD device, as exemplified by the work of MONAHAN,BOCKER,BROMLEYand LOUIE [I9751 and BROMLEY,MONAHAN,BOCKER,LOUIEand MARTIN [ 19771. Although this method is one-dimensional, PRAIIT[ 19751 showed that convolution/correlation of two-dimensional signals can be carried out in one dimension by proper conversion. Thus it may be possible to conceive of a system where a video signal is used to modulate the light source in a time integrating scanner, and where the output signal is connected to a video monitor. We must note, however, that a typical low resolution video signal contains about 250 000 picture elements (“pixels” or “pels”). Indeed, to account for the flyback (time intervals between successive line scans) in the video frame, perhaps 300000 total pixels must be handled by the correlator. This poses severe practical difficulties. The Psaltis two-dimensionalscanning correlator An interesting solution to the 2-D problem was offered by PSALTIS[ 19841. In his system (Fig. 5) multiple light sources (LEDs) are used. A system of cylindrical and spherical lenses images this vertical array, through an acoustooptical modulator (which acts as the moving transparency), onto a twodimensional array of detectors. Equation (19) now becomes
where m = I, ..., M ;
n
=
I, ...,N ;
5 are defined with eq. (19). The detector array in this system is a two-dimensional CCD device with N columns of M elements each. The (n, m)-th. detection element is located at coordinates (t,, qm). The LED array spacing and the vertical magnification of the lens system are set so the spacing between the images of the LEDs equals the spacing between the detection elements. Horizontally, each LED image is magnified to fill an entire CCD row. After each video line passes through the z and
406
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
+ELECTRON1 C MEMJRY
ACOUSTO-OPT1 C DEVl CE
REFERENCE IRAAGE TOP VIEW
Fig. 5 . A modified time integrating scanner for two-dimensional correlations. From PSALTIS [1984].
acousto-optical modulator, the CCD is clocked, so the stored (integrated) charge at each cell (t,, qm) is shifted down to a new location (l,, qm- ,); the bottom CCD detector now is shifted into a readout row, which is sent out once for each video frame. These shifts are equivalent to vertical motion of the detector array. Since the signal on each LED is independent of that of other LEDs, we get an effective vertical time integrating scanning. To calculate the signal that comes out of the CCD, we use
,
gn,m =
{
g(t, urn), if 1 < n < N otherwise, 0,
and 1 G m G M ,
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
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In the above equation tpixe,is the time spent on a resolution element in the video signal and tline is the line scan time, including the flyback time. Thus
where tn,m =
ntpixel + mtline .
This is a two-dimensional, semi-continuous correlation relation. The Psaltis correlator has electronic interface for both input functions; it is not necessary (like with many other optical correlators) to change any physical component to get a new transfer function, we only have to load these values into the digital memory that controls the LED array. This simplifies the interface with electronic systems and facilitates the use of transfer function libraries. The performance of this system is limited by the availability of suitable electrooptical devices such as LED arrays, video components, and acousto-optical modulators. With present state-of-the-art components PSALTIS[ 19841 estimates the space-bandwidth product of the input image to less than 10002and that of the transfer function to about loo2.
2.1.3. Radon space processors The Radon space (BARREIT[ 19841) is an n-dimensional space made of the complete set of the (n - 1)-dimensional projections from an n-dimensional Cartesian space. Specifically, if we start with two-dimensional space, we get a set of one-dimensional projections as defined in eqs. (12) and (13) in § 1.3. The coordinates in the 2-D Cartesian space are (x, y ) and those of the Radon space are (0, x). There are several applications where it is desirable to obtain the Radon transformation of a given 2-D function. Even more often we need to get the inverse Radon transformation. Following eq. (13) we shall use the notation 9-l { p g s ( x ) } = g ( x , y). Most of the approaches to computing the inverse Radon transform are based on the central slice theorem. If we take the Fourier transform of g ( x , y) in any rotated Cartesian coordinates ( x s ,ye), we get
where
408
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,B 2
(fe,x,f,,)are rotated in Fourier (spatial frequency) space, and &{* * * } is the one-dimensional Fourier transform operator. Stated verbally, eq. (22) says that if we “slice” the Fourier representation of a two-dimensional function, along any line that goes through the origin of Fourier space [ ( f x , f , )= (0, O)], we obtain the one-dimensional Fourier transform of the projection of the original function in a direction perpendicular to the “slice”. We shall now investigate some methods for obtaining Radon and inverse Radon transformations optically.
2.1.3.1. Optical Radon transforms Possibly the most widely used method of getting a Radon transform of a two-dimensional object (typically a two-dimensional mathematical slice of the three-dimensional human body) is tomography. In its simplest form we use a parallel, sheetlike, uniform beam of radiation (typically X-ray) and measure the transmitted irradiance through the object. Ignoring second order effects resulting from scattering, non-monochromaticity of the (X-ray) radiation, and other factors, the transmitted irradiance we measure is zcmt(x) = ‘in exp[ -
J
or ln(zin) - ln[‘out(x)l
=
Y(x, y ) dy]
1
(23)
Y ( X , Y ) dy = p ~ O ( x 0 ) 9
where y ( x , y ) is the local transmission coefficient and ,y0(xe) is its Radon transform. In transaxial tomography we obtain a set of such projections for a slice of an inspected object (typically a medical patient) and try to reconstruct y(x,y) from these - carry out the inverse Radon transform. Some optical methods for doing that are discussed in the next subsection. Here we shall discuss more methods for obtaining the direct (not inverse) transform. The Radon transformation can be regarded as a set of convolutions .go(xo) = g(x9 Y ) * [6(x)1
(24)
and similarly for 8 # 0. BARRETT[ 19821 demonstrated a system where a line of light, implementing the convolution kernel 6(xe), scans a reflective or transmitting object in the ye direction while 8 changes with time. The reflected (or transmitted) light flux is proportional to pge(xe). Alternatively, in a system presented by GINDIand GMITRO [ 19841 an image rotating prism is used to rotate the input image, after which an optical system containing cylindrical optics implements the 6 ( x ) point spread function by inserting a large amount of astigmatism into the image. The rotation does the 0 scan.
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
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2.1.3.2. Optical inverse Radon transfoms The problem of inverse Radon transforms is particularly interesting because of its direct relation to tomographic reconstmction. There are several known algorithms for reconstruction from projections (see, for example, HERMAN [ 19791). The most suitable algorithm for incoherent optical implementation so far is the back projection algorithm. Following BARRETT[ 19821, we use the central-slice theorem of eq. (22) to arrive at the back projection algorithm; we use the convention of “semi-polar” coordinate system in Fourier space: the frequency “radius”, p = (f: + h2)lI2, will be allowed to take both negative and positive values, but the angle 0 will be limited to values from 0 to w. If we have the set of projections of a given object, ,go(x,), we know from the central-slice theorem that the 2-D Fourier transform of the original object along the double radius peis 4 {pg,(x,)} = G,(p,). We can now integrate over 0 to obtain the 2-D Fourier transform G(f,, f,).We recall that the area of a frequency-space element in polar (or semi-polar) coordinates is df, df, = i p / dpd0, so g(x,Y) = =
fl-WLf,N
j_Y,j
G ( L , A>exp [2nj(xL + Ys,)I d~ d~
O3
- w
= =
SPI
ipl .G,(p) exp[2wj(xpcose
Ion j do
O3
+ ypsine)] d e d p
I ~ , ~I , ( p ) e x p ~ w j wdp l
--OD
’
Unfortunately, 4- {I p I } is a functional rather than a well-behaved function. In practice we can take advantage of the fact that pgs(xs) is band limited, so if we take out of 4- {I p i } the high spatial frequencies (where the Fourier transform of pg, is zero anyhow), the result of the convolution would not change. Thus we may use a band-limited version of $-’{ IpI}. For example,
where f,,,
is the spatial frequency bandwidth ofpg,(x,). We note that this form
410
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
tv, I 2
0 Fig. 6. A back projection convolution function. Only one side (the function is symmetric) is shown. From BARRETI. and SWINDELL[1977].
is an approximation and, hence, is not unique. Figure 6 shows the shape of bf,=(xe) of eq. (26). The reconstruction “recipe”, namely, the Jiltered back projection algorithm, is as follows: (1) Convolve (filter) all the projections, .g,(x,), for 0 = 0, . .., n, with bf,&,). (2) “Back-project” the filtered .g;) = * bfma projections. To do this we (a) smear each .g; along the y, axis or, mathematically, convolving each of these with 6(x,); (b) for 8 = 0, . . . , n we add all the “smeared” .g; * 6(x,) to the (x,y) plane. The TAT processor
GELUK [ 19791 has built an opto-electronic scanning system, named by him transverse analog tomography (TA T ) that uses the back projection algorithm directly. The TAT processor, depicted in Fig. 7, uses a CRT to display, as a single line, the .ge(Xs) projection for one value of 0 at a time. This image is picked (through fibre optics faceplate) by a modified image intensifier tube and moved by deflection coils on that tube across its output screen; curved mirrors are used to smear the image in they, direction while being scanned (convolved) by a mask in the x , direction. The light that comes through the mask is picked by a photomultiplier, giving .gL(X,) as a function of time. Actually, since bfmax
v, § 21
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
41 1
DETECTION
ION
b
CONVOLUTION A N 0 TIME BASE EXPANSION DEFLECTION
C
TARGET
SCAN CONVERTER BACKPROJECTION
Fig. 7. The transverse analog tomography (TAT) system. (a) getting tomographic projections; (b) convolution of the projections with the b,mm function; (c) back projection of the filtered projection on a storage CRT. From GELUK[1979].
is bipolar, two processing channels are used. (The use of two processing channels is discussed in 0 3.1.) Finally, a storage tube CRT collects all the filtered back projection and displays the resulting image.
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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The UCT processors BARRETTand SWINDELL [ 19771 have demonstrated several variations of a differentoptical method for implementing the filtered back projection method. They termed their approach uncomputerized tomography (UCT). In the UCT approach the complete set of projections,g, is recorded on X-ray photographic film.The format they used allocates parallel lines for the Ovalue. For the object g(x, y ) = 6(x - xo, y - y o ) we get on this film a sine-shaped curve, with an amplitude of ( x i + yZ)ll2 and phase tan- (yo/xo).For a complicated object the film image is a superposition of many such sine curves (luckily, X-ray film response is close to the logarithmic response desired for tomography). Barrett and Swindell called this record a sinogrum. In the UCT approach we essentially correlate this sinogram, filtered with the bfmm function, against sine curves. This was shown to be mathematically equivalent to the back projection algorithm. Two methods of correlation were demonstrated. In one, shown in Fig. 8, the sinograms are wrapped on a drum. The circumference of the drum equals precisely the period of the sine curves, so the sine curves are converted by this wrapping process into ellipses. For each point (xo,y o ) there is one rotation
I
Fig. 8. The uncomputerizedtomography (UCT) drum processor. From BARRETTand SWINDELL (1977).
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
413
angle (a, in Fig. 8) and one tilt angle (+in the figure) that make the ellipse look like a line. They used dual scanning, slowly on the tilt t,b and rapidly on the rotation a,.The cylinder, or drum, is transilluminated from inside. Like with the Geluk TAT processor, two channels are necessary. The light behind each mask is detected by a photomultiplier, and the difference between the two signals goes to a spirally scanned display. A different geometrical configuration that provides similar results, also shown by BARRETTand SWINDELL [ 19771 is depicted in Fig. 9. Here a sine curve “slit” was recorded on a long film. The amplitude of the sine curves varies slowly along the film. The sinogram is imaged onto this film, and the transmitted light is detected by a photomultiplier. As the film moves, the effective phase [representing tan - (xo/yo)] changes rapidly, whereas the amplitude [representing(x: + $)‘/’I changes slowly. The final display is done, as before, on spirally scanned CRT. Again, two subtracted processing channels are necessary, as shown in Fig. 9. In some of these systems the convolution was
’
P
SINOGRAM
PROCESS I NG
(c)
Fig. 9. The uncomputerized tomography (UCT) film processor. From BARRETTand SWINDELL [1977].
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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done by the slit itself(as shown in the figure), whereas other systems used OTF synthesis (see 0 2.5). 2.1.4. Sequential processorsfor general linear transfomations
The general form of a continuous linear transformation in N dimensions is
or for finite, discrete “vectors”
where “1
=
0,..., M ,
m,
=
0, ..., M2 - 1
- 1
... mN = 0, ..., MN - 1 .
For two dimensions these equations get the form of eqs. (3) and (4),of 0 1.3, respectively. For the 1-D, discrete case this is a vector-matrix multiplication formula where
[.
go
g=
...
gL-
and I-
, 1
G=
...
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
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moving riiirror
PD
detector. array
'\\
'\
\
mash
Fig. 10. A multichannel time integrating correlator configured for general linear transformations. This system is similar to the time integrating correlator of Fig. 4; however, several parallel transparencies are used, but there is only one detector-integrator per channel, giving only the 5 = 0 term of the output signal. The image of the transparency is moved (using an oscillating mirror) rather than the transparency itself.
Figure 10 shows one approach to computing this matrix-vector product optically. In this approach BOCKER[ 19741modified the BROMLEY [ 19741time integrating correlator; he noted that each element G , of the product z g , K I : , , is actually the correlation of g, with K,;, for the given m at = 0. Thus the Bocker system is similar to the time integrating correlator of Fig. 4 but has only one detector-integrator per channel at t = 0. To get G , for all m,he replaced the single transparency of Fig. 4 with M parallel transparencies and used M detectors-integrators (terms of Fig. 4). Bocker, following Bromley, used an imaging lens and an oscillating mirror so he could scan much faster than by moving the transparency. The mirror was mounted on a galvanometric drive that produced a saw-tooth motion, synchronized with the input signal.
c
2.1.4.1. The CCD processor One drawback of the system of Fig. 10 is its limited speed because of mechanical motion. To obtain faster processing, the mechanical moment of inertia of the mirror must be made smaller. This moment of inertia is related to at least the third power of the size of the mirror. However, a smaller mirror would degrade both the light efficiency and the diffraction-related, space-bandwidth product of the imaging lens. In later work MONAHAN,BOCKER,BROMLEYand LOUIE [1975] and
416
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
Y
4
k=O
1
2
I
(N- 1)
n
‘I OUTPUT
Fig. 1 1 . Charge movement in a CCD area device (TDI mode). From BROMLEYand LOUIE[1975].
MONAHAN,
BOCKER,
BROMLEY, MONAHAN, BOCKER,LOUIEand MARTIN[ 19771 showed a system using a modified CCD to do “electronic scanning”; the movement of photoinduced charges inside an area sensing CCD is used. Figure 11 shows how the charges inside the CCD are moved. There are two major cycles in this device: frame cycle and pixel (cell) cycle. At the beginning of each frame cycle the charges in all cells are effectively zero. Then, in each pixel cycle the charges move one cell to the right. During each pixel cycle photo-induced charge accumulates at each cell. At the Zth cycle (from the beginning of the current frame cycle) the kth cell contains charge equal to the sum of that induced on it at the current pixel cycle, that induced on the k - 1 cell at the I - 1 cycle, and so on. At the last pixel cycle of the frame cycle the last cell ( N - 1 in Fig. 11) of the mth row contains a charge that is equal to the temporal sum of the photo-induced charges on the mth row. This charge is transferred to the vertical readout column at the right and sent out (note that this mode of operating a CCD device, known as the time delay and integration or TDI mode, is different from the normal, imaging mode). To take advantage of this signal propagation pattern, Monahan, Bromley and their colleagues overlayed a mask (or transparency) on the CCD itself, as shown in Fig. 12. This device offers fast response (there are no moving mechanical parts) and compact size. Later versions of the CCD processor were
v 9
§ 21
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
417
Y
Fig. 12. A CCD vector-matrix multiplier: (a) a modulated LED; (b) some optics to spread the light from the LED evenly; (c) a mask containing (as horizontal lines) a set of M transparencies (see Fig. 1 1 and the text); (d) a CCD. From MONAHAN,BOCKER,BROMLEYand LOUIE [ 19741.
sufficiently small to be mounted as a component on an electronic printed circuit. We note, however, that the precision of the CCD processor depends heavily on the pixel-to-pixel uniformity of the custom-made CCD. The uniformity that is necessary is appreciably higher than for standard, imaging CCDs. 2.1.4.2. Double scanning processors Previously, in Fig. 4,we saw a simple time integrating processor where one signal enters the system as a temporal modulation of a light source and another signal is entered as variations of the spatial transmittance of a moving transparency; this device can be used for correlations and for vector-matrix multiplication. Another configuration, shown in Fig. 13, uses moving transparencies for both input signals. For simplicity we shall assume that the transparencies contain discrete data elements at spatial intervals A, that both transparencies move at the same speed u, and that the period of the detector-integrator array (in both spatial directions) is A. Each transparency has N “tracks” of data; the width of each track is A. We count tracks on the vertically moving transparency from left to right and on the horizontally moving transparency from top to bottom. Intervals on the vertical track are counted
418
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
rv, § 2
Fig. 13. Simplified double scanning processor. Uniform, not modulated, light passes through both transparencies, then is detected and integrated by an array of detectors-integrators.
from bottom up and on the horizontal one from left to right. Each detector-integrator carries two indices, beginning with (0,O) for the top left one, then ( N - 1,O) for the top right one, and finally (N - 1, N - 1) for the bottom right detector-integrator. Thus at t = 0 detector (0,O)sees flux t,(O, 0) th(O,O); t,(k, I) is the transmittance of the Zth cell in the kth track on the vertical transparency, etc.; for simplicity we assume that the input flux density is 1 unit per detector per unit time of A/u. At time t in A/u units the detector (n,rn) sees flux 4jr(n, rn) 4jr(n,m) =
t,(n, t - rn) th(t + n, rn) ,
so that the accumulated signal in that detector-integrator is
S(n, rn)
=
c f
4jt(n, m) =
c t,(n,
t
- rn) th(t + n, rn) .
(30)
I
Equation (30) has the same form as a set of scalar products (termed also dot product or inner product) of two vectors g 0 h = C g , h , = gTh, where superscript T stands for transposed. Now, matrix-matrix multiplicationcan be expressed as (for N x N matrices) a set of N 2 dot products. If AT* is a (horizontal) vector composed of the Zth row of the matrix [A] and the vector B*,k is the kth column of the matrix [B], then the (1,k)th element of [c]= [ A ] [Bl, C,,k, is A,,*#B*,,. We now have a device that does N 2 dot products in parallel, during one integration period. Can we devise transmission functions, th and t, so we get
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
419
the right dot products, as needed for matrix-matrix multiplication? The answer is yes; moreover, there is more than one such arrangement. One solution, namely the systolic “architecture”, is due to KUNG[ 19801. As an example, we bring it here for [ A ] b, where [ A ] is a 3 x 3 matrix and b is a 3 element vector (following our convention from Fig. 13):
[il,lf2” i2,2I’; i3,3) ’3.1
t,
=
’L2
‘1.3
,
’2.3
t, =
1:) bl
.
(31)
We note that th is, essentially, [ A ] rotated by 45 degrees; t,, has (2M - 1)2 cells for an M x M matrix, and t, has 2M - 1 cells for an M element vector. A version of the algorithm for matrix-matrix multiplication was also developed. An alternative is the SPEISER and WHITEHOUSE[ 19811 engagement algorithm. Here we use
i ! : , ) [:::: B1,3
’l,3
’1,2 ’2,3
‘1.1 ‘2,2
‘2,l
’3,3
’3.2
’3,l
)
’2.1
9
tu
=
‘3.2
(32)
to get the matrix-matrix product [ A ] [ B ] .Now we need (2M - 1) x M cells for M x M matrices. The computed result, in both algorithms, is available after 2M - 1 time units. It is obvious that, if we need to prepare transparencies for both inputs in advance, the double scanning processor would be quite useless. We must use [ 19841demonstrated processors using “real-time transparencies”. GUILFOYLE either the systolic or the engagement algorithms with acousto-optical transducers as the “transparencies”. Acousto-optic transducers are quite fast, but they can be used only with one-dimensional data, so Guilfoyle needed 2 N of these in his system. BOCKER,CAULFIELDand BROMLEY[1983] built systems that used two liquid crystal spatial light modulators instead. These are slower but two dimensional. Both systems are envisioned as attached processors of some digital electronic system. Some further information on spatial light modulators can be found in the review by TANGUAY and WARDE[ 19831. Both GUILFOYLE [1984] and BOCKER[1984] used schemes that rely on
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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correlation to provide digital multiplication ($4.5.1). Their systems are thus optical digital matrix-matrix multipliers. The above mentioned systems use lenses to re-image the transducers or modulators onto each other and then to the detector-integrator array. SOFFER, OWECHKO, MAROMand GRINBERC [ 19861 built a compact processor where electro-optical modulators, which are addressed electrically, are physically piled one on top of the other like a sandwich, and they obtained a very fast, very compact structure. Several groups are also building electronic processors that use the same algorithms (see, for example, SYMANSKI [ 19821). It remains to be seen whether electronics or optics best exploit these architectures.
2.2. SHADOW CASTING
One of the simplest optical processing configurations is shown in Fig. 14A. To see how this simple shadow casting correlator works, we first assume a totally opaque mask with a pinhole in its center; we get a “camera obscura”, that is, a pinhole camera. If Zin(x,y) and lout(x,y) are the irradiance distribution at the
,
inpiif
tput
Fig. 14. Shadow casting correlators. (A) A simple shadow casting correlator. Input: the plane where the input spatial distribution is presented; typically, it is a self-luminous object such as a CRT screen or a y-ray source; mask: a transparency carrying the second spatial function to be correlated; output: a plane, typically containing some area detector (photographic film, for example) where the system output is expected. (B) A collimated shadow casting correlator. Source: a spatially incoherent, diffuse, light source; collimator: a lens with a focal length F ; musk I: a transparency containing the first function to be correlated; musk 11: a transparency carrying the second spatial function; lens: a second lens, also with focal length F ;together with the collimator it images the source on the output plane; ourput: an area detector such as a photographic film.
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421
input and output planes, respectively, simple geometrical optics gives Zout(x,y ) cc Zin[ - (so/si)x, - (so/si)y]. (x and y are in-plane coordinates, whereas so and siare defined in Fig. 14A). Similarly, for a point object at the center of the input plane and some arbitrary transmittance distribution, z(x, y), for the mask we have I,,, cc z[ (si+ so)x/si,(si + so)y/si].Thus, for the general case
With proper scaling we get the desired correlation (or convolution) operation. Systems based on the configuration of Fig. 14A are discussed in several textbooks, including MERTZ[ 1965 J and ROGERS[ 19771, as well as in patents (for example, FOMENKO [ 19651). The method has found particular acceptance for medical diagnosis via nuclear medicine (a y-ray imaging technique) as discussed, for example, by BARRETT[ 19721, in chapter 8 of BARRETT and SWINDEI 19811 and in 34.2 below. A variation on the preceding theme is the collimated shadow casting system of BERGER[ 19571. Referring to Fig. 14B, let us now assume, for a moment, that the source plane is illuminated at a single point only, (x,, y,); the two lenses form an image of the source plane onto the output plane - a point at (x,, - y o ) = ( - x, , - y,). This point image is attenuated by the transmittances of mask I and mask 11. If zI(x,y ) and zI,(x,y ) are the transmittance functions of the masks, respectively, the total attenuation factor is j J zI(x, y ) zII(x- x,d/F, y - y,d/F) d x dy. In the absence of scattering and diffraction, light from any (x, ,y,) can go only to the corresponding (x, ,y o ) ; thus, when the irradiance distribution at the source is Zs(x,, y,), 5 = x,d/F = - x,d/F and q = ysd/F = - yod/F, we get for the irradiance at the output plane ZO(L
a) =
Z S ( L
rl)
JS
TII(X, Y ) z,,(x
-
5, Y - v) d x dY
the system aperture
= [TI
6 z111L .
(34)
Compared with the simple shadow casting correlator, the collimated correlator offers the following features: (1) Both input functions are to the same scale; and (2) the resulting correlation function can be multiplied by any desired
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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two-dimensional function. However, the following facts should also be observed: (1) the collimated shadow casting correlator is much more complex than the simple correlator; (2) Fig. 14B shows that both input functions must be trqsparencies; and (3)it uses refractive optics (which are not available for X-ray and pray radiation). The telecentric shadow casting correlatoris similar to the collimated correlator, but has no collimator lens, and the input may be self-luminous rather than a transparency. It offers the advantage of similar scale for both input functions without the penalty of having to use transparencies for both.
2.2.1. Performance of shadow casting processors The simplicity of shadow casting correlators is very attractive; our analysis of them, however, is based so far on the geometrical approximation, where the wavelength, R, is assumed to satisfy R ---t 0; we shall now try to get a rough estimate on how small A has to be. We shall use an approximation of the angular spectrum concept (see, for example, GOODMAN [ 19681). If we have a transparency and illuminate it with a plane coherent wave front, which, when entering the transparency, has uniform complex amplitude uin, it will leave the other side with a complex amplitude distribution of u,,,(x,y) = uin(x,y)z,(x,y); z, is called the amplitude transmittance of the transparency and is related to its irradiance (or “intensity”) transmittance z, which is the transmittance we usually refer to here for incoherent light, by z = I z, . ’1 (We note that z, is complex, whereas z is non-negative real, so in general, z, cannot be recovered from z.) For simplicity we shall this can be shown to be the optimistic case for the folassume that z, = lowing analysis. Using an inverse Fourier formula, we can write u d x , v> = J J Uf,,f,>exp P7lj(f,x + f,y)I df, df,. NOW,each of the terms U(f,,f,) exp[2nj(f , x , f , y ) ] for a given (f,,f,) describes a plane wave, propagating at angle cosines ;if, and R f , for the x and y directions, respectively. This mathematical separation of a complicated wavefront into many plane waves, each moving in slightly different angles, is known as the angular spectrum of the wavefront. As the wavefront propagates through space, each component of its angular spectrum changes phase. We can calculate the new phase from the angle cosines R and the propagation distance. We can then insert the new phase into the integral of the former paragraph and get the complex amplitude of our wavefront at the new plane, u,(x,y).
&;
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
For estimating the damage caused by diffraction, however, it is not necessary to go through the details of the preceding methodology. The mask is both space and spatial-bandwidth limited; the size of the smear that is caused by diffraction in the simple correlator is roughly Afmaxso, where f,,, is the maximum spatial frequency of the amplitude transmittance z,(x, y ) of the mask. To make this blur insignificant, we must have, 1fmaxs0 6 l/fmax, so
Similarly, for the collimated and telecentric processors we have the following: fiax
1 6 -. AF
Although we have ignored many fine details, including the effect of non-monochromatic light and the possibility of nonuniform phase in the complex amplitude transmittance at the masks, the results of eqs. (35) and (36) are useful as guidelines; more detailed analysis may show that for many actual systems the situation is slightly worse. To see the implications of this result, let us take typical values of 1 z 0.5 pm (visible light) and so or F of about 10 cm. We get f , 6 4.5 mm- In other words, to get 500 lines resolution across the output plane (home TV quality), our mask must be much larger than 10 x 10 cm2. On the other hand, for y-ray or X-ray radiation, where 1is in the order of a few Angstroms or less, diffraction clearly poses no problem. The simplicity of shadow casting processing, particularly the simple correlator (Fig. 14a), which contains no refractive or reflective optics, clearly wins out for application involving these wavelengths. Applications of such correlators are described in 3 4.2. Another case where shadow casting can be useful is where the mask to output distance (so) in the simple correlator can be quite small. Some suggestions for digital processors using this version of shadow casting correlator are discussed in 5 4.5.
’.
2.3. THE LENSLET ARRAY PROCESSOR
Nature, when looking for an “optimal” imaging system, could not find a single, good-for-everything solution. Both single-aperture (like the human eye) and multiple-aperture (like the eyes of insects - see HORRIDGE[ 19771, for example) systems are widely deployed. The advantages of using multiple lenslets were first recognized by LIPPMANN
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[1908a,b], who devised a method for true 3-D display, namely integral photography. Using a lenslet array, he obtained many images of a scene from differentvantage points, which he later projected back and obtained a true 3-D display. This method is the "grand-daddy'' of all auto-stereoscopic displays. In the imaging devices described above, we had many almost identical optical channels. We may also try to build devices that are less regular. Lenslet arrays can be considered as devices that compress four-dimensional information on two-dimensional surfaces, as shown by GLASER[ 1982bl. Devices for rearranging data are very useful; some early work on this subject was done by BRAGC[ 19441 and later by MAUREand BROOME[ 19721 and by FERRISand MELZER[1965], among others. The general form of a twodimensional, discrete, linear transformation is N-1 N - 1
Gn,m=
C kC gl,kKl,k;n,m, =O
1=0
n,m
= 0,
..., N - 1,
(37)
where g is the input function, G is the output function, and K is the transformation kernel. There are two possible procedures to compute the right-hand side of eq. (37). In the direct algorithm we calculate terms of G in a sequential manner (one term after the other)
DO N = 0 TO NUMBER-1; DO M = 0 TO NUMBER-1; OUTPUT (N,M) = INPUT (*,*) END ; END ;
*
KERNEL (*,+',N,M);
In the back-projection method the entire G array is formed in parallel during a single scan of g,,k
OUTPUT (*,") = 0; DO L = 0 TO NUMBER-1; DO K = 0 TO NUMBER-1; OUTPUT (*,*) = OUTPUT ('k,") + INPUT (L,K) * KERNEL (L,K,*,*); END ; END ; We use PL/I notation, where "*" indices denote "for all possible values" of that index; NUMBER = N, L = 1, M = m, N = n, INPUT = g, OUTPUT = G , and KERNEL = K as defined in eq. (37). Both methods can be implemented with a lenslet array processor (LAP), as shown in Fig. 15. The direct processor shown in (A) consists of an input
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base function K,,k;m , n . Each detector integrates and measures the light from one such mask to obtain the value of a term of Gn,m. The back-projection processor of Fig. 15B consists of an input planeZ, which coincides with a transposed base functions mask M (shown separately in Fig. 16B) and an adjacent lenslet array L. Since each mask/lenslet, which is illuminated by an area in the input plane, is effectively a miniature optical projector, the overall device acts as an array of such projectors. These projectors are aligned so that their projected images (of the individual
Fig. 17. Demonstration of the direct LAP. (A) An input spatial signal; (B) the mask (an 8 x 8 Walsh Hadamard transform biased to obtain non-negative values); (C) the output, that is, the cell average of B multiplied by images of A; (D) digital inverse Hadamard transform of the result of C. After GLASER[1980].
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L MO
I
A IML
0
B Fig. 15. Two lenslet array processor (LAP) configurations: (A) the direct processor; (B) the back-projection processor. From GLASER[ 19801.
planeZ, a lenslet array L, a base functions mask M, and a detector array D located at the output plane. Each lenslet of the array L forms one image of the irradiance distribution of the input plane, representing the input function & k . The base functions masks, shown in Fig. 16A, are located at the combined image plane of the lenslet array so that the light coming through the base function mask located at (n,m ) is the product of the input function g/,k by the I
. K M M
...
Fig. 16. Masks for the LAP: (A) base functions mask for the direct processor; (B) transposed base functions mask for the back-projectionprocessor. From GLASER [ 19801.
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transposed base functions) will overlap and be in sharp focus at plane 0,which is the output plane. A detector array D measures the light at this plane. Each point at the output plane receives light from all corresponding points of the transposed base functions with radiance proportional to the local value of gl,k. The total irradiance at each point of the output is a weighted sum of the terms Kl+; m , n , thereby implementing the back-projection procedure. A form of the back-projection LAP was described by JACKSON [ 19741. The direct LAP was demonstrated by GLASER [ 19801. Figure 17, shows some early results from a LAP. So far we have described how a LAP constructed from ideal components that were perfectly assembled behaves. Actual optical devices, however, have to perform in the presence of errors. Since the LAP is an analog rather than a digital device, any error in its components or construction will contribute to the output signal. In the next subsection we analyse the performance capability of such non-ideal LAP. 2.3.1. Pet$ormance analysis Following GLASER[1982a,b] we may classify the effects that cause degradation in the performance of the LAP into the three following groups: (1) Loss of definition in the images that are formed by the lenslets, (2) errors in the mask, and ( 3 ) coupling failure between the masks and the images that are formed by the lenslets. We shall deal with each group separately. In the following discussion we use q for the period of the images formed by the lenslets (which nearly but usually not precisely equals the period of the lenslet array itself); p for the size of one element (or pixel) inside each image or base-function transparency; N for the number of elements in each dimension of Gn,m and gl,k (thus there are N 2 elements in G and g and N 4 elements in Kl,k;),; and F for the focal length of each lenslet (we assume that F is much smaller than the distance between the lenslet array and the input plane, so F approximately equals the distance between the lenslets and the mask plane). 2.3.1.1. Optics The performance we need from the lenslets of the LAP is fairly modest, compared with, for example, photographic lenses. A spatial bandwidth product of few hundreds is usually more than adequate. On the other hand, it is clearly very difficult to produce a large array of multi-element, individually assembled and aligned lenses. In a practical device each lenslet has a single optical
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element. The image may be formed on its rear surface, where the mask is mounted and the detectors glued to the mask. Diffraction sets an ultimate limit to the achievable image quality. As our lenslets are set in a rectangular grid, their apertures are rectangular. For simplicity we will assume them to be square. For a given period of the array, q, the dimensions of each aperture cannot be larger than q x q. The pupil function is, for the most optimistic case, therefore P(x, Y ) = rect(x/q) rect(Y/q) .
(384
The optical transfer function (OTF) of a lenslet in incoherent light with dominant wavelength of 1 is
where
Here, the special functions rect (u) and A(u) are, respectively, the rectangular and the triangle functions, as defined by GOODMAN [ 19681. If we define the resolution of the lenslet to be the point where the value of the transfer function approaches 50 % , the resolution of a diffraction-limited lenslet is f o . We note that q is also the maximum field size of an individual lenslet. The space bandwidth product of a diffraction-limited lenslet becomes
For example, a lenslet used in visible light in an array of q = 1 mm can have at the most B = 200 for F = 4 m m and B = 80 for F = 1Omm. Analyzing the aberrations is more complicated, and we must reference a particular design. From SMITH[ 19661 for example, we note that the diffraction limit on the space bandwidth product is proportional to 42/F, whereas the geometrical optics limit is proportional to F/q3.Once a particular basic design is chosen, we may find the value of F that will balance diffraction and aberrations. For the small values of F used in lenslet arrays, it is possible for simple (single element) lenslets to achieve diffraction limited performance.
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2.3.1.2. The mask For a general linear transformation the mask should have a transmission function given by the transformation base (or inverse base) functions. Most transformation kernels, however, are bipolar (or complex) and cannot be directly implemented as transmittances in a noncoherent system. Some of the potential approaches are the use of a DC bias and the splitting of the kernel into positive and negative (or three for complex) kernels. Generally, the use of multiple kernels gives superior performance over that of biased kernels, at the expense of added complexity. This topic is the subject of § 3. Many, although not all, transformations will require continuous transmittance values at the mask. The material from which the mask is prepared has some limited dynamic range and exhibits some error in the resulting transmittance values. The dynamic range R of the mask material is
where T,,, and Tminare the maximum and minimum attainable transmittance values, respectively, We now assume a worst-case situation where g1,&= E, ( E << 1) for 1 = I, and k = k,, and gL,&= 1 otherwise. We also assume a matching worst-case kernel KI,k;m,n = 6,,Io bk,ko,where hi,,is the Kronecker delta, which equals zero unless j = i when it gets the value of 1. We expect an output value of Gm,+= E ; however, because of the limited dynamic range of the mask, “zero” kernel elements actually transmit 1/R of the light, so we get an erroneous result of RGm,n= ( N 2 - 1)/R + E E N 2 / R + E. Thus the error in the output for this worst case is A RGn,mE N 2 / R .
(39b)
This gives us the condition for acceptable error in the worst case
NGJR.
(39c)
For example, if R = lo4 (typical of high dynamic range, continuous tone photographic materials), then ZV< 100. We note that if two channels (for bipolar signals) are used, we can arrange this error to be self-compensated. Since the dynamic range of continuous tone photographic materials is inferior to that attainable with hard clipping (binary) materials, it is desirable to use the latter. If our kernel is continuous, we can use area modulation instead of transmittance-value modulation. Each pixel is divided into many sub-
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2
elements, each of size s x s, where all except v are opaque. The equivalent transmittance of that pixel is thus vp2/sz.This would “cost” us higher necessary resolution in the mask. However, equipment and materials for creating very high resolution binav patterns are readily available, as exemplified by VLSI technology. 2.3.1.3. Mask-lenslet alignment Errors in the alignment between the lenslet array and the mask can appear in each of the six (three linear and three angular) possible degrees of freedom. Of these possible misalignments, the longitudinal linear error and two of the angular errors will contribute both to image defocus and distortion. Since image I
translation
rotation
+++ magnification
keystoning
Fig. 18. Mask-lenslet alignment errors. From GLASER[1982a].
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
43 1
defocus is easy to detect and correct, these are of lesser practical consequence than the other alignment errors. The three remaining errors are magnification periodicity errors (Fig. 18), related to minor longitudinal (defocus) errors and/or to an error in the location of the input plane, in-plane rotation, and in-plane translations. Ideally we would like to have perfect registration between the coordinates of the individual masks and those of the images formed with the lenslets. Of course, perfect alignment is not usually achievable; by calculating the effect of [ 1982]), we can different errors on individual mask-image pairs (see GLASER obtain expressions for the allowable tolerances rotation (radians)
1 4N2
PG-
1 magnification (dimensionless) p < 4N2 4
x translation (length units)
,z
<
y translation (length units)
,z,
< 4 f 4i N ‘
~
4fiN ~
Table 1 gives some sample values for these tolerances.
2.3.1.4. Other error sources There are several other potential errors. One such problem is illumination nonuniformity. This can be caused by the “cos4”-law and by vignetting. For simple lenses the cos4-law simply states that an image of a uniform Lambertian object has an irradiance distribution of the form I,, c0s4 0, where 0 is the angle between a point in the image, the nodal point of the lens, and the optical axis. Because each lenslet “sees” the input plane from a different location, this error varies both within the field of each lenslet and between lenslets. For N = 64 this [ 19821). error can be kept to about 0.6%, which is usually acceptable (GLASER For applications employing larger values of N or stricter standards, compensation can be provided by modifying, for example, the transmittance distribution of the mask.
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TABLE1 Alignment tolerances for the LAP. From GLASER [1982a]. ~~
~
Number N of elements
Period q (mm) of images formed by lenslets
Errors"
8
16
32
64
128
256
P 0.125
P
0.004 13' 2.2
n.a.b
n.a.
ma.
ma.
ma.
0.004 13' 5.5
0.001 3'21" 2.1
0.0002 0'50" 1.3
ma.
n.a.
ma.
0.004 13' 11
0.001
3'21" 5.5
0.0002 0'50" 2.1
0.00006 0'13" 1.3
n.a.
n.a.
0.004 13' 22
0.001 3'21" 11
0.0002 0'50" 5.5
0.00006 0'13" 2.1
0.00002 0'03" 1.3
n.a.
0.004 13' 44
0.00 1 3'21" 22
0.0002 0'50"
0.00006 0' 13" 5.5
0.00002 0'03" 2.7
0.004 13' 88
0.001 3'21"
0.00006
0.00002 0'03" 5.5
T
P 0.250
P T
P 0.500
P
r 1.oo
P P
r
P 2.00
P T
P 4.00
P T
a
44
11
0.0002 0'50" 22
0' 13" 11
0.000004 0' 0.8" 1.3
0.000004 0' 0.8"
2.7
p: Magnification error (dimensionless). p: Rotation error (minutes' seconds" of arc). T: Translation error (micrometers). ma.: Not applicable.
2.3.2. Special conjiprations In the former subsections we analyzed the case where both the n and the m indices of the output array G and the I and k indices of the input array g span the same range 0, . .. ,N - 1. This is analogous to one-dimensional linear transformations where the transformation kernel is a square matrix. We define here non-square linear transformations as those where the number of elements in G is not equal to the number of elements in g. Another special class of linear transformation is quasi-convolution-this may be written as a "square" transformation where only those elements in the kernel for which I - n and k - m
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have relatively small values are non-zero. We now look into what special considerations and/or special geometry apply to these cases. 2.3.2.1. Non-square transformations Equation (37) now gets a modified form as follows: N-1
Gn,m=
N-I
C C gi,kKi,k;n,m, n , m = O , . . . , M - 1, 1=0 k=O
MZN.
(41)
4 4$N’
(43)
The “aspect ratio” of the transformation, A , is
M
A = -
”
We get (GLASER [ 19821) a revised form of eq. (40): p<-
1
4AN2 ’
p<-
1 4AN2 ’
7,
<
4 4&N’ ~
Ty <
~
There is little interest in transformations where there are more output than input elements. The usual case is where A c 1 ; here the p and p tolerances are less severe than for the square case of eq. (40). There is no change in the effect of diffraction or optical aberrations compared with the square case. If we use, however, the back-projection scheme with non-square transformation and A < 1, each lenslet needs to resolve M x A4 elements, so we get a relaxation on the required performance of individual lenslets. Thus we can choose between direct LAP with M 2 lenslet, each resolving N 2 elements, or a back-projection one with more (N’) lenslets with poorer resolution. 2.3.2.2. Quasi-convolution For quasi-convolution eq. (37) gets the form n+L/2-1
m+Lj2-1
I=n-L/2
k=m-L/2
C
Gn,m =
g;,kKI,&;n,m, n,m=O ,... > N - L L < N (44)
where g;,k =
i
g[,&, if O , < I < N - 1, and O < k < N - 1 0, otherwise.
The term “quasi-convolution’’comes from the fact that we can defme a “shiftvariant PSF’, hl,&(v,p ) = K,,k;[ - ”, &. II and rewrite eq. (44) as a “convolution”
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
i ii iii
iii
.
[V,8 2
.....
iv
y”-
. ..__
I
i
Fig. 19. Quasi-convolution LAP. i, ii, iii and iv are areas i n the input plane that are imaged by the lenslets that array the same indices. I is the input plane, L is the lenslet array, and M is the mask. From GLASER [1982a,b].
with this h. Another way of looking at eq. (44)is as a two-dimensionalgeneralization of vector-matrix multiplication where the matrix is a band matrix. Figure 19 illustrates a configuration for implementing the quasi-convolution operation with a modified LAP. As each lenslet sees only a small part of the complete input spatial signal, resolution and translational alignment (T, and T,,) requirements are far less stringent than for the more general LAP.
2.4. SPECTRAL DISPERSION
The spectral mixture of a polychromatic light can become a tool for optical infomation processing. One may want to correlate optically the spectral composition of light. Alternatively, we can use the extra dimensionality of wavelength (color) to process spatial information.
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2.4.1. Wavelength spectrum correlators It is often necessary to analyze the spectral contents of some polychromatic light. One example is chemical analysis, where we wish to detect the presence of some chemicals by looking at their emission or absorption spectrum. Another example is the use of the reflection or transmission spectrum of thin layers as a diagnostic or control parameter in optical and semiconductor manufacturing. CAULFIELD and MUELLER [ 19841used a grating to convert the spectral distribution of the light into a spatial pattern. This pattern, as shown in Fig. 20, is transmitted by a mask that is matched to the spectrum of some specific material. STRAND [ 19841 used a similar concept to analyze thin layers by looking at the spectral composition of light that was reflected from them.
Grating
Detector Mask
Fig. 20. A simple spectrum recognition correlator: (a) grating spectroscope that displays the wavelength spectrum of incoming light on a mask; (b) the displayed spectrum; (c) a set of masks; (d) an assembly of masks and detectors. From CAULFIELD and MUELLER[1984].
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2.4.2. Dispersive processing of spatial information 2.4.2.1. Correlators Let us consider the optical system of Fig. 21. For simplicity we assume that we use light with a flat spectrum; that is, the amount of energy flux in any band 1,. ..,I + A1 is independent of the wavelength 1.We have some input image, illuminated by our flat polychromatic light. A lens collimates (produces an image at infinity) the light from this object, and a dispersive prism separates the light into its spectral components. A second lens images this light on a second plane. For wavelength 1we get a displaced image O,(x - a I , y), where a is the dispersive coefficient of the prism and O,(x, y ) is the irradiance distribution of the original input image. In the system of BARTELT[ 19791, depicted in Fig. 21, there is a transparency with transmittance distribution O,(x, y). (Note the inverted coordinates in this transparency plane in the figure.) The transparency will transmit, at wavelength I , irradiance distribution O,(x - ail, y ) O,(x, y). A light collecting device, such as a light guide or a fiber bundle, mixes all the light from the image, so we get f f O,(x - a i , y ) O,(x, y ) d x dy, which is a function of il only. This light may now be collimated and introduced into a second dispersive prism. This gives a one-dimensional spatial distribution
,
, . .
X
Fig. 21. Spectral-multiplexing correlator. From BARTELT [1979].
v, § 21
431
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
where b (the dispersion coefficient of the second prism) is defined by X'
b=-.
A
This is the correlation of 0, with 0, along the line y = 0. As the range of wavelengths used, AA, is made larger, and as the size of 0, and/or 0, becomes larger relative to the focal length of the lenses, the performance of the system w ill degrade. This is because of the dependence of the dispersion of prism on the wavelength and on the entrance angle. If we use identical prisms in both parts of the system, their wavelength-dependent variation would self-compensate. There is no simple way to compensate for the angle dependence of the dispersion, which would cause partial loss of shift invariance. If the focal length of the first (collimating) lens is more than about 10 times the size of the objects, the system should perform reasonably well. KINGSLAKE[ 19381 provides further information on prism-based systems and their analysis.
2.4.2.2. General one-dimensional linear transformations In a later work BARTELT[ 19811 extends his correlator into a general linear transformer. The system is depicted in Fig. 22. In plane ( x , , y , ) we have an irradiance distribution k ( x , ,y , ) with flat spectral distribution. This light is collimated by a lens, dispersed by a prism, and then re-imaged by another lens onto plane (x,, y2), giving a distribution that is a function of both space and wavelength: u,(x2, y,, A) = k(x,, y , - an). At the (x,, y 2 ) plane we also have a transparency with a transmittance function z(x,, y,) = 6(y,) g(x2), so the transmitted light irradiance is given, for y , = 0, by k(x,, - an) g(x,) and for
X1
'X2
4
4
p-
f +2f
--q-f
T
-x3
4
fI - f
--p-f-r2f-rf
I
Fig. 22. Spectral-multiplexing system for general one-dimensional linear transformation. From BARTELT [1981].
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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y, # 0 to zero. A third lens now collimates this light. At the ( x 3 ,y 3 ) plane we get mixed light, where the spatial distribution is lost: u3(x3,y3,A) = = k(x,, - an) g ( x 2 )dx,. We now put a slit (not shown in Fig. 22) in the ( x 3 ,y 3 ) plane, so we have an irradiance distribution u(x3,y,, A) 6(x3).A lens now collimates this light, a second prism disperses it and, with the help of another lens, the spectrum of this light is displayed on the (x4,y4) plane. We get u4(x4, Y4,
4 = u3( - aA, Y4, 2) =
since
s w,,
x4) g(x2) dx,
9
(46)
x4 = - a A .
Thus u4(x4)is the linear transformation ofg with the kernel k. We can also write g as a discrete vector (gl)= g and k as a discrete matrix (k1,J = [ k ] ;u4(x4) is the vector-matrix product ~ l k l , m g=l [Klg. The system of Fig. 22 does not use the full potential of wavelength multiplexing. The y4 axis is essentially without use. BARTELT[ 19811 shows how, by adding cylindrical lenses to a similar system, it is possible to regain this “lost” dimension and have a matrix-matrix multiplier. Unlike the dispersive correlator of Fig. 21, the linear transform system of Fig. 22 and its matrix-matrix multiplier variant do not need shift invariance or wavelength-shift invariance. Therefore distortions by the prisms can be easily compensated for. The information capacity of this system is limited only by the spatial resolution of the lenses, the spectral resolution of the prism (which can be as high as the spatial resolution of the lenses), and photometric (related to detector noise) considerations.
2.5. OTF SYNTHESIS
Perfect lenses, or lenses where the image is indistinguishable from the object (except for a change in scale and attenuation), are not physically realizable. Diffraction, and usually also focus errors and aberrations, cause degradation that is usually considered undesirable. For many applications the degradation in the image formed by most lenses can be adequately approximated as convolution. If gperfect(x,y ) is the irradiance distribution in the hypothetical “perfect” image, the actual image can be written as
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
439
and from Fourier theory we get
where
Here h(x, y ) describes the response of the lens to a point object - the case where gperfect(xr y ) = 6(x,y ) . It is therefore named the point spread function (PSF) of the lens. H ( f , , f , ) , the Fourier transform of h, is called the optical transfer function (OTF) of the lens. (Note that we assume incoherent illumination; both the OTF and the PSF defined above are the incoherent OTF and incoherent PSF; GOODMAN[1968], for example, shows the relation between coherent and incoherent OTFs and PSFs.) When one designs an imaging lens,one usually tries to make the image as close to perfect as possible -one wants h(x, y ) = 6(x, y ) and, hence, an OTF that is constant or flat. For information processing, however, we often want a certain h(x, y ) that is not equal to 6(x, y ) or H ( S , , 4 )that is not flat. One possible approach to optical information processing is thus OTF synthesis, suggested for incoherent llght by LOHMANN [ 1959,19621.OTF synthesis is the design of an optical system that has a prescribed OTF. It is useful to distinguish between geometrical optics OTF synthesis and difractive OTFsynthesis. Although both can be explained in terms of diffraction, only the fmt one can also be analyzed using the tools of geometrical optics. Another difference is that the former is assumed to be wavelength dependent only through the refractive dispersion of glasses; it is quite achromatic and can be used with white, or at least wide spectral band light. 2.5.1. Geometrical-optics OTF synthesis
This subsection discusses geometrical OTF synthesis, or the use of intentional aberrations to control the OTF of a lens. A simple case of OTF synthesis is the synthesis of a line-shaped PSF: h(x,y) = 6(x) or H ( f , , f,) = S(f,). Such OTF's are useful for detection of directional patterns, [1979] and for construction of as shown by DEBRUSand FRANCON vector-matrix multipliers, as demonstrated by GOODMAN, DIASand WOODY [ 19781, who used cylindrical lenses to effect a one-dimensional de-focus. BARRETT,CHIU,GORDON and PARKS[1978,1979] analyzed the general problem of how to obtain a desired arbitrary OTF by introducing intentional
440
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
X
X’
Fig. 23. Geometry for calculating the one-dimensional point spread function from aberrations PARKSand SWINDELL [1979]. introduced by a phase plate. From BARRETT,CHIU,GORDON,
aberrations. Following this work, we shall assume the configuration depicted in Fig. 23. An aberrating (or a “phase”) plate is added at the exit pupil plane of an aberration-freelens. We shall give here the analysis of the one-dimensional OTF case. A discussion of the more general 2-D case is found in BARRETT, CHIU,GORDON and PARKS[1979]. The phase plate can be characterized by a thickness function ~ ( x ) which , distorts the “perfect” wavefront by an amount of W(x) = (n - 1) ~ ( x )n, being the refractive index of the glass. If we now take a ray that, in the absence of a phase plate, would hit the image plane at x’ = 0, it would now be deviated by the phase plate and arrive, instead, at x’
dW dx ’
=f -
where f is the distance between the phase plate and the image plane and dtj/dx is the local slope of the phase plate [thus d{(n - l)q}/dx = d W/dx is its contribution to the tilt of a ray passing through it]. Now, assuming that the exit pupil is evenly illuminated (by a point object) with an irradiance of I,, we know that a “slice” of the phase plate from x to x + dx will let a flux of I , dx through. It would hit the image plane between x’ and x’ + dx’, where the irradiance is Z and, hence, the flux equals 1 I dx ’ 1, IZ(x’)dx’/ = IZ,(x)dx/
(494
and
-’
I , d2 W Z(x’) = z, 7= - f Idx21
‘
v, § 21
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
441
We now define a normalized PSF: p ( x ’ ) = I(x’)/Io, so eq. (49b) can be rewritten as
In the general case, where there may be more than one point x at the pupil that contributes light to the point x ’ at the image plane, we should modify eq. (49c) to the form P(X’) =
c
(4 9 4
all x such that x ‘ =f(d W / d r )
It is difficult to invert eq. (49d); however, we may restrict ourselves to one-to-one solutions where eq. (49c) suffices. This is the case when d W/dx is monotonic (i.e., d2W/dxZdoes not change sign) over the full range of x. This method can be extended to two dimensions, although to my knowledge, it was implemented in a practical system only for a 1-D H . Since the method is achromatic and has high light efficiency, it is very attractive for many applications. At present its wide utilization is restricted by problems in computing the required q(x,y ) from the desired H (a problem which is likely to be resolved as computer and software technology advances), and the difficulty of manufacturing arbitrary surfaces to optical precisions. The latter problem may be insignificant where mass production by molding of optical plastics is practical. For small to medium production sizes it will probably continue to limit the use of this method in the future. 2.5.2. Difractive OTF synthesis In the former subsection we discussed some of the methods, and the problems, associated in using controlled aberrations to get a desired incoherent OTF (or PSF) from an optical system. An alternative approach that has attracted much attention from many researchers is the use of dzractive OTF synthesis. In this approach one starts with a well-corrected optical system and then uses a mask, or transparency, to modify its pupil function. The connection between the pupil function of an optical system and the incoherent PSF and OTF has been known since ABBE’S[1873] theory of the microscope. The notion of modifying the pupil intentionally to obtain a desired PSF had, apparently, to wait for quite some time (see, for example, the early works of LOHMANN[ 1959,19621).
442
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
tv, 8 2
Let us assume a lens with a focal length F. For simplicity we shall assume that the object is at infinity, so that the image is at the focal plane. We shall also assume that the exit pupil of the lens is at the rear principal plane, at a distance Fin front of the focal plane. Coordinatesin the pupil plane are denoted by (X, Y) or X and at the image/focal plane by (x,y ) or x. Assuming a single point object and monochromatic illumination, the perfect image is just 6(x, y). For the pupil function, or the complex-amplitude transmittance of the pupil, we use p(X, Y). The complex amplitude distribution at the image plane Upoint(X9 v), is given by
p(X)exp[ - j ( 2 n / A F ) X . x ] dXdY,
(50a)
where K, is some constant. is therefore The irradiance distribution at the focal plane gpoint(x)
where K 2 and K3 are constants, and
Thus if we now illuminate the system with some spatially incoherent object that corresponds to a perfect image gperfect(x), the actual image that we shall get, gW, is g(x) = gperfect(x)* h(x) 3
where
h(x) = gpoint(x) *
(50c)
Thus from the pupil function p ( X ) we obtained the PSF h(x) and the OTF H(r) and the OTF H(f)= 9 { h ( x ) } ,
'
h(x) = K3 1 91{ p ( - X ) } 12 = K3 9-{ p ( - X ) 8p( - X ) } ,
N f )= [P@PI
(-W
).
(504
From this result it is easy to see that, for spatially incoherent light, (1) the PSF is always non-negative real, and (2) the OTF is a geometrical autocorrelation of a bounded function.
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
443
LUKOSZ[ 19621 investigated the mathematical implications of these properties on the realizability of arbitrary OTFs and PSFs. Clearly, to get a bipolar real or complex valued PSF, and/or to obtain a high-pass filtering, one must use some indirect representation or an off-axis PSF, where there is an on-axis part of the PSF that, hopefully, would fall far enough from the area of interest. The process of OTF synthesis w ill be considered now. We want to realize some given PSF, so we start by inverting eq. (50b), upoint(x)= K~
ej ~ ( x .)
(514
Usually an overall intensity factor in the output is of little concern; we may drop the K4factor during the practical synthesis process. cp(x) can be quite arbitrary and its selection is part of the synthesis process. We now use the inverse transform to calculate the pupil function
=
k4[ F{ h(x)} * F{ d Vex)} ] ,
( 5 1b)
where p(f) is the pupil p(X)scaled to the f coordinates. We note that the choiceLf q ( x ) is related to the size of the pupil. In other words, the spatial frequency bandwidth of exp &)I, plus that of h(x) must be no larger than the available pupil size. It is usually not helpful to choose cp(X) = constant either; if the pupil area is small, the effective f-number of the system is low and its radiometric (light energy flux) efficiency is low. Another problem with OTF synthesis is the physical realization of the pupil function. Making a transparency with complex (amplitude) transmittance function is possible but very difficult. We would prefer, whenever possible, the use of non-negative real pupil functions. For these functions all we have to do is to prepare a transparency with an irradiunce transmittance function of [p(X)I2, and suppress any phase distortions. (For example, we can produce the desired transmittance distribution on a photographic plate, coat it with index matching extra gelatin, and then cover it with flat glass.) We must note that, although our object emits spatially incoherent light, light at the pupil plane is not spatially incoherent. The problem of pupil synthesis is closely related to the subject of computergenerated holography. Indeed, our synthesized pupil transparency is a Fraunhofer hologram of the PSF. Two notable methods for computergenerated holography are the detour phase method, originated by LOHMANN and PARIS[ 19671, which gives a binary non-negative real hologram, and the
444
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§ 2
kinofom of LESEM,HIRSCHand JORDAN [1969], which is a phase-only hologram. Detailed review of these and other methods of computer holography can be found, for example, in DALLAS [ 19801. One pupil synthesis technique that is not directly related to computer holography is the ofsetpupilmethod of STONER[ 1978bl. If the pupil functionp(f) is made of two offset pupils, p",, ( f ) and p 2 ( f ) , then
-
p ( f ) = p 1 ( f - ifc)+ p 2 ( f
+
-
if,> 3
where 1 fc=-Xc, 1F and $Xc is the offset of each of the subpupils from the center of the pupil plane. From eq. (50d) we get
W f )= p@'p = [([I
@PI)
@&l(f-fc>
+ "1
+ (pz @p2)l(f) + [;t)2@)PlI(f+fC).
(52b)
(We placed the coordinates for each of the correlation terms out of the square brackets.) The three terms are separated in spatial frequency space if the bounds of each of the p 1 and p 2 subpupils are smaller than f 4 If, 1. The first term, centered on s = 6,is of n6 interest to us. Each of the other terms, centered around f = ff,, contains the cross correlation of the two subpupils. Actually, since our pupil is real, it is easy to see that the second and third terms of eq. (52b) are 180" images of each other. In the image plane the three terms of the PSF h(x) that correspond to the three terms of H(f)would not, of course, be spatially separated. STONER [ 1978a,b] gives the following receipt on how to separate them: 1. Multiply the input image gi(x)by a grating (simply image it on a physical grating),
+
gi,c(x)= gi(x) [ 1 + cos (2 nfc x)I
9
which, in Fourier space, gives G i , c ( f ) = y{gi,c(x)} = gi(J)* +[Wf)+ $'
$Gi(f)
+ aGi(f - S,) + !Ci( s + S,).
(534
2. Using the offset pupils OTF synthesis system, we concolve gi,cwith the PSF of the system go,c(x) = gi,c *
4-' { H )
7
v, § 21
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
445
or, in Fourier space, G o , c ( f ) = F{go,c(x>}= G i , c ( f )H ( f ) =
{ + G i [ p lBf1
+ ~ 2 @ ~ 2 I ) ( f )+
+ { b G i [ f 2 B f J ) ( f +f c ) .
{ i G i [ p ,BfzI)(f-fc)
(53b)
If gi(x)does not contain any spatial frequencies higher than if,1, there are no cross terms. 3. The second (or third) term in eq. (53b), which we shall call G o ( f ) = 8 { g o ( x ) } ,contains the information we want. Although it does not overlap the other terms in Fourier space, it certainly does in x space. To extract it from the other terms, we can, for example, use a video camera, converting g0,,(x) into a temporal video signal; the demodulation can then be done electronically. The procesy is closely related to the spatial carrier method of representing bipolar or complex spatial signals with incoherent light, due to LOHMANN [ 1977al. It is discussed in § 3. The effective PSF of the system, he&) = 8' { p l @ p , } , can be bipolar or even complex valued. For example, it is possible to do high-pass filtering with this method. The synthesis of p l ( X ) and p 2 ( X ) from the desired H , , ( f ) (the inversion of the second term in eq. (5 Id) allows much flexibility -complicated effective OTF can be constructed with binary (two levels only, namely, opaque and clear) pupils. However, this synthesis process may not always be routine and simple. 2.5.3. Holographic incoherent OTF synthesis A special case of dsractive OTF synthesis is holographic OTFsynthesis. This method can be considered diffractive OTF synthesis with a hologram as the pupil mask. A different way of looking at it is as a system containing a display hologram that is reconstructed with an extended, spatially incoherent, source of light. In the following section we shall use both views interchangeably. Holographic OTF synthesis has been demonstrated both with coherent (VANDERLUGT [ 19641) and incoherent (LOHMANN[ 19681, LOWENTHAL and WERTZ[ 19681, and MALONEY[ 1971a,b,c]) illumination. Figure 24 depicts incoherent holographic OTF synthesis in its simplest form. A hologram is recorded (Fig. 24a) using diffuse, coherent illumination. The reference beam diverges from a point at the plane of the object. If we now illuminate this hologram (after development) with light from an extended,
446
Fig. 2
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§ 2
Recording and using a hologram for optical transfer function synthesis in incccirent light
- a simplified diagram. (a) Recording the hologram (this step is done with coherent light): a
transparency of a "model" (spatial signal) M is illuminated through a diffuser D using coherent light. Light from the same laser is diverged (using a lens-pinhole assembly) at a point R that is on the same x-y plane as the model transparency M. A holographic recording plate P is located at another plane, X-Y, which is parallel to the plane of the model. (b) Convolution is done in incoherent (but quasi-monochromatic) light. The developed hologram from step (a) is re-indexed
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441
incoherent (but quasi-monochromatic) source located about the original center-of-divergence of the reference beam, the reconstructed (virtual) image will be convolved (in irradiance) with the irradiance distribution of the readout source (Fig. 24b). If, instead, the readout source is located about the original position of the object (Fig. 24c), we get the correlation of the irradiance distributions of the readout with that of the object. 2.5.3.1. Non-paraxial and chromatic efects In the analysis of holographic incoherent OTF synthesis we shall use the following simplifying conventions and symbols : (a) We name the spatial signal from which the hologram is taken the model and denote it by gm(x,y). The spatial signal represented by the irradiance distribution of the extended incoherent source that is used in the convolution or correlation process is named the input, denoted by g,(x,y). The resulting spatial signal is called the output g,(x, y). Symbols associated with the model, input, and output spatial signals are subscripted with m, i and 0,respectively. (b) The input/output planes (which, in the absence of extra output optics are the same plane) will be named the x-y plane. The plane of the hologram is the X - Y plane. (c) One convolution is treated, except where noted otherwise, since correlation can be expressed as a convolution with another model. (d) We shall assume an on-axis coordinate system, with the holographic reference coming from ( x , y ) = (0,O). The off-axis shift will be assumed to be a part of the definition of the spatial signals. The reference source (used in the recording step) will be assumed to be on the optical axis (in contrast with Fig. 24). (e) We shall use, for most of our analysis, single point model and input spatial signals (Fig. 25a); model: g&,
y ) = b(x - x,, y - ym>,
(544
in its original location in the X - Y plane. An extended, spatially incoherent source that has spatial irradiance distribution proportional to the “input” spatial signal is now located at plane x-y at the location of the reference source (Rin (a), above). The holographic reconstruction is a virtual image, centered around the origin 0 of the x-y plane, the irradiance distribution of which is proportional to the “output” spatial signal, the convolution of the model with the input signals. (c) Convolution is also incoherent. The configuration is similar to that of the convolution (b, above), except that the input source is now placed at the origin of the x-y plane and the output appears around the former location (R in (a), above) of the reference source. From GLASER
[1986].
448
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
a
tx t
1
[V,3! 2
b
Fig. 25. Two geometries for holographic incoherent OTF synthesis. (a) Finite distance geometry. Input/output/model planex-y is at a distance b from the hologram plane X-Y. xm, xi and x, are locations of points in the model, input and output spatial signals, respectively. (b) Infinite geometry. By using a collimating lens with focal length the x-y plane is “thrown” to infinity. The location of points in the x-y plane may now be described by angles (from the optical axis z ) Urn,Ui, and U , as shown for the X-z plane and V,, q, and V, for the (not shown) Y-z plane. From GLASER[1986].
input: gi(x,y ) = 6(x - xi, y - y i ) .
(54b)
Clearly, the ideal output that should result from these also consists of a single point: go, ideakx, Y ) = 6(x
- xo, Y - Y O )
3
(54c)
where x, = xi+ x, ,
yo
=
yi
+ y, .
(f) For collimated systems (Fig. 25b), where the hologram sees the x-y plane at infinity, the use of coordinates at this (infinite) x-y plane is
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
449
meaningless. We thus use angles V,, Vi, and U,, measured from the optical (z) axis, for the direction of light coming from points of the model, input, and output, respectively, on the x-z plane and V,, q,and V, on the y-z plane. The ideal convolver should satisfy eq. ( 5 4 ) precisely. In practice the output spatial signal for impulse model and input functions eqs. (54a) and (54b) is never an impulse function (eq. 54c) because of aberrations and diffraction. We must take eq. (54c) as an approximation and redefine (xo, yo) as the centroid (or “center of gravity” of g,,(x, y)
In the presence of errors or distortion eq. (54c) may not be satisfied, even in this redefined meaning of eq. (55). In the paraxial approximation (Ixil + Ix,( 4 b in Fig. 25a) and “really” monochromatic light it is easy to see that eq. (54c) is correct to the limit set by the finite size of the hologram. This section deals with departures from this “standard” for systems where paraxial approximation and/or monochromaticity do not apply. Chromatic eflects When a hologram is reconstructed at a wavelength other than the recording wavelength, change of scale and possibly aberrations result. If the readout light is not quite monochromatic, the mixture of different magnifications results in blur. Assumingparaxial approximation, let us consider a hologram (for our holographic incoherent OTF synthesis system) that was recorded at a wavelength 1 , and is reconstructed with input at wavelength ,Ii. [The reference source is
450
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
rv, 8 2
at ( x , y ) = (0, O).] The output point will be at x,
=
xi
Ai +x,,
Am
4
Yo =Yi + - Y m *
(56)
Am
Now, if the input spatial signal is not quite monochromatic and has spectral bandwidth of A&, centered on the recording wavelength of &, the worst case chromatic smear size (Ax,, max, Ay,, ma) would become
Practically, most holograms are of the offset reference type (LEITHand UPATNIEKS [ 1962]), and the centroid of the model spatial signal is located at some point (x,,~, 0), where x , , ~# 0. The value of x , , ~ that will prevent overlap of different reconstruction terms depends on the type of bias that is used and the way the hologram is reconstructed. For optically recorded holograms tflat are reconstructed with spatially extended source gi(x,y), we get
where W,,, and Wi,x are the width (in the x dimension) of the (non-zero part of the) model and input spatial signals, respectively (see, for example, ABRAMSON [ 19811). In these terms the size of the chromatic smear is dictated by the size of the expected output spatial signal. Nonchromatic eflects The extensive work that was done by several researchers on the application of lens design methodology to the analysis of holography and, in particular, [ 19671, and LATTA holographic optical elements (MEIER[ 19651, CHAMPAGNE [ 19711, to list a few) may also be used to analyze the behavior of holographic incoherent optical transfer function synthesis systems. However, because of the relative simplicity of our problem, and for the sake of clarity, direct analysis from basic principles will be used here. Let us consider some holographic grating, recorded with our g,(x, y ) pointobject model of eq. (54a). At a given point at the hologram plane, the light arrived with a wave vector Kk,where ( K ' m (= l/A. Similarly, the reference beam arrived at the same point with wave vector K.The spatial frequency of
v, 8 21
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
45 1
the holographic grating k, is the difference between the projection of KL on the X - Y plane, kk, and that of K,, k,: we denote it by k,. We now reconstruct this hologram with ourgi(x, y ) point-object input of eq. (54b). At the same point it has a wave vector Ki with an X- Y projection ki.The reconstructed (output) wavefront would have, at this same point, a wave vector KOwith an X-Y plane projection k,, satisfying
For the two-dimensional case, where K,,,, &, and KY,*and, hence, also k,,,, k,,, are zero (the three wave vectors are in the X-z plane), eq. (59) reduces to sin (U,)
=
sin (Ui) + sin (Urn).
For the collimated case of Fig. 25b, k, gram. Thus k, = kk and K , = K k .
(60) =
0 for the entire surface of the holo-
Blur Referring again to Fig. 25a and using eq. (59) with the simple point model, point input spatial signals of eq. (54),we can use ray tracing to get a spot diagram for the output spatial signal. Since we use semi-geometrical approximation, the ideal expected pattern is a mathematical point; the output pattern is spread because of aberrations. The size of the output spot for this case is inversely proportional to the spatial resolution of the system. From the shape of the spots (see GLASER[ 19861) we can also infer the aberrations that are responsible to the loss of resolution. Here the most severe aberrations are usually astigmatism and field curvature. Fig. 26 maps the relative total RMS error (both blur and distortion) as a function of the positions of the model and input points for typical system parameters. Details of the calculation of this [ 19861. relative error are given in GLASER There is one special case that can be guaranteed to be aberration free and, hence, to give the best resolution possible with a given aperture (hologram) size: the infinite, or collimated, case of Fig. 25b. A point input/point model case with collimation is translation invariant at the X-Y hologram plane. Thus, by translational symmetry, output rays must also be perfectly collimated and so free of any blur-causing aberration. A collimated holographic incoherent OTF synthesis system is thus preferable to a noncollimated one. However, to get collimated input and model, a lens must be used between the physical input/model plane and the hologram. Another lens must also be used after the hologram if one wants to obtain the
452
[V, 5 2
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
INPUT
0
20
0
1.5
0
1 6
0
1 4
0
1 2
0
( 0
0
on
0
0 8
0
0 4
0
0 2
0
0 0 - 0
2 0
LECENO.
-0
ERROR
-0
1 4
0 8
-0
0
0 2
0 . 2
0 . 1
1
10
0
i 6
MOOEL
= 7
0
0 4
1
o
E-12
0
5
o
Fig. 26. Relative total error in a typical noncollimated OTF synthesis system. Both distortion and blur are included and given in percents relative to the maximum distance of x, x,, or expected distance ofx, from the axis. The worst case of either x, parallel to xi or orthogonal to each other is shown; model and input refer to x,,, and xifor the parallel case, and x, and yi for the orthogonal case. All values are given for the case where b in Fig. 25a equals 1. The hologram is circular with a diameter of 0.5 6. From GLASER[1986].
output spatial signal at some finite location. Assuming both lenses to be diffraction limited is not always sufficient. Distortion and loss of sh$t invuriunce High performance lenses are usually designed to produce tangent projection; that is, if an object point is at a distance x from the optical axis at the focal plane, its light would be collimated to a beam at angle U from the axis, given by
where f is the focal length.
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453
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
This is clearly in conflict with eq. (60); for example, in the two-dimensional (X-z plane) case we expect to have (assuming input and output lenses of the same focal length, hence unit system magnification) x, = xi + x, so from eq. (61) we would expect to have tan(U,) = tan(Ui) + tan(U,); this agrees with eq. (60) only when (I Uil + I UJ)4 1. The discrepancy between the tangent relation of eq. (61) and the sine relation of eq. (60) was named by GLASER [ 19861 the sin-tun (for sine-tangent) error. Figure 27 gives some computed results for the sin-tan error, using the same system parameters as in Fig. 26 (see GLASER [ 19861for computational details). It shows that, since for collimated systems there are no blur-causing aberrations, their performance is significantly better than noncollimated systems.
0
2 0
0
1 8
0
1 8
0
1 4
0
12
0
10
0
08
0
08
0
0 4
0
02
0
0 0 -0
1 0
-0
0 7
0
1 8
UODEL
LEGEND:
ERROR
0 . 1
I
]
1 . 0
0 . 2
0
s
I ] 2 . 0
Fig. 27. Relative distortion in a typical collimated OTF synthesis system. Both distortion and blur are included and given in percents relative to the maximum distance of x, xi, or expected distance ofx, from the axis. The worst case of either x, parallel to xi or orthogonal to each other is shown; model and input refer to x, and xi for the parallel case, and x, andy, for the orthogonal case. All values are given for the case wheref in Fig. 25b equals 1. The hologram is circular with a diameter of 0.5J Note the reduction in the amount of error compared with the noncollimated [1986b]. example of Fig. 26, although both examples used similar parameters. From GLASER
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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2.5.3.2. Optimization of dffractive (and holographic) OTF synthesis So far we have described holographic incoherent OTF synthesis from a researcher’s point of view: “How would a given system behave?” In contrast, our attitude in this part of the work is the engineer’s: “How can I make a system perform in a given way?’ We shall consider techniques that help holographic incoherent OTF synthesis systems get improved performance in space-bandwidth-product (which includes both low blur and less distortion), monochromacity requirements, and compactness. We noticed that collimated systems (like Fig. 25b) seem to offer several advantages over noncollimated (Fig. 25a) systems, including (1) no blurcausing aberrations, (2) insensitivity to translational positioning errors, and (3) system performance is not directly affected by the effective f-number. This latter feature leads to greater photometric efficiency (more light comes to the detector) and makes speckle (from diffuse holographic recording) sufficiently small to be insignificant. Collimated systems also have some disadvantages, including (1) an extra lens (of sufficient quality) is necessary, and (2) no total freedom from errors is achieved when conventional lenses (offering tangent projection) are used. It seems that for most applications, the advantages of collimated systems greatly outweigh their disadvantages. The following discussion will deal mostly with the collimated approach. Off-axis systems With very few exceptions most holograms use the off-axis scheme to allow separation between the reconstruction, pass-through, and conjugate reconstruction. Figure 28 illustrates some holographic incoherent OTF synthesis systems where the hologram is of the off-axis type. All examples are of the collimated type, and in all the output imaging lens (that must be used to image the virtual output of the system onto some electronic unage pickup device or a photographic film) is included. The most simple (conceptually) off-axis holographic incoherent OTF synthesis system is the one shown in Fig. 28A. The off-axis convolution( or correlation) image appears off the center of the field of view of the output imaging lens (Loin Fig. 28A), whereas the lens itself shares the same physical axis with the rest of the system. We note, however, that in order to provide sufficient resolution and freedom from vignetting, the output lens must be corrected over an angular field that is substantially larger than that of the output image itself. Besides forcing us to use a lens with much higher performance than we should have otherwise, this approach also makes the sin-tan error more severe. Some alternatives to the approach of Fig. 28A are shown in Fig. 28B through D.
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
455
A
E
I
rH W
c
Fig. 28. Collimated systems with an off-axis-reference hologram. In all parts of this figure, I is the input plane, Li is the input lens (collimator), H is the hologram, Lo is the output (imaping) lens, and 0 is the output plane. The systems shown are: (A) a single axis system; (B) a bent-axis system; (C) a grating compensated system, G is a grating; (D) a wedge-compensated system; W is a wedge. From GLASER[1986].
Figure 28B depicts a “bent” system that is similar to many practical implementations of coherent spatial filtering. (Note that many of the problems addressed here also exist theoretically in coherent spatial filtering; however, since in coherent spatial filtering systems the required precision in the location of the
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[v,
2
spatial filter is inversely proportional to the f-number of the Fourier lens, diurneter/F,and the angular field is about equal to the f-number, most workers preferred using systems with very small angular field. In contrast, incoherent OTF synthesis have no similar problem, so one would prefer using high f-number, wide angular field, and optics to improve the photometric efficiency of the system, and decrease its physical size; problems of theoretical value only for coherent systems become design issues in incoherent ones.) Assuming that the tilt of the hologram (0, in Fig. 28B) is zero, a collimated beam that enters it on axis and diffracted by 0, would also change its width from wi (width on input) to w, (width on output), w,
=
wi cos 0,.
(62)
Now, if any optical system changes the width of a parallel beam, it must also change the angular size of an image in infinity, following the equation (for example, KINGSLAKE [ 19831)
(63)
wiAUi = w,AU,,
where AUi and AU, are the angular extents of the image on input and output, respectively, and A U 1 for both. So we get
+
This angular magnification effect happens only in the plane common to the axes of the input lens and output lens (the plane of the paper in Fig. 28B). Thus we get anamorphic imagery. To avoid this problem, we must ensure that diffraction at the hologram would not change the width of a beam incoming on the axis. This is possible if we tilt the hologram by 0,
Note that the hologram must also be recorded using the same tilt. Instead of physically bending the axes of the system, we may insert some optical device to bend the light. In Fig. 28C we show the use of a diffraction grating for this purpose, whereas in Fig. 28D the use of an optical wedge is depicted. In either case the bending angle (of either the grating or the wedge) exactly compensates for the off-axis angle of the hologram. Both the hologram and the compensation grating of the configuration of Fig. 28C pass some undsracted light. Even if we assume high diffraction efficiencies for both, such as about SO%, we would still be left with a small
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
457
percentage of the input light that would pass through both and would be imaged by the output lens right on its optical axis. This is not a tolerable situation for many applications. To avoid this, we must increase separation between the hologram and the grating and put some physical stop between them. To avoid vignetting, we must use a larger aperture on either the input or the output lens (or both) as we increase the distance between the hologram and the grating. The system wdl be free from anamorphic imaging if the hologram and grating are on parallel planes. We also note that the compensation grating also compensates for the contribution of the off-axis reference angle to the sin-tan error. The wedge system of Fig. 28D does not have the preceding problem with undiffracted light, although care must be used to ensure that the hologram-wedge system does not cause astigmatism. When we consider polychromatic light, there is a significant difference between these two systems. The grating can compensate for the off-axis-angle contribution to the chromatic effect. The wedge will increase the chromatic blur. Control of the sin-tan error
We noted earlier that the sin-tan error, resulting from the combination of tangent projection lenses with the “sine arithmetics” of diffraction, can degrade the performance of dsractive OTF synthesis systems. This does not cause the system to exhibit distortion - the tangent projection of the output lens compensates perfectly for that of the collimatinglens. The hologuam, or pupil mask, however, does “see” a distorted input spatial signal. At the edge of the field this distortion makes an image of the object look different from that of the model; hence we get degraded correlation output - the system is no longer shift invariant. The offset angle of the reference makes the problem more severe. By using a compensating grating (Fig. 28C), the contribution of the offset angle to the sin-tan error becomes zero. In such a system we must consider only the angular size of the input field itself. There are several ways to confront this problem, such as the following: (1) Use longer focal length, and get smaller angles so that sin (U)- tan (U) will become lower than the resolution of the spatial signals. Figure 27, or a similar diagram, can be used to estimate the sin-tan error and select a suitable focal length. This solution leads, however, to larger optical systems. ( 2 ) Design sine projection lenses. Such lenses have been used for other applications (they are called “orthoscopic projection” lenses). This is the best solution, although it may not be economical for some applications.
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V, § 2
( 3 ) Segment the hologram, so that each part contains a distorted model, compensating the sin-tan error for one part of the entire field (if the hologram is not at the system aperture stop, each part of the hologram sees only a part of the entire scene).
2.6. INTERFEROMETRIC METHODS
We have noted in the Introduction that one does not usually expect to see interferencefringes with spatially incoherent light. The exception to this rule are shearing interferometers, where the incoming wavefront is replicated and made to interfere with itself. One such interferometer, the Fourier interferometer (Fig. 29) was introduced by MERTZ[ 19651 and KONJAEV[ 19671. Incoherent (although quasi-monochromatic) light from the input plane is collimated by a collimator lens, so that the rest of the system sees the input plane at infinity. The light then goes into a modified Michelson interferometer (see Fig. 1 for comparison). In this modified Michelson interferometer the two mirrors are replaced by roof prisms. One prism is rotated by an angle a (the rotation axis is perpendicular to the nonreflective face of the prism); a may take any value within the range 0 < a < (42). Light that was reflected from both prisms is recombined by the beam splitter and then detected at the output plane. For simplicitylet us assume, for now, the simplest object of all:a single point source, located at coordinates ( x , y ) in the input plane. The collimator lens has a focal length F so it converts the position ( x , y ) into a collimated beam at direction (u, u), where (for x < F and y 4 F) u = x/F and u = y/F. If this
Fig. 29. A Fourier interferometer.
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PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
459
collimated beam is reflected by a plane mirror, its new direction would become ( - u, - u). When, however, this beam is reflected by a roof prism, as shown in Fig. 30a, its new direction would be (u, - u), where u is the direction of the vertex of the roof prism. It is convenient to use a beam-angle diagram, as shown in Fig. 30a, where each beam is shown as a point in the (u, u) mathematical plane. We shall use the vector p for the values of u and u of the incoming beam and use the vector rp for the reflected beam
Now, let us find what the dflerence is between the directions of the reflected waves from the two roof prisms (Fig. 29). In Fig. 30b one reflected beam, , p , is given by eq. (66), but the other beam, reflected by the rotated prism and designated ,.p, must be calculated in the tilted coordinate system ( u ' , u'). It is convenient to use vector notation for these calculations; eu is a unit vector in the direction of the u axis and e,. is the unit vector of the u' axis. We can now
angle diagram
0
incoming outgoing
1
' P
Fig. 30. Beam redirection by roof prisms: (a) a single roof prism (top and side view) and its beam-angle diagram; (b) the beam-angle diagram for the Fourier interferometer of Fig. 29. Because of the beam splitter and the tile a of one of the proof prisms, we get two reflected wavefronts for each incoming wavefront.
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[V,§ 2
We now calculate the components of A r p in the (u, u ) plane explicitly. We substitute e,. = [shu]
e, = [ 3 ,
cos a!
and get
][
A r p = I s i n a ! [o 1 1 0
coscl s i n a ! ] t ] -sins cosa!
The first matrix is a reversal matrix, whereas the second one is a rotation matrix. Neither one changes the length of the vector they operate on. The 2 sin cl factor gives the ratio between \ A.p I and I p 1 . Now, since both beams came originally from the same input point, assuming that their path difference is small enough, they can interfere. Their interference pattern is ~(X,,t,Y0"t)
= 1-
+ 1-
9
(68)
where I-(xout, You,) a cos[24k*Xout) + 91 and
Yout
In eq. (68) 1- is some constant bias and 9 is the phase (path) difference between the two interferometer arms. In the more general case the input plane has some irradiance distribution g(x, y). Each point at the input phase is coherent only to itself, so cross-terms contribute only to I - . We generalize eq. (68), getting
'r].
where we recall that
p =
[:I
=
F Y
This equation (except for the constant term I - ) has the form of a Fourier
v 7
§ 21
46 1
PHYSICAL APPROACHES TO NONCOHERENT OPTICAL PROCESSING
relation. Indeed, if we set $ = 0, we get the real part of the Fourier integral, whereas for $ = - 4 2 we get its imaginary part. The interferometry can also be modified so it implements the Hurtley trunsform, as shown by BRACEWELL, BARTELT,LOHMANNand STREIBL[ 19851. The Hartley transform is defined as
1 1
00
H M x , Y>>=
-00
g(x, Y ) cas [24 f x x + f,Y)I d x dY
9
(70)
-00
where cas[p]
=
sinp + cosp = J z c o s ( b
+ an).
So if we set $ = n/4 in eq. (69), we get the Hartley transform. One particular advantage of the Fourier interferometer over other Fourier transform systems is the ease with which we can “zoom” through the scale factor of the Fourier space. We can change a in eq. (67d) from almost zero to 4 2 continuously, affecting a scale range of a few hundreds to one. On the other hand, the Fourier interferometer, like most interferometers, is sensitive to vibration and misalignments; $, the phase, must be kept within a few cycles and be stable within about n/50 - an optical path difference no larger than a few pm and sub-pm stability; this precision requires a stable work area and devices such as piezoelectric transducers.
2.7. DIRECT, PARALLEL, VECTOR-MATRIX MULTIPLICATION
GOODMAN, DIASand WOODY [1978] demonstrated how simple it is to produce a noncoherent system that carries out fully parallel, one-dimensional linear transformations. Figure 3 1depicts one variation of this idea. A line array of sources (for example, LEDs) is imaged by a cylindrical lens (the “vertical” lens in the figure) onto a transparency, or mask. This lens provides imaging only in they dimension; the light distribution impinging on the mask is independent of x. If the intensity of the source located at yk is gk and the transmittance of the musk at (x, ,y k ) (note that the axes x and y in different planes, in the figure, follow the inversions caused by the lenses) is z , , ~ , there is an irradiance distribution I,,kcoming out from the mask,
The second lens now images the x axis and smears they axis; a detector at x,
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
-,
input source array
[V, 8 2
output detector. array
Fig. 3 1. The Goodman vector-matrix multiplier.
would see irradiance distribution of Id,,,m , Idet, m
cc
Im,k = k
Tm,kgk
9
k
which is precisely the vector-matrix product of the vector g s ( g k ) with the matrix [TI = (z,,~). The performance of the Goodman vector-matrix multiplier is limited only by the resolution obtainable with cylindrical lenses and by the available source and detector arrays. Cylindrical lenses are easier to design than spherical lenses because they have no “skew” rays. They are, however, more difficult to manufacture and assemble with high precision. It is possible also to obtain the one-dimensional smearing needed for the Goodman scheme without using cylindrical lenses at all. For example, it is possible to use a stack of planar waveguides (microscope cover glasses immersed in a high-index liquid, for example) or several linear fiber bundles as shown by PERIand GOODMAN [ 19791. Such arrangements tend to be more compact than the cylindrical lens version.
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463
4 3. Bipolar and Complex-valued Spatial Signals Incoherent light is linear in irradiance. For information processing this is unfortunate because irradiance is (unlike the complex amplitude) real and non-negative. Bipolar-real, and sometimes complex-valued, spatial signals are common to many processing algorithms. Even a simple problem, such as, for example, restoration of linear-motion blur, calls for convolution with a bipolar function. Several suggestions have been made on how to represent a bipolar or complex function by a real function or functions. (In fact, this is the basic problem in holography.) Several basic approaches include the following: (1) Intensity: addition of constant level (dc bias) that is larger than the absolute value of the most negative data (ROGERS[ 19771). This method is not directly usable for complex valued data but can be combined with the other methods for use with complex-valued data. (2) Temporal encoding, where the time dimension is shared, somehow, between several non-negative spatial signals that are used to represent bipolar-real or complex-valued signals. ( 3 ) Space encoding: use of several non-negative channels. For example, bipolar data can be presented with negative and positive parts. These logical channels also can be physically separate, or they can share a single physical channel. (4) Frequency encoding: use of spatial carrier, similar in a sense to that of off-axis holography (LOHMANN [ 19771). Method 1 (dc biasing) is usually considered to be a bad choice; adequate dynamic range (signal-to-noise ratio) is harder to achieve in analog systems than bandwidth, so that it is more appropriate to use methods which employ bandwidth rather than dynamic range. Both methods 3 and 4 use some of the bandwidth (either by use of several processing channels or by sharing one channel) to gain bipolar or complex-valued capabilities. Each method uses this bandwidth differently and has different requirements on input pre-processing, intermediate function representation, and post-processing. Since many optical processing methods do not really use the available temporal bandwidth of optical processing, method 2 seems to be most attractive. Unfortunately, there are technical difficulties in its practical implementation. 3.1. MULTIPLE CHANNEL SYSTEMS
A simple way of representing some bipolar spatial signal g(x,y) with non-negative real signals is to define two signals g + (x, y ) and g - (x, y), such that
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For complex-valued signals two pairs of non-negative channels (four channels) can be defined. However, as shown by GOODMAN and WOODY[ 19771, it is possible to use three, rather than four, non-negative real signals for representing a complex-valued one:
To use this multi-channel spatial signal systems, we do the desired mathematical operation on the component signals using an incoherent optical system, then combine the results to obtain the desired bipolar, or complex-valued, output. We now assume that both input spatial signals, g(x, y ) and h(x, y), are bipolar or complex valued. The case when one input is non-negative real will be treated as a special case. Addition: s = g + h and subtraction: d = g - h
(Note that subtractions are possible with incoherent light only with the help of electro-optical devices such as SLMs.) Multiplication:p = g x h
(75b')
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Equations (75c) require only six multiplications, whereas eqs. (75b) require nine. If the “cost” of multiplications is significantly higher than that of additions/subtractions, this is a significant saving. Convolution: c = g * h =g+* h + +g- * h - , =g+* h - +g- * h + ;
C+ C_
co = go * ho + g, c1 = go * h , + g, c2 = go * h2 + g,
* hz + g2 * hi * ho + g2 * hz * h , + g2* ho . 9
9
Again, we can use the GOODMAN and WOODY [ 1977a,b] shortcut, go,1 = go + g,
ho,1
=
g0.2 = go + gz 9
7
ho + h 1 ,
ho,z = ho + h2
,
g1.2 = g1 + g z 9
hi,z
=
hi + hz ;
then
* h1.z + go * ho - h1 * g1 - hz * gz * h0,l - go * h, - h, * &?I + h2 * gz c2 = g0.2 * h 0 , Z - go * ha + hl * g1 - h2 * gz. co
= g1,2
9
c1
= go,1
Y
TWOother operations, namely division and correlation, can be regarded as special cases of multiplication and convolution, respectively. 3.1.1. Optimal choice of the component spatial signals
It is easy to see that the representations of eqs. (72) and (73) can be made unique by the condition that g+g- = 0 or gog,g2= 0, respectively. This condition, however, often results in the spatial bandwidth of the component spatial signals being larger than that of the original spatial signal. For example, the spatial bandwidth of sinx is much lower than that of (sinx), = (sinx + Isinx1)/2. This general property of non-negative signals was acknowledged by LUKOSZ[ 19621. MAIT[ 19841 related this problem to this specific issue. Thus, unless we have an abundance of spatial bandwidth, the choice of g+
g-
={ ={
if g > o ,
g,
0 , otherwise,
0,
if
g>o,
- g , otherwise
,
(77)
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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):
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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
BIPOLAR AND COMPLEX-VALUED SPATIAL SIGNALS
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[ 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].
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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,).
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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
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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.
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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
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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,
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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.
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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
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- 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
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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.
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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.
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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,
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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
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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].
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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
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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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
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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].
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
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
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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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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.
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INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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:
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
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
INFORMATION PROCESSING WITH SPATIALLY INCOHERENT LIGHT
[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<1/2, PNOR(X) =
0 , otherwise .
The two binary input lines to the gate are summed to yield a single, three-level signal whose value is equal to the number of input lines with binary value 1. This signal then goed to the nonlinear function, which produces a binary output. In an optical implementation the binary value 0 is represented by a low intensity and the value 1 is represented by a high intensity. The addition can be done by optically superimposing the signals on the input lines. With a detector that integrates the signal over the input spot, this can have the effect of adding the intensity levels regardless of the coherence properties of the light. We note that arbitrary binary circuits can be implemented using many gates of one type, such as NOR gates. The routing (or wiring) part can then be implemented by a linear transform. This is followed by a point-nonlinear transform to implement, many times, eq. (105). Optical logic systems based on complex amplitude, phase, and polarization have been suggested.
v, I 41
APPLICATIONS OF INCOHERENT PROCESSING SYSTEMS
BEAM
501
DEVICE
LENSLET ARRAYJ
IINTERCONNECTIONS
MASK
Fig. 42. The digital lenslet array processor ( D I L A P ) . The back of the nonlinear device (for example, an SLM or an array ofphotodiodes and LEDs) is imaged via mirrors by the lenslet array into a mask. The mask selects which pixels from the back (output) of the nonlinear device would be visible to each of the pixels at the front (input) side. The nonlinear device then produces the and PERELMUTTER [ 19861. desired logical operation. From GLASER
In a recent work by GLASER and PERELMUTTER [1986] a lenslet array processor, as described earlier in this article, was fitted with a mask that provided the routing necessary to carry out a “one bit full adder” using NOR gates. To obtain real-time operation, one can add some feedback mirrors and an electro-optical threshold device, as shown in Fig. 42.
Shadow casting methods for digital processing A particular combination of incoherent optics and digital processing that received appreciable attention is the use of shadow casting for digital logic. As noted, for example, by GERRITSEN and VERBER [ 19841, digital cellular automata, a processing architecture suitable for many image processing problems, can be easily implemented using convolution with a nonlinear operation. Cellular machines are computers that are made of an array of small, relatively primitive, processing elements (PE) rather than a single (or few) central processing unit (CPU) as in conventional digital computers. Using a similar line of thought, several groups worked on convolution-based digital processors, where the convolution is done with a shadow casting configuration. Because the three planes in the simple shadow casting system of Fig. 14A can be very close to each other, adequate resolution with visible
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[V,§ 5
(and even near IR) light is possible despite diffraction. MAURE[ 1972J describes [ 1983a,b] ageneral purpose system based on these lines. TANIDAand ICHIOKA and KOZAITISand ARRATHOON[1985] showed how a variety of more complex logic operations can be implemented with shadow casting.
5 5. Concluding Remarks In the Introduction we noted that for some potential applications of signal and image processing no available digital electronic computer is sufficient. These computers are too slow and their answer would come when it is no longer of use. Other applications have been demonstrated in experimental laboratories on digital computers, but despite all recent rapid progress the bulk, complexity, and cost of suitable electronic hardware would postpone their practical use for quite a while. Optical processors that try to solve the first group of problems are, necessarily, all out machines. Those which try to solve problems of the second group must compete against available or about-to-be-introducedsilicon technology. Very clearly, a different research philosophy should be applied to each of these groups. Work on optical solutions to the first, “impossible” class of problems is naturally more glamorous. Typically techniques such as bistable nonlinear optical devices and coherent light are used. In contrast, attempts to compete with more down-to-earth problems should use that part of optics which is mature and well understood, that is, incoherent optics. Throughout this chapter it was seen how complicated mathematical operations can be executed rapidly with very simple systems, using many optical principles with which Huygens himself would have probably felt at home. The simplicity and availability of incoherent optical processors are what makes them so attractive for applications where reliability and reasonable cost must be combined with very high performance. Another advantage of incoherent signal/image processors is interfacibility -they use the same kind of light that is available from a multitude of sources. When the system input is an outside image, it can be entered directly to many incoherent optical processors, and with simple, reliable interface such as a CRT to others. In two papers listed in the bibliography two respected workers disagree about the future of optical information processing. KEYES[1985], on one hand, claims that optics is doomed “in the light of computer technology”; as optics tries to catch up, silicon technology will keep advancing ahead. LOHMANN
VI
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503
[ 19831 on the other hand, does not seem to share that spirit. Who is right? One
should read the fine letters in both papers. Keyes’ optical computing is different from that of Lohmann. Although only time will tell whether Keyes’ pessimism regarding coherent optical switching is justified or not, processing schemes that exploit primarily the extreme parallelism of optics have a clear advantage over hard-wired electronics. Such optical systems are already in use for a few applications, and we are likely to see more of them in the future.
Acknowledgment The author is an incumbent of the Sir Charles Clore Career Development Chair.
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AUTHOR INDEX A AARTSMA, T.J., 74, 99 ABBE,E., 441, 503 ABEBE,M., 141, 160 ABITBOL,C., 154, 155, 159 ABRAMOWITZ, M., 246, 378 ABRAMSON, N., 450, 503 AGARWAL, G.S., 64,65,66,99,230,231,233, 234, 235, 236,239, 387 AGRAWAL, G.P., 53, 99, 138, 139, 163, 269, 336, 340,341, 342, 343, 344, 352, 357, 362, 378, 382, 383 AKHIEZER, N.I., 202, 215, 231, 378 ALCOCK,A.J., 144, 159 AL-JUMAILY, G.. 144, 145, 147. 162 ALLARD,N., 63, 79, 99 ALLEN,L., 230, 378 AL’TSHULER, G.B., 302, 378 ALUM,KH.P., 146, 159 ANAFI,D., 270, 328, 329, 345, 347, 350, 378, 384, 386 ANAN’EV,Yu.A., 168, 189, 196,270,274,276, 287, 288, 321, 329, 340, 354, 378, 379 ANDERSON,D.B., 26, 36 ANDERSON,D.Z., 118, 120, 161 ANDREWS,F.A., 161, 123, 160 ANDREWS, J.R., 99,44,68, 70, 72, 73, 74, 75, 76, 101 AOKI,H., 17, 36 APANASEVICH, P.A., 75, 99 ARCESE,P.H., 481,482,485, 503 ARIMOTO,A., 153, 164 ARITOME,H., 17, 36 ARMITAGE, J.D., 503 ARMSTRONG, J.A., 42.47.49, 99 ARNAUD,J.A., 175, 179, 196, 379 ARNAUTOV, G.P., 112, 159 ARNETT,K., 65, 100 ARONOWITZ, F., 138, 159 ARQUIE,L., 153, 159
ARRATHOON, R., 167, 189,244,269,304,354, 385, 498, 502, 503, 506, 507 ARSENAULT, H.H., 109, 162,487, 503 ASHBY,D.E.T.F., 125, 126, 159 ASHLEY,P.R., 32, 36 AUGUST, R.R., 26, 36 AVIZONIS, P.V., 250, 257, 258, 260, 262, 263, 265, 269, 354, 380, 383
B BACHER,G.D., 151, 161 BAIRD,K.M., 110, 156, 159 BALASHOV, I.F., 214,215, 216, 217, 379 BALHORN,R., 108, 159 BALLIK,E.A., 107, 162 BALSAMO,S.R., 139, 161 BAR-JOSEPH,I., 150, 160 BARKER,L.M., 121, 160 BARONE,S.R., 168,249, 379 BARRETT,H.H., 399,407,408,409,410,412, 413,421,439,440,469,492,493,496, 503, 504, 505, 506, 507, 509 BARSACK,E., 496, 509 BARTELT,H.O., 436,437,438,461,504 BASOV,N.G., 150, 160 BASSAN,M., 124, 160 BAUES,P., 179, 379 BAY,Z., 155, 160 BAYLIS,W.E., 78, 101 BEKEFI,G., 126, 162 BBLANGER,P.A., 288,386 BEL’DYUGIN, I.M., 287,288,289,292,296,379 BELIC,M., 344, 352, 382 I.M., 270, 379 BELOUSOVA, BENNETT,S.J., 108, 160 BERENBERG, V.A., 214,215,216,217,379 BERGER,F.B., 421, 504 BERGER,J., 111, 160 BERGH,R.A., 141, 160
512
AUTHOR INDEX
BURCH,J.M., 172, 173, 174, 177, 182,381 BERGSTEIN, L., 168,250,257,290, 379 BERMAN, P.R.,49,61,62,78,92,93,96,97,99,BURES,J., 143, 162 BURNS,W.K., 141, 160 101 BUROW,R., 131, 163 BERNABE, M.L., 260,328,329, 383, 385 BUSH,K.A., 196,288,300,311,314,317,323, BERTOLOTTI, M., 152, 160,230,240,379 325, 384 BETZ,A.L., 134, 135, 136, 162, 164 BUTCHER,P.N., 43,99 BHUSHAN,B., 131, 160 BUTTER,C.D., 143, 160 BILLING,H., 119, 120, 160, 163 B u n s , R.R., 250,257,258,260,262,263,264, BIRCH,K.G., 129, 160 269,354,380,383 BJORKHOLM, J.E., 57,99 BLASZUK,P.R., 270, 387 BYER,R.L., 158, 160 BLOEMBERGEN, N., 41,42,43,44,45,47,48, 49,51, 52,58,67,69, 70,71,81,82,83, 84, C 85,86,87,88,90,91,92,93,94,95,99,100,CABRERA,S.D., 506 101 CADWALLENDER, W.K.,114, 162 BLOK,H., 227,379 CAHILL,R.F., 141, 160 BLUESTEIN, L.I., 496,504 CAMUS,J., 12,36 BOCKER, R.P.,405,415,416,417,419,504,508 CARLSTEN, J.L., 58,99 Bo~~~~,A.R.,44,51,52,58,69,70,71,82,87, CAROME,E.P., 143, 160 99, 101 CARRB,P., 130, 160 BOND,W.L., 107, 162 CARREIRA, L.A., 43,99 BORD~, CH.J., 52,92,93, 100 CASASENT,D., 484,505 BORN,M., 4,36,171,177,178,222,231,379, CASON,C., 322,380, 384 394, 504 CASPERSON, L.W., 288,380 BOSSEL,H.H., 123, 160 CATHEY,W.T., 114, 160 BOSTROM, R.C.,1 1 1, 164 CAULFIELD, H.J., 419,435, 504 BOUCHIAT, M.A., 86,99 CEGLIO,N.M., 17,37 BOULANGER, Yu.D., 112, 159 CEREZ,P.,109, 160 BOURDET,G.L., 111, 160 CERVANTES, M., 146,147, 162 BOYD,G.D., 167,216,379 CHABAY, I., 43, 100 BOYD,J.T., 26, 36 CHABRAN, C., 154, 155, 159 BOYD,R.W., 55,63,80,95,99,100 CHALASINSKA-MACUKOW, K., 487,503 BRACEWELL, R.N., 461,504 CHAMPAGNE, E., 13, 36 BRAGG,L., 424,504 CHAMPAGNE, E.B., 450,504 BRANGACCIO, D.J., 130, 160 CHANG,T.H.P., 17,36 BRAUNECKER, B., 504 CHANG,W.S.C., 32, 36 BRILLET,A., 109, 116, 160 CHAW,H.H.M., 13, 36 BROMLEY, K.,405,415,416,417,419,504,508 CHAVEL,P.H., 395,504,507 BROOKS111, E.D., 120, 161 CHEN,L.W., 250,269,380 BROOME,B.J., 424,508 CHENAUSKY, P.P., 195,196,230,266,270,328, 380, 381 BROSSEL,J., 86,99 CHERNEN’KII, V.I., 229,380 BROWN,N., 108, 109, 111, 160, 164 BROWN,R.M., 106, 161 CHERNOV, V.N., 270, 321,379 CHESTER,A.N., 269,321, 354,355, 380,384 BRUNING, J.H., 130, 131, 160 BRYNGDAHL, O.,149, 160 CHIN,T.,506 CHIU,M.Y., 439,440,469,503, 504,505 BUCARO,J.A., 142, 143, 160, 161 BuczEK,C.J., 195,196,230,266,270,380,381 CHO, S.H., 250,260,269,380 CHODZKO,R.A., 221,270,328, 380 BUHOLZ,N.E., 111, 161 CHOW, c., 504 BUINOV,G.N., 14,36 CHOW,W.W., 139, 160 BULABOIS, J., 504
AUTHOR INDEX
CIDDOR,P.E., 108, 111, 160 CILIBERTO, S., 124, 160 CIRKEL,H.J., 229, 380 CLARK,R., 12, 36 CLUNIE,D.M., 125, 160 COANE,P.J., 17, 36 COCHRAN, J.A., 202,203, 380 COFFEY,B.J., 344, 352, 382 COLE,J.H., 142, 143, 161 COLLIN,J.H., 335 COLLIN,R.E., 380 COLLINS,S.A., 179, 380 CONNORS, C., 115, 162 COOK,D.D., 152, 160 COOK,H.D., 112, 161 COOPER,J., 54, 64, 65, 99, 101 COPSON,E.T., 244, 380 CORKE,M., 143, 162 COTTRELL, T.H.E., 168, 386 COURANT, R., 202,276, 380 COURJON,D., 504 CRAMER, S., 43, 100 CRANE,R., 114, 129, 131, 160, 164 CRANEJR., R., 130, 161 CREATH,K.,153, 160 CRONIN-GOLOMB, M., 151, 152, 162, 164 CROSS,E.F., 270, 328, 380 CRUZ,C.H.B., 260, 381 CULSHAW, B., 143, 161 CULSHAW, W., 112, 160 CUMMINS, H.Z., 120, 164 D DAGENAIS, M., 44, 51, 52, 58, 68, 69, 72, 99, 101 DAHLQUIST, J.A., 112, 160 DAHLSTROM, C.E., 109, 161 DAINO,B., 230,240, 379 DALLAS,W.J., 444, 504 DANDRIDGE, A,, 142, 143, 160, 161, 162 DANILOV, O.B., 270, 379 DAN’SHCHIKOV, E.V., 270, 380 DARBY,R.A., 123, 160 DARDY,H.D., 143, 160 DAVIES,D.E.N., 143, 161 DAVIS,D., 138, 163 DAVIS,J.L., 141, 160, 161 DAVIS,R.L., 26, 36 DAVIS,W.C., 114, 160 DAY,G.W., 110, 161
513
DE ARA~JJO, C.B., 53, 101 DE BREE,P., 74, 99 DE KOO, N.P., 227, 379 DE SOUZA,P.D., 143, 160 DEBRUS,S., 439, 505 DECOLA,P.L., 76, 101 DEFERRARI, H.A., 123, 160 DELONE,N.B., 46, 99 DEMAREST, F.C., 122, 164 DEMEESTER,G.D., 493,496, 503, 509 DEMERS,J.G., 288,382 DEMTRODER, W., 157, 158, 161, 162 DENIS,H., 35, 37 DENTE,G.C., 221,227,260, 328, 380, 382 DEVIJVER, P.A., 480, 505 DEYOUNG,T.F., 114, 162 DIAS,A.R., 439,461,491, 506 DICK,B., 46, 52, 65,66,72,76,95,96,99 DICKE,R.H., 82,93, 100, 101 DIRAC,P.A.M., 42, 100 DLOTT,D.D., 80, 101 DOHERTY,V.J., 131, 160 DOLASH,T., 484, 508 DORBAND, B., 131, 161 DORSCHNER,T.A.,139, 140, 161, 164 DOWNER,M.C., 44,69, 70, 71, 82, 87,99 DREVER,R.W.P., 118, 120, 161 DRUET,S.A.J., 42,52,54,55,92,93, 100, 101 DUCUING,J., 42,41,49,99 DUFFY,R.M., 108, 160 DUKES,J.N., 112, 113, 161 DUNCAN, M.D., 158, 160 DUNN,M.H., 336, 383 DUPPEN,K., 66, 101 DURRET,R.H., 229, 381 DURST,F., 121, 161 DUTTA,N., 64, 100 DYMSHAKOV, V.A., 270, 380 DYSON,J., 112, 161
E EASTONJR., R.L., 496, 505, 509 EBERHARDT, F.J., 123, 161 EBERLY,J.H., 64, 100 EESLEY,G.L., 53, 101 ELLIOTT,D.S., 65, 100 ELSSNER,K.-E., 131, 163 ENGEL,A., 12, 36 ENOMOTO, S., 23, 36 ERDBLYI,A., 244, 380
514
AUTHOR INDEX
ERICKSON, E.F., 106, 161 ERICKSON, J., 506 ERKILLA, J.H., 328, 384 EVENSON, K.M., 110, 161 EWART,P., 68, 86, 100 EZEKIEL,S., 139, 140, 141, 160, 161, 163
F FADER,W.J., 329, 380, 384 FALK,J., 329, 385 FALLER,J.E., 112, 164 FAN,G.J., 230, 380 FARHAT, N.H., 490,505 FATEEB,V.A., 12, 36 FAYER,M.D., 80, 101 FEDOSEJEVS, R., 229, 386 FEINBERG, J., 151, 161 FEIOCK,F.D., 355, 380 FEIT, M.D., 260, 269, 381 FELSEN,L.B., 250,260,269,322,354,380,385 FERGUSON, J.B., 108, 161, 163 FERGUSON, T.R., 221,227, 381 FERRIERE, R., 15, 36 FERRIS,R., 424 FESHBACH, H., 202,235,383 FINK,D., 221,227, 381 FIRESTER,A.H., 4.36 FIRTH,W.J., 191, 196, 387 FISCHER,A., 158, 161 FISCHER,B., 151, 164 FISHER,A.D., 114, 161 FISHER,R.A., 42, 100 FLANDERS, D.C., 17, 37 FLECKJR., J.A., 260,269, 381 FLUDE,M.J.C., 112, 161 FLYTZANIS, C., 49, 100 S.M., 421, 505 FOMENKO, FORD,G.M., 120, 161 FORGHAM, J.L., 187,276, 277, 278, 383 FORWARD, R.L., 118, 161, 163 FOSTER,M.C., 270, 387 FOURNIER, A., 35, 37 Fox, A.G., 167, 180, 354, 355, 381 FRADIN,D.W., 270, 328, 381 FRANCON, M., 439, 505 FRANKEN, P.A., 42, 100 FRANTZ,L.M., 131, 161 FREED,C., 109, 161 FREEMAN, R.H., 270, 328,329, 345, 378, 386
FREIBERG,RJ., 195, 196, 230,266, 270, 328, 380, 381 FRIEDMANN, H., 66, 100, 101 FROMZEL',V.A., 196, 383 FUJIMOTO, J.G., 54, 55, 57, 59,95, 100, 101 FUJITA,T., 4, 16, 17, 36 S., 35, 37 FUJIWARA, FURMAN,A,, 484,505
G GABOR,D., 490,492, 505 GALLAGHER, A., 79, 101 GALLAGHER, J.E., 130, 160 GALLION,P., 154, 155, 159 GAMBLE,T., 484, 508 GAMO,H., 192,206,208,210, 212, 213, 381, 386 GANIEL,U., 288,381 GARCIA,H.R., 270,329, 345, 386 GARDNER, J.L., 158, 161 S., 230, 378 GATEHOUSE, GATZKE,H.W., 340, 383 GAY,J., 134, 161 GAYLORD, T.K., LO, 36 GELUK,R.J., 402,410,411, 505 GERARDO, J.B., 126, 161 GERCK,E., 260, 381 GERRARD, A., 172, 173, 174, 177, 182, 381 GERRITSEN, F.A., 501, 505 GIACOBINO, E., 96,99 GIALLORENZI, T.G., 141, 142, 143, 160, 161, 162 GIANINO,P.D., 506 GIERULSKI, A., 77, 101 GILLARD,C.W., I l l , 161 GILLES,I.P., 143, 161 GILLILAND, K.E., 112, 161 GINDI,G.R., 408, 505, 507 GIRARD,F., 12, 36 GLASER,I., 424,425,426,430,431,433,434, 447, 448,451,452, 453,455, 471,472,475, 476,483,484,485,486,489,495,496, 501, 505, 506 I.M., 202,215,231, 378 GLAZMAN, GMITRO,A.F., 408,469, 504, 505, 506 GODON,P., 35, 37 GOEDGEBUER, J.P., 15, 36 GOODMAN, J.W., 186,355,381,396,398,422, 428,439,461,462,464,465,471,491,495, 505, 506, 508
515
AUTHOR INDEX
GORDON, G.B., 112, 113, 161 GORDON, J.P., 216, 379 GORDON, S.K., 108, 161,439,440, 503, 505 GORI,F., 230,240, 379, 381 GORLITZ, D., 477, 506 Goss, L.P., 43, 99 GOTO,K., 16, 36 GRANEK, H., 270, 381 GREEN,H.S., 169, 381 GREIVENKAMP, J.E., 505, 506 GRIESER, D., 484, 508 GRINBERG, J., 420, 508 GRINDI,J., 493 GRISHMANOVA, N.I., 270, 276, 288, 379 GROSSO,R.P., 130, 131, 161, 164 GROUILLET, A.M., 35, 37 GRYNBERG, G., 53, 58, 59, 60, 61, 77, 100 GRZANNA, J., 131, 163 GUHA,S., 329, 385 GUILFOYLE, P.S., 419, 500, 506 GURSEL, Y., 118, 120, 161 GUSTAFSON, T.K., 41, 54, 55, 101 H HALL,J.L., 108, 116, 156, 157, 160, 161 HALMOS, M.J., 154, 161 HALMOS, P.R., 212, 381 HAMBENNE, J.B., 139, 160 HAMILTON, M.W., 65, 100 HANES,G.R., 109, 161 HANLE,W., 77, 100 HAPPER, W., 86,91, 100, 101 HARDY, A,, 150, 160, 196, 288, 381, 386 HARIHARAN, P., 128, 129, 131, 132, 161 HARTMANN, S.R., 68, 96, 100 HATAKOSHI, G., 16,26, 32, 36 HAUCK,R., 504 HAUS,H.A., 140, 161 HAWKINS, R.T., 156, 157, 162 HAYES, C.L., 114, 160 HAYES,J., 153, 161, 328, 381 HAYSLETT, C.R., 129, 162 HECKENBERG, N.R., 126, 161 HEDIN,V.A., 187,276,277,278, 383 HELLER, D.F., 506 HELLWARTH, R.W., 141, 161 HENRY, C.H., 107, 161 HERCHER, M., 106, 162 HERELD, M., 118, 120, 161 HERMAN, G.T., 409, 506
HEROLD,H., 126, 162 HERRIOTT, D.R., 107, 130, 160, 162 HERZIGER, G., 12,36 HESSELINK, W.H., 74, 100 HETHERINGTON, W., 43, 100 HEURTLEY, J.C., 210,213, 381 HILBERT, D., 202,276,380 HILL,A.E., 100 HILL,L.W., 42 HILLER,W.J., 123, 160 HILLMAN, L.W., 63,95, 100 HIRSCH,K.M., 4, 12, 36 HIRSCH,P.M., 444,507 HOCHSTADT, H., 202,204, 381 HOCHSTRASSER, R.M., 44,46,52, 65, 66,68, 70, 72, 73, 74, 75, 76, 95, 96, 99, 101
HOCKER, G.B., 143, 160, 162 HOFFMAN, P., 328, 381 HOGGAN,S., 118, 120, 161 HOLLENBACH, R.E., 121, 160 HOLLY,S., 130, 163 HOLMES, D.A., 329, 385 HOLTZ,M., 139, 140, 161, 164 HOOPERJR., E.B., 126, 162 HOPF,F.A., 144, 145, 146, 147, 149, 162 HOPFIELD, J.J., 490, 506 HOPKINS, G.W., 127, 163 HOPKINSON, G.R., 112, 162 HORNER,J.L., 506 HORRIDGE, G.A., 423,506 HORWITZ, P., 250,251,252,253,254,256,260, 261,262,266,269,
322, 381
HOUGH,J., 118, 120, 161 HOUZEGO, P.J., 17, 36 HSU, Y.-N., 487, 503 HUANG,C.-C., 131, 162 HUBER,D.L., 58, 100 HUDSON,B., 43, 100 HUIGNARD, J.P., 151, 163 HUNSPERGER, R.G., 26, 36 HUNT,B.R., 506 HURWITZ, H., 221,381 HUTCHINGS, T.J., 139, 160,229, 381
I ICHIOKA, Y., 502, 506, 509 IGA,K., 4, 36 ILLUECA, C., 15, 36 IMAI,M., 142, 162
516
AUTHOR INDEX
INDEBETOUW, G., 403,468,485, 506 INGELSTAM, E., 105, 162 INOHARA, S., 14, 36 INOUE,H., 26, 36 ISAEV,A.A., 270,276,381, 387 ISHIDA,Y.,42, 53, 80, 101 ISYANOVA, E.D., 302, 378 ITOH,K., 124, 125, 142, 163, 164
J JACKSON, AS., 427, 506 JACKSON, D.A., 142, 143, 162 JACKSON, D.J., 95, 100 JACOBS,E.D., 229, 381 JACOBS,S.F.,108, 115, 116, 168 JADOT,J.P., 35, 37 JAHODA,F.C., 126, 162 JASUA,T.S., 116, 162 A,, 107, 109, 116, 157, 161, 162, 163 JAVAN, JEPHCOTT,D.F., 125, 126, 159 JOHNSON,G.W., 114, 162 JOHNSON, M.A., 134, 135, 162 JONES,D.G.C., 230, 378 JONES,R.C., 221, 225, 382 JONES,R.E., 143, 164 JONES,R.W., 322, 380 JORDAN JR., J.A., 4, 12, 36,444, 507 T.F., 202, 382 JORDAN, JORTNER, J., 65, 101 JOURNET,A., 134, 161 JUNCAR,P., 158, 162 K KACHRU,R., 68, 96, 100 KAFRI,O., 506 KAHANE,A,, 157, 162 KAHN, W.K., 167, 181, 249, 382 KALISH,E.N., 112, 159 KAMEGAI, M., 122, 162 KAMIYA, T., 15, 16, 36 KARAMZIN, Yu.N., 269, 352, 382 KAWSEV, V.B., 302, 378 KARBOWIAK, A.E., 215, 382 KATZIR,Y.,150, 160,475,483,484,485,486, 489,495,496, 505, 506 KAZARYAN, M.A., 270,216, 381, 387 KELLER,J.B., 258, 382 KELLMAN, P., 405, 506
KELLY,D.H., 506 KELLY,M.J., 157, 163,477 KERN,D.P., 17,36 KERR,I., 120, 161 KERSEY,A.D., 143, 162 KES, P.H., 95, 100 KEYES,R.W., 502,506 KHALEEV, M.M., 196, 383 A.M., 227,229, 387 KHROMYKH, KIELKOPF,J., 63, 79, 99 KILLPATRICK, J., 138, 162 KINGSLAKE, R., 437,456, 506 KIST, R., 109, 164 KIT, I.E., 14, 36 KLAUMINZER, G., 43, 100, 101 KLINGER,J.H., 112, 162 KNOX,J.D., 109, 162 KOBAYASHI, K., 30,32, 37 KODATE,K., 15, 16, 36 H.,10,36,167,182,365,379,382 KOGELNIK, KOLIOPOULOS, C.L., 131, 160 KONEV,Yu.B., 269, 352, 382 KONJAEV, K.V., 458, 506 Koo, K.P., 143, 162 G., 167,229, 382 KOPPELMAN, KORONKEVITCH, V.P., 12, 36, 112, 159 KORZHENEVICH, I.M., 288, 382 KOSUGE,K., 14,36 KOVAL'CHUK, L.V., 189, 354, 379, 382 KOVAL'CHUK, Yu.V., 146, 159 KOWALSKI, F.V., 156, 157, 162 KOYAMA, J., 4, 14, 16, 17, 30, 32, 35, 36, 37 KOZAITIS,S., 502, 506, 507 KRAINOV, V.P., 46,99 KRAMER,S.D., 43, 100 KRASINSKI, J., 63,95, 100 KREUZER, J.L., 270, 328, 383 KRUPKE,W.F., 183,270, 321, 382 KUJOORY, M.A., 493, 507 KULLMER, R., 158, 161 KUNASZ,C.V., 64,99 KUNG,H.T., 419, 507 KUNZMANN, H., 108, 159 KURNIT,N.A., 157, 163 KUROSAWA, K., 142, 164 KUSHNIR,V.R., 198, 382 KUWAYAMA, T., 14, 36 KWON,O., 129, 162 KWONG,S.-K., 151, 152, 162, 164 KYUMA,K., 143, 162
AUTHOR INDEX
517
L LOUIE,A.C.H., 405,415,416,417, 504, 508 LA TOURETTE, J.T., 168, 382 LOUISELL,W.H., 169,269,340,344,352,357, LACROIX,S., 143, 162 382, 383 LAM,J.F., 80, 85, 100 LOVBERG, R.H., 111, 160 LAMBJR.,W.E.,49,99,101,329,332,385,386 LOWENTHAL, S., 109, 162, 395, 445, 504, 507 LAMBROPOULOS, P., 64, 100 LUCHINI,P., 269, 383 LANDAU,H.J., 250, 382 LUKOSZ,W., 443,465, 507 LANDAU, L.D., 62, 100 LUNHAM, S.D., 288, 380 LANGE,S.R., 153, 161, 162 LUTHER,G.G., 155, 160 LANGE,W., 77, 78, 79, 101 LYNCHJR.,R.T., 41,42,43,49, 51,52,67,95, LANZL,F., 477, 506 99, 100 LAPIERRE, J., 143, 162 LATHAM,W.P., 221,227, 260, 269, 327, 382 L A ~ AJ.N., , 450, 507 LATTA, M., 129, 163 M LATTA,M.R., 14, 36 MACEK,W., 138, 163 LAVAN,M.J., 114, 162 MAGNUSSON, R., 10,36 LAVIGNE, P., 288, 382, 383 MAISCHBERGER, K., 119, 120, 160, 163 LAX,M., 169,269,336,340,341,342,343,344, MAIT,J.N., 465, 466, 467, 507 352, 357, 362, 378, 382, 383 MAITLAND, A., 336, 383 A,, 222, 382 LE FLOCH, MAK,A.A., 196, 383 LE NAOUR,R., 222, 382 MAKER,P.D., 43, 100, 148, 163 F.V., 270, 380 LEBEDEV, MALEIN,A., 126, 159 F., 108, 159 LEBOWSKY, MALLOY,T.B., 43,99 LEE, L.-S., 158, 162 MALONEY, W.T., 445, 508 LEE, P.H., 157, 163 MAMYAN, A.K., 281,288,289, 292, 296, 379 LEE, S.A., 120, 156, 157 MANDEL,L., 108, 163,231,232,238,239,383 LEEB,W.R., 138, 162 MANI, S.A., 322, 386 LEFEVRE, H.C., 141, 160 MARCUSE,D., 329,383 LEISTNER,A.J., 131, 132, 161 MAROM,E., 420, 508 LEITH,E.N., 450,473,493, 507 MARON,Y.,122, 163 LENORMAND, J.M., 222, 382 MARTIN,R.D., 405,416, 504 LEO, I., 496, 504 MASON,S.B., 221,270, 328, 380 LESEM,L.B., 4, 12, 36,444, 507 MASSIE,N.A., 130, 163 LEUNG,K.M., 153, 162 MATSUMOTO, H., 112, 163 LEVENSON, M.D., 42,43,45, 53, 100, 101 MAURE,D.G., 424,502,508 LEVIT,A.L., 302, 378 MCALLISTER,G.L., 269, 355, 386 LI, T., 167, 180, 182, 354, 355, 365, 381, 382 MCCARTHY, N., 288, 382, 383 LIAO,P.F., 57, 99 MCCARTHY,R.J., 269, 340, 383 LIEPMANN, T.W., 145, 147, 162 MCCULLOUGH,A.W., 270,328, 383 LIFSHITZ,E.M., 62, 100 MCFARLANE, R.A., 80, 85, 100 LIN, S.C., 141, 162 MCILRAITH,T.J., 158, 163 LIPPMANN, M.G., 423, 507 MCKNIGHT,W.B., 169,382 LISITSA,V.S., 62, 100 MCLEOD,J., 128, 163 LIU, L.S., 112, 162 MEERS,B.J., 118, 120, 161 LIZET,J., 35, 37 MEIER,G.E.A., 123, 160 LOHMANN, A.W., 12, 36, 439, 441, 443, 445, MEIER,R.W., 450, 508 461,463,466,473,474, 502, 503, 504, 507 MELLING,A., 121, 161 LOOMIS,J.S., 328, 381 MELZER,A., 424 LOTEM,H., 41,43,49, 51, 67, 95, 99, 100 MENAKER,M., 489, 506
518
AUTHOR INDEX
MENGERT,P.H., 481,482,485,503 N NAGEL,J., 269, 354, 383 MERMELSTEIN, M.D., 143, 160 NAITO,K., 26, 36 MERTZ,L., 421, 458, 508 MIDDLETON, S.P., 112, 161 NAKAJIMA,H., 154, 155, 159 Y.,14, 36 MIDWINTER, J.E., 147, 164 NAKAMURA, NAMBA,S., 17,36 MIELENZ,K.D., 112, 161 S.I., 150, 160 NASH,F.R., 152, 160 MIKHAILOV, MILAN,C., 157, 163 NAYAK,N., 66,99 MILES,R.O., 143, 160 NELSON,R.D., 130, 163 MILLER,E.L., 493, 507 NESTOR,J.R., 43, 101 MILLER,H.Y., 168,187,260,262,264,265,386 NEWMAN,D.J., 202,203, 204, 383 MILLER,L.R., 118, 163 NEWTON,G.P., 118, 120, 161 MILONNI,P.W., 376, 377, 383 NIEMAN,B., 13, 36 H., 133, 163 MIRELS,H., 270, 380 NIEUWENHUIJZEN, MIRONOV, A.B., 150, 160 NISENOFF,M., 148, 163 MIZRAHI,V., 59, 62, 100 NISHIDA,N., 14, 36 MLYNEK, J., 77,78, 79, 101 NISHIHARA,H., 4,14,16,17,23,30,32,35,36, MOE,G., 86, 101 37 MOELLER, R.P., 141, 143, 160, 163 NORTHAM, D.B., 270,322, 384 MOLCHANOV, v.YA., 221,225, 227, 383 NUNOSHITA, M., 143, 162 MOLLOW,B.R., 58,62, 100 M0~~~~~,M.A.,405,415,416,417,504,508 MONCHALIN, J.-P., 157, 163 0 OBRIEN,D.P., 196, 387 MONTALTI, F., 152, 160 ODA,T., 124, 163 MOORE,D.T., 114, 162 MOORE,G.T., 269, 340, 383 OGLAND, J.W., 328,383 MOORE,J.T., 396, 508 OHASHI,T., 142, 162 MORENCY, A.J., 270,381 OHTSUKA,Y., 123, 124, 125, 142, 162, 163 MORET-BAILLY, J., 157, 163 OIKAWA,M., 4,36 MORGAN,S.P., 202,203, 204, 383 OKULOV,A.Yu., 150, 160 OLEARY,S., 68, 86, 100 MORQUE,A., 34, 37 OLIVEIRA, F.A.M., 53, 101 MORRIS,M.B., 158, 163 MORRIS,R.H., 108, 161, 163 OLSSON,A., 117, 163 OMONT, A., 54, 101 MORSE,P.M., 202, 235, 383 ONEIL, B.D., 187, 276,277,278, 383 Moss, G.E., 118, 163 ONEILL,E.L., 212, 383 MOSSBERG, T.W., 68.96, 100 MOTTIER,F.M., 130, 163 ONO, Y., 14, 36 OREB,B.F., 131, 132, 161 MOTTIER,P., 32, 34, 35, 36, 37 ORSZAG,A.G., 111, 160 MUELLER, P.F., 435, 504 MUKAMEL, S., 55, 59, 62, 63, 79, 80, 95, 99, OSE, T., 142, 164 loo, 101 OSTROVSKAYA, G.V., 146, 159 O'SULLIVAN, M.S., 157, 162 MULLEN,L.O., 110, 161 OUDAR,J.L., 52,53, 54, 72, 80, 101 MUMOLA, P.B., 270,328,383 OUGHSTUN,K.E.,175,191,196,198,217,269, MUNLEY,A.J., 118, 120, 161 270,273,276,288, 300, 302, 311, 314, 317, MUNNERLYN, C.R., 129, 163 320, 322,323,325, 327, 329, 343, 344, 345, MURPHY,W.D., 260, 328, 383 346, 347,350, 352, 354, 362, 363, 372, 378, MURRAY,J., 116, 162 383, 384 MURTY,M.V.R.K., 128, 163 OVCHINNIKOV, V.M., 302, 378 MUSTAFIN,K.S., 14, 36 OWECHKO, Y,, 420,508 MYERS,O.E., 4, 36
AUTHOR INDEX
P PAEK,E., 490, 505 PALMA,G.E., 329, 384 PALMER, E.W., 112, 161 PAO,Y., 109, 162 PARENT,M., 143, 162 PARIS,D.P., 12, 36, 443, 507 PARK,W., 86, 101 PARKS,R.E., 328, 381,439,440, 503 PAUL,J., 158, 160 PAXTON,A.H., 269, 303, 328,384 PEDERSEN, R.J., 270,380 PEGG,D.T., 66, 101 L., 501, 505 PERELMUTTER, PERI,D., 462, 508 PERKINS, J.F., 269, 322, 355, 380, 384 PERSHAN, P.S., 42,47,49, 99 PERSONS, C.E., 509 PETERS,C.W., 42, 100 PETERSON, D.G., 112, 160 PETLEY,B.W., 111, 163 PETRASH, G.G., 270,276,381, 387 PETROVA, I.M., 270,276,288, 379 PHILLIPS,E.A., 270, 322, 384 PINARD,J., 158, 162 PINARD,M., 53,60, 100 PINE,AS., 92, 101 PIZZURO, V.F., 114, 160 PLUMMER JR., W.W., 221,270,328,380 POLE,R.V., 14, 36, 157, 163 POLLONI,R., 195, 196,288, 384 POON, T.-C., 403,468, 506 POST,E.J., 138, 163 POTTIER,L., 86,99 Pozzo, P.D., 195, 196,288, 384 PRATA,A,, 490, 505 PRATT,W.K., 397,405, 508 PRENTISS, M.G., 139, 140, 163 PRIEST,R.G., 142, 143, 161, 162 PRIOR,Y., 44,51,52,54,55,58,59,62,65,69, 82, 99, 100, 101 PSALTIS,D., 405,406, 490, 505, 508 PUGH,J.R.,120, 161 PULLICINO, M., 157, 163 PUSCHERT, W., 123, 163
R RABINER, W., 496 RABINOWITZ, P., 168, 382 RADER,P., 496
519
H., 151, 163 RAJBENBACH, RAMSDEN,S.A., 144, 159 RASHLEIGH, S.C., 142, 143, 161, 163 RATNER,A.M., 288,382 RAUTIAN,S.G., 93, 101, 270, 276, 381, 387 RAYMER, M.G., 58.99 RAYNOR,F.A., 126, 159 REILLY,J.P., 270, 322, 384 REMESNIK, V.G., 12, 36 RENARD,S., 35, 37 RENSCH,D.B., 269,270,355,378,384 RHODES,W.T., 466,469, 507, 508 RICHARDSON, J.M., 496, 508 RICHARDSON, M.C., 229, 386 RIGROD,W.W., 196, 336, 337, 339, 385 RINKER,R.L., 112, 164 RIOS LEITE,J.R., 53, 101 ROBERTS,R.B., 112, 163 ROBERTSON, H.J., 270,328,383 ROBERTSON, N.A., 120, 161 ROCK,N.H., 125, 160 RODIONOV, A.Yu., 354, 382 ROEHRS,F.S., 270, 380 ROGERS,G.L., 421,463,496, 508, 509 ROGOVIN,D., 269, 354, 383 ROLAND,J.J., 138, 139, 163 ROPARS,G., 222, 382 ROSENFELD, D.P., 130, 160 ROSENTHAL, A.H., 138, 163 L.J., 70,81,82,83,84,85,86,87, ROTHBERG, 88, 89, 90, 91, 93, 95, 99, 101 ROUSSEL,G., 157, 163 ROWLEY, W.R.C., 108, 163 RUDENKO, V.N., 118, 163 RUDIGER,A., 119, 120, 160, 163 RUDOLPH,D., 13, 15.36 RYAZANOV, A.V., 270, 380 S SAHEKI,T., 15, 16, 37 SAKURAI, Y., 112, 163 SALCEDO, J.R., 80, 101 SALIMBENI, R., 157, 163 SALVI,T.C., 269, 303, 382, 384 SAUMANN,H., 144, 163 SANDERS, G.A., 139, 140, 163 SANDERS, R.L., 139, 140 SANDERS, V.E., 139, 160 SANDERSON, R.L., 257,269,385 SANFORD, N.M., 157, 162
520
AUTHOR INDEX
SANTANA,C., 250, 288, 322, 354, 385 SARGENT,M., 49, 101, 139, 160,330,385 SASAKI,I., 123, 124, 163 SASAKI,O.,124, 163 SATO,T.,124, 163 SAVAGE,C.M., 148, 163 SAVRASOVA, M.I., 14,36 SAWCHUK,A.A., 131, 161,508 SAZHIN,M.V., 118, 163 SCHAFER,M.V., 496 SCHAFFER,F.P., 229,380 SCHAHAM, M., 130, 163 SCHAWLOW, A.L., 156, 157, 158, 162 SCHEITERER, E., 138, 162 SCHEK,I., 65, 101 SCHELLEKENS, P.H.J., 126, 164 SCHEPS,R., 79, 101 SCHIFFNER, G., 138, 162 SCHILLING,R., 119, 120, 160, 163 SCHILS,G.F., 488,508 SCHMAHL,G., 13, 15, 36 SCHMIDT,E.,206,385 SCHMIDT,H.,144, 163 SCHNUPP,L.,119, 120, 160, 163 SCHOU, R., 77,78,79, 101 SCHRAM,D.C., 126, 164 SCHULZ,W.E., 66, 101 SCHULZ-DUBOIS,E.O., 338, 385 SCHWIDER, J., 131, 163 SCULLY,M.O., 49, 101, 139, 160,330,385 SEE,Y.C., 329,385 SEINO,S., 112, 163
SHIN, S.Y., 250,260,269, 380 SHIONO,T.,35, 37 SHOROKHOV, O.A., 340, 379 SHORTHILL, R.W., 140,164 SHOUGH, D., 115, 116, 162 SIEGMAN,A.E., 80, 101, 107, 134, 163, 167,
168, 170, 181, 182, 185, 186, 187, 189, 191, 194, 196,244,256,260,262,264,265,269, 270,271,272,273,288,303,330, 332,333, 350, 351, 352,353, 354,355, 356,359,360, 364, 366, 367,369, 378, 385, 386 SIGELJR., G.H., 142, 143, 160, 161, 162 SILBERBERG, Y.,150, 160 SIMONOFF,A.M., 329,385 SINCLAIR,D.C., 168,386 SINGH, S.,64,65,99 SKENDEROFF, C.,405, 508 SKOLNICK,M.L., 230, 380 SKROTSKII, G.V., 221,225,227,383 SLAYMAKER,P.A., 196, 288, 300, 311, 314, 317, 323, 325,384 SLICHTER,C.P., 54,65, 101 SMITH, E.W., 54, 101 SMITH,H.I., 17,37 SMITH,I.W., 139, 140, 161, 164 SMITH, P.W., 106, 163 SMITH,S.J., 65, 100 SMITH,W.I.B., 126, 161 SMITH, W.J., 428,508 SMITHERS,M.E., 327,382 SNYDER,J.J., 158, 163 SOARES,O.,14,37 S~~1~0~0~,V.N.,287,288,289,292,296,379 SOBEL’MAN,I.I., 93, 101 SETTE,D., 230, 240, 379 SOFFER,B.H., 420,508 SHACK,R.V., 127, 163 SOLIMENO,S.,269, 383 SHAMIR,J., 154, 161 SOLOV’EV,V.S., 288, 382 SHARLAI,S.F., 302,378 SOMMARGREN, G.E., 122, 130, 131, 164 SHATAS,R.A., 269, 355, 384 SONG, J.J., 43, 53, 100, 101 SHAVER,D.C., 17,37 SOOY,W.R., 183,270,321, 382 SHAW,H.J., 141, 160 SOUMA,H., 42, 53, 80,84, 101 SHCHEGLOV, V.A., 328,386 SOUTHWELL,W.H., 221, 227, 269, 288, 354, SHEEM,S.K., 143, 162, 163 386 SHELLAN,J.B., 328, 385 SPEARS,D.L., 136, 164 SHEN,Y.R., 42,45,47,49, 52, 53, 54, 72, 80, SPEISER, J.M., 419,499, 508, 509 SPENCER,M.B., 329,386 94,99,101 SHENG,S.C., 196,288,385 SPERO,R.E.,118, 120, 161 SPINHIRNE,J.M., 270,328,329,345,347, 350, SHERMAN,G.C., 186,355,385 SHERMAN, R.C., 484,508 378, 384, 386 SHERSTOBITOV, V.E., 168,189, 196,269,270, SPIRIDONOV, J.M., 270 V.V., 379 274,287,321,323, 340,354, 379, 382, 385 SPIRIDONOV,
AUTHOR INDEX
SPIRO,T.G., 43, 101 SPOLACZYK, R., 131, 163 SPRAGUE, R.A., 114, 164 STATZ,H., 139, 140, 161, 164, 345, 386 STEEL,D.G., 80,85, 100 STEEL,W.H., 128, 164 STEGUN,LA., 246, 378 STEIER,W.H., 269, 355, 386 STEINBERG, G.N., 270,328, 383 STEPANOV, A.A., 328, 386 STEPHENS,R.B., 112, 161 STEPHENS, W.E., 496, 509 STETSON,K.A., 113, 164 STEVENSON, W.H., 121, 164 STOICHEFF,B.P., 157, 162 STONER,W., 444, 509 STOREY,J.W.V., 136, 164 STRAND,T.C., 435, 509 STREIBL, N., 461, 504 STREIFER,W., 192, 206, 208, 210, 213, 230, 250, 257,269,290, 381, 385, 386 STROUDJR., C.R., 63, 95, 100 H., 144, 163 STROWALD, STUMPF,K.D., 130, 164 STUS, YU.F., 112, 159 SUDA,S., 14, 36 SUGAMA,S., 14, 36 SUHARA, T., 14, 30, 32, 35, 36, 37 M., 4,37 SUSSMAN, SUlTON, E.C., 136, 164 SUlTON, G.W., 322, 386 SVELTO,O., 195, 196, 288, 384 SVENTSITSKAYA,N.A., 168,189,196,270,276, 288, 379 SWANTNER, W., 329, 386 SWEENEY, D.W., 488, 508 SWINDELL, w., 402, 410, 412, 413, 492, 503, 505, 506, 509 L., 405, 508 SYMANIEC, J.J., 420, 509 SYMANSKI, SZIKLAS,E.A., 186, 269, 270, 271, 272, 273, 350, 351, 352, 353, 355, 364, 366, 367, 369, 378, 386 SZOKE,A., 58, 99, 157, 163 SZUDY, J., 78, 101
T TAI, S., 143, 162 TAKASAKI, H., 108, 164 TAKEI,T., 15, 16, 37
52 1
TAKENAKA, H., 15, 16, 36 TAM,A., 86, 101 TAMIR,T., 26, 37 TAMURA,P.N., 493, 507 TANAKA,S., 26, 32, 36 TANG,C.L., 117, 163, 345, 386 TANG,H., 91, 100 TANGUAY JR, A.R.. 419, 509 TANIDA,J., 502, 506, 509 TANIGUCHI, N., 14, 36 TANNER,L.H., 144, 164 TARAN,J.-P.E., 42,52,54,55,92,93,100, 101 TARASYUK, V.G., 112, 159 TATSUMI,K., 15, 16, 37 TATSUNO,K.,153, 164 TEETS,R.E., 64, 101, 157, 162 TEICH,M.C., 134, 164 TERHUNE,R.W., 43, 100, 148, 163 THOMAS,J.E., 157, 163 THOMPSON, B.J., 114, 130, 164 TICKNOR,A.J., 496, 505, 509 TIMMERMANS, C.J., 126, 164 TOMITA, A., 144, 145, 147, 162 TOMOV,LV., 229, 386 TOSCHI,V., 495,496, 505 TOWNES, C.H., 116, 134, 135, 136, 137, 162, 164 TREACY,E.B., 168, 386 TROFIMOV,N.P., 270, 379 TROMBINI,E.M., 481,482,485, 503 H.P.,74,75,76,99, 101 TROMMSDORFF, TRUAX,B.E., 122, 164 TSUBOKAWA, M., 125, 163 TSUKERMAN, V.G., 12, 36 TSUKIJI, M., 108, 164 TURNER,E.B., 221,270, 328, 380 TVETEN,A.B., 142, 143, 160, 162 TYLER,G.A., 221,227, 386
U UDD, E., 141, 160 ULRICH,R., 140, 164 UMEDA,N., 108, 164 UMEGAKI,S., 26, 36 UNDERWOOD, K.L., 328, 381 UPATNIEKS,J., 450, 473, 493, 507 URY, I., 151, 152, 162 USTYUGOV, V.I., 196, 383 UTTAM,D., 143, 161
522
AUTHOR INDEX
V VAINSHTEIN,LA., 167, 386 VALETTE,S., 32, 34, 35, 36, 37 VALI,V., 111, 140, 164 VAN ROOY,D.L., 4, 12, 36 VANDERLUGT,A., 445,48 1, 509 VERBER,C.M., 484,508 VERBER,P.W., 501, 505 VERDEYEN,J.T., 126, 161 VERKERK,P., 53,60, 100 VETKIN,V.A., 227,229, 387 VILLARUEL,C.A., 141, 160 VINOKUROV,G.N., 189, 269, 274, 287, 323, 379, 385
VON DER OHE, W., 131, 161
WILSON-GORDON,A.D., 66, 100, 101 WINKLER,W., 119, 120, 160, 163 WINOCUR,J., 229,381 WINSOR,H.,143, 164 WISNER,G.R., 270, 387 WITTKE, J.P., 82,93, 101 WOLF, E., 4, 36, 169, 171, 177, 178,222,230, 231,232,233,234,235,236,238,239, 379, 381, 383, 387, 394, 504 WOLFELSCHNEIDER,H.,109, 164 WOODY,L.M., 439, 461, 464, 465, 471, 491, 505,506 WRIGHT,E.M., 191, 196, 387 WYANT, J.C.,129,130,131,160,162,164,328, 381 WYNNE,J.J., 95, 100
W WALSH,C.J., 111, 164 WARD,H., 118, 120, 161 WARD,J.F, 54, 101, 161 WARDE,R.E., 108, 114, 160, 419 WARNIAK,J.S., 108, 163 WEINER,M.M., 322, 386, 387 WEINREICH,G., 42, 100 WEINSTEIN,LA., 250,387 WEITEKAMP,D.P., 66, 101 WELFORD,W.T., 509 WELLS,J.S., 110, 161 WERTZ,A., 445,507 WEST, E.J., 143, 160 WHITCOMB,S.E., 119, 120, 161 WHITE,A.D., 115, 130, 160 WHITE,J.A., 155, 160 WHITE, J.O., 151, 164 WHITEHOUSE,H.J., 419,499, 508, 509 WHITELAW,J.H., 121, 161 WHITFORD,B.G., 110,164 WHITTAKER,F., 68.96, 100 WIERSMA,D.A., 66, 74,99, 100, 101 WIERSMA,P., 477, 509 WILLSON,J.P., 143, 164 WILSON,D.C., 108, 160, 163 WILSON,D.T., 493,496, 509
Y YAJIMA,T., 42, 53, 80, 84, 101 YAKOVLENKO, S.I., 62, 100 YARIV,A.,53,101,143,149,151,152,162,164 YEE, T.K., 41, 54, 55, 57, 59,95, 100, 101 YEH, P., 151, 164 YEH, S., 62,78, 101 YEH, Y., 120, 164 YODERJR., P.R., 131, 164 YORK,G., 79, 101 YOSHINO,T., 142, 164 YOUNG,M., 475,483,484,485, 506 YURATICH,M.A., 64, 101 Z
ZARAGA,F., 195, 196,288, 384 ZEMSKOV,E.M., 287,288,289,292,296,379 ZEMSKOV, K.I., 270, 387 ZERNIKE, F., 147, 157, 163, 164 ZHOU WANZHI,F., 128, 131, 161, 164 ZINGERY,W.L., 229, 381 Zou, Y.H., 90,91,93, 99 ZUBAROV,I.G., 150, 160 ZUCKER,H., 287,387 ZUMBERGE,M.A., 112, 164
SUBJECT INDEX A
- shift, 121
Abbe number, 22 ABCD rule, 175 acousto-optic modulator, 155 Airy function, 10 pattern, 310 angular spectrum, 422 aperture apodization, 287
dressed atom model, 59, 60
-
electron-beam lithography, 17 writing, 15, 17 encoding, spatial, 468, 470 -, temporal, 468
B
F
Bloch vector, 48 Bohr magneton, 78 B r a g cell, 123 Brillouin scattering, stimulated, 150, 151
Fabry-Perot cavity, 109, 228 etalon, 106 Faraday cell, 139 fast Fourier transform, 273, 355-357, 359, 360, 364, 366, 368 Fourier-Bessel expansion, 358 transformation, 359 -transfarm, 398, 496 , discrete, 494 four-wave mixing, 43, 52, 53, 58, 61, 64, 66, 67, 79, 80, 83, 92, 94, 95 Fraunhofer approximation, 306 Fredholm integral equation, 231, 235, 242 Fresnel diffraction, 395 ---Kirchoff diffraction integral, 171, 172, 178, 179 - lens 3-8, 11-16, 20-22, 29, 34, 35 - number, 188-190,246-248,260,261, 263, 270,273,276,298, 310, 311, 316, 317, 326, 350, 352 -waveguide, 26,28, 32 , gradient index, 29 - zone lens, 15-17 , micro, 12, 15 plate, 4
C
cavity Q factor, 334 central slice theorem, 407 chirp-Z algorithm, 495, 496 transform, 496 coherence, complex degree of spectral, 232, 233 -, dephasing-induced, 41,44, 56 - function, mutual, 231, 233 - length, 394 -, spatial, 105, 152, 230, 394 -, temporal, 105, 106, 394 - time, 394 coherent antistokes Raman resonance, 72, 92 - Stokes - -, 71-73 convolution, 397, 401 - theorem, 398 cross-spectral density, 231, 233, 238 D diffraction efficiency, 9, 22 Doppler broadening, 8 1, 82 ---free spectroscopy, 42
E
---
-_--
---G
Gabor zone plate, 13 Gardner transformation, 359, 372 geodesic lens, 28
524
SUBJECT INDEX
geometrical optics approximation, 340 gravitational waves, detection of, 118
H Hankel transform, 358-361, 363 Hanle resonance, 91, 92 --, collision induced, 90, 93 --, pressure induced, 77 Helmholtz equation, 169, 232, 357, 364 heterodyne detection, 133 Hilbert-Schmidt kernel, 231 - space, 212, 215 hologram, 14, 448, 449, 456, 458 holography, 149,463,493
Larmor precession frequency, 78, 90 laser, dye, 105 -, helium-neon, 105, 107, 110, 116, 126, 129 linewidth, 106, 108 modes, 106 -, semiconductor, 105 linear system, 396 , shift invariant, 397 Liouville equation, 42, 48, 62, 65 Luneburg lens, 28
-
-M
Mercer expansion, 232, 233, 239 Michelson-Morley experiment, 116
I impulse response, 398 incoherent light, spatially, 391 integrated circuits, 12 - optics, 26 interferometer, 112 -, Fabry-Perot, 15, 115, 125, 155-158, 229 -, Fizeau, 158 -, laser-feedback, 125, 126 -, Mach-Zehnder, 142, 146, 150 -, Michelson, 113, 118, 121, 124, 153, 156, 393, 458 -, phase-conjugate, 148, 150 -, second-harmonic, 144, 147 -, Twyman-Green, 129 -, Young, 393, 394 interferometry, 105, 11 1 -, frequency modulation, 117 -, heterodyne spatial, 132 -, laser Doppler, 120-122 -, two-wavelength, 117, 144
N nonlinear optical mixing, 45, 52, 95 spectroscopy, 43 numerical aperture, 3, 8, 9
0
optical communications, 26
- information processing, 392, 497 - processing, incoherent, 391, 395,400 - transfer function, 428, 439, 442, 443, 445, 447, 454, 456,457
P paraxial approximation, 189 -equation, 171, 331, 365 photorefiactive material, 12 - oscillator, 151 point spread function, 398 Poynting vector, 169 Psaltis correlator, 407
J
Jones calculus, 222
- matrix, 221 - vector, 223
Q quasi-convolution, 432, 433
R K Kirchhoff approximation, 357
L Lamb dip, 108 Landau-Zener curve crossing formula, 62
Rabi frequency, 46 Radon space, 399,407,409 - transformation, 399,407,408,493,496 Raman resonance, 92, 94 ray transfer matrix, 173-175, 182, 197 Rigrod’s method, 342 rotation sensor, interferometric, 138, 139
SUBJECT INDEX
S
tomography, 408,478
sampling theorem, 367, 399 scanning electron microscope, 17, 19 Schawlow-Townes relation, 107 Schmidt expansion theorem, 206 function, 206, 210, 211 second-harmonic generation, 42 signal processing, 26, 398 spatial frequency modulation, 470 stationary phase approximation, 244
-, transverse analog, 410
-
T tetradic T matrix formalism, 63 tomographic reconstruction, 409
-, uncomputerized, 412 transfer function, 402 two-photon absorption, 57
U unstable resonator, 167, 327, 340
z Zeeman coherence, 91
- resonance, 90, 91, 95
525
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CUMULATIVE 1M)EX - VOLUMES I-XXIV ABELES,F., Methods for Determining Optical Parameters of Thin Films ABELLA,I. D., Echoes at Optical Frequencies ABITBOL, C. I., see J. J. Clair AGARWAL, G. S., Master Equation Methods in Quantum Optics V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion AGRANOVICH, ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers AMMANN, E. O., Synthesis of Optical Birefringent Networks J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations ARMSTRONG, in Lasers J. A,, Hamiltonian Theory of Beam Mode Propagation ARNAUD, BALTES,H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images BARRETT, H. H., The Radon Transform and its Applications S., Beam-Foil Spectroscopy BASHKIN, BECKMANN,P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns BEVERLY111, R. E., Light Emission from High-Current Surface-Spark Discharges BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements BOUMAN, M. A., W. A. VAN DE GRIND,P. ZUIDEMA,Quantum Fluctuations in Vision BOUSQUET, P., see P. Rouard BROWN,G. S., see J. A. DeSanto W., H. PAUL,Theory of Optical Parametric Amplification and OscillaBRUNNER, tion BRYNGDAHL, O., Applications of Shearing Interferometry BRYNGDAHL, O., Evanescent Waves in Optical Imaging BURCH,J. M., The Meteorological Applications of Diffraction Gratings H. J., Principles of Optical Data-Processing BUTTERWECK, CAGNAC,B., see E. Giacobino CASASENT, D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition
11, 249 VII, 139 XVI, 71 XI, 1 IX, 235 IX, 179 IX, 123 VI, 211 XI, 247 XII,
1
I, 67 XXI, 217 XII, 287 VI, 53 XVIII, 259 XVI, 357 IX, 1 XXII, 77 IV, 145 XXIII, 1 xv, 1 IV, 3 1 XI, 161 11, 73 XIX, 21 1 XVII, 85 XVI, 289
528
CUMULATIVE INDEX
CEGLIO,N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and ApplicaXXI, 287 tions XIII, 69 J. L., see W. M. Rosenblum CHRISTENSEN, XVI, 71 CLAIR,J. J., C. I. ABITBOL,Recent Advances in Phase Profiles Generation XIV, 327 CLARRICOATS, P. J. B., Optical Fibre Waveguides A Review COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, 1 XV, 187 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy COLOMBEAU, B., see C. Froehly XX, 63 COURTES,G., P. CRUVELLIER, M. DETAILLE,M. SA~SSE, Some New Optical xx, 1 Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects XI, 223 CREWE,A. V., Production of Electron Probes Using a Field Emission Source x x, 1 CRUVELLIER, P., see C. G. Courtbs XIII, 133 Light Beating Spectroscopy CUMMINS, H. Z., H. L.. SWINNEY, XIV, 1 DAINTY,J. C., The Statistics of Speckle Patterns XVII, 1 DANDLIKER, R., Heterodyne Holographic Interferometry XII, 101 DECKERJr., J. A., see M. Hanvit VII, 67 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters IX, 31 DEMARIA,A. J., Picosecond Laser Pulses DESANTO,J. A., G. S. BROWN,Analytical Techniques for Multiple Scattering XXIII, 1 from Rough Surfaces xx, 1 DETAILLE, M., see G. Court& X, 165 DEXTER,D. L., see D. Y. Smith XII, 163 DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XIV, 161 DUGUAY,M. A., The Ultrafast Optical Kerr Shutter VII, 359 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons ENGLUND,J. C., R. R. SNAPP,W. C. SCHIEVE,Fluctuations, Instabilities and XXI, 355 Chaos in the Laser-Driven Nonlinear Ring Cavity XVI, 233 ENNOS,A. E., Speckle Interferometry XXII, 341 FANTE, R. L., Wave Propagation in Random Media: A Systems Approach I, 253 FIORENTINI, A., Dynamic Characteristics of Visual Processes IV, I FOCKE,J., Higher Order Aberration Theory Measurement of the Second Order Degree of CoheFRANCON, M., S. MALLICK, VI, 71 rence FRIEDEN,B. R., Evaluation, Design and Extrapolation Methods for Optical SigIX, 311 nals, Based on Use of the Prolate Functions FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE, Shaping and Analysis of PicoXX, 63 second Light Pulses VIII, 51 FRY,G. A., The Optical Performance of the Human Eye I, 109 GABOR,D., Light and Information 111, 187 GAMO,H.,Matrix Treatment of Partial Coherence XIII, 169 A. K., see M. S. Sodha GHATAK, XVIII, 1 Graded Index Optical Waveguides: A Review GHATAK, A., K. THYAGARMAN, XVII, 85 E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy GIACOBINO,
-
CUMULATIVE INDEX
529
IX, 235 GINZBURG, V. L., see V. M. Agranovich 11, 109 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media XXIV, 389 GLASER,I., Information Processing with Spatially Incoherent Light GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction IX, 281 Theory of Elastic Waves VIII, 1 J. W., Synthetic-Aperture Optics GOODMAN, XII, 233 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XX, 263 HARIHARAN, P., Colour Holography XXIV, 103 HARIHARAN, P., Interferometry with Lasers XII, 101 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry X, 289 HELSTROM, C. W., Quantum Detection Theory VI, 171 HERRIOTT,D. R., Some Applications of Lasers to Interferometry HUANG,T. S., Bandwidth Compression of Optical Images x, 1 JACOBSSON, R., Light Reflection from Films of Continuously Varying Refractive V, 247 Index 111, 29 Apodisation JACQUINOT, P., B. ROIZEN-DOSSIER, W., B. P. STOICHEFF,Generation of Tunable Coherent Vacuum-UltraJAMROZ, XX, 325 violet Radiation IX, 179 JONES,D. G. C., see L. Allen KASTLER,A,, see C. Cohen-Tannoudji v, 1 XX, 155 KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy IV, 85 KINOSITA,K., Surface Deterioration of Optical Glasses KOPPELMANN, G., Multiple-Beam Interference and Natural Modes in Open VII, 1 Resonators 111, 1 KOTTLER,F., The Elements of Radiative Transfer IV, 281 KOTTLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory VI, 331 KOTTLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory I, 211 KUBOTA,H., Interference Color XIV, 47 LABEYRIE, A., High-Resolution Techniques in Optical Astronomy XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XVI, 119 LEE, W.-H., Computer-Generated Holograms: Techniques and Applications Recent Advances in Holography VI, 1 LEITH,E. N., J. UPATNIEKS, XVI, 1 LETOKHOV, V. S., Laser Selective Photophysics and Photochemistry VIII, 343 LEVI,L., Vision in Communication LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch V, 287 of Physical Optics XXI, 69 LUGIATO,L. A., Theory of Optical Bistability XXII, 1 MALACARA, D., Optical and Electronic Processing of Medical Images VI, 71 MALLICK,L., see M. FranGon 11, 181 MANDEL,L., Fluctuations of Light Beams XIII, 27 MANDEL,L., The Case for and against Semiclassical Radiation Theory E. W., Gradient Index Lenses XI, 305 MARCHAND, Optical Films Produced by Ion-Based TechMARTIN,P. J., R. P. NETTERFIELD, niques XXIII, 113
530
CUMULATIVE INDEX
MASALOV,A. V., Spectral and Temporal Fluctuations of Broad-Band Laser XXII, 145 Radiation XXI, 1 MAYSTRE, D., Rigorous Vector Theories of Diffraction Gratings x v , 77 MEESSEN,A., see P. Rouard VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting Quasi-ClassicalTheory of Laser RadiaMIKAELIAN, A. L., M. I. TER-MIKAELIAN, VII, 231 tion XVII, 279 A. L., Self-Focusing Media with Variable Index of Refraction MIKAELIAN, Surface and Size Effects on the Light ScatteMILLS,D. L., K. R. SUBBASWAMY, XIX, 43 ring Spectra of Solids MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design 1, 31 MOLLOW, B. R., Theory of Intensity Dependent Resonance Light Scattering and XIX, 1 Resonance Fluorescence V, 199 MURATA,K., Instruments for the Measuring of Optical Transfer Functions VIII, 201 MUSSET,A,, A. THELEN,Multilayer Antireflection Coatings XXIII, 113 R. P., see P. J. Martin NETTERFIELD, XXIV, 1 H., T. SUHARA, Micro Fresnel Lenses NISHIHARA, XV, 139 OKOSHI,T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image Holographic Methods in Plasma OSTROVSKAYA, G. V., Yu. I. OSTROVSKY, XXII, 197 Diagnostics XXII, 197 YU. I., see G. V. Ostrovskaya OSTROVSKY, XXIV, 165 K. E., Unstable Resonator Modes OUGHSTUN, xv, 1 PAUL,H., see W. Brunner PEGIS,R. J., The Modem Development of Hamiltonian Optics 1, 1 VII, 67 PEGIS,R. J., see E. Delano J., Photocount Statistics of Radiation Propagating through Random and PERINA, XVIII, 129 Nonlinear Media V, 83 PERSHAN, P. S., Non-Linear Optics IX, 281 J., see K. Gniadek PETYKIEWICZ, V, 351 PICHT,J., The Wave of a Moving Classical Electron XVI, 289 PsALrIs, D., see D. Casasent RISEBERG,L.A., M. J. WEBER,Relaxation Phenomena in Rare-Earth LumiXIV, 89 nescence VIII, 239 RISKEN,H., Statistical Properties of Laser Light XIX, 281 RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy 111, 29 ROIZEN-DOSSIER, B., see P. Jacquinot ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical XIII, 69 Aberration Measurements of the Human Eye XXIV, 39 ROTHBERG, L., Dephasing-Induced Coherent Phenomena IV, 145 Optical Constants of Thin Films ROUARD,P., P. BOUSQUET, XV, 71 ROUARD,P., A, MEESSEN,Optical Properties of Thin Metal Films
CUMULATIVE INDEX
53 1
IV, 199 RUBINOWICZ, A,, The Miyamoto-Wolf Diffraction Wave XIV, 195 RUDOLPH,D., see G. Schmahl xx, 1 SAYSSE,M., see G. Courtts VI, 259 SAKAI,H., see G. A. Vanasse XXI, 355 SCHIEVE,W. C., see J. C. Englund SCHMAHL, G., D. RUDOLPH,Holographic Diffraction Gratings XIV, 195 The Mutual Dependence between Coherence ProSCHUBERT, M., B. WILHELMI, perties of Light and Nonlinear Optical Processes XVII, 163 SCHULZ,G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces XIII, 93 XIII, 93 J., see G. Schulz SCHWIDER, X, 89 SCULLY,M. O., K. G. WHITNEY,Tools of Theoretical Quantum Optics I. R., Semiclassical Radiation Theory within a Quantum-Mechanical SENITZKY, XVI, 413 Framework XV, 245 SIPE,J. E., see J. Van Kranendonk X, 229 SITTIG,E. K., Elastooptic Light Modulation and Deflection XII, 53 SLUSHER,R. E., Self-Induced Transparency VI, 21 1 SMITH,A. W., see J. A. Armstrong X, 165 SMITH,D. Y., D. L. UEXTER,Optical Absorption Strength of Defects in Insulators x, 45 SMITH,R. W., The Use of Image Tubes as Shutters XXI, 355 SNAPP,R. R., see J. C. Englund SODHA,M. S., A. K. GHATAK,V. K. TRIPATHI,Self Focusing of Laser Beams in XIII, 169 Plasmas and Semiconductors v, 145 STEEL,W. H., Two-Beam Interferometry STOICHEFF,B. P., see W. Jamroz XX, 325 STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere IX, 73 STROKE,G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy 11, 1 XIX, 43 SUBBASWAMY, K. R., see D. L. Mills XXIV, 1 SUHARA, T., see H. Nishihara SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser XII, 1 Beams SWEENEY, D. W., see N. M. Ceglio XXI, 287 VIII, 133 SWINNEY, H. H., see H. 2. Cummins TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets XXIII, 63 XVII, 239 TANGO,W. J., R. Q. TWISS,Michelson Stellar Interferometry V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation TATARSKII, in a Randomly Inhomogeneous Medium XVIII, 207 C. A,, see H. Lipson TAYLOR, V, 287 TER-MIKAELIAN, M. L., see A. L. Mikaelian VII, 231 THELEN,A., see A. Musset VIII, 201 B. J., Image Formation with Partially Coherent Light THOMPSON, VII, 169 THYAGARAJAN, K., see A. Ghatak XVIII, 1 TONOMURA, A,, Electron Holography XXIII, 183
532
CUMULATIVE INDEX
XIII, 169 TRIPATHI,V. K., see M. S. Sodha TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and 11, 131 by Spatial Frequency Filtering XVII, 239 TWISS,R. Q., see W. J. Tango VI, 1 J., see E. N. Leith UPATNIEKS, XVIII, 259 UPSTILL,C., see M. V. Berry USHIODA,S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in XIX, 139 Solids XX, 63 VAMPOUILLE,M., see C. Froehly VI, 259 VANASSE,G. A., H. SAKAI,Fourier Spectroscopy XXII, 77 VAN DE GRIND,W. A,, see M. A. Bouman I, 289 VAN HEEL,A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE,Foundations of the Macroscopic ElectromagneXV, 245 tic Theory of Dielectric Media XIV, 245 VERNIER,P., Photoemission XIV, 89 WEBER,M. J., see L. A. Riseberg IV, 241 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings XIII, 267 WELFORD,W. T., Aplanatism and Isoplanatism XVII, 163 WILHELMI, B., see M. Schubert X, 89 WITNEY,K. G., see M. 0. Scully WOLTER,H., On Basic Analogies and Principal Differences between Optical and I, 155 Electronic Information X, 137 WYNNE,C. G., Field Correctors for Astronomical Telescopes YAMAGUCHI, I., Fringe Formations in Deformation and Vibration Measurements XXII, 271 Using Laser Light VI, 105 YAMAJI,K., Design of Zoom Lenses T., Coherence Theory of Source-Sue Compensation in Interference YAMAMOTO, VIII, 295 Microscopy XI, 77 YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques XXIII, 227 Yu, F. T. S., Principles of Optical Processing with Partially Coherent Light XVIII, 207 ZAVOROTNYI, V. U., see V. I. Tatarskii XXII. 71 ZUIDEMA, P., see M. A. Bouman
