PROGRESS IN OPTICS VOLUME XXIII
EDITORIAL ADVISORY BOARD L. ALLEN,
Brighton, 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 XXIII
EDITED BY
E. WOLF University of Rochester. N . Y . . V.S.A.
Contributors J . A. DESANTO, G . S. BROWN K. TANAKA, P. J. MARTIN, R. P. NETTERFIELD A. TONOMURA, F. T. S. YU
1986
NORTH-HOLLAND AMSTERDAM . O X F O R D . N E W Y O R K . T O K Y 0
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I. I1 .
C O N T E N T S O F V O L U M E 1(1961) THEMODERN DEVELOPMENT OF HAMILTONIAN OPTICS.R . J . PEGIS . .
1-29
WAVE OPTICSAND GEOMETRICAL OPTICS I N OPTICALDESIGN.K .
31-66 MIYAMOTO. . . . . . . . . . . . . . . . . . . . . . . . . . . . I l l . THEINTENSITY DISTRIBUTION A N D TOTAL ILLUMINATION OF ABERRATIONFREEDIFFRACTION IMAGES. R . BARAKAT . . . . . . . . . . . . . . 67- 108 IV . LIGHTA N D INFORMATION. D. GABOR. . . . . . . . . . . . . . . . 109- 153 V. ON BASIC ANALOGIESA N D PRINCIPAL DIFFERENCES BETWEEN OPTICAL AND ELECTRONIC INFORMATION. H. WOLTER. . . . . . . . . . . . . 155-210 VI . INTERFERENCECOLOR.H. KUBOTA . . . . . . . . . . . . . . . . . 21 1-251 VII . DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES. A . FIORENTINI . . . 253-288 A . c. s. VAN HEEL . . . . . . . . . . 289-329 VIll MODERNALIGNMENTDEVICES.
C O N T E N T S O F V O L U M E I 1 (1963) I.
RULING.TESTING A N D USE OF OPTICAL GRATINGS FOR HIGH-RESOLUTION SPECTROSCOPY. G. W. STROKE. . . . . . . . . . . . . . . . . . . 1-72 I1 . THEMETEOROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS. J. M. 73-108 BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. DIFFUSION THROUGH NON-UNIFORM MEDIA.R . G. GIOVANELLI . . . . 109-129 IV . CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS A N D BY SPATIAL FREQUENCY FILTERING. J . TSUJIUCHI. . . . . . . . 13 1- 180 OF LIGHTBEAMS. L . MANDEL. . . . . . . . . . . . V . FLUCTUATIONS 18 1-248 VI . METHODSFOR DETERMINING OPTICAL PARAMETERS OF THINFILMS.F. ABELBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C O N T E N T S O F V O L U M E I 1 1 (1964) 1. THEELEMENTS OF RADIATIVE TRANSFER. F. KOTTLER . . . . . . . . I1 . APODISATION.P. JACQUINOT.B . ROIZEN-DOSSIER . . . . . . . . . . I l l . MATRIXTREATMENT OF PARTIAL COHERENCE. H. GAMO . . . . . . .
1. 11. 111.
C O N T E N T S O F V O L U M E IV (1965) HIGHER ORDERABERRATION THEORY. J . FOCKE . . . . . . . . . . . APPLICATIONSOF SHEARING INTERFEROMETRY. 0. BRYNCDAHL .
. . .
249-288
1-28 29- 186 187-332
1-36 37-83 85-143 145-197 199-240
SURFACE DETERIORATION OF OPTICAL GLASSES. K. KINOSITA. . . . . OPTICAL CONSrANTS OF THINFILMS.P. ROUARD.P. BOUSQUET . . . . THEMIYAMOTO-WOLF DIFFRACTION WAVE. A . RUBINOWICZ . . . . . . V. v1. ABERRATIONTHEORYOF GRATINGS A N D GRATING MOUNTINGS. W. T. WELFORD. . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . DIFFRACTION AT A BLACK SCREEN.PARTI: KIRCHHOFF’S THEORY.F. KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . . .
281-314
CONTENTS O F VOLUME V OPTICAL PUMPING. C. COHEN.TANNOUDJI. A . KASTLER. 1. I1 . NON-LINEAR OPTICS.P. S. PERSHAN. . . . . . . . . I11 . TWO-BEAM INTERFEROMETRY. W. H. STEEL . . . . . .
1-81 83-144 145-197
IV .
24 1-280
(1966) . . . . . . . . . . . . . . . . . . . . .
VI
INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFER FUNCTIONS. K. 199-245 MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . LIGHTREFLECTIONFROM FILMSO F CONTINUOUSLY VARYING REFRACTIVE 247-286 INDEX. R . JACOBSSON . . . . . . . . . . . . . . . . . . . . . . . DETERMINATION AS A BRANCHOF PHYSICAL VI . X-RAYCRYSTAL-STRUCTURE OPTICS, H . LIPSON. c. A . TAYLOR . . . . . . . . . . . . . . . . . . 287-350 CLASSICAL ELECTRON. J . P l C H T . . . . . . . 351-370 VII . THEW A V E O F A MOVING I V.
I.
C O N T E N T S O F V O L U M E VI ( 1 9 6 7 ) RECENTADVANCESI N HOLOGRAPHY. E . N . LEITH. J . UPATNIEKS. . . .
LIGHT BY ROUGHSURFACES. P. BECKMANN. . . . . . MEASUREMENT O F THE S E C O N D O R D E R DEGREEO F COHERENCE M . FRANCON. S. MALLICK . . . . . . . . . . . . . . . . . . . . . . O F ZOOM LENSES.K . YAMAJI. . . . . . . . . . . . . . . . IV . DESIGN SOMEAPPLICATIONS OF LASERSTO INTERFEROMETRY.D . R . HERRIOTT. V. STUDIES O F INTENSITY FLUCTUATIONS IN LASERS.J . A . VI . EXPERIMENTAL ARMSTRONG.A . W . SMITH . . . . . . . . . . . . . . . . . . . . . V I I . FOURIER SPECTROSCOPY. G . A . VANASSE. H . SAKAI. . . . . . . . . . AT A BLACK SCREEN. PART11: ELECTROMAGNETIC THEORY. VIII . DIFFRACTION F. KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. 111.
SCATTERING O F
.
1-52 53-69 71-104 105-170 1 7 1 -209 211-257 259-330 331-377
C O N T E N T S O F V O L U M E VII (1969) I.
MULTIPLE-BEAM INTERFERENCEA N D NATURAL MODES IN OPEN RESONATORS.G. KOPPELMAN . . . . . . . . . . . . . . . . . . . 11. METHODS O F SYNTHESIS FOR DIELECTRIC MULTILAYERFILTERS.E . DELANO.R . J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . FREQUENCIES. I . D . ABELLA . . . . . . . . . . 111. ECHOESA N D OPTICAL WITH PARTIALLY COHERENT LIGHT. B . J . THOMPSON IV . IMAGE FORMATION QUASI-CLASSICAL THEORY OF LASERRADIATION. A . L. MIKAELIAN. M . L. V. TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . VI . THEPHOTOGRAPHIC IMAGE. s. O O U E . . . . . . . . . . . . . . . . VII . INTERACTION OF VERY INTENSELIGHT WITH FREEELECTRONS.J . H . EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-66 67-137 139-168 169-230 23 1-297 299-358 359-415
C O N T E N T S O F V O L U M E VIII (1970) SYNTHETIC-APERTURE OPTICS. J . W . GOODMAN. . . . . . . . . . . I. I 1 . THEOPTICAL PERFORMANCE O F THE HUMANEYE.G. A . FRY . . . . . H . Z . CUMMINS. H . L. SWINNEY. . . . Ill . LIGHT BEATINGSPECTROSCOPY. ANTIREFLECTION COATINGS. A . MUSSET.A . THELEN. . . IV . MULTILAYER STATISTICAL PROPERTIES OF LASERLIGHT. H . RISKEN . . . . . . . . V. O F SOURCE-SIZE COMPENSATION IN INTERFERENCE VI . COHERENCE THEORY MICROSCOPY. T. YAMAMOTO . . . . . . . . . . . . . . . . . . . . H . LEVI . . . . . . . . . . . . . . . . V I I . VISIONI N COMMUNICATICN. V I I I . THEORY O F PHOTOELECTRON COUNTING. c. L. MEHTA . . . . . . . .
1-50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
C O N T E N T S O F VOLUME I X (1971) 1.
GAS LASERS A N D T H E I R APPLICATION TO PRECISELENGTHMEASUREMENTS. A . L. BLOOM . . . . . . . . . . . . . . . . . .
1-30
VII
LASERPULSES,A. J. DEMARIA. . . . . . . . . . . . . 11. PICOSECOND THE TURBULENT ATMOSPHERE,J. w . 111. OPTICAL PROPAGATION THROUGH STROHBEHN. . . . . . . . . . . . . . . . . . . . . . . . . . . O F OPTICALBIREFRINGENT NETWORKS,E. 0. AMMANN. . . IV. SYNTHESIS V. MODELOCKINGI N GAS LASERS,L. ALLEN,D. G . C. JONES . . . . . . v . M. AGRANOVICH, v. L. VI. CRYSTAL OPTICS WITH SPATIAL DISPERSION, GINZBURG. . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. APPLICATIONS OF OPTICAL METHODSIN THE DIFFRACTIONTHEORY OF ELASTIC WAVES,K. GNIADEK, J. PETYKIEWICZ . . . . . . . , . . . . DESIGN A N D EXTRAPOLATION METHODSFOR OPTICAL VIII. EVALUATION, SIGNALS, BASEDON USE O F THE PROLATEFUNCTIONS,9 . R. FRIEDEN .
31-71 73-122 123-177 119-234 235-280 281-310 311-407
CONTENTS OF VOLUME X (1972) BANDWIDTH COMPRESSION OF OPTICALIMAGES,T. S. HUANG . . . . . T H EUSE OF IMAGE TUBES AS SHUTTERS,R. W. SMITH . . . . , . . . TOOLSOF THEORETICAL QUANTUM OPTICS,M. 0. SCULLY,K. G . WHITNEY TELESCOPES, C. G . WYNNE . . F I E L DCORRECTORS FOR ASTRONOMICAL OPTICAL ABSORPTIONSTRENGTH OF DEFECTS I N INSULATORS,D. Y. SMITH,D. L. DEXTER . . . . . . . . . . . . . . . . . . . . . . . VI. ELASTOOPTIC LIGHTMODULATION A N D DEFLECTION, E. K. SITTIG . . . V I I . QUANTUM DETECTIONTHEORY,C. W. HELSTROM . . . . . . . . . . 1.
11. 111. IV. V.
1-44 45-87 89- 135 137- 164 165-228 229-288 289-369
CONTENTS OF VOLUME XI (1973) MASTEREQUATION METHODSIN QUANTUMOPTICS, G. s. AGARWAL, . I N FARINFRARED SPECTROSCOPIC TECHNIQUES, 11. RECENTDEVELOPMENTS H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . 111. INTERACTIONOF LIGHTA N D ACOUSTICSURFACEWAVES,E. G. LEAN. . IV. EVANESCENT WAVES I N OPTICAL IMAGING, 0.BRYNGDAHL. . . . . . V. PRODUCTION OF ELECTRONPROBES USING A FIELDEMISSIONSOURCE, A.V. CREWE . . . . . . . . . . . . . . . . . . . . . , . . . . . v1. HAMILTONIAN THEORY OF BEAMMODE PROPAGATION, J. A. ARNAUD , VII. GRADIENT INDEX LENSES,E. W. M A R C H A N D . . . , . . . . . , . . 1.
1-76 77-122 123-166 167-221 223-246 247-304 305-337
C O N T E N T S O F V O L U M E XI1 (1974) 1. 11. 111.
IV. V. VI
SELF-FOCUSING,SELF-TRAPPING, A N D SELF-PHASEMODULATIONOF LASERBEAMS,0. SVELTO . . . . . . . . . . . . . . . . . . . . . SELF-INDUCED TRANSPARENCY, R. E. SLUSHER. . . . . . . . , . . . MODULATION TECHNIQUES I N SPECTROMETRY, M. HARWIT,J. A. DECKER JR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS,K. H. DREXHAGE . . . . . . . . . . . . . . . . . . . . . . . . .. . . THEPHASE TRANSITION CONCEPT AND COHERENCE I N ATOMICEMISSION, R . G R A H A M. . . . . . . . . . . . . . . . . . . . . . . . . . . BEAM-FOIL SPECTROSCOPY, S. BASHKIN. . . . . . . . . . . . . . .
1-51 53- 100 101-162 163-232 233-286 287-344
C O N T E N T S O F V O L U M E XI11 ( 1 9 7 6 ) I.
O N THE VALIDITYOF KIRCHHOFF’S LAWOF HEATRADIATIONFOR A BODY I N A NONEQUlLlDRlUM ENVIRONMENT, H. P. BALTES . . .. . ..
..
1-25
Vlll
II. 111.
IV. V. VI.
THECASE FOR A N D AGAINST SEMICLASSICAL RADIATION THEORY, L. MANDEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-68 OBJECTIVE A N D SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS OF THE HUMANEYE,W.M. ROSENBLUM, J. L. CHRISTENSEN. . . , . . . 69-91 INTERFEROMETRIC TESTING OF SMOOTHSURFACES,G. SCHULZ,J. SCHWIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93- 167 SELFFOCUSING OF LASER BEAMS I N PLASMAS A N D SEMICONDUCTORS, M. S. SODHA,A. K. GHATAK, V. K.TRIPATHI. . . . . . . . . . . . 169-265 APLANATISMA N D ISOPLANATISM, W. T. WELFORD . . . . . , . . . . 267-292
C O N T E N T S O F V O L U M E XIV (1977) SPECKLE PATTERNS, J. c. DAINTY . . . . . . . . . TECHNIQUES IN OPTICAL ASTRONOMY, A. LABEYRIE. RELAXATION PHENOMENA IN RARE-EARTH LUMINESCENCE, L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . IV. THEULTRAFASTOPTICAL KERRSHUTTER, M. A. DUGUAY. . . . . . . V. HOLOGRAPHIC DIFFRACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . P. J. VERNIER . . . . . . . . . . . . . . . . . . . V I . PHOTOEMISSION, VII. OPTICAL F I B R E WAVEGUIDES-AREVIEW.P. J. B. CLARRICOATS . . . .
I.
THESTATISTICS OF
11. 111.
HIGH-RESOLUTION
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 ) 1.
II. 111.
IV.
v.
THEORY OF OPTICAL PARAMETRIC AMPLIFICATION A N D OSCILLATION, w. BRUNNER,H. PAUL . . . . . . . . . . . . . . . . . . . . . . . . OPTICAL PROPERTIES OF THINMETALFILMS, P. ROUARD,A. MEESSEN. PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI. . . . . . . . . . . . . QUASI-OPTICAL TECHNIQUES OF RADIOASTRONOMY,T. W. COLE . . . FOUNDATIONS OF THE MACROSCOPICELECTROMAGNETIC THEORYOF DIELECTRIC MEDIA,J. V A N KRANENDONK, J. E. SlPE . . . . . . . . .
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) I. 11.
LASERSELECTIVE PHOTOPHYSICS A N D PHOTOCHEMISTRY, V. S. LETOKHOV RECENT ADVANCES I N PHASE PROFILES GENERATION, J. J. CLAIR, c. I.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71-1 17 COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES A N D APPLICATIONS, W.-H. LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-232 IV. SPECKLE INTERFEROMETRY, A. E. ENNOS . . , . . . . . . . . . . . 233-288 v. DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL RECOGNITION,D. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . . . . . 289-356 v1. LIGHT EMISSIONFROMHIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLYI11 . . . . . . . . . . . . . . . . . . . . . . . . . 357-411 V I I . SEMICLASSICAL RADIATION THEORYWITHIN A QUANTUM-MECHANICAL FRAMEWORK, I. R. SENITZKY. . . . . . . . . . . . . . . . . . . . 413-448 111.
ABITBOL.
1-69
IX
C O N T E N T S OF VOLUME XVII (1980) 1.
II. Ill. IV. V.
HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. DANDLIKER . . . . 1-84 DOPPLER-FREE MULTIPHOTON SPECTROSCOPY, E.GIACOBINO, B. CAGNAC 85- 162 THEMUTUALDEPENDENCE BETWEENCOHERENCE PROPERTIES O F L I G H T AND NONLINEAR OPTICAL PROCESSES, M. SCHUBERT, B. WILHELMI . . 163-238 MICHIELSON STELLAR INTERFEROMETRY, W. J. TANGO, R. Q.T w I ~. ~. 239-278 SELF-FOCUSING MEDIA WITH VARIABLE INDEX O F REFRACTION,A. L. MlKAELlAN . . . . . . . . . . . . . . . . . . . . . . . . . . 279-345
.
C O N T E N T S OF VOLUME XVIII (1980) GRADEDINDEX OPTICALWAVEGUIDES: A REVIEW, A. GHATAK, K. THYAGARAJAN . . . . . . . . . . . . . . . . . . . . . . . . . . 1-126 11. PHOTOCOUNT STATISTICS OF RADIATION PROPAGATING THROUGH RANDOMAND NONLINEAR MEDIA,J. P E g l N A . . . . . . . . . . . . 127-203 111. STRONG FLUCTUATIONS IN LIGHTPROPAGATION IN A RANDOMLY INHOMOv. u. ZAVOROTNYI . . . . . . . . 204-256 GENEOUS M E D I U M , v. 1. TATARSKII, IV. CATASTROPHE OPTICS: MORPHOLOGIES O F CAUSTICS AND THEIR D I F FRACTION PATTERNS, M. v. BERRY,c. UPSTILL . . . . . . . . . . . . 257-346
I.
C O N T E N T S O F VOLUME XIX (1981) I. 11 111.
THEORY O F INTENSITY DEPENDENT RESONANCE L I G H T SCATTERING AND RESONANCEFLUORESCENCE, B. R. MOLLOW , . . . . . . . . . . . . SURFACE AND S I Z E EFFECTSO N THE LIGHTSCATTERING SPECTRA O F SOLIDS,D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . . . LIGHT S C A n E R l N G SPECTROSCOPY O F SURFACE ELECTROMAGNETIC WAVES IN SOLIDS,
IV V.
s. USHlODA . .
. .
. .
. . . .
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. . .
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. .
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. .
. . .
. . .
.
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45-137
. . . . .
139-210 21 1-280
.
281-376
PRINCIPLES O F OPTICAL DATA-PROCESSING, H. J. BUTTERWECK . . THEEFFECTSO F ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY,F. RODDlER
1-43
.
.
C O N T E N T S O F VOLUME XX (1983) I
S O M E NEWOPTICAL D E S I G N S FOR ULTRA-VIOLET BlDlMENSlONAL DETECTION O F ASTRONOMICAL OBJECTS, G . COURTZS, P. CRUVELLIER, M.
. . .
11. 111.
IV. V.
.
. . . . . ,. . c. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE . . . . . . . . . . . . . . . . . . . MULTI-PHOTON SCATTERING MOLECULAR SPECTROSCOPY. s. KlELlCH . COLOUR HOLOGRAPHY, P. HARIHARAN . . . . . . . . . , . . . . .
63-154 155-262 263-324
GENERATION O F TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION, JAMROZ,B. P. STOICHEFF . . , . . . . . . .
325-380
DETAILLE, M. SAYSSE . . . . . . . . . . . SHAPING AND ANALYSIS O F PICOSECOND L I G H T PULSES,
w.
.
. .
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. ..
1-62
X
C O N T E N T S O F VOLUME X X I ( 1 9 8 4 ) I. 11. 111.
IV.
V.
RIGOROUS VECTOR THEORIES O F DIFFRACTION GRATINGS, D. MAYSTRE . THEORY O F OPTICAL BISTABILITY, L. A. LUGIATO. . . . . . . . . . . THERADONTRANSFORM AND ITS APPLICATIONS, H. H. BARRET-T . . . Z O N E PLATE C O D E D IMAGING: THEORY AND APPLICATIONS, N.M. CEGLIO, D. W. SWEENEY . . . . . . . . . . . . . . . . . . . . . . . . . . FLUCTUATIONS, INSTABILITIES AND C H A O S IN T H E LASER-DRIVEN NONLINEAR R I N G CAVITY, J. c. ENGLUND, R. R. SNAPP, w. c. SCHlEVE . . .
1-68 69-216 217-286 287-354 355-428
CONTENTS O F VOLUME X X I I ( 1 9 8 5 ) 1. 11.
Ill. IV.
V. VI.
AND ELECTRONIC PROCESSING O F MEDICAL IMAGES, D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . MALACARA QUANTUM FLUCTUATIONS IN VISION, M. A. BOUMAN,w. A. VAN D E G R I N D , P. ZUIDEMA. . . . . . . . . . . . . . . . . . . . . . . . . . . SPECTRAL AND TEMPORAL FLUCTUATIONS O F BROAD-BAND LASER RADIATION, A. V. MASALOV . . . . . . . . . . . . . . . . . . . . HOLOGRAPHIC METHODSOF PLASMA DIAGNOSTICS, G. v. OSTROVSKAYA, Yu. I. OSTROVSKY . . . . . . . . . . . . . . . . . . . . . . . . . FRINGE FORMATIONS IN DEFORMATION AND VIBRATION MEASUREMENTS USING LASERLIGHT, I. YAMAGUCHI . . . . . . . . . . . . . . . . WAVEPROPAGATION I N RANDOMMEDIA:A SYSTEMS APPROACH, R. L. FANTE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OPTICAL
1-76 77-144
145-196 197-270 271-340
341-398
PREFACE As the contents of science journals and the programs of technical meetings
amply attest, there is no shortage at the present time of interesting new developments in optics and in related fields. This volume of PROGRESS IN OPTICS presents five review articles covering some of these developments. The articles deal with the theory of multiple scattering from rough surfaces, the use of Gaussian brackets in optical design, the production of optical films by ion-based techniques, electron holography and optical processing with partially coherent light. Roughly one half of the volume is concerned with theory and the other half with practical problems. I would like to use this opportunity to welcome as new member of the Editorial Advisory Board of this series Professor Franco Gori of the University of Rome, who is well known for his fine contributions to modern optics. EMILWOLF Department of Physics and Astronomy University of Rochester Rochester, NY 14627, USA May 1986
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CONTENTS I . ANALYTICAL TECHNIQUES FOR MULTIPLE SCATTERING FROM ROUGH SURFACES by JOHNA . DESANTO(GOLDEN. COLORADO)
and
GARY
s. BROWN(CHATMAN. NORTHCAROLINA)
1. INTRODUC~ION . . . . . . . . . . . . . . . . . . . . . 2. DEFINITION OF ROUGHNESSA N D SURFACE STATISTICS . . . . 2 .I Rayleigh (vertical) roughness criterion . . . . . . . . . . 2.2 Horizontal roughness and fractals . . . . . . . . . . . . 2.3 Additional statistical properties . . . . . . . . . . . . . 3. SINGLESCATTERING. . . . . . . . . . . . . . . . . . . 3.1 Flat interface . . . . . . . . . . . . . . . . . . . . 3.2 Perturbation theory . . . . . . . . . . . . . . . . . . 3.3 Tangent plane models . . . . . . . . . . . . . . . . . 3.4 Kirchoff approximation . . . . . . . . . . . . . . . . 4. MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCA'ITERING . 4.1 Scalar (acoustic) case . . . . . . . . . . . . . . . . . 4.2 Electromagnetic case . . . . . . . . . . . . . . . . . 4.3 Electromagnetic waves and angular spectra . . . . . . . 5. LIPPMANN-SCHWINGER EQUATION AND DIAGRAMS. . . . . 6 . SCATTERED FIELD STATISTICS . . . . . . . . . . . . . . . 7. k-SPACE FORMALISM-RANDOM SURFACE . . . . . . . . . . 7.1 Connected diagram method . . . . . . . . . . . . . . 7.2 Stochastic Fourier transform . . . . . . . . . . . . . . 8. SMOOTHING . . . . . . . . . . . . . . . . . . . . . . . 8.1 Coordinate space formulation . . . . . . . . . . . . . 8.2 Relation between smoothing and diagram methods . . . . 9 . OTHER MULTIPLE SCATTERING APPROACHES. . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . .
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3 6 6 8 10 11 11 14
17 19 22 22 26 28 32 36 39 39 45 50 51 53 56 59
I I . PARAXIAL THEORY IN OPTICAL DESIGN IN TERMS O F GAUSSIAN BRACKETS by KAZUOTANAKA (TOKYO. JAPAN) 1 . INTRODUCrlON
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2 . GAUSSIAN BRACKETS . . . . . . . . . . . . . . . . 2.1 Definition of Gaussian brackets . . . . . . . . . . 2.2 Characteristics of Gaussian brackets . . . . . . . 3. G E N E R A L I Z E D G A U S S I A N CONSTANTS (GGC'S) . . . . 3.1 Definition of GGC's . . . . . . . . . . . . . . . 3.2 Various formulae of GGC's . . . . . . . . . . . .
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65 67 67 69 72 72 73
XIV
CONTENTS
4 . PARAXIAL THEORY OF A HOMOGENEOUS OPTICALSYSTEMBY MEANS OF GGC's 4.1 Paraxial ray trace . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conjugate relation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Magnifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Cardinal points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Lagrange-Helmholtz invariant . . . . . . . . . . . . . . . . . . . . . 4.6 Other miscellaneous formulae . . . . . . . . . . . . . . . . . . . . . 4.7 Physical meaning of GGC's . . . . . . . . . . . . . . . . . . . . . . 5. PARAXIAL THEORY OF A N INHOMOGENEOUS OPTICAL SYSTEMBY MEANS OF GGC's 5.1 Paraxial ray transfer . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lagrange-Helmholtz invariant . . . . . . . . . . . . . . . . . . . . . 5.3 Equivalent homogeneous model by means of GGC's . . . . . . . . . . . . 6 . PARAXIAL THEORY OF A GAUSSIAN BEAMOPTICAL SYSTEM BY MEANS OF GGC'S 6.1 Gaussian beam parameters . . . . . . . . . . . . . . . . . . . . . . 6.2 Propagation of a complex beam parameter . . . . . . . . . . . . . . . . 6.3 Gaussian beam parameters by means of GGC's . . . . . . . . . . . . . . 6.3 Complex beam invariants . . . . . . . . . . . . . . . . . . . . . . . 7. FEATURES OF PARAXIAL THEORY BY MEANS OF GGC'S . . . . . . . . . . . . 8 . APPLICATION EXAMPLES OF PARAXIAL THEORY BY MEANS OF GGC'S . . . . . 8.1 Critical point and singular point of the zoom equation . . . . . . . . . . . 8.2 Equivalent homogeneous model of a gradient-index singlet . . . . . . . . . 8.3 Adjustment of complex beam parameters . . . . . . . . . . . . . . . . 9. CONCLUDING NOTE . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX A . NUMERICAL CALCULATION OF GGC'S . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 75 77 79 80 83 84 85 86 87 88 89 92 93 94 94 96 97 98 98 101
103 105 106 107 107
Ill . OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES by P. J . MARTINand R . P . NElTERFlELD (SYDNEY, AUSTRALIA)
I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. ION-SURFACE INTERACTION PHENOMENA . . . . . . . . . . . . . . . . . 2 I Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. I Preferential sputtering . . . . . . . . . . . . . . . . . . . . . . 2.2 Energies of sputtered particles . . . . . . . . . . . . . . . . . . . . . 2.3 Atomic mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 Ion reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Ion trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Spike phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. INFLUENCEOF ION BOMBARDMENT ON FILMPROPERTIES . . . . . . . . . 3.1 Structure of thin films . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Ion Bombardment effects on film structure . . . . . . . . . . . . . . . 3.3 Adhesion and stress . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Compound formation . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . IION-BASED METHODSO F THIN-FILM DEPOSITION . . . . . . . . . . . . . 4.1 Vacuum evaporation . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Reactive evaporation . . . . . . . . . . . . . . . . . . . . . . .
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115 117 118 119 122 123 124 125 126 128 128 132 134 135 137 137 137 139
CONTENTS
4.2 Ion plating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Ion-assisted deposition (IAD) . . . . . . . . . . . . . . . . . . . . . 4.4 Sputter deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Planar diode sputtering . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Magnetron sputtering . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Ion-beam sputtering . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ion-beam deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Ionized cluster beam deposition (ICB) . . . . . . . . . . . . . .
XV
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5 . OPTICAL THIN FILMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Silicon dioxide . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Aluminium oxide . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Titanium dioxide . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Zirconium dioxide . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Cerium dioxide . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Tantalum pentoxide . . . . . . . . . . . . . . . . . . . . . . . 5.2 Carbon films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Other materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Multilayer coatings . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Materials mixing . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Optical waveguides . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 High-power laser coatings . . . . . . . . . . . . . . . . . . . . 5.4.5 Protective layers . . . . . . . . . . . . . . . . . . . . . . . . . .
6. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139 140 142 142 143 144 146 146 148 148 148 149 150 158 162 166 167 172 173 173 174 175 176 177 177 178 178
IV . ELECTRON HOLOGRAPHY by AKIRATONOMURA (TOKYO.JAPAN)
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. COHERENCE OF A N ELECTRONBEAM . . . . . . . . . . . . . . . . . . . . 2.1 Time coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spatial coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. ELECTRONHOLOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . 3.1 In-line holography . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Off-axis holography . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . APPLICATIONS OF ELECTRON HOLOGRAPHY . . . . . . . . . . . . . . . . . 4 . I Correction of electron lens aberration . . . . . . . . . . . . . . . . . . 4.2 Measurement of thickness distribution . . . . . . . . . . . . . . . . . . 4.3 Observation of microscopic magnetic fields . . . . . . . . . . . . . . . . 5. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
185 187 188 191 194 194 201 204 204 206 211 217 218
XVI
CONTENTS
V . PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT bv F. T . S . Yu (UNIVERSITY PARK,PENNSYLVANIA) 1. INTRODUC~ION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COHERENT PROCESSING. . . . . . . . . . . . . . . . . . . . 2. PARTIALLY 2.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Spatially partially coherent processing . . . . . . . . . . . . . . . 2.1.2 Achromatic optical processing . . . . . . . . . . . . . . . . . . . 2.1.3 Band-limited partially coherent processing . . . . . . . . . . . . . . 2. I .4 Achromatic partially coherent processing . . . . . . . . . . . . . . 2.2 White-light processing . . . . . . . . . . . . . . . . . . . . . . . . . PROPAGATION. . . . . . . . . . . . . . . . . . . . . . . . 3. COHERENCE 3.1 Propagation of the mutual intensity function . . . . . . . . . . . . . . . 3.2 General formulation . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Coherence requirement for image correlation . . . . . . . . . . . . . . . 3.3.1 Temporal coherence requirement . . . . . . . . . . . . . . . . . 3.3.2 Spatial coherence requirement . . . . . . . . . . . . . . . . . . . 4. TRANSFER FUNCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Temporal coherence . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Spatial coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. NOISEPERFORMANCE. . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Noise performance under temporally partially coherent regime . . . . . . . 5.1.1 Noise at the input plane . . . . . . . . . . . . . . . . . . . . . 5.1.1.1 Weak phase noise . . . . . . . . . . . . . . . . . . . . . 5.1.1.2 Amplitude noise . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Noise at the Fourier plane . . . . . . . . . . . . . . . . . . . . 5.1.2. I Weak phase noise . . . . . . . . . . . . . . . . . . . . . 5.1.2.2 Amplitude noise . . . . . . . . . . . . . . . . . . . . . . 5.2 Noise performance under spatially partially coherent regime . . . . . . . . 5.2.I Noise at the input plane . . . . . . . . . . . . . . . . . . . . . 5.2.2 Noise at the Fourier plane . . . . . . . . . . . . . . . . . . . . 6. SOURCE ENCODINGA N D IMAGE SAMPLING . . . . . . . . . . . . . . . . . 6.1 Source encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Image sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 An application of source encoding . . . . . . . . . . . . . . . . . . . REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . 7. CONCLUDING . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES
223 224 225 228 229 229 230 231 233 233 236 238 238 241 245 247 248 251 252 254 254 256 258 259 259 261 263 266 267 267 270 271 273 273 274
E. WOLF,PROGRESS IN OPTICS XXIII 0 ELSEVIER SCIENCE PUBLISHERS B.V.. 1986
I
ANALYTICAL TECHNIQUES FOR MULTIPLE SCATTERING FROM ROUGH SURFACES BY
JOHN A. DESANTO Center for Wave Phenomena, Maihemaiics Deparimeni Colorado School of Mines Golden, Colorado 80401, USA
GARYS. BROWN Department of Electrical Engineering Virginia Polytechnic Insiiiuie and Siaie Universiiy Blacksburg, Virginia 24061, USA
CONTENTS PAGE
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3
$ 2 . DEFINITION OF ROUGHNESS AND SURFACE STATISTICS . . . . . . . . . . . . . . . . . . . . . . . . $ 3 . SINGLE SCATTERING . . . . . . . . . . . . . . . . .
6
$ 1. INTRODUCTION
11
$ 4. MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING . . . . . . . . . . . . . . . . . . . . . 22
$ 5. LIPPMANN-SCHWINGER EQUATION AND DIAGRAMS
32
. . . . . . . . . . $ 7. k-SPACE FORMALISM-RANDOM SURFACE . . . . $ 8 . SMOOTHING . . . . . . . . . . . . . . . . . . . $ 9. OTHER MULTIPLE SCATTERING APPROACHES . . REFERENCES . . . . . . . . . . . . . . . . . . . . .
36
$ 6. SCATTERED FIELD STATISTICS
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39
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56
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8 1. Introduction All real surfaces are rough. The only questions are those of degree and type of roughness. The degree of roughness depends on both the geometry and the wavelength of the incident probe. The types of rough surfaces are generally classified into periodic rough surfaces such as diffraction gratings and nonperiodic surface variability which includes random rough surfaces. We will restrict our attention here to the theoretical treatment of the latter class of surfaces, although, as will be seen, much of the development is in terms of a stochastic surface and stochastic equations which in principle hold true for an arbitrary non-stochastic surface as well. Review papers on periodic surface [ 1984a1, DESANTO[ 1981al) are available as are several scattering (MAYSTRE books on the subject (PETIT[ 19801, HUTLEY[ 19821 and WILCOX[ 19841). There are theories which treat combined surfaces having both deterministic [ 19631) and which treat the randomness as and random parts (KURYANOV vertical variability (BROWN[ 1983a1, DESANTO[ 1981bl or as horizontal variability of deterministically defined elements (TWERSKY[ 19571, BIOT [1968] and TOLSTOY[1984]. Papers on rough surface scattering span a multitude of disciplines. They include classical optics (FANO[ 19411, TOIGO, MARVIN, CELLIand HILL[ 1977]), classical acoustics (RAYLEIGH[ 19451, [ 19751, ZIPFELand DESANTO[ 1972]), interpretation of radar WATERMAN [ 19701, VALENZUELA [ 1968]), electromagnetic scattering returns (BARRICK (BROWN[ 19781, DESANTO[ 1974]), the study of physical properties of solid [ 19771). geophysical scattering surfaces by atomic beam scattering (GOODMAN problems (HILL and WUENSCHEL [ 19851, DESANTO[ 19731, Raman and Brillouin scattering from acoustic phonons in metals (BENNETT, MARADUDIN and SWANSON [ 1972]), and light scattering from dielectrics or the interaction of photons with the electron gas of a metal (CELLI,MARVINand TOIGO [ 1975]), to mention but a few areas. The problem can also be treated as a random potential scattering problem as in quantum mechanics (SHENand [ 19801) and we will discuss this in some detail later. Examples MARADUDIN of rough surfaces on the macroscopic geometrical level include terrain (MOORE [ 19701) and the sea surface (VALENZUELA [ 19781, LONGUET-HIGGINS [ 19831); and on a microscopic level thin films (BOUSQUET, FLORY and ROCHE
4
MULTIPLE SCAlTEKING FROM ROUGH SURFACES
[I. 8 1
[ 19811) and coatings (AMES,HALLand BRAUNDMEIER [ 19821). There are two classic books on the subject of rough surface scattering, each of which treats the surface from a different point of view. The book by BASSand FUKS[ 19791 concentrates on small height perturbation theory and the book by BECKMANN and SP~ZZICHINO [ 19631 treats the surface with emphasis on the Kirchhoff approximation. Both areas are discussed further in the text. There are also many review papers (e.g. SHMELEV [ 19721, MCGINNand SYKES[ 19801). On first encountering this vast array of seemingly diverse literature, there appears to be a corresponding diversity in methodology. This is not so. For example, if we ask for exact formulations, there are only two, the integral equation approach and what can be generally called the modal expansion method, For the former method using Green’s theorem we derive an integral equation on the field or full Green’s function for the problem. The Green’s function approach is more useful since it permits us to treat many different incident fields. For the latter method we expand in some complete set of functions and use the boundary conditions to generate equations on the expansion coefficients. Both methods are valid for an arbitrary functionally rough surface. The random nature of the boundary plays no intrinsic role. For a random boundary a major question arises as to how to solve the above formalisms. The equation and boundary conditions are linear in the field. The dependence on the surface height is in general nonlinear. The result is a nonlinear functional equation. The first approach to solution, a classical approach, is to solve the method for a given surface representation or member of the ensemble of realizations of the surface and then to average. Here, what is meant by the word “solve” is crucial. This approach has been most fruitful when approximate solutions to the problems are first developed, and then averaged. This has also been the most common approach to the problem. Examples include the use of asymptotic methods such as perturbation theory (BASS and FUKS [ 1979]), Kirchhoff approximation (BECKMANNand [ 19631) and variational methods (ANDREOand KRILL[ 19811). SPIZZICHINO Alternatively, a possible exact solution proceeding along these lines is the so-called Monte Car10 method. Here a statistical ensemble of solutions is created one at a time using the statistical ensemble of surface realizations. That is, to each surface realization there corresponds an exact (and almost always numerical) solution. The procedure is analogous to solving finite body scattering problems, here applied to a surface of infinite extent. As the surface realization is varied using its statistics, the ensemble of fields is created, any moment of which can then be formed and averaged. As is obvious, this is highly computer-intensive and has not been extensively studied, although MAYSTRE
1 9 0
11
INTRODUCTION
5
[ 1984bl demonstrated a “short-range coupling” phenomenon using this type of approach. The latter phenomenon enables one to characterize the scattering using an effective surface horizontal dimension scale and can lead to rigorous truncation methods for computational solutions. The second approach, and the one we shall emphasize, we term the modem approach. In this approach the exact stochastic integral equations are used to develop equations on the averages of the field quantities. The latter can also be made exact but, in order to be solved, must be approximated. In contrast to the classical approach the random nature of the problem is exploited ab initio. Examples of this technique include the connected diagram (DESANTO [ 1981b]), smoothing (WATSONand KELLER[ 19841, BROWN[ 1984a]), stochastic Fourier transform (BROWN[ 1982a]), probabilistic (CHOW[ 19781) and self-consistent (CHITANVIS and LAX[ 19851) approaches. Some of these methods have also been used in problems of wave propagation in random media (FRISCH[ 19681). The advantageofthe modem approach is that it yields simpler equations to solve. The disadvantageis that truncation techniques need to be employed to make the solution tractable and there is a difficulty in physically interpreting the neglected terms. We will discuss both advantages and disadvantages in subsequent sections. The following sections comprise a mathematically oriented review of rough surface scattering theory. In Q 2 the basic concepts of vertical and horizontal scales of roughness are introduced and a very succinct review of the kinds of surface statistics required in subsequent analyses is provided. The common single scattering approximations that are used in scalar scatteringproblems are reviewed in Q 3. These include the flat interface, perturbation theory, tangent plane method and the Kirchhoff approximation. The basic integral equations which form the foundation for nearly all multiple scattering formalisms are introduced in Q 4. This is done first for the scalar case and then for the electromagnetic or vector case. The section is concluded by introducing the angular spectral approach which unifies most of the approximations introduced in 8 3. In Q 5 the specific approaches presented in Q 4 are generalized by deriving a general Lippmann-Schwinger equation for the rough surface scattering problem. A brief discussion of scattered field statistics is presented in Q 6 along with comments on the difficulty associated with finding the probability density function for the scattered field. In 8 7 specific k-space formalisms are presented for dealing with multiple scattering for randomly rough surfaces, e.g. the general three dimensional Fourier transform approach and the stochasticFourier transform technique. In Q 8 the method of smoothing as applied to the rough surface multiple scattering situation is treated and
6
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. 5 2
compared with the results obtained from a diagram approach. The chapter is concluded in 8 9 with a necessarily brief discussion of other recent multiple scattering approaches which have not been discussed otherwise. Because of space limitations, this review is necessarily abbreviated. However, an extensive list of applicable references is provided so that the interested reader can easily find more material on the topics covered herein.
9 2. Definition of Roughness and Surface Statistics 2.1. RAYLEIGH (VERTICAL) ROUGHNESS CRITERION
The most commonly used definition of roughness is due to RAYLEIGH [ 19451 and is called the Rayleigh roughness criterion. Essentially it treats the plane wave or ray incident on a rough surface as being scattered specularly and modulated only in its phase by the vertical roughness elements. The geometrical picture is illustrated in Fig. 1. For a wave front incident at angle 0 defined from the vertical, the additional path length A$ (phase modulation) traversed by ray (2) relative to that path length for ray (1) which would specularly scatter from a flat surface in the absence of the (random) roughness h is given by A$ = 2xk
=
2hk cos 0 = 2hkz,
(2.1)
where k is the wavenumber ( = 241, where 1 is the wavelength). The rms phase modification is = ~ ~ U C O S2Z. ~ =
(2.2)
Fig. I . Rough surface element used in the definition of the Rayleigh roughness parameter.
I , § 21
DEFINITION OF ROUGHNESS AND SURFACE STATISTICS
7
where the brackets indicate the average over the statistical ensemble of surfaces, o is the r m s height, and Z the Rayleigh roughness parameter. The Rayleigh roughness criterion is that the surface is “smooth” if A $ < n/2 (ocos 6/L< or C < n/4)and otherwise “rough”. Here h is treated as a zero mean random process, so ( h ) = 0. The specific mathematical evaluation of the ensemble average is an integral over the probability density of surface heights, p(h), given by .r
*
(h) =
J
hp(h)dh.
--sc
For ensemble averages of products of surface height functions we require joint height probability density functions (BROWN[ 1982b1). We will discuss this further later in this section. KINSMAN[ 19831 provides an excellent source reference for the statistical description of a surface which is random both in time and space. We confine our attention in this review to spatial randomness. Specific measurements of spatial roughness parameters can be found in the papers by BENNETT[ 19761, CHURCH, JENKINSON and ZAVADA[ 19791, and ARCHBOLD and ENNOS[ 19721. In the above definition we have implicitly defined a reference surface, such that h coincides with one coordinate of a three-dimensional coordinate system. We generally choose this as the z-direction. The two-dimensional vector x, = ( X J ) is the orthogonal (transverse or horizontal) vector in the reference surface to the point at which h is measured, thus h = h(x,). We sometimes, for convenience in illustrating results, restrict ourselves to a one-dimensional dependence on x alone. We can always choose this reference surface such that expression (2.3) vanishes, i.e. so that we have the above-mentioned zero mean random process. The second moment of h,
is then a measure of the deviation of the process from its mean value. For a Gaussian distribution of surface heights, given by
it is easily verified that ( h )
=
0 and (h2)
= a’.
The Fourier transform of the
8
MULTIPLE SCATTERING FROM ROUGH SURFACES
probability density function
p(k)=
f
exp( - ikh) p(h) dh
--oo
is called the characteristic function for h and as we will see it occurs quite frequently in rough surface scattering. As an example of this vertical roughness effect, the z-dependent part of the field scattered from an undulating surface of this type can be represented as P ( z ) = R exp(ik,z
+ id+),
(2.7)
where R is the Rayleigh-Fresnel reflection coefficient (BREKHOVSKIKH [ 19601). The average or coherent scattered field is proportional to (@““(z)) = R exp(ik,z) (exp(id+)),
(2.8)
where A+ is given by eq. (2.1). For a Gaussian distribution of surface heights the ensemble average of the phase variability is just the two-way Fourier transform of p(h): (exp(id+))
=
3
exp(2ihkz) p(h) dh = exp( - 2Z2).
(2.9)
--OD
The surface undulation thus requires us, in the mean, to replace R in the coherent scattered field by R exp( - 2Z2), a multiplication by an additional decay function depending on frequency, incidence angle, and rms height. The coherent scattered intensity is thus modified by the factor exp( - 4 2 3 . No horizontal surface variability is present, only vertical variability. This important coherent scattering result seems to have been first published by AMENT[ 19531 and ECKART[ 19531.
2.2. HORIZONTAL ROUGHNESS AND FRACTALS
An alternate criterion for surface roughness can be developed using the horizontal variability of the surface. First, define the normalized correlation function C of the surface roughness:
where x,
= (x,y)
is the horizontal two-vector transverse (t) to the vertical
1, § 21
DEFINITION OF ROUGHNESS AND SURFACE STATISTICS
9
scattering direction z. In defining C we have invoked the concept of statistical homogeneity or statistical translational invariance of the surface. These are statistics which do not vary over the horizontal extent of the surface and are therefore independent of the coordinate origin. Nonhomogeneous surface statistics significantly complicate the surface scattering problem and will not be considered here. A useful correlation function example is the Gaussian C(x,) = exp( - x2/L; - y 2 / ~ : ) ,
(2.11)
where we have included the possibility of transverse anisotropy (L,# L,,). Here Lx and L,,are the correlation distances in the x and y directions, respectively. The correlation distances are usually defined from the l/e values of the correlation function. Other differential and integraldefinitions of the correlation distances are possible (BASSand FUKS[ 19791). The latter shift the correlation scales to smaller and larger values, respectively. The correlation function thus provides a measure of how far one must spatially separate two stochastic variables before they are uncorrelated. Next, define the spectral function as the Fourier transform of the correlation function :
S(k,) = 0'
ss
C(x,) exp( - ik, * x,) dx,.
(2.12)
This provides an indication of how the various contributions to the mean squared height are distributed in the spatial frequency domian, i.e. in k,-space. In one dimension assume the surface height satisfies the Lipschitz condition Ih(x + A) - h(x)l A" (2.13)
-
for small A. The value a is called the Lipschitz exponent. If a = 0 the surface is discontinuous. If a = 1 the surface is differentiable. For 0 < a < 1 the surface is continuous but not differentiable, i.e. it is a fractal. It can be shown (ROTHROCK and THORNDIKE [ 19801) that the spectral function for a surface of this type behaves asymptotically as
-
S(K) K p ,
(2.14)
where p = - 2a - 1 is the spectral exponent and K is the Fourier transform variable conjugate to A. For - 3 < p < - 1 the surface is a fractal which is a surface of a different character of (horizontal)roughness than the differentiable surfaces where p 5 - 3. A simple physical example appears to be the ice surface. JAKEMAN [1982] for example has considered the case of Fresnel scattering by a corrugated random surface with fractal slope.
10
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. 8 2
2.3. ADDITIONAL STATISTICAL PROPERTIES
Just as in the case of surface height, one can also generate a corresponding density function and moments for surface slopes (ahlax and ahlay), curvatures (a2h/ax2,a2h/axay, a2h/8y2),etc. It may also be necessary to treat joint height and slope statistics with density p(h, a,h) where 8, is the transverse (x, y ) gradient. If h and a,h are statistically independent then P(h, 8th) = Pdh) P2(ath),
(2.15)
i.e. the joint density can be factored into the product of two marginal densities. If h and dth are uncorrelated then (hdth)
=
0.
(2.16)
Statistical independence is a much stronger condition than being uncorrelated (PAPOULIS [ 19651). For a jointly Gaussian surface eqs. (2.15) and (2.16) are equivalent and all the higher surface derivatives have marginal densities which are also Gaussian (LONGUET-HIGGINS [ 19571). For surface scattering, correlation functions for surface height derivatives are also important. These can be obtained from the surface height correlation function defined in eq. (2.10) using the following general result (PAPOULIS [ 19651)
-
where pt = x, - x , . The mean squared value of the ith order surface height derivative is related to the 2ith order derivative of the correlation function evaluated at zero separation as (2.18) The existence of the second moments of the surface height derivatives is determined by the differentiability of the surface. Alternatively this can be expressed as the existence of certain moment integrals of the spectral function. For example, the existence of the function defined in eq. (2.18) is guaranteed if
r
--x
k;'S(k,) dk,
1,
I 31
SINGLE SCA"ER1NG
I1
is finite. For spectra with finite support, all surface height derivatives exist in a mean square sense. Because of the intimate relationship between the mean squared surface derivatives and the behavior of S(k,) as I k,1 + co we must exercise caution in using particular functional forms for S(k,) which are not realistic.
0 3. Single Scattering We define single scattering as those representations of the scattered field which are local in space or, if nonlocal, only depend on a single surface integration. For the flat interface locality is obvious. In perturbation theory the coherent field is local although the reflection coefficient is defined nonlocally via a single surface integration as is the incoherent field. Both tangent plane and Kirchhoff models involve a single surface integration. In the far field locality is recovered although the resulting scattering amplitudes include a single surface integration. This is true for Bragg scattering also. We have already presented one single scattering result for the Rayleigh roughness criterion in 5 2.
3.1. FLAT INTERFACE
The simplest single scattering result occurs when the rough interface is flat. The methodology, while simple and well known, illustrates general principles used in more complicated formulations for rough surfaces. For the scalar (say acoustical) case in two dimensions with plane wave incidence the problem is illustrated in Fig. 2. The fields in each region satisfy a Helmholtz equation with wave numbers k, = w/c,, where o is the circular frequency and c, is the sound speed. The first general principle is that the field in region V ,, @, , can be decomposed into incident (in) and reflected (r) fields (@ is the velocity potential for the scalar case): @](X,z) =
P ( X ,
z) +
W(X,
z),
(3.1)
where P ( x , Z) = Ai exp [ik,(x sin Oi - z cos Oi)],
(3 * 2)
+ z cos O,)].
(3.3)
and @ ( x , Z) = A, exp [ik,(x sin Or
12
MULTIPLE SCA~TERINGFROM ROUGH SURFACES
Fig. 2. Plane wave scattering from a flst interface separating media of different densities 6 and sound speeds c,.
The transmitted field in V2 is Q2(x,z) = A, exp [ik2(x sin 0, - z cos &)I.
(3.4)
The second general principle is that at z = 0 it should not matter what x-value we choose since we have plane waves of infinite extent. That is, we have translational invariance in the x-direction. So the coefficientsof x in each plane wave phase must be equal, i.e.
k, sin 0, = k, sin 0,
= k2 sin 0,.
(3.5)
We conclude that the scattering is specular (Oi = Or). Further, the right-hand equality in eq. (3.5)is Snell's law. Later we will demonstrate that, for a random surface, if its statistics are homogeneous (statistical translational invariance in the plane), the coherent scattered wave is specular. Snell's law however cannot be invoked for the statistical case. The thud general principle is the requirement of continuity of the pressure and normal velocity at the z = 0 interface. These conditions are @dX9
a -
az
0) = P@2(X,01, P = P2/P19
a
@,(x,0 ) = - a22(x,0).
(3.6)
(3.7)
aZ
These yield the Rayleigh-Fresnel reflection (R) and transmission (T)
1 . 5 31
SINGLE SCA'ITERING
13
coefficients given by p cos p cos
ei - ( K 2 - sin2Oi)li2 - p cos Oi - Kcos 8, oi + ( K -~ sin2ei)1'2 p cos ei + K cos e, '
2 cos oi pcosS, + ( K 2 - sin28i)1/2'
(3.8)
(3.9)
where K = k 2 / k , .These boundary conditions also hold true at a stochastic rough interface [see eqs. (4.13)and (4.14)].There are simple limiting examples such as the Dirichlet boundary condition ( p = 0, R = - 1) and the Neumann boundary condition ( p = 00, R = - 1). If we define the medium impedance Zi = pjcj(cosej)-', where el = Oi, 0, = O,, and picj is the normal or wall impedance then R has the form of a transmission line reflection coefficient
R
= (Z2
- Z , ) / ( Z 2 + ZI)*
(3.10)
An impedance boundary condition often used (which however violates Snell's law) is to set 0, = 0, to get
R, = (cosei - ~/p)/(cose,+ ~ / p ) .
(3.11)
The above results can also easily be found for the electromagnetic problem
(BREKHOVSKIKH [ 19601). In this case eqs. (3.6) and (3.7) are the continuity of the appropriate tangential and normal electromagnetic field components. For TE (TM) polarization p is the ratio of permeabilities (dielectric constants) of the two media. In addition it is easy to show that, on the interface, the fields are given by @I
(x, 0) =
2Ai cos ei exp(ik,x sinei), p cos + K cos 6,
(3.12)
and
ao, (x, 0) -- - 2 a i k , K cos ei COS e, exp(ik,x sinei). p cos ei + K cos e, az
(3.13)
For a Dirichlet surface ( p + O ) the derivative term in eq. (3.13)is double the z-derivative of the incident field evaluated at z = 0. For the Neumann problem ( p + "c) the function on the surface is double the value of the incident plane wave evaluated at z = 0. This is the fourth general principle, the doubling of the
14
MULTIPLE SCAlTERING FROM ROUGH SURFACES
[I, 5 3
incident field value at the boundary. For any surface nearly flat this must be true.
3.2. PERTURBATION THEORY
For a stochastic rough surface z = h(x,), we discuss the perturbation solution in powers of the height h for the Dirichlet boundary condition (BASSand FUKS [ 19791) given by @(x,) = 0,
(3.14)
where x = (xt, h(x,)). Since h is stochastic so is the boundary condition. In addition, however, it is local in space as was the flat surface boundary condition. We first expand the boundary condition about h = 0 in a Taylor expansion and drop terms of order h2 and higher. Then eq. (3.14) becomes
a@
@(x,, 0 ) + h 7(x,, 0) = 0. az
(3.15)
To this order in h, the normal derivative term is approximated by the z-derivative (this is clearly violated for surfaces with steep slopes). We express @ as the sum of a coherent or mean field ( @) and an incoherent or fluctuating field u, @ = (@)
+ u,
(3.16)
where the bracket average represents the average over the ensemble of surface realizations. (The field statistics are fully discussed later, in 8 6). In addition we have a zero mean random process ( h ) = 0 and note that from eq. (3.16), ( u ) = 0. Substituting eq. (3.16) into eq. (3.15) and taking the average yields (Q)
+ ( h au/az)
=
0,
(3.17)
which is a boundary condition at the flat surface h = 0 relating the coherent and incoherent fields. Further taking the average of eq. (3.15) and subtracting the result from eq. (3.15) yields (with some approximations) a second boundary condition on the incoherent field: u+h
a az
-
(@)
=O.
(3.18)
Since u is a solution of the homogeneous Helmholtz equation and, in perturbation theory, we have a Dirichlet boundary condition at a flat interface,
1,s 31
I5
SINGLE W A T E R I N G
u, off the boundary, can be written using a Rayleigh diffraction formula (BORN
and WOLF [ 19801): (3.19) where GY is the free space Green's function with wave number k,, viz. GY(x,x,) = [ 4 4 x - x o l ] - ' exp[ik, Ix - x,l] = GY(x - x,,).
(3.20)
Under the integral we can use eq. (3.18), and we can differentiate the lhs with respect to z, multiply by h, average, set z = 0 and use eq. (3.17) on the resulting lhs. The result is, with r, = x; - x,, given by
where the correlation function C and rms height 0 were defined in eq. (2.8). Equation (3.21) is a boundary condition solely on the coherent wave, but whereas the stochastic boundary condition was local, this condition is nonlocal. In the flat surface limit 0 = 0 and we recover locality. We have also used the statistical translational invariance in the correlation function. Note that in the limit L 4 x (C(r,)-, 1) locality is not recovered in perturbation theory. The coherent wave in eq. (3.21) is evaluated on a flat plane. We can thus decompose the coherent field as the sum of an incident wave plus scattered wave given by
(@(n))
=
-
exp [i(k, x, - k,z)]
+ R exp [i(k,
x,
+ k,z)]
(3.22)
with the dispersion relation k: + k,2 = k:, where k, = k, cosei. R is again the reflection coefficient. Substituting the result into eq. (3.21) and solving for the reflection coeficient yields R=
[Z(k,) ~ 0 - ~11 8 [Z(k,) cos + 11 '
~
(3.23)
where C(r,) exp(ik, r,) dr,. (3.24)
16
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I,§ 3
In the flat surface limit we have the Dirichlet reflection coefficient R = - 1. The effective impedance at the flat interface is given by p,c,l(k,). We can use eq. (3.22) to calculate the coherent intensity. The incoherent intensity can be found from eq. (3.19) where we use eq. (3.18) under the integral. The correlation function effectively limits the integrals to the neighborhood of r, = 0. The result can be written as (3.25)
where the geometry is defined in Fig. 3 and S is the spectral function defined in eq. (2.12). Here, the transverse scattered and incident wave vectors are defined by kf‘
=
k , ( x , - X; ) [(x, - X; )2 + z2]- I”,
kp
=
k , X; [xi’
(3.26)
and
+ z; ] - ”~ ,
(3.27)
so that in first-order perturbation theory, the average incoherent scattered intensity depends only on the single surface spectral component given by the difference between incident and scattered wave vectors. Note that the o2 term is contained in the definition of S. Equation (3.25) is a more general result. For example, it illustrates the Bragg resonance phenomenon in the sense that the incoherent intensity is only responding to a single spectral line. It can be shown that resonant peaks occur in the spectral function for out-of-plane scattering (see Fig. 4) at values 2k, sin 0, sin(@/2), which is the Bragg resonance condition (WRIGHT[ 19661). For $ x 0 we get L! = ( X t , Z ) x o = (Ot,Zo) 1 I
+X;* Fig. 3. Geometry and symbols for surface scattering from source x,, to receiver x via a shallow rough surface.
1. I 31
17
SINGLE SCATTERING
-
”
X-
Fig. 4. Geometry for B r a g scattering out of the plane of incidence.
a small Bragg line but this is overwhelmed by a much larger specular return. Whether the small line can be resolved depends on the receiver beam width. It, however, we look in backscatter, @ = n, the Bragg peak itself is still absolutely small, but there is no coherent scattered field present, and the line can be easily identified. The line position then specifies the resonant surface periodicity. Note that up to now in perturbation theory we have only characterized the surface in terms of its vertical variability. The horizontal variability is in the spectral function as in eqs. (2.11) and (2.12) and limits of eq. (3.25) can be found for large and small horizontal variability, as well as for grazing and non-grazing scattering (BASSand FUKS[ 19791). In addition, the method can be extended to other perfectly reflecting boundary value problems, transmission problems (KUPERMAN [ 1975]), and rough walled waveguide propagation (KRYAZHEV, KUDRYASHOV and PETROV[ 19761). More generalized “smoothed” boundary conditions of the type used here can also be derived [ 19741). (WENZEL
3.3. TANGENT PLANE MODELS
For the tangent plane models we begin with Green’s theorem to formulate the Helmholtz integral representation (BORNand WOLF [ 19801) on the field @,
@(x) = G ~ ( xxg) ,
+
a
[ G ~ ( xx, i ) 7@(xi)
an
(3.28)
18
MULTIPLE SCAlTERING FROM ROUGH SURFACES
[I, I 3
written in terms of its surface value @(xi) and normal derivativeon the surface. Gy is the free space Green’s function (incident field) and the source x, and field x coordinate values are as they appear in Fig. 3. For the Dirichlet boundary value problem, the field, decomposed into incident (point source) and scattered components, evaluated on the surface is
(3.30)
we take over the flat surface result in the sense that locally the surface is treated as planar but with a tilt (or tangent angle), hence the name tangent plane. On this tangent plane the normal derivative is approximately double the normal derivative of the incident field. Using these results in eq. (3.28) we fmd that (3.31) which we refer to as version (A) of the theory. Version (B) again uses Green’s theorem but now on QSc and Gf, the former field satisfying the sourceless, homogeneous Helmholtz equation. Using the same boundary conditions eqs. (3.29) and (3.30) we now find that
+ Gy(x:,
x,,)
a an
1
Gy(x, x i ) dS’ .
(3.32)
Version (C) uses Green’s theorem applied to aSc and a Green’s function G which vanishes on the surface: C(X, X i ) = 0.
(3.33)
Since everything is nearly planar, G can be approximately written as G(x, x ’ ) = Gy(x, x ’ ) - Q(x, x’),
(3.34)
where @,‘ is the image Green’s function. On the surface the normal derivative
SINGLE SCATTERING
19
of G is approximately double the source value, (3.35) to yield the final version of the tangent plane theory: (C) ~ ( x =)2
5 j[ A
Gy(x, x:)
an’
1
Gy(x:, x,) d S ’
(3.36)
Approximately evaluating the normal derivative and defming three-dimensional scattering and incident wave vectors as
’,
(3.37)
k’” = k,(x’ - x,) Ix’ - xol - I ,
(3.38)
ksc = k,(x - x ’ ) J X- X ’ I and
the three versions of the tangent plane approximation are
ss ss ss
(A)
P C ( x ) = 2i
(B)
@““(x)= i
and (C) @”(x)
=
- 2i
(kin.A‘) Gy(x, x:) Gy(x:, x,) d S ’ ,
(3.39)
(kin - k”‘)-A’ Gy(x,x:) Gy(x:,x,)dS’,
(3.40)
(ksc A‘) GY(x, x i ) Gy(x:, x,) d S ’ ,
(3.41)
where A‘ is the unit normal to the surface. It can easily be seen that (A) + (B) = 2(C). In general these three versions yield different results, but in the backscatter case k”‘ = - k’”they agree. Version (B) due to ECKART [ 19531 is the only version which is reciprocal, i.e. it remains invariant under the interchange k”‘ H - kin.However, this seems to be of no particular advantage in computations.
3.4. KIRCHHOFF APPROXIMATION
Again we begin with the Helmholtz integral representation of the scattered field as in version (B) of the tangent plane model:
(3.42)
20
[I. $ 3
MULTIPLE SCATTERING FROM ROUGH SURFACES
Originally the Kirchhoff theory referred to the result found by postulating both the field and its normal derivative on the surface. As generally used, these quantities are described using the tangent plane approximation. For plane wave incidence they are @(x,) = (1
a
-
an
+ R ) @in(Xs),
(3.43)
@(x,) = i( 1 - R) (kin * ri) Qin(xS).
(3.44)
We thus again have a local approximation with R being the reflection coefficient. In addition, as used, the Kirchhoff approximation usually includes a far-field approximation. The full expansion is in inverse powers of k,L, where L is a characteristic transverse surface distance. (See Fig. 5.) For fured L this is also a high-frequency approximation, and in the limit k, -+ co it is referred to as geometrical optics. In addition it has been shown for apertures (WOLF and MARCHAND[ 19641) that the difference between the Kirchhoff and the exact solutions is a boundary wave. Using the surface values and the far-field approximation we can write (r = Ix I ) after some algebra that @"(x)
= A(O,, 6,) exp(ik1r)/4nr,
(3.45)
where the scattering amplitude A is defined as for a one-dimensional surface: A(I3,, 0,)
=
ik,
exp( -ik,x'cr, - ik,h(x')&)
(6
a
d"):
Idx'. (3.46)
f
-L
N
Fig. 5 . Geometry for the KirchhofF approximation for scattering from a one-dimensional surface.
1.
I 31
SINGLE SCAWERING
21
Here ro = sin O2 - sin 0, ,
(3.47)
Po = cos 0, + cos 0, ,
(3.48)
and a = sin 6,
+ sin 8, + % R ,
(3.49) (3.50)
Function A will recur in our multiple scattering development later. Here it is expressed in terms of incident and scattering angles, but it can also be interpreted as a phase modulation (due to the surface roughness) amplitude spectrum or suface interactionfunction. It is the term which represents the dynamical properties of the surface interaction. Unlike the perturbation theory results which occurred in powers of the surface height, here the surface height occurs in the exponential. This can be interpreted as apartial summation of the full power series expansion and as such has a wider range of validity than the truncated power series. The multiple scattering results for the connected diagram expansion treated later will be an expansion in powers of A. Function A contains both height, h, and slope, dhldx’, terms. Provided we are not near both grazing incidence and reflection (so that Po # 0) we can remove the slope terms via a partial integration. This is easily seen to be equivalent to the replacement
provided the integrated term vanishes. In this case the resulting amplitude term comes outside the integral and can be interpreted as a kinematical modification of the surface interaction. This will lead to what we term the vertex function in 7. If the surface integration is over a finite region, and the integrated term does not vanish, its end point contributions can be included. The advantage of the partial integration will become evident later for the random surface, since we will only have to treat the statistics of the heights rather than the statistics of both heights and slopes. Also the ensemble average over the surface realizations of the partially integrated scattering amplitude is just the characteristic function of probability theory (PAPOULIS [ 19651) related to eq. (2.9).
22
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. 8 4
8 4. Mathematical Preliminaries for Multiple Scattering 4.1. SCALAR (ACOUSTIC) CASE
As we observed in the previous section, many theoretical approaches begin by using a variation of Green’s theorem. The most general development along these lines for the scalar case assumes the rough surface h(x,) is an interface dividing space into upper (V,, z > h) and lower (V,, z h) half spaces. Each region is filled with a material having density p, and sound speed c, (or wavenumber k,), where j = 1,2. The corresponding electromagneticmaterials are dielectrics with permittivities 5 and permeabilities p,. The electromagnetic development is the same as the scalar development provided the polarization does not change under the scattering process. The Green’s functions in each region satisfy the inhomogeneous Helmholtz equations
-=
where j = 1,2, am = (a/ax, a/ay, a/az), repeated subscripts m indicate summation from 1 to 3, x = (x, y, z ) = (x, ,x2, x3) E 5 and x, is the source point. The free-space Green’s functions G; satisfy eq. (4.1) everywhere in space. Explicitly they are G;(x,x,)=
[4njx-xol]-’exp[ik,Ix-x,l]
=
G,O(x-xo).
(4.2)
Green’s theorem applied in V , with the source point in V , yields G,(x’, x ” ) O(Z’ - h(x;))
+ akGy(x’, xs) nm(xt) Gl(xs, x”)],
where x, by
=
(4.3)
(x, ,h(x,)) is a point located on the surface and 8 and nm are defined
and
which is a vector in the surface normal direction. Note that it is not the unit
1.5 41
MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING
23
vector. In eq. (4.3) if the argument of 8 is positive, the field point is in V , , if negative it is in V2. Green’s theorem in V, correspondingly yields
Equation (4.6) evaluated for z’ E V , (so that the lhs vanishes) is called the extended boundary condition (WATERMAN[ 1965, 19711) or the extinction theorem (PATTANAYAK and WOLF [ 19761). It is that boundary condition which leads to the cancellation of the incident field. The relation between the two versions has been given by AGARWAL [ 19761. Both eqs. (4.3) and (4.6) are integral representations in the sense of explicitly defining the functional values of the field at point x ’ in terms of the values of the field and its normal derivative on the surface. Integral equations for these unknown surface field values can be found by taking the limit as the field point x ’ approaches the surface, and taking into account the jump discontinuity of the double layer potential term proportional to the normal derivative of the free-space Green’s functions. These can be found either by classical means (KELLOGG [ 19531) or by using distribution theory (DESANTO[ 1981bl). For the latter they are d;G,O(x’, x)nm(xt) = fP”’(x’, X) - $E(z’ - z ) ~ ( x ;- xt),
(4.7)
where E(Z‘- z) =
qZ’- z) - qz- z’)
(4.8)
and P ( J ) ( ~ ’ ,= ( 2 4 - 3
dkexp[ik-(x’ - x)] G J ( k ) P ~ ) ( k ) n , ( x , ) , (4.9)
with (4.10) where P stands for the Cauchy principal value.
24
[I, 5 4
MULTIPLE SCAlTERlNG FROM ROUGH SURFACES
dx,[Gi(xi, x,) ~,G,(x,, ~”)n,,,(x,)
+ :P(’)(x:,
(4.12)
xs)GZ(xs,x”)].
These surface limits from each region are related by continuity conditions at the interface given by Gl(xs,x”)
= PGz(x,,x”),
Nl(X,,X”)
= N,(x,,x”),
P=
(4.13)
P2IP1,
(4.14)
where (4.15)
$.(x,, x ” ) = d,,,Gj(~,,~ ” ) n , ( x , ) .
If we define N , = N and G, = G, eqs. (4.11) and (4.12) become ~ G ( x ~ , x=” )G:(x~,x”) -
ss
dx,[G~(~~,x,)N(X,,x”)
+ +P(’)(x:,x,)G(x,,
x”)],
(4.16)
and
+ :P-’P(~)(x:, x,)G(x,,x”)],
(4.17)
which are coupled integral equations for the surface values G and N. Solution of these equations then yield the field values from eqs. (4.3) and (4.6). The equations can be solved as coupled integral equations per se (KITTAPPAand KLEINMAN[ 19751, KRESSand ROACH[ 1978]), or the extended boundary condition can be used with an assumed field (and hence surface field) expansion technique to yield one integral equation for either the overall reflection or [ 19831). As an example of the latter, the transmission coefficient (DESANTO generalized reflection coefficient R can be written as the solution of the integral
1,
I 41
MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING
25
equation of the first kind: B ( x ~X ," ) =
ss
K(&, x,) R(x,, x " ) dx,,
(4.18)
where the functions B and K are known functions which depend only on the free-space Green's functions for the two media and their normal derivatives (DESANTO [ 19831). K arises from the Ansatz that the total Green's function in the upper region can be written as
G,(x', x " ) = G ~ ( x x' ", ) +
ss
G ~ ( x x,)R(x,, ', x " ) dx,.
(4.19)
In the limiting cases (see below) p-+ LC (Neumann, acoustically hard, TM boundary condition) and p 0 (Dirichlet, acoustically soft, TE) the single eq. (4.18) reduces to the standard integral equation results (NOBLE[ 19621, BOLOMEY and TABBARA [ 19731). In the limit of vanishing height the fact that R equals the standard Rayleigh-Fresnel reflection coefficient (BREKHOVSKIKH [ 19601) is derived. This is in contrast to perturbation theory where the latter is assumed. Reduction to a single integral equation for multilayered media has also been accomplished (MAYSTRE[ 19781, MARX[ 19821). There are other possible local boundary conditions for perfectly reflecting surfaces. They are, on the surface, -+
G
=
N
=O
ZN
+C
=
0
0
Dirichlet, Neumann, Impedance,
whose electromagnetic analogues are, respectively,TE polarization, TM polarization, and the Leontovitch type impedance condition (BREKHOVSKIKH [ 19601). Formally, eqs. (4.16) and 4.17) define an exact solution to the general scalar interface problem. Once the surface values G and N are determined, the field values can be determined from eqs. (4.3) and (4.6). MITZNER[ 1966a1, for example, treated the case of media of greatly different densities and solved the equations in an approximate perturbation series. The equations can also be used as a starting point for T-matrix calculations (WATERMAN[ 1969, 19711) or, since they are still stochastic equations in coordinate space, for the smoothing method (WATSONand KELLER[ 1984]), discussed in detail later. In general, these or equations analogous to them form the basis for any multiple scattering approach to random surface scattering.
26
MULTIPLE SCA'ITERING FROM ROUGH SURFACES
[I, § 4
By appropriate scaling, we can relate the Green's functions we have defined to electromagnetic field components. For the TE case we have GI = ( p l ) - ' E l , and G, = (p2)- 1E2y,where E.,, is the only electric field component and the p, are the permeabilities of the media. The p in eq. (4.13) is the ratio of permeabilities p2/pI.Equations (4.13) and (4.14) then correspond to the continuity of tangential electric and magnetic field, respectively. For the TM case we identify G I = 'HI,,,G, = (.5,)-'H2,, and p = E J E ~ , the ratio of dielectric constants. H , . is the only magnetic field component and eqs. (4.13) and (4.14) then correspond to the continuity of tangential magnetic and electric fields, respectively (DESANTO [ 19831).
4.2. ELECTROMAGNETIC CASE
The scalar results can be applied to the electromagnetic problem provided there is no polarization change in the scattering process. This is only true if the incident electromagnetic field component is parallel to the generator of the surface. A simple example is the case where the surface does not vary as a function of y. In general this is not so, a polarization change occurs, and thus the full electromagnetic problem requires a separate development. For simplicity we limit the development here to scattering from perfect electrically conducting surfaces. We also treat the problem conventionally, i.e. using vector analysis. The case of a perfect magnetic conductor using index notation was derived by DESANTO[ 19741. The surface h(x,) divides all space into a perfectly conducting medium (z < h) and free space (z > h). The electric (E"') and magnetic (H"') fields scattered from the surface can be found from the current J (which is related to the true surface current J, = J [ 1 + ( c ~ J z -) ~I / *]) induced on the surface from the following equations (VAN BLADEL[ 19641):
H S C ( x ' =) 8' x E"(x')
=
ss
J(x,) Go(x',x,)dx,,
iqok, '8' x H " ( x ' ) ,
(4.20) (4.21)
where G" is the free-space Green's function defined in eq. (4.2) with k, = k,, ylo = ( po/~o)1'2 is the characteristic impedance of free space in terms of the free space permeability ( p o ) and permittivity (to),and where we quote the integral term in Cartesian coordinates as we did for the scalar case. If x ' is in the far field or Fraunhofer zone of the scattering area (BREKHOVSKIKH [ 19601) the
1. I 41
21
MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING
equations simplify to (VANBLADEL[ 19641, BROWN[ 1983b1)
H ” ‘ ( x ’ )z iCo(x’, 0 ) ksc x and E ” ( x ’ ) z - k,
I
@”‘
ss
J(x,) exp( - iksc*x,)dx,,
x H”‘(x’),
(4.22)
(4.23)
where k”“is the vector pointing in the scattering direction defined by eq. (3.37), with k, replaced by k,,. The above equations determine the scattered fields once the current is known. To find the current, let x ’ approach the surface as in the scalar case. Use the boundary conditions: (a) the continuity of the total tangential electric field across the surface (for the perfectly conducting case the total tangential field vanishes on the surface), and (b) the discontinuity in the total tangential magnetic field across the surface which is equal to the surface current. The resulting integral equations were first derived by FOCK[ 19461, and later by MAUE[ 19491, and they are principal value integrals on the swface. In modern terminology (YAGHJIAN[ 19811) they are called the electric field integral equation (EFIE) and magnetic field integral equation (MFIE) since this indicates from which field they are derived. The EFIE has the form of a first kind integral equation n(x;) x ~‘“(x;)= (icoEg)-ln(x;) x
ss
[kiJ(x,)- a,da]
x
W ,x,)
dx,,
(4.24)
with E’” the incident field, and where n is the vector version of the normal defined in eq. (4.5): n(x;)
=
(4.25)
2 - a;h(x;);
and the MFIE is a second kind integral equation
J(x~= ) 2 4 ~) ;x H ’ ” ( x 6 )+ 2n(x( ) x
ss
J(x,) x dG(x;, x,) dx,, (4.26)
where Hi“ is the incident magnetic field. For perfectly conducting surfaces either of eqs. (4.24) or (4.26) provide the necessary integral equation to solve for the current induced on the rough surface (BROWN [ 1982a1). Since the MFIE is a second kind equation with a well developed theory we place greater emphasis on it.
28
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. 8 4
For an interface separating non-perfectly conducting media we must deal with coupled vector integral equations for the equivalent electric and magnetic currents induced on the surfaces (JONES [ 19791). The scalar analogy consists in treating the field (or Green's function) and normal derivative on the interface as discussed.
4.3. ELECTROMAGNETIC WAVES AND ANGULAR SPECTRA
In the previous section we formulated the exact integral equations for the electromagnetic scattering problem. There are similarities with the scalar case as well as distinct differences. We will discuss both in this section. In addition, we will discuss the angular spectral development of the exact equations as well as approximation methods for both the integral equations and the angular spectrum. It is more convenient to do this following the exact development, whereas in the scalar case we treated the approximations first in 5 3. The Kirchhoff approximation essentially estimates the electric and magnetic currents induced on the surface only in terms of the tangential incident field components and the local constitutive properties of the surface (SANCER [ 19691). For a perfectly conducting surface the Kirchhoff approximation for the electric current on the surface, J , is 2n x Hi",the Born term in eq. (4.26). This is similar to the scalar case where we also noted a doubling of the tangential incident field on the surface. For electromagnetic waves the Kirchhoff approximation and the tangent plane assumption are generally treated as synonymous, although there are notable exceptions. For example, HARRINGTON [ 19591used the tangent plane approximation to simplify two totally different scattering formalisms, and the results obtained are identical to eqs. (3.39) and (3.40). BROWN[ 1983~1noted that one can also recover the results in eq. (3.41) in this way. Further, the physical optics approximation is simply the Kirchhoff or tangent plane approximation modified by an approximate accounting for shadowing (BECKMANN[ 19651). We can expect the Kirchhoff/tangent plane approximation, as applied to electromagnetic scattering, to produce accurate results only when the Rayleigh roughness parameter is large and the surface contains no spatial frequencies on the order of the wavelength (resonance region) or smaller than the electromagnetic wavelength (BROWN[ 1985a1). Furthermore, it has also been shown (BARRICK [ 1968a1) that, with the above assumptions, it makes no difference whether one first carries out the spatial integration in eq. (4.22) and then averages the result (KODIS[ 19661, BARRICK[ 1968b]), or reverses the order
1, I 41
MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING
29
of integration and averaging (ISAKOVICH [ 19521). For a perfectly conducting random surface, the result obtained from the above operations is independent of polarization, i.e. it is a scalar. When the surface has finite conductivity, we obtain a polarization-dependent result (SANCER [1969]). This is simply a consequence of the change in polarization which occurs in reflection from a tilted, finite conductivity, planar surface (MITZNER[ 1966b], FUNG [ 19661). In addition, as mentioned above, shadowing has been included in the Kirchhoff approximation (WAGNER [1967], SMITH[1967], SANCER[1969], BROWN[1980]), but its validity is rather limited (BROWN[ 1984b1). Boundary perturbation methods can be used for surfaces having small height and slopes. The many diverse methods used (RICE [1951], BASS and BOCHAROV [ 19581, MITZNER [ 19641, WRIGHT[ 19661, BURROWS [ 19671 and SWIFT[ 197 11) are all based on the requirement that the total tangential electric field vanishes on a perfectly conducting interface or, for finite conductivity, is continuous across the interface. For backscattering from a perfectly conducting random surface, perturbation theory predicts the following polarization dependence for the surface scattering cross section per unit scattering area, go (PEAKE[ 19591, WRIGHT[ 19661): o ~ =H 1 6 ~ k :C O S 6 ~ S(2kp),
(4.27)
gtV= 16nk:
[ 1 t sin*B]*S(2k:),
(4.28)
g t H = 0;"
0,
(4.29)
=
where S is the two-dimensional surface height spectral function defined in 8 2.2. The Bragg resonance phenomenon is evident in the argument of S.In the above, H(V) refers to horizontal (vertical) or TE (TM) polarization, respectively, with the first subscript denoting the incident field polarization and the second subscript denoting the polarization used to sample the scattered field. 0 is the angle between the incident wavenumber kin and the z-axis. If we ignore the previously stated limitations for the use of the Kirchhoff approximation and instead extend it to the case of small height and slope, we find that it predicts okH = with both equal to the value given by eq. (4.27), and no cross-polarization (VALENZUELA, WRIGHTand LEADER[ 19721). Aside from the fact that this agreement of different polarization results is probably nothing more than happenstance (VALENZUELA,WRIGHTand LEADER[ 1972]), it is still of interest to inquire as to why the perturbation limit of the Kirchhoff approximation differs from the pure perturbation result, especially since the two approximations agree in the scalar case to order (k,a)
30
[I, I 4
MULTIPLE SCATTERING FROM ROUGH SURFACES
in the fields or (k,a)’ in power. This is a difficult question since the two techniques have so little in common and it is not obvious as to the degree of approximation contained in each. It is possible, however, to discuss the approximations in a common framework (BROWN[ 1985a]), using the twodimensional (Weyl) spectral representation (see eq. (5.7)) for the Green’s function in eq. (4.20). The latter is evaluated on the surface, so both up and down going waves must be included. The scattered field from eq. (4.21) can be written as
E s c ( x ’ )= iqok,
I
3’ x d‘ x Z(x‘),
(4.30)
where
x exp (ik, x,) dk, .
(4.3 1)
The up(A ) - and down(A - )-going plane wave spectral amplitudes are given by +
A * (kt,z ’ ) = 4K
ss
J ( x ) [ 1 f sgn (z‘
-
h ) ] exp [ T iKh
-
ik, x, ] dx, , (4.32)
where sgn is the signum function and K = (ki - k:)”’. We have that A + vanishes for z’ < hmin, A - vanishes for z’ > h,,,, and in the region hmin< z’ < h,,, both amplitudes are required. The complete functional dependence on the surface height is contained in these amplitudes. Equations (4.30)-(4.32) illustrate three distinct means for computing Esc above the rough surface. We will discuss two of these (BROWN[ 1985a1). First, if we know J, then we can compute A * from eq. (4.32) and hence E”“from eqs. (4.30) and (4.31). Alternatively, we could directly compute E”” from eqs. (4.20) and (4.21). This is just what we do in the Kirchhoff approximation, except that here we would use an alternative estimate of the current rather than the Born term in eq. (4.26), the MFIE. The second method involves a direct determination of A for z‘ > h,,,. Then, in this region, E”’ can be computed directly from eqs. (4.30) and (4.31). This is exactly the approach in the boundary perturbation approximation. To see this, ignore the need to include A - in the region hminc z’ < h,,,. This is the Rayleigh hypothesis approximation (DESANTO [1981a]). Next, take the point of observation x ’ to the surface and ignore the dependence of A on h, i.e. set A (k,,z’ ) = A + (k,). +
+
+
s
1, 41
MATHEMATICAL PRELIMINARIES FOR MULTIPLE SCATTERING
31
The signum function in eq. (4.32) is equal to one. Then set the tangential component of the total electric field on the surface to zero, this yields (BROWN [ 1985a1) n ( x ; ) x /?"(xi) = -ikoqon(x;) x P(x;),
(4.33)
where
(4.34) and k = k, '(k, + KI). Finally, we assume a perturbation expansion for the amplitude as (4.35) where A + ( n ) ( k t=) O(hi(dth)'), i
+j
=
n,
(4.36)
and 0 is the order symbol. Substituting the result in eqs. (4.33) and (4.34), equating like powers of the expansion parameters and solving for each A + w ) via Fourier transforms yields the boundary perturbation result in eqs. (4.27)-(4.29) to order (k,a)' in power. The angular spectrum approach thus clearly illustrates all the approximations inherent in the boundary perturbation technique. In addition, it can be shown that this approach is equivalent to solving the EFIE, eq. (4.24), by perturbation theory using the Rayleigh hypothesis. This permits us to relate the two basic single scattering approximations, the Kirchhoff/tangent plane and perturbation methods, to the asymptotic solutions of the MFIE and EFIE, respectively, the two fundamental integral equations of electromagnetics. If we could solve both equations exactly, both solutions would agree. However, since we are using two asymptotic approximations which rigorously have no common region of validity, it is not surprising that we obtain different results. Finally, we should point out that surfaces having both large and small scales of roughness have been treated by an appropriate combination of the Kirchhoff/tangent plane and boundary perturbation approximations (FUKS [ 19661, WRIGHT[ 19681, BROWN[ 19781, MCDANIEL and GORMAN [ 19831). For such surfaces, the Kirchhoff portion dominates in and about the specular scattering direction while a slightly altered form of the boundary perturbation
32
MULTIPLE SCATTERING FROM ROUGH SURFACES
11. I 5
result is the primary contributor in other directions. The interaction between the two approximate solutions produces the slight modification in the boundary perturbation result. In essence this modification is a consequence of the tilting of the small scale B r a g scatterers by the large scale surface undulations.
6 5. Lippmann-Schwinger Equation and Diagrams There is an alternate analytical development based on an analogy with quantum mechanical potential scattering (PATTANAYAK and WOLF [ 19761, GARCIA,CELLI and NIETO-VESPERINAS [ 19791, SHEN and MARADUDIN [ 19801, NIETO-VESPERINAS and GARCIA [ 19811). Our previous development treated an interface problem with point source incidence. Here, for the scattering of a massive particle from a rough hard wall, the total wave function (or field) @ satisfies the time-independent SchrOdinger equation (dm am
+ kT1@(XI=
u(x)
(5.1)
where for a rough hard wall the non-central potential is
U(x) =
0,
z > h(x,)
(upper region), co , z < h(x,) (lower region) .
For plane wave incidence the solution of the above problem can be written in terms of the free-space Green's function as
I'":[
=
@'"(x) -
jj
GY(x - x')U(x')@(x')dx',
where the lhs is equal to @ if x is in the upper region and vanishes identically if x is in the lower region. Equation (5.3) is the Lippmann-Schwinger integral equation. The function GY is given by eq. (4.2), and @'" is the incident plane wave @"(x) = exp[i(c.x, - K'z)],
where
The integral term in eq. (5.3) is the scattered field.
(5.5)
c
I. 51
LIPPMANN-SCHWINGER EQUATION A N D DIAGRAMS
33
Next we expand GY in a two-dimensional (Weyl) spectral representation (DEVANEY and SHERMAN [ 19731) as
and in addition express the integrand as W)@(X) =
s- 'f(x,)
- h(x,)),
where S is the surface area which is illuminated and is inserted for normalization purposes. Then, if we impose the (acoustically soft) boundary condition @(xt,h(x,)) = 0,
(5.9)
we can derive an integral equation for the source function f: Qiin(xt,h(x,))
=
SS
K(x,, xi )f(xl) dxl ,
(5.10)
S
where the kernel K is given by
(2 4
3
s
K - ' e x p [ i k , ( x , - x l ) + i K l h ( x , ) - h(x;)I]dk,. (5.11)
In addition, using the Weyl expansion in eq. (5.3) it is possible to write the scattered field as an angular spectral representation, provided f is known and provided that the z-coordinate is always greater than the maximum height (so the absolute value disappears). It is given by
M(k,) exp(ik, x,
@"(x) = @(x) - @'"(x) = --oo
+ iKz) dk,,
(5.12)
--oo
where the scattering amplitude is given by M(k,) =
jjj(x:)exp[ -ik;x; (2 x ) ~ K S
~
- iKh(xi)]dx;.
(5.13)
In principle this method is straightforward, but eq. (5.10) is a twodimensional integral equation of first kind and cannot be solved directly. Instead NIETO-VESPERINAS and GARCIA [ 19811 used the extinction theorem
34
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. 8 5
(5.4) for z values less than the minimum value of h to create an additional equation. The latter is solved term by term in perturbation theory of powers of the height h for the source function J The method is thus limited in application to small values of the Rayleigh roughness parameter, and thus inadequately describes the scattering from strongly rough surfaces (Z>1). It is also possible to derive the above results using the Helmholtz integral theorem (BORN and WOLF [ 19801) for plane wave incidence. For this latter case, the boundary condition (5.9) fixes the source function f as the normal derivative of the field on the surface (scaled by S). The Lippmann-Schwinger integral equation approach was first used by ZIPFEL and DESANTO[ 19721 for an acoustically hard (TM electromagnetic) surface, and was written in Fourier transform space (k-space). A summary of the method for the scalar case can be found in DESANTO[1981b], with extensions to the full electromagnetic (DESANTO[ 19741) and elastic (DESANTO [ 19731) cases. Essentially the method amounts to Fourier transforming the two-dimensional coordinate-space representation, still explicit in the height variable (and viewed as analogous to a z-coordinate), into threedimensional Fourier transform space off the energy shell. That is, while the integral equation is being solved, the three Fourier transform coordinates (propagation vector) are treated as unconstrained by the usual energy constraint or the dispersion relation provided by the Helmholtz equation. In particular the z-components of the incident and scattered wave vectors are not treated as being equal to K until after the integral equation is solved. To illustrate, the Fourier transformed integral equation for the scattered part of the Green’s function is the solution of the three-dimensional Lippmann-Schwinger integral equation (DESANTO [ 198lb]) T ( k ’ ,k ” ) = V ( k ’ ,&’)A(&’ - k ” )
+
111
Y ( k ’ ,k ) A(k’ - k ) Gy(k) T(k, k ” ) dk.
(5.14)
Here V and A are the vertex and interaction functions defined by (5.15) where q = 1 (all slopes included ) or q = 0 (no slope terms), P represents the Cauchy principal value, A(k) =
11
exp ( - ik . x,) dx,
(5.16)
LIPPMANN-SCHWINGER EQUATION AND DIAGXAMS
35
and GY(k)is the Fourier transform of the free space Green's function. Equation (5.14)is thus analogous to the Lippmann-Schwinger equation for potential scattering but here for a non-central potential given by the product VA. The scattering amplitude T can be formed from r by going on-shell, viz., T(k; ,k , ) = [@:)-
' r ( k ' ,k)]
Ik; = K',kZ =
-K
*
(5.17)
Equation (5.14)can also be interpreted diagrammatically as illustrated in Fig. 6. There is an analogy with diagram expansions for wave propagation problems (FRISCH [ 19681)but in the latter the variability lies in the index of refraction term in the equation, which can be thought of as an amplitude variability. Here the surface variability is in the phase, as illustrated in eq. (5.16). BASS and FUKS[ 19791 also present a diagram expansion method for the surface scattering problem, but again the surface variability is treated as an amplitude variability in a perturbation theory in powers of surface height. As we illustrated in the definition of Rayleigh roughness criterion, the phase variability is the simplest lowest order effect on the scattering. Physically, the Green's functions describe wave propagation and interaction. For example, eq. (5.14)describes the transition from an initial (propagation)
Fig. 6. Diagram notation associated with scattering from an arbitrary surface; (a) propagator; (b) vertex; (c) interaction; (d) full scattering amplitude; (e) representation of the integral equation (eq. (5.14)); and (0 first three terms in the Born expansion of eq. (5.14).
36
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I? 8 6
state represented by the wave vector k” to a final state given by k’.This can occur either directly via the single interaction Born term VA, or indirectly through the full range of intermediate states k which, since they are continuously varying, are integrated over. This is another illustration of the fact that in order to solve for the on-shell values of r, we must first solve eq. (5.14) at all the continuously varying off-shell values k and then set the vertical wave number components on-shell. Finally, the multiple scattering multiple interaction interpretation of eq. (5.14) is the same as that for any integral equation. Here it is represented as a Born series expansion in powers of VA with A representing the single interaction dynamics and V the single interaction kinematics. Note that since the height function is exponentiated in A, even a single interaction represents a partial summation of terms occumng in a perturbation expansion in powers of the height. Also note that if the surface height is a random variable, then eq. (5.14) is a stochastic equation for the scattered field. In principle, we can compute for this equation the ensemble average of any power of r once we know the ensemble average of any power of the surface interaction A. This illustrates the generality of the integral equation approach, and the direct relation between the statistics of the surface and those of the field moments. A further generality is provided by the fact that an integral equation similar to eq. (5.14) occurs in the electromagnetic (DESANTO[ 19741) and elastic (DESANTO [ 19731) problems, where they occur as coupled vector and tensor equations, respectively.
0 6. Scattered Field Statistics We discussed the statistics of the randomly rough surfaces in 5 2. Here we discuss the statistics of the field scattered by a random surface for plane wave or point source incidence. (We have already briefly treated the fluctuating field statistics in perturbation theory in § 3.) We treat specifically the vector electromagnetic case although the results apply to the scalar case as well. The scattered field E”‘ at a point x not on the surface can be related to its discontinuity across the surface AE”‘ by an integral relation of the form
Esc(x)=
j1
K(x, x,) d E S C ( x sdS, )
(6.1)
where K describes how a point discontinuity on S propagates from the surface point x, to x. The discontinuity can be related to surface field values or normal
1. I61
SCATTERED FIELD STATISTICS
31
derivatives as in Q 4. Ideally we want to obtain from eq. (6.1) the probability density function for E", p(ESC(x)),and the joint density for Esc(xx,)and ESC(x2), i.e. p(E"(xn,),E""(x,)).This is very difficult because the discontinuity is seldom known a priori, and must usually be determined from an integral equation. Even when the functional dependence of AE""on the random surface properties is known, it is usually not obvious how to transform these properties through the integral relation (6.1) to obtain p(E"(x)) (DAVENPORT and ROOT [ 19581). Little progress in this direction has been made. Instead, what is usually done is to consider E"(x) to be composed of a sum of elemental scattering events, i.e.
where the point source Green's function is defined by eq. (3.20). By ascribing various probability densities to the A, and the $, we can obtain a number of different probability density functions for E"", its amplitude, or its phase. BECKMANNand SPIZZICHINO [ 19631 present an excellent discussion of this methodology and its limitations. If N is very large and (E"") negligibly small in the direction of observation, then E""will be Gaussian distributed, 1 EscI will be Rayleigh distributed, and the phase of E"" will be uniformly distributed over ( - K, K ) . When ( E"') is not negligibly small in the direction of observation, we must add a constant (non-random) term to eq. (6.2) to account for the coherent scattered field. In this case when N is large, the density for E"' is Rician. There are two problems associated with this technique for finding the density function. The first is that only in very special cases we can relate the A, and $, to the actual surface roughness (KODIS [ 19661). The second is that the approach does not provide any insight into the physical mechanisms responsible for the non-Gaussian behavior of p ( E S Cfrequently ) observed in practice, [ 19511). Such especially for near grazing backscatter (KERRand GOLDSTEIN departures are generally attributed to the illumination of a relatively small number of scatterers, observation of the scattered field in the Fresnel zone, or the presence of a few very dominant scattering elements (DAINTY [ 19761, JORDAN, HOLLINSand JAKEMAN [1984]). The result is that we cannot determine the dependence of E"" on the surface statistics from our severely limited analytical knowledge of p(E""). We can, however, determine the statistical moments of the scattered field using a fundamental definition from probability theory (PAPOULIS [ 1965]), which states that the mth statistical moment of a function u(h) of the random
38
MULTIPLE SCATTERING FROM ROUGH SURFACES
variable h is given by (u"(h))
=
j
u"(h)p(h) dh.
--at
The integral expression for Esc in eq. (6.1) provides the means for applying eq. (6.3). In eq. (6.1) we know K and its functional dependence on the random surface. We assume that we can develop a geometric transformation to convert the integration over the random surface to one over a non-random surface, i.e. (6.4) d S = T ( x , , w)dx,, where T is a function of the non-random coordinate x, and specific random properties of the surface denoted by w. Thus eq. (6.1) becomes
W x r )=
ss
[ K ( x ,x,) ~ ( x , ,w)lAESC(xs) dx,,
(6.5)
where everything inside the square brackets is known. The average or coherent scattered field is ( E"'(x)) =
s
E s c ( ~p(62) ) d62,
where the random properties w are a subset of 62. The variance of the scattered field or the incoherent scattered power is ( I Es'(x) - (E"'(x)) I 2 , =
s
I ESc"()- (E"'(x)) I'p(62) d62.
(6.7)
The symbol 62 denotes the full random surface properties on which K T and AE'" depend. In general, we know which surface properties KT depends on, but we know the explicit dependence of AE"" on the surface properties only in comprises the bulk of the very simplified limits. The determination of AESC(xs) random surface scattering problem. Note also that eq. (6.7) requires joint surface statistics. This is obvious once we square the scattered field using eq. (6.5). It is clear that, using eq. (6.5) we can form expressions for the spatial correlation functions for the scattered field ( E"'(x,). E " ( x 2 ) ) ,
(6.8)
the scattered power (IW x l )1
I E"'(x2) I )
9
(6.9)
1.
I 71
k-SPACE FORMALISM-RANDOM SURFACE
39
and, in principle, for any order moment of the scattered field. In practice, however, we are limited by the number of integrations we can perform as discussed in 8 2.3. This is the major drawbacK to using an integral expression such as eq. (6.5) for the scattered field. The remainder of this review paper will be primarily devoted to computing the mean and variance of the scatterred field. Where possible, extensions to higher-order moments will at least be noted.
0 7. &-Space Formalism-Random Surface 7.1. CONNECTED DIAGRAM METHOD
We have already written the scattering amplitude for a hard surface in Fourier transform or &-space in eq. (5.14). This can be generalized to an interface separating media of different density (DESANTO[ 1981bl). The equation is stochastic and we will discuss its comparison to the smoothing approximation in the next section. Here, we indicate the connected diagram expansion solution for the ensemble average of the first and second moments of eq. (5.14). The electromagnetic (DESANTO[ 19741 and elastic (DESANTO [ 19731) cases have also been done. In eq. (5.14) we define the integral operator symbolically as L=
!!!
V@d&.
(7.1)
Then, eq. (5.14) can be written symbolically as
f = VA + LAf,
(7.2)
which has a formal solution by iteration given by
r =n c= O ( L A ~ V A . i z
(7.3)
The interaction term A contains the only explicit height variable. We assume now that this is stochastic. If we define the bracket average ( * * ) as the average over the ensemble of surface realizations, then, in order to find, say, the coherent scattered field ( f) we must know the ensemble average of products of interaction functions of the form
40
MULTIPLE SCAlTERING FROM ROUGH SURFACES
[I, § 7
which are related to the n-point characteristic function of the surface. The general case has been done for a surface satisfying multivariate homogeneous Gaussian statistics (ZIPFEL and DESANTO [ 19721). Essentially, one is able to advantageously exploit the translational invariance of the surface statistics to evaluate one (two-dimensional) integral in the ensemble average of the product. The latter yields a conservation of overall transverse momentum or wavenumber. It is simplest to see this using examples. For homogeneous Gaussian height statistics the ensemble average of a single interaction is ( A ( k , ) ) = A , ( k , ) = (2nI2 W,t)a(k,J,
(7.5)
where ~ ( k , =) exp( - a2k,2/2).
(7.6)
The delta function in eq. (7.5) illustrates the conservation of transverse wavenumber. The corresponding scattered term in eq. (7.3) is thus specular. This follows from the statistical homogeneity or translational invariance of the surface statistics. The Gaussian in eq. (7.6) is the surface characteristic function, and is related to the single scatter Rayleigh roughness result in Q 2. For the ensemble average of two interactions we cluster decompose as (HUANG[ 19631)
(A(k,)A(k,)) = Al(kl)Al(kZ) + A , @ , , k,),
(7.7)
where A,@,,
k2) =
(24* W
I t
+ k2t)B(kl,)B(k2,)RZ(kI,
k2z)
(7.8)
and
(7.9) Here Cis the correlation function defined by eq. (2.10). Again the delta function in eq. (7.8) illustrates the conservation of transverse momentum. Note that in perturbation theory the correlation function occurred in the amplitude while in eq. (7.9) it is exponentiated. This again illustrates the concept of partial summation. If we expand the exponential of the correlation function in eq. (7.9) in a Taylor expansion and keep only the first two terms, R , is proportional to the spectral function. Note also that asymptotically the integrand in eq. (7.9) vanishes.
1. I 71
41
k-SPACE FORMALISM-RANDOM SURFACE
For the three interaction approximation the cluster decomposition is )A(k2)A(k3
) = A 1 (&l
1( k 2 ) A 1
(k3
k3) + Al(k2)442(kl,
f
Al(kl)A,(k,,
f
Ai(ki)A2(ki, k2) f A3(k1, k2, k3),
k3)
(7.10)
which is useful in comparing this approximation to the smoothing approximation in 5 8. Again A , as well as the general term A , (ZIPFELand DESANTO [ 19721) for an n-cluster decomposition contains a delta function for overall transverse momentum conservation. In addition, the integrand in each A,,(n 2 2) can be shown to asymptotically vanish (so that for example it can be Fourier transformed). It will be shown in 8 8 that the smoothing approximation does not have this property for third- and higher-order interaction moments. In each n-point interaction average only the A , can be interpreted as a “connected” term involving an irreducible n-point interaction. For example, in eq. (7.10) the first four terms on the rhs consist of a product of three single interaction terms and the sum of products of single and double interaction terms. A , is the only true three-point interaction term. Using these results the coherent scattered field ( r ) can be found using eqs. (7.2) and (7.3). The result of eq. (7.3) can be partially r e s u m e d to an integral equation in k-space, which is
(T(k‘,k”)) = M ( k ’ , k ” ) +
jj
W k ’ ,k)G%k) ( r ( k ,k”)) d k (7.11)
with the “mass operator” M defined in analogy to the term used for wave propagation in random media (FRISCH[ 19681):
c M’(k’,k ” ) X
M ( k ‘ ,k ” ) =
, = I
X
L’-
’ VA,,
(7.12)
j= 1
where each M, is related to A,. The diagram interpretation of A , and the two lowest order connected diagrams and functional forms for M , and M2 are illustrated in Fig. 7. Since each A, and hence each Mi term contains an overall momentum conserving delta function so does ( r ) and it can be factored as
(T(k’, k“))
().;
=7 ( K ’ )2
T(ki,k:’) 6(k; - kJ‘).
(7.13)
111
The resulting one-dimensional singular integral equation for the scattering
42
MULTIPLE SCATTERING FROM ROUGH SURFACES
(a 1
Fig. 7. Statistical diagrams associated with scattering from a Gaussian distributed random surface; (a) connected multipoint interaction term with several momentum lines tied together; (b) two lowest-order connected diagrams in the mass operator and their functional correspondence.
amplitude T is in scaled variables
etc.): (7.14)
where W is given by an infinite series of terms (7.15) with ui related to A., and (7.16)
If we define the scattering amplitude T,,,, as the solution of the truncated version of eq. (7.14)
&-SPACE FORMALISM-RANDOM SURFACE
43
where W, is the truncated sum of eq. (7.15)stopping a t j = m,and, in addition, set W, = 0 we compute the on-shell intensity
I Tlo(l, - I)/* = exp( -4Z2) analogous to the result in
(7.18)
4 2. When m
Tll(t',5 " ) = U l ( t ' , 5 " ) +
=n =
5
u,(5'9
1 we have the integral equation
5)K(t) T,,(5, t")d5,
(7.19)
--iu
where u , ( t ' , t")= exp[
-fz2(('- 5")2].
(7.20)
Equation (7.19)has been solved numerically (DESANTOand SHISHA [ 19741) and its intensity along with that of eq. (7.18)and other models is compared to experimental data in Fig. 8. The ( x ) data are from CLAY, MEDWINand WRIGHT[ 19731. The experimental distribution was slightly non-Gaussian (for these wave tank measurements) and its Fourier transform was used to form the intensity in curve (b). It produces too deep a dip near Z = 1 and for large roughness falls off too fast. Curve ( f ) are combined averaged electromagnetic data from BEARD [1961] and curve(e) the single scatter eikonal approximation derived elsewhere (BROWN and MILLER[ 19741, DESANTO[ 1981bl). It uniformly overestimates the coherent return, but does serve as an upper bound on the coherent return. Data (0)are from NUMRICH [1979], involving a controlled experiment on a surface with known homogeneous Gaussian statistics. Curve (a) is the single scattering result eq. (7.18).It fails at about Z = 1, the Rayleigh roughness limit, but does produce a weak lower bound on the data. Curve (c) results from the Fourier transform of the shadow modified probability density function in CLAY,MEDWINand WRIGHT[1973].It contains the free parameter C"(O), proportional to the correlation function of surface slopes at zero separation. The latter cannot be measured directly. Its value extrapolated from separated measurements is about two orders of magnitude different from that necessary to produce the curve tit (c). Curve (d) is the intensity formed from T I in eq. (7.18)which is a multiple scattering result (DESANTO[ 1981bl). It fits the controlled experimental data (0)well out to about Z = 2 and then falls off faster than the data. The conclusion is that it is necessary to use multiple scattering to properly explain the coherent scattering data. Additional data are available from ZORNIG [ 19791. In addition, it is possible to form the ensemble averages of higher-order
44
MULTIPLE SCATTERING FROM ROUGH SURFACES
Fig. 8. Coherent specular intensity versus the Rayleigh roughness parameter for various theoretical models and experimental data. Curves are explained in the text.
moments of r, e.g. the second moment ( I T ) which leads to the Bethe-Salpeter equation. Corresponding diagram methods are available for these second moments for scalar (ZIPFELand DESANTO[ 1972]), elastic (DESANTO [ 19731) and electromagnetic (DESANTO [ 19741) problems. These serve to simplify the structure of any order approximation but they involve multiple integrals and have not been numerically evaluated as yet. An extension of this surface interaction expansion has been made to rough walled waveguide [ 1985b1). problems including a spatially varying index of refraction (DESANTO
I . I 71
k-SPACE FORMALISM-RANDOM SURFACE
45
7.2. STOCHASTIC FOURIER TRANSFORM
The previous discussion illustrated a k-space formalism valid as we transformed from coordinate space to Fourier transform space and useful for either deterministic surfaces (see eq. (5.14)), or for averaging over an ensemble of stochastic surfaces (see eq. (7.11)). Here we describe an alternate and different k-space formalism which applies only to the random surface problem (BROWN [ 1982al). It is called the stochastic Fourier transform approach, since the transforms are with respect to the stochastic variables which appear in the governing integral equation. There are two key steps in the approach. The first is to decide which stochastic variables are involved in the averaging process. The second is to note that the average of a stochastic variable can be written as the convolution of the stochastic Fourier transforms of the variable and its probability density function. We demonstrate the method for electromagnetic scattering from a perfectly conducting randomly rough surface. We use the MFIE defined by eq. (4.26) but eliminated the J, component by noting that n . J = 0, so that J, = J, ah/dx + J, ahlay. The resulting equation on the transpose of the twocomponent vector (Jx,J,) is
I ( &)= J'"(x:)+
J J -K ( x ~
- x,, h' - h, dlh', d,h)J(x:)dx,,
(7.21)
where we have explicitly written the dependence of the matrix 3 on the heights h' = h ( x ; ) and slopes. Its components are (7.22)
(7.23)
(7.24)
and (7.25)
46
[I, 8 7
MULTIPLE SCATTERING FROM ROUGH SURFACES
If we define the integral operator Lt
=
jj
(7.26)
dxt
equation (7.21) in operator form becomes
J = J i n + L,EJ
(7.27)
and a formal average is
( 7 ) = (5’”)+ L , ( E J ) .
(7.28)
We are immediately faced with the hierarchy problem for random equations (FRISCH [ 19681) and we must formulate an argument to find ( 5 ). We first need to know the stochastic variables on which 7 depends. Unfortunately, this is rigorously given by a solution of eq. (7.27) which we do not know. We do know, however, that for a randomly elevated or inclined planar surface the integral term in eq. (7.27) vanishes, the Born result is exact and most importantly it is local, i.e. the current is determined from the incident field and surface properties at a single point on the surface. This is just the single scattering result. The introduction of more complicated surface structure requires the inclusion of the integral term in eq. (7.27), i.e. the inclusion of multiple scattering, which means that 5 becomes nonlocal, and hence it depends on the incident field and surface properties in some neighborhood of a given point x,. If we assume the surface to be differentiable to any order then any surface property, e.g. the pth derivative of h at the point x,, = x, + ax,, can be expressed using a Taylor series expansion about x, as (TAYLOR[1955]) (p = 0, 1,2, . . .)
d”h(x,
+ hx,) = Ph(X,) +
“ c =,
1 (hx, * m! -
I
(7.29)
Ph(X,).
Although this is purely formal we can thus write the functional dependence of
5 as J(x,) = J(h, d,h, a:h,.
. . ,x,, x, + hx,).
(7.30)
and its average requires the single-point joint probability density function for h, @,I d:h, .. .. The average is thus
(J(x,)) =
jj ’
.’
J(h, d,h, a:h,
. . . ,x,, x, + bx,)
x p1(h, d,h, a:h,. . . ) dh dd,h dd:h..
..
(7.31)
47
k-SPACE FORMALISM-RANDOM SURFACE
Note _ _ that the lhs of eq. (7.3 1) only depends on x,. Similarly, to average the term
K J in eq. (7.27) we need the two-point joint probability density function P A h ' , h, dl h ' , ad, . .., k),
(7.32)
where, for homogeneous statistics, p 2 depends only on the horizontal distance between the two points, 6xt = x i - x,. This completes the first step in the process, the identification of the stochastic variable dependence. The second step, the introduction of the stochastic Fourier transforms, is more direct. We write eq. (7.3 1) as the convolution integral (at zero separation) of the two transforms
x P l ( - P o , {-flnt})dflod{flnt},
where {fl,,,} represents the set of all transformed functions are defined by
[
p,,
(7.33)
rn = 4 . 2 , . .., n, and where the
1
pjt d{h dh d{d:h},
x exp - i&h - i j = 1
(7.35)
which are simply the Fourier transforms with respect to the stochastic variables. We are thus able to form an integral equation on the stochastic transform (BROWN[ 1982a1) by multiplying eq. (7.27) with
7
4,
exp[ -ikoh - i
1 kj;d{h j = 1
1
,
(7.36)
averaging the resulting equation, converting the averages to convolutions in Fourier transform space and noting that the average scattered field is specular for homogeneous surface statistics, so that we can derive the fact that CI
7 = j(ko,{ k n t } )exp [ik$. x,].
(7.37)
48
MULTIPLE SCATTERING FROM ROUGH SURFACES
Here G i is the partial derivative of Go with respect to h, the tilde refers to the full transform, and
where q
=
1 ( = x), q = 2 ( = y ) . The symbol P2 represents
r: = a 2 ( k , - P , B - B 0 , { k n t } , { - B , t } ) e X P ( - i ~ . ~ x t )
(7.42)
and the tilde over the Green’s function denotes the stochastic Fourier transform (from h to Po). Since J, is known in terms of J, and Jy we can also find
(7.43) With the knowledge of j we can determine any statistical moment of the scattered field (BROWN[ 1983b1). Equation (7.38) is a single infinite-dimensional integral equation of first kind for the stochastic Fourier transform of the current induced on the random surface. It is infinite-dimensional since we have no apriori knowledge as to which stochastic variables are important and which we can dismiss. In contrast
1.
I 71
k-SPACE FORMALISM-RANDOM SURFACE
49
to the diagram method which generates a hierarchy of integral equations, the stochastic Fourier transform technique provides a means of formally increasing the solution accuracy by increasing the dimensionality of a single integral equation. In spite of the different approaches it is possible to compare the diagram and stochastic transform methods in certain limiting cases. In the limit k,L -, cc, with k, being the wavenumber of the incident field and L the surface correlation distance, both methods predict the same results for scattering from a randomly elevated plane. For k,L -,0, both methods reduce to the same integral equation for the scattered field which is only specular when either k, + 0 or b--,0. However, if 0 # 0 and L -,0 both methods fail to predict the perfectly reflecting nature of the surface, since both require a well defined normal at all but a countably infinite set of surface points (BROWN [ 1985b1). When L --, 0 the surface normal does not exist in a mean square sense because the variances of the surface slopes are infinite. Fortunately, this case does not occur in the real world and is of little practical consequence. The stochastic Fourier transform approach is still largely formal in that no attempt to solve eq. (7.38) has been made. However, it has been useful in that eq. (7.37) has been derived and this yields a factorization into an explicit dependence on the transverse coordinates. This form has been used to investigate the validity of the far field approximation in rough surface scattering (BROWN[ 1983b]), and to determine the scattering consequences of two-point surface statistics for which decorrelation does not imply statistical independence (BROWN [ 1982b1). The latter example is particularly interesting because it gives rise to a component of the incoherent scattered power which exists only in the specular scattering direction. The same methodology as developed above can be used to derive an integral equation of the second kind of the average scattered field (BROWN[ 1985b1). In this procedure, we use a conditional probability density function rather than the two-point joint density in eqs. (7.39)-(7.41). However, since it is the transformed current which is essential to the computation of higher-order scattered field moments, we must go through a deconvolution operation to obtain the transformed current. Finally, we should note that it has recently been stated that the dimensionality of eq. (7.38) could be reduced to the three associated with the height and the two slope components (BROWN[ 1985bl) and this would still yield an exact result. In view of the discussion as to why an infinite number of surface derivatives are needed, it is clear that a three-dimensional form may not, in general, be exact. Conversely, we cannot say with absolute certainty that such
50
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. I 8
a dimensionality reduction is in error. Since the three-dimensional form of the equation leads to the same asymptotic results as the infinite-dimensionalresults as L .+ 00 and L + 0, we should properly call the three-dimensional form an Ansatz. That is, it can neither be proved nor disproved but it does lead to the proper asymptotic result in the two known limits. Hopefully, future numerical studies of the three-dimensional form of eq. (7.38) will clarify this issue. One final point about the utility of the stochastic Fourier transform approach is that, with the aid of eq. (7.37), we can show that the incoherent scattered power vanishes as L -+ 0 (BROWN[ 1985b1)regardless of the surface roughness IT. This result was previously obtained in the limit of small k,a (NIETOand GARCIA[ 19811). Since conservation of energy demands that VESPERINAS the coherent or specularly scattered power approach unity (for unity incident power), the surface is acting as a pure reflective surface. Thus, we now know the asymptotic behavior of the scattering process as k,L + co and k,L + 0.
8 8. Smoothing For linear wave theory the smoothing method was developed to treat wave propagation through continuous random media (random volume problems) (BOURRET[ 19611, KELLER[ 19623, FRISCH[ 19681). It has recently been adapted to the rough surface problem for the scalar case with a variety of boundary conditions (WATSON and KELLER [1984]) and to the electromagnetic case of a perfectly conducting random surface with arbitrary surface statistics (BROWN[ 1984a1)which will be discussed in 5 8.1. DESANTO[ 19861 has given the relation between the smoothing and connected diagram method, and this will be treated in 3 8.2. WATSONand KELLER[ 19841 showed that in the limit of small surface height and slope the smoothing approach agreed with a number of other approaches. Both they and BROWN[ 1984al showed that smoothing leads to an algebraic result for the average scattered field rather than an integral equation which is obtained in the random volume problem. These results are new and for the most part formal because of multiple integrations involved and the uncertainty of convergence of the formal expansions we present (BROWN[ 1982b1). The method itself is a technique for obtaining the average of a function which satisfies a Fredholm integral equation of the second kind. It can be classified as a projection technique wherein we solve for the projection of a function, here the average of the function, rather than the full function.
1. § 81
SMOOTHING
51
8.1. COORDINATE SPACE FORMULATION
We demonstrate the method for electromagnetic scattering from a perfectly conducting, randomly rough, surface by using eq. (7.21). Multiply it by F(h(xi ), 8;h(x; )) = exp [ - ik;h(x; ) - ik;, 8; h(xi )],
(8.1)
because by setting k; = ky and k;, = 0 we can easily obtain the vector components of the average scattered field due to J, and Jy,and from a non-zero transverse part, the J, contribution (BROWN[ 1984a1). The result is a matrix equation
f ( x ; ) =pyx;)
and
+ L&;
- X,>f(X,),
(8.2)
-
-
C(X: - x,)
=
KF(h(x;) - h(x,), 8;h(x;) - dth(x,)),
(8.4)
with an obvious definition for f"' and f ( x , ) . The operator L, is defined as the integration over x,, as in eq. (7.26). Next, write the unknown function f as the sum of an average value (f)A Ef and a zero mean fluctuating part Sf, where (Sf) = 0,
f = Ef + bf.
(8.5)
Since multiple averaging is involved it is convenient to replace the bracket average ( * . ) by the symbol E which operates on any random variable occumng to its right. Substituting eq. (8.5) into eq. (8.2) yields
EI
+ ST= f i n + L,GE~+ L,E~].
(8.6)
Taking the average of eq. (8.6) and subtracting the result from eq. (8.6) yields
6f
=
B + L,[Z - E E ] bf,
(8.7)
where we have defined =fin
- Efin+ L,[E
-
EE]Ef,
which is expressed solely in terms of the incident and average fields. Assuming that B is known, a formal iterative solution of eq. (8.7) is (8.9)
52
[I. § 8
MULTIPLE SCATTERING FROM ROUGH SURFACES
which, when substituted into the average of eq. (8.6), yields Ef
=
Ef,' t L , E Z Q [ f " - E f " ]
+ L,EEQEf.
(8.10)
This is an integral equation for E l in x,-coordinate space. The z = h coordinate has been averaged out. For homogeneous surface statistics we know the Ef is specular (BROWN1982a, 1983b1) and hence we can factor its coordinate dependence Ef
=
(8.11)
R(k;, kit) exp(ik?. x; ) = R(k;, ki,)x'.
If we substitute eq. (8.11) into eq. (8.10) the result for R is algebraic in analogy with the calculation of the reflection coefficient in perturbation theory, eq. (3.23). The result is
R
=
XI" t [ ? - ( a ' ) - l L t E l l ~ a ] - I ( a ' ) - l L , E l r f ' " ,
(8.12)
where a = exp (ik;" . x,). The z-component of R is obtained from the fact that the surface current is tangential to the surface (BROWN[ 1984a]), i.e. R,(k', kit) = i[aR,.ak;, + a~,/ak;,]
(8.13)
and this is why we introduced the transverse component in F. Finally, the average scattered magnetic field at point x above the surface is a plane wave in the specular direction (ksc= k;" - ik:"), ( H " " ( x ) ) = -(2cosO,)-'
kcx R(kl =
-kl,,k2,
=
O)exp[ik"'.x]. (8.14)
The variance of the scattered field or any other moment of it can, in principle, be determined from eqs. (8.9), (8.11) and (8.12). To see the advantage of the smoothing method, note that if the fluctuating part of 7 is approximately equal to its Born scattering value, Sf x B-from eq. (8.7), we obtain the n = 0 or first-order smoothing result by setting Q = 7 in eq. (8.12). An alternate means of obtaining this first-order result is to solve eq. (8.2) by iteration and assume a hierarchy breaking approximation of the type E
[LE]nf,nz rr = 0
2 [LEE]"Ef",
(8.15)
n=O
which is equivalent to assuming that the mth order multiple scattering may be approximated by the average of m multiple scattering operators acting on the average of the single scattering "current" Ef'". If we then resum this result we get the first-order smoothing result. Thus, first-order smoothing sums an
1. § 81
53
SMOOTHING
infinite number of multiple scattering contributions, but only by approximating each contribution. This will be discussed further in 5 8.2. The above discussion details how the smoothing method achieves its results. To obtain a general idea of why it should be accurate to treat the n = 0 term only, we solve eq. (8.2) iteratively and average the result to obtain ET= E
f [Lz]"f'",
(8.16)
,, = 0
which involves an infinite number of interactions between both average terms and zero mean fluctuating terms. Even if the fluctuating part of is adequately given by g , eq. (8.16) still comprises an infinite series of interacting average terms. The smoothing method overcomes this problem by summing all the interactions between average terms and, if sf is accurately given by B, smoothing yields an exact closed form result for Ef. Smoothing recognizes that in iteratively solving for one iterates on both 'E and 87 and this causes problems even when' 6 is approximately B,because one still has an infinite series. Smoothing overcomes this by only solving for b7, and then selfconsistently solving for ET in a non-iterative manner. Since the only infinite series in the solution for Ef results from the iterative series of bf we obtain a closed form solution for Ef when b7 (and its series solution) shrinks to 3.This is the advantage of smoothing relative to a standard iterative solution. Unfortunately, it is not always obvious when bf is accurately given by and this is why it is difficult, in general, to establish the range of parameter variation for which the resulting series converges. Finally, the additional real problem with all our development is the computational difficulty involved. Even the lowest-order term involves averaging over the random surface height and a transverse spatial integration, the same integrations which appeared in the stochastic Fourier transform approach. WATSON and KELLER[1984] avoided them but only by restricting their analysis to the small height and slope limit. The connected diagram approach (DESANTO [ 1981bl) does not encounter these integrals either but has its own unique computational difficulties. Future research on smoothing should concentrate on computational simplifications in the series of terms and somehow taking advantage of the algebraic nature of the result.
7,
s,
8.2. RELATION BETWEEN SMOOTHING A N D DIAGRAM METHODS
In this section we treat the relation between the method of smoothing and the connected diagram method presented in 5 7.1. Although the method of
54
MULTIPLE SCATI'ERING FROM ROUGH SURFACES
[I, § 8
smoothing was developed in coordinate space, as presented in Q 8.1, the methodology can be applied to any stochastic equation and is applied here in k-space. As also discussed in Q 8.1, the method of smoothing is a partial summation of multiply scattered terms. So is the connected diagram method and we will compare the two methods term by term (DESANTO [ 19861). Begin with eq. (7.2) where we write r as the sum of an average or coherent term plus a fluctuating part r=Er+
sr,
(8.17)
where E 6 r = 0. Substitute eq. (8.17) into eq. (7.2) to get ET+
Sr= VA + L A [ E r + br].
(8.18)
Take the ensemble average of eq. (8.18), which is E T = VEA + L E A E r + LEAST,
(8.19)
and subtract eq. (8.19) from eq. (8.18) to yield
Sr=T + L [ A - E.41 Sr, where
(8.20)
r is
-
r = V[A- EA] + L[A - E A ] E r
(8.21)
and is independent of the fluctuating part. A formal solution by iteration of eq. (8.20) is (8.22)
which yields the fluctuating part solely in terms of the mean field. Next, substitute eq. (8.22)into eq. (8.19) and use eq. (8.21). Combining terms, we can write aJ
1 [ L A - LEA]"EI',
E T = M s+ L E A
(8.23)
n=O
which is the smoothing integral equation on the mean field with the smoothing mass operator defined by
M " = VEA + LEA
X
C
[ L A - LEA]"V[A - E A ] .
(8.24)
n=O
We compare the first three mass operator terms in the expansion for the smoothing method defined by Mi",j = 1,2, 3, with the corresponding terms in
1, § 81
55
SMOOTHING
the mass operator for the connected diagram expansiongiven by eq. (7.12). The first term from eq. (8.24) is Mi
=
(8.25)
VEA = V A , ,
where the latter equality follows from eq. (7.5). This is the same as M , from the connected diagram result, eq. (7.12). The second term, for n = 0 in eq. (8.24) is
Mi
=
LEAV[A - E A ] = LV[EAA - EAEA] = LVA,,
(8.26)
where the latter equality follows from eqs. (7.7) and (7.5). Again from eq. (7.12) this is seen to equal M,, the second term in the mass operator expansion for the connected diagram result. The third term, n = 1 in eq. (8.24), is
Mi
=
LEA[LA - L E A ] V [ A - E A ] ,
(8.27)
which can be written in Fourier transform notation as
M W ,k") =
I
J J J J J W ' ,k , ) G Y ( W V ( k I ,k,)GY(k,) x V ( k 2 , k " ) A ~ ( k ' - k l , k l - k ~ , k ~ - k dkz, " ) d k(8.28) ~
where A ; is given by four terms As(m19
m2, m 3 )
=
EA(m1)A(mZ)A(m3)
- EA (ml
(%IEA
(m3)
- EA(m,)Jwm,)A(%)
+ EA (m,)EA(m z )EA (m3).
(8.29)
Comparison with eq. (7.10) illustrates that eq. (8.29) differs from the connected diagram result A , in two respects. It neglects the initial/fmal state twointeraction correlation term EA(m,)A(m3)EA(m2),and has the opposite sign for the three single-interaction correlations. Higher-order terms can be computed and they also differ. Thus, the smoothing and connected diagram methods agree to second order in the interaction approximation and differ beyond that. One advantage of the connected diagram expansion was that the integrands in each connected tern vanished asymptotically so that their respective Fourier transforms existed. Although we have written the smoothing method in transform space and thus formally defined the respective Fourier transforms in each term of the smoothing mass operator, their actual existence is open to question in this
56
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I, 0 9
regard. Note finally that in comparing mass operator terms, we have not included the integral terms in the integral eqs. (7.1 1) and (8.23). This is because it makes sense to only compare respective Born term since, if we included the integral terms, both methods would fully resum to the same result in any given interaction order.
0 9. Other Multiple Scattering Approaches In addition to the techniques described in the previous section, there are a number of other rigorous formalisms for treating the rough surface multiple scattering problem. Many of these are in a relatively early stage of development in that they require further computational work to quantitatively assess the effects of multiple scattering; a situation which is also the case with the methods discussed in $8 7 and 8. In fact, this is a caveat which can be applied to the entire field. FURUTSU [ 19831 has recently introduced a technique using a Green’s function defined on a flat reference plane and determined by a surface impedance. The surface impedance is obtained by an exact transformation of the boundary condition on the true surface to the reference surface. The transformation, in turn, is given in an operator form which simplifies considerably in the large or small surface height limits. Of particular note in this work is the attention given to satisfying conservation of power at all levels of approximation. This leads to the introduction of a new tangent plane approximation and a perturbation result which agrees with the classical result (WRIGHT[1966]), only when the relative dielectric constant of the rough surface is large compared to unity. FURUTSU has drawn extensively from his earlier work on ground wave propagation and the random volume scattering problem. WINEBRENNER and ISHIMARU [ 19851 have considerably amplified a technique first set forth by SHENand MARADUDIN[ 19801, and their work has led to a much better understanding of basic boundary techniques. For a periodic surface, they show that both the classical Rayleigh-Rice and the extinction theorem based techniques lead to essentially the same results when solved by conventional power series like perturbation methods. In view of the simplicity of the extinction theorem approach and the inherent use of the Rayleigh hypothesis in the Rayleigh-Rice method, it is apparent from their work that the accuracy limiting factor is the power series like perturbation solution, common to both methods. They use a perturbation series in the phase of the unknown
OTHER MULTIPLE SCATTERING APPROACHES
57
field and obtain a significantly improved result. They investigate the reason for this improvement and show that the phase perturbation approach comprises a partial summation of the power series-like terms. For the periodic surface, they show that this method can be successfully extended into the so-called resonant region of scattering, i.e. where k,a 1 and a is the amplitude of the surface undulations. SHENand MARADUDIN [ 19801 have shown the power of the phase perturbation approach in dealing with the average scattered field for the randomly rough surface. ITO [ 19851 has recently developed a technique which is based on the use of an effective surface impedance and, in some respects, is similar to FURUTSU’S approach. ITO’S method specifically applies to the scalar scattering case with either Neumann or Dirichlet boundary conditions and small roughness height and slope. He generates a nonlinear Dyson equation for the average Green’s function using the Novokov-Furutsu theorem and employs this result to define the effective surface impedance. The integral equation describing the surface impedance is solved via iteration to second order and the results are interpreted in terms of energy exchange between scattered and evanescent waves. ITO also showed that energy is conserved to second order in the surface impedance and numerical results are presented for a one-dimensional rough surface. Since ITO’S reflection coefficient result remains well behaved near grazing incidence for the Neumann boundary condition, it is an improvement on power series like perturbation techniques which become singular in this limit. NAKAYAMA and co-workers (NAKAYAMA, OGURAand MATSUMOTO [ 19801, NAKAYAMA, OGURAand SAKATA[ 1981a,b] and NAKAYAMA, MIZUTANI, OGURAand HAYASHI [ 19841)have used a probabilistic approach to develop a theory for scattering from slightly rough, Gaussian distributed rough surfaces. The key to their approach is the characterization of the scattered field by an infinite series of Wiener-Hermite differentials having nonrandom coefficients which, in turn, follows from the fact that the scattered field is a nonlinear functional of the random surface height (WIENER[ 19581). Satisfaction of the boundary conditions on the surface leads to a hierarchy of equations for the nonrandom coefficients which are solved by truncation at order three and higher. Such a truncation is valid for sufficiently small surface roughness. The authors couch their results in terms of an equivalent surface impedance, because this gives insight into the loss of coherent or specularly scattered energy through the dual processes of incoherent scattering and surface wave generation. Conservation of energy is used to estimate the range of surface roughness height for which their solution is valid. BAHAR[ 1981a,b, 19821 has developed a full wave approach to the rough
-
58
MULTIPLE SCATTERING FROM ROUGH SURFACES
[I. § 9
surface scattering problem which is so named because of its explicit representation of the scattered field, the surface wave, and the lateral wave contributions. The method is basically a spectral approach which leads to coupled differential equations for the forward and backward wave amplitudes. These amplitudes have been obtained, to date, only at the neglect of multiple scattering. However, the technique is, in principle, general enough to encompass multiple scattering. An alternate spectral formalism has recently been developed by DESANTO [ 1985al. It is based on a technique originally developed for periodic surfaces (DESANTO[ 1975, 1981al) and is here generalized to include both arbitrary surfaces and transmission problems. The theory is exact and the derivation is quite simple in that it used only Green’s theorem and a class of auxiliary functions, which are the continuous analogue of the Bloch functions used in the periodic surface case. Explicit representations of the spectral amplitudes of the scattered and transmitted fields resulting from plane wave incidence are derived. They are expressed as integrals over the boundary values, and the amplitudes are demonstrated to satisfy a flux conservation condition. The equations are valid for any surface height and are the exact version of the approximate Rayleigh-Fano equations (CELLI,MARVINand TOIGO[ 19751, TOIGO,MARVIN,CELLIand HILL [1977]) which are valid in perturbation theory for shallow surfaces. CHITANVIS and LAX [ 19851 have recently presented a Green’s function approach centered around the Lippmann-Schwinger equation and a random potential describing the rough surface effect. This was discussed in 5 5. The authors have generalized the development to include acoustic, electromagnetic and elastic “potentials”, as well as the usual quantum mechanical results. From this, the authors derive a Lippmann-Schwinger equation on the average or specular field in terms of the self-energy in analogy with multiple scattering theories in condensed matter physics. The equations are in coordinate space and for the total rather than for the scattered field, as was the discussion in 3 5 5 and 7. Otherwise, the self energy is analogous to the mass operator introduced in 3 7. A hierarchy of approximations is introduced for the self-energy term in analogy to the cluster expansion described in Q 7. The authors developed a self-consistent approximation which retains all finite-order correlations of the potential. The net effect is one of renormalizing the interior propagator or Green’s function. As an example, all possible first- and second-order correlations are summed, and an approximation developed to treat the case of a large potential or, when L < a, where L is the transverse correlation distance and a the rms height. The latter is particulary important for the incoherent intensity
11
REFERENCES
59
and the authors develop a Bethe-Salpeter formalism to treat this case. We have discussed the surface scattering problem as just that, a surface problem rather than a volume scattering problem. There are essentially two characteristic approaches to treating the surface scattering as a volume scattering problem. RAHMAN and MARADUDIN [ 19801 replaced the self-edge region around the surface by a flat layer of thickness greater than the distance between the maximum and minimum surface heights. They treated this region as having a mean index of refraction or dielectric constant. The flat layers then contain wave fields with a known spectral expansion. The amplitudes of the expansion are found via continuity conditions at the flat interfaces, and the unknown dielectric constant and layer thickness are found via self-consistency. They demonstrated the splitting of the surface plasmon dispersion relation into two branches due to this surface “roughness” effect. The other type of volume problem is illustrated by ABARBANEL [ 19801. Here, one maps the rough surface into an equivalent and hence random index of refraction, and then treats the problem as one of wave propagation in a continuous random medium. Finally, multiple scattering theories based on the two major single scattering approximations discussed in 5 3, perturbation theory and the Kirchhoff approximation, have also been developed. BASS and FUKS[ 19791 discussed the multiple scattering version of boundary perturbation theory, complete with a diagram interpretation. LISZKAand McCoy [ 19821developed the Kirchhoff approximation into an iterative series solution of the exact surface integral equation. The work is based on earlier work by MEECHAM[ 19561. Their conclusion is that there are convergence difficulties associated with this iteration and although a method of renormalization is suggested for a numerical solution to converge, no analytic method of renormalization was obvious. References ABARBANEL, H. D. I.,1980, J. Acoust. SOC.Am. 68, 1459. AGARWAL,G. S., 1976, Phys. Rev. D14, 1168. AMENT,W. S., 1953. Proc. IRE41,142. Ament remarks that the result was derived independently
by Pekeris and MacFarlane during the war, but he provides no references. AMES, G . H., D. G . HALLand A. J. BRAUNDMEIER, 1982. Opt. Commun. 43, 247. ANDREO,R. H.,and J. A. KRILL,1981, J. Opt. SOC.Am. 71,978. ARCHBOLD,E., and A. E. ENNOS,1972, Opt. Acta 19, 253. BAHAR,E., 1981a, Radio Sci. 16, 331. BAHAR,E., 1981b. Radio Sci. 16, 1327.
BAHAR,E., 1982, IEEE Trans. Antennas Propag. AP-30, 712. BARRICK. D. E., 1968a, Proc. IEEE 56, 1728. B A R R I C K , D. E., 1968b, IEEE Trans. Antennas Propag. AP-16, 449.
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[I
BARRICK, D. E., 1970, Radio Sci. 5,647. BASS,F. G., and V. G. BOCHAROV, 1958, Radiotekh. Elektron. 3, 251. BASS,F. G., and 1. M. FUKS.1979, Scattering of Waves from Statistically Irregular Surfaces (Pergamon, New York). BEARD,C. I., 1961, Trans. IRE Antennas Propag. AP-9, 470. BECKMANN, P. and A. SPIZZICHINO, 1963,The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York). BECKMANN, P., 1965, IEEE Trans. Antennas Propag. AP-13, 384. BENNETT, B. L., A. A. MARADUDIN and L. R. SWANSON, 1972, Ann. Phys. 71, 357. BENNETT, 3. M., 1976, Appl. Opt. 15, 2705. BIOT,M. A., 1968, J. Acoust. SOC.Am. 44, 1616. BOLOMEY, J.-C., and W. TABBARA, 1973, IEEE Trans. Antennas Propag. AP-21, 356. BORN,M., and E. WOLF, 1980, Principles of Optics (Pergamon, New York). BOURRET, R. C., 1961, Report No. 215 (Hughes Research Laboratory, Malibu, CA). and P. ROCHE,1981, J. Opt. SOC.Am. 71, 1115. BOUSQUET, P., F. FLORY L. M., 1960, Waves in Layered Media (Academic Press, New York). BREKHOVSKIKH, BROWN,G. S., 1978, IEEE Trans. Antennas Propag. AP-26, 472. BROWN,G. S., 1980, IEEE Trans. Antennas Propag. AP-28, 788. BROWN,G. S., 1982a, IEEE Trans. Antennas Propag. AP-30, 1135. BROWN,G. S., 1982b. Radio Sci. 17, 1274. BROWN,G. S., 1983a, IEEE Trans. Antennas Propag. AP-31, 5. BROWN,G. S., 1983b, Radio Sci. 18, 71. BROWN,G. S., 1983~.IEEE Trans. Antennas Propag. AP-31,992. G. S., 1984a. IEEE Trans. Antennas Propag. AP-32, 1308. BROWN, BROWN,G. S., 1984b, Radio Sci. 19, 1461. BROWN,G. S., 1985a, Wave Motion 7, 195. BROWN,G. S., 1985b, IEEE Trans. Antennas Propag. AP-33, 48. R. M., and A. R. MILLER, 1974, Report 7704 (Naval Research Laboratory, Washington, BROWN, DC). BURROWS, M. L., 1967, Can. J. Phys. 45, 1729. and F. TOIGO,1975, Phys. Rev. B11, 1779. CELLI,V., A. MARVIN CHITANVIS, S., and M. LAX, 1985, Suppl. Prog. Theor. Phys. (Japan) 80.40. CHOW,P. L., 1978, SIAM J. Appl. Math. 35, 235. E. L., H.A. JENKINSON and 3. M. ZAVADA, 1979, Opt. Eng. 18, 125. CHURCH, CLAY,C. S., H. MEDWINand W. M. WRIGHT,1973, J. Acoust. SOC.Am. 53, 1677. DAINTY, J. C., 1976, in: Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland, Amsterdam) pp. 1-46. DAVENPORT, W. B., and W. L. ROOT, 1958, Random Signals and Noise (McGraw-Hill, New York). DESANTO,J. A., 1973, J. Math Phys. 14, 1566. DESANTO,J . A., 1974, J. Math. Phys. 15, 283. DESANTO,J. A., 1975, J. Acoust. SOC.Am. 57, 1195. DESANTO,J. A., 1981a, Radio Sci. 16, 1315. J. A., 1981b. in: Multiple Scattering and Waves in Random Media, eds P. L. Chow, DESANTO, W. E. Kohler and G . C. Papanicolaou (North-Holland, Amsterdam) p. 123. DESANTO, J. A., 1983, Wave Motion 5, 125. DESANTO,J. A., 1985a, J. Opt. SOC.Am. A 2 , 2202. DESANTO, J. A., 1985b, Wave Motion 7, 307. DESANTO,J. A,, 1986, J. Math. Phys. 27, 377. DESANTO,J. A., and 0. SHISHA,1974, J. Comput. Phys. 15, 286. DEVANEY, A. J., and G . C. SHERMAN, 1973, SIAM Rev. 15, 765.
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ECKART,C., 1953, J. Acoust. SOC. Am. 25, 566. FANO,V., 1941, J. Opt. SOC.Am. 31, 213. FOCK,V. A., 1946, J. Phys. USSR 10, 130. FRISCH, U. 1968, in: Probabilistic Methods in Applied Mathematics I , ed. A. T. Bharucha-Reid (Academic Press, New York) p. 75. FUKS,I. M., 1966, Izv. VUZ Radiofiz. 9, 876. FUNG,A. K., 1966, Proc. IEEE 54, 395. K., 1983, IEEE Proc. F, 130, 601. FURUTSU, GARCIA, N., V. CELL1 and M. NIETO-VESPERINAS, 1979, Opt. Commun. 30,279. GOODMAN, F. 0..1977, CRC Crit. Rev. Solid Mater. Sci. 7, 33. HARRINGTON, R. F., 1959, IRE Trans. Antennas Propag. AP-7, 150. 1985, Geophysics 50,90. HILL,N. R., and P. C. WUENSCHEL, HUANG,K., 1963, Statistical Mechanics (Wiley, New York). HUTLEY,M. C., 1982, Diffraction Gratings (Academic Press London). ISAKOVICH.M. A., 1952, Zh. Eksp. Teor. Fiz. 23, 304. ITO, S., 1985, Radio Sci. 20, 1. JAKEMAN, E., 1982, J. Opt. SOC. Am. 72, 1034. JONES,D. S., 1979, Methods in Electromagnetic Wave Propagation (Clarendon, Oxford). JORDAN, D. I., R. C. HOLLINSand E. JAKEMAN, 1984, Memo no. 3656 (United Kingdom Royal Signals and Radar Establishment, Malvern, UK). J. B., 1962, Proc. Symp. Appl. Math. 13, 227. KELLER, KELLOGG,0. D., 1953, Foundations of Potential Theory (Dover, New York). KERR, D. E., and H. GOLDSTEIN, 1951, in: Propagation of Short Radio Waves, ed. D. E. Kerr (McGraw-Hill, New York) ch. 6. KINSMAN. B., 1983, Wind Waves (Dover, New York). KITTAPPA. R., and R. E. KLEINMAN, 1975, J. Math. Phys. 16, 421. K o ~ l s R. , D., 1966. IEEE Trans. Antennas Propag. AP-14, 77. KRESS,R., and G. F. ROACH,1978, J. Math Phys. 19, 1433. F. I., V. M. KUDRYASHOV and N. M.PETROV,1976, Sov. Phys. Acoust. 22, 211. KRYAZHEV, KUPERMAN, W. A., 1975, J. Acoust. SOC.Am. 58, 365. KURYANOV, B. F., 1963, Sov. Phys. Acoust. 8, 252. LiszKA, E. G., and J. J. McCoy, 1982, J. Acoust. SOC.Am. 71, 1093. LONGUET-HIGGINS, M. S., 1957, Philos. Trans. Roy. SOC.A249, 321. LONGUET-HIGGINS, M. S., 1983, Proc. R. SOC.London A389, 241. MARX,E., 1982, J. Math. Phys. 23, 1057. MAUE,A. W., 1949, Z. Physik 126, 601. MAYSTRE, D., 1978, J. Opt. SOC.Am. 68, 490. MAYSTRE, D., 1984a. in: Progress in Optics, Vol. XXI,ed. E. Wolf(North-Holland, Amsterdam) p. 1. MAYSTRE, D., 1984b, J. Opt. (Paris) 15, 43. 1983, J. Acoustic, SOC.Am. 73, 1476. MCDANIEL, S. T., and A. D. GORMAN, MCGINN, A,, and J. SYKES,1980, Rep. No. AERE-R9768, (United Kingdom Atomic Energy Authority, Harwell, UK). MEECHAM, W. C., 1956, J. Rat. Mech. Anal. 5, 323. MITZNER, K. M., 1964, J. Math. Phys. 5, 1776. MITZNER, K. M., 1966a, J. Math. Phys. 7, 2053. MITZNER, K. M., 1966b, Radio Sci. 1 (New Series), 27. MOORE,R. K., 1970, in: Radar Handbook, ed. M. I. Skolnik (McGraw-Hill, New York) ch. 25. NAKAYAMA, J., H. OCURAand B. MATSUMOTO, 1980, Radio Sci. 15, 1049. NAKAYAMA, J., H. OGURAand M. SAKATA,1981a, Radio Sci. 16, 831. NAKAYAMA. J., M. SAKATA and H. OGURA,1981b, Radio Sci. 16, 847.
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NAKAYAMA, J., K. MIZUTANI, H. OGURAand S. HAYASHI,1984, J. Appl. Phys. 56, 1465. NIETO-VESPERINAS, M., and N. GARCIA, 1981, Opt. Acta 28, 1651. NOBLE,B., 1962, in: Electromagnetic Waves, ed. R. E. Langer (Wisconsin University Press, Madison, WI). NUMRICH, S. K., 1979, Scattering of Acoustic Waves from Randomly Rough Surfaces, Ph.D. Thesis (American University, Washington, DC). PAPOULIS, A., 1965, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York). PATTANAYAK, D. N., and E. WOLF, 1976, Phys. Rev. D13,913. PEAKE,W. H., 1959, IRE Nat. Conv. Record 7, 27. PETIT,R., ed.. 1980, Electromagnetic Theory of Gratings (Springer, Berlin). RAHMAN, T. S.,and A. A. MARADUDIN, 1980, Phys. Rev. B21, 2137. RAYLEIGH, J. W. S.,1945, The Theory of Sound (Dover, New York). RICE,S.0..1951, Comm. Pure Appl. Math. 4, 351. ROTHROCK, D. A., and A. S. THORNDIKE, 1980, J. Geophys. Res. 85, 3955. SANCER,M. I., 1969, IEEE Trans. Antennas Propag. AP-17, 577. SHEN,J., and A. A. MARADUDIN, 1980, Phys. Rev. 822,4234. SHMELEV, A. B., 1972, Sov. Phys. Usp. 15, 173. SMITH,B. G., 1967, IEEE Trans. Antennas Propag. AP-15,668. SWIFT,C. T., 1971, IEEE Trans. Antennas Propag. AP-19, 561. TAYLOR, A. E., 1955, Advanced Calculus (Ginn, Boston). TOIGO,F., A. MARVIN, V. CELLI,and N. R. HILL, 1977, Phys. Rev. B15, 5618. TOLSTOY, I., 1984, J. Acoust. SOC.Am. 75, 1. TWERSKY, V., 1957, J. Acoust. SOC.Am. 29. 209. VALENZUELA, G. R., 1968, Radio Sci. 3, 1057. VALENZUELA, G. R., 1978, Boundary-Layer Meteorol. 13, 61. VALENZUELA,G. R., J. W. WRIGHTand J. C. LEADER,1972, IEEE Trans. Antennas Propag. AP-21, 536. VAN BLADEL,J., 1964, Electromagnetic Fields (McGraw-Hill, New York). WAGNER,R. J., 1967, J. Acoust. SOC.Am. 41, 138. WATERMAN, P. C., 1965, Proc. IEEE 53, 805. WATERMAN, P. C., 1969, J. Acoust. SOC.Am. 45, 1417. WATERMAN, P. C.. 1971, Phys. Rev. D3, 825. WATERMAN, P. C., 1975, J. Acoust. SOC.AM. 57, 791. WATSON,J. G. and J. B. KELLER,1984, J. Acoust. SOC.Am. 75, 1705. WENZEL,A. R., 1974, J. Math. Phys. 15, 317. WIENER,N., 1958, Nonlinear Problems in Random Theory (MIT Press, Cambridge, MA). WILCOX,C. G.. 1984, Scattering Theory for Diffraction Gratings (Springer, New York). WINEBRENNER, D., and A. ISHIMARU, 1985, Radio Sci.20, 161. WOLF,E. and E. W. MARCHAND, 1964, J. Opt. SOC.Am. 54, 587. WRIGHT,J. W., 1966, IEEE Trans. Antennas Propag. AP-14, 749. WRIGHT, J. W., 1968, IEEE Trans. Antennas Propag. AP-16, 217. YAGHJIAN,A. D., 1981, Radio Sci. 16,987. ZIPFEL,G. G., and J. A. DESANTO,1972, J. Math, Phys. 13, 1903. ZORNIG, J. G., 1979, in: Ocean Acoustics, ed. J. A. DeSanto (Springer, Heidelberg) p. 159.
E. WOLF, PROGRESS IN OPTICS XXIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1986
I1 PARAXIAL THEORY IN OPTICAL DESIGN IN TERMS OF GAUSSIAN BRACKETS BY
KAZUOTANAKA Corporate Technical Planning & Operation Center. Canon Inc. 9-4, Shimomaruko 2-chome. Ohta-ku, Tokyo 146. Japan
CONTENTS PAGE
. . . . . . . . . . . . . . . . . . . . GAUSSIAN BRACKETS . . . . . . . . . . . . . . . . . GENERALIZED GAUSSIAN CONSTANTS (GGC'S) . . . .
$ 1 . INTRODUCTION
2. 3.
§ 4. PARAXIAL THEORY OF A HOMOGENEOUS OPTICAL
SYSTEM BY MEANS OF GGC'S
67 72 75
. . . . . . . . . . . . .
§ 5. PARAXIAL THEORY OF AN INHOMOGENEOUS OPTICAL
SYSTEM BY MEANS OF GGC'S
65
. . ........
. .
86
.
§ 6. .PARAXIAL THEORY OF A GAUSSIAN BEAM OPTICAL 92 SYSTEM BY MEANS OF GGC'S . . . . . . . . . . . . . § 7 FEATURES OF PARAXIAL THEORY BY MEANS OF GGC'S
97
§ 8 APPLICATION EXAMPLES OF PARAXIAL THEORY BY 98 MEANS OF GGC'S . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . , ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . APPENDIX A. NUMERICAL CALCULATION OF GGC'S . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . NOTE ADDED IN PROOF . . . . . . . . . . . . . . . . . § 9 CONCLUDING NOTE
.
105
. 106
. . .
107 107 111
8
1. Introduction
In the sense of physics, the paraxial theory is characterized by the statement that it treats the rays located near the optical axis which have infinitesimally small incident angles (see, for example, KINGSLAKE [ 19781). In the formalism of mathematics, the paraxial theory can be viewed as the collineation theory (see, for example, BORNand WOLF[ 19701). Within the framework of optical design, the paraxial theory, in other words, Gaussian optics, is usually employed to express all the problems in which the first-order effects of an optical system are considered. The paraxial theory is particularly significant for the analysis and/or synthesis of an optical system, since it determines the fundamental configuration. In their textbook on optics, JENKINS and WHITE[ 19571, have presented the paraxial theory based on the equation
where a and b respectively denote the object and the image position, and f' is known as the focal length of the optical system. However, this relationship of imaging is not convenient to analyze and/or synthesize some complicated optical systems such as a zoom system, which is one of the most interesting and prominent themes encountered at the present time in the field of lens design. To not only scrutinize various paraxial properties of an optical system but also to construct an intricate optical system, several approaches to formulate the paraxial theory have been made up to the present. [ 19631 constructed a tool named the Y - 7 diagram, and TAKEDA DELANO [ 19801 proposed a similar nomographic method. Applications of the Y - 7 diagram were carried out, for example, by BESENMATTER [ 1977a-e, 1978a,b, 1979, 1980a,b], SLEVOGT[ 19701 and STAVROUDIS [ 19821. The matrix method was studied by SMITH [1945], HARROLD[1954], HALBACH [ 19641, RICHARDS [ 19641, ARSENAULT [ 1980bl and others, and was summarized by BROUWER [ 19641 and by GERRARD and BURCH[ 19751. It was extended to a non-symmetrical system by ARSENAULT [ 1979, 1980a1, RODIONOV [ 19811 and ATTARD [ 19841, to laser beam propagation by
66
PARAXIAL THEORY IN TERMS OF G A U S S I A N BRACKETS
[II, $ 1
GERRARD and BURCH[ 19751 and others, and to distributed lens-like media by GERRARD and BURCH[ 19751 and others. The factorization of the matrix was investigated by CASPERSON[ 19811, FOG [ 19821, ARSENAULTand MACUKOW [ 19831, MACUKOW and ARSENAULT [ 19831 and TANAKA [ 1983e, 1984bl. The matrix method is one of the most familiar means in the paraxial theory. Applications are found in many papers, such as those by WOOTERSand SILVERTOOTH [ 19651, ARNAUD[ 19691, KUSAKAWA [ 19751, ARSENAULT [ 19791, C ~ R E and Z FELDER [ 19831 and others. Gaussian brackets were first introduced into optics by HERZBERGER [ 1943c, 1952, 19581, reviewed by ZIMMER[ 19671 and were generalized by TANAKA [ 1981a, 1982f, 1983b,d, 1984al with the help of the theory of continued fractions (see, for example, PERRON[ 19131 and SHIBATA[ 19331). The Gaussian brackets method was extended by TANAKA to an inhomogeneously configured optical system [ 1983e, 1984b], and to a Gaussian beam optical system [ 1984c,e]. Also, TANAKA[ 1984fl discussed the relationship between Gaussian brackets and the cumulant, which is employed in the theory of two-pair terminal networks and was proposed by HERRERO and WILLONER [ 19661. Modified Gaussian brackets, which are not Gaussian brackets in the sense as defined by HERZBERGER[1943c, 1952, 19581, were proposed by BERGSTEIN[ 19581. The use of Gaussian brackets in the analysis and synthesis of optical systems was worked out by PEGISand PECK[ 19621, MINAMI[ 19721, TANAKA [ 1977, 1979a-c, 1980, 198lb,c, 1982a-e,gyh, 1983a,c,f, 1984a,d,e], TANAKAand TAKESHI[ 19821, and SMIRNOV [ 19831. Modified Gaussian brackets were and MOTZ [ 1962a-cl. used by BERGSTEIN[ 19581, and by BERGSTEIN There are still other methods of treating paraxial theory: the Wigner distribution function was discussed by BASTIAANS[ 1978, 19791; Lie algebra was applied by DRAGT[ 19821; the canonical operator was adopted by SHAMIR [ 19791, by NAZARATHY and SHAMIR[ 1980, 1982a,b] and by NAZARATHY, HARDY and SHAMIR [ 19821; group theory was used by STAVROUDIS [ 19721; and graph theory was employed by WANG[ 1979, 1982, 1984, 19851. Among several treatments of the paraxial theory, the Gaussian brackets method is considered to be one of the most versatile and well-known methods to investigate various paraxial properties of the optical system. This paper reviews Gaussian brackets defined on the basis of the theory of continued fractions, and summarizes the paraxial theory formulated with these Gaussian brackets for both homogeneous and inhomogeneous optical systems
11. § 21
GAUSSIAN BRACKETS
67
and also for the Gaussian beam optical system. Some examples of the application of the Gaussian brackets formulation to the analysis and synthesis of the optical system are also presented. This paper is not a mathematical treatise in the rigorous theorem-proof sense. Rather, an attempt is made to show the usefulness of the Gaussian brackets method as a practical tool for lens design. Throughout this paper, rays are assumed to travel from left to right. The sign convention employed is the same as that used in a typical text book, such as that by SMITH [ 19781. Therefore, a distance is positive when measured from left to right or measured upward from an optical axis, and an angle is positive when a clockwise rotation turns a line from the direction of the optical axis to the direction in which the light travels. The refractive index is positive when a ray travels in the left-to-right direction. After reflection, when the ray travels from right to left, the refractive index is taken as negative.
8 2. Gaussian brackets Our starting point presented in this section is the description of Gaussian [ 1982f, 1983b, 1984al definition of Gaussian brackets brackets. TANAKA’S based on the theory of continued fractions (see, for example, PERRON [ 19131 and SHIBATA [ 19331) is described. It can be considered as the generalization of HERZBERGER’S ideas and theories [ 1943c, 1952, 19581. Also, several formulae in regard to the Gaussian brackets are collected from the works by HERZBERGER [ 1943c, 1952,19581,HERRERO and WILLONER [ 19661,ZIMMER [ 19671, and TANAKA [ 1981a, 1982f, 1983b, 1984al. Those interested in the historical aspect of Gaussian brackets should consult the publications by HERZBERGER [ 1943c, 19521. 2.1. DEFINITION OF GAUSSIAN BRACKETS
Let us consider a continued fraction given by
and let C,,/B,,be the nth convergent of eq. (2.1). Then it follows that
68
P A R A X I A L THEORY IN TERMS OF GAUSSIAN BRACKETS
[II, § 2
...
... ... . The denominator of the nth convergent can be expressed in a recurrent form, and
L
b,B,-
B,
=
I
+ C , B , - ~ , n I 1, n
1,
=
0,
n = -1.
B, can be written with the help of Muir's symbol as B,, =
c2, C 3 r * * * c, 6,,62,b3,. . ' , b n 9
(2.4)
In the case where every partial numerator of eq. (2.1) is equal to following simplified expression to denote B , is introduced: Bn = [ b l , 623 b3, . . ., b n I *
+ 1, the (2.5)
The notation for B, in eq. (2.5) is simply the notation of a Gaussian bracket having elements b , , b,, ..., b,. Generalizing the above reduction, the definition for Gaussian brackets is derived. A Gaussian bracket, whose elements consist of a set of numbers or functions, ai, ai I , ai+2 , . , . , uj - I , a,, is written in the form +
'G,= [ a i , a i + l , a i + 2 , . . . , a j ~ l , ~ j l and is defined by the recurrent expression 'Gi- la, + i G j - 2 , i s j , i = J + 1, i=j+2.
(2.6)
11, § 21
G A U S S I A N BRACKETS
69
The first line of eq. (2.7) is equivalent to the case having plural elements of Herzberger's definition; the second line corresponds to Herzberger's empty bracket, and the third, namely, 'C, is equal to zero when i equalsj + 2, is newly provided. The third equation is necessary for the first equation to hold when i=j. The second and the third lines of eq. (2.7) are considered as a formal definition, and are of practical value for the reduction rules. Therefore, the Gaussian bracket given by eqs. (2.6) and (2.7) is regarded as the generalization of the denominator of the nth convergent of a continued fraction, whose every partial numerator equals unity. For example, the Gaussian brackets for up to four elements are obtained as 'Gi-2 =
0,
'Gi-l
=
1,
'G,
= a',
' G j +I
= u,u,+
jGi+2
=
+ 1,
a,aj+ l 0 , + 2
+ a' + a j + 2 ,
i C i +=3a , a i + l a i + 2 a i ++3a,ai+I
+ aiai+, + ~ , + ~ a+~ 1.+ ,
(2.8)
2.2. CHARACTERISTICS OF GAUSSIAN BRACKETS
There are various interesting properties of the Gaussian brackets, and we enumerate them in a series of relevant formulae. The proof of each of these formulae can be carried out by the principle of mathematical induction, but will not be discussed further here. (1) The Gaussian bracket can be written by means of a determinant:
0 'GI =
(2.9)
0
70
111, § 2
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
(2) The Gaussian bracket can be expressed with the help of the Euler-Minding formula: ' G j = aiai+,..*aj 1+
1
i,j-1
1
+
LJ-2
1 -
k-zm
akak+l
akak+l
1
i.j-3
+I--
k - z m < n akak+l
am+1am+2
1 am+lam+2
1
1
+
..*).
an+2an+3
(2.10) (3) The Gaussian bracket is reversible: [ai,ai+ I , a,+,,
. . . ,aj-
ajl
=
[aj,aj- ',.. ., a,+ ail.
(2.11)
(4)The Gaussian bracket is a linear function of any partial brackets: iGk
- iG.J+'Gk
-
J
+
' G1 -. 1
(2.12)
j+2Gk.
( 5 ) The Gaussian bracket can be developed as a linear function with respect to each of its arbitrary elements:
li
ail+ ' G k jGk-2
,Gk=
+ i+2Gk,
(2.13) (2.14)
'Gk- lak,
Gj-
, J +
Gj-
IJ+'GkU,
'Gkaj
+ jGj - 2 J + 'Gk + ,Gj - ' j + ' G k , + [ai,a,+ 1 , . * .,U j - 2 ,
aj- 1
+ Uj+ 1, Uj+2,..
(2.15) - 9
a,].
(2.16)
(6) The Gaussian bracket can be differentiated with respect to each of its arbitrary elements: (2.17) (7) The Gaussian brackets obey a determinant formula:
(2.18) In the special cases, eq. (2.18) is reduced to
(2.19)
11, § 21
71
GAUSSIAN BRACKETS
(2.20)
(2.21) (8) The Gaussian bracket, iGk, having a single zero element, is written by the Gaussian Bracket containing (k - i - 1) elements: [ a i , a i + I , . . . ? a j - 1 9o , a j + l , ...,a,]
..,aj-2,
= [ U i , ui+ I , .
aj-
1
+ a,+
1, aj+2, *
9
41-
(2.22)
(9) The Gaussian bracket in which zero-valued elements are existing alternatively, is reduced as follows: 0, ..., 0, a,+,,]
[U,,O,Ui+,,
=
+ a i + , + ... + ai+2,,
ai
[a,,O,a,+,, 0 , . . .,a,+,,, 01 = 1,
[O, ai+ , 9 0 9
aj+3,
a . 0
9
0, a j + , n +
1
(2.23) (2.24)
1 = 1,
(2.25)
[O,ai+l,~,ai+,,...,O,ai+,,+l,~l =O.
(2.26)
(10) The Gaussian bracket whose elements are all negative, is written as [ -ai,
- a j + l , * . *-,a k ]
k - i + 1 = even, [ a i , a i +I , . . . , a k ] , k - i + 1 = odd, j E { i , i + 1,..., k}.
[ai,ai+I , .
=
{-
a,>O,
. . ,a,],
(2.27)
(1 1) The Gaussian bracket in which elements are symmetrical, iG, = [at, ai + 1 . . ., a,], 9
ai = akr aj+ 1 = ak-
1,
ai+2
=
ak-2,.
. .,
(2.28)
is reduced to
+ (,Gj- 1)2, = (i + k - 1)/2, k - i + 1 = even,
iGk = (‘Gj)’ j
(2.29)
or ‘Gk = CiGjj = (i + k)/2,
aj + 2‘Gj- i G j - 2 ,
k - i + 1 = odd.
(2.30)
[II. 8 3
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
72
(12) The Gaussian bracket whose number of elements is odd, obeys the following relationship : X[aj, aj + 1 % . .,
+ 2"l =
[Xu,, aj + I / X , Xu,+ 21
. . ., Xaj + 2n I,
(2.3 1)
and the Gaussian bracket consisting of an even number of elements obeys [ai, a,+ I r
. .., a,+ 2" + 1 I
=
[xu,,a'+ l/x, Xaj+ 2 , .
a .
9
a'+ 2"
+I
/XI. (2.32)
These formulae are of great advantage in discussing the problems expressed in terms of Gaussian brackets.
8 3.
Generalized Gaussian Constants (GCC's)
In preparation for the formulation of the paraxial theory in terms of Gaussian brackets, this section presents Generalized Gaussian Constants (GGC's) together with some related formulae. Generalized Gaussian constants are written with Gaussian brackets whose elements consist of a set of constituent parameters of an optical system, namely, the powers of components and reduced distances between successive components in the optical system. The article described in this section is based on the publications by TANAKA [ 1982f, 1983b, 1984al. 3.1. DEFINITION OF GGC's
Let @i be the power of the ith component and e,! be the reduced distance between the ith and (i + 1)th components. Here, a component means a lens surface in a thick lens system, or an element of a thin lens in a thin lens system. Let us consider a set of powers and negative reduced distances arranged as in the optical system: $19
- e i 9 $ 2 9
- e h 9 . . . 9
-e;-l?+kv
-ei,+k+,,*..
*
(3.1)
The GGC's for the sub-system consisting of the ith, (i + l)th, ...,Jth components are expressed with Gaussian brackets whose elements consist of a portion of eq. (3.1), and they are defined as
11,s 31
GENERALIZED GAUSSIAN CONSTANTS
13
In the expression of 'Bj, the powers of the ith and the jth components are omitted. Also for 'Aj and 'Dj, the jth power and the ith power are omitted, respectively. Nothing is omitted for 'C,. Applying the definition of the Gaussian brackets given in eq. (2.7) to the GGC's whose first term is the power, namely, 'Aj and 'Cj, one obtains their recurrence relations, and
i s j i=j + 1
(3.3)
In a similar manner the recurrence relations for 'Bj and 'Dj, whose first term is el!,are found to be:
' D .=
{';qj + 'Dj-
i< j , i=j.
(3.4)
3.2. VARIOUS FORMULAE OF GGCs
Among the various formulae relating to the Gaussian brackets presented in expansion (2.15), the differentiation (2.17), and the determinant formula (2.21) are the most versatile equations in the analysis and synthesis of optical systems. Applying these formulae to the GGC's, one obtains some useful relationships. (1) The expansions of the GGC's with respect to one of the powers or distances, are expressed as follows:
3 2.2, the
'A, =
'Bk =
JBk$J + 'cJ1 ' B , + 'AJ 'A,, '+lA,e; + 'AJ ' A , + 'C, lBk,
-'c
J +
J +
JBk(b, 4- '0,- 1 'Bk + 'BJ 'Ak, -'DJJ+'A,e; + ' B J J + l A k+ ' D J J + l B k ,
14
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
'Ck =
iAj .jDk$j + iq. - JDk +- 'Aj ck 3 - i q J + l Ck e'i + i A j J + l C k + i C , j + l D k ,
{ {
J +
iB,JDk$j
'Dk =
+ 'Dj-1 j Dk + iB.i+Ick
7
- ~ D ~ J k+ e'i ~+ci gJ . J + 1 c , + i ~ ~ j + 1 ~ , .
(3.5)
(2) The differentiation of the GGC's with respect to the power or the distance of an arbitrary elements, is written as follows:
(3.6)
(3) The identities for the determinant for the GGC's are presented as follows :
By using eq. (3.6), we have obtained the relation among the GGC's,
Some other relations concerning the GGC's are obtained by applying the Gaussian bracket formulae in 8 2.2 to the GGC's.
11, $41
PARAXIAL THEORY OF A HOMOGENEOUS OPTICAL SYSTEM
75
4 4. Paraxial Theory of a Homogeneous Optical System by Means of GCCS
This section is devoted to a formulation of the self-contained paraxial theory for orthogonally arranged and homogeneously configured optical systems. Paraxial ray tracing, conjugate relation, magnifications (lateral, longitudinal and angular),cardinal points (nodal, focal and principal), and Lagrange-Helmholtz invariant are generally expressed in terms solely of the GGC‘s. On the basis of the formulation of the paraxial theory, the physical meaning of GGC‘s is also described. Sections 4.1-4.4 are based on the publications by HERZBERGER [ 1943c, 1952, 19581, ZIMMER[ 19671 and TANAKA[ 1983d, 1984al. Section 4.5 is derived from the works by HERZBERGER [1943a,b] and TANAKA[1981a, 1984aI. Section 4.6 is cited from TANAKA’S papers [ 1983d, 1984al. Section [ 1943a1, GER4.7 is based on the investigations discussed by HERZBERGER RARD and BURCH [ 19751, NAZARATHY and SHAMIR[ 1982al and TANAKA [ 1983d, 1984al.
4.1. PARAXIAL RAY TRACE
When a paraxial ray passes through an optical system, we have to consider only two types of processes in order to determine its progress. They are refraction/reflection and transfer, as shown in Fig. 1. The paraxial ray refraction/reflection at the ith component, whose power is given by $i,is expressed as
I -TH COMPONENT
(a) Fig. I . Fundamental paraxial process. (a) Ray
(b)
refractionireflection.(b) Ray transfer
I6
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
[II, I 4
where ai and a! designate the reduced paraxial ray angles before and after the refraction/reflection, and hi denotes the paraxial ray height at the ith component. For a thick lens system, the power is defined by (n; - ni)/ri,where r, is the curvature of the ith surface and nl! and ni mean the refractive indices before and after the ith surface. Equation (4.1) is valid not only for refraction but also for reflection without any modifications. For a thin lens system, the power is given by INl!, wherefl! is the focal length of the ith thin lens. The paraxial ray transfer from the ith to the (i + 1)th component, when their reduced distance is el!, is written as
Applying eqs. (4.1) and (4.2) in the sequence in which a ray passes through, and arranging the product by using the associate properties of the matrix and the definition of GGC‘s, the relation between the ray entering into the ith component and the ray emerging from the (i + 1)th component, is obtained by [$i,
- ei! 1 -ei!,$i+ll
\
1
) (hl) ,
[-ei!,$i+11
ai
- el!
[$i,
Using the ray traveling procedure described above, the ray propagation from the ith component to the jth component as shown in Fig. 2 can be derived. The relation between the ray entering into the ith component and the ray emerging from the jth component is obtained as
(:)
=
(‘Ai f B i ) ( h i ) iCi ‘0, ai
,
I-TH
(4.4) J-TH COMPONENT
Fig. 2. Ray path from the ith component to the j t h component.
11. § 41
71
PARAXIAL THEORY OF A HOMOGENEOUS OPTICAL SYSTEM
The relation between the rays entering into the ith and the jth component is specified by
The relation between the ray emerging from the ith component and the ray entering into the jth component is given by
(4.6) Finally, the rays emerging from the ith and the jth component are related by
(4.7) By using eqs. (4.4)-(4.7),various paraxial properties of the optical system can be systematically investigated by means of GGC‘s.
4.2. CONJUGATE RELATION
Here, we discuss the points of location of an object and its image. We treat an optical system consisting of k components (k 2 1) as shown in Fig. 3. OPTICAL SYSTEM I
1-ST
u
\
K-TH COMPONENT
IMAGE
=o
“k Fig. 3. Conjugate relation. The rays drawn with solid lines and broken lines respectively indicate axial and oblique rays.
78
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
[II, 8 4
Applying eq. (4.4) to the whole system, the ray entering into the 1st component and the ray emerging from the kth component are related by
When we trace a ray backwards, the matrix equation (4.8)can be inverted with the help of the determinant formula (3.7), and it yields
):(
=(?Ck
'") (%>.
lAk -
(4.9)
The lens back e; is defined as the distance measured from the kth component to the image, and the object distance e; is given by the distance from the object to the 1st component. They are respectively obtained as
(4.10) and
(4.11) The axial points, which satisfy eqs. (4.10)and (4.1I), are called each other's conjugate points, and it means that all the rays emitted from the object unite at the image. By supposing the object and the image as the 0th component and the (k + 1)th component, respectively, eqs. (4.10)and (4.11)can be rewritten as
(4.12) and
(4.13) respectively. Equations (4.12)and (4.13)are equivalent to " B k +I =
0,
(4.14)
which gives the general expression for the conjugate relation in terms of the GGC's.
11. S 41
PARAXIAL THEORY OF A HOMOGENEOUS OPTICAL SYSTEM
19
In the case where the object and the image are both located at infinity, the imaging relation is derived as the limit of eq. (4.14), when both the object distance and the lens back simultaneously tend to infinity, and
--. "Bk+ = 'C,
lim
e;,. r ;
I
=
(4.15)
0.
x
4.3. MAGNIFICATIONS
When an optical system is analyzed, there are three types of magnification that we have to consider so as to determine the imaging properties. They are lateral, longitudinal, and angular magnifications. Applying eq. (4.6) to the conjugate points, the relation between the ray emerging from the object and the ray reaching at the image is expressed by OBk + I )
):(
(4.16)
OD, With the help of the identity given by eq. (3.7), eq. (4.16) can be rewritten invertedly, and (4.17) Employing the ray trace eqs. (4.16) and (4.17), and the conjugate relation (4.14), the lateral magnification defined by the ratio of the image height to the object height is expressed by =
h,
+
l/ho = ' A ,
+
, = l/"Dk.
(4.18)
Applying eqs. (4.16) and (4.17) to an axial ray ho = hk+ = 0, the angular magnification given by the ratio of the actual ray slope angle at the image to that of the ray at the object point is written as (4.19) where nl, and n; are the refractive indices in the object and image spaces, respectively. The longitudinal magnification defined by the derivative of the actual lens back with respect to the actual object distance is given by Q =
d(e'n') _ -n; JOBk+Jae;, - n; 'Ak+ -3d(ehnA) n; aoBk+Jae; n; ODk
(4.20) a
80
111, § 4
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
From eqs. (4.18)-(4.20), the three types of magnification are seen as not independent of one another, but are related, and
p=
(4.21)
scy.
When the whole system is telescopic, the angular magnification tends to .I
'
=
no -
n;
n' _1. IDk --0
(4.22)
n; ' A ,
It is also possible to obtain magnification of the sub-system or the component. For example, the lateral magnification of the sub-system consisting of the ith to jth components is obtained as
ji,,= j,8,
* * *
+
8.= ai/a! =
'i
- 'Cke;+ 'A,
- iilck k
+ j +
IA ,
-
~~
(4.23)
1
' + ' Ak +
I
When the object is set at infinity, the lateral magnification of the ith component tends to ICiI p. = -
,
'Ci
(4.24)
4.4. CARDINAL POINTS
The paraxial properties of an optical system can be characterized by six cardinal points, as shown in Fig. 4. The conjugate points, where the lateral magnification is equal to unity, are named the principal points. They are the front principal point in the object space and the rear principal point in the image space, respectively. With the help of eq. (4.18), the distance measured from the first component to the front principal point sp and the distance between the kth component and the rear principal point sb are respectively derived as (4.25)
81
PARAXIAL THEORY OF A HOMOGENEOUS OPTICAL SYSTEM
f‘
1-ST COMPONENT
&
J
si
>
I
<
SP
COMPONENT
Fig. 4. Cardinal points. F and F’ respectively signify front and rear focal points, P and P’ respectively denote front and rear principal points, and N and N’respectively mean front and rear nodal points.
and (4.26)
The conjugate points, where the angular magnification is equal to unity, are called the nodal points. By using eq. (4.19), the distance measured from the first component to the front nodal point located in the object space sN, and the distance from the kth component to the rear nodal point in the image space &, are respectively given by (4.27)
and (4.28)
The conjugate point of an infinitely distant object is the rear focal point in
82
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
[It, 0 4
the image space. A back focal length designated by the distance between the kth component and the rear focal point, is given by (4.29) An effective focal length in the image space, which is the distance measured from the rear principal point to the rear focal point, is obtained as
(4.30) Supposing the rear focal point as the (k rewritten as ' A , + , = 0.
+ 1)th
component, eq. (4.29) is (4.31)
By using a method similar to the derivation of eqs. (4.29) and (4.30), the front focal length sF, and the effective focal length for the object space f, are respectively given by (4.32) and 1 I=(".) -
a, a;=o
'C,
(4.33)
Supposing the front focal point, which is the conjugate point with respect to the image located at infinity, as the 0th component, eq. (4.32) tends to (4.34)
(IDk = 0.
Examining the above derived definitions, we notice that the six cardinal points are not independent of one another, but are related by: SF
- sp
=f9
s;
- s;
=f
SN - SF =
(4.35) ',
(4.36)
(n;/n;)f',
(4.37)
sk - s; = (n;,/n;)f.
(4.38)
PARAXIAL THEORY OF A HOMOGENEOUS OFTICAL SYSTEM
I L § 41
83
From eqs. (4.30) and (4.33), the power of the whole system is defined as (4.39)
4.5. LAGRANGE-HELMHOLTZ INVARIANT
If two mutually independent rays are traced through an optical system, and the data of two rays are indicated by (hi,a,, a,!)...(hi,ui,a,!) and (%,,iii, 5;).. . iij, Z,!), as shown in Fig. 5, the expression
(zj,
hiUi -
h,Zi
-
=
*
'
=
h.a. - h.Z. J J J J
(4.40)
is invariant across any component in the optical system. This relationship is called the Lagrange-Helmholtz invariant, and plays an important role in geometrical optics. Substituting the ray transfer equation given in eq. (4.5) and a corresponding equation for the ray specifed by the bar sign into eq. (4.40), and arranging it with the help of eq. (3.7), we have the LagrangeHelmholtz invariant in terms of the GGC's, and (4.41)
Similar expressions to eq. (4.41) are derived as follows: (4.42)
1-ST
I-TH
Fig. 5. Lagrange-Helmholtz invariant for a homogeneous optical system.
84
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
111, § 4
(4.43)
and
In the case where the sub .system consisting of the ith to jth components is telescopic, all the invariants expressed with the GGC's, eqs. (4.41)-(4.44), are reduced to iA.j
=
1.
(4.45)
4.6. OTHER MISCELLANEOUS FORMULAE
By dealing with the previous results, we can derive some other useful relationships between paraxial values and construction data of the optical system. (1) The reduced distance between the ith and the (i + 1)th component is expressed in terms of the powers and lateral magnifications of the components:
=
(1 - P i ) / $ / + ( P i + 1 - ')/($i+
I P i + 1).
(4.46)
(2) The power of the ith component is written in terms of the reduced distances between two successive components and the lateral magnifications of the components:
(4.47)
PARAXIAL THEORY OF AN INHOMOGENEOUS OPTICAL SYSTEM
11. § 41
85
(3) The power of the sub-system consisting of the ith to the jth component, is obtained in terms of the powers and the lateral magnifications of the components:
(4.48) (4) The power of the sub-system is also expressed in the form:
(4.49) [
n=i
n=i
4.7. PHYSICAL MEANING OF GGC's
In order to clarify the physical meaning of the GGC's, let us suppose that one of the four quantities, 'A,, 'B,, 'q, and 'C,, becomes zero. From eq. (4.4), when 'A, = 0 is satisfied, the equation for h, tends to h, = 'B,a,, and h, is proportional to a,. This means that all the rays entering into the ith component with the same angle a, will emerge from the same point in the jth component. This is equivalent to the statement that the jth component is set at the rear focal point of the sub-system consisting of the ith to ( j - 1)th components. This is also understood by considering eq. (4.31). When 'B, = 0 the equation for h, becomes h, = 'A,h,. This means that all the rays entering into the ith component at the same height will emerge from the same point in thejth component. The ray heights at the ith component and the j t h component are proportional to each other, and the proportional constant, namely, the lateral magnification, is given by 'A,. This is equivalent to the statement that the ith and the jth components are conjugate with each other. This is also deduced from eq. (4.14). If 'C, = 0, the equation for a,' is reduced to a,' = 'D,a,. This means that the
86
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
[II, § 5
parallel rays entering into the ith component will emerge from the jth component in a new direction still parallel to one another. The entering ray angle at the ith component and the emerging ray angle from the jth component are proportional to each other, and the proportional constant ~(n,’/n,),is given by ‘0,.This is equivalent to the statement that the sub-system consisting of the ith to jth components, is afocal. This is also clear from eqs. (4.30) and (4.33). When ‘0, = 0 we then have a,‘ = ‘C,h,. This means that all the rays entering into the ith component at the same height will leave thejth component keeping the same angle with the optical axis. This is equivalent to the statement that the ith component is located at the front focal point of the sub-system consisting of the (i + 1)th to the jth component. This is also proved by using eq. (4.34). Now, let us see the relationships between the paraxially traced values and the GGC’s for some special cases. In the case where the initial paraxial values at the ith component are given by h, = 1 and a, = 0, the GGC‘s ‘AJ and ‘C, become h, and a;, respectively. When h, = 0 and a, = 1 are simultaneously satisfied, this is the independent are equal to hJ and a;, respectively. condition to the above case; ‘B, and ‘0,
6 5. Paraxial Theory of an Inhomogeneous Optical System by Means of GGC’s In this section the analysis of the paraxial theory of an inhomogeneous optical system is presented. In this case, we assume that the distributions of the refractive indices are rotationally symmetrical around the optical axis. Differential equations governing a ray transfer in the gradient index medium are presented, together with the associate Lagrange-Helmholtz invariant. A homogeneously configured model, which has equivalent paraxial properties of ray transfer in an inhomogeneous medium, is described by using the factorization of the transfer matrix and the GGC‘s. The study of the paraxial theory of inhomogeneous optical systems has been made by many workers, such as BUCHDAHL[ 19681, SANDS[ 19711, MOORE [1971], and HARRIGAN [1984], and their works have been reviewed by MARCHAND [ 1973, 19781. A different approach to the study has been made by TANAKA [ 1983e, 1984b], and some of the main results will be described in this section.
11, § 51
PARAXIAL THEORY OF AN INHOMOGENEOUS OPTICAL SYSTEM
81
5.1. PARAXIAL RAY TRANSFER
The distribution of a refractive index in an optical system is expressed by n’(x, h) = no(x) + n,(x)h2 + n,(x)h4
+ n,(x)h6 + . ’ , *
(5.1)
where x means the distance along the optical axis and h is the height measured perpendicularly from the optical axis to the associate point. Figure 6 shows a ray path from an input plane to an output plane, which planes are separated by the distance x. Analogous to eq. (4.6), the ray heights and reduced inclination angles at both planes are related by a generic form
where the elements in the 2 x 2 matrix are not the GGC’s but the functions of the index distribution. The initial values for these functions are determined as A ( 0 ) = 1,
B(0) = 0,
C(0) = 0,
D(0) = 1.
(5.3)
Applying Fermat’s principle to this case and taking the first-order approximation into account, A(x) and B(x) are derived as two particular solutions of the second-order differential equation :
Let X i , i c { 1 , 2 } , be two particular solutions of eq. (5.4), then C ( x ) and D(x) are obtained as the solutions of the first-order differential equation: d K.(X) + n,(x) - X,(x) = 0, (5.5) dx
X
OUTPUT INPUT PLANE PLANE Flg. 6. Ray transfer in a rotationally distributed index medium.
88
[It5 5
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
or d -
dx
Yi(X) + 2n1(x)X,(x) = 0.
Equations (5.4) and (5.6), can be rewritten as a set of simultaneous first-order differential equations with three unknown variables:
dx d.-Z 1 = -2n,(x)Z2 dx
(5.7)
In the case where the initial conditions are given by Z,(O)
=
0,
ZI(O)= 0,
Z2(0) = 1,
the solutions of eq. (5.7) become Z2(X) = A(x),
Z,(x)
=
C(X).
(5.9)
When the initial conditions are Z,(O)
= -
Z,(O)
l/no(O),
=
0,
ZI(0)
=
1,
(5.10)
the solutions of eq. (5.7) tend to q x ) = B(x),
Z1(x) = D(x).
(5.11)
By evaluating the initial value problem derived above, the paraxial ray transfer in the distributed index medium is determined.
5.2. LAGRANGE-HELMHOLTZ INVARIANT
-
When two mutually independent rays are traced through an inhomogeneous medium, and the data of two rays are designated by (h(O), a(0)) (h(x), a(x)) and (E(O), ii(0)) ($(x), Z(x)) as shown in Fig. 7, an expression similar to eq. (4.40),
-
a(x) Z(x) - 5(x) h(x)
=
a(0)5 (0 ) - a(0)h(O),
(5.12)
11, § 51
PARAXIAL THEORY OF AN INHOMOGENEOUS OPTICAL SYSTEM
89
Fig. 7. Lagrange-Helmholtz invariant for an inhomogeneous optical system.
becomes invariant at any point in the medium. This corresponds to the Lagrange-Helmholtz invariant for the rotationally distributed index medium. By evaluating the Wronskian of eq. (5.4) or calculating an integral (5.13)
or (5.14)
we can rewrite eq. (5.12), with the consideration of eq. (5.3), in the form a(x) h(x) - Z(x) h ( x ) = A(x) D ( x ) - B ( x ) C(X) a(0)h(0) - a(0)h(0)
=
1.
(5.15)
Corresponding to eq. (4.42), eq. (5.15) gives the Lagrange-Helmholtz invariant for the inhomogeneous medium in terms of A ( x ) , B(x), C(x), and D(x).
5.3. EQUIVALENT HOMOGENEOUS MODEL BY MEANS OF GGC's
Let us consider an optical system in which we have plural refractive surfaces and the spaces between two successive components are filled with gradient index media.
90
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
[II, § 5
Here, the radius of curvature of the ith surface is denoted by ri. The actual distance measured from the ith surface to the (i + 1)th surface is designated by dl!. The refractive index of the medium between the ith and the (i + 1)th surface is expressed by nl!(x, h) = no, i
+ n1,i(x)h2 + n2,i ( ~ ) h +4
( ~ )
* *
,
(5.16)
where the origin of the x coordinate is set at the vertex of the ith surface. By employing eq. (4.1), the ray refraction at the ith surface is given by (5.17) where
$i
means the power of the ith surface and is defined by (5.18)
By using eq. (5.2), the ray transfer from the ith surface to the (i + 1)th surface is written as: (5.19)
Now, let us consider another optical system consisting of homogeneous media. For the notation, all quantities associated with this homogeneous system are indicated by placing an asterisk over the appropriate symbols. By using eq. (4.6) the ray propagation from the r*th to the (i + 2)*th surface is expressed by (5.20)
where "A,*, "B:, "C:, and "D,* are the GGC's for this homogeneous system. Equating ray transfer matrices in eqs. (5.19) and (5.20) and using the Lagrange-Helmholtz invariant (4.42), we have $+ '
I =
Ci(df!), (5.21)
Then the 2 x 2 transfer matrix in eq. (5.19) can be seen as the product of three
11,
I 51
PARAXIAL THEORY OF AN INHOMOGENEOUS OPTICAL SYSTEM
91
matrices, and
~ i ( d i ! )Bi(di!)) - (1 -‘;TI) (1 ( C i ( d ; ) Di(d;) 0 1 $?+1
0) (1 1
0
-el
*) .
(5.22)
1
Equation (5.22) implies that the ray transfer in an inhomogeneous optical system is equivalent to a combination of two ray transfers and one ray refraction in a homogeneous optical system as shown in Fig. 8. The powers of the i*th and the (i + 2)*th component in the homogeneous system are respectively equal to the powers of the ith and the (i + 1)th surface in the inhomogeneous system. The intermediate power $+ : in the homogeneous system is introduced by the ray-bending properties due to inhomogeneity. In the case where the index distribution is given by an axial form, the decomposition of the matrix described above is not necessary, since Ai(d,!) = Di(dl!) = 1, Ci(d;) = 0, and B,(d; ) # 0 are simultaneously satisfied, and the ray transfer matrix tends to an upper triangular matrix. Then, we have the power
$i*+ 1 = Ci(d,!)= 0,
(5.23)
Fig. 8. Equivalent model. (a) Inhomogeneous configuration. (b) Homogeneous configuration.
92
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
and the reduced distances ef *
=
1
-B,(d,!) =
dx,
0
(5.24)
e,‘,*, = 0.
By using the factorization of the transfer matrix, an optical system consisting of inhomogeneous media separated by k surfaces can be modified into a homogeneous optical system having (2k - 1) components without any changes of paraxial properties. Therefore, the paraxial theory for the homogeneous optical system in terms of the GGC’s discussed in 8 4 can be employed for an inhomogeneous optical system.
8 6. Paraxial Theory of a Gaussian Beam Optical System by Means of GGC’s This section deals with the propagation of a Gaussian beam through an optical system, which consists of plural refractive components orthogonally arranged with respect to an optical axis. It is supposed that the diameter of the optical system is so large as not to significantly obscure the Gaussian beam. At first, the Gaussian beam parameters are summarized. Then, the transformation of a complex beam parameter due to the optical system is uniquely determined in terms of the GGC’s. Complex beam invariants, related to the Lagrange-Helmholtz invariant, are also presented. Section 6.1 is a summary of the articles by KOGELNIK[ 1965a,b]. Section 6.2 is based on the works by TANAKA[ 1984c,e] and KOGELNIK[ 1965a,b]. Section 6.3. is quoted from the investigations by TANAKA [ 1984c,e], and Section 6.4 is based on the articles by TANAKA[ 1984c,e] and MARUYAMA [ 19841. The present theme, the passage of a Gaussian beam through an optical system in the realm of paraxial theory, has been studied also by SUEMATSU and and FUKINUKI [1965], DESCHAMPSand MAST [1964], TIEN, GORDON WINNERY [ 19651, STEIER[ 19661, GERRARD and BURCH[ 19751, HERLOSKI, MARSHALL, and ANTOS[ 19831, BRAUN[ 19841, and others, and reviewed by KOGELNIK and LI [ 19661, ARNAUD[ 19731, and TSIBULYA, CHERTOV and SHERESHEV [ 19771.
11, I61
PARAXIAL THEORY OF A GAUSSIAN BEAM OPTICAL
SYSTEM
93
6.1. GAUSSIAN BEAM PARAMETERS
The property of a Gaussian beam is characterized by four quantities: the spot size at the beam waist wo,the spot size at the arbitrarily positioned reference plane w, the distance measured from the reference plane to the beam waist z, and the radius of the curvature of the wavefront at the reference plane R , as shown in Fig. 9. They are not independent of one another, but are related by w2 = wgz{ 1 + (Az/7rwgz)2},
(6.1)
+(KW~/AZ)~},
(6.2)
R
= Z{ 1
where A is the wavelength and n denotes the Ludolph number. Equations (6.1) and (6.2) are respectively rewritten as
+ (KW~/AR)’) = R{ 1 + ( A R / K w ~ ) ~ } - ’ .
W$=
z
w’{ 1
(6.3) (6.4)
The distance between the beam waist and the reference plane and the beam waist size can be combined in a complex beam parameter defined by q=z
+ (nw;/A)j,
(6.5)
where j is the imaginary unit. The reciprocal of eq. (6.5) is written in terms of R and w2, and l / q = 1/R - ( A / n w 2 ) j .
(6.6)
w Fig. 9. Gaussian beam parameters.
94
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
PI, 8 6
6.2. PROPAGATION OF A COMPLEX BEAM PARAMETER
We consider the propagation of a Gaussian beam through an optical system consisting ofk components, as shown in Fig. 10. Let us introduce two reference planes. One is placed in the input space, and another in the output space. It is not necessary that they are each other's conjugates. For notation, the beam parameters associated with the output space are indicated by placing a prime mark over the appropriate symbols. Let the reference planes in the input and the output space be the 0th and the (k + 1)th component, respectively. By employing eq. (4.6), we have the GGC's, ' A k +1 , OBk+I , 'Ck,and ODk, which specify the paraxial ray propagation from the input to the output reference plane. Then, the transformation of the complex beam parameter by the optical system is expressed by:
Therefore, the propagation of the complex beam parameter through the optical system is expressed by the analogous relations to eq. (4. lo), which gives the paraxial conjugate relationship of the image formation. 6.3. GAUSSIAN BEAM PARAMETERS BY MEANS OF GGC's
By using eqs. (6.7) and (6.8),and decomposing the complex beam parameter into a real part and an imaginary part with the help of eqs. (6.5) and (6.6), the
INPUT REFERENCE
OPTICAL SYSTEM I
1-ST
K-TH
'
OUTPUT REFERENCE BEAM PLANE
-1-
Fig. 10. Gaussian beam propagation through an optical system.
PARAXIAL THEORY OF A GAUSSIAN BEAM OPTICAL SYSTEM
11, § 61
95
Gaussian beam parameters are uniquely determined in terms of the GGC's. They are, for example, expressed as (6.9)
(6.10)
(6.11) (6.12)
I ;ing the same procedure as was employed in the da..ration of eqs. (6.9)-(6,12), the output space Gaussian beam parameters as the function of w and R, and the input space Gaussian beam parameters as the function of those of the output space, can also be obtained in terms of GGC's. In the following, some special cases are studied. When two reference planes in both spaces are conjugate with each other and the beam waist is set at the input reference plane, by using eqs. (4.14), (4.18), and (4.39), the output Gaussian beam parameters are (6.13)
(6.14)
R ' = -P ,
(6.15)
@
(6.16)
w ' 2 = (PWo)2.
Let us suppose that the input and the output reference plane are respectively positioned at the front and the rear focal plane of an optical system, and the beam waist in one space is set at the corresponding focal plane. Then, the beam waist in another space is located at another focal plane. By employing eqs. (4.29), (4.31), (4.32), and (4.34), the GGC's become IAk+
=
OD, = 0,
'c, =
- l/OB,+, = @.
(6.17)
96
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
111.
86
Then, the Gaussian beam parameters in both spaces are given by z = z' =
(6.18)
0,
(6.19)
R=R'==,
(6.20) When the optical system is afocal, the reference planes are conjugate with each other, and the beam waists in both spaces are on the reference planes, by using eqs. (4.14), (4.22), and (4.39); the GGC's are given by l/'Ak+ 1
=
=
(ni/n;)yy
OBk+ 1 = ' c k =
0.
(6.21)
Then, the Gaussian beam parameters become z
=
z'
=
(6.22)
0,
R=R'=x, w';
=
w12
=
(6.23)
(4/& )2 (wo/l')2= ( M n ; l2(Wh)'.
(6.24)
From eqs. (6.13)-(6.24), we have physically useful relationships between Gaussian beam parameters and paraxial properties of an optical system.
6.4. COMPLEX BEAM INVARIANTS
Here, we derive another relation between the quantities pertaining to the input and the output space. When an optical system is given, its GGC's are uniquely determined in terms of the construction data of the optical system. Now, we introduce the coordinate transformation of complex beam parameters, and new parameters are defined by
Q=
' C k q 4-
Q' =
ICkq' -
'Dk, IAk+
(6.25)
I'
Multiplying Q and Q', and arranging the product with the help of eqs. (3.7) and (6.7),we have a complex beam invariant in the form of QQ'
= - ( ' A k+ I OD, - ' B k
+
I
'C,) = - 1 .
(6.26)
By a procedure similar to the derivation of eq. (6.26), and by setting another
FEATURES OF PARAXIAL THEORY
I I . 71 ~
97
coordinate transformation for the reciprocals of the complex beam parameters as Q*
=
Q*'
'A h + I + O E A
= - 0 0k +
+
1/(71
(6.27)
"Bk + I I q ' ?
we then have another expression of the complex beam invariant, and Q*Q*'
=
- ( ' A k + l o D k - O B k + l l C k ) =- 1 .
(6.28)
From eq. (4.42) and eqs. (6.26) and (6.28), it can be noticed that the complex beam invariants correspond to the Lagrange-Helmholtz invariant of the homogeneously configured optical system. As a special case, in which the reference planes in both spaces coincide with the focal planes in the respective spaces, the complex beam invariants are reduced to 1 QQ' = ( 1 qq' = = ff'. (6.29) OEk+I)' ('Cd2 Q*Q*' ~
The resulting relationship is an expression similar to the Newton equation in the paraxial theory of the homogeneous optical system.
4 7. Features of Paraxial Theory by Means of GGC's In this section some advantages of the paraxial theory formulated with the GGC's for the analysis/synthesis of optical systems are described on the basis of the article by T:\NAKA[ 1983fl. The application of various formulae concerning the Gaussian brackets (relating, for example, expansion, differentiation, etc.) to the paraxial values written with the GGC's, enables us to analytically and systematically analyze and/or synthesize optical systems. Since the Gaussian bracket is defined by a simple recurrent form, the numerical calculation of various paraxial values is easily performed even if the optical system is complicated. With the paraxial theory formulated with traced values, namely, ray heights and ray inclination angles, some paraxial values cannot be calculated, when the amount of the denominator of the expression of the paraxial value becomes zero. For example, 6 given in BEREK[ 19301 is impossible to be evaluated for the optical system in which the traced axial ray height becomes zero at some
[II, 8 8
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
98
component. With the Gaussian brackets we can avoid such a case, because the paraxial values are expressed only with the construction data of optical systems, which really do exist.
6 8. Application Examples of Paraxial Theory by Means of GCC‘s This section presents some application examples in regard to the paraxial theory formulated with the GGC’s to show the usefulness of the Gaussian brackets’ method. The studies on the critical point and the singular point of a zoom equation, on the numerical calculation of an equivalent homogeneous model, and on the adjustment of complex beam parameters are discussed. Various applications concerning the paraxial theory constructed with the Gaussian brackets are also found in several publications, such as PEGISand PECK [ 19621, MINAMI[ 19721, TANAKA [ 1977, 1979a-c, 1980, 1981b,c, and TAKESHI[ 19821 and 1982a-e,g,h, 1983a,c,f, 1984a,d,el, TANAKA SMIRNOV [ 19831.
8.1. CRITICAL. POINT AND SINGULAR POINT OF THE ZOOM EQUATION
The first application is the analysis of a mechanically compensated zoom system. Let us consider a k-component zoom system as shown in Fig. 1 I, in which the ith and the j t h components ( 1 5 i < j S k) are axially movable to vary the lateral magnification of the whole system, while the distance between the object and the image is always kept unchanged. By using eq. (4.14), the zoom equation governing the displacement of the components is iniplicitly expressed by z(x,, x i )
= OBk+ I =
[ -eA,
-ei,.
.., - ( e , ! - ,
+ xi), $i,-(el
- xi),.
. . . , - (e;- I + x,), $, - (ej - x,), . . . ,$k, -eA]
=
.. 0,
(8.1)
where x, and x, denote the amounts of displacement of the ith component and the jth component, respectively. With the help of the differentiation formula (3.6), the partial derivatives of eq. (8.1) with respect to the amounts of displacement are expressed in terms
11. § 81
99
APPLICATION EXAMPLES OF PARAXIAL THEORY
1- ST
I-TH
J-TH
K-TH
OBJECT
(b) t-
"0 Fig. I I . The mechanically compensated zoom system, in which the ith component and the jth component are relatively movable to each other to provide zooming effects. (a) Initial state. (b) Zooming state.
of GGCs, and
+ ODi(
-(+
'q&+
+
j+
1q i+
lAk+
(8.4)
By employing the lateral magnification (4.23), eqs. (8.2)-(8.4) can be rewritten as
100
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
respectively. Using eq. (8.5) and the corresponding equation involving xi, we have the derivative of x, with respect to x i , and
Our first problem is to find out the critical point of the movement of the jth component with respect to that of the ith component. In a case where IP;I=1
and
IpilZl
(8.9)
are simultaneously satisfied, a critical point is introduced to xj. By employing the Maclaurin expansion, the zoom equation in the vicinity of the critical point is approximately written by 1 a2x. J
x =
-
’
Xi’
2 ax:
(8.10) When j
=
i
+ 1 is satisfied, eq. (8.10) is reduced to (8.11)
Therefore, in the neighbourhood of the critical point, the zoom equation is approximately expressed by the parabolic form. The next problem is to determine the singular point of the zoom equation. In a case where
/?,I
= lP,I =
1,
(8.12)
is satisfied, the derivative (8.8), becomes indeterminate. This indicates the existence of a singularity. From the fact that the zoom movement of com-
11. §
XI
APPLICATION EXAMPLES OF PARAXIAL THEORY
101
ponents must be continuous, the Hessian of the zoom equation should be non-zero and negative:
s2z
d2Z
1
dx,dx,
ax;
I
-.
Hessian
.
=
Then, applying the Morse lemma to this case, the zoom equation in the neighbourhood of the singular point is expressed by
=
Whenj = i
PI+ I , , -
lP14?Xf +
Qs
1
P,+ I . , ~~
~
I
B,
2
xI
’
(8.14)
+ 1 is satisfied, eq. (8.14) is reduced to
Z(X,, x,
+
1)
2
= p,$),xi
+ $$+I ~.
x:+
, = 0.
(8.15)
Pi+ I
Therefore, the zoom equation at the singular point becomes a node, and the zooming locus is smoothly continuous. X.2. EQUIVALENT HOMOGENEOUS MODEL OF A GRADIENT-INDEX SINGLET
The second example is the numerical determination of the homogeneous model, which has equivalent paraxial properties of a gradient-index singlet. The configuration data of the lens that we are now concerned with, are given by r,
=
90.,
r,
=
-4O.,
d ; = 20., nl, = l . , n;(x, h ) = 1.55 x (0.0025~+ 1.)2 - 0.00015h2, n; = 1..
(8.16)
I02
P A R A X I A L THEORY IN TERMS OF G A U S S I A N BRACKETS
By using eqs. (5.7)-(5.1 l), we have the elements in eq. (5.2), and A , ( d ; )=
0.963955,
B I ( d ; )= - 12.138364, C , ( d ; )=
0.005927,
D , ( d ; )=
0.962764.
(8.17)
To check the values in eq. (8.17), we calculate the Lagrange-Helmholtz invariant for inhomogeneous media eq. (5.19, and (8.18)
Al(d;)Dl(d;) - B,(d;)Cl(d;) = 1 .
By using eq. (5.18), the powers of the front surface and the rear surface are respectively obtained as $1 = $: = (n;(O.,0.) - nh)/rl = 0.006111, $2
=
$;
=
(n; - n ; ( d ; ,O.))/rz
=
0.017722.
(8.19)
Substituting eq. (8.17) into eq. (5.21), we have the power due to the ray-bending property of a gradient-index and its position in the equivalent homogeneous model : $; = C , ( d ; )= 0.005927, e; *
=
(1. - D l ( d ; ) ) / C l ( d= ; ) 6.282831,
e;*
=
( 1 . - A l ( d i ) ) / C l ( d ;=) 6.082000.
(8.20)
By checking the Lagrange-Helmholtz invariant of the homogeneous model (4.43), we have ' A3* ID*3 - ' B3* 1c* 3 = I., where -e;*, $ ,;
'A:
=
[$:,
B ,*
=
[ -e;*,$T, -e;*]
'C:
=
[$:,
I
-e;*, $ ,;
ID* = [ - e ; * , $,:
-e;*]
-e;*, $f]
-e;*, $:]
(8.21) = = -
0.889776, 12.138363,
=
0.027579,
=
0.747650.
(8.22)
Finally, we have the equivalent homogeneous configuration whose data are indicated by tagging the appropriate symbols with an asterisk. By employing eq. (8.22) and the analysis described in 5 4, the paraxial properties of the lens given in eq. (8.16) can be discussed.
1 4 5 81
103
APPLICATION EXAMPLES OF PARAXIAL THEORY
8.3. ADJUSTMENT OF COMPLEX BEAM PARAMETERS
The third application is the synthesis of an optical system, the adjustment of complex beam parameters of a Gaussian beam by controlling constitutional parameters. Let us consider an optical system consisting of k components (k 2 2) as shown in Fig. 12. The problem, which we are now concerned with, is to determine two component powers, $iand @, (1 S i <j 5 k), which must fulfil the given input complex beam parameter l / q and the output beam parameter l/q’ without any changes of the other parameters in the optical system. As given in eq. (6.8), the complex beam parameters in the input space and the output space are governed by (8.23) The complex beam parameters can be expressed as
(8.24) Substituting eq. (8.24) into eq. (8.23) and decomposing it into a real part and an imaginary part, we have ‘ A k +]q,? + OEk+ - IC,(q,?q,*’ - q y q t ’ ) - OD,q,*‘ 1.4,
+
q* - ‘C,(q,*q*’ + q1*’qi*)- OD,q*’
=
0,
=
0,
(8.25) (8.26)
W --
OUTPUT SPACE Fig. 12. Adjustment of the input and output complex beam parameters by changing the powers of the ith component and the jth component. a’i
a’j
104
P A R A X I A L T H E O R Y IN TERMS
(11, § 8
OF G A U S S I A N BRACKETS
respectively. Equations (8.25) and (8.26) can be viewed as the simultaneous equations with respect to qj and qj. Applying the expansion formula (3.5) to the GGC‘s in eqs. (8.25) and (8.26) and arranging them to express qj in explicit form, we respectively have
- {A2q,* + B2 ?I
=
- C2(q,*q,*’ - qi*qi*’) -
D2q,*’ }
(8.27)
’
+ B , - C,(q,*q,*’ - qi*q*’) - D d ’
(8.28)
(8.29)
Equating the right-hand sides of eqs. (8.27) and (8.28) and arranging them by using the complex beam invariant (6.26), we have a simplified equation, and - (‘Aj)2(q,*2
+ qr2)q:’ - (oBj)2q,?’ - 2q,*qi*’
+(’Dk)2(q,?’2
‘A:Bj
+ qi*’2)q? + ( j B k +l)2qi* - 2q,*’qfJB,+
=
0. (8.30)
Equation (8.30) is an implicit equation of $ j . Expanding eq. (8.30) by using eq. ( 3 . 9 , and arranging it in a power series of r#~~,we finally have
xq; + Y q j + z= 0, where
(8.31)
11. I 91
U
105
CONCLUDING NOTE
+ 4:’2)q: - (,Bk+ 1)’q,* + 2q:’q:
= -(JDk)2(4,*’2
A,,
=
‘A,’B,,
A,,
=
‘C,- I ‘B, + ‘A, ‘ + ‘ A , ,
B,,
=
oBlrBJ,
B,,
=
OD,- I ‘B, + OBI ‘+‘A,.
’Bk+ 1 ’ D k ,
(8.32)
By using eqs. (8.31) and (8.27) or eqs. (8.31) and (8.28), we can fulfil the given conditions such as the input complex beam parameter l / q and the output beam parameter l/q’.
4 9. Concluding Note A summary of one of the useful methods for the analysis/synthesis of an optical system in lens design has been presented. The method is based on the concept named “Gaussian brackets”. Gaussian brackets are derived as the denominator of the nth convergent of a continued fraction, whose every partial numerator is equal to unity. The Generalized Gaussian Constants (GGC’s) are written with the Gaussian brackets, whose elements consist of constitutional parameters of an optical system. The GGC’s have a clear physical meaning, and are useful to formulate paraxial theory. The paraxial theory formulated with the GGC‘s has been developed to deal with homogeneously and inhomogeneously configured optical systems, and Gaussian beam optical systems. Various paraxial properties of an optical system, such as those of paraxial ray tracing, conjugate relation, magnifications, cardinal points, LagrangeHelmholtz invariant, the homogeneous model equivalent to the ray transfer in a gradient index medium, and the transformation of Gaussian beam parameters, have been derived in terms of the GGC’s. The usefulness of the Gaussian brackets’ formulation has been shown with some application examples. The author believes that Gaussian brackets provide one of the most versatile tools for optical engineers and will contribute to the development of future novel optical systems in various fields of application. Also the author hopes that the Gaussian brackets’ formulation will be applied not only to other types of optical systems such as a decentered optical system, but also to the aberration theory.
106
PARAXIAL THEORY IN TERMS OF GAUSSIAN BRACKETS
SI, 59
Acknowledgements The author is deeply grateful to Professor J. Tsujiuchi of the Tokyo Institute of Technology and Professor E. Wolf of the University of Rochester for giving the author the opportunity to write this review. The author also wishes to express his gratitude to Emeritus Professor K.P. Miyake of the University of Tsukuba for his invaluable suggestions on this article and to Emeritus Professor T. Ose of the University of Tokyo for his continuous interest in this theme. The author acknowledges the support of Canon Inc., who made it possible for him to pursue this work. Special appreciation is due to Dr. H. Mitarai and Dr. Y. Matsui, both of Canon Inc., for their encouragement. Thanks are also extended to Dr. M. Takashima of Canon Inc., who read and commented on the entire manuscript.
TABLE A. I Program list for numerical calculation of GGC's. SUBROUTINE GGC (POWER,DSTNC,I ,J,A,B,C,D) C C C
C C C
NUMERICAL C A L C U L A T I O N OF G E N E R A L I Z E D G A U S S I A N CONSTANTS , J ) ,B( I ,J) , C ( I ,J) ,D( I ,J) POWER(N) : POWER OF THE N-TH COMPONENT D S T N C ( N j : REDUCED D I S T A N C E BETWEEN THE N - T H AND (N+1 ) - T H COMPONENTS I M P L I C I T REAL*B(A-H,O-Z) DIMENSION POWER(I:J),DSTNC(I:J) A - 1 .DO0 B=O. DO0 C - 0 . DO0 D = l .DO0 IF(1.GT.J) GO TO 2 NO=I 1 C=C+A*POWER( NO) D=D+B*POWER(NO) I F ( N 0 . E Q . J ) GO TO 2 A=A-C*DSTNC( NO) B-B-D*DSTNC ( N O ) NO=NO+l GO TO 1 2 RETURN END
111
REFERENCES
107
Appendix A. Numerical Calculation of GGC’s When we have a series of powers and negative reduced distances of an optical system given in eq. (3.1), the GGC‘s are defined by the recurrent formula expressed in eqs. (3.3) and (3.4). The program list written with FORTRAN 77 for the calculation of GGC‘s is presented in table A. 1. When this program is used as a sub-routine program, a set of optical component data and the sutlixes must be provided in the main program. By the call of the following external procedure,
CALL GGC(POWER(I),DSTNC(I),I,J,A,B ,C,D),
(A. 1)
we have the values of GGC‘s, namely, ‘Aj,‘Bj, ‘Cj,and ‘Dj,in A, B, C, and D, respectively.
References ARNAUD, J. A,, 1969, Degenerate optical cavities, Appl. Opt. 8. 189. J. A., 1973, Hamiltonian theory of beam mode propagation,in: Progress in Optics. ARNAUD, Vol. XI, ed. E. Wolf (North-Holland, Amsterdam) p. 247. ARSENAULT, H. H., 1979, The rotation of light fans by cylindrical lenses, Opt. Commun. 31,275. ARSENAULT, H.H., 1980a, A matrix representation for non-symmetrical optical systems, J. Opt. (Paris) 11, 87. ARSENAULT, H. H.,1980b, Generalization of the principal plane concept in matrix optics, Am. J. Phys. 48, 397. ARSENAULT, H. H., and B. MACUKOW, 1983, Factorization ofthe transfer matrix for symmetrical optical systems, J. Opt. SOC.Am. 73, 1350. ATTARD, A. E., 1984, Matrix optical analysis of skew rays in mixed systems of spherical and orthogonal cylindrical lenses. Appl. Opt. 23, 2706 and 3740 (errata). BASTIAANS,M. J., 1978,The Wigner distribution function applied to optical signals and systems, Opt. Commun. 25, 26. BASTIAANS, M. J., 1979, Wigner distribution function and its application to first-order optics, J. Opt. SOC.Am. 69, 1710. BEREK, M.,1930, Grundlagen der praktischen Optik-Analyse und Synthese optischer Systeme (Walter de Gruyter, Berlin) p. 26. BERGSTEIN,L. 1958,General theory of optically compensated varifocal systems, J. Opt. SOC.Am. 48,154. BERGSTEIN, L., and L. MOTZ, 1962a, Two-component optically compensated varifocal system, J. Opt. SOC.Am. 52, 353. BERGSTEIN,L., and L. MOT& 1962b,Three-component optically compensated varifocal system, J. Opt. SOC.Am. 52, 363. L., and L. MOTZ,1962c, Four-component optically compensated varifocal system, BERGSTEIN, J. Opt. SOC.Am. 52, 376. BESENMATTER, W., 1977a, Das Delano-Diagramm des Vario-Glaukar-Objektivs,Optik 47, 153.
I08
P A R A X I A L THEORY IN TERMS OF GAUSSIAN BRACKETS
[I1
BESENMATTER,W., 1977b, Das Apertur-Diagramm des Vario-Glaukar-Objektivs,Optik 47,381. BESENMATTER,W., 1977~.Analyse des Vignettierungsverhaltens der Vario-Objektive mit Hilfe des Delano-Diagramms, Optik 48, 289. BESENMATTER, W., 1977d, Analyse des Aberrationsverhaltens der Vario-Objektive mittels Delano- und Apertur-Diagramm, Optik 49, I . BESENMATTER, W., 1977e, Das Delano- und Apertur-Diagramm der Vario-Objektive mit viergliedrigem, afokalem Vorsatz, Optik 49, 325. BESENMATTER, W., 1978a. Lageabweichungen einzelner Glieder von Vario-Objektiven, Optik 51, 147. BESENMATTER, W., 1978b, Analyse der primitren Wirkung asphiirischer Fliichen mit Hilfe des Delano-Diagramms, Optik 51, 385. BESENMATTER, W., 1979, The optical level s y s t e m 4 zoom projection lens with new features, Opt. Acta 26, 1377. BESENMATTER, W., 1980a. Zur Stabilisierung einfacher Variatoren mittels Asphgren, Optik 57, 123. BESENMATTER, W., 1980b, Designing zoom lenses aided by the Delano diagram, Proc. SPIE 237, 242. BORN,M., and E. WOLF, 1970, Principles of Optics, 4th Ed. (Pergamon, Oxford) p. 151. BRAUN, G., 1984, Optik GauDscher Biindel far eine fotometrische Anordung, Feingeriitetechnik 33, 173. BROUWER, W., 1964, Matrix Methods in Optical Instrument Design (Benjamin, New York) p. 20. BUCHDAHL, H. A,, 1968, Optical Aberration Coefficients (Dover, New York) p. 305. CASPERSON, L. W., 1981, Synthesis of Gaussian beam optical systems, Appl. Opt. 20, 2243. CEREZ,P., and R. FELDER, 1983, Gas-lens effect and cavity design of some frequency-stabilized He-Ne lasers, Appl. Opt. 22, 1251. DELANO, E., 1963, First-order design and the y , j diagram, Appl. Opt. 2, 1251. DESCHAMPS, G. A., and P. E. MAST,1964, Beam tracing and applications, in: Quasi-Optics, ed. J. Fox. (Polytechnic, Brooklyn) p. 379. DRAGT.A. J., 1982, Lie algebraic theory of geometrical optics and optical aberrations, J. Opt. SOC.Am. 72, 372. FOG, C., 1982, Synthesis of optical systems, Appl. Opt. 21, 1530. GERRARD, A., and J. M. BURCH, 1975, Introduction to Matrix Methods in Optics (John Wiley, London). HALBACH, K., 1964, Matrix representation of Gaussian optics, Am. J. Phys. 32, 90. HARRIGAN, M. E., 1984, Some first-order properties of radial gradient lenses compared to homogeneous lenses, Appl. Opt. 23, 2702. HARROLD, J. H., 1954, Matrix algebra for ideal lens problems, J. Opt. SOC.Am. 44,254. HERLOSKI, R., S. MARSHALL and R. ANTOS,1983, Gaussian beam ray-equivalent modeling and optical design, Appl. Opt. 22, 1168. HERRERO, J. L., and G. WILLONER. 1966, Synthesis of Filters (Prentice Hall, New Jersey). HERZBERGER, M., 1943a, A direct image error theory, Quart. Appl. Math. 1, 69. HERZBERGER, M., 1943b, Direct methods in geometrical optics, Trans. Am. Math. SOC.53,218. HERZBERGER, M., 1943~.Gaussian optics and Gaussian brackets, J. Opt. SOC.Am. 33, 651. HERZBERGER, M., 1952, Precalculation of optical systems, J. Opt. SOC.Am. 42, 637. HERZBERGER, M., 1958, Modern Geometrical Optics (Interscience, New York). JENKINS, F. A., and H. E. WHITE, 1957, Fundamentals of Optics, 3rd Ed. (McGraw-Hill, New York) p. 33. R., 1978, Lens Design Fundamentals (Academic Press, New York) p. 40. KINGSLAKE, KOGELNIK,H., 1965a. Imaging of optical modes - Resonators with internal lenses, Bell Syst. Tech. J. 44,455.
"I
REFERENCES
10')
KoGELNIK, H., 1965b. On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation, Appl. Opt. 4, 1562. KOGELNIK, H..and T. LI, 1966, Laser beams and resonators, Appl. Opt. 5, 1550. KUSAKAWA,T., 1975. Paraxial theory of zoom systems, in: Lens Design Seminar ofJOERA (Jpn. Opt. Eng. Res. Asoc., Tokyo) p. 1 (in Japanese). B., and H. H. ARSENAULT, 1983, Matrix decompositions for non-symmetrical optical MACUKOW, systems, J. Opt. SOC.Am. 73, 1360. MARCHAND, E. W., 1973, Gradient index lenses, in: Progress in Optics, Vol. XI, ed. E. Wolf (North-Holland, Amsterdam) p. 305. E. W., 1978, Gradient Index Optics (Academic Press, New York). MARCHAND, S., 1984, Gaussian beams and their transformation, Kogaku (Jpn. J. Opt.) 13,294 MARUYAMA, (in Japanese). MINAMI, S., 1972,Thin lens theory of zoom systems, Kogaku (Jpn. J. Opt.) 1,329 (in Japanese). MOORE,D. T., 1971, Design of singlets with continuously varying indices of refraction, J. Opt. SOC.Am. 61, 886. NAZARATHY, M. and J. SHAMiR, 1980, Fourier optics described by operator algebra, J. Opt. SOC. Am. 70, 150. NAZARATHY, M. and J. SHAMIR, 1982a. First-order optics - a canonical operator representation: lossless systems, J. Opt. SOC.Am. 72, 356. 1982b. First-order optics: operator representation for systems NAZARATHY, M. and J. SHAMIR, with loss or gain, J. Opt. SOC.Am. 72, 1398. NAZARATHY, M., A. HARDYand J. SHAMIR,1982, Generalized mode propagation in first-order optical systems with loss or gain, J. Opt. SOC.Am. 72, 1409. PEGIS, R. J., and W. G. PECK,1962, First-order design theory for linearly compensated zoom systems, J. Opt. SOC.Am. 52, 905. O., 1913, Die Lehre von den KettenbrUchen (Teubner, Leipzig). PERRON, RICHARDS, P. I., 1964, Conventions in matrix optics, Am. J. Phys. 32, 890. RODIONOV, S. A,, 1981, Matrix technique of Gaussian optics near an arbitrary ray, Opt. Spectrosc. 50, 53 1. SANDS,P. J., 1971, Inhomogeneous lenses, 111. Paraxial optics, J. Opt. SOC.Am. 61, 879. S H A M I R , J., 1979, Cylindrical lens systems described by operator algebra, Appl. Opt. 18,4195. SHIBATA, H.. 1933, Theory of Continued Fractions (Iwanami, Tokyo) (in Japanese). H., 1970, Hilfsmittel Wr die anschauliche Analyse von optischen Systemen, Optik 30, SLEVOGT, 431. SMIRNOV, s. E., 1983, Cakulation of a system of two thin components having a specified value of image curvature, Sov. J. Opt. Technol. 50, 26. SMITH, T., 1945, On tracing rays through an optical system, Proc. Phys. SOC.(London) 57,286. SMITH, W. J., 1978, Image formation: Geometrical and physical optics, in: Handbook of Optics, ed. W. G. Driscoll (McGraw-Hill, New York) 2-4. STAVROUDIS, 0.N., 1972, The Optics of Rays, Wavefronts, and Caustics (Academic Press, New York) p. 294. STAVROUDIS, 0. N., 1982, Modular Optical Design (Springer, Berlin) p. IS. STEER, W. H., 1966, The ray packet equivalent of a Gaussian light beam, Appl. Opt. 5, 1229. SUEMATSU,Y., and H. FUKINUKI, 1965, Analysis of the idealized light waveguide using gas lens, J. Inst. Electron. Commun. Eng. Jpn. 48, 1684 (in Japanese). TAKEDA, M., 1980, New diagram for nomographic design and analysis of paraxial optical systems, J. Opt. SOC.Am. 70, 236. TANAKA, K., 1977, Method of focusing a catadioptric system by movement of the front-member, KOGAKU (Jpn. J. Opt.) 6, 155 (in Japanese). TANAKA, K., 1979a. Allgemeine GauDsche Theorie eines mechanisch kompensierten ZoomObjektives, Opt. Commun. 29, 138 and 30,458 (errata).
I10
PARAXIAL THEORY I N TERMS OF GAUSSIAN BRACKETS
[I1
TANAKA, K., 1979b. Paraxial theory of zoom lens having built-in range extender, in: Digest of the Annual Meeting of the Institute of Television Engineers of Japan (Inst. TV Eng. Jpn., Hamamatsu)4-2, p. 93 (in Japanese). TANAKA, K., 1979c, Focusing system for zoom lenses, Oyo Buturi 48,950 (in Japanese). TANAKA, K., 1980,Focusing system with constant angular field ofview for the use ofzoom lenses, Oyo Buturi 49, 1102 (in Japanese). K., 1981a. Lagrange-Helmholtz’s invariant and optical invariants in terms of Gaussian TANAKA, brackets, Optik 58, 351. TANAKA, K., 1981b. Classification of mechanically compensated zoom lenses and paraxial analysis of the novel zoom type, Kogaku (Jpn. J. Opt.) 10, 356 (in Japanese). TANAKA, K., 1981c, General paraxial analysis of Bravais system, Optik 60,73. TANAKA, K., 1982a. Paraxial analysis of zoom lens with built-in range extender I: Thin lens configuration and thick lens system, Optik 60,209. TANAKA, K., 1982b. Paraxial analysis ofzoom lens with built-in range extender 11: Built-in range extender for the use of zoom lens with three relay parts, Optik 61, 79. K., 1982c, Paraxial analysis of zoom lens with built-in range extender 111: Thin lens TANAKA, configuration by taking into consideration of exit pupil position, Optik 61, 163. TANAKA, K., 1982d, Paraxial analysis of mechanically compensated zoom lenses. 1. Fourcomponent type, Appl. Opt. 21, 2174 and 3805 (errata). TANAKA, K., 1982e, Paraxial analysis of zoom lens with built-in range extender V: Adjustment of back focal distance, Optik 62, 15. TANAKA, K., 1982f, On Gaussian brackets and Gaussian constants, Optik 62, 211. TANAKA, K., 1982g, Allgemeine GauDsche Theorie eines mechanischen kompensierten ZoomObjektivs; 2. Zoom-Objektiv aus (n - 2) beweglichen Gliedern, Opt. Commun. 43, 315. TANAKA, K., 1982h, Paraxial analysis of mechanically compensated zoom lenses. 2. Generalization of Yamaji type V, Appl. Opt. 21, 4045. TANAKA, K., 1983a. Paraxial analysis of mechanically compensated zoom lenses. 3. Fivecomponent type, Appl. Opt. 22, 541. TANAKA, K., 1983b. Gaussian brackets and their application to paraxial theory. I. Gaussian brackets and generalized Gaussian constants, Optik 64, 13. TANAKA, K., 1983c, Allgemeine GauDsche Theorie eines mechanischen kompensierten ZoomObjektivs. 3. Kritischer Punkt und singdarer Punkt einer Zoom-Gleichung, Opt. Commun. 45, 231. TANAKA, K., 1983d, Gaussian brackets and their application to paraxial theory. 11: Paraxial theory in terms of generalized Gaussian constants, Optik 64, 89. TANAKA, K., 1983e, Paraxial theory of rotationally distributed index media, in: Technical Digest of the 4th Topical Meeting on Gradient-Index Optical Imaging Systems (Jpn. SOC.Appl. Phys., Opt. SOC.Am. and ICO, Kobe) A5, p. 30. TANAKA, K., 1983f, Paraxial analysis on zoom lens having built-in range extender by means of Gaussian brackets, Kogaku (Jpn. J. Opt.) 12, 484 (in Japanese). TANAKA, K., 1984a. Paraxial theory of mechanically compensated zoom lenses by means of Gaussian brackets, Doctorate thesis (Tokyo Institute of Technology, Tokyo). TANAKA, K., 1984b, Paraxial theory of rotationally distributed-index media by means of Gaussian constants, Appl. Opt. 23, 1700 and 3265 (erratum). TANAKA, K., 1984c, Analysis and synthesis of Gaussian beam optical system, Optik, 67, 345. TANAKA, K., 1984d, Zoom system for Gaussian beam, Optik 68, 7. TANAKA, K., 1984e. Analysis and synthesis of Gaussian beam optical system in: Conf. Digest of the 13th Cong. of the ICO (Sci. Coun. Jpn. and Jpn. SOC.Appl. Phys., Sapporo) B8-I, p. 678. TANAKA, K., 1984f, Extended definition of cumulants and its application to ladder-type networks, ntz Archiv 6, 207.
111
NOTE ADDED IN PROOF
Ill
TANAKA, K., and K.TAKESHI, 1982, Paraxial analysis of zoom lens with built-in range extender IV: Modular construction, Optik 61, 365. TIEN,P. K.,J. P. GORDONand J. R.WINNERY, 1965, Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media, Proc. IEEE 53, 129. TSIBULYA, A. B., V. G. CHERTOV and A. B. ~HERESHEV,1977, Spatial structure of laser beams and geometrical optics, Sov. J. Opt. Technol. 44, 633. WANG,S.,1979, Matrix and flow-graph methods in misalignment laser systems, J. Hangzhou Univ. 6, No. 3, 42 (in Chinese). WANG.S.,1982, Propagational problem ofoptical beams through an asymmetric inhomogeneous medium, Sci. Sinica A25, 72. WANG,S.,1984, Flow-graph analyses for increments of third-order aberration in decentered optical systems, J. Hangzhou Univ. 11, No. 2, 190 (in Chinese). WANG,S.,1985, Matrix methods in treating decentered optical system, Opt. Quant. Elect. 17, 1. WOOTERS,G.. and E. W. SILVERTOOTH, 1965, Optically compensated zoom lens, J. Opt. SOC. Am. 55,347. ZIMMER, H.-G.. 1967, Geometrische Optik (Springer, Berlin) p. 45.
Note added in proof After completing the article in December 1984, the following works in regard to Gaussian brackets have been carried out. T ~ K A H A STAJIMA HI, and M I N A M[I9851 I and TANAKA [1985a,b, 1986el employed the Gaussian brackets in the analysis of the novel optical systems. T.ANAKA [ 1985~.1986a,c] formulated paraxial theory for a slightly decentered optical system by using the GGC's. TANAKA [1986b] derived the paraxial skew invariants expressed with the GGCs and T A N A K[A 1986dI also proved the conjugate-shift invariant in terms of the GGC's. TAKAHASHI, S.,A. TAJIMAand S . MINAMI,1985, Compact zoom lens design using aspherical surfaces, in: Technical Digest of the International Lens Design Conference (SPIE, Opt. SOC. Am. and KO,Cherry Hill, NJ) WA2. TANAKA, K., 1985a. Analysis on novel range extender for zoom lenses by means of Gaussian Brackets, in: Technical Digest of the International Lens Design Conference (SPIE, Opt. SOC. Am. and ICO, Cherry Hill, NJ) WA3. K., 1985b, Paraxial analysis on novel configuration of zoom lens with built-in range TANAKA, extender by means of Gaussian Brackets, Proc. SPIE 554, 371. K., 1985c,Paraxial theory ofdecentered optical system by means of Gaussian Brackets, TANAKA, J. Opt. (Paris). 16, 251. K., 1986a. Paraxial theory of optical system having slightly displaced component(s) by TANAKA, means of Gaussian Brackets, Optik, 72, 125. TANAKA K., 1986b. Paraxial skew ray-tracing and paraxial skew invariants by means of Gaussian Brackets, Optik, 72, 149. K., 1986~.Paraxial theory of an optical system with plural decentered sub-systems by TANAKA, means of Gaussian Brackets, Proc. SPIE 655, to be published. TANAKA, K.,1986d.Conjugate-shift invariant in terms ofGaussian brackets, Appl. Opt. 25,1531. K., 1986e, A novel configuration of zoom lens, Optik, to be published. TANAKA,
This Page Intentionally Left Blank
E. WOLF, PROGRESS IN OPTICS XXIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1986
I11
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES BY
P.J. MARTINand R. P. NETTERFIELD CSIRO.Division of Applied Physics Sydney. Australia 2070
CONTENTS PAGE
. . . . . . . . . . . . . . . . . INTERACTION PHENOMENA . . . . . .
$ 1 . INTRODUCTION . . .
115
$ 2. ION-SURFACE
117
$ 3 . INFLUENCE OF ION BOMBARDMENT ON FILM PRO-
. . . . . . . . . . . . . . . . . . . . . . . $ 4 . ION-BASED METHODS OF THIN FILM DEPOSITION . $ 5. OPTICAL THIN FILMS . . . . . . . . . . . . . . . . $ 6 . CONCLUSIONS . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . PERTIES
. .
128 137
. 148
.
177
. 178 . 178
8 1.
Introduction
Since the inception of vacuum deposited optical coatings in the 1930s, continued efforts have been made to improve their properties to meet the increasing demands of optics. Initially, antireflective, high-reflectance, and beamsplitter coatings of a few layers were all that was required. Today, the output power of laser systems is often limited by the failure of coatings. Laser gyroscopes require mirrors with reflectances in excess of 99.99 percent and filters are expected to remain very stable for extended periods of time. The purist’s view of a multilayer coating is that it consists of homogeneous layers with parallel and perfectly smooth boundaries whose properties are the same as the bulk starting materials. Using such a model it is a simple task to calculate the spectral characteristics of different multilayer systems. In practice, the properties of vacuum deposited thin films usually fall far short of this ideal. Interfaces are often rough, adhesion to the substrate and to neighbouring films can be poor, the stress in the layer can be so high as to change the optical figure of the substrate or to cause if to peel off, and the optical properties of the film can be substantially different from those of the bulk material. An even more serious problem arises when the finished coating is exposed to the atmosphere as many of the film properties change depending upon the relative humidity. RITTER [1981] has summarized all the properties of thin films which are important in optical applications and reports the difficulty in reproducing them accurately since they are dependent on the coating deposition conditions. The influence of the process parameters on film properties is shown in table 1. Two dots represent a strong dependency, one dot an established dependency, and a dot in parentheses indicates a possible dependence. It can be seen, for example, that the refractive index of the film depends on the starting material, whether a glow discharge is used, on the evaporation rate, the residual gas pressure, the angle of incidence of the vapor stream on the substrate, and the substrate temperature. Increasing efforts have been made by the manufacturers of deposition systems to control as many of the process parameters as possible, so that the film properties are at least reproducible and it is hoped that they can perhaps be optimized. The effect of ions was not seen as a very important parameter in 1981, apart
I I6
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[Ill, § 1
TABLE1 Influence of process parameters on film properties. (RITTER[1981].) F,lm Prop,,ly
Substrate SubsIralo Starling
Glow
Malerial
~aierlol
Discharge Method
0
.
Cleaning
-
R elr acl#ve Index
Evsp.
Pressure
Rale
0.
0.
v
0.
~
Substrate ~ ~ Temp
~
0.
(4 Geomelric
1
Thlchness
I
,
I
I
.I
Adherence Hardness
.I
I
I
1
I
..I
I
0
.I
0
I
I
I
..I
.I
I
I
.I
.I
I
I
.I
.I
I
0
0.
0
0.
0
.
0
.
0
0.
0
0.
0
0.
I
0.
0
.I
0
.I
0 0
from the glow discharge cleaning of substrates to increase the film adhesion. AUWARTER [ 19601 first reported that optical absorption might be reduced in oxide films prepared by reactive evaporation if ionized oxygen, rather than neutrals, was used. Some improvements in the properties of optical films prepared by ion plating were reported by HENDERSON[1977] and REID, MACLEOD, HENDERSONand CARTER [1979], and for sputtered films by COLEMAN [ 19741. A glow discharge after deposition has sometimes improved film properties but in the case of MgF,, as in sputtering, problems were encountered. More significant improvements for oxide films have been demonstrated by HEITMANN [ 1971al and EBERT[ 19821, who developed apparatus to spray low-energy oxygen ions at the surface of the growing film. ALLEN[ 19821 confirmed that films produced with ions had lower absorptance and this helped to explain why these layers had higher laser damage thresholds. Only minor changes in packing density were reported. Improvements in the properties of optical films were observed when more energetic ions were used to bombard the growing film. MARTIN,MACLEOD, NETTERFIELD, PACEYand SAINTY[ 19831 showed that the microstructure of TiO,, SiO,, and ZrO, could be modified using 600 eV ions and that films with very high packing densities which did not absorb moisture could be produced.
,
111. § 21
ION-SURFACE INTERACTION PHENOMENA
117
Further work by ALLEN[ 19831, MCNEIL,BARRON,WILSONand HERRMANN [ 19841and MARTIN,NETTERFIELD and SAINTY[ 19841confirm these findings. For ZrO, films bombarded with oxygen ions, it was found that a stable layer with bulk-like properties was produced provided the ion-to-depositingmolecule arrival ratio was above a certain threshold. These films could be produced on unheated substrates or even on a highly porous underlayer. At last it appeared that reproducible films may be deposited under relaxed process control, provided the arrival ratio is above the critical point. Up to the present time the technique has produced films with very high packing densities for almost all the oxide materials used in optical coatings, but the technique has not been extended to other materials. The influence of ions, therefore, has become a very important process parameter in the deposition of optical films. Its importance has also been recognized in other areas of thin film technologyin recent times. WEISSMANTEL [ 19811 has used ion-based techniques for synthesizing films. TAKAGI [ 19821 has used accelerated ion-clusters to modify substantially the growth kinetics for many materials, and GREENE [ 19831 has made significant advances in 111-V compound synthesis using such techniques. This rapid increase in the importance of ion-based deposition of thin films has prompted the writing of this review. The emphasis in the next two sections is on detailing the physical processes associated with an energetic ion striking a surface and on the influence of ion bombardment on the film structure, adhesion and chemical composition. Section 4 gives a description of the various deposition techniques which use ions as a process control parameter. There is little optics in $8 2, 3 and 4, but they are a necessary background to the remainder of the review which details the results which have been published. A discussion of the oxide materials makes up the bulk of $ 5 since most successful applications of ion-based techniques to optical films have been made for these materials. However, the considerable interest in hard carbon films for infra-red and protective applications necessitated its inclusion in this review. Although coatings used in solar energy photo-thermal applications are an application of optical films, we have not included them in this review, since it is a large field and requires separate treatment.
0 2. Ion-Surface Interaction Phenomena Ions play a key role in thin-film deposition and influence the substrate, substrate-film interface, film structure and film composition. Ions can transfer
118
[III, § 2
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
energy, momentum, and charge by varying degrees depending upon the specific deposition technique employed. In all cases, however, a complex variety of processes occur simultaneously during energetic ion or neutral particle interactions with solid surfaces. There have been several comprehensive reviews of ion-surface interactions (KAMINSKY [ 19651, CARTER and COLLIGON [ 1968]), and only the following processes will be discussed in relation to thin film deposition and modification: sputtering, ion reflection, penetration and trapping, and chemical effects, and these are illustrated in Fig. 2.1.
2.1. SPUTTERING
One of the principal effects during ion-surface interaction is the ejection of material from the substrate/target surface (sputtering). The energetic primary particle (ion, neutral, or molecule) loses its energy in a series of elastic and inelastic collisions with target atoms. The recoiling target atoms initiate secondary collisons until the energy is dissipated within a volume of the target. The event is termed the collision cascade. Where the target surface intersects the cascade, energy and momentum are transferred tG surface atoms which may be ejected if the energy is sufficient to overcome the surface binding energy. The parameter of interest is the sputtering yield S, which represents the number of target atoms sputtered per incident particle. EMITTED PARTICLES
LOW ENERGV INCOMING ION
REFLECTED PARTICLES
IMPLANTATION, CHEMICAL EFFECTS,
Fig. 2.1. General ion-surface interaction processes.
111. § 21
ION-SURFACE INTERACTION PHENOMENA
1 I9
The theoretical model of SIGMUND[ 19691 has been highly successful in predicting the value of the sputtering yield for medium- to high-energy ions ( 10-100 keV) incident on single-element target materials. Generally, in ion-based thin film formation, the ion energies are below a few keV and usually less than 1 keV. Sigmund has proposed a sputtering yield S for ion energies E, below 1 keV and larger than the surface binding energy U,: S = 0.3
(”> E”, u,
MiM a (Mi + M ) 2 Mi
where Mi and M are the atomic masses of the incident ion and target respectively, and a(M/Mi)is a numerically tabulated function varying from 0.15 to 1.5. SEAH [1981] has compared this model with the measurements of WEHNER[1975] and OECHSNER [1975] for 500eV argon ions. When corrections were made for target contamination, the results closely agreed with Sigmund’s model. A more general model was developed by BOHDANSKY, ROTHand BAY[ 19801 to predict low-energy ion sputtering ofmonotomic solids. The model takes into account a threshold energy for sputtering. Eth,given by E,h
=
u,/y(l - y ) ,
=
U, 8(Mi/M)2’5. M i / M > 0.3,
Mi/M< 0.3, (2.2)
where y = 4MiM/(Mi+ M)’. The sputtering yield S for M i / M < 1 is given by S
=
6.4 x 10- MySl’E;/4(1 - E*-
‘)’I2
(2.3)
where E* = Eo/Eth.The sputtering yield is plotted against the normalized ion energy, in Fig. 2.2. The most complete set of experimental sputtering yields available has been compiled by ANDERSENand BAY [ 19801, covering ions over the energy range 0.1 to 100 keV. The situation is far less satisfactory for the sputtering of compound targets. There exists no universal sputtering theory and only limited experimental data are available. 2.1.1. Preferential sputtering Ion bombardment of a compound target surface results in a change of surface composition. This phenomenon is usually referred to as preferential sputtering and is a consequence of the different sputtering yields of each surface con-
I20
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
0
NORM. ION ENERGY
KE =
0
I
E'
Fig. 2.2. Normalized sputtering yield as a function of normalized ion energy(eV). (BOHDANSKY, ROTH and BAY [1980].)
stituent. The degree of preferential sputtering depends upon projectile energy, atomic masses and surface binding energies and can be significant for light ions such as He incident on Ta,O, (TAGLAUER [ 19821). The general result is that the surface is depleted ofthe lighter element until a steady-state value is reached. Lower energies and lower ion masses tend to increase the depletion. If either parameter is varied, a new steady-state surface composition is reached. TAGLAUER has interpreted this effect quantitatively in terms of the maximum energy transfer +
Tmax =
YE,.
(2.4)
In the case of compounds with large differences in mass, a higher fraction of
111, § 21
ION-SURFACE INTERACTION PHENOMENA
121
the He energy is deposited into the sub-lattice containing the lighter element and the effect becomes more pronounced at lower primary energies. At energies around the threshold for sputtering, not enough energy can be transferred to the high mass constituents to remove it from the surface, whereas the low mass atoms are eventually sputtered. Table 2 shows the relative equilibrium surface concentrations C,(e.g. Ta/O) of various compounds after 1 keV He and Ar bombardment. If can be seen that preferential sputtering effects under these conditions are quite marked for dielectric type materials such as Ta,O, and A1,0,. Oxide systems generally become depleted of oxygen under ion bombardment. HOLMand STORP[ 19771 have observed oxygen reduction of oxidized Mo, W, Nb, Ta, Ti, Zr, Si and Bi. Many measurements of preferential sputtering of compounds have been reported, and a compilation of available data of metal alloys as well as oxides has been made by COBURN[ 19791. KELLYand LAM [ 19731 have attempted to reconcile oxide sputtering measurements with the surface binding energy and have found that for Al,O,, MgO, Nb205, SiO,, Ta,O,, TiO,, UO, and ZrO, the results are in accordance with Sigmund’s theory, i.e. S is proportional to l/Uo where Uo is the surface binding energy of the compound. Attempts have also been made to predict bombardment-induced structural changes in compounds. Certain oxides undergo a crystalline-to-amorphous transition under ion impact and/or are reduced to a lower oxide phase. NAGUIB and KELLY[ 19751 have proposed two models, one based on a thermal-spike approach and the other on a bond-type criterion. In the thermal-spike model the ion impact creates a small disordered region equivalent to a liquid which cools rapidly (10- ” to 10- s), and crystallization begins when the tempera+
+
+
TABLE2 Ratio of the equilibrium surface concentrations C, of various compounds after 1 KeV He + and Ar bombardment. (TAGLAUER [ 19821.) +
Compound Ta205 TaC
wc
Ah0, Be0 TIN TIC TiB,
3.0 3.1 2.8 1.2
1.03 1.3 I 1.01 5 1.3
I22
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
“11, § 2
ture falls below the melting point. In the bond-type approach it is assumed that covalent solids which can accommodate substitutional disorder will be more easily amorphized than ionic solids which cannot tolerate substitutional disorder, even to a small extent. The degree of ionicity is therefore a measure of the degree of amorphization to be expected under ion impact. Both models are successful in predicting structural changes in oxides subjected to ion bombardment.
2.2 ENERGIES OF SPUTTERED PARTICLES
Film properties are largely determined by the energies of the depositing atoms. In conventional evaporative methods this may be only about 0.1 eV, whereas for sputtering the energies are considerably higher. The energy distributions of neutral particles sputtered during an ion beam-solid interaction have been investigated by SCHWEER and BAY[ 19821 and by BAY,SCHWEER, BOGENand HINTZ[ 19821 under a wide range of conditions. Figure 2.3 shows ENERGY (eV) I
00)0.5
1 2
L 6 8 10 15 20 25
30 LO 50 60 70 80
100 120
UO 6 0 180 200
I 1
TI-TARGETS
8keV OlkeV r25keV A05 x
He* He* D*
D*
u Vel oci t y r e solu t Io n
VELOCITY (km/sec 1 Fig. 2.3. Velocity distributions of sputtered neutral Ti atoms. (BAY,SCHWEER, BOGEN and HINTZ [1982].)
111,
8 21
ION-SURFACE INTERACTION PHENOMENA
I23
that the peak in the energy distribution shifts slightly with primary particle mass and energy and that the distribution is not determined by the surface binding energy alone. The maximum in the energy distribution is approximately at 1 eV with a tail extending to values in excess of 100 eV. The degree of ionization of the sputtered atoms depends strongly upon the bombarding conditions, in particular upon the state of the target surface. The presence of oxygen in the sputtering chamber leading to the formation of surface oxides can enhance the secondary ion yield by orders of magnitude. The ionized fraction is typically 0.01 and the average energy is greater than that of the neutral component. The peak energy is approximately 5-10 eV, again with a high energy tail. The sputtered flux will also contain a small percentage of molecular neutrals and ions and various complexes of metal-oxide molecules, again depending on the target conditions. (WITTMAACK [ 19761). The energy of the sputtered atoms at the substrate surface is determined by the energy loss to the residual gas in the deposition chamber. MEYER,SCHULLER and FALCO [ 19811have calculated the energy distributions of Nb and Cu atoms sputtered by low energy Ar bombardment and the effect of argon gas pressure as a function of distance away from the target surface. Their results show that the energy of the sputtered atoms approaches the thermal energy of the sputtering gas within a few tens of centimetres. Figure 2.4 shows that the energy distribution of sputtered Cu atoms arriving at the substrate could be adjusted by varying the sputtering conditions, affording greater control over depositing atom energy than is possible with evaporation. +
2.3. ATOMIC MIXING
Ion-based film deposition techniques usually produce greatly increased adhesion at the film-substrate interface. Broad interfacial regions of slowly changing composition have been reported and the mechanisms thought most responsible for interfacial mixing have been reviewed by CARTERand ARMOUR [ 1981 1. One possible mechanism is due to the thermal interdiffusion of coating and substrate atoms if the substrate reaches a sufficiently high temperature during film growth. Bombardment of the substrate by energetic particles may enhance the diffusion by creating mobile lattice defects. Radiation enhanced diffusion can then promote deep penetration of the coating into the substrate. A second mechanism is based upon the concept of cascade mixing. Initially collisions occur between recoiling substrate and coating atoms and later, as the film thickness increases, between film atoms alone resulting in mixing. The rate
124
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
200
0
2000 I000
0
0
10 20 30 ENERGY ( e V )
40
Fig. 2.4. Change in Cu energy distribution with distance from target. Distances are: (a) 0 cm, (b) 3 cm, (c) 6 cm, (MEYER, SCHULLER and FALCO[1981].)
at which interfacial mixing occurs has been estimated by CARTER and ARMOUR [ 19811 to be approximately the same as that of the rate of sputter erosion.
2.1. ION REFLECTION
Ion reflection is an important parameter in film formation where, for example, a directed ion beam is used to sputter a target. The particle flux at the substrate will contain target species, primary ions implanted in the sputter target and subsequently resputtered, and ions backscattered from the target surface. It is possible for light ions to be backscattered from heavier target atoms, for example H e + , Ne+ , and Ar+ ions from Cu and Au surfaces. The energy of the backscattered ion, E , ,can readily be calculated from binary collision theory (CARTER and COLLIGON [ 1968]), assuming the ion collides only once with a
I I I , § 21
125
ION-SURFACE INTERACTION PHENOMENA
target atom:
"=[ EO
cos 0 5 (p' - sin' 6)'12 l+p
where p = M J M , and 0 is the scattering angle. The differencebetween the measured and calculated energy of a backscattered ion indicates the presence of inelastic effects such as collisional excitation. Detailed estimates of backscattered fluxes are made difficult by neutralization processes. Low-energy ions near a solid surface have a strong probability of undergoing charge exchange processes with the surface (HAGSTRUM [ 19541). These processes can occur on both the incoming and the outgoing (reflected) trajectories, but the greatest degree of neutralization occurs during the collision event (MACDONALD and MARTIN[ 19811). ECKSTEIN and VERBEEK[ 19791 have performed detailed calculations and measurements of light-ion reflection. The study was related to fusion research and therefore was restricted to H , D ,T , and He but it does show that considerable backscattering occurs at low energies. For example, the reflection coefficient for He+ on W increased from 0.4 at 4 KeV to 0.85 at 15 eV. +
+
+
+
2.5. ION TRAPPING
If a low-energy ion of 50-1000 eV (typical energies in most coating processes) is incident on a solid surface, it can penetrate and eventually come to rest. The ion range in the solid is strongly energy dependent, and for energies greater than 10 keV, the implant depth can be accurately calculated (LITTMARK and ZIEGLER [ 19801). An approximation is 1 nm/keV. Very little data are available for low-energy ion ranges. The theoretical predictions presently available are based upon computer models such as that of B I E R S A ~ Kand HAGGMARK [ 19801. HARPER,CUOMO and KAUFMAN [ 19821 have suggested an expression for the depth of damage of low-energy ions, which corresponds to the penetration depth I of the form
I = 0.11 w,~ ; ' 3 / ~ , ( 2 ; 1 4 + 2:'4)*,
(2.6)
where I is the depth in nm, W, and P,are the atomic weight and specific gravity of the target element, E, the ion energy in eV, and Zi and 2,the atomic numbers of the ion and target materials. The equation is an empirical fit to the data of ASPNESand STUDNA [ 19801 and a typical value of I is 6 nm for 700 eV Ar on Ge. +
126
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[Ill. 5 2
Once the ion has come to rest in the target it may occupy an interstitial position or replace a lattice atom. The foreign atom may also be captured at a grain boundary or combine chemically with the host lattice. In some instances foreign atoms may cluster and form gas bubbles. CARTER, ARMOUR, DONNELY, INGRAM and WEBB[ 19801 have shown that the degree of gas trapping that occurs is strongly dependent upon the ion-target combination. Sufficient defects must be created in the target material to trap the incident ions. Light ions such as H e + create very few defects, since the maximum energy transfer to the target atom is relatively small and the defect production rate is also quite small even at higher energies. In the case of argon bombardment, the entrapment probability increases rapidly with ion energy. Figure 2.5 shows the entrapment probability q as a function of ion energy for Ar and Kr incident on borosilicate glass. In most cases maximum trapping probability occurs at 1 keV. +
+
2.6. SPIKE PHENOMENA
Structural changes in materials bombarded by energetic ions can be attributed to spike phenomena. BRINKMAN [ 19541 has suggested that when the energy
= I0.6 LL LL w 0
u
1-
0.41 P 4 In
! 0.2
0=1.75X1D ' ions/cm2 1 Arpon 2 Krypton
E.eV Fig. 2.5. Trapping probability ( 9 ) as a function of incident Ar' and Kr' ion energy for glass targets. (CARTER, ARMOUR, DONNFLY, INCRAM and WEBB [1980].)
111.8 21
127
ION-SURFACE INTERACTION PHENOMENA
of the incident particle and knock-on particles decrease to the point that the formation of defects is no longer possible, the rate of energy loss may be sufficient to cause melting of the solid around the track. The molten regions are described as displacement spikes and may recrystallize in a different orientation. The concept of a thermal spike was first discussed by SEITZand KOEHLER [ 19561). In this model the energy of the bombarding ion is transferred to the film without displacing film atoms. The temperature in a small volume of lo3 to lo4 atoms then rapidly rises to about lo4 K. The duration of the spike is typically 10 - I s and overlapping of spikes may occur. The structural state of the surface after bombardment has been shown by NAGUIBand KELLY[ 19751 to depend upon the ratio T,/T,, where T, is the crystallization temperature and T , the melting point of the material. When TJT, is smaller than 0.3 the surface remains or becomes crystalline, and when larger than 0.3 the surface remains or becomes amorphous. The model accurately predicts the behaviour of almost all materials for which information is available. Table 3 shows the crystal structure of a number of optical materials following ion bombardment.
TABLE3 Crystal structure and stoichiometry of optical materials following ion bombardment. Amorphous (Am), crystalline (Cr), stoichiometric (St), temperature of crystallization (TJ, melting point (T,,,). (ARer NAGUIBand KELLY[1975].) Material
Crystal structure
TJT,
Structure following ion impact
SiO,
Hexagonal Hexagonal Cubic Tetagonal Cubic Hexagonal Cubic Monoclinic Tetragonal Hexagonal Hexagonal
0.57 0.43
Am Am Cr Am, St Cr Cr Cr Am, St Am, St Cr Cr
,41203 ,41203
TiO, Ti0 Ti,O, ZrO, Nb205
Ta205
ZnS ZnSe
-
0.35
0.27 0.42-0.49 0.38-0.46
-
128
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, § 3
0 3. Influence of Ion Bombardment on Film Properties 3.1. STRUCTURE OF THIN FILMS
Common features of condensation and formation of thick films were first [ 19691 for metal and oxide deposits. identified by MOVCHAN and DEMCHISHIN Three characteristic structural zones were found which were determined by the temperature of the substrate T and the melting point T, of the material being deposited. The boundary temperatures T I and T2 were found to be 0.3 and 0.45-0.5 T, for metals and 0.22-0.26 and 0.45-0.5 T, for the oxides. Each zone has a well defined structure. At a substrate temperature below T I ,films comprise tapered columns which have increasing diameters with temperature and also characteristic, domed structures. The zone above T I is characterized by columnar grains with a smooth matt surface and the third zone above T, is characterized by a polyhedral structure. The mechanical properties of the films are ultimately determined by the structural features of the relevant zone. The zone concept has been extended, as shown in Fig. 3.1, to sputtering conditions by THORNTON [1974, 19751 who added a third coordinate to account for the influenceof the working-gas pressure. The model was developed essentially from data obtained for 25 pm thick deposits of metallic films. The TRANS I T ION STRUCTURE
COWMNAR GRAINS
POROUS S CONS IS T I CRY STALL BY VOIDS
Fig. 3.1. Three-dimensional zone structure model. (THORNTON [ 1974, 19751.)
111.5 31
INFLUENCE OF ION BOMBARDMENT ON FILM PROPERTIES
I29
zone structure was found to depend upon the interplay of atomic shadowing, adatom diffusion, and surface and volume recrystallization. A macrostructure (open grain boundaries) results when adatom diffusion is insufficient to overcome shadowing effects and microstructure (closed-type grain boundaries) results from surface and volume recrystallization. The pressure range investigated was 0.1-3 Pa of Ar. A typical example of highly tapered columnar growth is shown in Fig. 3.2 (MUNZand HOFMANN[ 19831). The instability of optical coatings, upon exposure to a humid atmosphere, is attributed to the structure of the films. Under typical deposition conditions Pa, substrate temperature: 30-300 "C, rate: 0.5-5 nm/s) a (pressure: dielectric thin film will generally develop a grain columnar structure (zone 1 and zone 1 1 type). Water from the atmosphere is then adsorbed throughout the film by capillary action. The process is largely irreversible and has plagued optical film development for many years. Attempts have been made to calculate the refractive index changes associated with water vapor adsorption (OGURA, and HIRAGA[ 19751). SUGAWARA, MACLEOD The mechanism of atom by atom columnar growth has been modelled by DIRKSand LEAMY[ 19771 as shown in Fig. 3.3. Their model assumed that an incoming molecule sticks at its place of impact on the substrate and then relaxes
Fig. 3.2. Example of gross columnar structure in thin Alms. (MUNZ and HOFMANN[ 19831.)
130
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
Fig. 3.3. Computer simulation ofan A o z B,, hard disk alloy "deposited at (a) a = 30". and (b) 1 = 60". (DIRKS and LEAMY[ 19771.)
a=30°
a=30°
T=lOK
~ = 3 0 0 ~
a=30°
T=287K
a=30°
T=317K
Fig. 3.4. Computer simulation of the effect of substrate heating on the void density of growing thin films for vapor flux angles of incidence of 3 0 ° , and 60" (facing page). (MULLER[1985].)
111.
0 31
INFLUENCE OF ION BOMBARDMENT ON FILM PROPERTIES
131
into the nearest triangular coordination location formed by two previously deposited atoms. The model predicts the formation of micro-co!umns caused by atomic self-shadowing, and a column orientation with respect to the substrate surface according to the tangent rule: tanr
=
2 tanj,
(4.1)
where x is the angle of the incident vapor stream and fi the angle of inclination of the columns. Detailed observations of the microstructure of thin-films have been made by MESSIER[ 19821. MULLER[ 19851 has recently simulated the effects of substrate temperature on the void density of growing films. It was assumed that an atom subject to a thermal fluctuation must overcome an energy barrier in order to jump from site to a neighbouring site fi' . The threshold energy is given by
where AJD,ND,are the coordination numbers. Here $I is the bonding energy and
132
OPTICAL FILMS PRODUCED BY ION-BASEDTECHNIQUES
[III, 0 3
Q is the activation energy for surface diffusion. The thermal energy fluctuation is given by E =
K.rln(1 - R ) ? ,
(3.2)
where y = ,-a/.', and the particle jumps whenever E > AEB-8,.The results of the model are summarized in Fig. 3.4, which shows simulations for vapor incidence angles of 30 and 60 at different temperatures. At low temperatures atomic shadowing leads to a porous columnar microstructure. As the temperature increases, atom mobility is enhanced and the film packing density increases to a maximum. Figure 3.5 shows the dependence of the film packing density on T for vapor flux angles of 30 and 60 . O
O
O
O
3.2. ION BOMBARDMENT EFFECTS ON FILM STRUCTURE
One of the earliest demonstrations of film modification by ion bombardment was reported by MATTOXand KOMINIAK [1972]. Thick films of Ta were deposited using a planar dc diode system. When the substrate was biased to attract positive ions it was found that columnar growth could be interrupted. The film density was increased from 14 g cm- to 16.3 g cm-3, close to the bulk value of 16.6 g cm- for tantalum. The crystallite size also decreased with increasing ion bombardment. A similar result was also found for Be by BUNSHAHand JUNTZ [ 19721, who biased the substrate negatively to attract
z Y
v -
0.4t
a=30°
A
~ 6 0 "
TEMPERATWIE IKI
Fig. 3.5. The calculated dependence of film packing density on substrate temperature for vapor flux angles of incidence of 30" and 60", (MULLER[1983].)
111, § 31
133
INFLUENCE OF ION BOMBARDMENT ON FILM PROPERTIES
positive ions from an electron-beam evaporation source. A marked refinement of grain structure was observed and attributed to localized temperature spikes at the vapor-solid interface due to ion bombardment. Mono-energetic ion sources were first used in thin-film deposition by DOBREVand MARINOV[1973], MARINOVand DOBREV[1977], and MARINOV [ 19771,who investigated the effects of 1-10 keV argon-ion bombardment on the growth of silver, gold, cadmium, and cobalt films. Changes were observed in crystallographic structure and some preferential orientation was detected. The ion bombardment was found to enhance the surface mobility of adatoms and crystallites and to accelerate nucleation. Similar studies were also made by BABAEV, Ju, BYKOVand GUSEVA [ 19761 on the condensation of Zn on Cu,O, and Sb condensation on NaCl under the influence of ion bombardment (Ar' and Ne' ; 100 eV to 3 keV, O-lO-' A rn-,). It was concluded that in addition to increased adatom mobility and increased nucleation rate, ion bombardment also causes the development of nucleus orientation which is enhanced by substrate orientation. Ion-beam-induced texture formation during and after thin-film deposition has been reviewed by DOBREV[ 19821.The main conclusion was that under the action of ion beams at normal incidence, a fibrous ( 1 1 0) texture is formed in face centred cubic metal films deposited on amorphous substrates and the texture remains stable at high temperatures. The anisotropy of incident ion 1.o
I
I
NO IONS
a
400
500 WAVELENGTH (nm)
I
600
700
I
0,IONS
b
400
500
600
TOO
WAVELENGTH (nm)
Fig. 3.6. Transmittance over the visible spectrum of a ZrO, film deposited: (a) without ion assistance, and (b) with oxygen-ion assistance. Solid iine are data in vacuum and dashed line are and SAINTY[1984].) data in air. (MARTIN,NE'ITERFIELD
I34
OPTICAL FILMS PROOUCED BY ION-BASED TECHNIQUES
“11, § 3
penetration and the propagation of focussed impacts along the most closely packed ( 1 1 0 ) crystallographic directions were proposed as the most likely reason for the preferred ( 1 1 0 ) orientation. The effect of ion bombardment on the microstructure of growing films can be inferred from comparisons of spectral transmittance curves of layers immediately after deposition and on venting to a humid atmosphere, as shown in Fig. 3.6. The effect of water absorption into the voids of columnar films prepared without ion assistance can be seen by the shift of the curve to longer wavelengths. Densely packed ion-assisted films which do not absorb water, were first reported by MARTIN,MACLEOD,NETTERFIELD, PACEY and SAINTY [ 19831. The substantial modification by ion bombardment of the film microstructure can be seen by comparing the electron micrographs in Fig. 3.2 and Fig. 3.7.
3.3. ADHESION AND STRESS
A comparative study of gold films prepared by various deposition methods by CHOPRA[ 19691 revealed that films deposited by sputtering were more
Fig. 3.7. The microstructure of a film produced under the same conditions as in Fig. 3.2 but with the addition of ion bombardment. (MUNZand HOFMANN[1983].)
111,
I 31
INFLUENCE OF ION BOMBARDMENT ON FILM PROPERTIES
I35
adherent than evaporated films and that adhesion was independent of the deposition rate. Adhesion of a film to a substrate is strongly dependent on the state of the substrate surface and sensitive to the presence of surface contamination or oxides. The adhesion is generally enhanced by increasing the number of nucleation sites. The average arrival energy at the substrate is higher for sputtered atoms than for evaporated atoms and this is usually the reason for enhanced adhesion. Higher arrival energies can lead to the sputtering of loosely bonded substrate contaminants prior to film coverage and also of weakly bonded film atoms during growth. Several studies have been made of the influence of ion irradiation on film adherence. FRANKS,STUARTand WITHERS[ 19791 found that the adhesion of gold to various substrates could be improved by depositing the film and simultaneously irradiating with 5 keV argon ions. Slight improvements in the adhesion of gold to silicon and to glass were observed by SALEMand SEQUEDA [ 19811 when the substrates were pre-irradiated with oxygen ions. However, substantial improvements in gold adhesion to copper and glass were reported by HERRMANN and MCNEIL[1982] for oxygen and argon precleaned and ion-assisted films. MARTIN,SAINTY and NETTERFIELD [ 19841found that the scratch-test loadings for oxygen-assisted films could be increased from around 2 to 2000 g before film failure was evident for gold on silicon and glass. Under these loadings the substrate was damaged before the film was removed. Ion bombardment also influences the stress of films. HIRSCHand VARGA [ 19781 found that both the adhesion and stress of germanium films were influenced during argon-ion assisted deposition. Adhesion was increased and stress reduced once a critical ion density was reached. The mechanism thought to be responsible for the effect was the thermal-spike process discussed in 5 2.6. CUOMO, HARPER,GUARNIER, YEE, ATTANASIO,ANGILELLO,WU and [ 19821 also observed variations in stress in thin films deposited HAMMOND under ion bombardment. It was found that the stress in Nb films could be changed from tensile to compressive when the deposition temperature was raised to 400 " C and a sufficiently high argon ion flux was directed at the growing film.
3.4. COMPOUND FORMATION
DUDONIS and PRANEVICIOUS [ 19761 showed the influence of oxygen-ion bombardment on the properties of vacuum-evaporated thin films. Aluminium and SiO were evaporated by electron-beam hezting at a constant rate of 0.5
I36
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[I14 § 3
to 2 nm s - I in a vacuum of 10 - Pa. The growing film was bombarded with 5 keV oxygen ions, In the case of SiO, above an ion dose of 10’’ cm - 3 , the film composition was found to be SiO,. The bombardment of A1 with oxygen ions indicated a dielectric oxide formation at a dose of lo2’ cm - ’. Later experiments by PRANEVICIOUS [ 1979) demonstrated that the A1203phase was formed at oxygen ion doses greater than 6 x cm- 3 , but the higher doses resulted in bubble formation in the film. Similar experiments with N: resulted in homogenous AIN film growth. Irradiation of a growing SiO film varied the refractive index depending upon the concentration of ions. Table 4 shows the dependence of the refractive index n on the concentration of implanted oxygen. The highest dose results in a film with the index of SO,.
TABLE 4 Dependence ofthe refractive index of SiO films on the concentration of implanted oxygen. (PRANEVICIOUS [ 19791.) ~
Concentration of implanted oxygen (cm-’)
Refractive index of SiO film
0
1.85 1.66 1.55 1.48 1.46
2x
1019
2 x 1022 6 x lo2* I x 1023
The bombardment of a continuously deposited film by reactive or inert-gas ions in a reactive-gas atmosphere has the effect of stimulating a surface chemical reaction. GRIGOROV, MARTEVand TZATSOV[ 19791 observed that if a titanium film was bombarded by 1 keV argon ions in a nitrogen atmosphere the capture coefficient and sorption ratio could be increased seven times. Using this technique TIN., films were produced with x greater than 1.15. The dependence of the sticking coefficient and the sorption ratio on ion mass and energy was found to correlate with the number of vacancies and interstitials (Frenkel pairs) generated by the ion-surface interaction. The presence of ions has recently been shown to play an important role in the synthesis of materials such as oxynitrides (HEITMANN[ 197lb]), Si,N, (WEISSMANTEL [ 1976]), and AIN (HARPER,C u o ~ and o HENTZELL[ 19831).
111, I41
ION-BASEDMETHODS OF THIN FILM DEPOSITION
137
3.5. SUMMARY
The effects of ion bombardment on growing films have been summarized by [ 19821: TAKAGI (a) Enhancement of the surface mobility of adatoms. (b) Stimulation or acceleration of the nucleation, the growth of the nuclei, and the coalescence at the initial stage of film formation. (c) Creation of activated sites that stimulate the nucleation process. (d) Development of nucleus orientation. (e) Recrystallization of the deposited film. (f) Increase in the bonding energy between the deposited film and substrate. ( g ) Decrease in film stress. (h) Stimulation of sorption by ion-induced radiation damage leading to an enhanced chemical reaction for reactive species.
0 4. Ion-Based Methods of Thin-Film Deposition As we have seen in 3 3, the bombardment of a growing film with energetic particles can substantially modify the properties of the film. The degree of modification depends upon the kinetic energies and flux of the particles involved which are largely determined by the film deposition technique employed. The typical energy ranges of ions for ion-based methods is shown in Fig. 4.1. The following is a summary of the most commonly used techniques.
4.1. VACUUM EVAPORATION
In conventional evaporation the source material is heated until it evaporates and the vapor condenses on the substrate mounted directly above the source, Ion Plating and
Conventional Vacuum Evaporation
.- - - - - - - - - - - - - I
I
Sputter Ion Beam Deposition Deposition
-
.
- - - - - - - - - - - - - - - !- .
109
104
1 101 102 KINETIC ENERGY(eV1
1 0 3
Fig. 4.1. Energy range typical of the preparation methods for film formation. (TAKAGI [ 19821.)
138
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, 5 4
as shown in Fig. 4.2A. The process takes place in a vacuum of about Pa and the substrate may be heated. Electron-beam evaporation is frequently employed for greater control over deposition rates and source temperatures. The energy of the depositing atoms and molecules is about 10 - to 1 eV and a very small number may be ionized through thermal dissociation or interaction with the electron beam. Even with an ionized fraction of only 0.01 percent, CHOPRA [ 19641observed that the application of an electric field of 100 V cm - I was sufficient to influence the nucleation rate. -
w
i
p 0P ’
RADIANT HE4TER
SUBSTRATE
\
CRUCl0LE
SUBSTRATE
c
C Fig. 4.2. Experimental arrangements for evaporation-based deposition: (A) electron-beam evaporation, (B)activated reactive evaporation ARE and enhanced ARE, (C) ion plating, and ( D ) ion-assisted deposition IAD.
111, § 41
ION-BASED METHODS OF THIN FILM DEPOSITION
139
4.1.1. Reactive evaporation In the case of oxide-film deposition, the vacuum system is usually backfilled with oxygen to a pressure of about 2.5 x lo-, Pa. Under these conditions the process is termed reactive evaporation. Film properties are then influenced not only by evaporation rate, crucible starting material, and substrate temperature but also by oxygen pressure. Reactive evaporation is employed to form a compound by reaction between metal vapor and the reacting gas (e.g. Ti and 0, to form TiO,). In such a process three factors influence compound formation: (a) an adequate supply of reactant, (b) collisions between reactant species, and (c) reaction between colliding reactant species (BUNSHAHand RAGHURAM [ 19721). The probability of collision between reactant species increases with increasing partial pressure. The probability of reaction upon collision can be enhanced by activating one of the species: for example, ionization of the oxygen by an electrical discharge during Ti deposition. The process is then called Activated Reactive Evaporation (ARE) and is illustrated in Fig. 4.2B. In ARE a high flux of metal atoms is provided by an electron-beam evaporation source. Gas is introduced into the chamber to a pressure of about 10 ' Pa. A probe is mounted above the source and biased positively 100 to 200 V. The probe attracts electrons from the source and initiates a plasma discharge which activates evaporating atoms and background gas. The compound formed in the vapor phase is then deposited on the substrate. The substrate may be heated by a radiant heater to a controlled deposition temperature. The process has been successfully used to synthesize oxides, nitrides, and carbides at deposition rates of around 4 pm min - I . At lower deposition rates, electron-beam power must be reduced and the discharge sustained by an auxiliary electrode above the source. The deposition rate may then be reduced to 0.03 pm min- I , if required. BUNSHAH and RAGHURAM [ 19721 describe this modification as enhanced ARE and it is shown in Fig. 4.2B.
4.2. ION PLATING
The ion plating process' was developed by MATTOX[ 19641 and combines evaporation with a glow discharge, as shown in Fig. 4.2C. Ions are produced by thermally evaporating material in the region of a 2000-5OOOV inert gas discharge. A fraction of the ionized atoms are then accelerated across the dark
140
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, § 4
space to the cathode which is usually the substrate. Each ion experiences many collisions before reaching the substrate and loses its energy in charge-transfer collisions. Estimates by TEER[ 19771 show that the cathode is bombarded by 20 energetic neutrals for each ion leaving the edge of the dark space. The average energy of the ions arriving at the cathode is 30 eV and the average energy of the neutral particles is 135 eV. The ion flux is sufficient to sputter clean the substrate and influence film formation. One other advantage over conventional evaporation is the ability of an ion-plating system to coat all sides of the substrate. The gas pressure is sufficiently high to scatter the depositing atoms and thus increase throwing power. Ion plating is routinely used to deposit metal coatings on metal substrates and also to synthesize compounds such as carbides and nitrides by replacing the inert gas with a reactive species.
4.3. ION-ASSISTED DEPOSITION (IAD)
A significant advance in the technique of reactive evaporation has been made by the introduction of directed ion beams (Fig. 4.2D). It was originally propos--CAP
-'ION
OUTLET PLATE
DISCHARGE 'VESSEL
-
HOLLOW CATHODE
-HV-
02-
SUPPLY
SUPPLY
Fig. 4.3. Simple discharge source. (EBERT119821.)
111, § 41
141
ION-BASED METHODS OF THIN FILM DEPOSITION
ed by AUWARTER[1960] that the reactivity of the residual gas could be increased by ionization and the first practical demonstration of the effectiveness of the idea was made by HEITMANN [ 1971al. Material was evaporated from a crucible and the growing film subjected to bombardment from a discharge tube. Films of SiO,, SiO,N,,, and TiO, with low absorption and a high refractive index were successfully deposited by this technique. A simpler discharge source was constructed by EBERT[ 19821 consisting of a single arm of the HEITMANN source but operating at both positive and negative voltages (Fig. 4.3). The most successful results were obtained using negative ions and electrons. The most recent development in ion-assisted deposition has been to use a KAUFMAN type ion gun (KAUFMAN [ 1978]), as shown in Fig. 4.4. This source is a high-intensity ion gun which can operate over the range 30 to 1500 eV with inert and reactive gases. MCNEIL,BARRON,WILSONand HERRMANN [ 19841 have analyzed the flux from the KAUFMANsource operated with oxygen gas using mass- and energy analysis and found that the 0; : 0 ratio is approximately 3, and the energy spread is on the order of 10 eV. The energy and intensity of the beam can be controlled, and output current densities of 200 A m - 2 can be achieved. The high current densities enable the substrate surfaces to be sputter cleaned prior to film deposition, and films to be removed +
-,-,Permsable Malarial
A 1 2 3 1 5
0
cm Fig. 4.4. High-intensity ion source. (KAUFMAN, READER and ISAACSON [ 19761.) (Copyright American Institute of Aeronautics and Astronautics).
142
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III,
84
if necessary (HERRMANN and MCNEIL[ 19811). The effects of surface charging in the case of insulating substrates can be minimized by electron-emitting filament mounted in the ion beam or close to the substrate. The area of the substrate subjected to ion bombardment is dependent upon the ion gun aperture, beam divergence, and source-to-substrate distance. Typical beam divergence is 10". Typically the operating pressure of the chamber is approximately 2 x 10- Pa due to gas leakage through ion-gun grids, but the mean free path of 0.5 m is sufficiently great that the ions lose little energy in collisions with residual gas atoms. The technique is highly directional and therefore limited to line-of-sight geometries, but affords considerable control over the properties of the growing films and will undoubtedly see greater use particularly in materials synthesis. 4.4. SPUTTER DEPOSITION
Sputtering is a versatile technique for depositing thin films of virtually any element, compound, or alloy with a high degree of control. The phenomenon of sputtering, described briefly in 0 2, has several advantages in film deposition. There is no direct heating of the material as in evaporative based methods and therefore no reaction between the source and crucible takes place. The deposition rate is linearly dependent upon the bombarding flux, whereas in evaporation there is an exponential dependance of rate on source temperature. The geometry of the system is also not limited to any particular direction. The bombardment may take place in high vacuum (10 - 5-10 - Pa) or in a plasma (10- '-10 Pa) and the distinction between an ion-beam method and a plasma-based process is made according to the pressure in the vacuum vessel. 4.4.1. Planar diode sputtering In this configuration a low pressure negative glow discharge is maintained between the cathode target material and the substrate anode as shown in Fig. 4.5.Secondary electrons are created at the target surface by ion bombardment, accelerated in the cathode dark space, and enter the negative glow as primary electrons with sufficient energy to ionize the gas atoms and sustain the discharge. The sputtering sources are made from high purity materials usually bonded to a water cooled electrode. The applied power is generally dc for metal targets and rf (13.56 MHz) for non-conducting targets. The discharge voltage is typically 100-4000 V. The most commonly used sputtering gas is Ar but
143
ION-BASED METHODS OF THIN FILM DEPOSITION
I l l . $41
CATHODEDARK
SPUTTERED ATOMS
ION
' ELECTRON-ELECTRON EMISSION
ANODE
6 Fig. 4.5 Schematic of diode sputtering.
considerable work is in progress to reactively sputter deposit compound films from elemental targets using reactive gases such as O,, N,, or H, in combination with Ar. The relatively high operating pressures necessitate throttling of the vacuum pump by means of a variable valve between the chamber and the pumping station. An unbiased substrate will be subjected to positive ion bombardment with particle energies in the range 5-30 eV (THORNTON [ 1983]), and further bombardment is often induced by negatively biasing to 500 V to promote film modification during growth. 4.4.2. Magnetron sputtering
The deposition rates obtainable in a planar-diode system are limited by the sputtering-gas ionization efficiency.The application of a magnetic field parallel to the cathode surface can restrict the primary electrons to paths close to the cathode, as shown in Fig. 4.6. This trapping of electrons near the target surface results in an increased ionization efficiency of the working gas and hence a greater sputtering rate of the material to be deposited. A secondary result of this arrangement is that the sputtered flux itself may be ionized during passage through the high-density plasma trapped above the target, and consequently will further influence the film growth at the substrate. This type of sputtering
144
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[Ill, 8 4
/
ELECTRON PATH
SPUTTERING REGION
Fig. 4.6. Schematic of magnetron sputtering.
arrangement is referred to as a magnetron source and has been configured in several different ways (THORNTON and PENFOLD [1978], DANILINand [ 19781. SIRCHIN
4.4.3. Ion-beam sputtering
In ion-beam sputtering, energetic particles generated by an external source are used to bombard a target as shown in Fig. 4.7. Material is sputtered from the target and condenses on the substrate which may be heated. The main advantages are that (a) the growing film is not exposed to high ion fluxes or large fluxes of electrons, (b) the deposition may be carried out under highvacuum conditions, (c) the ion beam may be directed through a certain aperture or focussed to a defined shape, and (d) the depositing atoms have mean energies of 10-20 eV, since they do not experience collision with the residual gas. Film growth rates are determined by the sputtering yield and maximum ion flux obtainable at a given energy. The development of high-power ion sources gun have led to deposition rates in excess of 1 pm h such as the KAUFMAN depending upon the ion-target combination and source-substrate distances. Film properties can be substantially modified as in ion-assisted evaporation
'
ION-BASED METHODS OF THIN FILM DEPOSITION
< SUBSTRATE
A<
WATER
SINGLE BEAM SUBSTRATE
>
=A?
:
ION BEAM 2
\ I\
(INERT OR REACTIVE)
AGAS
ION BEAM 1
ION SOURCE
(INERT OR REACTIVE)
b< WATER DUAL BEAM Fig. 4.7 Schematic of single and dual ion-beam sputtering
145
146
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
"11, § 4
by bombarding the growing film with a flux of particles from a second source in a dual ion beam deposition configuration, as shown in Fig. 4.7. This method has been extensively used by WEISSMANTEL [ 19761 to both modify the structure of growing films and to synthesize compounds. Diamond-like carbon or i-carbon films have been prepared by sputtering carbon and irradiating the growing film with inert gas ions. Amorphous hydrogenated silicon (a-Si : H) and Si,N, have been synthesized by sputtering pure Si and irradiating with hydrogen or nitrogen ions.
4.5. ION-BEAM DEPOSITION
Ion-beam deposition refers to the process in which the depositing atoms are fully ionized, accelerated, and deposited on a substrate as shown in Fig. 4.8. The deposition energy can be adjusted by substrate bias, and substantial modifications of film properties have been achieved particularly in the case of diamond-like carbon films (AISENBERG and CHABOT[ 19711). Low porosity, hard, highly-adhesive films with low stress have been successfully deposited, but the main limitation is the relatively low deposition rates achievable. 4.5.1. Ionized cluster beam deposition (ICB)
This technique was developed by TAKAGI, YAMADAand TAKARA [ 19821 and is based upon film growth using macro-aggregate (cluster) deposition and is shown in Fig. 4.9. Vaporized metal clusters are formed by adiabatic exGAS
1 SUBSTRATE
SPUTTER OR GAS DISCHARGE ION SOURCE
Fig. 4.8 Schematic of ion-beam deposition process.
111, Q 41
ION-BASED METHODS OF THIN FILM DEPOSITION
147
1SUBSTRATE
ACCELERATING
SOURCE MATERIAL Fig. 4.9. Schematic of ionized cluster beam (ICB) deposition.
pansion through a nozzle leading to the deposition chamber. The clusters (typically 500-2000 atoms) are singly ionized by electron bombardment in an electrode assembly mounted over the crucible and subsequently accelerated towards the substrate. If should be mentioned, however, that the growing film is also subjected to a bombardment by neutral clusters, atoms, and ions. Furthermore the size of the ionized clusters, particularly in the case of Ag, has been shown by KUIPER,THOMAS and SCHOUTEN [ 19811 to be only about 25 atoms. Upon impact the ionized cluster is broken up and each atom has an average energy E = eV,/N, where e is the electronic charge. V, is the acceleration voltage and N the number of atoms in the cluster. The average energy of the depositing atoms can then be varied by changing the value of V,. Typical values of E are 0.1-10 eV. Reactive ICB deposition is possible by introducing oxygen, nitrogen, or other gases through a nozzle close to the metal-vapour source. The gas may be ionized with the metal vapour so that reaction occurs and compound films may be grown. The main advantages of the ICB method are: (a) surface cleaning of the substrate by cluster bombardment resulting in high adhesion, (b) surface heating due to the high-energy deposition leading to crystalline films, and (c) modification of film growth parameters as with ionassisted techniques. Among the types of film deposited by ICB are BeO, PbO, ZnO, a-Si : H, MgF,, and metal films. The maximum deposition rate is 10 nm min ' (Si) with a uniformity of k 10% over 0.1 m2 area. ~
148
OFTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
8 5.
[III, B 5
Optical Thin Films
Ion-based thin-film deposition techniques are used to deposit materials for a wide range of applications. We restrict ourselves in this review to those materials which find their greatest application in optical devices. Dielectric oxide films are an important class of materials since they form the basis of most thin-film filters and, therefore, are discussed in some detail. Carbon films are presently being studied by many researchers and show great promise for many optical applications because of their very attractive properties. Ion-based techniques are essential for the preparation of diamond-like carbon. Some aspects of materials mixing and synthesis are covered, and also a few examples of the applications of optical thin films prepared by ion-based techniques are given.
5.1. OXIDES
5.1.1. Silicon dioxide
Silicon dioxide is the material most frequently used in combination with TiO, in multilayer interference stacks and is also an important dielectric in the electronic industry. Consequently this material has been prepared by various techniques extending from simple evaporation of silica to oxygen ion implantation into silicon. A number of problems are usually associated with silica deposition, i.e. porous films, variable index, and substrate temperature effects. When SiO, is deposited by evaporation as an alternating layer with TiO,, electron-microscopic observations reveal that the gross columnar structure always observed in TiO, is absent in the SiO, layer (GUENTHER [ 1984]).The low packing density of SiO, is influenced to a small degree by substrate temperature. Evaporation in ultra-high vacuum by electron-beam evaporation yields packing densities of 95 percent and 98.7 percent at substrate temperatures of 50 ’C and 250 ‘C, respectively (RITTER [ 19721). REALE[ 19781reports values of 88 percent and 98 percent at the same temperatures. HEITMANN [ 1971al has measured a value of 93 percent for room temperature low-energy IAD. We see, therefore, that unity packing density is not achieved even at substrate temperatures of 250 O C. ALLEN [ 1982, 19831 has investigated the effect of ion species and ion flux on the refractive index for IAD silica (SiO, starting material). Bombardment with Ar or 0, increases the refractive index relative to that of conventionally +
+
111, I 51
149
OPTICAL THIN FILMS
deposited silica (no ions). The extinction coefficient was too low to measure at 550 nm. The absorptance at 1.06pm was also very low (9 x increasing to 2.1 x lo-' at 325 "C substrate temperature for a 345 nm thick film. Oxide reduction due to preferential sputtering, which increased absorption in TiO,, was not observed for silica films. Sputter deposited films do not exhibit significantly different indices to conventionally evaporated films (PAWLEWICZ, HAYSand MARTIN[ 19801). Figure 5.1 summarizes the optical measurements reported for ion-based silica deposition. The values vary considerably for most wavelengths from the bulk silica curve although many authors routinely report 1.46 for n at 550 nm. 5.1.2. Aluminium oxide
1
.
7
1
,
,
0
x 1.6w
a
.
1
,
,
,
l
,
,
,
1
,, 1,
, , ,
0
0
z w
>
6 a
1.5
A.
:
EW
A
A
4
K
1.4-
1'3
012 ' 013 ' Ol4
' 0:s ' 016
0!7 ' 018
'
019
1:O
l!l
'
Fig. 5.1. Summary of the refractive index as a function of wavelength of SiO, prepared by ion-based techniques. Symbols: (IAD) 0 ALLEN[1982], @ ALLEN[1983], 0 EBERT[1982]; (reactive evaporation) 8 ALLEN[1983]; (sputtering) D SUZUKI and HOWSON[1983], A PAWLEWICZ, MARTIN,HAYSand MANN[1982], V COLEMAN [1974], 4 PERVEEV, CHEREZOVA and MIKHAILOV [1977], Solid line fused silica values.
150
[III, 5 5
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
oxide are listed in the ASTM powder diffraction file. Of the seven evaporated materials studied by REALE[ 19781, A1,0, had the highest packing density of 95 percent, showed the least variation with substrate temperature and had a constant refractive index of 1.60 up to a substrate temperature of 300 C. Both rf and dc sputtering techniques are used in A1,0, deposition. Adherent, amorphous films were deposited to a thickness of 8 pm at 40 nm s - by [ 19771 using rf planar magnetron methods. The film stoichiometry NOWICKI was dependent upon the power density and approached A1,0, at 4 W cm-’, giving a refractive index of 1.63 at 546.1 nm. The films contained argon up to a maximum of 4.8 percent. Al,O, targets were sputtered in pure argon. No details of film absorptance were given in this work, but DESHPANDEY and HOLLAND [ 19821 have measured some values for films prepared by dc and rf sputtering of an aluminium target. It was possible to deposit 60 nm thick films with absorptances of less than 1.0 percent by dc magnetron sputtering of aluminium with a 10 percent 0,-Ar mixture. In the case of rf magnetron sputtering the absorptance increased to 6 percent but could be reduced at any particular power and deposition rate by increasing the oxygen content of the discharge. The film index was estimated at 1.61-1.62. PAWLEWICZ, HAYSand MARTIN[ 19801 have reported a mid-range index of 1.67 for AI,O, prepared by rf diode sputtering of aluminium in argon-oxygen mixtures. The films were thought to be stoichiometric, the structure cubic, and the refractive index significantly higher than that reported by other workers. Ion-assisted deposition has also been used by EBERT[1982] and BINH, NETTERFIELD and MARTIN[ 19841. The latter measured a refractive index of 1.65 and an extinction coefficient of 1.8 x 10 - at 633 nm. The dispersion of the refractive indices reported for some recent studies of AI,O, is shown in table 5. O
’
5.1.3. Titanium dioxide
Titanium dioxide is a hard, chemically resistant material, transparent in the visible and near infrared and with a high refractive index. Consequently this material is widely used in thin-film technology and has been deposited by a wide range of techniques. Titanium dioxide occurs in three main crystalline forms, rutile, anatase, and brookite, although the latter has not been observed in vacuum deposited thin films. The phase diagram of the Ti-0 system of ROY and WHITE[ 19721, reproduced in Fig. 5.2, shows the numerous oxide phases [ 19771 there exists an possible. As discussed by GERAGHTY and DONAGHEY indeterminate number of phases between Ti0 and TiO,, of which Ti,O, and
111,
I 51
151
OPTICAL THIN FILMS
TABLE 5 Refractive indices of A1,0, films produced by ion-based techniques. Wavelength
Refractive index
Refsa
(w) 0.25 0.30 0.40 0.50 0.54 0.63 0.80 1.oo 1 .oo 1.06 5.00 a
1.75 1.70 I .70 I .70 I .63 1.65 1.68 1.64 1.67 1.67 1.74
References: [I] PAWLEWICZ, HAYSand MARTIN[1980]. [2] VARASI,MISIANOand LASAPONARA[I9831 [3] NOWICKI [1977]. and MARTIN[ 19851. [4] BINH,NETTERFIELD
Ti,O, are the distinct phases, and TiO, (anatase) is the stable oxide phase at a temperature of 400 C. The most desirable oxide phase from the point of view of optical properties is rutile. It is not surprising then that the substrate temperature and the degree of oxidation of the film are critical parameters for TiO, deposition and that the optical properties will vary from technique to technique. A comprehensive study of reactive evaporation methods by PULKER, PAESOLDand RITTER[ 19761 has highlighted the problems encountered in TiOz deposition by conventional techniques. The refractive index is dependent upon the substrate temperature, the oxygen partial pressure during deposition, and the deposition rate. Successive evaporation of different starting materials results in different refractive indices of the deposited films, particularly for TiO, Ti,O, and TiO, starting materials. Reproducible results were obtained only when Ti or Ti,O, starting materials were reactively evaporated. In the Ti case rutile films resulted, whereas for Ti,O, the oxide phase depended upon the substrate temperature used. The variations in index with successive evaporations of the other source materials is attributable to their changing composition in the crucible during deposition, which in turn results in a varying vapor stream composition. Vapor stream analysis of Ti and Ti,O, evaporants revealed the presence of only Ti and T i 0 particles, respectively. A major
152
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, 5 5
a
p"
800-
I
W I-
640
-
a+v
480-
-
Y
320c+a
160-
,.
a+b
(TizO,
,/ c+v I 0.6
A 0.80
1.0
1.20
1.4(
.o 32
Ti;O, T i e d 1 ,
Fig. 5.2. The low temperature condensed phase diagram for the Ti-0 system. (ROY and WHITE [1972].)
problem with reactive evaporation is the reduced packing density of the deposited material, a parameter which can only be influenced by deposition at elevated substrate temperatures in order to enhance surface mobility of depositing atoms and molecules. Higher packing densities are achieved by sputtering techniques where greater mobility of surface atoms is possible due to their higher deposition energies. Considerable control over the oxide phase and grain size has been achieved by PAWLEWICZ and BUSCH[ 19791and PAWLEWICZ, MARTIN,HAYSand MANN [ 19821 using rf sputtering. TiOz phase compositions were varied from 40 percent rutile, 60 percent anatase to purely rutile. Rutile is obtained over a wide range of temperatures at high oxygen pressures and the corresponding grain size varies from 10 to 60 nm. The influence of grain size on the refractive index is difficult to separate from packing density effects, but is thought to increase indices by approximately 5 percent (SCHILLER, BEISTER,SIEBER,SCHIRMER and HACKER[ 19811).
111, § 51
153
OPTICAL THIN FILMS
In general, sputtering is the preferred method for depositing TiO, with reproducible properties although good results have been obtained using reactive rf biased ion plating. Using this technique SUZUKIand HOWSON[ 19831 have obtained high quality TiO, films (n = 2.49 at 633 nm) on water cooled glass substrates. However, the refractive index is sensitive to the deposition rate and oxygen partial pressure, as shown in Fig. 5.3. SUZUKIand HOWSON have also prepared Ti0, using dc magnetron sputtering and rf enhanced dc magnetron sputtering. In the latter technique rf bias is applied to the substrate. The index-pressure and rate-pressure relationships for these techniques are shown in Fig. 5.4. An advantage of these techniques is that the source material can be titanium which can be sputtered at a relatively high rate. Ion beams have been used by TAKIGUCHI, OGAWAand TAKAHASHI [ 19831 to deposit titanium oxides directly by sputtering metal targets in oxygen but their most successful application has been in IAD. HEITMANN[ 1971al evaporated Ti,O, with oxygen ion assistance. At 550 nm, the refractive index was estimated to be between 2.2 and 2.3 and found to depend slightly on deposition rate. The absorption coefficient at 633 nm was estimated to have an upper limit of 40 cm- and at 10.5 pm it was 10’ cm- Single crystal rutile reaches a comparable absorptance only at 11.6 pm. The discrepancy was assumed to be a structural effect as the IAD films were all amorphous. The experiments of HEITMANN were repeated in greater detail by KUSTER
’,
1
OD
1
’.
1
1
1
1
1
1
1
1
1
1
1
2.5-
1
-
(Y P) (D
I 4
-
x
-
a
-
W
E
g 2.0F
0 4
a U. W
a
-
X
-
-
1.5 0
-
50
100
150
DEPOSITION RATE &mi”)
Fig. 5.3. The refractive index at 632.8 nm of TiO, films prepared by ion plating for oxygen pressures of: 0 3 x lo-, Pa, A 5 x lo-, Pa, 0 7 x IO-’Pa; x Conventionally evaporated in 3x Pa. (SUZUKIand HOWSON[1983].)
I54
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
a
1
I
1
I
l
l
1
1
2.5-
-120 0
-1ooz
x
E
W
\
P
5
-8OO: I-
W
a a
2
I- 2.2V
-=O
a
a
LL W
-40
a
2
0
$
: W
2.0-
-20
1.5 1
I
I
I
I
2
3
4
5
l
l
l
l
~
10
PRESSURE ( x 10-1) Pa)
b
12.5
Y
-
-100 -1000
2.4 -
w
0
=z 2.3-
-
i=
V
re
2.2-
E
2.1-
-20-
- 200
10
PRESSURE IXIO-zPal Fig. 5.4. Refractive index and deposition rate of TiO, film versus oxygen pressure prepared by ( a ) dc magnetron sputtering, (b) rf enhanced dc magnetron sputtering. (SUZUKIand HOWSON [ 19831.)
and EBERT[ 19801 and EBERT[ 19821 using a refined HEITMANNion source. Both T i 0 and Ti,O, materials were used, and the effects of neutral oxygen, positive and negative ions and excited molecules on the absorption and refractive index were investigated over the substrate temperature range 50 to 325 "C. Their data are summarized in Fig. 5.5. Films with the lowest ab-
I55
OPTICAL THIN FILMS
WAVELENGTH hlnml Fig. 5.5. The dispersion ofTiO, indices for the starting materials-TiO, --- Ti,O, at substrate temperatures of 50°C and 325 "C.(KUSTER and EBERT[1980].)
sorption were produced with negative oxygen ion and electron bombardment for TiO, deposition, although Ti,O, starting material produces more reproducible films. This is, however, contrary to the findings of PULKER,PAESOLD and RITTER[1976] who demonstrated that Ti,O, or even Ti metal yield more reproducible optical properties. Further demonstrations of the successful applications of IAD were made by ALLEN[ 19821, also using a HEITMANNtype source. ALLENused negative ions (and electrons) and T i 0 as the starting material. The absorption coefficients at 1.06 pm are reproduced in table 6. X-ray diffraction measurements revealed these films to be amorphous. TABLE 6 Measured absorptance at 1.06 pm of reactively deposited titania films as a function of source current. (ALLEN[1982].) Source current
150 250 350
Measured absorptance
(4
Absorption coefficient (cm- ' )
1.9 x l o - ' 1.8 x 10-3 9.0 x 1 0 - 3
1992 29 160
I56
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
100
a
0 0
X
Argon
1
k
f
0 U
60
'
U 8 0 : / z w
40 Argon/Oxygen,A
L
0
4
t; 2 0 -
zIX
w 0
lo---.
I
oxygen
, -.
~
--
c
* - - - **--.# *---
:
@' -
/-a
Argon/Oxygen,B 1
I
, .
WAVELENGTHhm)
Fig. 5.6. (a) Variation of the extinction coefficient of titania using argon, oxygen, and mixed ions. E(argon) = 400 eV, y = 0.17; E(argon/oxygen,A) = 400 eV, y = 0.17; E(argon/oxygen,B ) = 300 eV, y = 0.1zE(oxygen) = 300 eV, y = 0.12. y is the ion-to-molecule ratio at the substrate. (b) Comparison of the refractive indices of IAD and conventionally deposited titania films. E(oxygen) = 300 eV, y = 0.12. (ALLEN(19831.)
ALLEN[ 19831 has recently compared the influence of argon and oxygen-ion assisted deposition using a KAUFMANNion source. In this work 300 and 400 eV positive ions were used. Figure 5.6 shows the effect of the ion bombardment on the optical properties for different ion-to-TiO-molecule ratios at the substrate. The best results were found for oxygen bombardment where very low
I I L § 51
OPTICAL THIN FILMS
157
values of k were obtained. Absorptance figures at 1.06pm are shown in table 7. Titania films were found to have the lowest absorptance for 300 eV oxygen ions at an ion-to-molecule ratio of 0.12which yielded a refractive index of 2.49. The effect of bombarding at very low energies on the optical properties of TiO, films has been studied by MCNEIL,Barron, WILSONand HERRMANN [ 19841.The film absorptance is reduced for the lower ion energy case although no figures on index or absorptance were given. It was found that the ion bombarded films had considerably less hydrogen and hence less water vapor penetration than films deposited without ions. All IAD studies of titania have shown that the deposited films are amorphous. These observations are consistent with earlier studies of bombardment-induced structural changes in solids. A TiO, sample subjected to ion bombardment (40keV, 10'3-1016 ions cm-') will revert to the lower oxide and KELLY[ 19751). Since film IAD growth is essentially a Ti,03 (NAGUIB layer by layer process with each layer ion bombarded, the situation is equivalent to post-bombardment of bulk oxides. The dispersion of the refractive index of TiO, obtained by the various techniques is summarized in Fig. 5.7.It is interesting to note that there is a considerable spread of data points, and very few results approach the optical properties of bulk rutile. IAD data points show a good reproducibility between various groups but in general fall short of the rutile values at most wavelengths. All the IAD data were taken from samples deposited on unheated substrates which generally leads to non-uniform coatings in conventional evaporation (due to a gradual heating of the film from radiation from the crucible during deposition). The IAD data are, however, in very good agreement with those of CHEREPONOVA and TITOVA[1979] for TiO, films deposited at 300°C, indicating that ion-assistance is equivalent to enhancing film atom mobility by substrate heating. TABLEI Variation in absorptance at 1.06pm with various source gases. (ALLEN[1983].) Source gas
Absorptance ( a x lo6)
Argon Argon and oxygen Argon and oxygen Oxygen
319000 65 000 41 300 514
I58
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, 8 5
3.0 2.9
-
2.8
-
x
xz 2.7$
2.6-
I-
0
2
2.5-
Y W
a
2.42.3-
WAVELENGTH (pm)
Fig. 5.7. Summary of the dispersion of refractive index ofTiO, produced by ion-based techniques. Symbols: (IAD) 0 ALLEN [1982], 0 ALLEN [1983], C, KUSTER and EBERT [1980], 0 HEITMANN[1971a]; (sputtering) A PAWLEWICZ,MARTIN,HAYS and MANN [1982], V SCHILLER, BEISTER,SIEBER,SCHIRMERand HACKER[1981], D SUZUKIand HOWSON [I9831 pC MATSKEVICHand CHERNYAVSKII[1979], 4 COLEMAN [1974], b PERVEEV, and IL’IN [1975], fl MOTOVILOV and RUDINA[1974], A VARASI, MIKHAILOV, MURANOVA and LASAPONARA [ 19831, 8 MISIANO,VARASI,MANCINI,SARTORI and LASAPONARA MISIANO [1983]; (reactive evaporation) 0 PULKER,PAESOLDand RIITER [1976], 0 RITTER[1966], MATSKEVICH and BAZHINOV [1977], E;1 CHEREPANOVA a n d T i r o v a [1979]. Solid line is the bulk rutile value.
5.1.4. Zirconium dioxide Zirconium dioxide also has a high refractive index and is a hard material, but is subject to considerable variation of refractive index with deposition condiFADEEVA and PERVEEV [ 19771, HIRAGA,SUGAWARA, tions (MURANOVA, OGURA and AMANO[ 19741). The packing density of ZrO, is considerably less than that of other dielectrics when deposited by evaporation. REALE[ 19781 gives a figure of 0.7 at 50 O C and 0.95 at 300 C substrate temperature with a corresponding rise in index from 1.80 and 2.15. PERVEEV, CHEREZOVA and MIKHAILOV [ 19771 report a 12 percent porosity for evaporated ZrO,. With such a poor packing density it is not surprising that wide variations in optical properties are reported. The growth of ZrOz can be substantially influenced by ion bombardment. GREENE, KLINGER, WELSHand SZOFRAN [ 19771 showed that rfbias-sputtered Y,O,-doped ZrO, films are modified by increasing the bias from - 40 to
111,
o 51
I59
OPTICAL THIN FILMS
The - 40 V biased specimens exhibited columnar microstructure, but when the bias was increased to - 60 V, the columnar structure was removed. The oxygen/zirconium ratio was also increased by increasing the bias. PAWLEWICZ and HAYS[ 19821 examined the microstructure of various ZrO, films prepared by rf sputtering also as a function of bias. The predominant crystal structure of ZrO, films prepared by sputtering Zr in an Ar-0, mixture, was monoclinic. At target power densities higher than 10 W ern-,, the substrate temperature increased to 750 K and the crystal phase changed from monoclinic to tetragonal. The cubic phase was observed for CaO- and Y,O,doped ZrO, films prepared with no bias. Bias sputtering of Zr0,-CaO results in preferential sputtering of Ca and a change back to a monoclinic phase. A detailed study of the modification of the optical and structural properties of dielectric ZrO, produced by ion-assisted deposition has been made by MARTIN,NETTERFIELD, SAINTY,CLARK,LANFORDand SIE [ 19831 and MARTIN, NETTERFIELD and SAINTY[ 19841. The effect of ion irradiation on the optical properties is clearly demonstrated by the observation of vacuum-air effects and water absorption, as shown in Fig. 3.5. Stable films which did not adsorb water were produced for a molecule-to-ion-arrival-ratio of 3.5 for 1200 eV 0, ions. The result is ascribed to the avoidance of microvoids which occur in normally evaporated films by the ion-assisted process. Ion-assisted deposition increases the packing density and prevents water penetration. Evidence for this mechanism is provided by profiling the films for water vapor. This is most readily achieved by detecting hydrogen (from water vapor) as a function of film thickness by nuclear reaction analysis. Figure 5.8 shows - 80 V.
20
40
60
80
100
120
140
DEPTH ( n m )
Fig. 5.8. Hydrogen depth profile in ZrO, films produced by: -0- evaporation.- x -argon IAD. (MARTIN,N E ~ E R F I E L and D SAINTY[1984].)
160
[III, § 5
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
hydrogen profiles for a film produced without ion bombardment and also by argon ion-assisted deposition. A substantial reduction in hydrogen, and hence water vapor content, is seen for the ion-assisted film. The crystal structure of ZrO, is considerably modified by ion-assisted deposition. Films deposited at room temperature in the absence of ions show no diffraction lines. Ion irradiation produces a face centred cubic structure when the ion current density is greater than 5 x 10 - A m - (ion : molecule arrival rate of 1 : 75, Fig. 5.9). Heated films exhibit only the monoclinic phase
’
JJ
’
OXYGEN (HOT)
DIFFRACTION ANGLE 28
Fig. 5.9. X-ray diffraction spectra for ZrO, Nms produced by argon and oxygen IAD at room temperature and heated substrates. (MARTIN,NEITERFIELDand SAINTY[ 19841.)
111.8 51
OPTICAL THIN FILMS
161
in the absence of ions but both a cubic and a monoclinic phase in the presence of ions. The crystallization effects closely follow the observations of NAGUIB and KELLY[ 19751 for post-irradiation of amorphous ZrO, by 2-35 keV Kr . They observed that ion impact promoted a crystallization of the cubic phase and proposed a thermal-spike mechanism (8 2.6). The critical ion current density required to promote the cubic phase can be estimated from the number N , of atoms participating in an activated re-arrangement process during the lifetime of a thermal spike. The equation used is that of SEITZand KOEHLER [ 19561. +
Nt = 0.0 16p(E,,/E~)~’~,
(5.1)
where pis a material constant in the range 1-10, E,, is the energy of the incident ion, and EQ is the activation energy for the transition into the cubic phase. Using NAGUIBand KELLY’S value of 2.34 eV for EQ and a fmed deposition rate of 0.8 nm s - I , a critical Ar current density of about 2 x 10 - A m - is then required at 600 eV incident ion energy to activate all the condensing atoms. It is not possible at this stage to attribute the main crystallizationmechanisms in IAD processes to thermal spikes alone since many ion-target combinations have yet to be studied. The amorphous-cubic phase transition temperature for ZrO, is 400 “C (KLECHKOVSKAYA, KHITSOVA,SAGITOVand SEMILETOV [ 1980]), which is far greater than the substrate is likely to achieve from radiative heating from the crucible during deposition. The changes in crystal structure and film packing density have a strong influence on the optical properties of ZrO,. Figure 5.10 shows the refractive index at 550 nm as a function of argon and oxygen ion current density. For argon, a vacuum-air variation effect is observed up to an index of 2.138 and then the refractive index decreases. This index reduction at high argon current densities is due to preferential sputtering of oxygen (5 2.1.1) and argon incorporation into the layer. Films produced by oxygen bombardment have a higher index of 2.19. The highest indices for both ion species are observed only when the substrate is heated and the film is bombarded with oxygen ions. An interesting effect was also found on the optical properties when the ion beam was switched either off or on during a deposition. Figure 5.1 1 shows the transmittance at 550 nm as a function of film thickness. In case (a) the film is deposited with 0; ions, and an index of 2.18 was measured. The deposition continued and the ion flux was switched off, which resulted in a reduced index of 1.9. The reverse case - no ions then ions on - shown in Fig. 5.1 l b almost reproduces the results in reverse. Ion-assisted deposition is shown to produce an immediate effect on the packing density of ZrO, and is sufficient to disrupt
I62
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
a
~
300 C
2.2
1
0
. 20
40
8 60
80
8 100
Ar CURRENT DENSITY (pAcm-2)
Fig. 5.10 The refractive index of ZrO, at 550 nm measured in vacuum and air as a function of and ion current density: (a) Argon ions, (b, facing page) oxygen ions. (MARTIN,NETTERFIELD SAINTY[1984].)
columnar growth even on a low packing density layer under continuous deposition conditions. The available data on the refractive indices of ZrO, are summarized in table 8. 5.1.5. Cerium dioxide Cerium dioxide has been prepared by sputtering (MISIANO and SIMONETTI [ 19771 and also by ion-assisted deposition by NETTERFIELD, SAINTY, MARTIN and SIE[ 19851. As with other oxides prepared by IAD, vacuum-ab changes
163
OPTICAL THIN FILMS
I
I
I
I A
b
300-c
2.:
2.
x
W
cl
I
W
: a 2_
2.1
f:W K
L I
1.
vacuum
1. 50
100
150
200
0 , ION CURRENT DENSITY (pA/cm*)
in the refractive index are observed until the ion current density reaches a certain value. Figure 5.12 shows the refractive index for 0; bombardment. A gradual decay from the maximum value of 2.4 is observed at high current densities which is thought to be a result of trapping of oxygen in the film and/or preferential sputtering effects. The mass difference between Ce and 0 is sufficiently great that preferential sputtering is to be anticipated. The extinction coefficient is shown in Fig. 5.13, where it is seen that ion irradiation at any energy increases the extinction coefficient relative to an unbombarded film. The packing density of films deposited without ions is only about 0.55. As with ZrO, the packing density can be improved with ions to unity. No crystallographic phase changes are observed although structural studies do indicate some preferred orientation effects, particularly for films heated to 300 "C. Table 9 summarizes the refractive indices of CeO, prepared by ion-based methods.
I64
[IIL § 5
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
1.0
I
1
0, IONS
0
I
1.0'
I
100
200
i
I
i
NO IONS
NO IONS
300
i
0, IONS
100
0
THICKNESS (nm)
i
200
300
THICKNESS (nm)
Fig. 5.1 1. Transmittance of ZrO, films at 550 nm as a function of thickness for depositions: (a) and SAINTY with ion-beam on, ion-beam off, and (b) beam-off, beam-on. (MARTIN,NETTERFIELD [ I9841 .)
TABLE 8 Refractive indices of ZrO, films produced by ion-based techniques. ~~~
Wavelength ( F )
Refractive index
Refs."
0.25 0.40 0.55
2.47 2.2 2.15 2.19 2.15 2.15 2.15 2.20 2.10
[ll
0.80 I .oo 1.06 1.11 a
PI PI PI PI PI [11 [31 (41
References: [I] PAWLEWICZ, HAYSand MARTIN[1980]. and SAINTY[1984]. [2] MARTIN,NETTERFIELD [1974]. [3] COLEMAN [4] GREENE, KLINGER,WELSHand SZOFRAN[1977].
165
OPTICAL THIN FILMS
0, ION CURRENT DENSITY (PA Cm+)
Fig. 5.12 The refractive index of CeO, at 550 nm as a function of oxygen-ion current density. (NETTERFIELD, SAINTY,MARTINand SIE [ 19851).
TABLE9 Refractive indices of CeO, films produced by ion-based techniques. Wavelength
Refractive index
Refs."
(w) ~
0.55 0.56 2.58 0.76 1.06 5.00 10.00 a
2.4 2.5 2.49 2.45 2.45 2.25 2.10
References: [ 11 NE-ITERFIELD, SAINTY,MARTINand SIE [ 19851. [2] COLEMAN [19741. [3] MISIANO, VARASI,MANCINI,SARTORI and LASAPONARA [1983].
166
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
Fig. 5.13 The extinction coefficientofCeO, at 550 nm. (NEITERFIELD,
SAINTY,MARTINand SIE
[ 19851.)
5.1.6. Tantalum pentoxide
The final dielectric oxide discussed in our materials review is tantalum oxide. Tantalum oxide is a high refractive index material that is useful as an alternative to titania for some applications. The material is readily prepared by sputtering either an oxide target or an elemental Ta target in an oxygen atmosphere. For oxide targets a 90 : 10 partial pressure mixture of argon and oxygen is usually and TOMLIN[ 19751, SITES[ 19831) for both diode and selected (KHAWAJA ion-beam sputtering. The percentage of oxygen is usually increased to at least 25 percent for Ta targets to reduce optical absorptance in the films. Ta205has also been deposited using electron-beam evaporation of Ta20, although considerable outgassing of the source material occurs and oxygen backfilling must be employed to minimize absorption. The optical properties are also sensitive to substrate temperature and deposition rate (HERRMANN [ 19811). Ion-assisted deposition has been used to obtain high index material (2.1 at 550 nm, BINH, NETTERFIELD and MARTIN[ 1984]), but no detailed investigations have yet been reported, although preferential sputtering effects are known to be
111, $51
167
OPTICAL THIN FILMS
important in low-energy bombardment of Ta,O, (8 2.1.1). Figure 5.14 shows the data reported to date of films prepared by various techniques.
5.2. CARBON FILMS
In the early 1970s it was found that ion-beam deposition could be used to deposit thin films of insulating carbon on room temperature substrates. The material was found to have remarkable properties and has since been the subject of intense investigation for many applications. AISENBERG and CHABOT [ 19711described this material as Diamond-Like Carbon (DLC) and listed the followingcharacteristics: (1) high transparency, (2) index ofrefraction greater than 2.0, (3) highly insulating, (4)harder than glass, ( 5 ) resistant to chemical attack, (6) diamond-like crystal structure, and (7) dielectric constant between 8 and 14. The method adopted by AISENBERG and CHABOT was a deposition of carbon ions produced in a coaxial ion source by argon sputtering of a solid carbon electrode. The positive ions produced were then accelerated by a negative potential applied to the substrate. The original experiments of l
2.7-
'
i
'
~
'
~
'
l
'
l
'
1
'
l
' I
'
l
'
l
-
'
-
2.6 - A
-
x w 2.50
E
w 2.4-
2 I-
2
2.3-
a
:2.2Y
2.1
-
2.01.9
Oll
0
-
D
v Q
-
O
lzl A
'I 0
4 012 ' 013 ' 014 ' 0:s ' Of6
a
' I
017 ' d . 8 ' of9
'I-
:=g
1 . d 2.0
l
I68
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[]I],
85
AISENBERG and CHABOT were later repeated by SPENCER, SCHMIDT, JOYand SANSALONE [ 19761 using a similar ion source to produce a beam of positively charged carbon ions of energy 50-100 eV. These measurements also showed the presence of polycrystalline cubic diamond, high resistivities 2 loi2R cm, and a refractive index of about 2.0. It has now been demonstrated that DLC films can be deposited by rf plasma decomposition from a hydrocarbon gas such as butane, low-energy carbon ion-beam deposition, ion plating and dual-ion-beam sputtering. The films are sometimes referred to as i-Carbon (i-C) to indicate that ions are used in the deposition process. This term was introduced by WEISSMANTEL, BEWILOGUE, DIETRICH, ERLER,HINNEBERG, KLOSE,NOWICKand REISSE[ 19801, who have studied the preparation and properties of i-C in some detail. However, some workers prefer the descriptive name amorphous hydrogenated carbon (a-C : H), particularly for films produced by plasma decomposition from hydrocarbons. Films prepared by this technique have variable optical constants depending upon the hydrogen content and are frequently found not to contain diamond-like crystallites. In contrast, films deposited by direct ion-beam deposition often show evidence of a diamond-like structure. DLC films are attractive since they are hard and have a high index and a low optical absorption at least in the infrared region. BUBENZER,DISCHLER, BRANDT,and KOIDL[ 19841 have tested DLC as a material for laser mirrors and found C 0 2 single-pulse damage thresholds for CO, laser pulses on AR-coated Ge up to 280 MW cm-2. The films were chemically resistant, stable to 250°C, and had a Vickers hardness of 18000HV compared to 12000 HV for diamond.
WAVELENGTH Inml
Fig. 5.15. Extinction coeficient versus wavelength for ion-beam-deposited carbon films grown with 300 eV C + ions. (MIYAZAWA, MISAWA,YOSHIDAand GONDA[1984].)
111, § 51
OWICAL THIN FILMS
169
The optical properties of a-C : H can be tailored to render it suitable for anti-reflectingGe. The optical absorptancein the 8 to 12 pm range is 250 cm - ' compared to about 10 cm- for ZnS. The absorptance in the visible region is
'
A 0
0
SPUllER DEPOSITED ONLY (1W A THICK) SIMULTANEOUSLY SPUTTER DEPOSITED AND ION BOMBARDED (1738A THICK) UNCOATED FUSED SILICA (D.762rtlm THICK)
WAVELENGTH, A
Fig. 5.16. Reflectance, absorptance and transmittance of thin diamond-like carbon films prepared by sputter deposition and ion-assisted-sputter-deposition techniques. (BANKSand RUTLEDGE [1982].)
I70
"11,
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
s5
Fig. 5.17 Transmission spectra of IBD on carbon film using: 300 eV C ion beam (44 nm), and evaporation (50 nm). ( M I Y A Z A W A , MISAWA, YOSHIDAand GONDA[1984].) +
relatively high and Fig. 5.15 shows the extinction Coefficient of a film deposited by 300 eV C . An example of the transmittance and absorptance variation in the visible for ion-beam sputtered films is shown in Fig. 5.16. The transmittance of these films is seen to be quite poor in the visible region, and is only on the order of 20 percent at 550 nm. Absorptance values are typically 70 percent throughout the visible region. Much higher transmittances are observed over the wavelength range 4 to 20 pm, and Fig. 5.17 shows a comparison between ion-beam deposited carbon (44nm) and evaporated carbon (50 nm) films. The optical properties of DLC films prepared by rf glow-discharge deposition and ion-beam sputtering have been accurately measured over the wavelength range 254 to 633 nm by KHAN,MATHINE, WOOLLAM and CHUNG[ 19831. The film thickness was approximately 120-150 nm and the optical properties were measured by ellipsometry. The n and k values for films deposited by both techniques were the same as shown in Fig. 5.18. It was found that annealing the films in H, at a temperature of 500 "Chad only a slight effect on the optical properties. Comparisons were also made with dc sputtered samples, and table 10 shows values of n and k measured at a few wavelengths. The lower +
TABLE10 Refractive index and extinction coefficient of dc sputtered a-C films at selected wavelengths. (KHAN,MATHINE, WOOLLAM and CHUNG[ 19831.)
(rtm)
Wavelength
Refractive index
Extinction coefficient
Thickness (nm)
0.44 0.55 0.63
1.40 f 0.07 1.37 f 0.06 1.33 + 0.05
0.167 f 0.004 0.167 f 0.006 0.128 f 0.008
66.4 f 5.7 66.4 f 5.7 66.4 f 5.7
111, § 51
OPTICAL THIN FILMS
4 -
171
a
SYMBOL SAMPLE NO. 0.1
Y = 2
8.3 @=4
x-5
C
,
I
-0.6
,
I
-
1
I
-0.01 250
b
350
450
550
WAVELENGTH X ( m )
650
75
Fig. 5.18 The dispersion of: (a) refractive index, and (b) extinction coeffrcient for glow-discharge and ion-beam sputtered carbon films as prepared and also aRer annealing in hydrogen at 500 " C . (KHAN,MATHINE, WOOLLAM and CHUNG[1983].)
I72
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, I 5
values of n and high k values for these samples indicated possible structural differences between DLC film prepared by ion-beam sputtering or plasma deposition and dc sputtered films. The main problem with DLC as an optical material is the high optical absorption in the visible region. The adhesion is poor on certain substrates such as copper and the films are often highly stressed. Films deposited by either dc or rf glow discharge decomposition of hydrocarbon gas can contain large amounts of incorporated hydrogen which result in a very high intrinsic compressive stress (NIR [ 19841) on the order of l O I 3 N m-,. The material is already finding applications in the military field, in particular as a scratch resistant ir transmitting coating on missile domes, and is being produced by the Hughes Corporation on an industrial scale by the dual ion beam sputtering technique. As the technology develops DLC will find wider applications as a protective and high-index, anti-reflective coating with good resistance to chemically aggressive gases. 5.3. OTHER MATERIALS
Only a few investigations have been made on other optical thin films produced by ion-based techniques. Silicon nitride has been synthesized by ion-beam sputtering (WEISSMANTEL, REISSE,ERLER,HENNY,BEWILOGUE, EBERSBACH and SHURER[ 19791) and by reactive sputtering (PAWLEWICZ, MARTIN,HAYS and MANN [1982]). Amorphous hydrogenated silicon is presently under intensive investigation for applications in solar devices, and ion beam techniques have successfully been used in its preparation (MARTIN, NETTERFIELD, SAINTYand MCKENZIE[ 19831). Tin oxide and indium-tin oxide are frequently prepared by sputtering, and a comprehensive review of these and other conducting transparent coatings has been made by VOSSEN [ 19771 and by DAWAR and JOSHI[ 19841. Epitaxial zinc selenide and zinc sulphide films have been prepared by JONES,MOOREand COTTON[ 19821. The films were produced by rf sputtering with reproducible refractive indices close to the bulk values. Attempts to deposit dense stable magnesium fluoride films on unheated substrates by sputtering have largely been unsuccessful. COLEMAN[ 19741 showed that sputtering of MgF, targets produced absorbing films due to dissociation. Films with acceptable optical properties were produced by sputtering in freon-14 but had extremely poor adhesion. KENNEMORE and GIBSON[1984] have reported deposition of hard, adherent MgF, films on ambient temperature substrates by low-energy argon-ion assistance. However, the films were found to contain substantial quantities of oxygen.
111, § 51
OPTICAL THIN FILMS
173
5.4. APPLICATIONS
5.4.1. Multilayer coatings
Reactive sputtering has been shown to be a promising method for the production of multilayer coatings. PAWLEWICZ, MARTIN,HAYSand MANN [ 19821 have successfully produced a number of filters with dielectric oxides, electrically conducting transparent indium-tin oxide, silicon nitride and amorphous hydrogenated silicon. Multilayer optical-edge filters of exceptional stability in humid atmospheres were fabricated with Si,N, and SiO, for use in the visible region (MARTIN,PAWLEWICZ, COULTand JONES[ 19841). The 50 percent transmittance point shifted by less than 2 nm after 15 months at 85 C in a relative humidity of 85 percent. Conductive multilayer antireflection coatings have been deposited by IAD by EBERT[1982]. The spectral reflectances of a three- and a four-layer broadband filter are reproduced in Fig. 5.19. Figure 5.20 (MARTIN,MACLEOD, NETTERFIELD, PACEYand SAINTY[ 19831) shows the effect of IAD on the stability of a ZrO,(H), SiO,(L) multilayer narrowband filter. The shift in the spectral transmittance of the evaporated filter on exposure to air is due to absorption of water vapor throughout the layers. When prepared by IAD the filter was stable to less than 0.5 nm. The stability is a consequence of film O
WAVELENGTH
[nm]
Fig. i:19. A three-layer antireflection coating and four-layer conductive antireflection coating [ 19821.) produced by ion-assisted-deposition.(EBERT
174
[Ill, § 5
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
400
500
600
WAVELENGTH (nm)
700
400
500
600
700
WAVELENGTH (nm)
Fig. 5.20 (a) Spectral transmittance ofa fifteen-layer filter produced by evaporation only ofZrO, and SiO, layers. Dashed line shows the effect of atmosphere on the optical properties. (b) Spectral transmittance of an eleven-layer filter produced by ion-assisted-deposition. No vacuum-to-air effects were observed. (MARTIN,MACLEOD,NETTERFIELD, PACEYand SAINTY [ 19831.)
densification by ion-assistance which also produces layers with higher refractive indices. Hence the 11 layer IAD filter had the same bandwidth as a 15 layer evaporated filter. 5.4.2. Material mixing Ion-based deposition techniques are well suited to mixing materials to obtain a range of optical properties. Sputter deposition is independent of the material melting points and under controlled conditions it is suitable for the deposition of stoichiometric dielectric films, as we have seen in $4.4, MOTOVILOV and RUDINA[ 19741 have deposited mixtures of tantalum, niobium, zirconium, and titanium oxides, with silicon dioxide by sputtering from two cathodes. The refractive index of a mixed layer is well approximated by the BRUGGEMAN [ 19351 equation. Material mixing can also be produced by reactive deposition. Silicon oxynitride SiO,N, can be synthesized by several techniques to vary the refractive index from 1.46 to 2.0. The most successful method used to date has been liquid-phase chemical-vapor deposition (KUIPER,Koo, HABRATEN and TAMMINGA [ 1983]), although promising results have been obtained by reactive
111, § 51
OPTICAL THIN FILMS
I75
sputtering (FRANKand MOBERG[ 1970]), reactive evaporation (ERIKSSON [ 1983]), and ion-assisted deposition (HEITMANN [ 197 la,b]). and GRANQVIST HEITMANN found a refractive index of 1.6 at 633 nm for freshly deposited films with N; ion-assisted deposition of SiO. However, the film index was reduced to 1.47, and the UV transmittance increased once the films were heated to 350 “ C for 9 hours, indicating the instability and incomplete reaction of oxynitride formation for these deposition conditions. Infrared measurements were used to estimate a film composition of about 14 percent Si,N,, the remainder being SiO,. The ability of ion-based techniques to synthesize compounds will undoubtedly find greater application in dielectric optical films, once the deposition conditions and operating parameters are established.
5.4.3. Optical waveguides The use of ion-based techniques in dielectric film deposition has resulted in films with better optical properties than those prepared by simple evaporation. For the fabrication of integrated optical circuits, it is necessary to deposit thin-film waveguides with low optical loss and free from scattering defects. Optically transparent materials with a high refractive index in the range 2-3 are of interest in planar optical systems design. They are required in the fabrication of the waveguide layers on semiconducting materials with high refractive indices and materials with non-linear electro-optic and acousto-optic properties, and in the fabrication of planar optics elements (lenses, prisms, reflectors etc.). A profile for ideal waveguide material has been formulated by PITT[ 19811: (i) Good refractive index selectivity with a precision of one part in lo4. (ii) Accurate control of film thickness, since this determines the energy propagation velocity in the guide. A precision of at least one part in lo3 per centimeter of guiding path is required. (iii) Low absorption loss at the excitation wavelength. (iv) Minimum scattering defects to achieve a total loss (absorption plus scattering) of less than 1 dB cm - . (v) A configurable geometry is required in order to define the device shape. Waveguides are fabricated by various techniques such as diffusion, sputtering, ion-implantation and chemical-vapor deposition. However, waveguides produced by normal evaporation are unsuitable because of the variations in refractive index and scattering. Guided wave losses can be reduced by increasing the substrate temperature and vary over the range 50 to 1 dB cm(KERSTEN, MAHLEIN and RAUSCHER [ 19751).
’
’
I76
OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
[III, 8 5
Waveguidesproduced by IAD show greater promise. Initial results by BINH, NETTERFIELD and MARTIN[ 19851 show a loss of 1.54 dB cm- * for A1,0, prepared by oxygen-ion assisted deposition on room temperature substrates. Sputtering is frequently used to deposit Ta,O, optical waveguides. INGREY, BOYNTONand MACLAURIN [ 19761 have deposited WESTWOOD, LIVERMORE, waveguides with losses < 1 dB cm- by dc diode sputtering of tantalum in 0,-N, mixtures. WESTWOOD and INGREY [ 19761 studied ion-beam sputtering of tantalum with oxygen and argon but obtained high loss films of 11 dB cmwhich was attributed to absorption rather than scattering. Other sputtered oxides have also been investigated for waveguide applications, including niobium pentoxide with a waveguide loss of less than 2 dB cm - (AAGARD [ 19751).
5.4.4. High-power laser coatings
A major area of interest in inertial-confinement fusion and molecular laser isotope separation programs is the development of damage-resistant optical coatings. Many materials have been investigated for laser damage resistance including most dielectric oxides. PAWLEWICZ and BUSCH [1979] and PAWLEWICZ, HAYSand MARTIN[ 19801 investigated the basic mechanisms of coating failure and the systematic raising of damage thresholds. Titania films have been studied and found to have a high threshold when in a glassy state (no long-range order), rather than polycrystalline state. The grain size is controllable if reactive sputter deposition is used. The highest threshold reported by the group for titania is 8.7 & 1.5 J ern-,, (1 ns pulse at 1064 nm over 2 mm beam diameter). Glassy Ta,O, optical coatings were found to be less damage resistant than TiO, and failed at 3.0 0.5 J cm-, under the same conditions. Transparent indium-tin oxide coatings also have high damage thresholds of 5-6 J cm - when prepared by rf reactive sputtering. MISIANO, VARASI, MANCINI, SARTORI and LASAPONARA [ 19831 have also used ion-beam and rf sputtering to prepare laser coatings. Damage thresholds were measured for Y,O,, CeO,, and TiO, under various laser conditions, and Y,O, was found to have a threshold twice that of TiO, while CeO, was intermediate. Ion-assisted deposition can also be used to improve damage resistance, as demonstrated by EBERT[1982] for multilayer coatings for use with ArF excimer lasers operating at 193 nm. A 35 layers broad-band BeO-SiO, multilayer mirror achieved a reflectanceof 99 percent and a damage threshold
111, I61
CONCLUSIONS
177
of 7.2 f 0.75 J cm - (15 ns pulse), an improvement of 4 and 10 times that of 'AI,O,-SiO, and AI,O,-MgF, multilayers, respectively. The precise mechanisms of film damage under laser irradiation have yet to be established and many factors such as film structure, uniformity, absorptance and surface defects are all influential to some degree. The use of sputtering and ion-assisted techniques has demonstrated that these methods can raise damage thresholds and will undoubtedly receive increasing attention in laser materials research. 5.4.5. Protective layers
Coatings are frequently used to protect thin films and substrates which are susceptible to chemical or abrasive degradation, a good example being metal layers on astronomical mirrors. DLC coatings have been developed for infrared applications and have been discussed in 5 5.2. COLE,MORAVEC,AHONEN and EHLERT [ 19831 have shown that ion-beam sputtered Y,O, films are much less permeable to water than evaporated coatings for protective layers on diamond turned aluminium mirrors. SAINTY, NETTERFIELD and MARTIN [ 19841have reported on the deposition of protective oxide layers on aluminium and silver mirrors by IAD. Thin films of A1,0,, SiO, and ZrO, were deposited on aluminum coated substrates under varying conditions. The substrates were immersed in a solution of NaOH and the etch rate was monitored by observing the optical transmission. Similar experiments were performed on silver coated substrates, this time by overcoating with either SiO, or ZrO, and etching in HNO,. In both sets of experiments it was found that the greatest protection against film etching was achieved with an overcoat layer of 120 nm ZrO, deposited with ion-assistance. The resistance to etching was increased almost two orders of magnitude over that of a conventionally evaporated protective film. This is explained as a consequence of the reduction of film porosity by ion-assisted deposition preventing penetration of film etchant to the metal layer. Such applications of IAD should find application in extending the lifetime of large astronomical mirrors which may be subjected to chemical attack.
4 6. Conclusions Thin-film coatings are an essential part of nearly all modern optical devices, and in many cases are the limiting factor in the performance of these instru-
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OPTICAL FILMS PRODUCED BY ION-BASED TECHNIQUES
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ments. Until recently the properties of thin films have been dependent on most of the deposition parameters, such as deposition rate, substrate temperature, and residual gas composition and partial pressures, which has resulted in films with properties often very different and inferior to bulk material and difficult to reproduce. In recent times increasing use is being made of ion-based techniques to overcome these difficulties. Film growth is now principally dependent on ion kinetic processes and these can be varied considerably. In the case of directed ion beams. the vaporization and condensation stages do not interact and the individual conditions such as arrival rates, energies, and species can be controlled independently. These techniques should offer the greatest opportunities for study of ion-based techniques in the future. There are a number of limitations at present in the ion sources themselves. More reliable ion guns are needed which will operate at lower energies with higher ion-current densities, and which will have extended lifetimes with reactive gases. Sputtering will also probably play a more prominent role in optical coatings. The film growth kinetics can be varied by adjusting the system pressure, the position of the substrate relative to the discharge, and by substrate bias. Since ions and energetic species have been found to produce significant effects (often beneficial) on the microstructure and stoichiometry of thin films, ion-based techniques will undoubtedly play an increasing part in their future development. Detailed computer simulations of film growth mechanisms and ion-surface interaction processes should lead to a better understanding of atom-by-atom film growth and its relationship to optical properties.
Acknowledgements We wish to thank K. H. Muller for permission to include early results of computer simulations of film growth and M. Tychsen for the careful preparation of the figures. References AAGARD,R. L., 1975, Appl. Phys. Lett. 27, 605. AISENBERG,S., and R. CHABOT,1971. J. Appl. Phys. 42, 2953.
S., and R. CHABOT,1973, J. Vac. Sci. Techno]. 10, 104. AISENBERG, ALLEN,T. H.,1982, Properties of ion-assisted deposited silica and titania films, in: Optical Thin Films, Los Angeles, ed. R.I. Seddon, SPIE Proc. 325.93. ALLEN,T. H., 1983. Ion-assisted deposition of titania and silica films, in: Proc. Int. Ion Eng. Congr.,Kyoto, Vol. 2, ed. T. Takagi (Ionics Co., Tokyo) p. 1305.
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E. WOLF, PROGRESS IN OPTICS XXIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1986
IV
ELECTRON HOLOGRAPHY BY
AKIRATONOMURA Advanced Research Laboratory, Hiiachi Ltd. Kokubunji. Tokyo.Japan
CONTENTS PAGE
.................... COHERENCE OF AN ELECTRON BEAM . . . . . . . . ELECTRON HOLOGRAPHY . . . . . . . . . . . . . . . APPLICATIONS OF ELECTRON HOLOGRAPHY . . . . . § 5 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . 8 1. 5 2. 8 3. 0 4.
INTRODUCTION
185 187 194 204 217 218
9 1. Introduction Interest in electron holography is presently on the upswing because of recent improvements in electron beam coherence that have brought the technology closer to practical realization. In short, electron holography is a two-step imaging method (Fig. 1): a scattered electron wave from an object is recorded on film as an interference pattern through overlapping a reference wave, and subsequently the incidence of a light wave onto this film (the “hologram”) produces wavefronts identical to the original electron ones. This imaging technique was conceived by GABOR [ 1949, 19511, in an effort to break through the resolution limit of electron microscopes. However, it was not possible to put this methodology to practical use at that time because there were no coherent
Electron
Hologram
Reconstructed image
Hologram
Conjugate image
Fig. I , Principles of electron holography: (a) electron hologram formation; (b) optical image reconstruction.
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[IV, § 1
electron or light beam sources, which are indispensable for holography. This situation changed with the advent of the laser in 1960. Holography then came into bloom in the field of light optics, though not much fruit was borne in the field of electron holography. Again the reason for this was the lack of a coherent electron beam. As a parallel development, the resolution of electron microscopes has been steadily improving since the days of Gabor’s invention. This technology has now arrived at the point where even single atoms can be observed (CREWE, WALLand LANGMORE [ 19701, UYEDA,KOBAYASHI, SUITO,HARADAand WATANABE, [ 19721, HASHIMOTO, KUMAOand ENDO[ 19781). However, the problem still remains that the resolution is determined by aberrations in the objective lens. No practical aberration-correction methods have been successfully developed yet. Gabor’s aim whereby single atoms in a molecule would be visualized at a resolution of 1 A is still an unrealized dream. This dream is not only of intellectual interest, however; its realization is becoming a practical necessity for the modem electronics industries. To cite an example, recent technologies have made it possible to control even a single atomic layer during the production of materials for electronic devices. Methods are accordingly being searched now to allow observation and measurement of these materials in terms of their atomic dimensions. Recently, a field-emission electron gun, which had first been developed by CREWE,EGGENBERGER, WALLand WELTER[ 19681 to use as a scanning electron microscope, has been employed as a coherent electron source. With this, the maximum number of interference fringes observable has increased up to 3000 compared to 300 with a conventional electron source (TONOMURA, M A T S U D A and KOMODA[ 19781). Due to this improvement, electron holography has taken a major step forward: the quality of the reconstructed image has become far better, with picture elements being more than 500 x 500 in number. Since then, new information has become available which was not previously accessible through the use of an electron microscope. New application fields have opened up even outside the area of high resolution observation, which was Gabor’s original aim. One example is interference electron microscopy, where microscopic magnetic lines of force and thickness contour lines are visualized in an electron micrograph. Instructive review papers on electron holography have already been published by HANSZEN[ 19821, MISSIROLI, Poxzi and VALDRE [ 19811, WADE [ 19801, ROGERS[ 19801, and ZEITLER [ 19791. In the present article, recent progress in the area of experimental research regarding electron holography will be reviewed. It needs to be stated, however,
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that, since this new field is currently in a state of rapid change, it is difficult at this early stage to describe accurately the present status of each part of it. Correctly predicting future developments is an even more difficult task.
8 2.
Coherence of an Electron Beam
The recent development of a field-emission electron beam has marked significant progress in the improvement of beam coherence. However, the resolution and quality of the reconstructed image are still limited during electron-hologram formation, though not in the optical reconstruction stage. The reason for this lies in the inferior coherence of an electron beam compared to that of a laser beam. These coherence characteristics of an electron beam will accordingly be discussed here briefly. For detailed theoretical arguments, the reader is referred to HAWKES[ 19781. The physical image of a light wave is familiar to us (i.e. rapidly oscillating electromagnetic fields). But what about an electron wave? An electron wave cannot really be interpreted physically. It is detected as particles. The particles are, so to speak, virtual images of an unobservable wave function Y and they are detected proportional to the probability density of the wave field I !PI 2 . The simplest form of the wave field is a sinusoidal wave, but an actual electron beam deviates far from this image. A single electron corresponds to a spatially localized wave packet (Fig. 2), as in the case of a single photon. The extension of a wave packet is very important in discussing interference phenomena such as holography, since the phase of a wave function is coherent only within a wave packet. Therefore,
I
Fig. 2. Wave packets corresponding to electrons.
188
ELECTRON HOLOGRAPHY
[IV, § 2
interference is possible when the shift of two partial waves of a wave packet is within the extension of the wave packet. Strictly speaking, interference fringes are observed as the time average of a probability density I Y I formed by each successively arriving electron. The coherence of an electron beam can be considered to consist of two elements: time coherence (longitudinal coherence) and spatial coherence (transverse coherence). Time coherence refers to the coherence in the propagation direction of a wave packet, which arises from the energy spread of the beam. Spatial coherence is related to the coherence in the plane perpendicular to the propagation direction, which comes from the finite source size. Actual factors determining the two kinds of beam coherence and values for coherence lengths will be discussed in the following sections.
2.1. TIME COHERENCE
The time coherence length is the maximum path difference under which interference is possible between two beams travelling along two different paths. This value is, so to speak, the average length of the wave packets. The length I, of a wave packet is given by the uncertainty AA in the wavelength I, or by the uncertainty eAE in the electron energy eE, as
The number n of wavelengths in a single wave packet is given by
Let us consider here what determines AA or A E in practical terms. Electrons in a metal originally take the form of wave packets, which are different depending upon the energy levels the electrons belong to. Only electrons at higher energy levels can be emitted from a metal. For example, the energy distribution of thermionic electrons follows a Maxwell distribution, which is only the higher part of the Fermi distribution of electrons in a metal (see Fig. 3). Therefore, thermionic electrons have approximately an energy spread kT, where k is the Boltzmann constant and T is the absolute temperature. At T = 3000 K, the energy spread amounts to 0.3 eV. Although this distribution agrees with experimental data, the energy spread of an electron beam
IV. § 21
I89
COHERENCE OF ELECTRON BEAM
0
5kT
Fig. 3. Energy distribution of electrons: (a) Fermi distribution of electrons in metal; (b) Maxwell distribution of electrons emitted from metal. N : number of electrons, E,: Fermi energy, @: work function.
accelerated to, say, 100 kV becomes larger than the theoretical value due to the Boersch effect (BOERSCH[ 19541). Measured energy spreads for various types of electron cathodes are listed in table 1. The energy spread eAE reaches 2 eV for the hair-pin type thermionic cathode conventionally employed in electron microscopes. A thermionic electron beam from a pointed cathode, developed by HIBI[ 19561, has an energy spread of 1 eV. The energy spread eAE decreases to 0.3 eV in the case of a field-emission electron beam, where it is not given by kT but by k ’ F (k’being a constant dependent on the work function, and F, the electric field at the metal surface). Time coherence lengths for these beams can be estimated as follows. According to LENZand WOHLAND [ 19841, the coherence lengths are derived from eq. (1) by substituting the measured energy spreads of electron beams for eAE. Calculated coherence lengths 1, and wave numbers n for a given coherence length are also listed in table 1. The coherence lengths 0.4-4 pm seem extremely short compared with those for laser beams, but this is primarily due to the short wavelengths of electron beams. Actually, no electron interference experiment has ever been restricted by a time coherence length. Only recently has this length been measured using special contrivances. M~LLENSTEDT and WadL A N D [ 19801 measured the coherence length of a 2.5 kV electron beam to be
-
-
-
I90
[IV. § 2
ELECTRON HOLOGRAPHY
TABLE1 Time coherence of electron heams. Cathodc type
Structure
Source size
Energy spread
(pm)
Wave number in a packet (n)
Time coherence length (It)
(ev)
Hair-pin
Pointed
Fieldemission
30pm
2
0.4
105
Ipm
I
0.7
2 x 105
0.3
4
1O6
00 A
0.12 pm using a Wien filter. Later, SCHMID [ 19841 obtained a value of 0.46 pm for a 35 kV electron beam using an electrostatic cylinder. It is interesting to consider here whether the coherence lengths are unchangeable or not. When an electron beam is accelerated, the wave number n increases proportional to E (eq. 2), but the wave length I decreases in a manner inversely proportional to \/E.Accordingly, it is clear that the length of a wave packet becomes larger proportional to JE. Furthermore, this length increases when the energy of the electron beam is filtered through a monochromator. A wave packet propagating in a free space gradually lengthens its extension, which is not the case for a wave train of light. This is because the electron velocity depends on the wavelength. The increase of l, is, however, too small to be detected experimentally. For example, this increase is even less than 1 8, when a 100 kV electron beam has travelled as far as 1 m.
IV, § 21
COHERENCE OF ELECTRON BEAM
191
2.2. SPATIAL COHERENCE
An electron wave packet, as described in the preceding section, is confined in its propagation direction. Wavefronts can be defined only within the limits of their extension. However, wavefronts extend infinitely in planes perpendicular to the propagation direction. This is because the velocity of an electron is assumed to be of uncertain magnitude but of certain direction. Any uncertainty in the propagation direction determines the width of a wave packet. The spatial coherence length of an electron beam is given by the average width of wave packets in the beam. The main experimental factor determining the coherence length is the source size or illumination angle of an electron beam. To assist in a physical understanding of the spatial coherence length, the simplest case is presented in Fig. 4. As can be seen in the figure, three electron beams propagate in slightly different directions (28: contained angle). Wavefronts perpendicular to the three directions are drawn so that they are in phase at point Poin the observation plane. Wavefront displacement among the three beams becomes larger as the distance from point Po increases. The displacement of two beams A and B, or A and C,amounts to half a wavelength at point P,. Definite wavefronts can be observed inside the region of radius Therefore, the width 1, of this region can be calculated as
$pop,.
An electron beam with an illumination angle of 28 can be seen to have approximately the spatial coherence length given by eq. (3). The coherence length may seem to become arbitrarily large with the selection
Electron beams € ip 4 W,I
Wavefronts
Observation plane
f,oiit displ~cementof three beams in slightly different directions
I92
[IV,§ 2
ELECTRON HOLOGRAPHY
of a small value for the illumination angle 28. Actually, though, this coherence length is restricted by the beam intensity required for observation. Therefore, the coherence length is determined by the characteristics of the electron beam, that is, the degree of intensity which can be obtained under a small illumination angle. These characteristics can be represented by the “brightness” B, which is conventionally used in electron optics. This is defined as B
=
(4)
i/n/J2,
where i and n j 2 are the current density and solid angle, respectively, for the electron beam. It can be proven from Liouville’s theorem that the value of B depends only on the kind of electron beam and the accelerating voltage, and that B is constant in any cross-section of an electron beam. Brightness values are listed for various electron beams in table 2. Moving from a conventional hair-pin type cathode to a pointed cathode, brightness increases about fourfold. The brightness of a field-emission electron beam exceeds the others by nearly a factor of 1000. Spatial coherence lengths increase with brightness, though the absolute values of coherence lengths have no important meaning. This is because electron interference fringes are always magnified by electron lenses for observation. The lengths depend greatly on this magnification. The maximum number N of observable interference fringes is, however, independent of such experimental conditions. N expresses faithfully the degree of spatial coherence of an electron beam. Consider a simple case, such as represented in Fig. 5, where electron interferencefringes are formed with an electron biprism, magnified m times, and subsequently recorded on film. The incident electron beam is assumed to have an infinite time coherence length. The conditions for recording the fringes are as follows. First, the spatial coherence length, 1/2pof the electron beam in the TABLE2 Spatial coherence of electron beams. Cathode type
Number of fringes
Brightness, B (A/cmZster)
observable Hair-pin Pointed Field-emission
5 x 105 2 x lo6 3 x 108
observed
150
100
300 4000
300 3000
IV, I 21
COHERENCE OF ELECTRON BEAM
193
Interference
Fig. 5. Electron interference fringes magnified and recorded on film.
specimen plane has to be at least two times larger than the width of the interference pattern :
where s is the minimum fringe spacing recordable on film.Then, the current density should be large enough to sensitize the film. If the charge density required for sensitizing the film is given by 1 and the exposure time is set to 1 s for simplicity, the current density in the specimen plane is m21 = B . no’.
(6)
Equations ( 5 ) and (6) lead to the maximum observable number of interference fringes:
The value of N is proportional to the square root of the brightness B, independent of experimental conditions such as magnification m. Calculated
I94
ELECTRON HOLOGRAPHY
[IV, 0 3
values for N are listed in table 2, together with reported experimental results. This table makes clear why field-emission electron beams are indispensable for electron holography.
8 3. Electron Holography In electron holography, a hologram of an object is formed with an electron beam, and the image is then reconstructed with a light beam. Several types of electron holography have been devised up to now, and interesting investigations have been reported which used a scanning transmission electron microscope (STROKEand HALIOUA[1972] and VENEKLASEN[1975]) or an electron diffractometer (BARTELL [ 19751)in the hologram-formation stage. Discussions here will be limited to electron holography using a transmission electron microscope, since this was Gabor’s original intent. When Gabor invented holography, he had to adopt in-line holography, where a coherent background is used as a reference wave, for two reasons. One is that there were no wave sources available then that had large spatial and time coherence lengths. The other was the lack of necessary electron-optical devices, such as beam splitters. Since in-line holography entails the problem that a reconstructed image is always disturbed by its conjugate image, it took more than ten years from the time of Gabor’s invention before the worth of holography was recognized through implementation of off-axis holography by LEITHand UPATNIEKS [ 19621, using a laser beam. In the following, principles and types of holography will be explained.
3 . 1 . I N - L I N E HOLOGRAPHY
Gabor first proposed the method shown in Fig. 6 . This later came to be known as the “projection method”. It involved the formation of a point focus using an electron beam. The diameter of the focus determines the resolution of the reconstructed image. A spherical wave from the point focus is disturbed by an object located near the focus. Most of the wave passes through it undisturbed. Both the disturbed and the undisturbed part are projected onto the hologram plane, and recorded on film as a hologram. If the amplitude of the electron wave is considered to be $ ( x , y ) at the hologram plane, $(x, y ) can be split into a background wave, $+,(x, y), and a
IV. $ 3 1
PRINCIPLES A N D IMPLEMENTATIONS
195
Fig. 6 . In-line holographic pro@tion method: (a) electron hologram formation; (b) optical image reconstruction.
disturbance wave (object wave) (PI ( x , y ) resulting from the object: W Y ) = $"(X,Y)
OI(X,Y).
(8)
The amplitude transmittance ( T ( x , y ) ) of this hologram is given by (I $(x, y ) l ) - )'I2, where y is the film contrast. That is, T(x,
v) = l$o I
- 4Y $o$? - 4 Y $$$I.
(9)
The hologram is then enlarged by the ratio A'/A( = m ) for light-to-electron wavelengths. The transmittance of the enlarged hologram is T ( x / m ,y/m). Subsequently, it is illuminated with a light wave $o(x/m, y/m), having wavefronts identical with those of the electron background wave except for a scaling up. The amplitude of the transmitted light is
If the intensity of the background wave 1 &, 1 is constant, the original wave + $I is reconstructed. The last term represents the conjugate wave which has a phase of the opposite sign. Disturbance of the wavefront reconstruction may occur in some cases, as will be discussed later. The appearance ol the
I96
[IV, B 3
ELECTRON HOLOGRAPHY
conjugate wave stems from the fact that the sign of the phase cannot be stored in an interference pattern. This projection method, however, was not practical for electron microscopes. In a word, it was not easy to form an electron beam with a focal point of a few A diameter. In this connection, the “transmission method” was proposed by HAINEand DYSON[ 19501, and was analyzed in detail by GABOR[ 19511. As can be seen in Fig. 7, an object is illuminated with a collimated electron beam. Since the effect of an object is assumed to be as small as with the projection method, waves tend to pass the object virtually without being affected in any way. Object and background waves overlap to form an interference pattern. The pattern is enlarged by electron lenses, and recorded on film as a hologram. This is, so to speak, taking an out-of-focus electron microscopic image using collimated illumination. Experiments regarding this method were carried out by HAINEand MULVEY [ 19521 and by HIBI[ 19561. Since the objects used in these experiments were large, reconstructed images were greatly disturbed by the conjugate images. An example of an electron hologram formed in a Fraunhofer dsraction plane is shown in Fig. 8. Small holes are dotted in a collodion film, and each hole produces concentric interference fringes like a zone plate. Equations (8)-( 10) also hold for this transmission method. The only slight (a)
Source
Lens
Obje
m
Source Lens Reconstructed
Fig. 7. In-line holographic transmission method: (a) electron hologram formation; (b) optical image reconstruction.
IV, § 31
PRINCIPLES AND IMPLEMENTATIONS
197
Fig. 8. Example of Fraunhofer in-line hologram.
difference is that the background wave $+,(x, y ) becomes a plane wave instead of a spherical wave. Twin images are formed in a mirror-symmetrical position with respect to the hologram. Until now, the hologram magnification m has been considered to be the wavelength ratio A'/A. But what happens if m is different from the wavelength ratio? The magnification of a reconstructed image then is no longer A ' / A , but has become m.Then, the position of a reconstructed image 2' is not given by (A'/A)Z,but by
In short, lateral and longitudinal magnifications are different from each other when m # 2'12. In-line holography restricts the size of an object, because the background wave has to function as a reference wave. Further restrictions are placed on the object for eliminating the effect of the conjugate image. Although trials of various techniques have been made, the most effective means found so far is [ 19651 for Fraunhofer holography, which was employed by THOMPSON
198
ELECTRON HOLOGRAPHY
[IV. § 3
measuring particle sizes. With this method, a hologram is formed in the Fraunhofer diffraction region of an object given by
where a is the size of the object. Under this condition, twin reconstructed images have no influence on each other. If one of the twin images is observed, the other image is completely blurred in this plane (Fig. 9). Simple calculations assure that the intensity of the blurred image is a small constant within the region of the reconstructed image. The electron version of this experiment was conducted by TONOMURA, FUKUHARA, WATANABE and KOMODA[ 19681, and several possibilities concerning electron holography could be seen to open up. Reconstructed images in this case are shown in Figs. 10-12. In these experiments,holograms were formed using a 100 kV electron beam (A: 0.037 A) from a pointed cathode, with the condition Z = 2 mm. Images were reconstructed using a He-Ne laser beam (A: 6328 A). The object in Fig. 10 is a smoke particle of ZnO which is obtained when Zn is burned in air. The shape is similar to that of the tetrapods used to construct breakwaters along a seacoast. The total size of the particle is 0.5 pm; consequently, condition (12) is not satisfied. Since the needles are thin enough, though, they are clearly reconstructed. The object in Fig. 11 is a MgO particle placed on a collodion film.The particle size is 1 pm, and so the disturbance from the conjugate image appears outside the reconstructed image. The disturbance fringes can be said to be Fresnel fringes of the conjugate image located just at the opposite side of the hologram. Small holes in the collodion film are clearly reconstructed. Contrast is enhanced by the selection of a large y value for the hologram plate. (See eq. 9.) Hologram
Light
k-
2'
+ -I 2'
Fig. 9. Effect of conjugate image on reconstructed image in Fraunhofer in-line holography.
PRINCIPLES A N D IMPLEMENTATIONS
199
Fig. 10. ZnO particle: ( a ) electron micrograph; (b) in-line hologram; (c) reconstructed image.
An example is shown in Fig. 12, where this method of contrast enhancement is applied to unstained biological specimens with a very low incoherent contrast. The object here is formed by tobacco mosaic viruses. Contrast in the reconstructed image is higher than that in the electron microscopic image, though the latter is printed with a high level of contrast. The resolution of reconstructed images obtained in this experiment was a few tens of A at best. Here, it would be opportune to examine what restricts the resolution. Evidently, the resolution does not deteriorate in the optical reconstruction stage, but rather in the hologram formation stage. Therefore, only the hologram formation of a point object is presented in Fig. 13. An in-line hologram of a point object is a zone plate. The zone plate consists of concentric
Fig. I I . MgO particle: ( a ) electron micrograph; ( b ) in-line hologram; (c) reconstructed image.
200
"V. § 3
ELECTRON HOLOGRAPHY
Fig. 12. Tobacco mosaic viruses: (a) electron micrograph; (b) in-line hologram;(c) reconstructed image.
interference fringes and ideally has an infinite diameter. The fringe spacing decreases with the distance from the center. That is, the radius r,, of the n-th circular fringe can be approximated by r,, = JGZ.
(13)
If an ideal zone plate formed with an electron beam is magnified by the wavelength ratio A'/A, and if the magnified zone plate is illuminated with a plane light wave, then the reconstructed image of a point object has an intensity distribution with a spread representing the order of magnitude of the wavelength A ' . Moreover, since an actual zone plate of diameter D' is limited in size, the
Electron Point beam object
Hologram
Zone plate
Fig. 13. In-line hologram of point object.
IV, 8 31
PRINCIPLES AND IMPLEMENTATIONS
20 1
image is further blurred. The spread is considered to be the same as in the case of a lens with an aperture diameter D ’ . Therefore, the image size d ‘ of a point object, or the resolution of a reconstructed image, can be given by
A’Z‘ d’ = 1.6 7 .
D
The value d’ refers to resolution in the reconstruction stage, but the same equation can be made to hold for an electron microscope by removing primes. In this case, the discussion does not involve primes, i.e., for an electron microscope. It should be noted here that the resolution d of a reconstructed image is nearly equal to the minimum fringe spacing ,/in the zone plate. From eq. (14), it can be seen that the conditions for improving resolution are: (1) small distance Z and (2) large diameter D of the zone plate. However, it is impossible to adopt a small value for Z due to the effect of a virtual image (eq. 12), except in the case of an extremely small object. The size of the zone plate is determined by the spatial coherence length A/2p of an illuminating electron beam. If D is equal to A / 2 p in eq.(14), the resolution is 3.2pZ. Substituting experimental values of 5 x lo-’ rad and 2 mm for p and Z, respectively, the resolution can be calculated to be 30 A. This value agrees with the experimental results. MUNCH[ 19751improved the resolution down to 10 A using a field-emission electron beam and fine gold particles as a subject.
3.2. OFF-AXIS HOLOGRAPHY
Experiments involving off-axis electron holography were first carried out by MOLLENSTEDT and WAHL[ 19681, as well as by TONOMURA [1969]. The former used an electron biprism (M~LLENSTEDT and DUCKER[ 19561) as a beam splitter, while the latter employed a single-crystalline film. Since the coherence conditions for an illuminating electron beam are more stringent than those for in-line holography, the quality of the reconstructed images was not as good as with in-line holography. Further investigations involving off-axis electron holography were made by TOMITA,MATSUDA and KOMODA[ 19721 and by SAXON[ 19721; here, image quality was improved. In the late 1970s, TONOMURA, MATSUDA and ENDO[ 1979al have used a field-emission electron beam to obtain reconstructed images which are almost comparable to electron microscopic images in resolution.
202
ELECTRON HOLOGRAPHY
IIV, § 3
Image formation in off-axis holography is essentially the same as that for in-line holography. Taking a hypothetical point as the object, the in-line hologram would be a zone plate. The off-axis hologram is also part of the zone plate, located far from the center. Since a zone plate entails lens action, in-line and off-axis holograms correspond to the central and peripheral regions of a lens, respectively. Any part of a lens involves the same lens action. The only difference between in-line and off-axis holography is the effect of the conjugate image. In off-axis holography, reconstructed image and its conjugate are formed separately in the f. 1st order diffracted beams, and they can be observed independently. Therefore, there are no such restrictions on an object as with in-line holography; off-axis holograms are usually formed in the Fresnel diffraction plane or in the image plane of an object. An experimental arrangement for off-axis electron holography is shown in Fig. 14. Initially, off-axis electron holograms are formed in an electron microscope, where an electron biprism is installed. A collimated electron beam illuminates an object located in one half of the object plane. The other half is for the reference wave. An image of the object is formed through the objective lens. The biprism, which is situated between the objective lens and its image plane, makes the image and the reference wave overlap. If the spatial coherence length 1/2/? of the illuminating electron beam is large enough to cover both object and reference waves, interference fringes are formed in the overlap
Fig. 14. Off-axis holography: (a) electron hologram formation; (b) optical image reconstruction.
IV,§ 31
PRINCIPLES AND IMPLEMENTATIONS
203
region. This interference pattern is enlarged by magnifying lenses and recorded on film as a hologram. An example of a Fresnel hologram is shown in Fig. 15. The object is formed by MgO particles. The hologram looks like a defocused electron micrograph. However, when a part of the hologram is enlarged as in Fig. 15b, it proves to consist of fine interference fringes. When the hologram is illuminated with a collimated laser beam, the image shown in Fig. 15c is reconstructed. The resolution of reconstructed images, of course, depends on various conditions, but it is essentially determined by the spacing of carrier fringes in a hologram in the off-axisholography case shown in Fig. 14. Up to now, lattice fringes of gold { 1 1 l} planes with spacings of 2.4 A have been reconstructed by TONOMURA, MATSUDA and ENDO[ 1979a1, while LICHTE[ 19841 has also reconstructed 3.4 A lattice fringes of carbon black. The resolution of reconstructed images has become comparable to that of electron micrographs with the use of a field-emission electron beam. One more interesting result worth mentioning here concerns the Fouriertransform holography carried out by LAUER[1984]. In this experiment, an object was located in one half of the back focal plane of an objective lens, and a weak scattering foil in the specimen plane was used as a beam scatterer. The resolution of reconstructed images in this case was 30 A.
Fig. 15. Example of Fresnel off-axis hologram: (a) hologram; (b) enlarged hologram; (c) reconstructed image.
204
[IV. § 4
ELECTRON HOLOGRAPHY
0 4. Applications of Electron Holography Up to this point, we have described optical reconstruction of an electron microscopic image by means of electron holography. In this section we shall look at new techniques and applications of electron holography that have become feasible as a result of optical processing of reconstructed images. Examples here are the correction of the aberration in an electron lens such as Gabor was aiming at, and interference electron microscopy, an application opening up entirely new possibilities.
4.1. CORRECTION O F ELECTRON LENS ABERRATION
An electron lens has the same function as an optical lens. But with a magnetic electron lens, the lens action is accompanied by rotation of a meridional plane. One decisive inconvenience is that there are no concave axial-symmetric magnetic lenses. For this reason, lens abberrations cannot be compensated for by combining concave and convex lenses, as with optical lenses. In reality, the spherical aberration of an objective lens has a tremendous influence on electron microscopic images. For example, an image of a crystalline object formed with a transmitted electron beam is often displaced by that formed with Bragg-reflected electron beams, even under an in-focus condition (Fig. 16). This is because Bragg-
Electron Lens
Object
Lens
Fig. 16. Effect of spherical aberration on image.
Image
IV, $ 4 1
APPLICATIONS
205
reflected beams pass through the peripheral parts of an objective lens, which may have shorter focal lengths due to spherical aberration. What is more, the point resolution of modem electron microscopes extends to less than 3 A, a value which is completely determined by spherical and chromatic aberrations in the objective lens. This value is larger than a wavelength-determined fundamental limit by two orders of magnitude. While the effect of chromatic aberration can be lessened by monochromatizing an electron beam, there is no means of coping with spherical aberration. If this spherical aberration can be compensated for, it will be possible to obtain a resolution of less than 1 A. Correction of aberration in an electron lens was experimentally carried out for coma aberration by SAXON[ 19721. Later, spherical aberration was compensated for by TONOMURA, MATSUDAand ENDO [ 1979b], in a manner which will be explained here in detail. Images that are optically reconstructed through use of electron holography are subjected to the aberrations of an objective lens in an electron microscope. This is because the aberrations disturb only the object wave and not the reference wave, since the latter wave only passes through the center of the objective lens. Several techniques can be considered for correcting spherical aberration in the optical reconstruction stage. Examples are: (1) an aberration-correction plate is used at the back focal plane of the image-forming lens, as Gabor proposed; (2) an aberration of opposite sign is applied to the image by employing a corresponding optical convex lens; (3) the aberration of the image is compensated for by that inherent in holography. In the experiment by TONOMURA, MATSUDA and ENDO[ 1979b], method (2) was adopted, which was originally proposed by WEINGARTNER, MIRANDE and MENZEL[ 19691. A concave lens was used as a correction lens in place of a convex lens, because of the simplicity of its optical reconstruction system. This is peculiar to holography, where an additional image with a negative aberration appears beside the reconstructed image. An optical system for spherical-aberration correction is shown in Fig. 17. The conjugate image is selected by an aperture from two reconstructed images. The spherical aberration of this image is compensated for by a convex lens, and an aberration-free image is produced. The magnitude of the spherical aberration to be corrected is given by
206
[IV, 8 4
ELECTRON HOLOGRAPHY
Laser
Hologram
Correction lens
Corrected image
Fig. 17. Optical system for spherical aberration correction.
where C, is the spherical aberration constant of the objective lens in the electron microscope, and C: is the spherical aberration constant for the conjugate image. The value of C, is 1.7 mm, and therefore C: is 240 m. This is necessary. The value of C: can value is so large that a demagnification of be finely adjusted by changing the demagnification rate. The reconstructed image of a fine gold particle is shown in Fig. 18a. Brag-reflected electron beams produced at the particle form two different images outside the in-focus image, due to the spherical aberration. Inside the two images, lattice fringes of gold { 1 1 1) planes are observed, although no crystal lattices actually exist there. In the corrected image shown in Fig. 18b, lattice fringes are observed inside the particle image, and provide correct information on the crystal structure of the particle. The lattices fringes of gold { 1 1 1). planes (d = 2.4 A) as well as the half-spacing fringes (d/2 = 1.2 A) can be observed. The resolution of the corrected image ought to be better than that of the original electron micrograph, but the actual image obtained is worse in quality due to granular noise. This originates from speckle noise in the optical reconstruction stage. In this case, though, granule size could conceivably be made small enough if the spacing between the hologram carrier fringes were to be reduced to a size much less than the required resolution.
-
4.2. MEASUREMENT OF THICKNESS DISTRIBUTION
Once an image is optically reconstructed by electron holography, the phase distribution can easily be displayed as an interference micrograph by employing optical means. Interference electron microscopy is thought to be particularly
APPLICATIONS
207
Fig. 18. Spherical aberration correction for fine gold particle: (a) reconstructed image; (b) spherically corrected image.
208
[IV, 0 4
ELECTRON HOLOGRAPHY
applicable to the measurement of the thickness distribution of an object, as can easily be imagined from an analogy with the optical case. The refractive index n of a non-magnetic object can be expressed as n = l + -?V, ,C L1;
where V, is the inner potential of an object. This means that when an electron beam penetrates an object, it is accelerated by V,, and the wavelength becomes a little smaller. If E = 100 kV and Vo = 20 V, then the value of n - 1 is 1 x The object thickness corresponding to a phase shift of one wavelength is 400 A. It is due to the small value of n - 1 that the sensitivity to object thickness is so poor even though the wavelength of the electron beam is so short. The optical reconstruction system for interference microscopy is shown in Fig. 19. A collimated laser beam is split into two beams travelling in different directions by a Mach-Zehnder type interferometer. These two beams, A and B, illuminate an image hologram, each producing three beams: a transmitted beam and two diffracted beams. Reconstructed and conjugate images appear by means of the diffracted beams. An interference micrograph is obtained when the transmitted beam of beam A and the reconstructed image of beam B are selected by an aperture, and overlap to interfere with each other in the observation plane. If reconstructed and conjugate images are selected and overlap, a two-times phase-amplified interference image can be obtained. When higher-order diffracted beams are used, an amplificationof more than [ 19701 reported an two times is possible. MATSUMOTO and TAKASHIMA amplification of 14x in an optical case. This amplification technique was introduced into electron holography by ENDO,MATSUDA and TONOMURA [ 19791.
Mach-Zehnder interferometer
Hologram
Interference image
Fig. 19. Optical system for interference electron microscopy.
IV, B 41
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209
An interference electron micrograph is shown in Fig. 20. The object is formed by MgO particles. It is impossible to determine the three-dimensional shape from the electron micrograph (Fig. 20a), but it becomes much easier from the interference micrographs (Fig. 20b,c), which represent contour maps of the object's thickness. The spacing between two adjacent fringes in Fig. 20b corresponds to change a in thickness of 500A. The phase distribution is amplified two times in Fig. 20c; as a consequence, a more detailed thickness distribution can be observed. This example shows that interference electron microscopy can be effectively used to identify the external shape of a fine particle. If a crystalline object is to be observed, Bragg reflections should not be excited. This is because the phase of the transmitted electron beam depends on the Bra= conditions, which constitutes an obstacle to absolute measurement of the electron phase. Conversely, electron interferometry can also be utilized for investigations of Bra= reflections (YADA, SHIBATAand HIBI, [ 19731, HANSZEN [ 19831). Recently, much smaller changes in thickness, reaching down to atomic scale, have been detected by TONOMURA,MATSUDA,KAWASAKI,ENDO and OSAKABE [ 19851. In this experiment, a different reconstruction method was employed which permitted a higher amplification rate. This technique was first proposed in the optical field by BRYNGDAHL [1969], and developed by MATSUDA, FREUND and HARIHARAN [ 19791. In this technique, a new hologram, that is phase-amplified two times, is ftrst formed from the original electron hologram, using an optical spatial filtering system (Fig. 21). This can be done by selecting only the 1st order diffracted beams with an aperture which has two holes situated at the back focal plane of lens L,. When this procedure is repeated n times, the amplification rate is 2". This method does
Fig. 20. Interference micrographs of MgO particles; (a) electron micrograph; (b) interference micrograph; (c) interference micrograph (amplification: x 2).
210
[IV, § 4
ELECTRON HOLOGRAPHY
Film
LZ Aperture
Ll Hologram Fig. 21. Optical spatial filtering set-up for forming amplified holograms.
not rely on the film’s nonlinearity, as did the method described in Fig. 19, and consequently the amplification rate may be raised as high as desired. However, a problem does exist as to whether it is possible to read out an extremely small phase shift on the order of &j A. An interference micrograph of MoS, film is shown in Fig. 22; this has been phase-amplified 24 times. Fringe steps can be perceived along three oblique
Fig. 22. Interference micrograph of MoS, film (amplification: x 24).
IV. § 41
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lines designated A, B and C in the micrograph. They correspond to one, three and five atomic steps on cleaved surfaces, respectively. One step corresponds to a thickness change of 6.2 A, which is one-half of the lattice spacing of the c-axis.
4.3. OBSERVATION OF MICROSCOPIC MAGNETIC FIELDS
Interference electron microscopy also provides information on magnetic fields; this is an advantage not appearing with light optics. This fact was first predicted by COHEN [ 19671and later by OLIVEI [ 19711. They emphasized that a ferromagnetic thin film presents a pure phase object to an illuminating electron beam, and that electron holography would enable observation of detailed magnetization distributions. Subsequently, it was shown in experi[ 19721 and by POZZIand MISSIROLI [ 19731 that the ments by TONOMURA phase distribution of an electron beam transmitted through a ferromagneticthin film provides information on the magnetic domain structure. The observation principle involved is presented in simplif'ied form in Fig. 23. The incident electron beam is deflected by a uniform magnetic field. When the beam is regarded as a wave, it is convenient to introduce the concept of a wavefront. Since wavefronts are perpendicular to rays in light optics, the relation is assumed to hold also for this case. The incident beam corresponds to a plane wave, and the deflected beam to an inclined plane wave. What kind of influences does a magnetic field exert on an incident wavefront? A magnetic field rotates the wavefront along a rotation axis determined by a magnetic line of force. Therefore, contour lines for the transmitted wave-
r1-b
a
Electron beam Magnetic field Wavefront
Contour Lines Fig. 23. Observation principle regarding magnetic lines of force with implementation of interference electron microscopy.
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front describe magnetic lines of force (WAHLand LAU[ 19791, TONOMURA, MATSUDA, ENDO,ARII and MIHAMA [ 19801). This is because the height of the wavefront is always the same along the rotation axis. Thus, the very simple conclusion is that the contour fringes in an interference micrograph represent magnetic lines of force, as viewed from the direction of an electron beam. In addition, a simple calculation shows that a constant magnetic flux of h / e ( = 4.1 x 10- l 5 Wb, where h is Planck‘s constant and e is the electron charge) flows between two adjacent contour lines. The amazing point here is that this relation holds for any particle beam having the electron charge e, whatever velocity it may have. All results obtained using “classical” methodologies also hold when quantum-mechanical treatment is applied. An example is shown in Fig. 24. The object is a fine cobalt particle. No contrast is observed inside the reconstructed image (Fig. 24a). On the other hand, contour lines appear in the interference micrograph (Fig. 24b). The contour lines follow in-plane magnetic lines of force, because the particle is uniform in thickness. They clearly show how magnetization rotates in such a fine particle. However, this contour map cannot help clarify whether the magnetization is clockwise or counter-clockwise. At this juncture, the direction can be decided as being clockwise from the interferogram (Fig. 24c); that is, the wavefront is either advanced or retarded depending on whether the magnetization direction is clockwise or counter-clockwise. The next example is a cross-tie wall in a permalloy thin film (TONOMURA, MATSUDA, TANABE, OSAKABE, ENDO,FUKUHARA, SHINAGAWA and FUJIW A R A [ 1982a1). The wall described here is a kind of 180” domain wall, but has a complex structure as shown in Fig. 25a. Although confirmations regarding
Fig. 24. Interference micrographs of plate-shaped Co particle: (a) reconstructed image; (b) contour map (amplification: x 2); (c) interferogram (amplification: x 2).
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Fig. 25. Cross-tie wall in permalloy film: (a) predicted domain wall structure; (b) interference micrograph.
this structure have already been made using several techniques, an interference micrograph (Fig. 25b) directly shows the predicted magnetic lines of force. Applications of interference microscopy are not restricted to observation of the magnetization distribution in thin films. Observation of spatially distributed magnetic fields (MATSUDA, TONOMURA, SUZUKI,ENDO,OSAKABE,UMEZAKI,TANABE,SUGITAand FUJIWARA [ 19821) is also possible since the magnetic flux flow can be observed using this method. An example is shown in Fig. 26. The object here is the propagation circuit of a “bubble” memory device. Magnetic fields from tiny horseshoe magnets propagate magnetic bubble domains situated below this circuit as a result of rotation of an applied in-plane magnetic field. The distribution is very important for accurate and smooth propagation of the magnetic bubbles. Magnetic lines of force originating from one end of a magnet are directed either to the other end of the same magnet, or to the end of the adjacent magnet. Magnetic-field strength can also be estimated everywhere throughout the micrograph, because a constant magnetic flux of h/2e is contained within two adjacent contour lines. Moreover, interference microscopy has been employed in practical applications concerning magnetic recording. Recorded magnetization patterns were
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ELECTRON HOLOGRAPHY
[IV,§ 4
Fig. 26. Interference micrograph of microscopic magnetic fields emitted from tiny horseshoe magnets in a bubble memory.
observed in detail as interference micrographs, and a recording of 0.15 pm has been confirmed (OSAKABE, YOSHIDA, HORIUCHI, MATSUDA,TANABE,OKUWAKI, ENDO, FUJIWARA and TONOMURA[ 19831, YOSHIDA,OKUWAKI, and OSAKABE, TANABE,HORIUCHI, MATSUDA,SHINAGAWA, TONOMURA FUJIWARA [ 19831). Magnetic lines of force in a permalloy thin fdm were also observed by MATTEUCCI, MISSIROLI and POZZI[ 19821, who used a crystalline thin film as an amplitude beam splitter. Since coherence conditions for the illuminating electron beam are less stringent in this case, electron holograms could be formed even with a thermionic electron beam from a pointed cathode. Up to this point, interference electron micrographs could be completely understood only in reference to the relevant electromagnetic fields. However, there are some cases where a consistent interpretation is not possible. One example is a ferromagnetic particle which has a rotational magnetization distribution, as in Fig. 24. Now a small hole is bored at the center of such a particle (Fig. 27). The wavefront transmitted through a particle without a hole is a circular cone, as shown in Fig. 27a. But how about a particle with a hole? Since there are no magnetic fields outside the particle, no forces should act on electron beams passing through space inside the hole and outside the particle. It is difficult to believe that the electron beams are physically influenced. Wavefronts in the two regions are likely to proceed similarly, but if they do,
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Particle
(a)
(b)
Fig. 27. Electron wavefronts transmitted through ferromagnetic particles: (a) particle without hole; (b) particle with hole.
discontinuity should result. If, on the contrary, wavefront continuity is adhered to, we have to accept that an unnoticed difference is produced in the wavefront level between the two regions. In fact, the latter is the right answer. Vector potentials exist even in regions without magnetic fields, and they interact directly with electron beams to shift the wavefront. When the observation principle was explained in Fig. 23, it was assumed that electron trajectories were deflected by electro-magneticfields and that the resultant wavefront was then determined as a surface perpendicular to the trajectories. This, however, is not really exact. In order to investigate the behavior of an electron wave, we have to rely on the Schrbdinger equation, which is not expressed through field strengths, but through potentials. An electric or a magnetic field is defined as a force acting on a charged particle. But when an electron is regarded as a wave, the concept of force has no direct meaning. Instead, potentials are introduced as a fundamental physical quantity which produces a change in the electron phase. Actually, the refractive index n can be expressed as
where e, m and t are the electron charge, mass and the unit vector of the electron trajectory, respectively. The refractive index depends on the direction of the electron trajectory. Moreover, there is an ambiguity regarding the vector potential A, due to the gauge transformation A -,A + grad x ( x : arbitrary function). A distribution of A for the ferromagnetic particle with a hole is shown in Fig. 28. Vector potentials flow up from the hole like a fountain, and then sink
216
“V, § 4
ELECTRON HOLOGRAPHY
-A
Phase Difference
1 Fig. 28. Vector potentials near ferromagnetic particle with hole (A: vector potential, B: magnetic field).
back to its bottom. Wavefronts inside the hole and outside the particle are displaced upward and downward, respectively. This strange phenomenon was theoretically predicted by AHARONOV and BOHM [1959], and is called the “Aharonov-Bohm effect”, or simply “AB effect” in their honor. This effect was experimentally confirmed just after its prediction, by CHAMBERS [ 19601, FOWLER, MARTON,SIMPSONand SUDDETH [ 19611, BOERSCH, HAMISCH, GROHMANN and WOHLLEBEN [ 19611 and MULLENSTEDT and BAYH [1962]. The significance of this effect has not, however, been fully recognized until recently, when the theory of gauge fields (YANGand MILLS[ 19541) was rediscovered as a basic principle of physics. It was stressed by WU and YANG[ 19751 that a vector potential is the simplest example of a gauge field, and that the AB effect is only direct experimental evidence that electromagnetism follows the theory of gauge fields. The AB effect has often been discussed since its prediction; in fact, its existence was denied by BOCCHIERIand LOINGER[ 19791. They asserted that the AB effect is purely of mathematical origin, and moreover questioned past experiments claiming that the electron beams passing through field-free regions had to be affected by the fringing magnetic fields from the finite solenoids or whiskers used in the experiments (ROY[ 19801).
IV,§ 51
217
CONCLUSION
~
- - ~
Fig. 29. lnterferogram showing phase distribution of electron beam passing through toroidal ferromagnet.
Interference electron microscopy was effectively applied to this problem, and the ambiguity concerning leakage effects was removed in research performed by TONOMURA, MATSUDA,SUZUKI,FUKUHARA, OSAKABE,UMEZAKI, ENDO, SHINAGAWA, SUGITAand FUJIWARA [ 1982bI. Instead of a solenoid or a whisker, a toroidal magnet (Fig. 28) was used whereby a magnetic flux circuit was configured. The existence of the AB effect was tested by measuring the phase difference between two electron beams passing through the two field-free space inside the hole and outside the particle. The phase distribution of an electron beam passing through the toroid is shown in Fig. 29. Nonexistence of the AB effect would mean that any fringe would follow a straight line inside the hole and outside the particle. The photograph reveals that a phase difference really exists between these two regions where there are no magnetic fields. Leakage fields from the toroid were quantitatively measured from the interference micrograph, and confirmed to be too small to influence the conclusion.
6 5. Conclusion Although the invention of electron holography dates back to 1949, it is only quite recently that applications of the technology have become of practical use due to the advent of a coherent field-emission electron beam.
218
ELEmRON HOLOGRAPHY
IVI
The original objective of electron holography, i.e. improvement of electron microscope resolution, has now been confirmed. The current outlook is that it will be possible to attain an ultra-high resolution of less than 1 8, by correcting aberrations. Furthermore, new applications have been developed which Gabor did not anticipate 35 years ago. One example is interference electron microscopy. In the field of optics, phase information has been effectively employed in various kinds of microscopy and measurement systems. However, electron phase information has rarely been utilized, other than in a few very interesting results with an electron biprism. (See MISSIROLI,POZZI and VALDRE[ 19811 and HIBI and YADA[ 19761.) Holographic interference microscopy displays phase distributions as contour fringes in an electron micrograph. This has opened up new practical applications, such as the observation of microscopic magnetic lines of force. Holographic techniques can be effectively used to detect very small phase shifts. Up to now, advances have been made to the extent that a phase shift of $ A can be detected in an ideal case. If the detection limit is improved to less than &A, which is approximately equal to the electron phase shift caused by a single atom, electron holography will become a powerful means to observe and measure microscopic objects in atomic dimensions. One concrete example of future possibilities lies in biological research. Biological specimens always need staining, since they are transparent and show no contrast in an electron micrograph. Here, electron phase information can be implemented for observing the structure of a biological specimen just as in optical phase-contrast microscopy, which was ingeniously demonstrated by UNWIN[ 19721. Heavy atoms for staining will then become only an artificial nuisance in the case of atomic-scale microscopy. Three-dimensional observation, a special feature of holography, will also be helpful, since all living molecules have three-dimensional structures, such as a spiral. Of course, many obstacles have yet to be solved before such a methodology is firmly established. One problem is radiation damage to a specimen. However, once such barriers have been overcome, the microscopic world will open up before our eyes, through the help of holographic techniques.
References AHARONOV,Y., and D. BOHM, 1959, Phys. Rev. 115,485. BARTELL,L. S., 1975, Optik 43, 373, 403. BOCCHIERI, P.,and A. LOINGER, 1979, Nuovo Cimento A51, 1.
1VI
REFERENCES
219
BOERSCH,H., 1954, Z. Physik 139, 115. BOERSCH,H., H. HAMISCH, K. GROHMANN and D. WOHLLEBEN, 1961, Z. Physik 165.79. BRYNGDAHL,O., 1969, J. Opt. SOC.Am. 59, 142. CHAMBERS, R. G., 1960, Phys. Rev. Lett. 5, 3. COHEN,M. S., 1967, J. Appl. Phys. 38, 4966. CREWE,A. V., D. N. EGGENBERGER, D. N. WALLand L. N. WELTER,1968, Rev. Sci. Instrum. 39, 576. A. V.. J. WALLand J. LANGMORE, 1970, Science 168, 1338. CREWE, ENDO,J., T. MATSUDAand A. TONOMURA, 1979, Jpn J. Appl. Phys. 18,2291. FOWLER, H. A,, L. MARTON,J. A. SIMPSONand J. A. SUDDETH,1961, J. Appl. Phys. 32, 1153. GABOR,D.,1949, Proc. Roy SOC.London A197,454. GABOR,D., 1951, Proc. Phys. Soc. 864,449. HAINE,M. E., and J. DYSON,1950, Nature 166, 315. HAINE,M. E., and T. MULVEY,1952, J. Opt. SOC.Am. 42, 763. HANSZEN, K.-J., 1982, Holography in electron microscopy, in: Advances in Electronics and Electron Physics, VoI. 59, ed. C. Marton (Academic Press, New York) p. I. HANSZEN, K.-J., 1983, Phys. BI. 39, 283. HASHIMOTO, H., A. KUMAOand H. ENDO, 1978, Single atoms in molecules and crystals observed by transmission electron microscopy, in: Proc. Int. Conf. on Electron Microscopy, Toronto, 1978, Vol. 3, ed. J.M. Sturgess (Microscopy Society of Canada, Toronto) p. 244. HAWKES,P. W., 1978, Coherence in electron optics, in: Advances in Optical and Electron Microscopy, Vol. 7, eds V. E. Cosslett and R. Barer (Academic Press, London) p. 101. HIBI,T., 1956, J. Electron Microsc. 18, 266. HIBI,T. and K. YADA, 1976, Electron interference microscope, in: Principles and Techniques of Electron Microscopy, Vol. 6, ed. M.A. Hayat (Van Nostrand, Amsterdam) p. 312. LAUER,R., 1984, Optik 66, 159. LEITH,E. N., and J. UPATNIEKS, 1962, J. Opt. SOC.Am. 52, 1123. LENZ,F., and G. WOHLAND,1984, Optik 67, 315. LICHTE,H., 1984, High resolution imaging by off-axis electron holography, in: Proc. Eur. Congr. on Electron Microscopy, Budapest, 1984, Vol. I , eds A. Csanldy, P. Rohlich and D. Szab6 (Program Committee, Budapest) p. 282. MATSUDA, K., C. H. FREUNDand P. HARIHARAN, 1979, Appl. Opt. 20,2763. MATSUDA, T., A. TONOMURA, R. SUZUKI,J. ENDO,N. OSAKABE,H. UMEZAKI,H. TANABE, Y. SUGITA and H. FUJIWARA, 1982, J. Appl. Phys. 53, 5444. MATSUMOTO, K., and M. TAKASHIMA, 1970, J. Opt. Soc. Am. 60,30. MATTEUCCI, G., G. F. MISSIROLI and G. POZZI,1982, Ultramicroscopy 8,403. MISSIROLI, G. F., G. POZZIand U. VALDRB, 1981, J. Phys. E14, 649. M~LLENSTEDT, G., and W. BAYH,1962, Phys. BI. 18, 299. M~LLENSTEDT, G., and H. DOCKER,1956, Z. Physik 145,377. M~LLENSTEDT, G., and H. WAHL,1968, Natunvissenschaften 55, 340. M~LLENSTEDT, G., and G. WOHLAND,1980, Direct interferometric measurement of the coherence length ofan electron wave packet using a Wien filter, in: Proc. Eur. Congr. on Electron Microscopy, The Hague, 1980, Vol. I , eds P. Brederoo and J. Van Landuyt (7th Eur. Cong. on Electron Microscopy Foundation, Leiden) p. 28. MUNCH,J., 1975, Optik 43,79. OLIVEI, A,, 1971, Optik 33. 93. OSAKABE, N., K. YOSHIDA, Y.HORIUCHI,T. MATSUDA,H. TANABE,T. OKUWAKI, J. ENDO, 1983, Appl. Phys. Lett. 42,746. H. FUJIWARA and A. TONOMURA, Pozzr, G., and G. F. MISSIROLI,1973. J. Microsc. (Paris)18, 103. ROGERS,J., 1980, Electron holography, in: Imaging Processes and Coherence in Physics, eds M.
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Schlenker, M. Fink, J. P. Goedgebuer, C. Malgrange, J. Ch. V16not and R. H. Wade (Springer, Berlin) p. 365. ROY,S. M., 1980, Phys. Rev. Lett. 44, 111. SAXON,G., 1972, Optik 35, 195, 359. SCHMID,H., 1984, Coherence length measurement by producing extremely high phase shifts, in: Proc. Eur. Congr. on Electron Microscopy, Budapest, 1984 Vol. I , eds A. Csanhdy, P. RBhlich and D. Szab6 (Program Committee, Budapest) p. 285. STROKE,G. W., and M. HALIOUA,1972, Optik 35, 50. THOMPSON, B. J., 1965, Jpn J. Appl. Phys. 4, Suppl. 1, 302. TOMITA,H., T. MATSUDAand T. KOMODA,1972, Jpn J. Appl. Phys. 11, 143. TONOMURA, A., 1969, J. Electron Microsc. 18. 77. TONOMURA, A., 1972, Jpn J. Appl. Phys. 11,493. H.WATANABE and T.KOMODA,1968, Jpn J. Appl. Phys. 7,295. TONOMURA, A., A. FUKUHARA, TONOMURA, A, T. MATSUDAand T. KOMODA,1978, Jpn J. Appl. Phys. 17, 1137. TONOMURA, A,, T. MATSUDAand J. ENDO, 1979a, Jpn J. Appl. Phys. 18,9. TONOMURA A,, T. MATSUDAand J. ENDO, 1979b, Jpn J. Appl. Phys. 18, 1373. TONOMURA, A., T. MATSUDA, J. ENDO,T.ARIIand K.MIHAMA,1980, Phys. Rev. Lett. 44,1430. K. TONOMURA, A., T. MATSUDA,H. TANABE,N. OSAKABE,J. ENDO, A. FUKUHARA, SHINAGAWA and H. FUJIWARA, 1982a, Phys. Rev. B25, 6799. TONOMURA, A.,T. MATSUDA, R. SUZUKI, A. FUKUHARA, N. OSAKABE, H. UMEZAKI,J. ENDO, K. SHINAGAWA, Y. SUGITAand H. FUJIWARA, 1982b, Phys. Rev. Lett. 48, 1443. 1985, Phys. Rev. Lett. TONOMURA, A,, T. MATSUDA, T. KAWASAKI, J. ENDOand N. OSAKABE, 54,60. UNWIN,P. N. T., 1972, Proc. Roy. SOC.London A329, 327. E. SUITO,Y. HARADAand M. WATANABE, 1972, J. Appl. Phys. 46, UYEDA,N., T. KOBAYASHI, 5181.
VENEKLASEN,L. H., 1975, Optik 44.447. WADE,R. H., 1980, Holographic methods in electron microscopy, in: Computer Processing of Electron Microscope Images, ed. P. W. Hawkes (Springer, Berlin) p. 223. WAHL,H., and B. LAU, 1979, Optik 54, 27. I., W. MIRANDB and E. MENZEL,1969, Optik 30,318. WEINGARTNER, Wu, T. T., and C. N. YANG,1975, Phys. Rev. D12,3845. YADA, K., K. SHIBATA and T. HIBI, 1973. J. Electron Micros. 22, 223. YANG, C. N., and R. MILLS,1954, Phys. Rev. 96, 191. YOSHIDA, K., T. OKUWAKI, N. OSAKABE,H. TANABE,Y. HORIUCHI.T. MATSUDA,K. 1983, IEEE Trans. Magn. MAG-19, 1600. SHINAGAWA, A. TONOMURA and H. FUJIWARA, ZEITLER, E., 1979, Electron holography, in: Proc. 37th EMSA Meeting, San Antonio, 1979, ed. G.W. Bailey (Claitor, Baton Rouge, LA) p. 376.
E. WOLF, PROGRESS IN OPTICS XXIII 0 ELSEVIER SCIENCE PUBLISHERS B.V., 1986
V
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT BY
F. T. S . Yu Electrical Engineering Department The Pennsylvania State University University Park, PA 16802. USA
CONTENTS PAGE
Q 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . .
223
Q 2. PARTIALLY COHERENT PROCESSING . . . . . . . . . 224 Q 3 . COHERENCE PROPAGATION
. . . . . . . . . . Q 4 . TRANSFER FUNCTION . . . . . . . . . . . . . . Q 5 . NOISE PERFORMANCE . . . . . . . . . . . . . Q 6 . SOURCE ENCODING AND IMAGE SAMPLING . .
. . . .
. . . .
. . . .
233 245
251 267
Q 7 . CONCLUDING REMARKS . . . . . . . . . . . . . . . 273
. . . . . . . . . . . . . . . . . . 273 . . . . . . . . . . . . . . . . . . . . . . . 274
ACKNOWLEDGEMENTS REFERENCES
0 1. Introduction Advances in quantum electronics have brought into use the infrared and the visible range of electromagnetic waves. The investigation of intense coherent light sources has permitted us to build more efficient optical systems for communication and signal processing. However, coherent optical systems are plagued with coherent artifact noise, which frequently limits their processing capability. As was noted by the late D. GABOR[1970], coherent noise is regarded as the number one enemy of modern optical processing. In addition to coherent noise within the systems, the coherent source is generally considered expensive, and the requirements for a processing environment can be very stringent, for example a dust-free environment and a heavy optical bench are two main requisites. Most of the optical processing systems to date have confined themselves to the cases of complete coherence or complete incoherence. However, a continuous transition between these two extremes is possible, in principle, and one then speaks of partially coherent processing (PCP). Historically, the earliest investigation of the property of partial coherence is due to VERDET [1865]. However, it was MICHELSON’S work [1890] that brought out the relationship between the visibility and the intensity distribution of an extended source. Nevertheless, the works of VAN CITTERT [ 19341 and ZERNIKE [ 19381 are considered the most significant developments in partial coherence theory. They were able to determine the degree of coherence at any two points on a screen illuminated by an extended source. Their method was further simplifiedby HOPKINS [ 1951,19531in connection with image formation in an optical system, but it was WOLF’Smutual coherence function [ 19551that set the foundation for the analytical theory of partial coherence. More specifically, the properties of the mutual coherence function are found to be adequate for determining the performance of any optical system, with diffraction effects taken into account. Mention must be made also of the work of T H ~ M P and~ ~ N WOLF [ 19571 and THOMPSON [ 19581, who demonstrated a two-beam interference technique to measure the degree of partial coherence. They have shown that, under quasi-monochromatic illumination, the degree of spatial coherence is dependent on the source size and on the distance between two arbitrary
224
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, $ 2
points. The degree of temporal coherence is, however, dependent on the spectral bandwidth of the light source. They have also illustrated several coherence measurements that are consistent with the Van Cittert-Zernike predictions. Communication theory, on the other hand, was originated by a group of mathematically oriented electrical engineers. However, the application of the communication aspect in optics has never been totally disregarded. The earliest application of communication theory to optical systems is due to GABOR [ 19461, and the concept was further developed by ELIAS,GREYand ROBINSON [ 19521, and ELIAS[ 19531. The very first application of communication aspects to optical information processing are, however, due to O’NEILL[ 19561 and to CUTRONA, LEITH,PALERMO and PORCELLO [ 19601. Since then, the application of communication theory to optical signal processing has reached a new plateau with its successful application to synthetic-aperture radar of CUTRONA, and VIVIAN[ 19661. The inventions of the complex spatial LEITH,PORCELLO filter by VANDERLUGT [ 19641, and the computer generated hologram by BROWNand LOHMANN [1966] are also important cornerstones of modern optical processing. The recent development of real-time spatial light modulators (GRINBERG, JACOBSON, BLEHA,MILLER,FRAAS,BOSWELLand MYER [ 19751, WARDE,FISHER, Cocco and BURMAWI [ 19781, Ross, PSALTIS and ANDERSON [ 19831) and partially coherent processing techniques (ROGERS [ 19771 and Yu [ 1983a1)have brought optical processing to a new height, and much attention has been focused on high-speed high-data-rate optical computing.
0 2. Partially Coherent Processing Although coherent optical processors can perform a variety of complex operations, coherent processing systems are usually plagued with coherent artifact noise. These difficulties have prompted us to look at optical processing from a new standpoint, and to consider whether it is necessary for all optical processing operations to be carried out by pure coherent sources. We have found that many types of optical processing can be carried out by partially coherent sources or white-light sources (LOHMANN[ 19771, RHODES[ 19771, LEITHand ROTH[ 19771, Yu [ 1978a1, STONER[ 19781, MORRIS and GEORGE [ 1980al). The basic advantages of partially coherent processing are: (1) it can suppress the coherent artifact noise, (2) partially coherent sources are usually inexpensive, (3) the processing environment is generally very relaxed, (4) par-
v. § 21
PARTIALLY COHERENT PROCESSING
225
tially-coherent systems are relatively easy and economical to operate, and ( 5 ) partially-coherent processors are particularly suitable for color image processing.
2.1. HISTORICAL BACKGROUND
The optical system in the partially coherent regime has been studied by BECHERERand PARRENT[1967], SWINGand CLAY[1967], THOMPSON [ 19691, and DUTTAand GOODMAN [ 19771. They have shown that there are difficulties in applying the linear system concept to the evaluation of the performance at high spatial frequencies. These difficulties are primarily due to the inapplicability of the linear system theory under partially coherent illumination. However, with strictly coherent illumination (i.e., spatially and temporally coherent), the optical system operates in a complex wave field. The output light field can be described by a complex amplitude convolution integral,
J J -c€
which yields a spatially invariant property of the complex amplitude transformation, where f(x, y ) is a coherent complex wave field that impinges at the input end of an optical system, and h ( x , y ) describes the spatial impulse response of the optical system. Equation (2.1) can be written in the following Fourier transform :
w,4 )
=
F(p7 4 ) H(p9
4 1 9
(2.2)
where G ( p , q), H ( p , q) and F ( p , q) are the Fourier transforms ofg(x, y), h(x, y ) and f ( x , y), respectively. Thus the effect of eq. (2.2) describes the operation of the optical system in the spatial frequency domain, where H ( p , q) can be referred to as the coherent or complex amplitude transfer function of the optical system. On the other hand, with strictly incoherent illumination, the system operates in intensity, such that the output irradiance is described by the following intensity convolution integral:
226
[v, 8 2
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
which yields a spatially invariant form of the intensity operation, where Zi(x,y ) = I f ( x , y ) I is the input irradiance and hi(x, y) = 1 h(x, y) I is the intensity spatial impulse response of the optical system. Similarly, eq. (2.3) can be written in Fourier transform form:
’
I(P, 4 ) = pi(P, 4 ) Hi(P, 41,
(2.4)
where Z(p, q), Fi(p, q ) and Hi(p, q ) are the Fourier transforms of Z(u, /I), f,(x,y), and hi(x, y), respectively, and Hi(p, q ) can be referred to as the incoherent or intensity transfer function of the optical system.
These normalized quantities should give a set of convenient mathematical forms and also provide a concept of image contrast interpretation. Since the quality of a visual image, to a large extent, depends on the contrast (or the relative irradiance) of the image, a normalized intensity function would certainly enhance the information bearing capacity. With the application of the Fourier convolution theorem to eq. (2.3), the following relationship can be written:
((P, 4) = F,(P, 4 ) fiii(P, 4 1 3
(2.8)
"3
§ 21
221
PARTIALLY COHERENT PROCESSING
where I?,(p,q ) is commonly referred to as the optical transfer function (OTF) of the optical system, and the modulus of 1 A( p, q) 1 is known as the modulation transfer function (MTF) of the optical system. We further note that eq. (2.7) can also be written as
sj
W p ' , q' ) H * ( P '
i?b,4) =
- P,q' - 4) dp'
--ic
(2.9)
j] -
IH(p',q')l2dp' dq'
U'
Upon changing the variables p" = p' - fp,q" following symmetrical form:
fl
H ( p " + i p , q"
Hi(p, 4 ) =
dq'
= q'
- $4, eq. (2.9) results in the
+ $ q ) H*(p" - i p , q"
- i q ) dp" dq"
-m
m
(2.10)
We note that the definition of OTF is valid for any linearly spatial invariant optical system regardless of whether the system is with or without aberrations. Furthermore, eq. (2.10) serves as the primary link between the strictly coherent and incoherent systems. There are, however, threefold limitations associated with the strictly coherent processing. First, coherent processing requires the dynamic range of a spatial light modulator to be about 100000 : 1 for the input wave intensity, or a photographic density range of 4.0. Such a dynamic range is often quite difficult to achieve in practice. Second, coherent processing systems are susceptible to coherent artifact noise, which frequently degrades the image quality. Third, in coherent processing, the signal being processed is carried out by a complex wave field. However, what is actually measured at the output plane of the optical processor is the output wave intensity. The loss uf the output phase distribution may seriously limit the applicability of the optical system in some applications. To overcome the drawbacks of coherent processing, it is useful to reduce either the temporal coherence or the spatial coherence, and there are several techniques available for this purpose. We will briefly discuss four frequently used methods. The first is based on a spatially partially coherent source, and
228
PRINCIPLES OF OFTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, 5 2
the other three methods utilize a broad spectral band light source (e.g., a white-light source) to perform complex wave field operations. 2.1.1. Spatially partially coherent processing Techniques of utilizing spatially partially coherent light to perform complex data processing have been proposed by LOHMANN[ 19771, RHODES[ 19771, and subsequently by STONER [ 19781. These techniques share a basic concept: The optical system is characterized by use of the point spreadfunction (PSF), which is not constrained to the class of non-negative real functions used in conventional incoherent processing. The output intensity distribution can be adjusted by changing the PSF, that is I(x’, Y’ ) =
ss
Wx, Y ) W x ‘ ,Y’ ; x, Y ) d x dy,
(2.11)
where h(x, y ; x ‘ , y ’ ) is the PSF, and y ’ ) and O(x, y ) are the image and object intensity distributions, respectively. If the pupil function of an optical system is P(a, fl), the PSF is equal to the square of the Fourier spectrum of the pupil function. Therefore, the output intensity distribution can be adjusted by selecting an adequate pupil function. A typical processing system, as proposed by RHODES, is shown in Fig. 2.1. This optical processing system, which is characterized by an extended pupil region, has two input pupil functions P,(a, j)and P2(cc,p). With the optical path lengths ofthe two arms ofthe system being equal, the overall system pupil function is given by the sum of P,(cr, 8) and P2(x, fl). If the path length in one arm is changed slightly, however (e.g. by ](XI,
I(x,y) Fig. 1. Two-pupil incoherent processing system.
v. § 21
229
PARTIALLY COHERENT PROCESSING
moving mirror M, a small distance), a phase factor is introduced in one of the component pupil functions, with the result P ( S 8) = PI(%P) + P,(a,
P ) exp[i$(a, PI].
(2.12)
It is evident that if the pupil transparencies P,(a, b) and P,(a, 8) are recorded holographically, arbitrary PSF's can then be synthesized using the 0" -180" phase switching operation. Thus, we see that this optical system is capable of performing complex data processing with spatially partially coherent light.
2.1.2. Achromatic optical processing An achromatic optical signal processing technique with broad-band source (e.g., white light) was proposed by LEITHand ROTH[ 19771 and LEITH[ 19801, as shown in Fig. 2. Lens Lo collimates the white-light point source S, which illuminates hologram R. The diffraction wavefront is spatially filtered by the aperture A, which removes the zero-order term and the lower sideband. The upper sideband is imaged onto the signal plane P; lens L, Fourier transforms the resulting wavefront; and observation in the transform plane is confined to the optical axis by a slit in the output planeO. The convolution of the demodulated input signal with the desired reference function is recorded by synchronously translating the signal and output films. Such a system, termed an achromatic system, thus has the flexibility of a coherent optical signal processor along with the potential for the noise immunity of an incoherent system. 2.1.3. Band-limited partially coherent processing A technique using a matched filter for operation with band-limited illumination was proposed by MORRISand GEORGE [ 1980a1, as shown in Fig. 3. The matched filter consists of a frequency-plane holographic filter, an achromatic-fringe interferometer, a color-compensating grating and an achro-
R A P Fig. 2. Achromatic optical processing system.
0
230
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
k-f + f
+
f+
f +f+
f +f+f
[v, $ 2
-I
Fig. 3. Band-limited partially coherent processing system.
matic doublet. The matched filter, a Fourier hologram, is made by recording the object spectrum and a collimated reference beam with an exposure wavelength &. The reference-beam angle is at 0, with respect to the object beam axis. In the correlation operation the object is illuminated using a broad-spectral source. The various spectral components are dispersed in angle due to the grating-like structure of the matched filter. A lateral dispersed correlation signal is generated in planec which is the conventional correlation plane. These spectral components are recombined by imaging the matched filter into plane d, where a compensation grating produces a color dispersion that is equal but opposite to that introduced at plane b. The color-corrected Correlation signal is observed in the output plane. However, the compensated signal is still wavelength-dependent, since the illumination wavelength changes the scale of the object spectrum at plane b. Insertion of a slit in plane c provides a convenient way to band-limit the correlation signal to AA. Band-limiting improves the signal-to-noise ratio (SNR) of the compensated correlation output. Broadband illumination can be used for automatic scale search and object size determination. Since spatially incoherent light as well as coherent light can be used to perform matched filtering, an extended white-light source is used together with a slit at planec to provide a band-limited partially coherent processing. 2.1.4. Achromatic partially coherent processing
An achromatic partially coherent optical processing technique with a whitelight source was introduced by Yu [ 1978a,b]. Figure 4 illustrates the processing system, where all the transform lenses are assumed achromatic. A high diffraction efficiency phase grating, with an angular spatial frequency po, is used at the input plane to disperse the input object spectrum into a rainbow
23 I
PARTIALLY COHERENT PROCESSING
source
'ro
Fig. 4. Achromatic partially coherent optical processing system.
color spectrum in the Fourier plane p,. Thus it permits a stripwise design of a complex spatial filter for each narrow spectral band in the Fourier plane. The achromatic output image irradiance can therefore be observed at the output plane P,. The advantages of this technique are that each channel (e.g., each spectral band filter) behaves as a partially coherent channel, while the overall output noise performance is primarily due to the incoherent addition of each channel, which behaves as if under incoherent illumination, Thus, this partially coherent system has the capability for coherent noise suppression.
2.2. WHITE-LIGHT PROCESSING
We now describe an achromatic partially coherent processing technique that can be carried out by a white-light source, as shown in Fig. 4. The partially coherent processing system is similar to a coherent processing system, except for the following: It uses an extended white-light source, a source encoding mask, a signal sampling grating, multispectral band filters, and achromatic transform lenses. For example, if we place an input object transparency s(x, y ) in contact with an image sampling phase grating, the complex wave field, for every wavelength A, at the Fourier plane P, would be (assuming a white-light point source) E ( P ,4 ; 4
=
{{
s(x, Y ) exp(ipox) exp [ - ~ ( P X + 4~11d x dy
=
S ( P - p0,41, (2.13)
where the integral is over the spatial domain of the input plane P,, (p, q ) denotes the angular spatial frequency coordinate system, po is the angular spatial frequency of the sampling phase grating, and S ( p , q ) is the Fourier
232
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,5 2
spectrum of s(x, y). If we write eq. (2.13) in the form of a spatial coordinate system (a, fl), we have (2.14)
where p = (2x/Af)a, 4 = ( 2 n / I f ) B ,and f is the focal length of the achromatic transform lens. Thus, we see that the Fourier spectra would disperse into rainbow colors along the a-axis, and each Fourier spectrum for a given wavelength I is centered at a = ( 1 f / 2 x ) p 0 . In complex signal filtering, we assume that a set of narrow spectral band complex spatial filters is available. In practice, all the input objects are spatial frequency limited; the spatial bandwidth of each spectral band filter H ( p , , 4,) is therefore (2.15)
where p , = ( 2 n/l,f)a, q, = ( 2 n/I,f)fl, 1, is the main wavelength of the filter, a, = ( I , f / 2 n) ( p o + Ap) and a2 = ( I , f / 2 n) ( p o - Ap) are the upper and lower spatial limits of H ( p , , q,), and Ap is the spatial bandwidth of the input image Y). Since the limiting wavelengths of each H ( p,, 4), are
4x9
Ad = I,,
~
” ,
Po - AP
and Ih = I,, Po - AP Po + AP ’ ~
(2.16)
its spectral bandwidth can be approximated by (2.17)
If we place this set of spectral band filters side-by-side and position them properly over the smeared Fourier spectra, the intensity distribution of the output light field can be shown to be (2.18)
where h ( x , y ; &) is the spatial impulse response of H ( p , , q,) and * denotes the convolution operation. Thus, the proposed partially coherent processor is capable of processing the signal in a complex wave field. Since the output
v, 8 31
233
COHERENCE PROPAGATION
intensity is the sum of the mutually incoherent narrow-band spectral irradiances, the disturbhg coherent artifact noise can therefore be immuned. It is also apparent that the white-light source emits all the visible wavelengths; the partially coherent processor is very suitable for color image processing.
8 3. Coherence Propagation In investigating the behavior of a partially coherent processor (PCP), it is necessary to establish a transformational relationship for the mutual intensity function which determines the degree of coherence under a partially coherent regime. We use WOLF’Stheory [ 19551 to develop a transformational formula for the mutual intensity function propagating through an ideal thin lens. We then apply this transformational formula to derive an operational formula for a PCP. 3.1. PROPAGATION O F THE MUTUAL INTENSITY FUNCTION
One of the most remarkable and useful properties of a converging lens is its inherent ability to perform a two-dimensional Fourier transformation of a complex wave field. Under partially coherent illumination, a thin lens takes a four-dimensional Fourier transformation of the mutual intensity function. Evidently, when a monochromatic plane wave passes through a thin lens, phase delay transformation takes place, characterized by
r(5,rl) = exp(ikrlA,) exp[ - ik(t2 + r12)/2fl.
(3.1)
Here rl is the refractive index of the thin lens, f is the focal length, A. is the maximum thickness of the lens, k = 2 n/A, A is the wavelength, and (5, q) is the spatial coordinate system of the thin lens.
do+--
d,
-4
Fig. 5. Partially coherent optical system.
234
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,$ 3
Let us now consider an object transparency, inserted at a distance do in front of the lens and illuminated by a spatially partially coherent light. If the mutual intensity function at the object plane is J , ( x , , y , ;x2, y 2 ) , as depicted in Fig. 5, then the mutual intensity function at the output (a, 8) plane can be shown as
where Cis an appropriate constant. This equation shows that the output mutual intensity function can be obtained by a four-dimensional integral equation. Up to this point we have disregarded the finite extent of the lens aperture. This approximation is accurate, if the distance do is sufficiently small to place the input transparency deep within the region of Fresnel diffraction with respect to the lens aperture. This condition is satisfied in the vast majority of problems of interest, particularly for optical processing. There are, however, two special cases worth mentioning: (1) For d, = f the output plane is located at the back focal plane. Equation (3.2) reduces to
235
COHERENCE PROPAGATION
Thus, except for a quadratic phase factor, the output mutial intensity function is essentially the Fourier transform of the input mutual intensity function. (2) For do = d , = f both the input plane and the output plane are located at the front and back focal length of the lens. The quadratic phase factor vanishes and eq. (3.3) reduces to
-m
(3.4) which is exactly a four-dimensional Fourier transformation, between the input and output mutual intensity functions. In partially coherent processing, an extended incoherent source is usually used at the front focal plane of a collimating lens to illuminate the input object transparency. Thus the mutual intensity function at the source plane (xo,yo) can be written as J(x&Y;,;x;,Y;)
=
y(xo,yo),
for x; = x i = xo, y;, = y; otherwise,
= yo,
where y(xo, yo) is the intensity distribution of the light source, and (xo,yo) is the spatial coordinate system of the source plane. If we assume that the input object transparency is located at the back focal plane of the collimator, then the mutual intensity function at the input plane reduces to m
JLXl
-
x,;y, - Y 2 ) =
jj
Y(XO9YO)
- x
which is essentially the Van Cittert-Zernike theorem (VAN CITTERT [ 19341, ZERNIKE [ 19381). It is evident that the mutual intensity function at the input plane is a spatial invariant function.
236
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, 5 3
3.2. GENERAL FORMULATION
With reference to the PCP of Fig. 4,the mutual intensity function at the input plane P I due to the source irradiance y(xo, yo; A) is J(x,,y,; x,,y2;4
=
ss
y(x0,yo;
4
where the integration is over the source plane Po. The mutual intensity function immediately behind the sampling phase grating can be written as
where the superscript * denotes the complex conjugate, and vo is the spatial frequency of the sampling grating. Similarly,the mutual intensity function at the Fourier plane P, can be written as
where the integration is over the source plane Po, and S(a, fl) is the Fourier spectrum of s(x,y). If we assume that a set of narrow spectral band spatial filters H,,(a,B) is inserted at the Fourier plane, the output mutual intensity function is J(x;,Y;;x;,Y;;4
xexp[
-i
271 -
Af
=
J’(x,,y,; x21y2; 4
(x;al+y;flI-x;a2-y;B2)
1
da,dfl,da,dB,.
Thus the corresponding output intensity distribution is
ss
~ , z ( ~ ’ ~ ~ ’ dxodyoy(xo,yo;l) ; ~ ) =
(3.10)
v. I 31
COHERENCE PROPAGATION
for
231
(3.11)
AL,, 5 A 5 Ah,,,
where I,, and Ah, are the lower and upper wavelength limits of H,(a,B). Let us denote S(A)and C(A)as the relative spectral intensity of the light source and the relative spectral response of the output detector, respectively, then the output irradiance resulting from each narrow spectral band filter is 1, + A1J2
In(x’,Y’) =
j jj
d x o d y o d ~Y(xo,yo; A)S(I)C(A)
1, - A1.,/2
f o r n = 1,2,..., N,
(3.12)
where A,, and 81, are the center wavelength and the bandwidth of the nth narrow spectral band filter H,(a,8). It is therefore apparent that the overall output irradiance would be the incoherent addition of I , , ( x ’ , y ’ ) ,that is (3.13) where N is the total number of the spectral band filters. Furthermore, if the image processing is a one-dimensional operation, then a fan-shaped spatial filter can be utilized, thus the overall output irradiance is reduced to the following form:
xexp[
-i
2 R
- (x‘a +y’B)
Af
238
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, $ 3
Now to what degree can the coherence requirements be relaxed without sacrificing the overall output image quality of the processing system? The answer is that the nature of the processing operation governs the degree of temporal and spatial coherence necessary to obtain satisfactory results. We apply the general formulation either of eqs. (3.12) or (3.13) to suit the specific need of the processing operation under consideration. In the following we shall discuss the temporal and spatial coherence requirements for image correlation (Yu, ZHANGand ZHUANG[1983]) as an example. For other processing operations, the coherence requirement can also be deduced by a similar approach. However, such a lengthy illustration is beyond the scope of present interest and is therefore left for the readers to investigate.
3.3. COHERENCE REQUIREMENTS FOR IMAGE CORRELATION
The spatial coherence effect on the correlation intensity for a rectangular object was evaluated by WATRASIEWICZ [ 19691, and the effect due to temporal coherence was recently analyzed by MORRIS and GEORGE[1980b]. Nevertheless, the temporal and spatial coherence requirements for a partially coherent system were recently discussed by ZHUANGand Yu [ 1982bl. 3.3.1. Temporal coherence requirement
To investigate the temporal effect on a partially coherent correlator, we assume a temporal partially coherent source [i.e. y(x,, y o ) = S(x,, y o ) ] . For simplicity, we let S(A) and C(A)be constant. The output intensity distribution of eq. (3.12) reduces to (3.15)
IA,(x’,y’; A)I2 d l ,
I ( x ’ , y ’ )= n= 1
A,
where P
P
x exp[ - i nf 2n
(MX’
1
+ B y ‘ ) dcrdp,
(3.16)
v. 5 31
239
COHERENCE PROPAGATION
where S(a - Afvo,/?) are the smeared Fourier spectra,
H,(a,/?I = S*(a - 3.nfvo, 81,
(3.17)
I,l is the center wavelength of the filter and the superscript * denotes the complex conjugate. We further note that eq. (3.15) can be approximated by A,(x',y';
A) 1: 3. s ( x ' , y ' ) * s ( x ' , y ' ) .
(3.18)
I
To investigate the effect due to temporal coherence, we assume a onedimensional sinusoidal function as the input object, that is
(x>
where qo is the angular spatial frequency of the input sinusoid, rect
{
1, l Y l _I w/2, 0, otherwise,
and W is the spatial extension of the input object. From eq. (3.18), we have
x rect(%){l
where W,
=
WA/A, and qh
=
+cos[qh(q-y')]}dq,
qoI/A,,. Thus at y'
=
(3.20) 0, eq. (3.20) reduces to
(3.22) A&
where AA,
=
\I,,
-
Id,\
240
[v,5 3
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
Figure 6 shows the variation of the normalized correlation peak as a function of the spectral bandwidth, for various spatial frequencies of the object. We see that the normalized correlation peak decreases as the spectral bandwidth A An increases, and it drops rather rapidly as the spatial frequency of the object increases. Figure 7 shows the variation of the normalized correlation peak as a function of the spectral bandwidth A&, for various object extensions. Again, we see that the normalized correlation peak decreases monotonically as AA,, increases, and it decreases very rapidly as the object extension increases. If we use the half-power-point criterion, the spectral bandwidth (i.e., the temporal coherence requirement) of the filter H,,(cr, fl) can be determined from Figs. 6 and 7. Some of the numerical results are tabulated in tables 1 and 2. Thus, for a relatively small and lower spatial frequency object, a broader spectral bandwidth of a spatial filter can be used. In other words, the higher the spatial frequency of the object or the larger the extension, the higher the degree of temporal coherence required. It is apparent that the degree of temporal coherence in the Fourier plane can be increased simply by increasing the sampling frequency vo of the grating. However, when vo is increased, larger
X
H
Y
Q
w
a z
0 ta
-I W
I1I CK 0 0
n w
N 1
Q
r
[L
0
z
0
20
40
60
80
100
SPECTRAL BANDWIDTH A X , ( i ) Fig. 6. Normalized correlation peak as a function of the spectral bandwidth, for various input spatial frequencies.
COHERENCE PROPAGATION
24 I
i .a
0.8
0.6
0.4
0.2
0
SPECTRAL BANDWIDTH
AAn
(i)
Fig. 7. Normalized correlation peak as a function of the spectral bandwidth, for different object extensions.
achromatic transform lenses may be needed. This is the price we pay to improve the temporal coherence. 3.3.2. Spatial coherence requirement We assume that the light source is a one-dimensional extended monochromatic source, that is (3.23) where rect
(E)4 {
1, lvol S A S / 2 , 0, otherwise,
and A S is the size of the extended source. Similarly, the output plane intensity
242
[v, $ 3
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
(3.24)
(3.25) and I, is the wavelength of the monochromatic source. Let us consider a one-dimensional Gaussian object, which is described as s ( y ) = exp ( - a2y2 ) .Then the output intensity distribution would be
(3.26)
TABLE1 Temporal coherence requirement as a function of the spatial frequency and I ,
=
40
W
- , where
2n
=
10 m m
5461 A.
Spatial frequency
5 (lines /mm) 2R
Spectral bandwidth A &(A)
I
3
5
7
10
25
50
100
540
180
99
85
58
22
11
6
TABLE2 Temporal coherence requirement as a function ofthe target extension W ,w h e r e 3 = 25 lines/mm 2n and I , = 5461 A. Spatial extension W(rnn1)
Spectral bandwidth AA,,(A)
I
2
3
5
10
20
30
220
99
15
46
22
I1
8
60
4.1
v. I 31
243
COHERENCE PROPAGATION
W
a a
0 0
n W
N
0
0.4
1.2
0.8
SOURCE
SIZE
I .6
2.0
AS(mm)
Fig. 8. Normalized correlation peak as a function of the source size, for a Gaussian spatial signal.
To determine the output correlation peak, we let y’ AS12
I(0) =
a’
[
exp 0
(-y]
af Lo
dyo.
=
0, that is
(3.27)
It is evident that I(0) depends on the size of the light source. Figure 8 shows the corresponding normalized correlation peak as a function of the source size. The correlation peak drops rather rapidly as the source size increases. We now consider the effect due to spatial frequency of the object. Again, we use a spatially limited sinusoidal grating of eq. (3.19) as the input object. For simplicity, we let qo = 2ny0/(x0f) and g = n/(Aof). At y ’ = 0, eq. (3.25) becomes SinkWYo) sin[WkYo - 24O)l A(O;yo) = gyo gyo - 24, t
sin t w v o + 24011 +- 1 gyo + 40 gyo + 240
244
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,
3
where qo is the angular spatial frequency of the sinusoidal grating, and W is its spatial extension. The output irradiance can be shown as AS12
(3.29) Since g is not related to xo, the spatial coherence requirement is evidently independent of the spatial frequency of the input object. However, the spatial coherence strongly depends on the extension of the object, as can be seen in Fig. 9. The normalized correlation peak monotonically decreases with increasing source size. The rate of the decrease is rapidly increased for larger extension of the object. Again, by using the half-power-point criterion, the
25
0
0.1
0.2
SOURCE SIZE
lines / rnm
0.3
0.4
0.5
AS(rnm)
Fig. 9. Normalized correlation peak as a function of the source size, for different object extensions.
v. I41
245
TRANSFER FUNCTION
spatial coherence requirement for various values of the object extension are tabulated in table 3, and it shows that the spatial coherence requirement increases rapidly, as the object extension increases. Finally, we remark that the solution of the coherence requirement is not restricted to the examples that are presented; it can, however, be evaluated in a similar manner for other processing operations.
0 4. Transfer Function The description of a transfer function for a linear spatially invariant optical system is an important concept. The techniques of using amplitude and intense sinusoidal objects to determine the transfer characteristic of a coherent and an incoherent optical system have been described by O’NEILL[ 19631 and GOODMAN [1968]. They used the system transfer function to evaluate the image quality of a coherent and an incoherent optical system. However, a strictly coherent or incoherent optical wave field can never exist in practice. In reality, a physical source cannot be a point, but is, rather, a finite extension that consists of many elementary radiators. We now evaluate the relationship between the MTF of an optical system and the spectral bandwidth of a spectral band filter. Let us consider a sinusoidal object transparency to evaluate the MTF of a partially coherent processor of Fig. 4. The overall input transmittance, which includes the diffraction grating, can be written as s(x) = (1 + CCOS2IlVX)(1 + cos211vox), (4.1) where v is the spatial frequency of the sinusoidal object and C is the contrast. We again use a one-dimensional notation. The output intensity distribution [i.e. eq. (3.12)] can be shown as (ZHUANG and Yu [ 1982b1) A,, + A A d 2 N
I ( x ’ ;AS) =
y(xo) IA,(x‘, x,; n= 1
1,
All2dx,
dA,
(4.2)
- A;.,,/2
TABLE 3 Spatial coherence requirement as a function of the target extension, where I ,
=
5461 A.
Spatial extension
Wmm)
1
2
3
5
Extended source size AS(mm)
0.28
0.14
0.09
0.07
10
0.03
20
0.015
50
0.006
246
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
where m
A,(x'; x,; A)
=
j ( ) * j + rect
a - AJv, *an
exp ( - i
[v. 8 4
:
ax') da
-m
m
j b ( -a_ l fx,+
+
1
v,)exp(
( it:)
rect a
- i 2n a x ' ) d a
exp
(-
i
ax') d a
-m m
-m
+
1
-i
rect(ait:)exp(
:
dx')do
-m m
*
j -m
=
{i exp (i + -C
4
:
xox') rect
[. (Af3 + [ ;(- -
exp 12n
+ 4-C exp
i2n
x exp( -i2nv0x').
(2)
1 (xoiy)
v)x' rect
____
v)xl]rect(y)}
(4.3)
In the above formula the asterisk denotes the convolution operation. In the following, we will utilize this equation to evaluate the MTF under the temporal and spatial coherence regimes.
247
TRANSFER FUNCTION
4.1. TEMPORAL COHERENCE
If we let the incoherent source be a point source [i.e., y(xo) = 6(xo)], eq. (4.3) reduces to c2 cos(2nvx’)rect IA,,(X’; A)I2 = + - rect 8 +
(g)
t
+ C2
cos(4nvx’)rect 8 A - A,, the output intensity distribution becomes
(4.4)
-
By letting A’
=
N n=
I -m
+ -c cos(2nvx’) rect A,, f v + A’fv 2 Act,,
( ) (AJlinA’v)] .
+ c2 - cos(4nvx’)rect 8
dA’
(4.5)
Since the MTF is defined as the ratio of the contrast of the output fundamental frequency signal to the contrast of the input signal, the MTF can be written as
MTF(v) =
II
for v <
”(2 + c 2 ) (Aa, - 2fvAn + fvA1,) , 4fvA1, + c2(Aa, - 2fv2, + f v n , )
for
Aan (21, + A W ’
A an A all < v< (24, + A4,)f (24, - A4,)f’
which is dependent on the spatial frequency of the diffraction grating vo, the spatial width of the filter Act,,, and the spectral bandwidth AA,, of H,(ct). If on utilizes Ax,, = fv,AA,,, then eq. (4.5) reduces to v AL,, for v < 0, 21, + AA,,
MTF( V )
( 2 + c’) ( v J I- 2 ~ 4 +, VAL,) 4vAL + c ’ ( v & ~ . - 2 ~ 4 +, vAA,)’
=
I
0.
for
vbL ~
21,
+ AAm
v b
4
21, - A l , (4.7)
248
PRINCIPLES OF OITICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,8 4
Ban= 4.5 mm I .o
0.8 LL
t-
z
06
0.4
0.2
Y
0
4
8
12
16
20
Spatial Frequency (lines/mm) Fig. 10. MTF as a function of input signal frequency v, for various values of spectral width Aln.
Figure 10 shows plots of MTF as a function of the input object frequency for various values of Al,,. Thus the MTF is not appreciably affected by A&,, except for some slight changes in frequency response. For example, as Al,, becomes broader a slight reduction in frequency response is expected. However, the system bandwidth is somewhat broader. Figure 11 shows the variation of the MTF as a function of v for various values of Aa,, for a given A l , , . We see that the system bandwidth is linearly related to Aa,,. In other words, an increase in Aa,, results in a wider system bandwidth. Figure 12 shows the dependence of the MTF on vo, and we see that the system bandwidth is linearly proportional to vo. In other words, a higher frequency grating has the advantage of achieving finer image resolution. However, this advantage is somewhat reduced with the use of larger achromatic transform lenses, which are generally more expensive.
4.2. SPATIAL COHERENCE
We assume that the light source is monochromatic, but of finite extent. The output intensity distribution can be shown to be
v, 8 41
249
TRANSFER FUNCTION
m
-w
+
-
c 162
[
rect
(A, T 2 A f v )
(A,
+
r.,,,>]} (2)
dxo
+ C cos(2nvx') -
4
7
x
(2)+
[red
-02
+CZ cos(4nvx') 8
T2AfJ] (2)
rect (A,
rect
7
dx,
T2,> (2)
rect (A,
dx,.
rect
(4.8)
-03
AXn= 3008
IL
I-
I
-
0
4
8
12
16
20
Spatial Frequency ( lines/mm) Fig. 1 1 . MTF as a function of input signal frequency width Amn.
Y,
for different values of the spatial filter
250
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
I
[v, 8 4
AXn=500%
I .o
0.8 LL
I-
E
0.6
0.4
0.2
0
4
12
8
16
20
Spatial Frequency (lines/rnrn) Fig. 12. MTF as a function of input signal frequency
Y,
for various values of the grating frequency
YO.
Thus, the MTF can be shown to be for 0 < A s < A a - 2Afv, 2(2 + c’) (AM - 2Afv) , for Acr - 2Afv < A s < A x , AS + c’(As + Acr - 2J.f~) MTF
=(
2(2 + c’) (ACI- Afv) , for Acr < As c Acr + 2Afv, 4 A a + c’(As + Ax - 2Afv) Ac~ 2Afv , ACr
for A a
0,
for A a < Afv.
+ 2Afv < As, (4.9)
From the above equations, one can see that the MTF decreases rather rapidly as the source size A S increases. Figure 13 shows the MTF as a function of the input spatial frequency for various AS. We see that the frequency response decreases quite rapidly as the A S increases. In other words, for a fixed filter size the smaller the source size used, the better the frequency response approaches that of a coherent case.
25 I
NOISE PERFORMANCE
V. § 51
I .o
t
Aan = 3.omm
0.0
0.4
\\
/AS
= 4.0mm
0.2
0
4
8
12
16
20
Spatial Frequency (lines/mm) Fig. 13. MTF as a function of input signal frequency v , for various values of the source size
AS.
Figure 14 shows the MTF as a function of the input spatial frequency for various sizes A a of the spatial filters. We see that the system bandwidth is linearly related to AM. Thus, the MTF is obviously limited by the filter bandwidth of the optical processor. We note that the increase of the filter bandwidth would also cause a reduction in temporal coherence, which in turn reduces the processing capabilities of the optical system. Nevertheless, an optimum processing capability of the optical system can still be obtained, with an appropriate MTF for a specific processing operation.
5 5. Noise Performance Partially coherent processors are known to perform better under noisy environments than their coherent counterparts. CHAVELand LOWENTHAL [ 19781and CHAVEL [ 19801have studied the relationship between the noise and coherence of a conventional optical imaging system. They have shown that the noise fluctuations caused by the pupil plane can be reduced considerably if broad band illumination is employed. They have also shown that the noise at the object plane due to defects other than the phase deviations, cannot be
252
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
0
4
8
12
16
[v,$ 5
20
Spatial Frequency (lines/rnm) Fig. 14. MTF as a function of the input signal frequency v , for various values of the filter width A a.
suppressed by partially coherent illumination. LEITHand ROTH [ 19791 have also studied the noise performance of an achromatic optical system. They analyzed the problem by introducing the concept of a three-dimensional transfer function to describe the noise suppression properties of the system. They demonstrated that such a system shows considerable noise immunity if a broad spectral light source is employed. Recently, we have studied the performance of a partially coherent processor under a noisy environment (Yu, SHAIKand ZHUANG[ 1984, 19851). 5.1. NOISE PERFORMANCE UNDER TEMPORALLY PARTIALLY COHERENT REGIME
The problem of coherent artifact noise in an optical processing system is intimately related to the coherence properties of the source. For temporally partially coherent (TPC) illumination, the spectral broad band source is assumed as a point [i.e. y(xo, yo) = 6(xo, y o ) ] . Thus the output intensity distribution due to H,,(a, p) would be
NOISE PERFORMANCE
2n
-i -(x‘u
+ y‘p)
253
(5.1)
The spectral bandwidth AA,, of H,,(or, 8) can be approximated as
The redundancy in the filtered signal is proportional to AL,, of the nth filter and also to the number of spectral band filters employed in the Fourier plane. It is evident that for a fixed spatial width of the filter either a lower spatial frequency of phase grating, or a shorter focal length of the transform lens should be used to improve the SNR at the output plane. However, a decrease in the grating spatial frequency also reduces the number of spectral band filters that can be placed in the Fourier plane, limiting the processing capability of the system. An appropriate relationship between the two parameters is given by (Yu [ 1978b1)
where A 1 and I are the spectral bandwidth and the mean wavelength of the light source, respectively, and A v, is the spatial frequency bandwidth of the input object in u-direction. Although the output SNR from each spectral band filter would decrease as vo increases, the problem can be offset by the use of a larger number of spectral band filters in the Fourier plane. Thus, the overall output SNR would improve considerably with an appropriate choice of the above mentioned parameters. Let us define the output SNR for the nth spectral band in a standard manner as SNRn(yO 2 E [ ~ ~ ( Y ’ ) I / ),~ ~ ( Y ’
(5.4)
where E[ . ] denotes the ensemble average, and G~(Y’
9 E [ ~ : ( . Y11’ - { E [ l n ( y ’)I}*.
(5.5)
Since we are interested in the comparisons of a partially coherent to a coherent system, we define the normalized SNR for the nth spectral band as __ A SNR =
SNR, for partially coherent illumination SNR (coherent)
(5.6)
254
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,
5
5.1.2. Noise at the input plane
We now evaluate the noise performance when the noise is present at the object plane. The output intensity distribution in a one-dimensional form due to the nth spectral band filter would be A, + AAJ2
In(Y’
j 1
=
1,- AAJ2
fl
(2
s ( y ) sinc[2nvc(y’ - y)l dy dJ9
- JC
(5.7)
where s ( y ) represents the one-dimensional input signal with noise, and where we have assumed that the nth spectral band filter is given by Hn(B) = rect(B/Wvc),
rect (BIB) 2
(5.8)
{
1, IBI SB9 0, otherwise,
and
is defined as the cut-off frequency of the optical system. Without loss of generality, the noise transmittance of the object plane can be written as S(Y) =
4.v) exp [ik$(Y)l + n ( y ) ,
(5.9)
where k = 2 4 1 and a ( y ) is a one-dimensional signal. The multiplicative exponential represents the noise due to a random thickness fluctuation of phase distribution $(y), and n ( y ) includes other additive amplitude noise sources at the input plane.
5.1.2.1. Weak phase noise To study the phase noise exclusively, we let n ( y ) = 0. Also, we assume that $ ( y ) is wide sense stationary with zero mean and variance o;, and that its probability density is identical throughout the recording medium (ONEILL [ 19631, USCINSKI [ 19771). The expressions for the moments are E[$(Y)l
=
0.
(5.10)
and E [ $ ( y , ) $(Y2)1
= 0;
exP(- lY, - Y 2 l / 4 ,
(5.11)
v. I 51
255
NOISE PERFORMANCE
where we have chosen an exponential form of the correlation, in which the quantity d is called the scale size of the irregularities, or more commonly the correlation distance. If we assume further that the phase noise is weak with a very small variance [i.e. k $ ( y ) G 11, the third- and higher-order moments can then be neglected in the Taylor series of expansion. This makes the problem tractable by reducing it to additive noise; however, the terms are not independent of the spectral bandwidth, and the output SNR improves slowly with an expanding spectral bandwidth. Experimental evidence suggests that the improvement of SNR would be more significant if higher-order terms could also be included in the calculations. Under these assumptions, the relevant moments for the phase noise may be written as
Let us assume that the object transmittance is a(y)
=
1
+ sin(2xBy),
where
SZ is the relative frequency of the input grating. By proper substitution and staightforward calculation, from eq. (5.7) we obtain E[l,(y’)] =
?T2
[+ 1
(F, -
.2)
A:
]
4a;5 AA,, , (AA/2)2
-
(5.14)
where
x sinc (2 n(y ’ - 3
1 exp ( - I y - J I / d ) dy d j ,
and
From the definition of eq. (5.4), the output SNR due to the nth spectral band
256
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
filter would be
[v,§ 5
. -
(5.15)
Note that, for a white-light source spanning the visible spectrum, the magnitude of the overall spectral bandwidth of the spectral band filters, that is A I = XAI,,, would.be as large as the mean wavelength 2 of the light source. The SNR for the strictly coherent case can readily be obtained by assuming the light-source characteristic to be in the form of a delta function, i.e. S ( I ) = 6 ( I - I ) , and carry out the necessary calculations. Here, the mean wavelength I,, of the nth spectral filter is assumed to be the characteristic wavelength of the coherent counterpart. Thus we conclude, using eq. (5.6), that the normalized SNR in the case of weak input phase noise for the nth spectral band filter is (5.16) It is evident that the normalized SNR would improve with the spectral bandwidth of the processor. Since the channels are not physically separated, the observed improvement in SNR is primarily due to the optical path difference for the wavelengths in the source spectrum. The effect is quite noticeable for broad spectral band filters. Numerical results based on eq. (5.16) are summarized in Fig. 15. The output SNR improves with the spectral bandwidth AAn; also, it is somewhat higher for shorter wavelengths I,,. We emphasize that eq. (5.16), obtained under simplified conditions to make the problem tractable, is an approximation. It shows only a trend of improvement in the SNR. The SNR of a partially coherent processor improves with increasing spectral bandwidth AA,, of the spatial filter H,(ct,b); also it is somewhat higher for shorter wavelengths. We expect that the result is more realistic if higher terms for the phase noise are included in the calculation. 5.1.2.2. Amplitude noise It is experimentally known that amplitude noise, e.g. noise due to film granularity, at the object plane behaves as a part of the object and that its size and shape do not change with the wavelength of the system. Since we are unable to distinguish the object from the amplitude noise at the object plane by varying the wavelength of the system, we expect to find no significant advantage in using a broad-band source. With @ ( y )= 0, the output intensity
v. I 51
NOISE PERFORMANCE
1175-
251
u; = .02%
for a coherent source can be written as (5.17) -m
Since the above expression is independent of the wavelength, the SNR for both a temporal partially coherent and a monochromatic source would have the same value. This implies, from the definition of eq. (5.6), that the normalized SNR will be unity for any spectral bandwidth of the system. If we assume that the additive noise is a stationary Gaussian process with mean m and a correlation function of the form Ein(vl) n(y2)l = of exp( - IY, - Y , I/b) + m 2 ,
(5.18)
where b is the correlation distance, then the third- and fourth-order moments may be expressed in terms of the second-order moments (THOMAS [ 19691). If we also choose some suitable functional form for the input object, the expression for the output SNR can be simplified to a larger extent, and can then be numerically evaluated. The results we have obtained with a ( y ) = 1 + sin(2 n a y ) show that the output SNR depends strongly on the mean value m of the input additive noise, as shown in Fig. 16.
258
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,
5
f
10 -
I
I
0000 0.025
I
0050 0075
I
0.100
I
0.125
Mean Value
I
-
0.150
rn
Fig. 16. Plot of SNR for additive Gaussian noise at the input plane as a function of its mean value.
Furthermore, for the case of amplitude noise, the separation of filter channels, for example, using a phase sampling grating at a later stage, will not by itself help to improve the noise performance of the optical system. However, when spectral band filters specifically designed to reduce the input noise are inserted at the Fourier plane, considerable improvement in the output SNR can be expected.
5.1.3. Noise at the Fourier plane Since photographic plates are commonly used to synthesize the spatial filters for optical processing, the phase fluctuations and other additive random defects of the filters significantly limit the processing capability of the optical processor. This is especially true for coherent optical processing systems. Thus, the noisy channel nth spectral band filter may be written as H,,(B)= {exp[ikqWI
+ n(B)} rect(B/v,~f).
(5.19)
The rectangular function represents the size of the filter in the /I-direction. Again, a one-dimensional object is considered, such that the amplitude trans-
v, 8 51
NOISE PERFORMANCE
259
mittance at the input plane is s ( x , y ) = a(y)exp(i2nv0x).
(5.20)
We note that, as long as a suitable spatial frequency is chosen for the phase sampling grating, the processor will achieve a desirable separation of channels along the a-direction. Under these conditions, the output intensity can be written as
where S ( 8 ) is the Fourier transform of the input object. 5.1.3.1. Weak phase noise
Under the weak phase noise approximation of eq. (5.19), and assuming the output normalized SNR can be calculated. Qualitative effects pertaining to the noise performance of the optical processing system are rather difficult to deduce from the result obtained by Yu, SHAIK and ZHUANG [ 19841. However, several numerical computations were made to develop a feelingfor the result. We find that the normalized SNR improves with the spectral bandwidth of the processor and that it is higher for shorter wavelengths, as shown in Fig. 17. We note that the comments of weak phase noise at the input plane can also be applied to the observations that we have obtained in this figure. Furthermore, contributions from separate spectral bands to the output intensity would have a smoothing effect on the phase noise and the overall SNR would show considerable improvement. Since the channels are physically separated by the action of the phase sampling grating at the input plane and encounter independent realizations of noise, approximately a - f o l d improvement in the output SNR would result, where N is the total number of narrow spectral band filters employed in the Fourier plane.
a ( y ) = 1 + cos(2 K@),
5.1.3.2. Amplitude noise Another source of noise in photographic filters is granularity. But available models for the grain noise are inadequate and are mathematically intractable. As before, we consider an example of additive Gaussian noise with mean m and a correlation function described by eq. (5.18). The desired size of the spectral spread Al,, over the nth spatial filter H,,(p) along the r-direction can be obtained by a suitable choice of the spatial
260
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
0 ; '
R
,021
= .5
- /
U
X = .45 p m
z Cn
[v,8 5
1.08
0.0
0.I
0.2
0.3
0.4
Bandwidth, A X
Fig. 17. Normalized SNR for the phase noise at the Fourier plane as a function of the spectral bandwidth for two different wavelengths.
frequency of the input phase sampling grating. The physical size of the narrow spectral band filters can be chosen to be of the order of several correlation distances. The corresponding spectral bandwidth of a channel can then be easily evaluated from eq. (5.2). Under these conditions, for a single filter channel, the noise is highly correlated at the Fourier plane and, for reasons discussed in 5 5.1.2.2, there is little improvement in the output SNR. But the channels in this case are corrupted by independent additive noise as they have been sufficiently separated from each other by the use of phase sampling grating at the input plane. Since all channels are carrying the same information with independent Gaussian noise, their superposition at the output plane results in an fi-fold improvement for the white-light processing system. Figure 18 shows the improvement in the output SNR as a function of the number N of fdter channels in the system. Thus, we see that the improvement in the output SNR depends on the extent of spectral decomposition at the Fourier plane. However, the number of channels for the optical system is restricted by eq. (5.3) and also by practical considerations in generating the narrow spectral band filters.
26 1
NOISE PERFORMANCE
I
0
4
8
12
16
20
Number of Channels, N
Fig. 18. Output SNR for the additive Gaussian noise at the Fourier plane as a function of the number of filter channels in the system.
5.2. NOISE PERFORMANCE UNDER SPATIALLY PARTIALLY COHERENT REGIME
In this section we discuss the noise performance of optical systems under spatially partially coherent (SPC)illumination. A SPC light source corresponds to many channels. The information about the object carried by each of these channels is superimposed incoherently at the image plane to give the net observable output irradiance. If the channels are physically separated when they hit the noise plane and if the noise is assumed to be spatially independent and wide sense stationary, the noise effects would generally cancel each other, thus providing considerable improvement in the output SNR of the optical system. The source plane Po and the Fourier plane P, (shown in Fig. 4), form a conjugate pair in the sense that the light source is imaged in the Fourier plane by lenses L, and L,. The ratio of the source image size to the spatial filter size may be defined as a parameter of spatial coherence. If the lenses L, and L, have the same focal length, the spatial coherencepurameterp, as defined by BORNand
262
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,5 5
WOLF [ 19641, for the optical system can be written as p
5 AS/AH,
(5.22)
where A S and AH are the source and the filter size, respectively. For p 4 1 we have strong spatial coherence. As it increases to infinity an extremely incoherent illumination results. A typical range for an SPC source is 0.2 < p < 0.7. The number of channels, or the space-bandwidth product, can be simply related to the spatial coherence parameter p. For convenience, if all focal lengths f for the achromatic transform lenses are chosen to be equal, then eq. (5.22) can be written as (5.23) where Aa is the input object area, and N, and Nf define the space-bandwidth product, or the number of channels, for the source plane and the Fourier plane, respectively. The one-dimensional expression for the output intensity distribution for the SPC optical system of Fig. 4 can be written as
As
-03
(5.24) where y(xo) is the spatial intensity distribution of the monochromatic source, S(a) and H(a) are the spectrum of the input signal transmittance and the spatial filtering function, respectively, and A S is the size of the light source. Also, without loss of generality, we have assumed that the spatial frequency of the phase sampling grating at the object plane is zero, since we would be concerned only with the monochromatic illumination. Equation (5.24) can be put in a more appropriate form if we define the following transformed coordinates: uo = ( l / A S ) x o ,u = (AH/Af)x, z = ( l / A H ) a ,and u’ = (AH/Af)x‘.Usingthese transformed coordinates and also assuming that y(xo) = rect(x,/AS), the output intensity function can be written as Z(u’) =
c
11
sinc[2np(u, - uz)] S ( U , ) S * ( U , ) x K(u’ - u I ) K*(u’ - ~
2 du, )
duz,
(5.25)
v. I 51
NOISE PERFORMANCE
263
where C is an appropriate constant and
H ( r ) exp( - i27cru) dz.
K(u)
(5.26)
-*
5.2.2. Noise at the input plane Consider the input transmittance in the form s(u) = a(u) exp [ik$(u)]
+ n(u),
(5.27)
where $ ( u ) and n(u) are the random phase fluctuations and the amplitude noise, respectively, a(u) is the input signal, k = 27c/A, and u = [AH/(Af)]x. The additive noise at the input plane behaves as if it were a part of the signal to be processed. All channels see the same “realization” of the amplitude noise process. Since we are dealing with a particular realization superimposed on the signal, no averaging effect for the additive noise is observed at the output image plane. We expect its behavior to show little dependence on the degree of spatial coherence as was the case with TPC illumination. A detailed analysis of this type of noise is given by CHAVEL [ 19801. It is therefore sufficient to say that SPC illumination does not help to improve the SNR of the optical system in this case. Experiments show that phase defects at the input plane can be handled effectively when spatially incoherent light is employed. Neglecting n(u) in eq. (5.27) and assuming that a(u) = 1 + sin(2nG?u), we have s(u) = [ 1
+ sin(2nblu)l exp [ik$(u)],
(5.28)
where (5.29) is defined as the input relative spatial frequency, v is the real spatial frequency and A H / i f is the cut-off spatial frequency of the optical processor. We further assume that the spatial filter is a rectangular function, that is, H ( r ) = rect(r). Therefore, K(u) =
s
- 1
exp( - i27cru) d r = C, sinc(2m),
(5.30)
264
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, 8 5
where C, is an appropriate constant and is ignored in the following discussion. Thus, from eq. (5.25) we write
~ ( ~ ’ ) = ~ ~ ~ ~ ~ ’ , ~ l , ~ z , p ) ~ ~ ~- IH) ~~ ~* )dul ~I ~}dUZ, 2 ) e x p { i ~
(5.31)
where
F(u’, u I , u2, p) = sinc[2np(u1- u,)] sinc[2n(u‘ - u , ) sinc[2n(u’ - u 2 ) ] .
(5.32) Assuming that the phase noise is weak (k$(u) < < 1) with zero mean, the exponential in eq. (5.3 1) can be expanded into a Taylor series, and a solution for eq. (5.32) can be obtained by neglecting higher-order terms. The relevant moments for this case are
a4441 =
(5.33)
0 9
and
E[$(u,)9(u,)l
=
k2a; exP(- lu,u,l/d),
(5.34)
where a; is the variance and d is the correlation distance for the phase noise. The SNR for the SPC illumination can now easily be calculated, i.e. (5.35) where n
n
(5.36) and G2(u’ =
I
F(U
3
U I u2, P ) d u l ) a * ( u 2 ) exp (
- I u, - u, 1 /d) du, du,
.
(5.37) When p e 1, the spatial intensity distribution of the source can be replaced by a delta function. Assuming a point source at the origin, we have y(x,) = 6(x,). The output irradiance can be appropriately modified to obtain the SNR for the coherent case. It can be shown that
(5.38)
v. I 51
NOISE PERFORMANCE
265
where
(5.39)
and G4(u’) =
jj
sinc[2n(u’ - u , ) ] sinc[2n(u’ - u,)]
x exp(- Iul - u21/d)du1du,.
(5.40)
The ratio of eqs. (5.35) and (5.38) gives the normalized SNR for weak phase noise at the output plane. Although the equations are complex, it is easy to see that the SNR would increase to infinity as the variance in noise is decreased to zero (i.e. the noise is eliminated from the system). A numerical evaluation of the results given above is summarized in Fig. 19. It can be seen that the normalized SNR increases as the requirements on spatial coherence are relaxed, that is, as p is increased. Note that the problem for a strong phase can be worked out following PARRY[ 1974, 19841, since a Gaussian character for the noise can be assumed.
I
X = .5rm
[z
z
1.150
.-
0) N
0
E
b z
1.10-
Coherence Parameter, p Fig. 19. Normalized SNR for the phase noise at the input plane as a function of the spatial coherence parameter.
266
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v,
5
5.2.3. Noise at the Fourier plane The results obtained for the input noise show that while a reduced degree of spatial coherence considerably attenuates the phase fluctuations, it is ineffective in reducing amplitude noise. However, we see that the situation is somewhat different for the noise at the Fourier plane. We assume that the noise at the Fourier plane is wide sense stationary Gaussian. Although this assumption is only approximate, it has been shown (CHAVEL and LOWENTHAL [ 19781) that the output SNR obtained under the more realistic non-Gaussian form is, in general, very close to or larger than that obtained in the Gaussian case. We also assume that the correlation distance for the complex noise is much smaller than the diffraction limited image of the source point at the Fourier plane. Under the foregoing assumptions, it can easily be seen that various channels in the system will be independent. They see different “realizations” of the complex noise at the Fourier plane. Each channel will be diffracted at the noisy Fourier plane, resulting in high-contrast noise fringes at the output image plane. Since the overall output irradiance is the superposition of images from each channel, the visibility of noise fringes decreases with increasing size of the source. More precisely, the output SNR for independent Gaussian channels increases as JN, where N is the total number of channels in the SPC system and is proportional to the area of the source. Except for some intensity attenuation, the amplitude noise at the Fourier plane can be eliminated by using an extended light source with p = 1 (typical incoherent source corresponds to lo6 channels). The number of channels that eventually contribute to the output image is constrained by the limiting aperture of the system. Usually this critical aperture for the system is determined by the filter size. Figure 20 shows a plot of the output SNR as a function of the spatial coherence parameter p. For p < 1, when the source size is smaller than the filter size, the output normalized SNR improves as JN,, where N, denotes the number of channels from the source. However, when the source size becomes larger than the filter size, i.e. p > 1, the system no longer supports the additional channels and the output normalized SNR stays at a constant level, JNf, where Nfis the total number of the independent filter channels.
-
261
SOURCE ENCODING AND IMAGE SAMPLING
V. § 61
0.0
. 0.5
I .o
I.5
20
Coherence parameter, p
Fig. 20. Plot of SNR for the complex noise at the Fourier plane as a function of the spatial coherence parameter.
8 6. Source Encoding and Image Sampling In this section we shall discuss a linear transform relationship between the spatial coherence (i.e. the mutual intensity function) and the source encoding (Yu [ 1983b1). Since the spatial coherence depends on the image processing operation, a more relaxed coherence requirement may be used for specific image processing operations. The concept of source encoding is to alleviate the stringent coherence requirement so that an extended source can be used. In other words, source encoding is capable of generating an appropriate spatial coherence for a specific optical signal processing application, such that the available light power from the source may be efficiently utilized.
6 . I . SOURCE ENCODING
We begin our discussion with Young’s experiment under an extended source illumination, as shown in Fig. 21. First, we assume that a narrow slit is placed in the source plane Po behind an extended monochromatic source. To maintain
268
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
Po
[v,
6
PI
Fig. 2 1. Young’s experiment with extended source illumination.
a high degree of coherence between the slits Q , and Q , at plane P,, the source size should be very narrow. If the separation between Q 1 and Q , is large, then a narrower slit size S , is required. Thus the slit width should be
where R is the distance between the planes Po and P I ,and 2h0 is the separation between Q , and 4,. Let us now consider two narrow slits S, and S , located in the source plane Po. We assume that the separation between S , and S, satisfies the following path-length relation r ; - r;
=
(rl - r,)
+ mA,
(6.2)
where the r’s are the respective distances from S, and S, to Q, and Q2,m is an arbitrary integer, and II is the wavelength of the extended source. Then, the interference fringes due to each of the two source slits S and S, will be in phase and a brighter fringe pattern will be seen at plane P,. To further increase the intensity of the fringes, one would simply increase the number of slits in appropriate locations in plane Po, such that the separation between slits satisfies the fringe condition of eq. (6.2). If the separation r is large, that is, if
,
v. § 61
269
SOURCE ENCODING AND IMAGE SAMPLING
R % d and R % 2h,, then the spacing d becomes
d=m
-.I R
(6.3) 2hO Thus, by properly encoding an extended source, it is possible to maintain a high degree of coherence between Q I and Q2, and at the same time, to increase the intensity of the fringes. To encode an extended source, we first search for the coherence function for a specific image processing operation. With reference to the optical processor shown in Fig. 4, the mutual intensity function at the input plane P, can be written as (BORNand WOLF[ 19641)
ss
J(x19xi) =
Y
~
~
,
~
~
~
~
o
~
~
,
~
~
(6.4) ~
~
where the integration is over the source plane Po, y(x,) is the intensity distribution of the encoding mask, and K(x,, xI)is the transmittance function between the source plane Po and the input plane P,, which can be written as
By substituting K(x,, x I )into eq. (6.4), we have
J(xl - x i ) =
JI
1
y(xo)exp[i2nGx0 (xI- x i ) dx,.
From the above equation we see that the spatial coherence and source encoding intensity form a Fourier transform pair, that is
and
J(xl - x i )
=
8-"6(x,)],
(6.8)
where 9denotes the Fourier transformation operation. It is evident that the relationship of eqs. (6.7) and (6.8) is the well-known Van Cittert-Zernike theorem (VANCITTERT[ 19341, ZERNIKE [ 19381). In other words, ifa required coherence is provided, then a source encoding transmittance can be obtained through the Fourier transformation. In practice, however, the source encoding transmittance should be a positive real quantity that satisfies the physically realizable condition 0 5 y(xo) 5 1.
(6.9)
,
,
~
l
270
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, 5 6
6.2. IMAGE SAMPLING
There is, however, a temporal coherence requirement for partially coherent processing. Since the scale of the Fourier spectrum varies with the wavelength, a temporal coherence requirement should be imposed, as has been discussed in § 3. If we restrict the Fourier spectra, due to wavelength spread, within a small fraction of the fringe spacing d of a narrow spectral band filter Hn(a,p), then we have
(6.10) where l/d is the highest spatial frequency of the filter, P , is the angular spatial frequency limit of the input image transparency, f is the focal length of the achromatic transform lens, and AA,, is the spectral bandwidth of H,,(a, p). The temporal coherence requirement of the spatial filter is, therefore,
(6.11) where A,, is the central wavelength of the nth narrow spectral band filter, and 2h0 = A,,f/d is the size of the input image transparency. To gain some feeling of magnitude, we provide a numerical example. Let us assume that the size of the input image is 2h0 = 5 mm, the central wavelength of the filter Hn(a,P) is A,, = 5461 A, and take a factor 10 in eq. (6.11) into consideration, that is
lOsA, A&=-. (6.12) hOP, Several values of the spectral width requirement AA,, for various spatial frequencies P , are tabulated in table 4.It is therefore evident that, if the spatial TABLE 4 Temporal requirement for H,(a,
b)
Angular spatial frequency limit
5 (lines /mm) 2R
0.5
1
5
109.2
21.8
20
100
Spectral bandwidth
A4,(h
218.4
5.46
1.09
"3
I61
SOURCE ENCODING AND IMAGE SAMPLING
27 I
frequency of the input image transparency is low, a broader spectral width can be used. In other words, if a higher spatial frequency is required, then a narrower spectral width is needed. Evidently, a narrower spectral spread A & corresponds to a higher temporal coherence requirement, which can be obtained by increasing the image sampling frequency po. However, if a higher image sampling frequency is used larger apertures may be required for the transform lenses in the optical system, which tend to be more expensive. Nevertheless, in practice, high quality images have been obtained with relatively low-cost lenses. 6.3. APPLICATION OF SOURCE ENCODING
It would be an exhaustive effort to discuss all the applications of the partially coherent processing. However, we shall illustrate a sample application utilizing the source encoding concept. Let us consider the image subtraction of LEE, YAO and MILNES[1970]. Since image subtraction is a one-dimensional processing operation, a one-dimensional fan-shaped diffraction grating should be utilized, as illustrated in Fig. 22. We note that the fan-shaped grating (i.e. White Light Line. source ,T(x)
Fig. 22. Partially coherent image subtraction processor. T(x) phase grating, L, image lens, L, collimated lens, L, and L, achromatic transform lenses, y ( y ) fan-shaped source encoding mask, G fan-shaped diffraction grating.
212
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
[v, $ 6
the filter) is imposed by the temporal coherence condition of eq. (6.1 1). Since image subtraction is a point-pair processing operation, a strictly broad spatial coherence at the input plane is not required. In other words, if one maintains a high degree of coherence between the corresponding image points to be subtracted at the input plane, then the subtraction operation can be carried out at the output plane. Thus, instead of using a strictly broad spatial coherence function, a reduced spatial coherence may be utilized: J ( y - y ' ) = S ( y - y ' - h,)
+ S(y - y ' + h,),
(6.13)
where 2h, is the main separation between the two input image transparencies. The source encoding function can therefore be evaluated through the Fourier transform of eq. (6.7): (6.14) Unfortunately, eq. (6.14) is a bipolar function that is not physically realizable. To ensure a physically realizable source encoding, we let the mutual intensity function be (Wu and Yu [ 19811)
where N % 1 a positive integer, and w 4 d. Equation (6.15) represents a sequence of narrow pulses that occur at every ly - y ' I = nho, where n is a positive integer, and their peak values are weighted by a broader sinc factor. Thus, a high degree of spatial coherence can be achieved at every point-pair between the two input image transparencies. By taking the Fourier transformation of eq. (6.15), the corresponding source encoding transmittance is N
y ( l y ( )=
C n= I
IY - ndl rect -,
(6.16)
W
where w is the slit width, d = (Aflh,) is the separation between the slits and N is the number of slits. Since y( 1 y / ) is a positive real function that satisfies the constraint of eq. (6.9), the proposed source encoding function of eq. (6.16) is physically realizable. Furthermore, the separation d is linearly proportional to L and the source encoding is therefore a fan-shaped function. To obtain lines of rainbow color spectral light sources for the image processing, we would utilize a linearly extended white-light source with a dispersive phase grating, as illustrated in
v, I 71
CONCLUDING REMARKS
273
Fig. 22. Thus, with the described broad band source encoding mask and a fan-shape sinusoidal filter, a subtracted image can be seen at the output plane.
8 7.
Concluding Remarks
We have reviewed the basic principles of optical processing under partially coherent illumination. The coherence requirement, the modulating transfer function and the noise performance of the optical processing system under a partially coherent regime have been discussed in detail. A number of important conclusions can be drawn concerning this technique: The most significant one is its capability of processing an image in a complex wave field as a coherent processor, and the ability to suppress coherent artifact noise as an incoherent processor. Another important aspect of partially coherent processing is the color image processing, for which the optical system is capable of exploiting the spectral content of the object. Finally, in contrast with its coherent counterparts, the partially coherent processor is generally economical and easy to operate. In spite of the flexibility of digital techniques, the partially coherent optical techniques offer the advantages of capacity, color, simplicity, and cost effectiveness. Instead of a confrontation, we expect a gradual merging of the optical and digital techniques, and it is to be hoped that the continued development of optical digital interfaces and various electro-optical devices will lead to a fruitful result of hybrid optical digital processing techniques, utilizing the strengths of both processing operations. Furthermore, it is the author’s belief that partially coherent processing is at the threshold of widespread application, and it is his hope that this article will provide a foundation and a guide to interested readers and researchers toward various imaginative optical processing applications.
Acknowledgements The support of the U.S. Air Force Ofice of Scientific Research and the Rome Air Development Center at Hanscom Air Force Base in the area of white-light optical processing is gratefully acknowledged.
274
PRINCIPLES OF OPTICAL PROCESSING WITH PARTIALLY COHERENT LIGHT
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AUTHOR INDEX A AACARD,R. L., 176. 178 ABARBANEL, 11. D. I., 59 AGARWAL,G. S., 23.59 AHARONOV,Y.,,216, 218 AHONEN,R. G., 177, 179 AISENBERG, s., 146, 167, 168, 178 ALBELLA,J. M.,167, 181 A L L E N , T . H . ,116, 117, 148, 149, 155, 156, 157, 158, 178 AMANO,S . , 158, 180 AMENT,W. S., 8, 59 AMES, G. H., 4, 59 ANDERSEN,H. H., 119, 179 ANDERSON,R. H., 224,274 ANDREO,R. H., 4, 59 ANGILELLO, J., 135, 179 ANTOS, R., 92, 108 ARCHBOLD,E., 6, 59 ARII,T., 212, 220 ARMOUR.D.G., 123, 124, 126, 179 ARNAUD,J. A., 66,92, 107 ARSENAULT. H. H., 65, 66, 107, 109 ASPNES,D. E., 125, 179 ATTANASIO, L. J . , 135, 179 ATTARD.A. E., 65, 107 AUWARTER, M.,116, 141, 179
B BABAEV.V. 0.. 133, 179 BAHAR.E.. 57, 59 BANKS,B. A., 169. 179 BARRICK. D.E., 3, 28, 59, 60 BARRON.A.C., 117, 141, 157, 181 BARTELL.L. S., 194, 218 BASS, F. G., 4, 9, 14. 17, 29, 35, 59, 60 BAS'I'IA4NS. M.J., 66, 107 B ~ YH. . L.. 119, 120, 122, 179, 181 BAYII, W., 216, 219
BAZHINOV,V. V., 158, 181 BEARD,C. I., 43, 60 BECHERER,R. J., 225, 274 BECKMANN,P., 4, 28, 37, 60 BEISTER,G., 152, 158, 182 BENNETT,B. L., 3.60 BENNETT,J. M., 7,60 BEREK, M., 97, 107 BERGSTEIN,L., 66, 107 BESENMATTER,W.. 65, 107, 108 BEWILOGWE, K.,168, 172, 182 BIERSACK,J. P., 125, 179 BINH,L.N.,150, 151, 166, 167, 176, 179 BIOT, M. A., 3, 60 BLEHA,W., 224, 274 BOCCHIERI,P., 216, 218 BOCHAROV,V. G., 29, 60 BOERSCH,H., 189,216,219 BOGEN, P., 122, 179 BOHDANSKY,J., 119, 120, 179 BOHM, D., 216,218 BOLOMEY,J. -C., 25, 60 BORN, M., 15, 17, 34, 65, 108, 261, 262, 269, 274 BOSWELL,D., 224,274 BOURRET.R.C., 50, 60 BOUSQUET,P., 3,4, 60 BOYNTON,R.J.. 176, 180 BRANDT,G., 168, 179 BRAWN,G., 92, 108 BRAUNDMEIER, A. J., 4, 59 BREKHOVSKIKH, L. M.,8, 13, 25, 26, 60 BRINKMAN, J. A,, 126, 179 BROUWER,W., 65, 108 BROWN,B. R., 224, 274 BROWN,G. S., 3, 5, 7, 27-31, 45,47-52 BROWN,R. M., 43, 60 BRUGGEMAN,D. A. G., 174, 179 BRYNGDAHL, O., 209, 219 BUBENZER,A.. 168. 179
278
AUTHOR INDEX
BUCHDAHL,H. A,, 86, 108 BUNSHAH,R. F., 132, 139, 179 BUKCH,J. M.,65, 66, 75, 92, 108 BURMAWI,M.Y., 224, 274 BURROWS,M. L., 29, 60 BUSCH,R., 152, 167, 176, 181 BYKOV,Ju.V., 133, 179 C
CARTER, G., 118, 123, 124, 126, 179 CARTER, M.J., 116, 181 CASPERSON, L. W., 66, 108 CELLI,V., 3, 32, 58, 60, 61, 62 CEREZ,P., 66, 108 CHABOT, R., 146, 167, 168, 178 CHAMBERS, R. G., 216, 219 CHAVEL, P.,251, 263, 266, 274 CHEREPONOVA, M.N., 157, 158, 179 CHEREZOVA, L. A., 149, 158, 181 CHERNYAVSKII, V. A,, 158, 181 CHERTOV, V. G., 92, 1 I 1 CHITANVIS, s.,5, 58, 60 CHOPRA, K. L., 134, 138, 179 CHOW, P. L., 5, 60 CHUNG, Y., 170, 171, 180 CHURCH, E. L., 6. 60 CLARK, G. J., 159, 180 CLAY,C. S., 43. 60 CLAY,J. R.,225, 274 COBURN, J. W., 121, 179 Cocco. D. M..224, 274 COHEN,M.S., 211, 219 COLE,B. E.. 177, 179 COLEMAN, W. J., 116, 149. 158, 164, 165, 172, 179
COLLIGON, J. A,, 118, 124, 179 COTTON,D. R., 172, 180 COULT,D., 173, 180 CREWE, A. V., 186, 219 CUOMO,J. J., 125, 135, 136, 179, 180 CUTRONA, L. J., 224, 274
D DAINTY, J. C., 37, 60 DANILIN, B. S., 144, 179 DAVENPORT, W. B., 37, 60 DAWAR, A. L., 172, 179 DEITCH. R.H., 167, 179 DELANO, E., 65, 108
DEMCHISHIN, A. v., 128, 181 DENNIS,J., 167, 181 DESANTO, J. A., 3, 5, 23.24, 25, 26, 30, 34, 36, 39,40,41,43, 44, 50, 53, 54, 58, 60, 62 DESCHAMPS, G. A,, 92, 108 DESHPANDEY, C., 150, 179 DEVANEY, A. J., 33,60 DIETRICH, D., 168, 182 DIRKS,A. G., 129, 130, 179 DISCHLER, B., 168, 179 DOBREV, D., 133, 179, 180 DONAGHEY, L. F., 150, 179 DONNELYN, S . E., 126, 179 DRAGT,A. J., 66, 108 DOCKER,H., 201,219 DUDONIS, J., 135, 179 DUTI-A,K., 225, 274 DYSON,J., 196, 219
E EBERSBACH, V., 172. 182 EBERT,J., 116, 140, 141, 149, 150, 153, 154, 155, 158, 173, 176, 179, 180
ECKART,C., 8, 19, 61 ECKSTEIN, W., 125, 179 EGGENBERGER, D. N., 186, 219 EHLERT,L. B., 177, 179 ELIAS,P.,224, 274 ENDO.H., 186, 219 ENDO,J., 201,203, 205, 208, 209, 212-214, 217, 219, 220
ENNOS,A. E., 7, 59 ERIKSSON, T. S., 175, 179 ERLER,H.-J., 168, 172, 182
F FADEEVA, E. I., 158, 181 FALCO. C. M.,123, 124, 181 FANO,V., 3, 61 FELDER,R., 66, 108 FISHER,A. D., 224, 274 FLORY,F., 3,4, 60 FOCK,V. A., 27, 61 FOG, C., 66, 108 FOWLER, H. A., 216, 219 FRAAS.L., 224, 274 FRANK,R. I., 179 FRANKS, J. H., 135, 175, 179 FREUND, C. H., 209, 219
AUTHOR INDEX
279
FRISCH, U., 5, 35,41,46, 50, 61 FUJIWARA, H., 212, 213, 214, 217, 219, 220 FUKINUKI, H., 92, 109 FUKS,1. M., 4, 9, 14, 17, 31, 35, 59, 60, 61 FUKUHARA, A., 198, 212, 217, 220 FUNG,A. K.,29, 61 FURUTSU, K., 29, 56, 57, 61
HARRIGAN, M. E., 86, 108 HARRINGTON, R.F., 28, 61 HARROLD, J. H., 65, 108 HASHIMOTO, H., 186, 219 HAWKES, P. W., 187, 219 HAYASHI, S., 57, 62 HAYS,D.D.,149, 150, 151, 152, 158, 159,
c
HEISIG,V., 167, 182 HEITMANN, W., 116, 136, 141, 148, 153, 154,
164, 172, 173, 176, 181
GABOR,D., 185, 196, 223, 219, 224, 274 GARCIA, N., 32, 33, 50, 61, 62 GEORGE, N., 224, 229, 238, 274 GERAGHTY, K. G., 150, 179 GERRARD, A,, 65, 66, 66, 75, 92, 108 GIALLORENZI, T. G., 167, 179 GIBSON, U.J., 172, 180 GOLDSTEIN, H., 37, 61 GONDA,S., 168, 170, 181 GOODMAN, F. 0.. 3,61 GOODMAN, J. W., 225,245, 274 GORDON, J. P., 92, 1 I 1 GORMAN, A. D., 31, 61 GRANQVIST, C. G., 175, 179 GREENE, J. E., 117, 158, 164, 179, 180 GREY,D. S., 224, 274 GRIGOROV, G. I., 136, 180 GRINBERG, J., 224, 274 GROHMANN, K., 216, 219 GUARNIER, C. R., 135, 179 GUENTHER, K. H., 148, 180 GUSEVA, M. B., 133, 179 H HABRATEN, F. H. P. M., 174, 180 HACKER, E., 152, 158, 182 HAGGMARK, L. G., 125, 179 HAGSTRUM, H. D., 125, 180 HAINE,M. E., 196, 219 HALBACH, K.,65, 108 HALIOUA,M., 194, 220 HALL,D.G., 4, 59 HAMISCH, H., 216, 219 HAMMOND, R.H., 135, 179 HANSZEN, K.-J., 186, 209, 219 HARADA, Y., 186, 220 HARDY, A,, 66, 109 HARIHARAN, P., 209. 219 HARPER,J . M . E., 125, 135, 136, 179, 180
155, 158, 175, 180
HENDERSON, E., 116, 180, 181, 182 HENNY,F., 172, 182 HENTZELL, H. T. G., 136, 180 HERLOSKI, R.,92, 108 HERRERO, J. L., 66.67, 108 HERRMANN JR, W.C., 117, 135, 141, 142, 157, 166, 167, 180, 181
HERZBERGER, M., 66.67, 75, 108 HIBI,T., 189, 196, 209, 218, 219, 220 HILL,N. R.,3, 58, 61, 62 HINNEBERG, H.-J., 168, 182 HINTZ,E., 122, 179 HIRAGA,R.,129, 158, 180, 181 HIRSCH,E. H., 135, 180 HOFMANN, D., 129, 134, 181 HOLLAND, L., 150, 179 HOLLINS, R.C., 37, 61 HOLM,R.,121, 180 HOPKINS,H. H., 223,274 HORIUCHI, Y.,214. 219, 220 HOWSON,R.P., 149, 153, 154, 158, 182 HUANG,K., 48, 61 HUTLEY, M. C., 3, 61
I IL’IN,V. V., 158, 181 INGRAM, D. C., 126, 179 INGREY, S. J., 176, 180. 182 ISAACSON, G. C., 141, 180 ISAKOVICH, M. A,, 29, 61 ISHIMARU, A., 56, 62 [TO,
s.,57, 61
J JACOBSON, A. J., 224, 274 JAKEMAN,E., 9, 37, 61 JENKINS, F. A,, 108
280
AUTHOR INDEX
J E N K I N S O NH. , A,, 6, 60, 65 JONES,D. S., 28, 61 JONES,J., 173. 180 JONES, P. L., 172, 180 JORDAN,D. I., 37, 61 JOSHI.J. C., 172, 179 JOY, D. C., 168, 182 JUNTZ,R. S., 132, 179
K KABAYASHI, M., 167, 182 KAMINSKY, M., 118, 180 KAUFMAN,H. R., 125, 141, 144, 156, 180 KAWASAKI, T., 209, 220 KELLER,J. B., 5, 25, 50, 53, 61, 62 KELLOGG,0. D., 23, 61 KELLY,R., 121, 127, 157, 161, 180, 181 KENNEMORE, C. M., 172, 180 KERR,D. E., 37, 61 KERSTEN,R. TH., 175, 180 KHAN,A. A,, 170, 171, 180 KHAWAJA,E. E., , 166, 167, 180 KHITSOVA, VI. I., 161, 180 KINGSLAKE.R., 65, 108 KINSMAN,B., 6, 61 KITTAPPA,R., 24, 61 KLECHKOVSKAYA, V. V., 161, 180 KLEINMAN, R. E., 24, 61 KLINGER, R.E., 158, 164, 179 KLOSE, S., 168, 182 KOBAYASHI, T., 186, 220 KODIS,R. D., 28, 37, 61 KOEHLER,J. S., 127. 161, 182 KOGELNIK,H., 92, 108, 109 KOIDL,P., 168, 179 K O M I N I A KG.J., , 132, 181 KOMODA,T., 186, 198, 201, 220 K o o , S. W.. 174. 180 K R ~ S SR.. . 24, 61 KRILL.J. A,, 4, 59 KRYAZHEV, F. I., 17, 61 KUDRYASHOV, V. M., 17, 61 KUIPER,A. E. T., 147, 174, 180 KUMAO,A., 186, 219 K U P E R M A NW. , A,, 17, 61 KLIRYANOV, B. F., 3, 61 KUSAKAWA, T., 66, 109 KUSTER.H., 153, 154, 155, 158, 180
LAM, N., 121, 180 LANFORD,W. A,, 159, 180 LANGMORE,J., 186,219 LASAPONARA, L., 151, 158, 165, 176. 181. 182
LAU, B., 212, 220 LAUER,R., 203, 219 LAX, M., 5, 58, 60 LEADER,J. C., 29, 29, 62 LEAMY,H. J., 129, 179 LEE, S. H., 271, 274 LEITH, E. N., 194, 219, 224, 229, 252, 274 LENZ, F., 189, 219 LI, T.. 92, 109 LICHTE,H., 203, 219 LISZKA,E. G., 59, 61 LITTMARK,V., 125, 180 LIVERMORE,F. C., 176, 180 LOHMANN,A. W., 224, 228, 274 LOINGER,A., 216, 218 LONGUET-HIGGINS, M. S., 3, 10, 61 LOWENTHAL,S., 251, 256, 274 M MACDONALD,R. J., 125, 180 MACLAURIN,B. K., 176, 180 MACLEOD,H . A . , 116, 129, 134, 173, 174, 180, 181
MACUKOW,B., 66, 107, 109 MAHLEIN,H. F., 175, 180 MANCINI,C., 158, 165, 176, 181 MA”, 1. B., 149, 152, 158, 172, 173, 181 MARADUDIN, A. A,, 3, 32, 56, 57, 59, 60, 62 MARCHAND,E. W., 20, 62, 86, 109 MARINOV,M. V., 133, 179, 180 MARSHALLS., 92, 108 MARTEVI. N., 136, 180 MARTIN,P., 173, 180 MARTIN,P. J., 117, 125, 133-135, 149, 159, 160, 162, 164, 165-167, 172-174, 176, 177
MARTIN,P . M . , 149-152, 158, 164, 172, 176 MARTINEZ-DUART, J. M., 167, 181 MARTON,L., 216, 219 MARUYAMA,S., 92, 109 MARVIN,A,, 3, 58, 60, 62 MARX,E., 25, 61 MAST, P.E., 92, 108 MATHINE,D., 170, 171, 180
AUTHOR INDEX
MATSKEVICH. L. L.. 158, 181 MATSUDA,K.. 209, 219 MA.lSIJD.4, T., 186, 201, 203, 205, 208. 209, 212-214,217.219, 220 MATSUMOTO,B., 57. 61 MAI'SUMOTO,K., 208, 219 G., 214, 219 MATTEUCCI, MATTOX,D. M.. 132, 139, 181 M A U L A . W., 27, 61 MAYSTRE,D..3-5,25, 61 McCoy. J. J., 59, 61 MCDANIEL,S. T., 31, 61 M C G I N N A,, , 4, 61 MCKENZIE,D. R., 172, 180 MCNEIL,J. R., 117, 135. 141, 142, 157, 180, 181
MEDWIN,H., 43. 60 MEECHAM,W. C., 59, 61 MENZEL.E.. 205, 220 MESSIER.R., 131, 181 M E Y E RK., , 123, 124, 181 MICHELSON, A. A,, 223, 274 M I H A M AK.. , 212,220 MIKHAILOV, A. v . , 149, 158, 181 MII.I.ER, .A. R., 43, 60 MILLER,L., 224, 274 MILLS,R., 216, 220 A. G., 271, 274 MIL-NES. M I N A M IS., , 66, 98, 109, 1 1 1 MIRANDt, w., 205, 220 MISAWA, s., 168, 170, 181 MISIANO. c., 151, 158, 162, 165, 176, 181, I82 MISSIROLI, G. F., 186, 21 I , 214, 218, 219 MI'IZNER, K. M., 25, 29, 61 M I Y A Z A W T., A , 168, 170, 181 MIZIJTANI, K., 57, 62 MOBLRG.W. L., 175, 179 Mbl.l.t.NSTEDT, G . , 189, 201, 216, 219 MOORE, D.,172, 180 MOORE,D. 'T,,86, 109 MOORE,R. K.. 3. 61 MORAVEC, T. J., 177. 179 M O R R I S . G . M.. 223. 229, 238, 274 Mo-rovri.ov. 0. A,. 158, 174. 181 M o r z , L.. 66, 107 MO~IIAN B. .A,, 128. 181 MIIl.LER. K. H., 130. 131. 132, 181 M i i ~ v t uT., , 196. 219 M~JNCH J.,. 201, 219
28 I
MUNZ.W. D., 129. 134. 181 MURANOVA, G. A., 158, 181 MYER,G., 224, 214
N NAGUIB,H. M., 121, 127, 157, 161, 181 NAKAYAMA, J., 57, 61, 62 NAZARATHY, M., 66, 75, 109 NETTERFIELD, R. P., 116, 117, 133. 134, 135, 150, 151, 159, 160, 162, 164, 165. 166. 167, 172, 173, 174, 176, 177, 179, 180, 181 NIETO-VESPERINAS, M., 32, 33, 50, 61, 62 NIR, D., 172, 181 NOBLE,B., 25, 62 NOWICK,H., 168, 182 NOWICKI,R. S., 150, 151, 181 NUMRICH,S. K., 43, 62
0 OECHSNER,H., 119, 181 OGAWA,S., 153, 182 OGURA,H., 61, 62 OGURA,S., 57, 129, 158, 180, 181 T., 214, 219, 220 OKUWAKI, OLIVE], A., 21 I , 219 O N E I L L ,E. L., 224, 245, 254, 274 OSAKABE, N., 209,212,213,214, 217. 219. 220
P PACEY,C. G . , 116, 134, 173, 174, 180 PAESOLD,G., 151, 155, 158, 181 PALERMO,C. J., 224, 274 PAPOULIS,A,, 10, 21, 37, 62 PARENT,G. B., 225, 274 PARRY,G., 265, 274 PATTANAYAK, D. N., 23,32, 62 PAWLEWICZ, W . T . , 149, 150, 151, 152, 158, 159, 164, 167, 172, 172, 176, 180, 181 PEAKE,W. H., 29, 62 PECK, W. G., 66,98, 109 PEGIS,R. J., 66, 98, 109 PENFOLD,A. S., 144. 182 PERRON,0.. 66, 67, 109 PERVEEV,A. F., 158, 181 PETIT, R.,3, 62 PETROV, N. M., 17.61 Plm, C. W.,175, 181
282
AUTHOR INDEX
PORCELLO, L. J., 224, 274 POZZI,G., 186, 211, 214, 218, 219 L., 135, 136, 179, 181 PRANEVICIOUS, PSALTIS,D., 224. 274 H. K., 151, 155, 158, 181 PULKER,
R RAGHURAM, A. C., 139, 179 RAHMAN, T. S., 59, 62 RAUSCHER, W., 175, 180 RAYLEIGH, J. W. S., 3, 6, 62 READER,P. D., 141, 180 REALE,G., 148, 150, 158, 181 REID, 1. M., 116, 181 REISSE, G., 168, 172, 182 RHODES,W. T., 224, 228. 228, 274 RICE, S. O., 29, 62 RICHARDS, P. I., 65, 109 R I ~ T E R E., , 115, 116, 148, 151, 155, 158, 181 ROACH,G. F., 24, 61 D. Z., 224, 274 ROBINSON, ROCHE,P., 3, 4, 60 S. A,, 65, 109 RODIONOV, ROGERS,G. L., 224, 274 ROGERS,J., 186, 219, 220 ROOT,W. L., 37, 60 Ross, W. E., 224. 274 ROTH,J., 119, 120, 179, 224, 229, 252, 274 ROTHROCK, D.A,, 9, 62 ROY,R., 150, 152, 181 ROY,S. M., 216, 220 RUBIO,F., 167, 181 RUDINA,0. G., 158, 174, 181 RUTLEDGE, S. K., 169, 179
S SAGITOV. S. I., 161, 180 SAINTY, W . G . , 116, 117, 133, 134, 135, 159, 160, 162, 164, 165, 166, 172, 173, 174, 177, 180, 181 SAKATA. M.,57.61 SALEM, J., 135, 181 M. I., 28, 29, 62 SANCER. SANDS, P. J., 86, 109 F. J., 168, 182 SANSALONE, SARTORI, P., 158, 165, 176, 181 SAXON, G., 201, 205, 220 S., 152, 158, 167, 182 SCHILLER,
G., 152, 158, 182 SCHIRMER, SCHMID,H., 190, 220 SCHMIDT,P. H., 168, 182 SCHOUTEN,W. J., 148, 180 SCHULLER, I. K., 123, 124, 181 SCHWEER, B., 122, 179, 181 SEAH,M. P., 119, 182 SEITZ,F., 127, 161, 182 S. A,, 161, I80 SEMILETOV, SEQUEDA,F., 135, 181 SHAIK,K. S., 252, 259, 275 SHAMIR,J., 66, 75, 109 SHEN,J., 3. 32, 56, 57, 62 A. B., 92, 11 I SHERESHEV, SHERMAN, G. C., 33, 60 H., 66, 67, 109 SHIBATA, K.,209, 220 SHIBATA, SHINAGAWA, K., 212, 214, 217, 220 SHISHA,O., 43, 60 A. B., 4, 62 SHMELEV, SHURER,C., 172, 182 SIE, S. H., 159, 162, 165, 166, 181 SIEBER,W., 152, 158, 182 SIGMUND, P., 119, 182 SILVERTOOTH, E. W., 66, 11 1 SIMONETTI, E., 162, 181 SIMPSON,J. A., 216, 219 SIRCHIN,V.K.,144, 179 SZTES,J. R., 166, 167, 182 SLEVOGT,H., 65, 109 SMIRNOV, S. E., 66, 98, 109 SMITH,B. G., 29, 62 SMITH,T., 65, 109 SMITH,W. J., 67, 109 E. G., 168, 182 SPENCER, A,, 4, 37, 60 SPIZZICHINO, STAVROUDIS, 0.N., 65,66, 109 STEIER,W. H., 92, 109 K., 167, 182 STEINFELDER, STONER,W., 224, 228, 274 STORP,S., 121, 180 STROKE,G. W., 194,220 J. , 167, 182 STRUMPHEL, STUART,P. R., 135, 179 STUDNA,A. A., 125, 179 SUDDETH, J. A,, 216, 219 Y., 92, 109 SUEMATSU, SUGAWARA, N., 129, 158, 180, 181 SUGITA,Y., 213, 217, 219, 220 SLIITO,E., 186, 220
AUTHOR INDEX
SUZUKI, K., 149, 153, 154, 158, 182 S U Z U K IR., , 213, 217, 219, 220 L. R., 3, 60 SWANSON, SWIFT,C. T., 23, 62 SWING.R. E., 225, 274 SYKES,J., 4, 61 F. R., 158, 164, 179 SZOFRAN.
T TABBARA, W., 25, 60 E., 120, 121, 182 TAGLAUER, TAJIMA, A., 1 I 1 TAKAGI, T., 117, 137, 146, 182 TAKAHASHI, S., I 1 1 TAKAHASHI, Y., 153, 182 TAKARA, H., 146, 182 M.. 208, 219 TAKASHIMA, TAKEDA, M., 65, 109 K., 66,98, 111 TAKESHI TAKIGUCHI, K., 153, 182 TAMMINCA, Y., 174. 180 H., 212, 213, 214, 219, 220 TANABE, K., 66, 67, 72, 75, 86. 92, 97, 98, TANAKA, 110
TAYLOR, A. E., 46, 62 TEER,D. G., 140, 182 TERUI,H., 167, 182 THOMAS, G. E, 147, 180 THOMAS, J. B., 257, 274 B. J., 197, 220, 223, 225, 274 THOMPSON, THORNDIKE, A. S., 9, 62 THORNTON, J.A., 128, 143, 144, 182 T I E N ,P. K., 92, 1 1 1 TITOVA, N. F., 157, 158, 179 TOICO,F.,3, 58. 60, 62 I., 3, 62 TOLSTOY. TOMITA, H., 201,220 S. G., 166, 167, 180 TOMLIN, TONOMURA, A., 186, 198, 201, 203, 205, 208, 209, 211, 212, 213, 214, 217, 219, 220 A. B., 92, 1 1 1 TSIBULYA, TWERSKY, V.. 3, 62 K. K., 136, 180 TZATSOV, U UMEZAKI, H., 213, 217, 219, 220 UNWIN,P. N. T., 218, 220 J., 194, 219 UPATNIEKS, USCINSKI,B. J., 254, 274 UYEDA,N., 186, 220
283
V ~ A L D R U., S , 186, 218, 219 VALENZUELA,G. R., 3, 29, 62 VAN BLADEL, J., 26, 27, 62 VANCITTERT,P. H., 223, 235, 269, 274 VANDER LUGT,A,, 224, 274 VARASI,M., 151, 158, 165, 176, 181, 182 VARGA, I. K., 135, 180 VENEKLASEN,L. H., 194, 220 H., 125, 179 VERBEEK, VERDET,M. E., 223, 214 VIVIAN,W. E., 224, 274 VOSSEN, J. L., 172, 182
W WADE,R. H., 186, 220 WAGNER,R. J., 29, 62 WAHL,H., 201, 212, 219, 220 WALL,D. N., 186, 219 WANG,S., 66, 111 WARDE,C., 224, 274 H., 198,220 WATANABE, WATANABE, M., 186, 220 P. C., 3, 23, 25, 62 WATERMAN, WATRASIEWICZ, B., 238, 275 WATSON, 3. G . , 5,25, 50, 53, 62 WEBB,R. P., 126, 179 WEHNER,G. K., 119, 182 WEINGKRTNER, I., 205, 220 WEISSMANTEL, C., 117, 136, 146, 168, 172, 182 WELLER,J. F., 167, 179 WELSH,L. B., 158, 164, 179 WELTER,L. N., 186, 219 WENZEL,A. R., 17, 62 WEST, E. J., 167, 179 WESTWOOD, W. D., 176, 180, 182 WHITE,H. E., 65, 108 WHITE,W. B., 150, 152, 181 WIENER,N., 57, 62 WILCOX,C. G., 3, 62 G . , 66, 67, 108 WILLONER, WILSON,S. R., 117, 141, 157, 181 D., 56,62 WINEBRENNER, J. R., 92, 1 1 I WINNERY, WITHERS, R. B., 135, 179 WIITMAACK,W., 123, 182 G., 189, 219 WOHLAND, D., 216, 219 WOHLLEBEN,
284
AUTHOR INDEX
WOLF,E., 15, 17, 20, 23, 32,34, 60, 62, 65, 108, 223, 233, 261, 262, 269, 274, 275 WOOLLAM, J. A., 170, 171, 180 WOOTERS,G., 66, 1I 1 WRIGHT,J. W., 16, 29, 31, 56, 62 WRIGHT,W. M., 43, 60 Wu. C. T., 135, 179 Wu, S.T., 272, 275 Wu, T. T., 216, 220 WUENSCHEL, P. C., 3, 61 Y YADA,K., 209, 218, 219, 220 YAGHJIAN, A. D., 27, 62 YAMADA,I., 146, 182 YANG,C. N., 216,220 YAO, S.K.,271, 274
YEE, D. S.. 135, 179 YOSHIDA,K., 168, 170, 219, 220 YOSHIDA.S., 181, 214 Yu, F. T. S., 224, 230, 238, 245, 252, 253, 259, 267, 272, 275
Z ZAVADA,J. M., 6, 60 ZEITLER,E., 186,220 ZERNIKE,F., 223, 235, 269, 275 ZHANG,Y. W., 238, 275 ZHUANG,S. L., 238, 245,252, 259. 275 ZIEGLER,J. F., 125, 180 ZIMMER,H.-G., 66, 67, 75, 111 ZIPFEL,G. G., 3, 34, 40, 41,44, 62 ZORNIG,J. G., 43, 62
SUBJECT INDEX A
E
activated reactive evaporation, 139 activation energy, 132 Aharonov-Bohm effect, 216, 217 amorphous hydrogenated silicon, 146 angular spectrum, 28
electron beam, field-emission, 192, 201 - gun, ---, 186 - lens aberration, 204 - micrograph, 199, 209 - microscopy, 204, 21 1 extinction theorem, 33, 56
B Bethe-Salpeter equation, 44 Bloch function, 58 Boersch effect, 189 Bragg reflection, 209 - resonance condition, 16 - scattering, I 1 brightness, 192 Brillouin scattering, 3 C carbon film, 167 cardinal points. 80, 82 coherence length, 189, 192 spatial, 188, 191, 248 -, temporal. 188, 224. 239, 247 communication theory. 224 connected diagram method, 39. 50. 53, 55 continued fraction, theory of, 66, 67 convolution theorem, Fourier, 226 corrclation function. 15. 40
-.
D
diffraction grating. 3. 245 dioxidc. cerium. 162 -. silicon. 148 -, titanium. 150 -. 7irconiuni. 158 Dirichlct boundary condition. 13. 14. 18, 57 - reflection coefficient, 16 Dyson equation. 57
F Fermi distribution, 188 filter, holographic, 229 matched, 229, 230 Fourier transform, stochastic, 5, 45, 47, 49,
-.
50
fractal, 8 Fraunhofer diffraction, 196, 198 - zone, 26 Fredholm integral equation, 50 Fresnel diffraction, 234 - scattering, 9 - zone. 37
c Gaussian beam, 92-97 -brackets, 66-74, 97, 98, 105 - constants, generalized, 72-77, 83, 85-87, 89. 90, 92, 94-99, 104, 105, 107 grating, color compensating, 229 -, sampling phase, 23 1 Green’s function, 4, 15, 18, 22, 23, 25, 26, 34, 35, 37, 48, 58 - theorem, 4, 17, 18, 23 gyroscope, laser, 115
H Helmholtz equation, 11, 18, 22 - integral representation, 17, 19 hologram, 185, 194-197, 199 computer generated, 224
-.
286
SUBJECT INDEX
Fourier, 230 Fresnel, 203 holography, electron, 185, 186, 194, 204, 208, 217, 218 -, Fourier-transform. 203 -, Fraunhofer, 197 in-line, 194, 202 -, off-axis, 201, 202 -, -,
Muir's symbol, 68 multilayer coating, I73 multiple scattering, 22 mutual intensity function, 234-236, 272
-.
N Neumann boundary condition, 57 Novokov-Furutsu theorem, 57 nucleation, 137, 138
I impulse response, spatial, 225 ion-assisted deposition, 140, 150, 161 ---based film deposition, 123, 137 ---beam deposition, 146 - plating, 139, 140 ionized cluster beam deposition, 146 inertial confinement fusion, 176 interferometer, achromatic fringe, 229 -, Mach-Zehnder, 208 isotope separation, laser, 176 K Kaufman gun, 144 Kirchhoff approximation, 4, 5, 19, 20, 28, 29, 59 k-space formalism, 39
L Lagrange-Helmholtz invariant, 75, 83, 86, 88, 89. 97, 102 Lens-like media, 66 Lie algebra, 66 Liouville's theorem, 192 Lippmann-Schwinger equation, 5, 32, 34, 35, 58 Lipschitz condition, 9 - exponent, 9 M magnification, angular, 79, 80 -, lateral, 79, 80, 84 longitudinal, 79 mass operator, 41 matrix method. 66 Maxwell distribution, 188 Maclaurin expansion, 100 metal-oxide semiconductor, 149 microscope, electron. 185, 186, 194 Monte Carlo method, 4 Morse lemma, 101
-.
0 optical coating, 115 - computing, 224 - imaging, 251 - processing, 223, 273 - thin film, 148 -waveguide, 175 oxide, aluminium, 149 -, titanium, 153
P paraxial ray tracing, 75 -- transfer, 87 - theory, 65, 66, 92, 97, 98 partial coherence theory, 223 partially coherent light, 228, 234 -- processor, 225, 233 -- source, 224, 227, 238, 257, 261 point spread function. 228 processing, achromatic optical, 229 -, color image, 225 -, data, 228, 229 -, optical information, 224 -, - signal, 224 -, partially coherent, 223, 224, 229-231, 235, 245, 251, 270 pupil function, 228
Q quantum electronics, 223
R Raman scattering, 3 Rayleigh diffraction formula, 15 --Fano equation, 58 --Fresnel reflection coefficient, 8, 12, 25 - hypothesis, 30, 31 --Rice theorem, 56 - roughness criterion, 6, 7, 1 I , 35 -- parameter, 7, 34
SUBJECT INDEX
rough surface scattering, 4, 5 Rudolph number, 93
S Schrodinger equation, 215 hell’s law, 12 spectral bandwidth, 240 spherical aberration constant, 206 sputtering, 118-121, 123 -, ion-beam, 144 -, magnetron, 143, 153 -, planar diode, 142
T tantalum pentoxide, 166 transfer function, 245 - _ , amplitude, 225
--, intensity, 226 --, modulation, 227 --, optical, 227 V Van Cittert-Zernike theorem, 235, 269 variational method, 4
W Weyl spectral representation, 33 Wien filter, 190 Wigner distribution function, 66 Wolfs mutual coherence function, 223 Wronskian, 89 2
zoom equation, 98, 100
- system, 98
287
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CUMULATIVE INDEX
- VOLUMES I-XXIII
ABELES,F., Methods for Determining Optical Parameters of Thin Films ABELLA.I. D., Echoes at Optical Frequencies C. I., see J. J. Clair ABITBOL. AGARWAL.G . S., Master Equation Methods in Quantum Optics AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers E. 0.. Synthesis of Optical Birefringent Networks AMMANN, J . A,, A. W. SMITH,Experimental Studies of Intensity Fluctuations ARMSTRONG, in Lasers A R N A U D 1. , A,, Hamiltonian Theory of Beam Mode Propagation BALTES,H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a Body in a Nonequilibrium Environment BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images BARRUT,H. H., The Radon Transform and its Applications BASHKIN, S., Beam-Foil Spectroscopy BECKMANN, P., Scattering of Light by Rough Surfaces BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns 111, R. E., Light Emission from High-Current Surface-Spark Discharges BFVERLY BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements , Z U I D E M AQuantum , Fluctuations in B O ~ J M AM. N , A,, W. A. VAN DE G R I N DP. Vision BOLJSQUET, P., see P. Rouard BROWN, G. S., see J. A. DeSanto B R U N N E W., R . H. PAUL,Theory of Optical Parametric Amplification and Oscillation BRYNGDAHL, 0.. Applications of Shearing Interferometry BRYNmAtiL, 0.. Evanescent Waves in Optical Imaging BoRCH. J . M.. The Meteorological Applications of Diffraction Gratings Bri I I I.RWt.CK, H. J., Principles of Optical Data-Processing CAGNAC, B., ree E. Giacobino CASASI-N I , D., D. PSALTIS, Deformation Invariant. Space-Variant Optical Pattern Recognition
I I , 249 VII, 139 XVI, 71 XI, 1 IX, 235 IX, 179 IX. 123 VI, 21 1 XI, 247 XII.
I
I, XXI, XII, VI.
67 217 287 53
XVIII, 259 XVI, 357 IX. I XXII, 77 IV, 145 XXIII. I
xv, IV, XI, II, XIX, XVII,
I 31 167 73 21 I 85
XVI, 289
290
CUMULATIVE INDEX
CEcLio, N. M., D. W. SWEENEY, Zone Plate Coded Imaging: Theory and Applications XXI, 287 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 Recent Advances in Phase Profiles Generation CLAIR,J. J., C. I. ABITBOL, XVI, 71 CLARRICOATS, P. J. B., Optical Fibre Waveguides A Review XIV, 327 COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, 1 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, 187 COLOMBEAU, B., see C. Froehly XX. 63 G., P. CRUVELLIER, M. DETAILLE, M. SAYSSE,Some New Optical COURT~S, Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects xx. 1 CREWE,A. V., Production of Electron Probes Using a Field Emission Source XI, 223 CRUVELLIER, P., see C. G. Courtes xx, 1 CUMMINS, H. Z., H. L.. SWINNEY, Light Beating Spectroscopy XIII, 133 J. C., The Statistics of Speckle Patterns DAINTY, XIV, 1 DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DECKERJr., J. A., see M. Harwit XII, 101 E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters DELANO, VII, 67 DEMARIA, A. J., Picosecond Laser Pulses IX. 31 DESANTO,J. A., G. S. BROWN,Analytical Techniques for Multiple Scattering from Rough Surfaces XXIII, 1 DETAILLE, M., see G. Courtis xx, 1 DEXTER,D. L., see D. Y. Smith X, 165 DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 DUGUAY, M. A., The Ultrafast Optical Kerr Shutter XIV, 161 J. H., Interaction of Very Intense Light with Free Electrons EBERLY, VII. 359 J. C., R.R. SNAPP,W. C. SCHIEVE,Fluctuations, Instabilities and ENGLUND, Chaos in the Laser-Driven Nonlinear Ring Cavity XXI, 355 ENNOS,A. E., Speckle Interferometry XVI, 233 FANTE,R. L., Wave Propagation in Random Media: A Systems Approach XXII, 341 FIORENTINI, A., Dynamic Characteristics of Visual Processes I, 253 FOCKE,J., Higher Order Aberration Theory IV, 1 FRANCON, M., S.MALLICK, Measurement of the Second Order Degree of Coherence VI. 71 FRIEDEN, B. R.,Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 311 FROEHLY, C., B. COLOMBEAU, M. VAMPOUILLE,Shaping and Analysis of Picosecond Light Pulses XX, 63 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I, 109 GAMO,H.,Matrix Treatment of Partial Coherence 111, 187 GHATAK, A. K., see M. S. Sodha XIII, 169 Graded Index Optical Waveguides: A Review XVIII, 1 GHATAK, A,, K. THYAGARAJAN, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy GIACOBINO, XVII, 85
-
CUMULATIVE INDEX
29 1
GINZBURG, V. L., see V. M. Agranovich IX, 235 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 11, 109 GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffraction IX, 281 Theory of Elastic Waves GOODMAN, J. W., Synthetic-Aperture Optics VIII, 1 R., The Phase Transition Concept and Coherence in Atomic Emission GRAHAM, XII, 233 HARIHARAN, P., Colour Holography XX, 263 XII, 101 HARWIT, M., J. A. DECKERJr., Modulation Techniques in Spectrometry HELSTROM, C. W., Quantum Detection Theory X, 289 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 JACQUINOT, P., B. ROIZEN-DOSSIER, Apodisation 111, 29 W.,B. P. STOICHEFF,Generation of Tunable Coherent Vacuum-UltraJAMROZ, violet Radiation XX, 325 JONES,D. G. C., see L. Allen IX, 179 A,, see C. Cohen-Tannoudji KASTLER, v, 1 KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy XX, 155 K.,Surface Deterioration of Optical Glasses KINOSITA, IV, 85 G., Multiple-Beam Interference and Natural Modes in Open KOPPELMANN, Resonators VII, I KOTTLER,F., The Elements of Radiative Transfer 111, 1 KOITLER,F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory IV, 281 KOTTLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory VI, 331 KUBOTA,H., Interference Color I, 211 A., High-Resolution Techniques in Optical Astronomy XIV, 47 LABEYRIE, LEAN,E. G., Interaction of Light and Acoustic Surface Waves XI, 123 XVI, 119 LEE,W.-H., Computer-Generated Holograms: Techniques and Applications Recent Advances in Holography VI, I LEITH,E. N., J. UPATNIEKS, LETOKHOV, V. S., Laser Selective Photophysics and Photochemistry XVI. 1 LEVI,L., Vision in Communication VIII, 343 LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 LUGIATO, L. A., Theory of Optical Bistability XXI, 69 XXII, 1 MALACARA, D., Optical and Electronic Processing of Medical Images MALLICK, L., see M. Francon VI, 71 MANDEL, L., Fluctuations of Light Beams 11, 181 XIII, 27 MANDEL,L., The Case for and against Semiclassical Radiation Theory XI, 305 MARCHAND, E. W., Gradient Index Lenses P. J., R. P. NETTERFIELD, Optical Films Produced by Ion-Based TechMARTIN, niques XXIII, 113 MASALOV, A. V., Spectral and Temporal Fluctuations of Broad-Band Laser Radiation XXII, 145
292
CUMLILATIVE INDEX
MAYSTRE. D.. Rigorous Vector Theories of Diffraction Gratings MEESSEN,A,, see P. Rouard MEtiTA. C. L.. Theory of Photoelectron Counting Quasi-Classical Theory of Laser RadiaMIKAELIAN, A. L.. M. I. TER-MIKAELIAN, tion MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction Surface and Size Effects on the Light ScatteMII.LS,D. L., K. R. SUBBASWAMY, ring Spectra of Solids K., Wave Optics and Geometrical Optics in Optical Design MIYAMOTO, MOLLOW,B. R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence MLRATA,K., Instruments for the Measuring of Optical Transfer Functions MUSSET,A., A. THELEN,Multilayer Antireflection Coatings N E ~ E R F I E LR. D ,P., see P. J. Martin OKOSHI,T., Projection-Type Holography OOUE,S., The Photographic Image G. V., Yu. I. OSTROVSKY. Holographic Methods in Plasma OSTROVSKAYA, Diagnostics OSTROVSKY, Yu. I., see G. V. Ostrovskaya PAUL,H., see W. Brunner PEGIS,R. J., The Modern Development of Hamiltonian Optics PEGIS,R. J., see E. Delano J., Photocount Statistics of Radiation Propagating through Random and PERINA, Nonlinear Media PERstim, P. S., Non-Linear Optics J., see K. Gniadek PETYKIEWICZ, PICHT,J., The Wave of a Moving Classical Electron PSALTIS,D., see D. Casasent RISEBERG, L.A., M. J. WEBER,Relaxation Phenomena in Rare-Earth Luminescence RISKEN, H., Statistical Properties of Laser Light RODDIER, F., The Effects of Atmospheric Turbulence in Optical Astronomy ROIZEN-DOSSIER, B., see P. Jacquinot ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye ROCIARD, F., P. BOUSQUET,Optical Constants of Thin Films ROUARD,P., A. MEESSEN,Optical Properties of Thin Metal Films RCJBINOWICZ, A,, The Miyamoto-Wolf Diffraction Wave RUDOLPH,D., see G. Schmahl SA'ISSE,M., see G. Courtes SAKAI,H.. see G. A. Vanasse SCHIEVE, W. C., see J. C. Englund
XXI. I XV. 17 VIII. 373 VII, 231 XVII, 279
XIX, 43 I, 31 XIX,
1 V, 199 VIII, 201
XXIII, 113 XV, 139 VII, 299 XXII, 197 XXII, 197 xv, I 1, 1 VII, 67 XVIII, 129 V, 83 IX, 281 V, 351 XVI, 289 XIV, 89 VIII, 239 XIX, 281 111. 29 XIII, 69 IV, 145 xv, 77 IV, 199 XIV, 195 xx, 1 VI, 259 XXI, 355
293
CUMULATIVE INDEX
SCtiMAHL, G . . D. RUDOLPH. Holographic Diffraction Gratings The Mutual Dependence between Coherence ProSC'tiuBERT. M., B. WILHELMI, perties of Light and Nonlinear Optical Processes Sctiui.z. G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces J., see G. Schulz SCHWIDER, Tools of Theoretical Quantum Optics Scut.i.v, M. O., K. G. WHITNEY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical SENITZKY, Framework SIPE,J. E., see J. Van Kranendonk SITTIG.E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self-Induced Transparency SMITH,A. W., see J. A. Armstrong SMITH,D. Y.,D. L. DEXTER,Optical Absorption Strength of Defects in Insulators SMITH,R. W., The Use of Image Tubes as Shutters SNAPP,R. R., see J. C. Englund V. K. TRIPATHI,Self Focusing of Laser Beams in SODHA,M. S., A. K. GHATAK. Plasmas and Semiconductors STEEL,W. H., Two-Beam Interferometry STOICHEFF, B. P., see W. Jamroz STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere S'rRoKE, G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy SUBBASWAMY, K. R., see D. L. Mills SVELTO,0.. Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams SWEENEY, D. W., see N. M. Ceglio SWINNEY. H. H., see H. Z. Cummins TANAKA, K., Paraxial Theory in Optical Design in Terms of Gaussian Brackets TANGO, W. J., R. Q. TWISS,Michelson Stellar Interferometry TATARSKII, V. I., V. U. ZAVOROTNYI, Strong Fluctuation in Light Propagation in a Randomly lnhomogeneous Medium C. A., see H. Lipson TAYLOR, TER-MIKAELIAN, M. L., see A. L. Mikaelian T H E L E N , A., see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see A. Ghatak TONOMURA. A,, Electron Holography V. K., see M. S. Sodha TRIPATHI, TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering Twiss, R. Q., see W. J. Tango UPATNIEKS, J., see E. N. Leith UPSTILL.C., see M. V. Berry
XIV, 195 XVII, XIII, XIII, X.
163 93 93 89
XVI, 41. XV, 245 X, 229 XII, 53 VI, 21 I X, 165 x, 45 XXI, 355 XIII, 169 V, 145 XX. 325 IX, 73
II, I XIX, 43 XII, 1 XXI, 287 VIII, 133 XXIII, 63 XVII, 239 XVIII, 207 V, 287 VII, 231 VIII, 201 VII, 169 XVIII, I XXIII, 183 XIII, 169
II, 131 XVII, 239 VI, 1 XVIII, 259
294
CUMULATIVE INDEX
USHIODA, S.. Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids VAMPOUILLE, M., see C. Froehly VANASSE, G. A,, H. SAKAI, Fourier Spectroscopy VAN DE GRIND, w. A,, see M. A. Bouman VAN HEEL,A. C. S., Modern Alignment Devices VAN KRANENDONK,J., J. E. SIPE, Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media VERNIER, P., Photoemission WEBER,M.J., see L. A. Riseberg WELFORD, W. T., Aberration Theory of Gratings and Grating Mountings WELFORD, W. T., Aplanatism and Isoplanatism WILHELMI, B.. see M. Schubert WITNEY, K. G., see M. 0.SCUllY H., On Basic Analogies and Principal Differences between Optical and WOLTER, Electronic Information WYNNE,C. G., Field Correctors for Astronomical Telescopes YAMAGUCHI,I., Fringe Formations in Deformation and Vibration Measurements Using Laser Light YAMAJI,K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy YOSHINAGA,H.,Recent Developments in Far Infrared Spectroscopic Techniques Yu, F. T. S . , Principles of Optical Processing with Partially Coherent Light ZAVOROTNYI, V. U., see V. 1. Tatarskii ZUIDEMA, P., see M. A. Bouman
XIX, XX, VI, XXII, I,
139 63 259 77 289
XV, 245 XIV, 245 XIV, 89 IV, 241 XIII, 267 XVII, 163 X, 89
I, 155 X, 137 XXII, 271 VI, 105 VIII, XI, XXIII, XVIII, XXII,
295 77 227 207 77
