Progress in Optics

PROGRESS IN OPTICS

VOLUME XIX

L. ALLEN,

M. FRANCON,

E. INGELSTAM,

K. KINOSITA,

A. KUJAWSKI,

A. LOHMANN,

M. MOVSESSIAN,

G. SCHmZ,

W. H. STEEL,

W. T. WELFORD,

EDITORIAL ADVISORY BOARD

Brighton, England

Pans, France

Stockholm, Sweden

Tokyo, Japan

Warsaw, Poland

Erlangen, Fed. Rep. Germany

Armenia, U.S.S.R.

Berlin, D.D.R.

Sydney, Australia

London, England

PROGRESS IN OPTICS V O L U M E XIX

EDITED BY

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

Contributors

B. R. MOLLOW, D. L. MILLS, K. R. SUBBASWAMY S. USHIODA, H. J. BU’ITERWECK

F. RODDIER

1981

NORTH-HOLLAND PUBLISHING COMPANY- AMSTERDAM * NEW YORK . OXFORD

NORTH-HOLLAND PUBLISHING COMPANY-1981

AU Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,

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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297 ISBN: 0 444 85444 4

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM . OXFORD

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PRINrED IN THE NETHERLANDS

I . I1 .

I11 .

IV . V .

VI . VII . VIII

I .

I1 .

I11 . IV .

V . VI .

I . I1 . I11 .

I . I1 . 111 . IV . V . VI .

VII .

I . I1 .

CONTENTS OF VOLUME 1 ( 1 9 6 1 )

THE MODERN DEVELOPMENT OF HAMILTONIAN Om~cs. R . J . PEGIS . . . WAVE O m c s AND GEOMETRICAL OPTICS IN OPTICAL DESIGN. K . MIYAMOTO . . . . . . . . . . . . . . . . . . . . . . . . . . . THE INTENSITY DISTRIBUTION AND TOTAL ILLUMINATION OF ABERRATION- FREE DIFFRACTION I ~ G E S . R . BARAKAT . . . . . . . . . . . . . . LIGHT AND INFORMATION. D . GABOR . . . . . . . . . . . . . . . .

AND ELECTRONIC INFORMATION. H . WOLTER . . . . . . . . . . . . INTERFERENCE COLOR. H . KUBOTA . . . . . . . . . . . . . . . .

MODERN ALIGNMENT DEVICES . A . C . S . VAN HEEL . . . . . . . . . .

ON BASIC ANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN OmCAL

DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES. A . RORENTINl . . .

CONTENTS OF VOLUME 11 (1963)

RULING. TESTING AND USE OF OPTICAL GRATINGS FOR HIGH-RESOLUTION SPECTROSCOPY. G . w . STROKE THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS. J . M . BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIFFUSION THROUGH NON-UNIFORM MEDIA. R . G . GIOVANELLI CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND

BY SPATIAL FREQUENCY FILTERING. J . TSUJIUCHI . . . . . . . . . . . FLUCTUATIONS OF LIGHT BEAMS. L . MANDEL . . . . . . . . . . . . METHODS FOR DETERMINING OPTICAL PARAMETERS OF THIN FILMS. F . A B a s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . .

CONTENTS OF VOLUME 111 (1964)

THEELEMENTSOFRADIATIVETRANSFER,F.K~T~LER . . . . APODISATION. P . JACQUINOT AND B . ROIZEN-DOSSIER MATRIX TREATMENT OF PARTIAL COHERENCE . H . GAMO

. . . . . . .

CONTENTS OF VOLUME IV (1965)

HIGHER ORDER ABERRATXON THEORY. J . FOCKE APPLICATIONS OF SHEARING INTERFEROMETRY. 0 . BRYNGDAHL SURFACE DETERIORATION OF O ~ I C A L GLASSES. K . KINOSITA OFTICAL CONSTANTS OF THIN FILMS. P . ROUARD AND P . BOUSQUET THE ~~NAMOTO-WOLF DIFFRACTION WAVE. A . RUBINOWICZ ABERRATION THEORY OF GRATINGS AND GRATING MOUNTINGS. W . T .

DIFFRACTION AT A BLACK SCREEN. PART I: KIRCHHOFF’S THEORY. F . KOTI-LER . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

. . . . . . . .

. . . . .

WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS OF VOLUME V (1966)

O ~ I C A L PUMPING. C . COHEN-TANNOUDJI AND A . KASTLER NON-LINEAR Omcs. P . S . PERSHAN . . . . . . . . . . . . . . . .

111 . TWO-BEAM INTERFEROMETRY. W . H . STEEL . . . . . . . . . . . . .

1-29

3 1-66

67-108 109-153

155-210 211-251 253-288 289-329

1-72

73-108 109-129

13 1-1 80 181-248

249-288

1-28 29-186

187-332

1-36 37-83

85-143 145- 197 199-240

241-280

281-314

1-81 83-144

145-197

IV.

V.

VI.

VII.

I. 11. 111.

IV. V. VI.

VII. VIIl

I.

11.

111. IV. V.

VI. VII.

I. 11. 111. IV. V. VI.

VII.

INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFER FUNCTIONS, K. MURATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGHT REFLECTION FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE INDEX, R. JACOBSSON . . . . . . . . . . . . . . . . . . . . . .

Opr~cs, H. LIPSON AND C. A. TAYLOR . . . , . . . . . . . . . , . THE WAVE OF A MOVING CLASSICAL ELECTRON, J. PICHT . . . . . , .

X-RAY CRYSTAL-STRUCTURE DETERMINATION AS A BRANCH OF PHYSICAL

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

RECENT ADVANCES IN HOLOGRAPHY, E. N. LEM AND J. UPATNIEKS . .

MEASUREMENT OF THE SECOND ORDER DEGREE OF COHERENCE, M. SCATTERING OF LIGHT BY ROUGH SURFACES, P. BECKMANN . . . . . .

FRANCON AND S. MALLICK . . . . . . . . . . . . . . . . . . . . DESIGN OF ZOOM LENSES, K. YAMAJI . . . . . . . . . . . . . . . . SOME APPLICATIONS OF LASERS TO INTERFEROMETRY, D. R. HERRIOTT . EXPERIMENTAL STUDIES OF INTENSITY FLUCTUATIONS IN LASERS, J. A. ARMSTRONG AND A. W. SMITH . . . . . . , . . . . . . . . . , . FOURIER SPECTROSCOPY, G. A. VANASSE AND H. SAKAI . . . . . . , .

F . K o ~ R . . . . . . . . . . . . . . . . . . . . . . . . . . . DIFFRACTION AT A BLACK SCREEN, PART 11: ELECTROMAGNETIC THEORY,

C O N T E N T S OF V O L U M E V I I ( 1 9 6 9 ) MULTIPLE-BEAM INTERFERENCE AND NATURAL MODES IN OPEN RES- ONATORS, G. KOPPELMAN . . . . . . . . . . . . . . . . . . . . .

A N O ~ R. J. PEGIS . . . . . . . . . . . . . . . . . . . . . . . ECHOES AT O ~ C A L FREQUENCTES, I. D. ABELLA . . . . . . . . . . IMAGE FORMATION WITH PARTIALLY COHERENT LIGHT, B. J . THOMFSON QUASI-CLASSICAL THEORY OF LASER RADIATION, A. L. MIKAELIAN AND

M. L. TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . THE PHOTOGRAPHIC IMAGE, S. OOUE , . . . . . . . . . . . . . . IIWERACTION OF VERY INTENSE LIGHT m FREE ELECTRONS, J. H. EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . .

METHODS OF SYNTHESIS FOR DIELECTRIC MULTILAYER FILTERS, E. DEL-

C O N T E N T S OF V O L U M E V I I I ( 1 9 7 0 ) SYNTHETIC-APERTURE Omcs , J. W. GOODMAN . . . . . . . . . THE O ~ C A L PERFORMANCE OF THE HUMAN EYE, G. A. FRY . . . . . LIGHT BEATING SPECTROSCOPY, H. Z . C W I N S AND H. L. SWINNEY . . MULTILAYER ANTIREFLECTION COATINGS, A. MUSSET AND A. THELEN . STATISTICAL PROPERTIES OF LASER LIGHT, H. RISKEN . . . . . . . . COHERENCE THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY, T. Y W O T O . . . . . . . . . . . . . . . . . . . VISION IN COMMUNICATION, L. LEVI . . . . . . . . . . . . . . . .

VIII. THEORY OF PHOTOELECTRON COUNTING, C. L. MEHTA . . . . . . . .

C O N T E N T S OF V O L U M E I X ( 1 9 7 1 ) I. GAS LASERS AND THEIR APPLICATION TO PRECISE LENGTH MEASURE-

MENTS, A.L. BLOOM . . . . . . . . . . . . . . . . . . . . . . .

199-245

247-286

287-350 351-370

1-52 53-69

7 1-104 105- 170 17 1-209

21 1-257 2 5 9-3 3 0

331-377

1-66

67-137 139-168 169-230

231-297 299-358

3 59-4 15

1-50 51-131

133-200 201-237 2 3 9-294

295-341 343-372 373-440

1-30

11. PICOSECOND LASER PULSES, A. J. DEMARIA . . . . . . . . . . . . 111. OF-TICAL PROPAGATION THROUGH THE TURBULENT ATMOSPHERE, J. W.

STROHBEHN . . . . . . . . , . . . . . . . . . . . . . . . . . . IV. SYNTHESIS OF O ~ C A L BIREFRINGENT NETWORKS, E. 0. AMMA” . . . V. MODE LOCKING IN GAS LASERS, L. ALLEN AND D. G. C. JONES . . . . VI. CRYSTAL Omcs WITH SPATIAL DISPERSION, V. M. AGRANOVICH AND V.

L.GINZBURG . . . . . . . . . . . . . . . . . . . . . . . . . . VII. APPLICATIONS OF OFTICAL METHODS IN THE DIFFRACTION THEORY OF

ELASTIC WAVES, K. GNIADEK AND J. PETYKIEWICZ . . . . , . . . . .

SIGNALS, BASED ON USE OFTHE PROLATE FUNCTIONS, B. R. FRIEDEN . . VIII. EVALUATION, DESIGN AND EXTRAPOLATION METHODS FOR OPnCAL

CONTENTS OF V O L U M E X (1972) I. 11. 111.

IV. V.

VI. VII.

I. 11.

111. IV. V.

VI. VII.

I.

11. 111.

IV.

V.

VI.

I.

BANDWIDTH COMPRESSION OF OmCAL IMAGES, T. s. HUANG . . . . . THE USE OF IMAGE TUBES AS SHUTTERS, R. W. SMITH . . . . . . . . TOOLS OF THEORETICAL QUANTUM Omcs , M. 0. SCULLY AND K. G. WHITNEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . FIELD CORRECTORS FOR ASTRONOMICAL TELESCOPES, C. G. WYNNE . . OFTICAL ABSORFITON STRENGTH OF DEFECTS IN INSULATORS, D. Y. SMITH AND D. L. DEXTER . . . . . . . . . . . . . . . . . . , . .

Q U A N T U M DETECTION THEORY, C. W. HELSTROM . . . . . . . , . , ELASTOO~C LIGHT MODULATION AND DEFLECTION, E. K. S I I T G , . .

CONTENTS OF V O L U M E X I (1973) MASTER EQUATION METHODS IN Q U A N T U M Omcs, G. S. AGARWAL . .

H.YOSHINAGA . . . . . . . . . . . . , . . . . . . . . . . . . RECENT DEVELOPMENTS IN FAR INFRARED SPECTROSCOPIC TECHNIQUES,

INTERACTION OF LIGHT AND ACOUSTIC SURFACE WAVES, E. G. LEAN . . EVANESCENT WAVES IN OPTICAL IMAGING, 0. BRYNGDAHL . . . , . . PRODUCTION OF ELECTRON PROBES USING A FIELD EMISSION SOURCE, A. v.cREwE . . . . . . . . . . . . . . . . . . . . . . . . . . . HAMILTONIAN THEORY OF BEAM MODE PROPAGATION, J. A. ARNAUD . GRADIENT INDEX LENSES, E. W. MARCHAND . . . . . . . . . . . .

CONTENTS OF V O L U M E XI1 (1974) SELF-FOCUSING, SELF-TRAPPING, AND SELF-PHASE MODULATION OF

LASER BEAMS, 0. SVELTO . . . . . . . . . . . . . . . . . . . . SELF-INDUCED TRANSPARENCY, R. E. SLUSHER . . . . . . . . . . . MODULATION TECHNIQUES IN SPECTROMETRY, M. HARWIT, J. A. DECKER JR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS, K. H. DREXHAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . THE PHASE TRANSITION CONCEF-T AND COHERENCE IN ATOMIC EMISSION, R.GRAHAM . . . . . . . . . . . . . . . . . . . . . . . . . . BEAM-FOIL SPECTROSCOPY, s. BASHKIN . . . . . . . . . . . . . . .

CONTENTS OF VOLUME XI11 (1976) ON THE VALIDITY OF KIRCHHOFFS LAW OF HEAT RADIATION FOR A BODY IN A NONEQUILEIRJUM ENVIRONMENT, H. P. BALTES . . . . . . . . .

3 1-7 1

73-122 123-177 179-234

235-280

281-310

311-407

1-44 45-87

89-135 137- 164

165-228 229-288 289-369

1-76

77-122 123-166 167-22 1

223-246 247-304 305-337

1-51 53-100

101-162

163-232

233-286 287-344

1-25

11.

111.

IV.

V.

VI.

I. 11. 111.

IV. V. VI . VII.

I.

11. 111. IV. V.

I.

11.

111.

IV. V.

VI.

VII.

I. 11.

THE CASE FOR AND AGAINST SEMICLASSICAL RADIATION THEORY, L. MANDEL . . . . . . . . . . . . . . . . . . . . . . . . . . . .

THE HUMAN EYE, W. M. ROSENBLUM, J. L. CHRISTENSEN . . . . . . . INTERFEROMETRIC TESTING OF SMOOTH SURFACES, G. SCHULZ, J. SCHWIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . SELF FOCUSING OF LASER BEAMS IN PLASMAS AND SEMICONDUCTORS, M. S. SODHA, A. K. GHATAK, V. K. TFUPATHI . . . . . . . . . . . APLANATISM AND ISOPLANATISM, W. T. WELFORD . . . . . . . . . .

C O N T E N T S OF V O L U M E X I V ( 1 9 7 7 )

THE STATISTICS OF SPECKLE PATERNS, J. C. DAINTY . . . . . . . . . . HIGH-RESOLUTION TECHNIQUES IN OFTTCAL ASTRONOMY, A. LABEYRIE . RELAXATION PHENOMENA IN --EARTH LUMINESCENCE, L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . THE ULTRAFAST O P ~ C A L KERR SHUTTER, M. A. DUGUAY . . . . . . . HOLOGRAPHIC DIFFRACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . PHOTOEMISSION, P. J. VERNIER . . . . . . . . . . . . . . . . . . ~ P ~ C A L FIBRE WAVEGUIDES-A REVIEW, P. J. B. CLARRICOATS . . .

OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATTON MEASUREMENTS OF

C O N T E N T S OF V O L U M E X V ( 1 9 7 7 )

THEORY OF O ~ C A L PARAMETFUC AMPLIFICATION AND OSCILLATION, W.

OPTICAL PROPERTIES OF THIN METAL FILMS, P. ROUARD, A. MEESSEN . PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI . . . . . . . . . . . . QUASI-OFTTCAL TECHNIQUES OF RADIO ASTRONOMY, T. W. COLE . . .

DIELECTRIC MEDIA, J. VAN KRANENDONK, J. E. SIPE . . . . . . . . .

BRUNNER, H. PAUL . . . . . . . . . . . . . . . . . . . . . . .

FOUNDATIONS OF THE MACROSCOPIC ELECTROMAGNETIC THEORY OF

C O N T E N T S OF V O L U M E X V I ( 1 9 7 8 ) LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY, V. S. LETOKHOV . . . . . . . . . . . . . . . . . . . . . . . . . . . RECENT ADVANCES IN PHASE PROFILES GENERATION, J. J. CWR, C. I. ABITBOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS, W.-H. LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . SPECKLE INTERFEROMETRY, A. E. ENNOS . . . . . . . . . . . . . .

TION, D. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . LIGHT EMISSION FROM HIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLY 111 . . . . . . . . . . . . . . . . . . . . . . . .

FRAMEWORK, I. R. SENITZKY . . . . . . . . . . . . . . . . . . .

DEFORMATION INVARIANT, SPACE-VARIANT OPITCAL PATIERN RECOGNI-

SEMICLASSICAL RADIATION THEORY WITHIN A QUANTUM-MECHANICAL

C O N T E N T S OF V O L U M E X V I I ( 1 9 8 0 ) HETERODYNE HOLOGRAPHIC INTERFEROMETRY, R. D~NDLIKER . . . . DOPPLER-FREE MULTPHOTON SPECTROSCOPY, E. GIACOBINO, B. CAG-

27-68

69-91

93-167

169-265 267-292

1-46 47-87

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

1-75 77-137

139-1 85 187-244

245-350

1-69

71-117

119-232 233-288

289-356

357-41 1

413-448

1-84

NAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85-161

111. THE MUTUAL DEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR OPTICAL PROCESSES, M. SCHUBERT, B. WILHELMI . . . 163-238

IV. MICHELSON STELLAR INTERFEROMETRY, W. J. TANGO, R. Q. Twtss . . . 239-277

MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . 279-345 V. SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION, A. L.

CONTENTS OF V O L U M E X V I I I (1980)

I. GRADED INDEX OFTICAL WAVEGUIDES: A REVIEW, A. GHATAK, K.

11. PHOTOCOUNT STATISTICS OF RADIATION PROPAGATING THROUGH RAN- THYAGARAJAN . . . . . . . . . . . . . . . . . . . . . . . . . . 1-126

DOM AND NONLINEAR MEDIA, J. PE~INA. . . . . . . . . . . . . . . 127-203 111. STRONG FLUCTUATIONS IN LIGHT PROPAGATION IN A RANDOMLY

INHOMOGENEOUS IMEDIUM, V. I. TATARSKII, V. U. ZAVOROTNYI . . . . 205-256

TIONPATTERNS, M. V. BERRY, C. UFSTILL. . . . . . . . . . . . . . 257-346 Iv. CATASTROPHE OPTICS: MORPHOLOGIES OF CAUSTICS AND THElR DIFFRAC-

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PREFACE

The present volume contains five review articles dealing with various topics of current interest in modern optics. Although four of the articles describe recent investigations relating to light-scattering, they deal with very different aspects of this broad subject. They cover topics such as resonance scattering, surface and size effects on Raman and Brillouin spectra of solids and effects of atmospheric light scattering on astronomical measurements. The remaining article deals with fundamentals of optical data-processing, a subject that is becoming of increasing importance for communication theory.

Department of Physics and Astronomy University of Rochester Rochester, N.Y. 14627

EMIL WOLF

February 1981


CONTENTS

I . THEORY OF INTENSITY DEPENDENT RESONANCE LIGHT SCATTERING AND RESONANCE FLUORESCENCE

by B . R . MOLLOW (BOSTON. MASSACHUSETTS)

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 . DESCRIITION OF THE ELECTROMAGNETIC FIELD . . . . . . . . . . . . . . .

RELAXATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The exact quantum statistical method

1.1 Survey of early work . . . . . . . . . . . . . . . . . . . . . . . .

3 . SCATTERING SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM WITH b D I A T N E

. . . . . . . . . . . . . . . . . 3.1.1 Coherent field solution in the Schrodinger picture 3.1.2 Comparison of theory with experiment 3.1.3 n-photon incident field; the dressed atom method 3.1.4 Coherent field solution in the Heisenberg picture

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

. . . . . . . . .

. . . . . . . . . 3.2 Pure state analyses . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Incident field statistics . . . . . . . . . . . . . . . . . . . . . . . .

4 . COLLISIONAL RELAXATION . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The impact approximation . . . . . . . . . . . . . . . . . . . . . . 4.2 Collisions of nonzero duration

5.1 The laser-coupled transition . . . . . . . . . . . . . . . . . . . . . 5.2 Transitions involving other states

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . 5 . MULTILEVEL ATOM WITH Two LASER-COUPLED STATES . . . . . . . . . . .

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

I1 . SURFACE AND SIZE EFFECTS ON THE LIGHT SCATTERING SPECTRA OF SOLIDS

by D . L . MILLS (IRVINE. CALIFORNIA) and K . R . SUBBASWAMY (LEXINGTON. KENTUCKY)

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . LIGHT SCATTERING FROM OPAQUE MEDIA AND FILMS OF FINITE THICKNESS;

QUALITATWE CONSIDERATIONS . . . . . . . . . . . . . . . . . . . . . . 3 . SURFACE AND GUIDED WAVE POLAMTONS 4 . LIGHT SCATTERING FROM SURFACE AND GUIDED WAVE POLARITONS

4.1 Derivation of the spectral differential cross section 4.2 Raman scattering from polaritons in thin crystals

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

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

4.2.1 Surface and guided wave polaritons in free-standing GaP films 4.2.2 Surface polaritons in a GaAs film on a sapphire substrate

. . . . . . . . .

5 . SURFACE AND SIZE E m m ON BRILLOUIN SCATTERING FROM ACOUSTICAL h O N O N S AND SPIN WAVES . . . . . . . . . . . . . . . . . . . . . . .

3 6

12

15 15 15 20 21 23 25 28 31 31 35 36 36 39 40

47

51 64 76 77 83 85 94

95

xiv CONTENTS

5.1 Acoustical phonons in opaque solids . . . . . . . . . . . . . . . . . 95 5.2 The scattering of light from spin waves on the surface of opaque ferromagnets

and in thin films . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 . LIGHT SCATTERING AS A MICROSCOPIC PROBE OF THE SURFACE REGION 124 7 . CONCLUDINGREMARKS . . . . . . . . . . . . . . . . . . . . . . . . . 134 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

. . . .

111 . LIGHT SCATTERING SPECTROSCOPY OF SURFACE ELECTROMAGNETIC WAVES IN SOLIDS

by S . USHIODA (IRVINE. CALIFORNIA)

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2 . SURFACE POLARITONS IN DIFFERENT GEOMETRIES . . . . . . . . . . . . . . 144

2.1 Single interface surface polaritons . . . . . . . . . . . . . . . . . . 145 2.2 Double interface surface polaritons . . . . . . . . . . . . . . . . . . 148 2.3 Guided wave polaritons (GWP) . . . . . . . . . . . . . . . . . . . 152

3 . RAMAN SCATTERING BY SURFACE POLA~UT~NS . . . . . . . . . . . . . . . 155 3.1 Basic concepts of Raman scattering . . . . . . . . . . . . . . . . . . 156 3.2 Raman scattering intensity and selection rule for surface polaritons 161 3.3 Experimental method . . . . . . . . . . . . . . . . . . . . . . . . 166

4 . EXPERIMENTALRESULTS . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Single interface modes (SIM) and the selection rule 172 4.2 Double interface modes (DIM) . . . . . . . . . . . . . . . . . . . . 180 4.3 Guided wave polaritons (GWP) . . . . . . . . . . . . . . . . . . . 185

5 . E m c r s OF SURFACE ROUGHNESS . . . . . . . . . . . . . . . . . . . . . 190 5.1 Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . 191 5.2 Experimental results and comparison with theory . . . . . . . . . . . . 194

6 . CONCLUDINGREWS . . . . . . . . . . . . . . . . . . . . . . . . . 202 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Appendix: DERIVATION OF m DISPOSAL RELATION FOR SURFACE POLARITONS AND

GUIDED-WAVE POLARITONS IN A DOUBLE INTERFACE GEOMETRY . . . 203 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

. . . .

. . . . . . . . . . .

IV . PRINCIPLES OF OF'TICAL DATA-PROCESSING

by H . J . B ~ R W E C K (EINDHOVEN, THE NETHERLANDS)

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 2 . FIELD THEORY OF O m c a SYSTEMS . . . . . . . . . . . . . . . . . . . 216

2.1 The data-processing mode . . . . . . . . . . . . . . . . . . . . . 216 2.2 The reciprocity theorem . . . . . . . . . . . . . . . . . . . . . . 220

3 . SYSTEM-THEORETICAL APPROACH TO COHERENT O ~ C A L SIGNAL PROCESSORS . 222 3.1 Input-outputrelationsinspace andfrequencydomain . . . . . . . . . 222 3.2 Cascades and inverse systems . . . . . . . . . . . . . . . . . . . . 225

4 . PART~WV COHERENT ILLUMINATION . . . . . . . . . . . . . . . . . . 227 4.1 Spectral treatment of partial coherence . . . . . . . . . . . . . . . . 227 4.2 Incoherent illumination . . . . . . . . . . . . . . . . . . . . . . 230 4.3 Coherent illumination . . . . . . . . . . . . . . . . . . . . . . . 231

5 . BASIC SYSTEM CONS- . . . . . . . . . . . . . . . . . . . . . . 232

CONTENTS XV

5.1 Single constraints . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Multiple constraints . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . 6.1 Physical systems . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Abstract systems . . . . . . . . . . . . . . . . . . . . . . . . .

6 . EXAMPLES OF PHYSICAL AND A~STRACX SYSTEMS

6.3 Cascades, inversions, and dualities of elementary systems . . . . . . . . 7 . OPERATIONAL NOTATION OF OFTICAL SYSTEMS AND BASIC CASCADE EQWA-

LENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 An operational notation . . . . . . . . . . . . . . . . . . . . . . 7.2 Cascade equivalences . . . . . . . . . . . . . . . . . . . . . . .

8 . OPERATIONAL ANALYSIS OF OFTICAL SYSTEMS . . . . . . . . . . . . . . . 8.1 Actual realizations of Fourier transformer and magnifier 8.2 Fourier filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Insertion of a modulator in front of a focus; aberration errors 8.4 Some phenomena in free-space propagation . . . . . . . . . . . . .

9.1 Equivalent “circuits” . . . . . . . . . . . . . . . . . . . . . . . 9.2 Modulators in L?G-systerns . . . . . . . . . . . . . . . . . . . . . 9.3 Systems containing cylindrical lenses . . . . . . . . . . . . . . . . .

10 . SHIFT-INVARIANT SYSTEMS: COHERENT VERSUS INCOHERENT ILLUMINATION . . 10.1 Coherent illumination . . . . . . . . . . . . . . . . . . . . . . . 10.2 Incoherent illumination . . . . . . . . . . . . . . . . . . . . . . 10.3 Low-pass filters . . . . . . . . . . . . . . . . . . . . . . . . .

11 . RELATED TOPICS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . .

9 . SYSTEMS COMPOUNDED OF LENSES AND SECTIONS OF FREE SPACE (G-SYSTEMS)

V . THE EFEECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY

by F . RODDIER (NICE. FRANCE)

1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . STATISTICAL PROPERTIES OF ATMOSPHERIC TURJWLENCE . . . . . . . . . .

2.1 Structure of turbulence . . . . . . . . . . . . . . . . . . . . . . . 2.2 Temperature and humidity fluctuations 2.3 Refractive index fluctuations . . . . . . . . . . . . . . . . . . . . 2.4 Dependence of C i with height and time . . . . . . . . . . . . . . .

3 . STATISTICAL PROPERTIES OF THE PERTURBED COMPLEX FIELD . . . . . . . . 3.1 Output of a thin turbulence layer . . . . . . . . . . . . . . . . . . 3.2 Multiple layers and thick layers . . . . . . . . . . . . . . . . . . . 3.3 Fourth order moments . . . . . . . . . . . . . . . . . . . . . . .

4 . LONG-EXPOSURE IMAGES . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Relation between the object and the image . . . . . . . . . . . . . .

4.3 Resolving power . . . . . . . . . . . . . . . . . . . . . . . . .

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

4.2 Expression for the optical transfer function

4.4 Application to Michelson’s stellar interferometry 4.5 Experimental measurements of the long-exposure transfer function

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

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

5 . SHORT-EXPOSLJRE IMAGES . . . . . . . . . . . . . . . . . . . . . . . 5.1 The image energy spectrum . . . . . . . . . . . . . . . . . . . . .

232 239 240 245 245 249 251

252 252 254 256 256 258 259 262 263 263 265 267 268 268 270 272 275 279

283 284 284 286 287 288 291 292 295 296 297 298 298 300 302 306 309 309

xvi CONTENTS

5.2 The aperture-synthesis approach . . . . . . . . . . . . . . . . . . 5.3 The probability density functions of stellar speckles . . . . . . . . . .

. . . . . . . . . . . . . 6.1 Speckle cross-spectra . . . . . . . . . . . . . . . . . . . . . . . 6.2 Effect of non-isoplanicity . . . . . . . . . . . . . . . . . . . . . . 6.3 The time evolution of speckles . . . . . . . . . . . . . . . . . . . 6.4 Effect of the exposure time on the image spectrum . . . . . . . . . . .

7 . O ~ C A L PATH FLUCTLJATIONS . . . . . . . . . . . . . . . . . . . . . . 7.1 Effect of a thin turbulent layer . . . . . . . . . . . . . . . . . . . 7.2 Multiple layers and thick layers . . . . . . . . . . . . . . . . . . . 7.3 The near-field approximation . . . . . . . . . . . . . . . . . . . . 7.4 Phase fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Angle-of-arrival fluctuations . . . . . . . . . . . . . . . . . . . . 7.6 Image motion and blumng . . . . . . . . . . . . . . . . . . . . .

8 . STELLARS~INTILLATION . . . . . . . . . . . . . . . . . . . . . . . . 8.1 First order statistics . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Second order statistics . . . . . . . . . . . . . . . . . . . . . . .

9 . A~PLICATIONS TO HIGH RESOLUTION IMAGING . . . . . . . . . . . . . . . 9.1 Classical methods . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Adaptive optics . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Mchelson interferometry . . . . . . . . . . . . . . . . . . . . . .

6 . EXPOSURE-TIME AND NON-ISOPLANICITY Emm

9.4 Speckle interferometry . . . . . . . . . . . . . . . . . . . . . . . 9.5 Image reconstruction . . . . . . . . . . . . . . . . . . . . . . .

10 . SEEING MONITORSAND S w TFSTING . . . . . . . . . . . . . . . . . . . 10.1 Seeing monitors . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Atmospheric soundings . . . . . . . . . . . . . . . . . . . . . . 10.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RF3WRENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 318 319 319 320 324 326 328 328 331 332 334 334 337 341 341 345 350 350 352 354 357 360 360 361 365 366 367 368

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 SUBJECTINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 CUMULATIVE INDEX- VOLUMES I-XIX . . . . . . . . . . . . . . . 389

E. WOLF, PROGRESS IN OPTICS XIX @ NORTH-HOLLAND 1981

I

THEORY OF INTENSITY DEPENDENT RESONANCE LIGHT SCATTERING AND RESONANCE FLUORESCENCE*

BY

B. R. MOLLOW

Department of Physics, Uniuersiiy of Massachusetts at Boston, Boston, Massachusetts 021 25, U.S.A.

* Supported by the National Science Foundation.

CONTENTS

PAGE

$ 1 . INTRODUCTION . . . . . . . . . . . . . . . . 3

$ 2. DESCRIPTION OF THE ELECTROMAGNETIC FIELD . . 12

$ 3 . SCATTERING SPECTRUM FOR A CLOSED TWO- LEVEL SYSTEM WITH RADIATIVE RELAXATION. . . 15

$ 4. COLLISIONAL RELAXATION . . . . . . . . . . . 31

$ 5 . MULTILEVEL ATOM WITH TWO LASER-COUPLED STATES. . . . . . . . . . . . . . . . . . . . 36

REFERENCES . . . . . . . . . . . . . . . . . . . 40

6 1. Introduction

When a strong monochromatic field is incident upon an isolated, stationary atom, important changes are induced both within the atom itself and in the way it interacts with other systems. If the incident field frequency w lies near the resonance frequency wl0 = ( E ~ - E o ) / l i for transitions between a particular pair of atomic states (1) and 10) with energies and E" , respectively, i.e., if

A = o - o , ~ < < w , (1.1)

then the field can induce appreciable oscillations both in the populations of the pair of coupled levels and in the amplitude of the oscillating, field-induced dipole moment if the field intensity is sufficiently high. These oscillations occur at the Rabi nutation frequency

0' = (R2+ A ');,

where R is defined in terms of the incident field amplitude E and the atomic dipole matrix element plo as

R = p Elh. (1.3)

The parameter R will be assumed throughout this article to obey the inequality

(thus ruling out harmonic generation), but to be otherwise unrestricted in value. The conditions (1.1) and (1.4) are sufficient to justify the resonance (or rotating wave) approximation, which retains only the coupliqg between terms of nearly equal frequency.

Important effects which are nonlinear in the incident field intensity arise when R is comparable in magnitude to or greater than both the detuning A and the (homogeneous) atomic relaxation rates. In that case the steady state population (which is achieved after the oscillations under discussion have damped out) of the upper level 11) becomes comparable to that of the lower level lo), and asymptotically equal to it in the limit of

3

4 RESONANCE LIGHT SCAlTEIUNG AND FLUORESCENCE [I, § 1

high saturation, f2 4 00 (KARPLUS and SCHWINGER [1948], DILLARD and ROBL [1969]). In addition, the steady state rate at which the atom absorbs energy from the incident field, considered as a function of the incident field frequency, is represented by a curve which is broadened relative to the one which represents the familiar weak field limit by an amount proportional to the parameter f2 (KARPLUS and SCHWINGER [1948]). The nutation, saturation and broadening effects under discussion, which have been familiar for decades in investigations of spin systems under magnetic dipole coupling, have in more recent years been extensively investigated in optical transitions under electric dipole coupling. The formal analogy between any two-level system and a spin one-half system of course makes the correspondence between the phenomena in the two different frequency regimes entirely understandable, apart from questions relating to the proper description at optical frequencies of the incident field (i.e., quantum vs. c-number) and of the radiative damping process under intense excitation.

At optical frequencies, however, there appears a class of phenomena which are not readily observable at lower frequencies. These relate to the properties (particularly the spectral properties) of the radiation emitted by the atom as it undergoes transitions from the state 11) down to the state lo), in the presence of the coherent incident field. In cases where n o transitions to the state 11) take place except those induced from the state 10) by the incident field itself, the process under discussion is simply one of intensity dependent resonance light scattering, since then all of the light which is emitted by the atom must first be absorbed from the incident field. When, on the other hand, energy-increasing transitions to the state 11) are induced by incoherent excitation processes, e.g., collisional, thermal, or broadband pumping, or when the system is initially prepared in the state 11) and a transient rather than a steady state process is observed, one may think of the phenomenon as consisting of the simultaneous operation of two processes, field-modified spontaneous emission and intensity dependent resonance light scattering. Strictly speaking, however, any distinction of this kind is at most a formal one, since the intense incident field may drive the atom back and forth between the two states 10) and 11) many times before a photon is emitted. In steady state, of course, the process is characterized by a single solution which is independent of initial conditions. One thus has n o choice but to abjure all perturbation theory notions based on linearity, and to solve the nonlinear problem in its entirety, with both the coherent applied field and

1,s 11 INTRODUCTTON 5

all relaxation processes (excitative, if these are present, as well as dissipative) fully represented.

When the field-coupled state of lower energy 10) is the ground state of the atom and the atomic relaxation is purely radiative, one is dealing with a problem of a particularly fundamental kind, for then the interaction between the atom and everything outside it is entirely electromagnetic. The solution of this nonlinear problem in closed form in a way which accurately incorporates effects of all orders in the incident field intensity has justly attracted great interest, particularly because of the means it affords of comparing the solution of a manifestly quantum mechanical problem (that of near resonance multiphoton scattering) to solutions which are based (or appear to be based) upon semiclassical methods.

In the case under discussion, that of light scattering with purely radiative damping, it is important to avoid a plausible error which oversimplifies the problem at the outset, namely that of thinking that the scattered field is determined by the expectation value of the induced atomic dipole moment, and that it is accordingly described in steady state by a 6-function spectrum. What is actually obtainable in this way is simply the expectation value of the scattered field, or its coherent part. This part is quite sufficient to determine the propagation characteristics of the (mean) incident field itself, in a medium consisting of many similar atoms, and hence is adequate for the treatment, e.g., of self-induced transparency (MCCALL and HAHN [1969]), or more generally for evaluating (linear or nonlinear) dielectric susceptibilities for the incident field. The mean value of the atomic dipole moment is also adequate to describe coherent multi-atom scattering processes such as photon echoes (ABELLA, KURNIT and HARTMANN [1964]). But in order to describe the single-atom, isotropic (apart from polarization effects) side scattering, one must take into account a further, fluctuating or incoherent part of the scattered field spectrum (MOLLOW [ 19691) whenever the incident field is strong enough to begin to saturate the transition. (The side scattering in an extended medium is simply the sum of one atom contributions in the limit treated in this article, that in which the atomic number-density is low enough to rule out cooperative effects.)

That the mean value ( p ( t ) ) of the atomic dipole moment does not determine the full scattered field spectrum is perhaps most evident in the limit of full saturation (0 + m), where ( p ( t ) ) approaches zero, whereas the intensity of radiation emitted by the maximally excited atom certainly cannot vanish. From the point of view of quantum mechanical scattering

6 RESONANCE LIGHT SCATERING AND FLUORESCENCE [I, 5 1

theory, too, it is clear that energy conservation requires that the frequency of a scattered photon must equal that of an incident photon only in the limit of weak incident fields, where the ongoing scattering process can be represented as the infrequent repetition of individual one-photon scattering events, each one of which conserves energy. When the incident field becomes intense enough so that in an elementary scattering event more than one photon is absorbed (and an equal number emitted), however, then though energy conservation imposes a restriction on the sum of the frequencies of the scattered photons in a process of a given order, it imposes no restriction upon the frequency of any one photon. In fact, the incoherent part of the spectrum is represented by a broadened function which breaks up, when the Rabi frequency 0’ becomes large compared to the atomic relaxation rates, into three equally spaced features, centered at w-L?’ , w, and w + 0 ‘ , respectively (RAUTIAN and SOBEL‘MAN [ 19611, APANASEVICH [1964]), with (generally unequal) widths proportional to the relaxation rates. Superimposed upon this incoherent spectrum is the 6-function which represents the coherent part of the spectrum. (The coherent part is also present in the side scattering, as it is in the familiar case of low intensity Rayleigh scattering, where it appears as a one atom effect because of the phase randomization which results from the uncorrelated positions of the atoms in the scattering medium.)

The appearance of three equally spaced spectral features separated by the frequency 0’ (the so-called dynamical Stark effect) is easily understood by picturing the atomic dipole moment as suffering an amplitude modulation at the frequency 0’ while it oscillates at the incident field frequency w ; indeed, that is exactly what would happen to the atom if it began, e.g., in its ground state and if no relaxation mechanism were present. The fact that the spectral features under discussion remain in steady state, after the amplitude modulation has been damped out by the atomic relaxation mechanism, may be understood by thinking of the process as then described by an ensemble of atomic systems, all with oscillating dipole moments undergoing amplitude modulation, but with the phase of this modulation randomly distributed over the ensemble, reflecting the incoherent nature of the relaxation mechanism.

1.1. SURVEY OF EARLY WORK

Early treatments of resonance light scattering which led to the intensity dependent dynamical Stark splitting of the spectrum were given .by

I , § 11 INTRODUCnON 7

RAUTIAN and SOBEL'MAN [1961], APANASEVICH [1964], BURSHTEIN [1965], NOTKIN, RAUTIAN and FEOKTISTOV [ 19671 and NEWSTEIN [ 19681.

RAUTIAN and SOBEL'MAN [1961] considered the problem of intensity effects in resonance fluorescence for the case of purely radiative damping, and under the assumption that through radiative transitions the system eventually decays out of the subspace of laser-coupled states (referred to as 10) and 11) throughout this article, whether or not 10) is the ground state) to states of lower energy. The spectrum of the radiation for the 11) +. 10) transition was then evaluated for the transient emission process which takes place after the atom is initially prepared in a specified atomic state (11) in the bulk of their analysis). Crucial to the validity of their treatment is the condition (NOTKIN, RAUTIAN and FEOKTISTOV [1967]) that the radiative transition rate (Einstein A-coefficient) K~~ for the transition in question be small compared to one of the radiative widths K~ or K , of the states 10) or I I), respectively (where K~ = x k K j k ) , i.e.,

This assumption is necessary in order to justify the basic method of calculation employed by RAUTIAN and SOBEL'MAN [1961], who, though assuming a multiphoton incident field, retain terms representing at most one photon in the emission field. (Actually RAUTIAN and SOBEL'MAN [1961] imposed the more restrictive condition K ~ ~ < < K ~ , in order to justify their choice of initial state.) Under the stated conditions, the calculation of the one-photon amplitudes is quite straightforward, since (NOTKIN, RAUTIAN and FEOKTISTOV [ 19671, MOLLOW [1975a]) the radiative damping process is then adequately represented by the addition of the imaginary terms -$ihKi to the state energies ci. Though RAUTIAN and SOBEL'MAN [1961] do not present their solutions in simple analytic form, their method of calculation appears correct for the limiting case they treat, and the graphs they present agree at least roughly with the exact analytic results obtained later. (There is n o 6-function term in the spectrum they find because the process is transient rather than steady state. If a weak incoherent excitation mechanism repopulated the lO)-ll) subspace at rate R, then the emission spectrum of the steady state which would result would be proportional to the transient one under discussion, but only in the limit R -+ 0. The coherent &function term in steady state would then be proportional to R2, and hence insignificant. See, for example, MOLLOW [ 19761 and 5 5.1 .)


The fundamental assumption of decay out of the laser-coupled subspace on which the work of RAUTIAN and SOBEL‘MAN [1961] is based is a rather restrictive one from an experimental point of view, since the results obtained would be directly applicable only if the repopulation rate into the laser-coupled subspace were so small as to excite only a vanishingly small fraction of the atoms present. Outside this limit, the repopulation mechanism itself would introduce a further broadening of the spectral lines, and other nontrivial spectral effects.

APANASEVICH [ 1963, 19641 attempted to calculate the emission spectrum by a method which allowed for nonradiative relaxation processes (including excitative ones) as well as radiative relaxation, and decay out of (and apparently repopulation into) the subspace of laser-coupled states. Though multiphoton effects are included for the incident field, it is not clear whether they are included for the scattered field as well. APANASEVICH [ 19641 correctly found the three-peaked structure produced by the Rabi modulation at frequency 0‘. His evaluation of two separate spectral functions, however, one for “scattering” and one for “emission” (corresponding, respectively, to the choice of 10) or 11) as an initial state) makes sense only if the system decays out of the lO)-ll) subspace, since only a transient process can depend upon initial conditions. Consistent with this interpretation is the absence of a 6-function in his solutions. On the other hand, a steady state does appear to be assumed in his two-level model, while the narrowness of the “scattering spectrum” in the weak field limit must be the consequence of an unstated condition (that of zero width) on the lower state. In general, one must say that though the curves obtained by APANASEVICH [ 19641 (particularly for the “scattering spectrum”) through numerical analysis of the rather detailed expressions found by APANASEVICH [1963] bear a rough resemblance to ones found through exact methods, the treatment of atomic relaxation, repopulation, equilibrium and coherence are not clear enough to permit ready comparison with the exact theory.

BURSHSTEIN [1965] treated the case of intensity dependent scattering by a two-level system for the case of incident radiation of fixed amplitude, but with phase which is instantaneously randomized (uniformly distributed) at random instants of time, thus providing the only relaxation mechanism in his model. His evaluation of the scattering spectrum, which he incorrectly identifies with the function which governs the response of the system to a weak probe field (MOLLOW [1972a]), nevertheless leads to the correct Stark splitting in the high intensity, on-resonance case. The

L511 INTRODUmION 9

weak field limit for the spectrum (which is actually well described by an Einstein B-coefficient calculation) has width proportional to the incident field intensity, in contrast to the strict &function which in the same limit describes the case in which a coherent field is incident upon a radiatively damped atom. The mean rate 7i1 of phase randomizations in the model of BURSHTEIN [1965] would have to be large compared to the radiative width to justify ignoring the latter, and the model is in fact implicitly represented by one-photon terms in the emission field.

The work of NOTKIN, RAUTIAN and FEOKTISTOV [1967] is devoted to an analysis of the same system as the one treated by RAUTIAN and SOBEL'MAN [1961], and for the most part in the same approximation, with the justification for the condition (1.5) being clearly established (though the justification for using a c-number incident field is not correctly stated, see § 2). The basic formal analysis treats multiphoton emission contributions of all orders, and develops the full solution as a power series (similar, in fact, to the one later found by MOLLOW [1975a]), which, however, is not summed by NOTKIN, RAUTIAN and FEOKTISTOV [1967] in any approximation. Only the first, one-photon term (which is shown to be identical to the solution found by RAUTIAN and SOBEL'MAN [1961]) is discussed, and is shown to represent the spectral density of radiation emitted at frequency u during the (1)- (0) transition in an interesting approximate form, proportional to the quantity

Lm d t 6' dtreiu(r-r*) g (1) ( t ; t ' ) + C.C.

(1.6)

in which p ( 0 ) is the initial atomic density matrix, and a,, and a:,, are the atomic transition operators

g"'(t; t ' ) = Tr {p(O)a:, (t'>a,dt)I,

a,,= lO>(ll; = Il>(Ol. (1.7)

The superscript 1 on the atomic correlation function in (1.6) designates the one-photon approximation, in which the effect of radiative damping is shown to be represented simply by the addition of imaginary terms to the state energies. The time dependence of the atomic operators in (1.6) is thus governed in this approximation by a non-Hermitian time development operator which however is completely expressible in terms of atomic variables when the incident field is described by a c-number (see § 2).

10 RESONANCE LIGHT SCAlTERING AND FLUORESCENCE [I, 8 1

NOTKIN, RAUTIAN and FEOKTISTOV [1967] also discuss the spectrum for radiative transitions from (or to) a laser-coupled state to (or from) another, uncoupled atomic state. Here again a one-photon approximation, which is adequate under the stated conditions, is adopted for the emission field, leading to results apparently in agreement with the appropriate limiting forms of those found by exact methods, in particular exhibiting the correct two peak structure with peak separation R’ (MOL- LOW [1972a, 1973bl).

The work of NOTKIN, RAUTIAN and FEOKTISTOV [1967], being based upon the same assumption (that of decay out of the laser-coupled subspace) as that of RAUTIAN and SOBEL‘MAN [1961], is similarly restricted in the range of its likely experimental applicability.

NEWSTEIN [1968] constructed a model in which the incident field- modified emission spectrum can be studied within a well defined context, and in which analytic solutions can be expressed in relatively simple, closed form, free of any ad hoc parameters save a single collision rate. He chose a two-level system driven by a prescribed c-number field, and assumed that the atomic relaxation was due to abrupt, instantaneously thermalizing collisions (KARPLUS and SCHWINGER [ 1948]), which were assumed to occur at a rate K , much greater than the radiative width of the transition, thus making it possible to ignore the latter. (For the same reason, the emission spectrum is actually determined in this model by one-photon terms in the emission field, though this fact need not be considered explicitly in the calculation.) The spectral density of the emitted radiation in steady state was correctly recognized (NEWSTEIN [ 19681) to be expressible quite generally and exactly in the form

m

g(v) = e’”‘g(t) dt

where the steady state density operator p and the Heisenberg transition operator a&), unlike the quantities in (1.6), act in the full state-space of the system of atom plus relaxation mechanism, not in the state-space of the atom alone.

The method NEWSTEIN [ 19681 used to evaluate the atomic correlation function g(t) in (1.8) was based upon a treatment of the collisional process which turned out to be fully accurate only in the important limit of high saturation. Also, it was shown by MOLLOW [1975a] that when an

I, B 11 WTRODUCITON 11

algebraic error in Newstein’s calculation is corrected, the result is reason- ably accurate in the limit of well-separated spectral lines (a’ >> K J , where it describes the sidebands quite accurately and correctly gives the integrated intensity (though not the detailed structure) of the central term. Except in these limiting cases, however, Newstein’s method is not fully adequate, and the expressions which result from it accordingly are inex- act. (They remain so even when the aforementioned algebraic error is corrected.) MOLLOW [1975a, Sec. 91 has shown that the method of NEWSTEIN [ 19681 is actually based upon a one-collision approximation, which ignores multi-collisional effects. The incompleteness of Newstein’s analysis is most evident from the absence of a coherent term in the spectrum he finds, and from the failure of his solution to reduce in the weak field limit to the one found by lowest order scattering theory. It should be emphasized that what Newstein has evaluated is not simply the incoherent part of the spectrum, i.e., one cannot simply add the coherent scattering term to his solution to get the correct result. To do that would be to overestimate the total scattering intensity, which is correctly given by Newstein’s formula. The coherent part of the spectrum was shown by MOLLOW [1975a] to be effectively incorporated within Newstein’s approximate formula, in a way which broadens the 6-function and fails to distinguish it from an incoherent term at line center. A fully accurate treatment of Newstein’s model, valid for arbitrary field strengths and degrees of saturation, is given by MOLLOW [1970]. It should be mentioned, finally, that the KARPLUS-SCHWINGER [ 19481 instantaneously thermalizing strong collision model is apparently not physically realistic at optical frequencies (CARLSTEN, SZOKE and RAYMER [1977]), where the elastic (dephasing) collision rate typically exceeds the inelastic rate by at least an order of magnitude (see 9 4.1). Thus the model under discussion here, like the others discussed in this section, must be understood as more useful for heuristic purposes than as affording a means of comparing theory with experiment.

NEWSTEIN [1968, 19721 has also discussed phase relations between emitted field components, as specified by nonstationary (i.e., non-cross- spectral) atomic correlation functions, useful for treating forward scattering. See also MOLLOW [1973a, b].

Not discussed in the above survey are works insufficiently accurate (e.g., that of BERGMANN [1967]) to yield both saturation effects and the correct form of the dynamical Stark splitting of the spectral lines. In all of the works that are discussed, the multiphoton character of the incident

12 RESONANCE LIGHT SCATTERING AND FLUORESCENCE [I, $ 2

field (though not of the scattered field) is fully taken into account (even when the incident field is represented by a c-number, as shown in 8 2), and the range of validity of the results is not restricted by any condition limiting the intensity of the incident field.

P 2. Description of the Electromagnetic Field

Inasmuch as the problem under discussion is one of multiphoton scattering theory, it is necessary to describe the incident field as well as the scattered or emitted field in fully quantum mechanical terms. A common way of approaching this problem is to choose a finite quantization volume V and to take all of the discrete modes to be initially unpopulated except the single mode which represents the incident field, which is chosen initially to have a definite number n of photons in it. If both n and V are allowed to approach infinity in such a way that n/V remains constant, thus preserving the field intensity, one can obtain accurate solutions by this method. (It is perhaps somewhat misleading to refer to the limit under discussion as “semiclassical”, a term which might better be reserved for the limit n+m, V=constant, i.e., the limit of infinite field intensity, where one knows a priori that a c-number description of the incident field is valid. On the other hand, the results obtained by taking n, V-+ m with n/V constant and finite do correspond exactly with those obtained through a c-number description of the incident field, a fact which one may gain some understanding of by recognizing that the assumed coherence of the incident field allows passage to the limit of infinite quantization volume, and hence infinite quantum numbers, even while the intensity of the incident field remains fixed.)

To treat scattering problems with infinite wave trains of nonvanishing spatial intensity, it is of course necessary in using the n-photon method to take the limit n, V --+ 00. Any calculation which leaves both n and V finite (and distinguishes, e.g., between n and n + 1) can at most be describing the problem of scattering from a finite wave train with exactly n photons in it - surely an artificial problem. Also, it is clear that a finite value of n cannot be taken to represent the number of photons in the laser cavity, since the irradiated atom is outside the laser, which of course contains other atoms which are not explicitly taken into account. Thus the limit n, V-00 in steady state scattering problems is, strictly speaking, necessary to give them clear physical meaning, and certainly does not represent a semiclassical approximation.

4 8 21 DESCRIPTION OF THE ELECTROMAGNETIC FIELD 13

(In the familiar cases where perturbation theory is applicable, on the other hand, one may evaluate, e.g., a scattering rate for a specific n-photon process, whose dependence on the incident field intensity I is known in advance to be I”, by holding n fixed and letting V approach infinity, thus letting I approach zero. N o such method has been successfully employed in the case of intensity dependent resonance light scattering, however, except in the weak field limit (SOKOLOVSKII [ 19701, MOLLOW [1975a]). It should be emphasized that in scattering processes of all orders, the photons are emitted, in the limits (1.1), (1.4) where the resonance approximation is justified, with frequencies which all lie near the atomic resonance frequency for the transition in question. There is thus no simple way of distinguishing by their frequency the spectral contributions of processes of different orders, which of course all take place simultaneously. It should also be mentioned that in view of the saturation of the transition, an nth order process cannot remain proportional to I” for all I , as is discussed in § 3.)

Another, and often more fruitful way of treating the electromagnetic field in single-atom scattering problems is to describe it entirely in terms of the correlation functions (GLAUBER [1963a]) which represent the expectation values of normally ordered products of field operators at specified points in space and time. The cross-spectral correlation function

which determines the field spectrum at any spatial point (r’ = r ) may be rigorously shown (MOLLOW [ 19691) in the scattering region to be obtainable, through the familiar formula for the classical dipole field, from the atomic correlation function (p ( - ) ( t ’ )p (+) ( t ) ) (where the superscripts are frequency signatures), thus justifying eq. (1.8).

The configuration-space method under discussion is particularly suitable for describing the incident field, since on the one hand only the value of this field at the position of the atom is physically significant at any instant of time, while on the other hand the incident field is conveniently regarded as a continuous function of position and time which may have infinite spatial and temporal extent. By far the simplest choice for the state of the incident field is a coherent state l{ak}) (GLAUBER [1963b]), which is an eigenstate of the field annihilation operator,

14 RESONANCE LIGHT SCATTERING AND FLUORESCENCE [I, 8 2

with the eigenvalue Er’(r, t ) a c-number quantity which formally resembles the positive frequency part of a continuous, freely propagating classical field but is otherwise unrestricted in form. By choosing the field eigenvalue at the position of the atom ( r = 0) to oscillate harmonically in time

(2.2) Er’(0, t ) = i E exp ( - i d ) ,

with E a c-number constant, one obtains the closest possible quantum mechanical counterpart to the classical concept of a monochromatic field with complete amplitude and phase stability. Alternatively, one may allow for a more complicated temporal development of the incident field simply by choosing a suitably modified c-number function E r ) ( 0 , f ) ,

while one may include the effect of incident field statistics, as in 8 3.3, by means of the P representation (GLAUBER [1963b], SUDARSHAN [1963]), which closely resembles a classical probability distribution. (Neither non- monochromatic time-dependence nor field statistics, it should be noted, are easily treated by the n-photon method.)

Calculations for the case of an initially coherent field state are greatly facilitated by the use of a theorem proved by MOLLOW [1975a], which is based upon a canonical transformation generated by the same unitary displacement operator (GLAUBER [1963b]) as the one that generates the coherent states from the vacuum state. As the result of this transformation, the initial state of the field can be represented as the vacuum state, while the c-number function

EJr, t ) = Er’(r , 1 ) + C.C.

must be added to the quantum mechanical field operator E(r, t ) , so that the latter now describes the scattered field alone.

It follows then that for an initially coherent field state, one may represent the incident field by a c-number function, provided that a quantum mechanical description is retained for the scattered field. Apart from the assumption of perfect coherence for the incident field, there is no approximation whatever in this method, in particular none involving the intensity of the incident field or the mean number of photons in it: It is the coherence of the incident field rather than its intensity which allows it to be represented by a c-number. (Contrary to appearances, the attenuation of the field is fully represented, and results from the destructive interference in the forward direction between the scattered field and the freely propagating incident field, precisely as in classical applications

I, 8 31 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 15

of the optical theorem. The action of the scattered field back on the atom is also taken into account, and gives rise to radiative damping.)

0 3. Scattering Spectrum for a Closed Two-level System with Radiative Relaxation

3.1. THE EXACT QUANTUM STATISTICAL METHOD

3.1.1. Coherent field solution in the Schrodinger picture

The reduced density matrix for the atom (obtained by taking the trace over all non-atomic variables) for a wide variety of relaxation mechanisms obeys equations of the form

where

u;k=(&j-&k)/fi, K j = C Kjk, K;k=K;;, (3.2) k

and E(t) = EJO, t ) is the c-number coherent state eigenvalue which represents the incident field at the position of the atom.

Not included in (3.1) is the quantum mechanical operator which represents the scattered field. The effect of this on the atom is incorporated into the damping parameters, which under purely radiative relaxation (MOLLOW and MILLER [1969]) are simply the Einstein A-coefficient (in rationalized units)

K;k = I p j k 1’ u7k/3rfic3 (3.3a)

and the familiar radiative transition width

K;k=i(K; + Kk), (3.3b)

with K ~ , as given by (3.2), just the radiative width of the state l j ) . The validity of these relations depends only upon the assumption that

during the convergence time of the damping integral from which they are obtained, the density matrix elements P;k ( t ) have approximately their


natural time dependence exp (- iwikt). As this convergence time is simply the reciprocal transition frequency w;' in the presently considered case of radiative damping, if the incident field couples only two levels 10) and ll), the necessary conditions for the validity of eqs. (3.1) and (3.3) are simply the conditions

(3.4)

which is equivalent to (1.1) and (1.4), and the innocuous condition wjk. The simple form (3.1)-(3.3) of the radiative damping relations is

applicable, then, as long as the resonance approximation is valid, i.e., as long as only near-resonance oscillations are induced. The incident field, in particular, may be strong enough to cause appreciable saturation, as long as it is not so strong as to cause population oscillations at a rate comparable to the atomic transition frequencies. (LEHMBERG [ 19701 has evaluated correction terms of order O/w to the Bloch equations (3.1) in a two-level model. Terms of the same order, it should be noted, arise when field-induced transitions to other levels are taken into account, giving rise, e.g., to frequency shifts (BLOCH and SIEGERT [1940]) and other effects. The two-level model for optical transitions in most cases represents a valid approximation only within the resonance approximation (3.4). GUSH and GUSH [1972] have nevertheless evaluated harmonic production in very intense fields within a two-level model.)

Since the states 10) and 11) are assumed to be the only states coupled by the laser field, the atom constitutes a closed two-level system if 10) is the ground state and the state 11) decays only to the state 10). In that case K~ = 0, K~ = K ~ ~ ' K , K { ( ) = K ; ~ = + K , and the steady state solution to the equations (3.1) for the four density matrix elements pl l , Flo(t)= p10 exp ( - i d ) , &( t ) = pol exp (iwt) and poO are (MOLLOW and MILLER r 19691)

(3.Sa)

(3.5b)

pol = p z ) and pOO= l - p l l , with O and A defined by eqs. (1.3), (2.2) and (1.1).

It follows at once (MOLLOW [1969]) from the relationship between the scattered field and the atomic dipole moment that the coherent part of the steady state scattering spectrum is

I , § 31 SPECTRUM FOR A CLOSED TWOLEVEL SYSTEM 17

while the total (coherent plus incoherent) rate of scattering of radiation is proportional to the population of the excited state,

g(v) dv/27~ = pl l . (3.7)

The latter relation is just what one would obtain by means of familiar perturbation theory methods, with the incident field playing no explicit role, though of course it establishes the population pll in the present case. Such methods are in fact a perfectly valid means of deriving eq. (3.7), subject only to the very same conditions (1.1) and (1.4) as those necessary to justify the treatment of radiative damping in eqs. (3.1) and (3.3)".

In the weak field limit (0 -+ 0), the solutions (3.5) are well approximated as

pll = Iplo12= $ 0 2 / ( A 2 + $ ~ 2 ) .

The spectrum is thus completely coherent in this limit, and is identical to the elastic scattering result predicted by lowest order scattering theory (WEISSKOPF and WIGNER [ 19301, WEISSKOPF [193 I]). (Observation of scattered field linewidth less than the natural width was reported by Wu, GROVE and EZEKIEL [1975] and GIBBS and VENKATJZSAN [1976].)

The ratio Ip,o)2/p11 of coherent to total scattering intensity falls from unity at low field intensity to zero at high intensity, where saturation is reached and the scattering is almost completely incoherent (MOLLOW [1969]).

MOLLOW [ 19691 considered the closed two-level radiatively damped atom in detail, and evaluated the scattering spectrum under quite general conditions, obtaining an exact formal solution for the initial transient regime and an exact analytical expression for the steady state case. The solution in both cases is valid for arbitrary field strengths and detunings, subject only to the restriction (3.4). The basic method of solution consists of an adaptation of the quantum fluctuation -regression theorem, developed by LAX [1968] to treat problems in quantum statistical mechanics. It

*The familiar simple perturbation theory derivation of the expression K P , , ( ~ ) for the (in general time-dependent) total rate of emission of photons is quite valid even in the presence of a coherent field strong enough to produce saturation. In that case the emission involves coherent multiphoton effects, the sum of whose contributions is nevertheless accurately given by the term in question, notwithstanding the simplicity of its derivation, and notwithstanding the fact that more complicated methods are necessary to evaluate the spectrum of the field. Failure to understand this point has been the source of much confusion in the literature on this subject.

18 RESONANCE LIGHT SCA‘ITERING AND FJAJORFSCENCE [I, 0 3

follows from this theorem that the four quantities

Rjk(t; t ’ ) =Tr {pu:O(t’)ajk(t)} , (3.8a)

where t 2 t ’ , t’ is fixed, and j , k = 0, 1, obey the same set of coupled equations as do the four density matrix elements pjk(t) = Tr {paik(t)}. The solution of these equations suhject to the initial conditions

Rll(t’; t ’ ) = 0 . Rlo(t‘; t’) = pll(t’),

Rol(t’; t‘) = 0, Roo(t’; t’) = pol(t’)

is straightforward, and leads to a relation (eq. (4.6) of MOLLOW [1969]) which expresses the atomic correlation function

Rlo(t; t ’ ) =Tr {pa:o(t’)u,o(t)}= g ( t ; t ’ ) (3.8b)

which determines the scattering spectrum both in the transient and steady state regimes in terms of specified functions of t - t ’ and the density matrix elements pll(t’) and po , ( t ’ ) . The latter quantities are of course obtained as the solutions to eqs. (3.1), subject to whatever initial conditions may apply. In steady state ( t ’ + w), the solutions (3.5) may be used, and the general formula for the closed two-level scattering spectrum under radiative relaxation was found to be specified by the symmetric function (MOLLOW [1969])

g(v)= g ’ ( v - w ) (3.9)

g ’ ( U ) = (&oI2 2‘7T6(V)f f i 1 1 K f 1 2 ( V 2 + $ f 1 2 + K2)/lf(iV)I2,

in which

If’(iv)12 = v2(v2-fl’2-$K2)2+ K2(4u2-$f12- A 2 - i K 2 ) 2

and plo and f i l l are given by eqs. (3.5).

intensity are retained, one finds (MOLLOW [1969]) When in the weak field limit terms quadratic in the incident field

K O n 2 1. (3.10) [(v - A ) ’ + ~ K ” ] [ ( v +A)2 +$K2]

The second term in this expression, which specifies peaks in the spectrum S(v) at u = w - A = wl0 and at u = w + A, is identical to the one obtained by means of a straightforward scattering theory calculation (SOKOLOVSKII [ 19701, MOLLOW [ 1975al) of the two-photon scattering

I, § 31 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 19

spectrum, in which two photons each of frequency w are absorbed from the laser field and two are emitted, the latter with energies which sum to 2 0 but which individually form the continuous spectrum shown. (As was mentioned above, the full spectrum (3.9) has never been obtained as a sum of n-photon processes for arbitrary 0. Indeed, it is clear from the phenomenon of saturation that in any such expansion the contribution made by a process of any fixed order II would have to eventually diminish with increasing 0 rather than remain proportional to 02”. The one- photon contribution in (3.10) is actually slightly overestimated; a small negative correction of order O4 having been omitted.)

The rather complicated function in eq. (3.9) takes on a somewhat simpler form when the dynamical Stark effect becomes pronounced enough to split the incoherent part of the spectrum into three relatively well separated components, centered at w, w - O’, and o +a’, respectively. This happens for O’>>K, in which case the spectrum is well approximated by the expression (MOLLOW [ 19691)

2s,AiNC g’(u) = Jp1,(2 27TS(u)+

v2+ s:

2uA+ 2uA- + + ( u + a’)’+ u2 (v - 0’)’+ u2’

(3.11)

in which the integrated intensities of the incoherent components are

AtNc = ~06/fl’2(0’2+A2)2 (3.12a)

A+ = A _ = 404/0’2(O’2+A2), (3.12b)

and the widths are

(3.13)

In the limit of intense incident fields (0 >> K, \ A I), the coherent part ot the spectrum is inappreciable. The integrated intensity of each sideband in this limit is equal to one half that of the central peak, A+ = A- = t, A, = a, while the (equal) widths of the sidebands are 50% larger than that of the central peak, u = $ K , S , = ~ K .

(The validity of the optical Bloch equations and of the spectrum found by MOLLOW [1969] was proved subsequently (MOLLOW [1975a]) by a method which did not rely upon an atom-field statistical factorization assumption.)

20 RESONANCE LIGHT SCATERING AND FLUORESCENCE [I, § 3

The exact result found by MOLLOW [1969] for the spectrum of intensity dependent resonance light scattering was the first to incorporate multiphoton effects in the scattered field (as distinct from the incident field) with even rough accuracy, all earlier analyses having either treated such effects incorrectly or else having treated only cases in which a one-photon approximation is adequate. Because of the purely electromagnetic nature of the process, the solution of this nonlinear problem in closed form is of special interest, particularly in view of the fact that the solution contains no undefined parameters whatsoever, the atomic relaxation being clearly expressed in terms of the Einstein A-coefficient for the transition. Finally, because of the absence of assumptions which are unduly restrictive or physically unrealistic (thus contrasting with all previous work), the result found by MOLLOW [1969] is not only interesting from a theoretical point of view, but is capable of being compared with experiment. (A calculation essentially identical to that of MOLLOW [1969] was performed by BAK- LANOV [1973], and a similar, though somewhat less exact calculation was performed by KAZANTSEV [1974].)

3.1.2. Comparison of theory with experiment

The first attempt to measure the spectrum of intensity-dependent resonance light scattering was made by SCHUDA, STROUD and HERCHER [ 19741. Utilizing atomic beam techniques to eliminate Doppler broadening and a tunable dye laser, they were able to measure the spectrum of light scattered by sodium atoms in a hyperfine component of the D, transition. Because of the choice of linear polarization of the laser field a strict two-level system was not achieved, due to Zeeman degeneracy. The measurements, which were carried out with the laser tuned both on and off resonance, clearly revealed a three-peaked structure, though not with sufficient accuracy to permit a quantitative comparison between experiment and theory.

More accurate measurements were subsequently made by WALTHER [ 19751, Wu, GROVE and EZEKIEL [1975], HARTIG, RASMUSSEN, SCHIEDER and WALTHER [1976] and GROVE, Wu and EZEKIEL [1977]. The first of these measurements to confirm the theory in a quantitative way were those of Wu, GROVE and EZEKIEL [1975], who, though using linearly polarized light and thus not achieving a strict two-level system, confirmed the 3: 1 ratio of central peak to sideband maximum spectral density

I, § 31 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 21

predicted by theory (MOLLOW [1969]) for the on resonance, strong field limit. Using circularly polarized light and thus achieving a strict two-level system, HARTIG, RASMUSSEN, SCHIEDER and WALTHER [ 19761 and GROVE, Wu and EZEKIEL [1977] were able to confirm the theory in greater detail (see also EZEKIEL and Wu [1978] and WALTHER [1978]). The measurements of GROVE, Wu and EZEKIEL [1977], in particular, show an impressive quantitative correspondence between theoretical and experimental curves in both the on and off resonance cases, with one such comparison being made, at low temperature, with no free parameters except an overall normalization constant. Of particular importance is the fact that the symmetry of the spectrum was confirmed with high accuracy.

The power-broadening and saturation predicted by eq. (3.5a) was confirmed experimentally by CITRON, GRAY, GABEL and STROUD [1977] and by EZEKIEL and Wu [1978]. In the latter experiment, an asymmetry of the absorption curve (not the scattering spectrum) was observed, possibly due to the cumulative effect of radiation pressure on the atoms.

3.1.3. n-photon incident field; the dressed atom method

In view of the discussion of 0 2 it is clear that the results obtained by taking the initial field state to have exactly n photons in a particular mode and then letting n and the quantization volume V approach infinity with n/V constant must be identical to the ones obtained for the case of an initially coherent field state. Calculations of the spectrum based upon what are effectively density operator techniques, with atomic radiative relaxation evaluated by the same Markoff type methods used to obtain the optical Bloch equations (3. l ) , and with the fluctuation-regression theorem used to obtain the spectrum, have been performed by OLIVER, RESSAYRE and TALLET [ 19711, who numerically evaluated the spectrum in the initial transient regime, and by CARMICHAEL and WALLS [1975, 1976a1, COHEN-TANNOUDJI [ 1975, 19771 and COHEN-TANNOUDJI and REYNAUD [1977a, b, 19781.

In all of the work just cited, the so-called dressed atom approach is used, in which the basis states for the system of atom plus incident field are chosen to be the particular linear combinations of the nearly degenerate states of the form 11, n) , 10, n + 1) which diagonalize the interaction term which couples the atom to the incident field. The counterparts of these superposition states in the formalism in which the incident field is

22 RESONANCE LIGHT SCATl73RING AND FLUORESCENCE [I, 5 3

coherent and thus represented by a c-number are the familiar states representing the atomic pseudospin, in the rotating wave frame, as parallel or antiparallel to the effective field, i.e.,

(3.14)

where

c, = [(l *AlO’)$.

The states I+) and I - ) diagonalize the purely atomic part of the Hamiltonian in the rotating wave frame, where they have energies - IhO’ and $Of, respectively. The energy fiw of a laser photon must be added to the rotating wave transition-energy to obtain the energy of the emitted photon in the laboratory frame. (In the dressed atom picture, there is an infinite hierarchy of doublets I*), separated by the energy Am.) In the limit of well separated spectral lines (O’>>K) the use of these states facilitates calculations greatly (except to the extent that the detailed structure, including the coherent part of the spectral lines, is important), as well as providing a useful picture of the emission process as occurring during transitions between the states in question. (In the c-number formalism these include transitions from a given state to itself.) By simply transcribing the optical Bloch eqs. (3.1) for the two-level radiatively damped system directly into the form specified by the basis set I + ) , one finds that transitions from I + ) to I - ) and from I - ) to I+) take place, respectively, at the rates per atom in each initial state (COHEN-TANNOUDJI [1975,1977], COHEN-TANNOUDJI and REYNAUD [1977a], MOLLOW [1977])

r+- =$(Of+ A)”/O’”, r-+ = $(Of -A)2/O’2, (3.15)

where r is the Einstein A-coefficient previously designated as K . The actual rate of such transitions is thus (COHEN-TANNOUDJI and REYNAUD [ 1977al)

A, = p++r+-, A- = p--r-+, (3.16)

where p++ and p-- are the respective atomic populations. These populations may be evaluated in steady state simply by noting that then A+=A-. The solutions for p++ and p-- thus obtained with the aid of (3.15) yield, upon substitution into (3.16), exactly the value for the sideband intensities A, found previously in eq. (3.12b). The width CT of

1,831 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 23

the sidebands is also obtainable immediately as simply the damping parameter for the off -diagonal density matrix element p++.

It should be emphasized that the method under discussion is useful only in the limit of well separated spectral lines (O'>>r) . More generally, all four density matrix elements ppy (where F, v = + , -) are coupled to one another, and the analysis is actually more complicated in the I + ) - \ - ) basis than in the lO)-ll) basis.

Analogous methods have proved useful (COHEN-TANNOUDJI and REYNAUD [ 1977b, c]) in treating many-level problems involving both (near) degeneracy and more than one incident field frequency. (POLDER and SCHUURMANS 119761 treated a degenerate case by the exact statistical method, using a coherent incident field with linear polarization. An interesting aspect of their solution is that while the coherent part of the spectrum (governed by the incident field polarization) is a strict S- function in steady state, the other polarization components, which form part of the incoherent spectrum, have zero width in the limit of weak incident fields, where they form part of the elastic scattering spectrum. Their width is nonzero as long as the field intensity is nonzero, however, and they are thus represened by a 6-function in a less strict sense than are the coherent components.)

3.1.4. Coherent field solution in the Heisenberg picture

The understanding of quantum mechanical radiative relaxation has been greatly facilitated by the development of operator radiation-reaction theory (ACKERHALT, KNIGHT and EBERLY [ 19731, BULLOUGH [1973], Ac- KERHALT and EBERLY [1974], ALLEN and EBERLY [1975], HASSAN and BULLOUGH [1975], SAUNDERS, BULLOUGH and AHMAD 119751, KIMBLE and MANDEL [1975a,b, 19761). By methods analogous to those used in classical theory, one finds that the positive frequency part of the full Heisenberg quantum mechanical electric field operator at the position of the atom may be written, subject to exactly the same conditions (essentially (3.4)) as those needed to justify the results of Q 3.1.1, as

(3.17)

where a radiative frequency shift has been omitted, K is again the Einstein A-coefficient, and EF(r, t ) is the initial, freely propagating part of the field operator. By using this relation for the case of an initially

24 RESONANCE LIGHT SCATITRING AND FLUORESCENCE [I, I 3

coherent field state, one obtains immediately the previously described form of the optical Bloch equations (ACKERHALT and EBERLY [1974], KIMBLE and MANDEL [ 1975a, b], SAUNDERS, BULLOUGH and AHMAD [1975]). The effect of the free field operator in these relations is simply to lead to the appearance of the c-number coherent state eigenvalue in its familiar role.

In attempting to verify the fluctuation-regression theorem by similarly evaluating the time development of the quantities in eq. (3.8), however, one encounters the serious difficulty of the possible noncommutativity between free field operators and full Heisenberg matter operators (AGAR- WAL [1974a, b], HASAN and BULLOUGH [1975]). The value of the free field-matter commutator was found quite generally by MOLLOW [1973d], and it was shown by MOLLOW [1975b] that the particular commutators involved in the evaluation of the time derivatives of the quantities in eq. (3.8) do vanish. This completed the proof of the quantum fluctuation- regression theorem in the Heisenberg picture, and thus established by a different method the validity of the spectrum found by MOLLOW [1969].

The treatment of KIMBLE and MANDEL [1976] was unusually careful, avoiding in particular the atom-field statistical factorization assumption. (In this respect their work was complementary to the Schrodinger picture analysis of MOLLOW [1975a].) In addition, their analysis exhibited the quantum mechanical nature of the spectral emission process in a particularly explicit way. Their method of calculating the spectrum, on the other hand, was based upon a solution of the four coupled equations for the quantities in (3.8) specified by the fluctuation-regression theorem, which in turn was proved with the aid of the theorem proved by MOLLOW [ 1 97 5 b] .

Further discussions of intensity-dependent resonance light scattering by Heisenberg picture methods were given by RENAUD, WHITLEY and STROUD [1976, 19771, who evaluated transient effects and the effect of finite observation time.

It should be emphasized that the solution for g ( t ; t ’ ) as obtained by MOLLOW [1969] is valid for all times t , t ‘ , the solution for t < t ’ being rigorously obtainable from the solution for t > t’ by interchanging t and t’ in the identity

Tr {pat(t)a(t’)> = [Tr {pa+(t‘)a(r>}]*, (3.18)

which of course holds under transient as well as steady state conditions. (The density operator p in (3.18) is evaluated in the Heisenberg picture

L 8 31 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 25

and is constant for that reason alone.) Also, the transient regime is fully described by eq. (4.6) of MOLLOW [1969], while the passage to steady state is a manifest property of the general derived solution, not an assumption made at the outset. (The existence of the steady state in the solution of MOLLOW [1969] follows simply from the fact that all of the roots of the polynomial f(s) have negative real parts.)

All of the analyses which evaluate the spectrum by solving the four coupled (optical Bloch) equations which are specified by the quantum fluctuation-regression theorem for the quantities in (3.8a) are described in this article as following from the quantum statistical method, independently of the means or of the rigor of the derivation leading to the coupled equations in question. This designation is not intended as a disparagement, for exceptionally rigorous justifications (MOLLOW [ 1975a1, KIMBLE and MANDEL [1976]) of the statistical method have been achieved. The existence of irreversible atomic damping and of the asymptotic statistical independence between the state of the atom and the state of the photons which were scattered by it in the distant past are really inescapa- ble physical facts, and their emergence within any theory should be regarded as a sign of its success rather than as an indication of its inexactness.

3.2. PURE STATE ANALYSES

Though the frequency spectrum of the scattered field is determined by the combined effect of multiphoton contributions of all orders, it is useful at least for formal purposes to obtain explicit solutions for the individual n-photon wave functions, which describe the scattered field in a detailed way. STROUD [1971, 19731 attempted to do this in an approximation which retained only one photon in the scattered field. Though exhibiting the dynamical Stark effect, his solution was not accurate enough to describe the spectrum in detail.

MOLLOW [1975a] achieved a rigorous and exact solution for the Schrodinger pure state vector I t ) which describes the joint, correlated system of atom and field, including multiphoton terms of all orders. The effect of atomic damping was shown to be obtainable from the rigorously derived relation

(3.19)

26 RESONANCE LIGHT SCATTERING AND FLUORESCENCE [I, 5 3

where E(+) is the Schrodinger picture quantum mechanical photon absorption operator at the position of the atom, to which the c-number coherent state eigenvalue EF'(t) must be added when the canonical transformation described in 0 2 is inverted. The relation (3.19), which is the Schrodinger picture counterpart of the Heisenberg operator radiation reaction equation (3.17), enables one to accurately represent the process of re-absorption of previously scattered photons (though not of the incident photons, which are accurately described by a c-number) by means of a damping parameter in the atom-multiphoton amplitude of any given order n, rather than in the form of a direct coupling to the amplitude of order n + 1. In fact, the re-absorption is accurately represented by the addition of the imaginary term -$ihK to the energy of the upper atomic state 11) in the joint atom-field amplitude in each order, a procedure which is to be sharply distinguished, e.g., from a one-photon calculation in which damping appears only in the purely atomic amplitudes, before any photons are emitted. It is the fact that terms of a given order are coupled only to terms of lower order which makes possible closed solutions for the atom-multiphoton amplitudes*, which were in fact achieved quite exactly and generally, and for all n, by MOLLOW [1975a].

With the aid of (3.19) MOLLOW [1975a] was able, without making an atom-field statistical factorization assumption, to prove the validity of the optical Bloch equations and the fluctuation-regression theorem, as well as to solve for the spectrum in a direct way (in Sec. 5 of MOLLOW [1975a]), which however might well be included under the heading of statistical theories, because of its close formal resemblance to the method of solution specified by the fluctuation-regression theorem. In addition, a species of reduced atomic density matrix &) ( t ) , corresponding to the presence of exactly n photons in the field, was evaluated, and shown to be well approximated for large n and t by the product of the steady'state atomic density matrix and the Poisson formula

P ( t ) = exp (- ~ j i , ~ t ) - ( K & t)"/n !, (3.20)

the latter representing the probability of exactly n photons being scattered by time t. (OLIVER, RESAYRE and TALLET [1971] performed a

*The amplitude for finding exactly n photons in the field at a given instant of time is quite different from the contribution made by an n-photon scattering process, for the former is the sum of contributions made by scattering processes of all order lower than o r equal to n. As was mentioned in §§2 and 3.1.1, no treatment of the multiphoton scattering process of fixed order n has ever been made in a way that incorporates saturation effects.

I, 5 31 SPECTRUM FOR A CLOSED TWO-LEVEL SYSTEM 27

similar calculation for the on-resonance case, and in the strong field limit obtained the same result. No attempt was made by them, however, to evaluate the spectrum corresponding to the terms in question.) MOLLOW [ 1975al showed that the n-photon contribution to the scattering spectrum is well approximated for large n and t as the product of the same Poisson formula and the previously found steady-state spectrum. In a more accurate approximation, the n-photon spectral contribution was shown by MOLLOW [1975a] to be represented by a function with time dependent widths, with the latter assuming their steady state values only during the limited time interval within which the Poisson probability P(”)( t ) is appreciable.

MOLLOW [1975a] was also able to evaluate the n-photon wave functions in configuration space, finding an exact and general result proportional to the quantity

n-1

sin @ ( t - rl/c) n sin $i(rj - rj+J/c (3.21) i = l

in which rn < * < rl < ct, fi is the complex quantity

d = [a’+ (A + & c ) ~ ] & ,

and ri is the distance of the jth photon from the atom. The vanishing of (3.21) at equal spatial arguments implies the existence of the photon antibunching effect, first predicted in resonance fluorescence by MOLLOW [1975a, Sec. IVA], worked out in greater detail by CARMICHAEL and WALLS [1976b] and by KIMBLE and MANDEL [1976, 19771, and confirmed experimentally by KIMBLE, DACENAIS and MANDEL [1977].

MOLLOW [1975a] showed that the emission process can be pictured in terms of an ensemble of pure states, each consisting of the atom and one photon. This picture leads to a particularly simple evaluation (MOLLOW [1978]) of the configuration-space wave functions (3.21), and to an expression (MOLLOW [1975a]) for the scattering spectrum of a form superficially very different from the one found by MOLLOW [1969], but identical to the one found by SWAIN [1975] by a direct summation of n-photon amplitudes. In fact, the solutions under discussion agree quite generally and exactly, notwithstanding their different forms. It should be mentioned, finally, that SMITHERS and FREEDHOFF [ 19751 have summed n-photon amplitudes in important limiting cases and have shown agreement with the exact theory.

[I, 8 3 28 RESONANCE LIGHT SCATTERING AND FLUORESCENCE

3.3. INCIDENT FIELD STATISTICS

When the incident field is not completely coherent, the effect of its fluctuations cannot be evaluated in any simple closed form for completely general types of fluctuations except in the limit of vanishing field intensity, where lowest order scattering theory applies, and the scattered field spectrum for the case of radiative damping is simply and generally the product of the incident field spectrum and the atomic response function ( A * + ~ K ~ ) - ~ (WEISSKOPF [1931], HEITLER [1954]). When saturation effects are important, on the other hand, the nonlinearity of the process makes the general solution far too complicated to be expressed in closed analytic form. There appear to be only two classes of cases which can then be solved, those of very slow and those of very fast field fluctuation.

It is easy to show that in the narrowband limit, in which the laser intensity I varies slowly, i.e., by small amounts during the atomic lifetime (though possibly appreciably over longer time intervals), the expectation value of any physical quantity that depends upon I in a known way when I is constant can be evaluated in the stationary case simply by averaging the known solution over the laser intensity probability distribution P ( I ) , obtained in a simple way (MOLLOW and MILLER [1969]) from the P representation (GLAUBER [1963b], SUDARSHAN [1963]) that describes the field statistics. In the case of resonance light scattering, it is therefore clear that sufficiently large, slow amplitude fluctuations will broaden the sidebands in the Stark-split spectrum to the point where they are no longer resolvable, a result that has been obtained by AVAN and COHEN- TANNOUDJI [1977] for the case of Gaussian fluctuations. The fact that incident field correlation functions of all orders are in general important in determining the response of the atom to the field has been emphasized by MOLLOW and MILLER [1969] and by AVAN and COHEN-TANNOUDJI [ 19771.

In the opposite, fast-fluctuation limit, the most familiar cases are those in which the field components themselves are governed by a stationary random process with correlation time short compared to all other relevant time scales. In the simplest such case, i.e., that of a fully chaotic field, the (rotating wave frame) positive frequency part of the field % = + ix2 has zero mean value, and and zz fluctuate independently and equally. The field spectrum 9(0) in such cases is broad enough to be considered constant, and the effect of the field is simply to induce transitions between the levels at a rate proportional to the product of 9 ( w l 0 ) and the Einstein

1,s 31 SPECl‘RUM FOR A CLOSED TWO-LEVEL SYSTEM 29

B-coefficient, thus leading to extra transition and damping terms in the optical Bloch equations. Straightforward use of the fluctuation-regression theorem then yields the emission field spectrum, which can also be evaluated for the case in which a coherent field acts along with the incoherent field under discussion. (In that case the fluctuation correlation time must be small compared to the Rabi period O’-* as well as small compared to the atomic lifetime if a more complicated analysis is to be avoided, as shown by SZIKLAS [1969] and discussed in D4.2.)

It is important to understand in this connection that since a broadband incoherent field with bandwidth Si excites atomic transitions at the rate

- ( l ~ ~ o . W f i l ~ ) / 6 i ,

while the rate for a coherent field is

- 1 k o ‘ & / f i 1 2 / K ’ ,

an incoherent field would have to have mean intensity much larger than that of a coherent field to induce transitions at the same rate. (The ratio of incoherent to coherent intensity would have to be - SJK’ >> 1.) Equival- ently, an incoherent field would have to have mean intensity ~ J K ’ times the threshold intensity for coherent field-induced saturation (i.e., for Stark splitting of the spectral lines) in order to appreciably broaden the spectral lines.

For this reason, at optical frequencies the effect of small (i.e., of limited excursion) rapid stationary fluctuations in the amplitude or phase of an otherwise coherent incident field is unlikely to be observable, since the fluctuating part of the field is then by hypothesis small compared to the coherent part. For such fluctuations to be observable, the intensity of the coherent field would have to exceed the threshold value for saturation by several orders of magnitude, to allow the smaller incoherent field to exceed the threshold value by at least one order of magnitude. Such intense coherent fields, however, would induce multilevel effects, and have not been used to date under cw conditions.

,The only other case of rapid fluctuation that appears to have been solved exactly is the one in which the incident field has a fixed amplitude, but suffers a rapid frequency-fluctuation leading to a continuous diffusion of its phase (GLAUBER [1965], PINCINBONO and BOILEAU [1967]), which eventually becomes uniformly distributed between 0 and 2 ~ . The case of rapid (essentially discontinuous) frequency fluctuation, which leads to continuous phase diffusion, should be sharply distinguished from the case

30 RESONANCE LIGHT SCATIXRING AND FLUORESCENCE [I, B 3

of rapid (essentially discontinuous) phase fluctuation mentioned above, where the phase is governed by a stationary random process of short correlation time. In the frequency fluctuation case under discussion, the fluctuating part of the frequency is governed by a stationary joint Gaus- sian random process, with correlation time short compared to all other relevant time scales ( K ’ - ’ , W’); but the value which is obtained for the phase diffusion rate b (which is also the incident field bandwidth) can have any magnitude relative to K’ or 0’. The solution for the effect of incident field frequency fluctuations upon the spectrum of the scattered light is not only important physically (since light beams from lasers operating well above threshold appear to be well described by the model in question), but represents a nontrivial mathematical problem of some difficulty, in particular one not in general solvable by modifying the damping parameters in the optical Bloch equations. Limiting cases have been correctly treated by ACARWAL [1976] and by EBERLY [1976]. The general solution for the case of radiative damping was obtained and examined in detail by KIMBLE and MANDEL [1977] and MANDEL and KIMBLE [1978a, b], and generalized to allow for collisions by AGARWAL [1978], who also considered a variety of related problems. Solutions have also been obtained by ZOLLER [1977, 19781, ZOLLER and EHLOTZKY [1977], and AVAN and COHEN-TANNOUDJI [ 19771. In the off-resonance, radiatively damped case (KIMBLE and MAN- DEL [1977]), asymmetries appear in the spectrum of the scattered field. The modifications are all of order b / K , however (where b is the field bandwidth), and so would have been inappreciable, e.g., in the experiments of GROVE, Wu and EZEKIEL [1977], in which the laser bandwidth was less than one per cent of the atomic linewidth.

It should be emphasized that except for the narrowband case (that of slow variation), the only exactly solvable cases involving incident field fluctuations that incorporate saturation effects appear to be those in which some field-related quantity varies very rapidly, i.e., with correlation time short compared to the atomic lifetime*. The only case of the latter kind that appears to have been solved with any rigor to date, apart from the simple one involving transitions induced by an incoherent broadband field, is the nontrivial problem of rapid frequency fluctuation, leading to continuous phase diffusion.

(The model of discontinuous large phase fluctuations discussed by BURSHTEIN [1965] and in 0 1.1 cannot without justification be treated by the same methods as those which describe the continuous phase diffusion * In a recent article P. ZOLLER [1979], however, has treated the case of a chaotic (Gaussian) field of arbitrary bandwidth.

COLLISIONAL RELAXATION 31 I, 8 41

model. The correlation time T~ for phase fluctuations in Burshtein’s model must be small compared to the atomic lifetime to justify omitting radiative damping effects, but is otherwise unrestricted in value. It is not clear how accurate is the method Burshtein has developed to treat nonlinear effects within his model, nor is it clear how the effect of radiative relaxation can be taken into account. ZUSMAN and BURSHTEIN [1971] have treated transitions to an adjacent level by a similar method.)

P 4. Collisional Relaxation

4.1. THE IMPACT APPROXIMATION

When collisions take place abruptly enough to be treated in the impact approximation, i.e., when the duration T~ of a collision (as distinct from the time between collisions) is small compared to both the Rabi period Of-’ and the (collision-shortened) atomic lifetime, it is possible to represent the effect of collisions by means of appropriate damping constants and transition rates in the Bloch equations (3.1). In the case of a closed two-level system with no energy-increasing collision-induced transitions ( K ~ = 0, K~ = K’O= K, K { O = K;), = K’) one may write (HUBER [1969], OMONT, SMITH and COOPER [1972])

(4.1) K = r + 01, K ’ = +( r -t QI + QE),

where r is the Einstein A-coefficient and QI and QE are the rates of inelastic (quenching) and elastic (dephasing) collisions, respectively.

The relations (4.1) are really intended to describe weak collisions, which change the state of an atom only slightly during a collisional event (VAN VLECK and WEISSKOPF [1945]). Weak collisions are always describable by second-order perturbation theory, which leads in the elastic case to a damping integral involving the autocorrelation function for a fluctuating (c-number) term Aw,,(t) in the atomic resonance frequency*, and in the inelastic case to a similar function involving quantum mechanical reser- voir variables (MOLLOW and MILLER [1969]). The correlation times in these autocorrelation functions represent the collision duration, and hence the evaluation of the integrals is modified outside the impact regime by the incident field, as discussed in 0 4.2, but the basic method remains applicable as long as the collisions are weak.

*The effect of fluctuations in the atomic resonance frequency is quite different from the effect of fluctuations in the incident field frequency.

32 RESONANCE LIGHT SCATTERING AND FLUORESCENCE [I, B 4

A strong collision, on the other hand, which makes an appreciable change in the state of the atom during a single collisional event, can in no case be represented by a perturbation theory calculation. If its duration is short enough to justify the impact approximation, however, then its effect may nevertheless be represented in many cases (KARPLUS and SCHWINCER [1948], BURSHTEIN [1965]) by means of appropriate terms in the Bloch equations. One may then regard the relations (4.1) as simply representing a convenient parametrization, e.g., in the model of KARPLUS and SCHWINGER [1948], QI = QE= K,, the mean collision rate. Strong collisions outside the impact approximation, on the other hand, are very difficult to treat by any method.

In the limit of weak incident fields, where lowest order scattering theory applies, the spectrum of the scattered light is given by the relation (HUBER [1969], OMONT, SMITH and COOPER [1972], MOLLOW [1973c])

and thus consists of two components, a coherent component which represents elastic scattering, and an incoherent component proportional to the rate QE of dephasing collisions. A noteworthy feature of the collision-modified spectrum is that it is in general asymmetric, even in the weak field limit.

When the intensity of the incident field becomes great enough to begin to cause saturation effects, the fluctuation-regression theorem enables one to calculate the scattering spectrum in a straightforward manner. A general solution for the closed two-level case which allowed for incoherent excitation processes was found by MOLLOW [1972b, eq. (4.9)]. In the limit of well separated spectral lines (Or >> K'), the solution in the presently considered case (in which the inelastic collisions are purely dissipative) may be approximated (MOLLOW [1977]) as in eq. (3.11), but with coherent and incoherent integrated intensities

lp1012 =$.R2A2/(qOZ+ A2)2 ,

AANc=~04[q202+ (27 - 1)A2]/Of2(702+

A+ = QOz((n'+A)[~(Or+A)-A] /Of2(~02+A2) , (4.3a)

A _ = ~ . R 2 ( O r - A ) [ ~ ( O f - A ) + A ] / O ' 2 ( ~ ~ 2 + A 2 ) , (4.3b)

so that the sum of coherent and incoherent components at line center is

A 0- = I PlO - l 2 + AiNC = a 0 2 / f l f 2 , (4.3c)

I , § 41 COLLISIONAL -TION 33

while the widths are

so= ( K ’ O ~ + K A ~ ) / R ’ ~ ,

u = [ K R ~ + K ’ ( R ~ + ~ A ~ ) ] / ~ O ’ ~ . (4.4)

The parameter q in the above relations is the decay constant ratio

q = K ‘ / K = $(I- + QI + QE)/(r + QI). (4.5)

The coherent plus incoherent integrated intensity A,, at line center, which may be called the Rayleigh component of the spectrum (CARLSTEN and SZOKE [1976]) is shown by eq. ( 4 . 3 ~ ) to be independent of q and thus independent of the type of relaxation process. The sideband coefficients A+ and A_, however, (which describe, for A > O , what may be called, respectively, the fluorescent and Raman contributions to the spectrum) depend importantly upon q and thus upon the type of relaxation mechanism. The two terms A+ and A- are equal only when q = (in particular for purely radiative relaxation), in which case the spectrum is symmetrical. More generally, i.e., if elastic collisions are present, then A+ # A_, and the spectrum is asymmetrical. In the latter case, the fluorescent term (-A+) is a Stark-shifted and intensity-modified form of the term which is centered at I, = wl0 in the low intensity limit (4.2), while the Raman term (-A-), which is inherently dependent upon multiphoton processes, is for all values of q proportional in the weak field limit to R4 (i.e., to the square of the incident field intensity), and consequently does not appear in the low intensity limit (4.2).

Experimental .results involving intensity-dependent resonance light scattering in a collisional environment have been reported by CARLSTEN and SZOKE [1976], PROSNITZ, WILDMAN and GEORGE El9761 and CARL- STEN, SZOKE and RAYMER [1977].

The work of CARLSTEN, SZOKE and RAYMER [1977] consisted of a careful investigation under a wide range of experimental conditions of the scattering of light from a high intensity pulsed laser by strontium vapor in argon buffer gas, with intensities and detunings high enough to place the Rayleigh and Raman terms well outside the Doppler profile. Though steady state conditions were not strictly achieved nor was most of the work carried out in the impact regime (thus making the decay parameters dependent on the detuning), an impressive degree of correspondence with theory (MOLLOW [1977]) was achieved. Noteworthy in this respect were the confirmation of the intensity dependence of the Raman (or “three- photon”) term A- as given by (4.3b), and the measurement of the

34 RESONANCE LIGHT SCATIZRING AND FLUORESCENCE [I, 8 4

quenching rate Qr even though the latter was small compared to the radiative width r.

The use of the basis states I * ) makes possible a simpler analysis (MOLLOW [1977]) of the limit of well separated spectral lines in the collision-modified case, by a suitable generalization of the methods developed in particular by COHEN-TANNOUDJI and REYNAUD [1977a] to treat the radiative case, as described in 5 3.1.3. In the collision-modified case the transition rates between the states I&) defined by eqs. (3.14) are (MOLLOW [ 19771)

which determine the steady state populations p++ and fi-- through the relation

f i++K+- = f i - - K _ + , (4.7)

while the spectrum, which is of course determined by radiative transitions only, may still be found from the relations (3.16) and (4.3~). Since the collision-modified transition rates (4.6) are not in general proportional to the purely radiative rates (3.15), eqs. (3.16) and (4.7) make it clear why the spectrum is in general asymmetrical (A+ f A-), the exception being the case QE = 0, for which K+- - r+-, K-+ - r-+.

A similar analysis of the limit of well separated spectral lines under collisional relaxation was subsequently carried out by COURTENS and SZOKE [1977], who also treated the spectrum for transitions to other levels (MOLLOW [ 1973bl) while retaining, however, the two-level model (4.1) for the relaxation process.

MOLLOW [1977] was able to show that the (well-separated) spectrum for the laser-coupled 11) + (0) transition itself is accurately described, even when collisional, radiative, or other incoherent mechanisms induce transitions to and from other levels, by the relations

in which the steady-state populations are determined by the simultaneous action of the laser field and all incoherent processes which act upon the many-level atom.

(No relaxation process was explicitly introduced into the analysis of GUSH and GUSH [1972], which was shown by MOLLOW [1975a] to yield

I, 8 41 COLLISIONAL RELAXATION 35

the same results as a collisional model rather than a radiative one, and moreover to incorporate coherent and incoherent effects within a single term, similar to the Rayleigh term discussed in this section.)

4.2. COLLISIONS OF NONZERO DURATION

Outside the impact regime, either the collision duration 7,. the detuning A, or the power-broadening parameter 0 is so large that the product 0’7, becomes appreciable (rather than small compared to unity, as it is in the impact regime), and the effect of the coherent incident field must be taken into account in calculating the effect of collisions, even in the weak field limit. The Bloch equations in the simple form (3.1) are then no longer applicable.

By simply incorporating the Rabi oscillation within the autocorrelation functions described in § 4.1, one may evaluate outside the impact regime the effect of both inelastic collisions (LEHMBERG [1970]) and elastic collisions (SZIKLAS [1969]). (Inelastic collisions, which at optical frequencies are in any case usually treatable in the impact regime, were represented by SZIKLAS [1969] by means of a fluctuating term in the incident field, a procedure which is valid only at infinite temperature.) Working directly with the states I + > defined by eqs. (3.14), Sziklas was able to obtain simple solutions for the relaxation parameters K + _ and cr which appear in the equations

d d

(4.9)

which are of precisely the form discussed in § 4.1 (though with K+- = K-+

because of the assumption of infinite temperature). The values found by SZIKLAS [1969] for K+- and (T correctly depend upon the value of 0’7, (where T~ is the collision duration or equivalent fluctuation-correlation time), and reduce to the values found in the impact approximation only in the limit 0 ’ ~ , + 0 . In the opposite limit, ~ ’ T ~ + M , the effect of the oscillations induced by the incident field, roughly, is to reduce the effect of (elastic) collisions.

It is important to understand that the basic equations (4.9) used by SZIKLAS [1969] are valid only in the limit of well separated spectral lines, 0’ >>K’ Outside this limit, terms coupling, e.g., p+- to p++ and p_-, which

36 RESONANCE LIGHT SCATERING AND FLUORESCENCE [I, § 5

were not considered by Sziklas, would have to be included. The justification for dropping these terms is not that the coupling coefficients in question are small, for in fact they are comparable to those for the terms which are retained; rather, it is the great difference (-0') between the oscillation frequencies of the terms in question that effectively decouples their time dependence. A misleading impression is conveyed by SZIKLAS [1969] that the Bloch equations in their familiar form (3.1) fail to be applicable whenever saturation effects are important. In fact, the Bloch equations are valid for arbitrary degrees of saturation, provided only that the impact approximation is valid (O'T,<< 1). In this limit, restoration of the omitted terms described above leads, upon transcription to the lO)-l1) basis, precisely the simple and familiar form of the Bloch equations, in particular with plo coupled only to itself by the relaxation mechanism, whereas in the I * ) basis, all four density matrix elements are coupled to one another. SZIKLAS [1969] is evidently concerned, however, with the limit of well separated spectral lines, and the expressions he finds, notwithstanding the misleading statement described above, are valid in the stated limit.

Presumably the spectrum of the scattered field in the limit of well separated spectral lines can still be evaluated by means of eqs. (3.16) even outside the impact regime (though with the populations p++ and p-- evaluated (SZIKLAS [ 19691) by methods that allow for the nonzero duration of the collision), since the radiative emission process is governed by the correlation time w-' , and hence is essentially instantaneous under the conditions assumed in this article.

When collisions are both strong and lie outside the impact regime, i.e., when a single collision changes the state of the atom appreciably and has duration which is no longer short compared to the Rabi period, then only rather complicated calculations, which take into account the interaction between the colliding atoms in a detailed way, are applicable (LISITSA and YAKOVLENKO [1975]).

0 5. Multilevel Atom with Two Laser-Coupled States

5.1. THE LASER-COUPLED TRANSITION

When incoherent relaxation and pumping processes induce transitions out of and back into the two-dimensional subspace lO)-ll) of laser- coupled states, the solution for the emission field spectrum for the 11) + (0) transition, though it is in fact obtainable by means of an entirely

I , § 51 MULTILEVEL ATOM 37

straightforward generalization of the methods which apply in the two- level case, rapidly becomes very complicated as the effect of other atomic states is taken into account. This complexity is not due primarily to the difficulty in solving the many-level Bloch equations (3.1) for the steady state density matrix elements, which appear as constant coefficients in the solution for the spectrum and can accordingly be regarded as adjustable parameters or measured experimentally if the direct solution for them proves too difficult to carry out. The primary source of the complexity is the fact that transitions to all of the levels in question must be included in the rime-dependent Bloch equations for the correlation functions of the form (3.8), which through Fourier transformation determine the form of the spectrum even in the limit r‘ + 00 where the coeficients in the solution are constant parameters.

There are special limiting cases, however, where the solution of many- level problems of this kind can be obtained exactly or else expressed entirely in terms of quantities which relate only to the states 10) and 11) themselves (including their full widths, which involve transitions to other levels only implicitly, or possibly including the steady-state repopulation rates into the states in question). One such solution, which describes the limit of well separated spectral lines, has already been presented in eq. (4.8). Another case is that of the KARPLUS-SCHWINGER [1948] instantaneously thermalizing collision model, for the case of a many-level atom at nonzero temperature, where the exact solution can be obtained by making the substitution 1 - 2pE) + 0:s - 0:’: (where #’ is the zero field thermal population of the state l j ) ) in the result of MOLLOW [1970].

Another solvable case is the one treated approximately by RAUTIAN and SOBEL‘MAN [ 196 13 and NOTKIN, RAUTIAN and FEOKTISTOV [ 19671, where there is decay out of the laser-coupled subspace \O)-\l), but little or no repopulation back into it. By using the quantum statistical method, MOLLOW [1976] solved this problem in full generality, in particular avoiding imposing the restrictive decay-constant condition (1.5) on which earlier work had depended, and thus allowing for the possibility that many photons are emitted during the 11) + 10) transition before decay out of the Il)-lO) subspace takes place. The solution under discussion (MOL- LOW [1976]) exactly describes the case of decay with no repopulation, when the atom is initially prepared in the state 10) or 11) and the transient emission process is then observed. From the general solution one may subtract the term

12

38 RESONANCE LIGHT SCAlTERING AND FLUORESCENCE [I, 0 5

which in the limits K~ -+ 0, K~ - K ~ ~ ) 0 is proportional to the coherent &function term which appears in the steady-state spectrum for the closed two-level system. Even for nonzero values of K~ and K~ - K~~ the term in question, though it has nonzero width, would nevertheless represent a coherent part of the (transient) emission spectrum if all of the (nearby) atoms were actually prepared in the same initial state (10) or 11)) at the same time; for during the transient emission process that followed, all of the atoms would have mean dipole moments that oscillated in phase with one another.

If, on the other hand, a weak incoherent pumping mechanism repopu- lates the lO)-ll) subspace, then in the steady state which results, the term under discussion is proportional to part of the incoherent scattering spectrum, in the limit of vanishing repopulation rate R. The broadened term which formerly represented a coherent transient effect represents an incoherent steady state effect in the present case, since the limit R -+ 0 implies that different atoms are excited to the l0)-(1) subspace at different times. Thus the full spectrum which characterizes the case of decay without repopulation, including the term which in that case describes a coherent effect, is proportional to the incoherent part of the spectrum in the case where steady state is produced by a weak repopulation mechanism. In the latter case the coherent part of the spectrum, which is always a strict 6-function in steady state, is simply given by eq. (3.6). Being proportional to R2 (since p l o is proportional to R ) , the coherent part is inappreciable compared to the incoherent part, as a consequence of the limiting assumption R -+ 0.

The range of applicability of the model under discussion here, though broader than that of RAUTIAN and SOBEL'MAN [1961] and NOTKIN, RAU- TIAN and FEOKTISTOV [1967] because of the absence of the restrictive condition (1.5), is still somewhat limited by the assumption R -+ 0. When R becomes appreciable, the spectrum is broadened and otherwise altered in a nontrivial way. (COOPER and BALLAGH [1978] have obtained closed solutions which include the effect of repopulation in the limit of weak incident field intensity.)

In the weak field limit, the solution found by MOLLOW [1976] reduces to the one found by OMONT, SMITH and COOPER [1972]. The result, which shows the effect of nonzero width of the lower level, is of somewhat more complicated form than the (erroneous) one given by HEITLER [ 1954, p. 1981.

GOODMAN and THIELE [1972] have treated the problem of light scattered by a many level system that undergoes unimolecular decay.

I, 8 51 MULTILEVEL ATOM 39

5.2. TRANSITIONS INVOLVING OTHER STATES

Transitions between a laser-coupled state and another, uncoupled state of the atom (with either state the initial state) show the effect of the laser coupling through what may be roughly described as a Stark splitting into two components of the laser-coupled state involved. The spectrum for such transitions has been found under quite general conditions by MOL- LOW [1972a, 1973bl. Unlike the more complicated solutions which describe emission during transitions between the two laser-coupled states themselves, the exact solutions for transitions involving another level can be expressed (MOLLOW [1972a, 1973bl) even when the repopulation rate is appreciable in a form which explicitly involves only quantities which refer to the transition in question, though the steady-state density matrix elements which appear as constant coefficients in the solutions can be directly evaluated only by solving the many-level Bloch equations. Like the other solutions which follow from the exact statistical method, the ones under discussion here implicitly involve coherent multiphoton emission effects (MOLLOW [1973b, Sec. IVB]) and are thus more general than the one-photon emission solutions of the same problem found by NOTKIN, RAUTIAN and FEOKTISTOV [1967]. The latter solutions, unlike those that follow from the exact statistical method, require even for transitions to another level that the system decay completely out of the laser-coupled subspace, and hence that when repopulation does take place, that the repopulation rate be so small as to make the steady-state probability of finding the atom in the laser-coupled subspace small compared to unity.

By contrast, in the solutions of MOLLOW [1972a, 1973b], the steady state probability Po,+ PI, is unrestricted, and the solutions are thus valid, e.g., even when the lower laser-coupled state 10) is the atomic ground state and has population of order unity. In the limit of weak incident fields, if 10) is the ground state and the relaxation processes are all dissipative, the emission spectrum for transitions from the upper laser- coupled state 11) to a third state l j ) of lower energy consists of two components (HUBER [1969], OMONT, SMITH and COOPER [1972], MOLLOW [ 1973b]), a fluorescent component centered at the spontaneous emission frequency u = olj for the transition in question, and a Raman component centered at the frequency I/ = o - wjo. The fluorescent component has width K ; , and has nonvanishing intensity only if elastic collisions are present, thus establishing an incoherent population in the state 11). In the Raman component, by contrast, the state 11) is simply an intermediate state in the two-photon transition sequence 10) + 11) + I j ) , and does not

40 RESONANCE LIGHT SCATTERING AND FLUORESCENCE [I

enter into the determination of either the width K;” or the frequency o - ojo of the spectral contribution. (This process has been investigated experimentally by ROUSSEAU, PATTERSON and WILLIAMS [ 19751.)

For more intense incident fields, the limit of well separated spectral lines for the \l)-+ lj) transition is well approximated as (MOLLOW [1973b])

2u-A- + 2U+A+ (I, - wlj -;A + $0’)” + U: ( v - wli -;A -;a’)’+ u?’

g(v) =

A, ,(1 *A/0’)T$(fioO- PI , )0 ’ /0 ’A , (5.1)

U , = ~ ( K J ~ + K ~ , ) T ; ( K ~ ~ - ~ l , ) A / 0 ’

The frequencies of the fluorescent and Raman terms ( - A + and A_, respectively, for A > 0) are thus (equally) moved away from each other by one half of the dynamical Stark shift O‘-A. The steady state populations in the above expressions for A,, as in the analogous eqs. (4.3a, b), must be determined in the presence of the coherent incident field and all of the relaxation processes which act upon the many-level atom. In the high saturation limit (0 >>A, K ~ , PI, = &,) the two terms in (5.1) have equal intensity A+ = A- =$PI ,, and equal width u+ = u- = &(KJ()+ K ; , ) .

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34 (1972) 5201.


E. WOLF, PROGRESS IN OPTICS XIX @I NORTH-HOLLAND 1981

I1

SURFACE AND SIZE EFFECTS ON THE LIGHT SCATTERING SPECTRA OF SOLIDS

BY

D. L. MILLS*

Department of Physics, University of California, Irvine, California 9271 7, U.S.A.

and

K. R. SUBBASWAMY

Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, U.S.A.

* Research supported by the U.S. Air Force Office of Scientific Research under contract No. F49620-78-c-00 19.

CONTENTS

PAGE

$1. INTRODUCTION . . . . . . . . . . . . . . . . 47

$2 . LIGHT SCATTERING FROM OPAQUE MEDIA AND FILMS OF FINITE THICKNESS; QUALITATIVE CON- SIDERATIONS . . . . . . . . . . . . . . . . . 5 1

Q 3. SURFACE AND GUIDED WAVE POLARITONS . . . . 64

$ 4. LIGHT SCATTERING FROM SURFACE AND GUIDED WAVE POLARITONS . . . . . . . . . . . . . . 76

5 5. SURFACE AND SIZE EFFECTS ON BRILLOUIN SCAT- TERING FROM ACOUSTICAL PHONONS AND SPIN WAVES . . . . . . . . . . . . . . . . . . . 95

$ 6. LIGHT SCATTERING AS A MICROSCOPIC PROBE OF THE SURFACE REGION. . . . . . . . . . . . . 124

$7 . CONCLUDING REMARKS . . . . . . . . . . . . 134

REFERENCES . . . . . . . . . . . . . . . . . . 135

fi 1. Introduction

The study of the inelastic scattering of light from solids has proved a powerful method of studying elementary excitations such as phonons, spin waves, the collective motions of charge carriers, and polaritons (see HAYES and LOUDON [1978]). The coupling tensors between light and these excitations are intimately related to those that control the phenomena of interest in non-linear optics. One thus obtains insight into the non-linear optical response of solid materials through study of the light scattering spectra. The field has grown to become one of the principal experimental techniques in condensed matter physics, after the appearance of highly monochromatic laser sources and associated detection techniques.

Most of the literature on light scattering from solids is concerned with samples of macroscopic dimensions that are also transparent to both the incident and the scattered radiation. One can then discuss the phenomenon as if this scattering event occurs in a medium of infinite spatial extent, with both the incident and scattered photons treated as plane waves.

In the past few years, both experimental and theoretical papers have been concerned with the scattering of light from opaque crystals, where as a consequence of the finite skin depth, the incident and scattered radia- tions sample only a small volume near the sample surface. Also, analyses of spectra of very thin films have appeared, along with theoretical discussions. Thus, we have in hand a substantial body of material that explores the effect of a surface or of finite sample size on the light scattering spectra of solids. The purpose of this article is to discuss these recent developments, with emphasis on certain theoretical methods that have proved useful to the present authors, and to others engaged in research in the area. While we shall also discuss the experimental data, a companion article places its primary emphasis on experimental studies of size and surface effects.

Before we direct our attention to the size effects mentioned above, we outline the key concepts useful in analyzing the spectrum of a medium of infinite spatial extent also transparent to the incident and scattered radiation. We illustrate with a description of the scattering of light from

47

48 LIGHT SCATTERING SPECTRA OF SOLIDS [II, § 1

the quantized lattice vibrations of a solid (phonons), though the reader will recognize the concepts we emphasize here are applicable to the scattering of light from a wide variety of elementary excitations in solids.

In Fig. l . l a , we sketch the phonon spectrum of a typical simple diatomic solid of cubic symmetry (KITEL [1971]). We give here the frequency of the various normal modes (plane waves for the infinite solid) as a function of wave vector. In the figure, we presume for simplicity that the wave vector is directed along a principal axis. One sees a transverse and a longitudinal optical phonon branch, labeled TO and LO, one of which (TO) is two-fold degenerate. The optical branches are characterized by dispersion relations wLO(Q) and wTO(Q) with frequency that remains finite as the wave vector Q approaches zero. For a typical crystal, the Q = 0 optical phonon frequencies (in units of 2 d A ) lie in the range from a few hundred cm-' to at most -2500 cm-'. One also has transverse and longitudinal acoustical phonon branches, labeled TA and LA, one of which (TA) is also two-fold degenerate for the special propagation direction assumed in Fig. 1. l a . The acoustical phonon frequencies wTA(Q) and wLA(Q) both vanish linearly with the magnitude of the wave vector Q, in the limit as Q approaches zero. Finally, note that the dispersion relations terminate at the boundary of the Brillouin zone, Q = Q B z =

da,, with a, the lattice constant of the crystal. With the typical value a,= 3 x lo-' cm, we have Q B ~ E 10' cm-'.

I I

W

Fig. 1 .1 . (a) Sketch of the phonon spectrum of a typical diatomic crystal. Here w is the frequency of the phonon, Q its wave vector, and Q,, the wave vector of a phonon at the boundary of the first Brillouin zone. (b) Sketch of the elementary process in which a photon

scatters from a phonon of wave vector Q.

11, 0 11 INTRODU~ION 49

A scattering event in which an incident phonon of wave vector k, and frequency wu is scattered inelastically is illustrated in Fig. 1.lb. The scattered photon has frequency o, related to that of the incident photon via the energy conservation condition

hw, = ho,, - hw(Q), (1.1)

where w(Q) is the frequency of the phonon responsible for the scattering. The wave vector k, of the scattered photon is found from

k, = k, - Q. (1.2)

In eq. (1.2), upon assuming that the incident photon is in the visible frequency range, we have for the magnitude of ko, Ikol = 10’ cm-’. Since the frequency shift is small, necessarily lkol = IkJ, so the wave vector of the phonon created in the scattering process is also the order of 10’ cm-’. This means, if QBz is the maximum phonon wave vector in Fig. 1 . l a (the wave vector at the boundary of the first Brillouin zone), one has Q / Q B z z

lo-’ for the phonon which participates in a light scattering event. Thus, on the scale of wave vectors relevant to Fig. 1.1 a. the wave vector of the phonon created in the scattering event is very near zero. Its wavelength is comparable to that of the incident radiation, which is very long compared to the lattice constant of the crystal. Thus, by means of the light scattering methods, one can probe only excitations with wavelength long compared to a lattice constant*.

The phonon created in the scattering process may be either an optical phonon, or an acoustical phonon. The frequency shift suffered by the photon differs by some three orders of magnitude in the two cases, by virtue of the vanishing of the acoustical mode frequencies at zero wave vector. For scattering from acoustical phonons, the frequency shift is typically a few tenths of a wave number, while it is typically a few hundred wave numbers for scattering from optical modes. In the laboratory, very different experimental techniques are required to detect scattering from optical phonons, and from acoustical phonons. In the former case, a grating spectrometer may be employed, and in the latter, a Fabry-Perot interferometer is required for the small frequency shift to be detected. Thus, while both processes seem very similar in nature to the theorist, to the experimentalist they appear very different indeed.

* This statement is true for all scattering processes in which a single quantum of crystal excitation is created. Higher order processes, such as scattering with participation of two phonons, may involve short wavelength modes, as long as the sum of the wave vectors of all participating elementary excitations is nearly zero.

50 LIGHT SCATTERING SPECTRA OF SOLIDS [II, 9: 1

A scattering process where an optical phonon is created, or more generally any inelastic event with frequency shift sufficiently large to be detected by a grating spectrometer, is referred to as a Raman scattering process, while an event with frequency shift in the range of one wave number or below, with the consequence that a Fabry-Perot device is required, is called a Brillouin scattering process. In the present article, we shall explore surface and size effects on both the Raman and the Brillouin spectra of crystals.

For any light scattering event, whether it takes place in an infinitely extended solid, a crystal of finite size, or even a small molecule, it is necessary that eq. (1.1) hold. This is just the statement of energy conservation, where the energy of the incident and scattered photons must differ by precisely that of the vibrational quantum created (or destroyed) in the scattering event. Of course, in a small system, it makes n o sense to characterize the vibrational quanta as having a well defined wave vector Q, but the various normal modes of the system have well defined vibrational frequencies and quantized energy levels.

The statement of conservation of wave vector, eq. (1.2), holds only if the propagation length of the radiation in the medium is a large number of wavelengths. Strictly speaking, it holds only in a perfectly transparent medium of infinite extent, since it is only then we may think of the incident and scattered light waves as perfect plane waves characterized by a precisely defined wavelength. A large portion of this article will be devoted to what may be learned when eq. (1.2) breaks down severely, either because the light is strongly absorbed by the substrate with the consequence that the propagation length is short compared to the wavelength, or because the scattering event takes place in a thin film with thickness the order of an optical wavelength.

The organization of this article is as follows. In § 2, we discuss a number of reasons why the light scattering spectrum of an opaque or finite crystal may differ from that of an infinitely extended transparent one. In § 3 and § 4, we explore in detail a specific example that has been the focus of considerable experimental and theoretical investigation. This is Raman scattering by polaritons in thin films, where surface and guided wave polaritons appear in the spectra. We shall understand this language in § 3, where the basic concepts are introduced from a qualitative point of view. Then, 04 is devoted to a detailed description of light scattering from these entities, while § 5 explores Brillouin scattering from spin waves on magnetic surfaces and in thin films, and also Brillouin scattering

11, P 21 OPAQUE MEDIA, FILMS OF FINITE THICKNESS 51

from phonons on surfaces and in thin films. The concepts developed in the special case examined in detail in § 3 and § 4 may be brought to bear on these examples directly. Finally, § 6 explores the scattering of light from surfaces of or interfaces between doped semiconductors. In the examples discussed up to this last example, the physical properties of the finite crystal film have been assumed quite identical near the surface as in the bulk. In doped semiconductors, the charge carrier density may be dramatically different near the surface or near an interface than in the bulk; in some instances the surface or interface profile is subject to experimental manipulation, for a given sample. A variety of material properties near the surface and in the bulk now differ dramatically, and the light scattering method provides a powerful probe of the surface region.

8 2. Light Scattering from Opaque Media and Films of Finite Thickness; Qualitative Considerations

The purpose of this section is to outline the principal notions that prove useful in the analysis of light scattering from samples either opaque to the incident radiation, or in the form of thin films. In both cases, the incident photon interacts with the excitation from which it scatters in a volume reduced by several orders of magnitude over that appropriate to transparent materials, where path lengths the order of a centimeter are commonly encountered. The emphasis in the present section is on qualitative considerations. We turn to a detailed and quantitative discussion of a particular physical situation in § 3 and (i 4. We begin here by examining first the case of the inelastic scattering of light from a material opaque to the incident radiation, then turn our attention to scattering from a thin film.

In Fig. 2.1, we illustrate schematically the geometry employed in the study of light scattering from a substrate opaque to the incident radiation. Incident light strikes the material, at an angle of incidence Oo measured with respect to the normal to the surface. Of course, there is a specularly reflected ray of light that also makes an angle O0 with the normal. The penetration depth of the incident radiation is the skin depth 6, which may be of the order of 1000 8, or 2000 8, for radiation with frequency beyond the absorption edge of a typical semiconductor, or as small as 150 8, or 200 8, for radiation incident on a metal.

While it is in the material, the light may scatter from a phonon, as

52 LIGHT SCATTERING SPECTRA OF SOLIDS [II, 0 2

INCIDENT LIGHT

REFLECTED INELASTICALLY LIGHT SCATTERED

LIGHT

Fig. 2.1. A diagram of a backscattering experiment from an opaque substrate. In addition to the specularly reflected beam, light scattered from excitations in the solid is deflected away

from the specular direction.

illustrated in Fig. 2.1, or from some other elementary excitation. The scattered light emerges not along the specular direction, but rather at an angle 8,f O0 from the normal, where the determination of 8, will be discussed below. The basic experiment that motivates the present article is the frequency spectrum of this off -specular radiation. Its experimental study is not simple. The basic problem faced by the experimentalist is that the signal is very weak. While one may direct 1017-101’ photons/sec onto the sample from the incident laser, in a typical experiment only a few photons/sec will arive at the detector, in the frequency regime of interest. The problem is to discriminate between the photons of interest, which suffer inelastic scattering, and the light in the wing of the laser line scattered elastically away from the specular direction by the roughness inevitably present on the best prepared samples. Despite these difficulties, since the pioneering study of the Raman spectra of optical phonons in metals by Parker and co-workers (FELDMAN, PARKER and ASHKIN [1968]; PARKER, FELDMAN and ASHKIN [ 1969]), the inelastic scattering of light from opaque media has been studied in a number of laboratories. We refer the reader to the companion article by USHIODA [1981] for a detailed discussion of the experimental techniques required for such studies.

We must begin our discussion of the experiment with an analysis of the elementary excitations of the semi-infinite crystal. Since the radiation samples only the near vicinity of the sample surface, features in the

11,s 21 OPAQUE MEDIA. FILMS OF FINITE THICKNESS 53

spectra can and often do reflect the influence of the surface on the excitation spectrum. To simplify our discussion, we confine our attention here to light scattering from phonons, and for the moment we assume further that the physical properties of the material within the skin depth are identical to those in the bulk. There is very great interest in utilizing the light scattering method to explore differences in material properties near the surface, and we shall discuss experiments that explore this question later in the present article.

A review of the influence of a surface on the lattice dynamics of crystals has been presented recently (WALLIS [1973]), and we shall require only a few general principles here.

In an infinitely extended material, the presence of translational invariance in three spatial directions is exploited to endow each phonon with a three dimensional wave vector Q; we utilized this in our discussion of scattering kinematics in D 1. For a semi-infinite crystal with a flat surface, translational invariance remains only in the two directions parallel to the surface. It follows that each normal mode has associated with it a two dimensional wave vector Qll which lies in the plane of the surface. If the 2 axis is normal to the surface, then Q, is a “good quantum number” for phonons in the infinitely extended crystal, but not for phonons in the semi-infinite crystal.

To appreciate the nature of the normal modes of the semi-infinite crystal imagine one “launches” a bulk phonon deep within the crystal, and lets it propagate to and reflect from the surface. For definiteness, suppose we launch a longitudinal-acoustical (LA) phonon. Let the wave vector of the phonon be Q, with QII the projection of Q on a plane parallel to the surface. The LA phonon can reflect off the surface; if Q, is the wave vector component of the initial phonon normal to the surface, then the reflected LA wave has wave vector -Q,. One can say that the surface has thus mixed two bulk excitations, one with wave vector component +Q,, and one with wave vector component -Q, normal to the surface. Both waves have precisely the same projection Ql, of the wave vector onto the plane of the surface, but the new entity cannot be characterized by a single value of Q,. A full analysis of the problem including appropriate boundary conditions at the surface shows that in general, not only is a wave of LA character reflected off the surface, but so are transverse acoustical (TA) phonons. Thus, the surface mixes together different phonon branches of the bulk, to produce a complex entity that is the normal mode of the semi-infinite crystal (see LANDAU and LIFSHITZ [1959]

54 LIGHT SCA’ITERING SPECTRA OF SOLIDS [II,6 2

p. 101). All waves combined to form this entity have the same wave vector component Qa parallel to the surface; this is insured by the translational invariance that remains. We denote the frequency of the resulting excitation by the symbol o,(Qll) where a, not simply Q,, is an index which labels the specific mode of interest.

The previous paragraph discusses the manner in which the surface “mixes” various bulk normal modes of the crystal, to lead to a new and complex entity that is the normal mode of the semi-infinite material. In addition, in the presence of the surface we find a new class of modes that have no counterpart in the infinitely extended medium, but owe their very existence to the presence of the surface. These are surface phonons, again each characterized by a well defined value of QII but with displacement fields that fall to zero exponentially as one moves into the interior from the surface. Perhaps the best known example is the Rayleigh wave, a surface acoustic wave known since the nineteenth century (see LANDAU and LIFSHITZ [1959] p. 107) and currently of great interest in device applications. This is an acoustical wave that emerges from the theory of elasticity, in which a particle executes elliptical motion, with the size of the ellipse an exponentially decreasing function of distance from the surface. If QII is the magnitude of the wave vector of the Rayleigh wave parallel to the surface, the distance required for the displacement in the Rayleigh wave to decay to zero is the order of Qi’. That is to say, the displacement field penetrates into the crystal interior a distance the order of the wavelength parallel to the surface.

In the range of frequencies allowed for optical phonons, one finds a wide variety of surface modes. Some, as in the case of the Rayleigh wave, have the property that the displacement field associated with the wave penetrates a distance the order of Q,’ into the material. These modes, often called surface polaritons, will be discussed in detail below. Others are microscopic in character, i.e. the displacement field is localized in the outermost few atomic layers of the material (see WALLIS [1973]). We know of no study of such a “microscopic surface optical phonon” by the light scattering method, though one clear example has emerged from the high resolution studies of inelastic electron scattering from the (1 11) surface of silicon (IBACH [1971]).

We next turn to a description of the kinematics of the scattering of light by the normal modes of a semi-infinite crystal. Once again, the key element of the discussion is that components of wave vector normal to the surface are no longer conserved, although components parallel to the

11, 5 21 OPAQUE MEDIA. FILMS OF FINITE THICKNESS 55

surface remain conserved. Thus, we can generalize eq. (1.1) and eq. (1.2). Let w,(Qll) be the frequency of a normal mode of the semi-infinite crystal: we examine a Stokes process where the photon loses energy after creating an excitation in the substrate. We use a slightly more explicit notation than in Q 1, and write the energy conservation relation in the form

hw(kJ = hw(ko) - hw, (QII), (2.1)

while now only components of wave vector parallel to the surface are conserved:

kf’ = kf’ - QII . (2.2)

If we consider scattering by a particular normal mode of the substrate, presumably k, and QII are known in eq. (2.1) and eq. (2.2). The direction and magnitude of k, requires the knowledge of three pieces of information, so eq. (2.1) and eq. (2.2) uniquely determine the direction and magnitude of the wave vector k,.

In many experimental configurations, to excellent approximation, all normal modes of the crystal with the same value of Qll (all surface and bulk waves with a particular QII) scatter the incident light into the same direction*. Suppose the incident radiation lies in the visible range of frequencies, where hw,(ko) is two or three electron volts. For typical solids, hw,(Qll) is at most 0.05 eV, so the factor Aw,(QII) may be dropped from the right hand side of eq. (2.1), which then reduces to the statement lksl= lko\. The direction of k, is then found from eq. (2.2), and the direction of k, is thus the same for all modes with a given value of Q,,.

We then have the following picture of the spectrum of radiation obtained in the backscattering experiment outlined in Fig. 2.1. For a fixed value of QII, a whole band of frequencies is allotted to the bulk waves associated with a given value of QII. In our example of an LA phonon reflected from the surface, the frequency of a plane wave excitation in the infinitely extended crystal is cl(Q;+ Q:);, where c1 is the longitudinal sound velocity. So in a continuum theory, the band of frequencies allotted to LA phonons with fixed wave vector Q,, extends from the lower bound c1Qr (Q, = 0) to infinity (Q, = m)t. At the same time, for fixed Q l l , there may be one or more surface modes with frequency outside the band of

*This breaks down for light scattered very near the specular direction, or for light scattered very close to the forward direction through a film.

Of course, in a proper lattice dynamical theory, Q must lie within the first Brillouin zone, so for any value of Q,,, there is an upper bound on Q,. This is the Q,, of 5 1.

56 LIGHT SCATERING SPECTRA OF SOLIDS [II, (i 2

frequencies allotted to the bulk modes. In our example, the frequency of the Rayleigh wave is c,Qll, where the velocity cR of the Rayleigh wave is smaller than either cl or the transverse sound velocity c,. The spectrum of scattered light thus contains lines produced by scattering from surface waves, and bands produced by scattering from the continuum of bulk excitations associated with the particular value of QII that defines the direction of the outgoing radiation.

To predict the shape of the band produced by scattering from the bulk excitations, one requires a detailed theory of the light scattering process (MILLS, MARADUDIN and BURSTEIN [197O]) and also of the influence of the surface on the excitation spectra of the material. We shall discuss theories which combine these elements in specific examples outlined below. The principal features that enter can be sketched intuitively, and we turn to such a simple discussion next.

We shall ignore the vector character of the light field for the moment, and focus our attention o n only its spatial form. Inside the medium, presumed to occupy the half space z > O , the light field associated with both the incident and scattered waves has the form

E,(x) = E, exp [ikll * xII+ iky’z - k:2’z], (2.3)

where kll is the wave vector parallel to the surface (real), and k,= ky’+ iky) is the complex wave vector normal to the surface. The magnitude of kll is ( w / c ) sin 8, where w is the frequency of the wave and 8, the angle the wave field makes with the normal to the surface, in the zlacuum outside the crystal. The complex wave vector k, is given by [ E W ~ / C ~ - kf]? with E the complex dielectric constant of the substrate and the complex square root is taken such that Im (k,) = ky’>0. The optical skin depth 6, defined as the distance the wave must penetrate into the substrate before the intensity of the radiation falls to l/e of its value at the surface, is given by 6 = (2ky’)-’.

We may rewrite eq. (2 .3) in the form

and thus regard the exponentially decaying wave in eq. (2.3) as a synthesis of pure plane waves, but with wave vector q1 normal to the surface spread over a range controlled by k:*’. If dq,f(q,) is the fraction of the wave field intensity stored in the piece with wave vector from q, to

11, § 21 OPAQUE MEDIA. FILMS OF FINITE THICKNESS 57

q1 + dq, perpendicular to the surface, then

1 k f (4,) = - 2Tr (q1-k1")2+(ky '

Now consider a scattering event, where an incident photon scatters from a phonon. We wish here to examine the simplest possible picture which incorporates the influence of the finite skin depth on the spectrum. Thus, we begin by considering scattering from a bulk phonon, and despite the analysis put forward earlier in this section, we ignore the influence of the surface on the nature of the normal modes. Later in the article ($4), when the Green's function approach of calculating light scattering spectra is presented, we shall have ample opportunity to see how this is incorporated into a proper theory, and how the spectrum is influenced by surface modifications of the bulk eigenmodes.

Let M(Qll, Q,) be the matrix element that describes the coupling of light to a phonon of wave vector Q = QII+ iQ, (where 2 is a unit vector normal to the surface). The intensity of the scattered light can be found by simply summing up the intensity for scattering from the individual plane wave components in eq. (2.4). If unessential factors are set aside, and scattering from only one phonon mode is considered, then with the direction of the outgoing light found from the kinematical conditions in eq. (2.1) and eq. (2.2), the intensity is proportional to

I (Q) = IM(QII, Qz)Iz a(hwo-hw,-hw(Q))

(2.6)

where fo and fs are the form factors in eq. (2.51, for the incident and scattered wave, respectively.

The integral on q, in eq. (2.6) is readily carried out, and if we further note, as remarked earlier, that all modes with a given QII scatter into the same solid angle to excellent approximation, to within a multiplicative constant A the scattering intensity per unit solid angle dS/dQ found by integrating on Qz, is given by

Here we have written the wave vector component normal to the surface

58 LIGHT SCATERING SPECTRA OF SOLIDS [II, P 2

of the incident and scattered fields as kF’= kYl’+ikf‘*’, and k‘,“’= kFL’+ iky’, respectively. For a back scattering experiment, as illustrated in Fig. 2.1, where the scattered wave propagates in the opposite direction from the incident wave, ky” in the denominator is a negative number, while kiol’ is positive. Results identical to eq. (2.7) have been derived rigorously elsewhere, within the framework of a Green’s function method that produces a proper expression for the prefactor A in eq. (2.7) (MILLS, MARADUDIN and BURSTEIN [1970]).

We now apply eq. (2.7) to various special cases. This will provide us with a feeling for the role strong absorption can play in affecting the shape of the light scattering spectrum.

Consider the scattering of light by optical phonons first, and suppose we consider a crystal where the matrix element M(Qll, Q,) is non-zero when the wave vector Q vanishes. This will happen in general for crystals which lack an inversion center, and possibly for a subset of the optical phonons at Q = 0 in a crystal with an inversion center present (HAYES and LOUDON [1978]). The Lorentzian factor in eq. (2.7) plays the role of constraining the wave vector mismatch AQl = k?”- k Y ” - Q, in the scattering process to be comparable to the inverse skin depth 6, assuming the skin depth of the incident and scattered radiation to be the same. If M(Qll, Q,) is finite at Q = 0, then it can be expected to vary only slightly over the small range of 0, values that enter the integral in eq. (2.7). Thus, we replace M(Q11, Q,) by its value at Q = 0, and a similar argument allows us to do this also for the phonon frequency w(Q). Then upon carrying out the integration on Q,, one finds

dS/dO = A IM(0)l’ 6(hwo- ha,- hw(0)).

We have a line spectrum identical in form to that seen in a transparent medium; a proper theory shows the prefactor A to be proportional to the skin depth 6. Thus, in this example, the effect of strong absorption is trivial. Its role is only to reduce the volume within which the scattering event occurs. This surely weakens the signal very substantially, to the dismay of the experimentalist, but the spectrum is unchanged in form or content from that appropriate to the transparent medium. Of course, the appearance of the delta function in eq. (2 .8) is a direct consequence of our neglect of the finite lifetime of the Q = 0 optical phonon. Proper inclusion of this replaces the delta function in eq. (2.8) by a symmetric Lorentzian, with width in frequency controlled by the phonon lifetime (MILLS, MARADUDIN and BURSTEIN [1970]).

11, 0 21 OPAQUE MEDIA, HLMS OF FINITE THICKNESS 59

Much more interesting to consider is the scattering from a phonon branch with symmetry such that lM(Qll, Q z ) 1 2 vanishes at zero wave vector. Two examples come to mind. One is the scattering from optical phonons forbidden by symmetry considerations to scatter at Q = 0. The second is scattering from acoustical phonons (Brillouin scattering). For- mally, an acoustical phonon with Q-0 describes a rigid body translation of the center of mass of the crystal (KITTEL [1971]). Such a rigid translation cannot alter the index of refraction of the crystal, and thus can not lead to time dependent fluctuations in it; we get no scattering from acoustical phonons of zero wave vector, and M(Qll, Q,) necessarily vanishes there.

If M(Q11, 0,) vanishes at zero wave vector, then for small values of Q, a Taylor series in the components of Q is appropriate to find the first non-vanishing term. We have seen that the phonon wavelengths encountered in the light scattering theory are always expected to be long compared to the lattice constant. Under these conditions, the Taylor series expansion is a quite satisfactory procedure.

In eq. (2.71, the value of QI1 is fixed by the scattering geometry and the kinematical constraint in eq. (2.1). However, Q, is integrated upon, and is unrestricted. We expect a term in Q, in the Taylor series expansion of M(Q11, Q,) quite generally. When this term is squared, it leads to a contribution proportional to Q l in the numerator of eq. (2.7). For large Q,, this is the dominant contribution to IM(QII, Q,)I” and the ratio of the numerator and the denominator in eq. (2.7) thus approaches a constant for large Q,. Hence isolating this dominant term gives

which leads to a spectrum qualitatively different in shape from that displayed in eq. (2.8). If the difference Aw = wo - w, lies in a range where the argument of the delta function vanishes, then we have (assuming w(Q) is an even function of Qz)

(2.10)

where the derivative in the denominator of eq. (2.10) is evaluated at that value of Q, for which the argument of the delta function vanishes.

We may obtain a feeling for the spectral shape predicted by eq. (2.10) with application to two examples. Consider first scattering by an optical

60 LIGHT SCATERING SPECTRA OF SOLIDS [II, 8 2

phonon “forbidden” by symmetry to scatter at Q = O . This case was examined in the original paper on backscattering from an opaque material (MILLS, MARADUDIN and BURSTEIN [1970]). For a cubic crystal the dispersion relation at long wavelength has the form w(Q) =

wo[l-P(Qao)2], where a, is the lattice constant and 6, a constant of order unity, which is generally positive. This dispersion relation gives

(2.11)

where O(x) is unity for positive values of its argument and zero for negative values. We have ignored the influence of the small term pa:Qf in deriving eq. (2.11).

The result in eq. (2.11) describes a spectrum which differs qualitatively from the symmetric Lorentzian displayed in eq. (2 .8) . We have a highly asymmetric line in eq. (2.11), which has a sharp onset at the frequency shift Aw = w, where scattering is first allowed, then tails off slowly as Aw moves into the allowed region. The square root singularity in eq. (2.11) is an artifact of our neglect of the finite lifetime of the optical phonon. Calculations which include the lifetime can be found in the original literature (MILLS, MARADUDIN and BURSTEIN [1970]). One finds a blunting of the divergence at Aw = wo, but the lineshape remains highly asymmetric with the long square root tail as displayed in eq. (2.11).

ANASTASAKIS and BURSTEIN [ 197 11 have observed Raman scattering from Raman-inactive optical phonons in 11-IV semiconductors of the anti-fluorite structure, under conditions similar to those presumed above. In their work, the skin depth was the order of 100 A, and the forbidden modes appear under resonant Raman scattering conditions (i.e. the incident laser photon frequency lies close to the fundamental absorption edge of the material) where the derivative (dM/dQ,), is expected to be very large (MARTIN [1971]). In Mg,Si and Mg,Ge, the line shape is difficult to analyze, since in one case (Mg,Si) the line is weak, and in the second (Mg,Ge) it lies very close to a strong Raman-allowed mode. In Mg2Pb, the “forbidden” lines are asymmetric, but they also appear on top of a strongly frequency dependent background. This experiment thus reports scattering very similar to that described above, though no clear conclusions about the line shape follows from the data.

It is interesting to note that we may expect strong scattering from phonons with large values of Q, (i.e. far into the Brillouin zone) near those points where the derivative (dw/dQ,), vanishes. In essence, the

11, 8 21 OPAOUE MEDIA, FILMS OF FINITE THICKNESS 61

electric fields in the medium acquire spatial Fourier components of large wave vector in the presence of absorption, and these large wave vector components may excite very short wavelength phonons, with the possibility of features in the spectrum from those regions of the dispersion curve (van Hove singularities in the one dimensional density of states) where (dw/dQ,), vanishes. We are unaware of any experimental observation of such features in a light scattering measurement.

The result in eq. (2.10) may be applied directly to scattering from acoustical phonons near the surface of an opaque material (BENNE~T, MARADUDIN and SWANSON [1972]). Here we have for the dispersion relation w(Q) = c,(Q;f+ Q:); with c, the velocity of sound for the particular phonon branch of interest, assuming the dispersion relation is isotropic for simplicity. Thus, in this case we are discussing Brillouin scattering. For a given scattering geometry, the spectrum begins at the frequency shift %QII, and extends on to higher frequencies. Eq. (2.10) then gives for this shape of the spectrum the result

(2.12)

Again we have a highly skewed spectrum, with a long high frequency tail that extends well into the allowed region of scattered frequencies. At the “onset” frequency Aw = c,QII, we have a square root singularity very similar to that in eq. (2.11).

There have been by now a sequence of experimental studies of Bril- louin backscattering from acoustical phonons on the surface of opaque media. The first observation of an asymmetric line shape with the form given in eq. (2.12) was reported by DIL and BRODY [1976]. Since then, very beautiful and complete studies of phonons on metal and semiconductor surfaces have been presented (SANDERCOCK [ 19781). A complete interpretation of the results requires a more careful treatment; most particularly one must take due account of the influence of the surface on the excitation spectrum of the material. The experiments show scattering from surface phonons (Rayleigh waves) not considered here, and features that give information on the polarization properties of the bulk waves as well as the influence of the surface on them. We turn to this topic later in the present article, where an outline of the proper theoretical treatment will be provided.

The remarks above outline the main qualitative features of the influence of a surface, combined with absorption of the incident radiation,

62 LIGHT SCATTERING SPECTRA OF SOLIDS [II ,52

on the light scattering spectrum of solid materials. On the basis of a simple calculation, we can see that strong absorption introduces spatial Fourier components of large wave vector into the light field in the medium. For scattering from bulk excitations, this leads to the breakdown of selection rules that apply to nominally transparent materials, and allows one to probe “forbidden scatterings”. The resulting spectra have skewed shapes that differ qualitatively from spectra taken in transparent materials. In addition, we have yet to discuss the rich variety of surface modes that may be explored in such experiments. We conclude this section with a brief comment on scattering from thin films.

Now suppose that we consider scattering from a thin film that has finite thickness L. Then, clearly there are two basic ways in which the above picture is altered, if the film is sufficiently thin.

First, we can interpret eq. (2.7) in the following fashion. If we ignore the difference between ky2’ and ky’ produced by the small frequency shift suffered upon scattering, then eq. (2.7) states that there is breakdown in wave vector conservation for wave vector components normal to the surface, with (Ak,)S- 1, where Ak, is a measure of the amount by which k y l ) - k p ” can deviate from zero in a scattering that contributes importantly to the spectrum. The non-linear interaction that is responsible for the scattering takes place in a small region of spatial extent 6 normal to the surface, and we expect the relation (Ak,)6 - 1 simply from this alone. In the language of non-linear optics, the skin depth 6 is the coherence length for the non-linear interaction. If we have a film of thickness L<<6 (such a film is described here as a “thin film”), then clearly the same reasoning shows it is L and not 6 that controls the breakdown of wave vector conservation normal to the surface. Eq. (2.7) is replaced by a form that gives us (Ak , )L - 1 in the scattering process. Thus, even for a material transparent to the incident radiation, if the scattering takes place in a sample prepared as a thin film, the wave vector component normal to the surface is not conserved, and we see light scattering spectra influenced strongly by this breakdown.

Also, in a thin film, the phonon spectrum is altered distinctly from that appropriate to the semi-infinite geometry. In the semi-infinite limit, any value is allowed for the wave vector Q, of the phonon normal to the surface. In eq. (2.7), this is expressed by the integration over a continuous range of values of 0,. In the film, the bulk phonons become standing waves in the direction normal to the surface, with allowed values of Q, = nr /L , where n is an integer. To determine the precise values of Q,,

11,s 21 OPAQUE MEDIA, FILMS OF FINITE THICKNESS 63

one must solve a boundary value problem for the film, as we do for an example in 5 3. The allowed values of 0, may always be written QF) =

n ( r + S , ) / L , where 6, is a phase shift that lies between 0 and T. As L decreases, then necessarily the separation between adjacent values of Q, increases inversely with L, as does the frequency separation between two nearly normal modes with the same value of QII . We have

Aw,,(Qli, QF’) = w(Q11, Q?+”) - w(Qll, QF)) = (ao/aQ,)(QF”’ - a?’), (2.13)

so Awn - 1/L. As soon as L becomes so small that Awn is greater than the inverse lifetime characteristic of the modes in question, then the quantization of the phonon wave vector in the direction normal to the film will manifest itself in the spectrum. The long asymmetric tails discussed above will no longer be smooth and monotonic, but will acquire structure with peaks at the frequencies allowed for the normal modes of the finite film. Several examples of such spectra now exist in the literature. While alternate methods may be used to excite standing wave acoustical resonances in films (along with standing spin waves in ferromagnets’or the guided wave polaritons discussed in 5 3), the light scattering method is the only one that allows the frequency response of the film to he probed over a broad, continuous range of excitation frequencies. In the light scattering experiment, the spectrum provides a continuous scan of Aw = wo - w,.

One may regard Aw as the frequency at which the film is driven by the non-linear interaction responsible for light scattering, and Akll = kr’- kf’ as the wave vector at which it is excited. Thus, one can scan both Aw and Akll in a continuous manner if desired. While only a few light scattering studies of standing waves of excitation in thin films have appeared at the time of writing, we believe the method has very great potential, with a unique capability for studying both interfaces and inhomogeneities in devices which incorporate thin films as essential elements.

The purpose of this section has been to outline, in simple qualitative terms, the influence of either strong absorption or sample size on the light scattering spectra of materials. We see that for samples of semi-infinite extent (thickness L large compared to the optical skin depth S ) , the breakdown of wave vector conservation normal to the surface allows one to see “forbidden scattering”, and can lead to highly asymmetric line shapes. In thin films (thickness L small compared to S ) , the breakdown of wave vector conservation becomes more dramatic, and one can probe standing wave resonances of the film/substrate structure. Finally, one has

64 LIGHT SCA'ITERING SPECTRA OF SOLIDS [II, 5 3

a complex variety of surface and interface waves that may be probed under these conditions; we have yet to discuss such waves in detail.

We turn next to a more detailed exposition of a specific example: light scattering by surface and guided mode polaritons in thin films. § 3 is a review of the properties of the modes that are studied by the light scattering method in § 4.

Q 3. Surface and Guided Wave Polaritons

The purpose of this section is to outline the properties of surface and guided wave polaritons in films, and in the semi-infinite geometry. We first begin by considering the nature of these modes in the simplest possible geometry: an infinitely extended, polar crystal of cubic symmetry. A more extensive and comprehensive review appears in a recent review article (MILLS and BURSTEIN [1974]) in which the properties of polaritons in anisotropic media are also reviewed, along with experimental methods of probing these waves.

Here we begin by considering the long wavelength optical modes of vibration in a crystal of cubic symmetry. If we consider a diatomic solid, then for wavelengths very long compared to the lattice constant, the relative displacement of the two sublattices u provides an excellent approximation to the normal coordinate. (For a detailed description of the dynamical properties of crystal lattices, see MARADUDIN, MONTROLL, WEIS and IPATOVA [1971].) If we regard the interatomic forces as short ranged only, then for a cubic crystal, as the optical phonon wave vector Q + 0, the Cartesian components u, of the relative displacement u obey the simple harmonic oscillator equation of motion

ii, + o:u, = 0,

where oo is the (triply degenerate) Q = 0 optical phonon frequency. The simple form in eq. (3.1) must be modified, if it is to be applied to

crystals with ionic character, such as alkali halides or the 111-V semiconductors. The relative displacement of the ionic sublattices will produce, in the long wavelength limit, a macroscopic electric field E that will react back on and drive the relative displacement of the sublattices. Thus, eq. (3.1) is modified to read

11, $31 SURFACE AND GUIDED WAVE POLARITONS 65

where is the reduced mass of the unit cell, and e* is the transverse effective charge, which describes the force on the sublattice that results from the macroscopic field E.

Now that the electric field has been added to the equation of motion for u, we require a method of calculating it. To do this, we must combine eq. (3.2) with Maxwell’s equations. We do this, by writing the displacement field D = ~ , E + 4 m e * u , where E , is the contribution to the dielectric tensor from electronic excitations, and the dipole moment per unit volume associated with the relative displacement of the sublattices is PL= ne*u. Here and throughout the present section, we are considering fields with frequencies very small compared to E,/h, where EG is the fundamental band gap. Then the electronic contribution to the polarizability may be described by introducing the constant E , in the constitutive relation between D and E.

In the above picture, Maxwell’s equations give a second relation between E and u:

V2E-V(V * E ) - E , .. 47rne* - E - - u=0 . C 2 C2

(3.3)

If we examine eq. (3.2) and eq. (3.3) for solutions in which both E and u exhibit the space and time dependence exp (iQ - x - iwt), then we obtain two sets of equations that link the Cartesian components of E with those of u. Eq. (3.2) becomes

( w ; - w 2 ) u , - - E , e* =0 ,

P

while eq. (3.3) gives

&,w2 4.rrne” C C2

Q,(Q-E)- Q’E, +2 E, +- u, = 0.

(3.4a)

(3.4b)

We may regard both u and E as normal coordinates for a “bare” lattice, and the electromagnetic field in the medium, with lattice motion clamped rigidly. When the lattice is allowed to move and to generate an electric field through its motion, the two normal coordinates couple, and the resulting normal mode coordinates of the system may be viewed as a mixture of the two original vector fields u and E. The resulting coupled mode is called a polariton. The terminology indicates that a deeper level of theory shows these modes to be quantized, with boson character familiar from phonon theory and the theory of the electromagnetic field ( HOPFIELD [ 195 81).

66 LIGHT SCATlZRING SPECTRA OF SOLIDS [II, § 3

From eq. (3.4a) and eq. (3.4b), we may obtain the dispersion relation of the polariton waves. The results are most easily summarized by introducing the dielectric function

4 7 ~ n e * ~ 1 E ( W ) = Em+--

/L &-w” (3.5)

Then the secular equation formed from eq. (3.4a) and eq. (3.4b) becomes very simply

(3.6)

that is, we have a doubly degenerate solution of the theory for which

C 2 Q 2 / W 2 = &(W), (3.7)

and also a solution with E ( w ) = O . The results above may be written down at once, from simple considera-

tions. We recognize eq. (3.5) as simply the frequency dependent dielectric constant of a medium with an electric-dipole active excitation of frequency w. Then eq. (3.7) is simply the well known dispersion relation of the (doubly degenerate) transverse electromagnetic waves that propagate through our isotropic crystal.

The frequency w for which E ( W ) = 0 is the frequency of a longitudinal normal mode for which E is directed parallel to Q. Then V x E vanishes identically, as does H (assumed equal to B here), and also V - D = E(o)V - E = 0. The non-trivial solution with Ef 0 requires the frequency to be such that ~ ( w ) vanishes. This frequency, wL, is given by

WL = ( E , / E , ) L k I , (3.8)

where E,= ~ , + 4 ~ n e * ~ / p ~ ~ is the static dielectric constant formed by setting w = O in eq. (3.5). The frequency wL is the lattice dynamical longitudinal optical (LO) phonon frequency of a solid, wo is that of the transverse optical (TO) phonon, and eq. (3.8) is the well known Lyddane- Sachs-Teller relation of solid state physics (KITTEL [1971]).

Thus, the polariton is nothing more than an electromagnetic wave which propagates in a dispersive dielectric; we have considered here one particular means of introducing strong dispersion into the dielectric response of the medium. This is through excitation by the electric field of an electric-dipole active optical vibration mode of the lattice. To use a term such as “polariton” to describe such a well known and common

11, § 31 SURFACE AND GUIDED WAVE POLARITONS 61

phenomenon is perhaps confusing. The term was introduced into the solid state literature (HOPFIELD [1958]) to have a meaning very different than that used presently and in the contemporary literature; it seems that it acquired its present sense somewhat later, possibly through a misunder- standing (see the discussion presented by BURSTEIN [1974]). The term is now used mostly in contexts where it is convenient to describe an electromagnetic wave explicitly as a coupled mode between the amplitude u of the physical entity that produces the polarization responsible for anomalous dispersion in the dielectric constant, and the electric field oscillations present with the excitation “clamped”. The utility of this approach has been discussed at length elsewhere (MILLS and BURSTEIN [1974]). But if we have no interest in the explicit composition of the wave, we may deal directly with the Maxwell equations applied to the dispersive medium.

In Fig. 3.la, we sketch the dispersion relation deduced from eq. (3.7) applied to a dielectric medium with the dielectric tensor in eq. (3.5). Note that we have a two branch dispersion relation. The upper branch approaches the LO phonon frequency wL as Q-0 (this is a transverse wave always!), and cQ/& as Q + m. The lower branch has wave vector that approaches infinity as the frequency w approaches wo from below, and the frequency o approaches c Q / E ~ as Q -+ 0 as illustrated.

From Fig. 3.la, one sees there is a “stop band” between wo and wL

w

WL

WO

Q

(a)

E iw)

Es

Em

0

(b ) Fig. 3.1. (a) The dispersion relation for bulk polaritons in an isotropic dielectric described by eq. (3.5) of the text, and (b) the frequency dependence of the dielectric constant

displayed in eq. (3.5).

68 LIGHT SCATIERING SPECTRA OF SOLIDS [II, B 3

within which bulk polaritons fail to propagate. The reason for this is illustrated in Fig. 3. lb , where the frequency dependent dielectric constant E ( W ) is plotted. Between wo and wL, E ( W ) is negative, and the wave vector Q that emerges from eq. (3.7) is purely imaginary. This stop band, where the crystal is perfectly reflecting at our level of approximation, will play a crucial role in the discussion below. The physical origin of the negative dielectric constant is that for frequencies above wo, the lattice polarization PL=ne*u is 180" out of phase with the driving E field. This is a fundamental property of the elementary harmonic oscillator. Between wo

and oL, PL is large and negative, so D = ~ ~ ) E + 4 m e * u becomes antiparallel to E.

Now suppose we add a surface to our problem, by considering a semi-infinite dielectric with dielectric constant given by eq. (3.5). In this case, we encounter solutions of Maxwell's equations with qualitatively new behavior: surface polaritons. In essence electromagnetic waves with electric and magnetic fields localized near the dielectric/vacuum interface, can propagate along the interface. These are for Maxwell's equations the analogue of the Rayleigh surface acoustic waves described briefly in (i 2. We turn to a description of the surface polaritons, for the simple case of the interface between vacuum and the isotopic dielectric.

Suppose the dielectric material lies in the half space z > 0, with surface in,the x-y plane. Then any electromagnetic disturbance in the vacuum z < 0 must have V . E = 0. We examine Maxwell's equations for solutions with E in the plane formed by the z axis, and a wave vector Qll in the x-y plane. If this solution describes fields localized near the surface, we must have

E%, t ) = E<[Qll-i(Qll/ao)i] exp (iQl,*xll+ aoz -iwt) (3.9)

for V-E to vanish. Also, the electric field in eq. (3.9) must satisfy the wave equation in vacuum (eq. (3.4b) with E , set equal to unity, and the terms in u missing). This requires a() to assume the value

= (a;- W * / C * ) ~ , (3.10)

where necessarily QII > w/c if the solution describes fields localized to the interface.

In the dielectric, a solution with V * E = 0 ( E must be divergence free for all frequencies, save w = wL where ~ ( w ) vanishes) and which is localized to the interface is

E>(x, t ) = E'[i)ll+ i(Qll/a)i] exp [iQe * XII - az - iwt], (3.1 1)

11, f) 31 SURFACE AND GUIDED WAVE POLARITONS 69

where now

= (Qf- E ( W ) O * / C ~ ) ; . (3.12)

For the solutions in eq. (3.9) and eq. (3.11), the magnetic field H is parallel to the surface. We are thus describing electromagnetic waves of TM character.

We may write down the solutions in eq. (3.9) and eq. (3.11) for any value of Qll and w. However, further constraints are imposed by the requirement that boundary conditions in the fields at the interface be obeyed. Conservation of tangential components o f E leads us to set E’ = E‘, and normal components of D are continuous only if

~ ( w ) = -a(Q\l, w ) / a o ( Q ~ ~ , w ) , (3.13)

where in eq. (3.13) we explicitly indicate that both a and a. are functions of QII and w.

We cannot satisfy eq. (3.13) for any arbitrary values of QII and w. Given QII, from eq. (3.13) we may determine the (possibly unique) frequency w.

Thus, we have a dispersion relation, w = w(QII), for the surface polariton contained implicitly in eq. (3.13). In all continuum theories of surface wave propagation, the dispersion relation emerges when boundary conditions are applied to appropriate solutions of the bulk equations that describe the medium on each side of the interface. In our case, rearrange- ment of eq. (3.13) leads to a simple, explicit form of the dispersion relation :

(3.14)

with a constraint on the allowed frequencies that follows from a quick glance at eq. (3.13).

The constraint on the surface polariton frequencies follows upon noting that we must always have both a(Qll, w ) and ao(Qll, w ) positive. Thus, the boundary conditions can be satisfied only in the frequency regime w,<

w < wL, where ~ ( w ) is negative. The surface polaritons always lie in the stop band described earlier, where no bulk polaritons can propagate.

The dispersion relation for the surface polaritons is sketched in Fig. 3.2a, where we see the features outlined in the above paragraphs. The band of surface polaritons does not extend throughout the entire frequency regime between wo and wL, but rather between wo and ws, where e(ws) = -1. Notice as Q,I is decreased, the dispersion relation terminates

70 LIGHT SCAlTERING SPECrRA OF SOLIDS [II, B 3

Q,, Q,,

(a 1 (b) Fig. 3.2. (a) The dispersion relation for surface polaritons on the interface between a dielectric and vacuum, for a case where the dielectric constant of the substrate is given by eq. (3.5). (b) A plot of the o, and a- branches of the surface polariton dispersion curves in

a finite film of thickness L.

abruptly when it strikes the “light line” o = cQII. This is dramatic and unusual behavior not found in the theory of bulk excitation spectra of solids, as far as we know.

We have considered here the simplest description of surface polaritons, namely those on an interface between an isotropic dielectric and vacuum. Examination of more complex geometries shows a rich and varied spectrum of surface modes can occur. We do not review these here, but rather direct the reader’s attention to an earlier review article (see MILLS and BURSTEIN [1974] section X) which explores this question in greater detail.

From the point of view of the present article the interest in surface polaritons is that they will appear in light scattering spectra of the opaque solid, when the backscattering technique discussed in § 2 is employed. Since these modes depend on the presence of the surface for their existence, in principle their study can provide us with detailed information about the electromagnetic response of the near vicinity of the surface. Before we treat the theory of light scattering from surface polaritons, we turn to the discussion of electromagnetic modes present in thin films. This can be done by straight forward extension of the above discussion.

We can see what to expect for surface polaritons in the finite film from simple physical considerations. First, from the form of the electromagnetic fields in eq. (3.9) and eq. (3.11), and the dispersion relation in eq. (3.14), one sees that in the semi-infinite geometry, the fields extend


into the substrate a distance the order of Qi’, just as in the case of the Rayleigh surface acoustic wave discussed in Q 2. Thus, for a film of thickness L, when QllL >> 1, there will be a surface polariton located on each film surface, with fields compacted to the interfaces so tightly that the modes propagate independently, to excellent approximation. As the wavelength increases to the point where QllL becomes of order unity, the two modes interact, repel each other with the consequence that we have two dispersion relations w+(Qll) and w-( QII).

To describe the situation explicitly, consider a film of thickness L with one surface in the x-y plane ( z = 0) and one along z = L. For z < 0, eq. (3.9) remains a solution of Maxwell’s equations, and here as before we confine attention to the regime QII > w/c so is positive. Above the film, where z > L, we have another solution localized near the interface:

E’(x, t ) = E’[Qll+i(Ql,/ao)2] exp [iQ - XII- aoz - i d ] . (3.15)

To proceed, we must synthesize a set of fields within the film and match boundary conditions as before. Within the film, the expressions in eq. (3.11) and eq. (3.12) remain solutions of Maxwell’s equations. We have also a second independent solution identical in form to eq. (3.11), except that a has the opposite sign. There is no reason to omit this second solution for the finite film; we rejected it before because its field becomes exponentially large as z -+ m. It is convenient to form from the fields just described the combinations E+(x, t ) and F ( x , t ) which have components with well defined parity under reflection through the mid plane of the film at z = L/2. Upon multiplying eq. (3.11) by exp(+aL/2), the second solution by exp (-aL/2) and forming the appropriate linear combinations, we have the two forms for 0 < z < L,

E(+)(x , t ) = E(+)[Qll sinh ( a [ z -&])-i(Qll/a)i cosh ( a [ z -iL])]

x exp [i(Qll * xIl - wt)], (3.16a)

E‘-’(s, t) = E(-)[dll cosh ( a [ z - iL]) - i(Qll/a)i sinh ( a [ z -iL])]

and

xexp [i(Q - xII-wt)]. (3.16b)

Each of these forms is to be matched to solutions above and below the film with the same parity under reflection through the midplane z = L/2. Thus, we match E‘+’(x, t ) , to the combination

E% t ) = El[i)ll+ i(Q,l/ao)21 exp [ - a d z - L)] exp [ i (q l - q- of)]; z >L, (3.17a)

12 LIGHT SCAWERING SPE(;TRA OF SOLIDS [II, F3 3

and

E X c , 2) = E 3 - Q 1 1 + i(Qll/adiI exp [+aozl exp [i(QII * q- 4 1 ;

z<O, (3.17b)

while E‘-’(x, t ) is matched to

E?(x, t ) = E_”Qll+ i(Qll/ao)fl exp [-ao(z - L)I exp [i(QII.xll- w t ) ] ;

z > L , (3.18a)

and

E ~ x , t ) = E2[~l,-i(Ql,/ao)f] exp [-aOz] exp [i(Qll-xll- w t ) ] ;

z<O. (3.18b)

Application of the boundary conditions leads to two relations that replace eq. (3.13). One comes from the E(+) mode, and one from the E(-) mode. If we call w+(Qll), w_(Qll) the dispersion relations that emerge from each, we have in place of eq. (3.13) the following pair of implicit dispersion relations:

(3.19a)

(3.19b)

In the limit a(Qll, w,)L >> 1, both the tanh and coth on the right hand side of eq. (3.19a) and eq. (3.19b) reduce to unity and we obtain from each a dispersion relation identical to eq. (3.14). Thus, we have two surface polaritons, one localized to each interface; actually the solution which emerges from the present mathematics has fields of the two degenerate waves combined to form waves with fields of well defined parity about the mid plane z = L/2. Given two degenerate modes, any desired linear combination of fields remains a solution of Maxwell’s equations and the boundary conditions, so one can always form a linear combination with high symmetry, if desired.

The dispersion curves of the w, and w- branches are illustrated in Fig. 3.2b. The trends illustrated may be appreciated by direct inspection of eq. (3.19b). At any value of Ql1, there is a repulsion between the two modes produced by overlap of their fields. This produces two distinct dispersion curves which become degenerate as the film thickness L + w. Also, as QII decreases and approaches w/c, then a,(Qll, wJ vanishes, to drive the


frequency to that value that makes &(a) become infinite for both branches, i.e. as QII decreases, both w+( QII) and w-( Ql,) must approach the TO phonon frequency w,). The dispersion relations obtained above were derived some years ago by KLIEWER and FUCHS [ 19661. In their paper, one finds a number of detailed plots of the dispersion relation. Valdez has carried out quantitative studies of the modes, for the geometry of interest in our discussion of light scattering (VALDEZ [1978] 0 4).

At the end of $2, we pointed out that in a finite film, in addition to surface waves of the kind discussed above, one also has bulk waves which in the thin film limit become standing wave resonances. A description of the standing wave resonances with p-polarized electric fields (TM modes) is already contained in eq. (3.19a) and eq. (3.19b). All we have to do is apply the equations to the relevant frequency domain, where E ( W ) is positive.

We can understand the regime of the w-Ql~ plane where the standing wave resonances with fields confined to the film arise by simple reasoning. A bulk polarition which propagates parallel to the x-y plane in the infinitely extended medium has the dispersion relation w = cQll/s(w)f [eq. (3.7)]. If we cant the wave vector out of the x-y plane, and keep the projection QII on the x-y plane fixed (hence increasing the magnitude of the wave vector), the frequency must increase. This is evident from Fig. 3.1. Now, to turn to the standing wave resonances, each may be regarded as synthesized from the fields of two bulk polaritons. Each bulk polariton has the same projection QII of its wave vector onto the x-y plane, but opposite wave vectors *Q, normal to it. As discussed in Q 2, the boundary conditions at the film surfaces are responsible for mixing the two modes. The remarks above lead us to examine eq. (3.19a) and eq. (3.19b) for solutions with fixed QII in the domain where

(3.20)

and where, as implied by the inequality in eq. (3.20), we have E ( w ) > 1. The left hand side of the inequality insures we are studying modes with fields that decay exponentially as we move into the vacuum either above or below the film (then a,(Qll, w ) is real), and the right hand side combined with the condition E(W) > 1 insures we are looking at modes for which the wave vector Q, inside the film is real.

We refer to the modes described above as guided wave polaritons. The electromagnetic fields associated with the waves have standing wave

74 LIGHT SCATTERmG SPECTRA OF SOLIDS [II, 5 3

character in the direction normal to the surface, as we have seen, but they propagate parallel to it and transport energy along the film as they proceed. For a film deposited on a substrate, a similar class of modes may exist, in frequency domains where the film dielectric constant is larger than that of the substrate. These are the modes that form the basis of the new integrated optics technology (TIEN [1977]). While integrated optic devices utilize guided waves with frequency in the visible, our interest here is in the nature of these modes in the infrared where E ( W ) has strong dispersion, in polar materials.

To describe the guided modes it is convenient to rearrange eq. (3.19a) and eq. (3.19b). Let wlp)(QII) be the frequency of the nth guided wave polariton of p-polarization, and with wave vector QII parallel to the film. Then introduce the wave vector QY’(w, Qll) normal to the film through the bulk polariton dispersion relation

- 2

(3.21)

Then in terms of QY), eq. (3.19a) and eq. (3.19b) become

QY’ tan [iLQY)]= + ~ ( w ) c w ~ ( Q , ~ , w ) , (3.22)

and

QY’ cot [iLQ?)]= -E(w) (YO(QI I , w ) . (3.23)

These equations remind one of the eigenvalue equations encountered in the quantum mechanical description of a particle confined to an attrac- tive, one dimensional square potential well. For a particular value of QII,

we may sweep out the dispersion relation of the TM guided waves from eq. (3.22) and eq. (3.23).

There are also guided wave polaritons with electric fields of s- polarization. We write down the form of the fields and dispersion relation of these waves, without detailed derivation. Again, we have modes of even and odd parity with respect to the mid plane z = L/2, so there are two transcendental equations to solve. One of these has the form

for a mode with electric field in the film given by

E(x, t ) = j?EO sin [Q,(z - $L)] exp [i(Qll - xII - wt)]. (3.25)


The second transcendental equation is then

QP’ tan (IQY’L) = +a0(QII, o) (3.26)

and in this mode the electric field within the film has the form

E(x , t ) = $Eo cos [Q,(z - $L)] exp [i(Qll - xII - w t ) ] . (3.27)

In Fig. 3.3, we sketch the dispersion relation of the various guided wave polariton modes. Again, the results in eq. (3.22), eq. (3.23), eq. (3.24) and eq. (3.25) can be found in the paper by KLIEWER and FUCHS [1966], along with a series of quantitative model calculations for the theory applied to LiF, while VALDEZ [1978] has studied the modes for parameters characteristic of GaP films.

This completes our discussion of surface polaritons in a semi-infinite dielectric, and the polariton eigenmodes of thin films, where one finds both surface modes and a rich spectrum of guided wave modes. The reader will appreciate that the concepts introduced here can be applied to a variety of situations, such as acoustic waves in the semi-infinite geometry and films, and spin waves in a ferromagnet. We turn next to the theory of light scattering from surface and guided wave polaritons, then to other physical systems where surface and size effects influence the light scattering spectrum importantly.

W

WL

WO

I I I

Fig. 3.3. A sketch of the guided wave polariton dispersion curves for the first few lowest modes below wo, and above oL. Note that for w > w , ~ there are no modes to the left of the light line

o = cQII, since the fields in the vacuum are no longer localized near the film.


0 4. Light Scattering from Surface and Guided Wave Polaritons

The light scattering method outlined in § 2 offers a powerful method of probing the surface and guided waves described in § 3. Here we outline a theoretical approach that has proved useful in analyzing the spectra. Some general comments may be useful before we begin.

A possible method is to proceed as follows. The various surface and guided waves discussed above are in fact exact electromagnetic eigenmodes for the finite film. We can introduce boson annihilation and creation operators for such modes, in a manner now standard in the discussion of elementary excitations of solids. Similarly, the incident and scattered photons may also be described by such a field theoretic structure, and the light scattering process may be described as a scattering of the laser photon off the quantized polariton normal mode of the film.

While such an approach is elegant, it is in fact extremely difficult to apply it properly to the conditions outlined in § 2. We saw there that the light scattering spectrum can be influenced very strongly by absorption of the incident light. The propagation of electromagnetic waves in a strongly absorbing medium is not readily discussed through use of the field theoretic method outlined in the last paragraph. Also, the surface and guided wave polaritons are exact, infinitely long-lived elementary excitations of the film only within a framework of a simple model for which the dielectric constant E ( O ) is purely real. The lattice motion that is responsible for the resonant term on the right hand side of eq. (3.5) is in fact damped by crystalline anharmonicity, even for a very pure crystal. Thus, in the infrared, E ( W ) acquires an imaginary part, with the consequence that the polaritons have a finite lifetime. The influence of this finite lifetime on features in the light scattering spectrum cannot be readily incorporated into a field theoretic treatment that begins by quantizing the normal modes through use of boson annihilation and creation operators. We know of no theoretical study based on the use of this operator formalism which fully incorporates into the treatment the absorption of both the incident and scattered radiation, and the finite lifetime of the excitations in the medium.

By proceeding in a very different manner, it is possible to calculate the light scattering spectra of films and of semi-infinite media in a manner that fully includes the effects mentioned above. One proceeds in a manner that appears semi-classical in nature, but which in our view is a fully rigorous procedure. One supposes that the elementary excitations in

11, § 41 LIGHT SCAlTERING FROM SURFACE 77

the medium, present by virtue of the finite temperature, modulate the dielectric tensor E,, to produce a fluctuating part &,,(x, t) that depends on both position and time. Fluctuations in the dielectric tensor then scatter light; this scattering may be described by solving Maxwell’s equations applied to a finite sample with a fluctuating dielectric tensor. (In principle, scattering can also be produced by fluctuations in the magnetic permeability xFV(x , t ) , although in most situations this is a small effect.) The method has the virtue of describing a photon incident on the sample from the outside, and the scattered wave field is also evaluated outside the medium. Thus, the theory may be used to calculate absolute scattering intensities, with influence of the impedance mismatch at the crystal surface included fully. Such theoretical spectra may be placed directly alongside data.

By this means one is led to a formula for the scattering efficiency rather similar in structure to the well known van Hove expression for neutron scattering from solids (see KITEL [1963] Ch. 19). It is the correlation function (~E,,,(x, t ) ~E,,,.(x’, t ’ ) ) that controls the form of the spectrum, after it is subject to the appropriate Fourier transform procedure. A model of the manner in which the excitations in the solid influence the dielectric tensor allow this correlation function to be expressed in terms of those that describe directly the fluctuating variables in the solid. In this section we first present the derivation of the scattering cross section. Then, for a particular example, we show how this approach can be developed, and present results of calculations based on it.

4.1. DERIVATION OF THE SPECTRAL DIFFERENTIAL CROSS SECTION

An excellent discussion of light scattering from infinitely extended media using the dielectric fluctuation approach may be found in LANDAU and LIFSHITZ [1960]. Here we are concerned with a medium of finite extent. The two relevant scattering configurations used in the study of surface and size effects are (i) the backscattering geometry illustrated in Fig. 2.1, and (ii) the forward scattering geometry illustrated in Fig. 4.1. In practice, at least one of medium 1 and medium 3 in Fig. 4.1 is vacuum. The derivation of the light scattering cross section for the two cases proceed along similar lines. The discussion below will emphasize the principles involved at the expense of generality. For more general deriva- tions the reader is referred to the original literature (MILLS, MARADUDIN


\ i z MEDIUM 3 ,

z=L

2 =o

Fig. 4.1. The forward scattering geometry for thin films.

and BURSTEIN [1970], MILLS, CHEN and BURSTEIN [1976]). Thus, in what follows we assume both medium 1 and medium 3 to be vacuum. This describes light scattering from free-standing crystalline slabs, a configuration that has been used extensively in experimental investigations.

We assume the crystal can be described, in the absence of fluctuations, by an isotropic complex dielectric constant E ~ ( o ) corresponding to the frequency of the incident optical radiation. Thus we can write the dielectric constant of the vacuum/crystal/vacuum three layer system in the absence of fluctuations as

z <o; E ( z ) = Eo=E1+iEZ, O < z < L ; (4.1) [:I z>L .

The incident radiation “sees” the excitations in the crystal through their modulation of the dielectric tensor of the crystal, ~E,,(x, t ) . Thus, in the presence of fluctuations, we may write the dielectric tensor of the system as

E,,(x, t ) = E ( Z ) s,, + e ( z ) w - - 2) ~E,,(x, t ) , (4.2)

where the unit step functions* e(x) are introduced to denote that the dielectric fluctuations are present only within the crystal, 0 < z < L. The electric field E ( x , t ) obeys the Maxwell equation

(4.3a)

11, P 41 LIGHT SCA’TTERING FROM SURFACE 79

with

(4.3b)

The dielectric tensor E,,(x, t ) is given in eq. (4.2). The usual boundary conditions of the continuity of the tangential components of E, and the normal component of D apply at the surfaces z = 0 and z = L. Viewing the fluctuations in the dielectric tensor as being the source terms causing light scattering, we rewrite the wave equation eq. (4.3a) as

with

L,@ = ~- a2 s,, [ v2--- &y,4”.]. ax, ax,

(4.4b)

We note that the fluctuations of the electric field occur at the frequency of optical radiation (- 10’’ Hz), while those of the dielectric tensor are comparatively slow ( - 1 O l 3 Hz). Hence we may ignore the time derivatives of SE,@(X, t ) on the right hand side of eq. (4.4a).

We now introduce the Green’s functions GaP(x, x’; t, t ’ ) that obey the equation

C L,.,G.,p(x, x’; t, t‘)= - 4 ~ 6 , ~ 6 ( ~ , x’) 6 ( t - t’). (4.5) Y

This Green’s tensor is subject to boundary conditions such that E obeys the appropriate electromagnetic boundary conditions mentioned above. Using this Green’s tensor we can cast eq. (4.4a) in the form of the following integral equation,

E, (x, t ) = E:’(x, t ) - - 1 4TC2

Id2$ [,‘dz’Idt’

Gap (x, x’; t, t ’ ) (x’, t ‘ ) azEi:f’ “’1. (4.6)

where xll denotes the projection of the position vector (x = xlI + zi) in the x-y plane. The function ET’(x, t ) is the solution of the corresponding homogeneous equation obtained from eq. (4.4a) by setting = 0. It corresponds to the specularly reflected or transmitted light, and has the

80 LIGHT SCAWERING SPECTRA OF SOLIDS [II, 5 4

form

EC’(x, t) = EC’(k(”’w,,; z ) exp { i k ~ ’ . x l l - iwot}, (4.7)

where kf’ is the projection of the incident wave vector in the x-y plane and wg is the frequency of the incident light. The explicit form of E~’(k‘’’wO; z) may be found in Appendix A of MILLS, CHEN and BURSTEIN [1976].

Since ?kap is first order in the amplitude of the excitation responsible for the scattering, we can obtain the electric field of the scattered wave E”’ to the same order by replacing E,(x, t’) on the right hand side of eq. (4.6) by EF’(x’, t’). This is the analog of the first Born approximation of quantum mechanical perturbation theory. The expression for the scattered field obtained from eq. (4.6), applies for forward scattering when z > L, 0 < z’ < L, and applies for backscattering when z < 0, 0 < z’ < L.

The derivation of the scattering cross section is now straightforward, although somewhat cumbersome algebraically. We introduce the Fourier transforms of the Green’s tensor which take account of the translational invariance in the x-y plane that remains,

Once again, for the explicit form of the Green’s tensor for the three layer system, we refer the reader to MILLS, CHEN and BURSTEIN [1976]. Next we calculate the Poynting vector and from it, the intensity of scattered radiation with propagation directions that lie within a solid angle dl2 about the direction of scattering (O,, c$J in the frequency range w, to w, + dw, viz., the spectral differential cross section

where

haPy(kf)w, I 2’) = Gap(klp,; z z ‘ ) E ~ ’ ( k ~ ’ w , ; 2’) . (4.9b)

Here kf’ and w, are, respectively, the wavevector projection in the x-y plane, and the frequency of the scattered radiation. The wavevector component parallel to the surface and the frequency, QII and 0, of the excitation responsible for the scattering obey the kinematic conservation

11, P 41 LIGHT SCA’ITERING FROM SURFACE 81

conditions

and

(4.10a)

(4. lob)

As discussed in § 2, the wavevector component normal to the surface is not subject to the conservation condition due to the finite extent of the scattering volume. Finally, the power spectrum of the excitations is contained in the Fourier transform of the dielectric tensor fluctuation correlation function,

9ap;a,,dQi@; zz ’ ) = d 2 q I d f

x exp {-iQll~rll+iRt}(S~,,(rllz, t ) S E , . , , ( ~ Z ’ , 0 ) ) . (4.11)

The brackets ( ) denote a statistical average corresponding to the prevail- ing thermodynamic state of the crystal. This completes the formal derivation of the cross section. Eq. (4.9) and eq. (4.11) are our central formulas. Before turning to the task of relating the dielectric fluctuations to the fluctuations of the relevant dynamic variables, we pause to comment briefly on the structure of eq. (4.9) leaving aside non-essential factors.

The incident electric field within the crystal (O<z<L) has the form (see MILLS, CHEN and BURSTEIN [1976])

E(0)(k(O)w II 0 . 9 z ) = [ E , e - i k p + E,e’ :““I. (4.12)

The first term corresponds to the incident wave transmitted through the lower surface, and the second term represents the wave reflected off the crystal-vacuum interface after it has been transmitted through the film. Furthermore the Green’s functions Gap that enter eq. (4.9) have the form for z > L , O<z’<L ,

G(kf)w S I . 22‘) = eik‘?z8(s)(kf)W s9 * z’) 7 (4.13a)

with

g(S)(kf)w s7 . z ) = Ae-ikp + Beiky)z . (4.13b)

As mentioned already, the explicit forms of the Green’s functions may be found in MILLS, CHEN and BURSTEIN [1976]. Then, in the language of quantum mechanical transition theory, the integral in eq. (4.9) has a form similar to the square of a “matrix element” (E‘O’ I SE 1 8‘“’) involving an effective interaction sandwiched between an initial state E‘O’ and a final


state @). As may be seen from eq. (4.12) and eq. (4.13b), there are four terms in the product E'o'8'"' and each term may be represented by a diagram, as illustrated in Fig. 4.2. Alongside each process is the phase factor which enters the contribution to the effective matrix element from the process. The processes in Figs. 4.2a,d involve forward scattering events while those of Figs. 4.2b, c involve backscattering events. Thus there are a total of sixteen terms for each integral appearing in eq. (4.9) and our theory includes all of them. The major drawback of some of the other approaches to light scattering from thin films (NKOMA [1975]) is that they retain only the process shown in Fig. 4.2a from the outset. As we shall see, this can lead to the exclusion of important features from the light scattering spectra.

We now return to the question of relating the dielectric fluctuations to the relevant dynamical variables (such as atomic displacements, etc.) in the crystal. We formally expand the dielectric tensor in powers of the normal mode coordinates u"(x, t ) of the excitations of the crystal, retaining only the linear term for a first order process. Thus, for the

L /

/

/

/

/

a) exp[-i(k';-$))z]

b) exp[-i(k':+kT)z]

c ) exp[i(k,+k,)z] 10) Is1

d ) exp[ i(k,-k,)z] I01 Is1

Fig. 4.2. Four fundamental scattering processes that contribute to the Raman cross section for forward scattering.

11, I41 LIGHT SCATERING FROM SURFACE 83

dielectric fluctuation we write

S&(X, t ) = c b;u"'(x, t ) . i

(4.14)

The phenomenological parameters bj are assumed to be the same as for the bulk material and assumed to be known. The correlation functions d(';) = ( u ( ~ ) u O ' ) ) may be determined through their well known relationship to retarded Green's functions (see ABRIKOSOV, GORKOV and DZYALOSHINSKI [1963]).

d('"(R) = i[n(O)+ l]{dg'(O+) - dg)(O-)} , (4.15)

where n(O) is the Bose occupation factor [exp (R/k,T)- 11-' and by R+ and 0-, we mean the frequency evaluated just above and just below the real axis in the complex R plane. Equivalently, the correlation functions may be determined from a response function formalism (see BARKER and LOUDON [1972]) identical in content to the method described here. In several cases of interest the correlation functions entering eq. (4.9) may be determined exactly and hence a direct comparison with experiments is possible. The rest of this section and the following section will deal with several examples where such concomitant studies have been made in recent years.

4.2. RAMAN SCATTERING FROM POLARITONS IN THIN CRYSTALS

The basic properties of the electromagnetic normal modes of a polar dielectric (polaritons) were discussed in 0 3. We now apply the considerations of the previous subsection (04.1) to the study of these normal modes through light scattering.

In a polar crystal (such as Gap) the coupling of light to the polar optical vibrations comes about through both the relative atomic displacement u, as well as the resulting macroscopic electric field E present in the crystal (see HAYES and LOUDON [1978] Ch. 4). Thus eq. (4.14) takes the explicit form

k p ( X , t ) = 1 a,,,u,(x, t ) + c b,pvEv(X7 t ) , (4.16)

where a,,, is the atomic displacement polarizability tensor and baPY is called the electrooptic tensor. When eq. (4.16) is substituted into eq. (4.11) we get the four correlation functions (uu), (uE) , ( E u ) and (EE) .

(I Y

84 LIGHT SCATIERING SPECTRA O F SOLIDS [II, 8 4

fhHowever, as discussed in § 3 the relative lattice displacement u and the electric field E are simply related [see eq. (3.4a)l. Thus, all the above four correlation functions may be expressed in terms of (EE) . The resulting expression for the dielectric tensor fluctuation correlation function is (SUBBASWAMY and MILLS [1977]; NKOMA and LOUDON [1975])

(4.17a) where

(4.17b)

and

d!$E’(Qllfl; zz ’ ) = I d2x11 I d t

x exp {-iQll .xII+ iOt}(E,(qz; t )Ep(0z’ , 0)). (4.17d)

Recall that the lattice contribution to the dielectric constant ~ ( f l ) has been defined in eq. (3.5). Also, as in 9 3, E , is the electronic contribution to the dielectric constant, e* the transverse effective charge, and n the number density of ions. Thus, for light scattering from polaritons, the problem of calculating the dielectric tensor fluctuation correlation function [eq. (4.11)] reduces to that of determining the correlation functions dlrEpE) of electric field fluctuations within the crystal. These in turn are related to the corresponding retarded Green’s functions [dgE)Iap through eq. (4.15).

Now, it can be shown (see MILLS, CHEN and BURSTEIN [1976] Appendix B, or ABRIKOSOV, GORKOV and DZYALOSHINSKI [1963] 928) that the electric field retarded Green’s functions [dFE)lap are identical to the electromagnetic Green’s functions Gap [defined in eq. (4.5) and eq. (4.8)] used in determining the scattered electric field, when continued analyti- cally into the complex frequency plane. As remarked earlier, these Green’s functions have been determined exactly and are conveniently tabulated in MILLS, CHEN and BURSTEIN [1976] Appendix A. When all the

11,s 41 LIGHT SCATTERING FROM SURFACE 85

above considerations are put together, we have a closed albeit lengthy, algebraic expression for the scattering cross section for light scattering from polaritons in thin cubic crystals. The evaluation of this expression on a computer is straightforward.

As remarked in P 1 we assume the various parameters (aupy, bapy, etc.) entering the theory have the same value near the boundaries as in the bulk of the crystal. Since all these bulk parameters for many crystals of interest (e.g., Gap, GaAs) are known, there are no free parameters in the theory and the theoretical and experimental spectra may be compared directly. The electric field correlation functions that determine the power spectrum have poles corresponding to all the polariton normal modes of a thin film discussed in Q 3. In addition, all effects arising from the breakdown of wavevector conservation and attenuation of light in the medium are fully incorporated into the theory. The particular features present in any Raman spectrum, of course, depend strongly on the scattering geometry, the polarization of the incident and scattered light, and the orientation of the crystal relative to the scattering directions. In the remainder of this section we shall present the results of theoretical calculations for Raman scattering from polaritons in free-standing GaP films (SUBBASWAMY and MILLS [1978]), and in GaAs films on sapphire substrates (MILLS, CHEN and BURSTEIN [1976]) and illustrate the various effects described hitherto. The corresponding experimental work has been carried out by Ushioda and coworkers (see USHIODA [1981]).

4.2.1. Surface and guided wave polaritons in free-standing GaP films

Gallium phosphide is a semiconducting crystal of the zinc-blende structure whose various properties have been studied extensively. At optical wavelengths (specifically, for 5 145 8, radiation) the real and imaginary parts of the dielectric constant have values = 13.32, and E~ = 0.009, respectively. This corresponds to a skin depth 6 = 33.3 Frn for normally incident light, while the wavelength of the incident light inside the crystal is A, = 1.9 p.m. Furthermore, the infrared active transverse optic mode has a frequency vT0 = 370 cm-' (where we have adopted the Raman spectroscopy convention of measuring frequencies in wave numbers, u,, = 6&27rc), which corresponds to a wavelength A, = 27 Fm. Thus for thin GaP films (thickness L 5 30 m) we may expect both the finite size and the attenuation of the light to have an important effect on

86 LIGHT SCA'ITERING SPECIXA OF SOLIDS [II, cj 4

their Raman spectra. We shall present results of theoretical calculations based on the formulation described above for GaP films of various thicknesses in three different geometries. These geometries are shown in Fig. 4.3 and we shall refer to them as geometry A, B and C , respectively. In each case the light is incident normally (along 2) on the film, and the scattered light is analyzed in the near forward direction. The legend in Fig. 4.3 gives the orientation of the polarization of the incident and scattered light relative to the crystal axes.

We emphasize that there are n o free parameters in the theory. Assum- ing that the coupling parameters are the same in the vicinity of the surface as in the bulk, all the constants entering the theory are known. After presenting the theoretical results we comment briefly on the experimental results. A detailed exposition of the experimental work may be found in the companion article by USHIODA [1981].

Geometry A. With the help of the tabulated forms of the Raman tensor for crystals of various symmetry (see HAYES and LOUDON [1978] Table 1.2) we determine that only the components qYY and ayyz enter eq. (4.16) in this geometry. Of course, the same is true of the electro-optic tensor bnpv. This means that components of atomic displacement u and electric field E parallel to 2 and 3 can scatter in this configuration. Thus, eigenvector components of the surface polariton along i can scatter, as can the i component of the p-polarized (TM) guided waves. The s- polarized guided waves contribute also.

Light lnciden t Light

GEOMETRY A

I v Y

GEOMETRY B GEOMETRY C

Fig. 4.3. Description of scattering configurations for the calculated spectra for GaP films.

11, B 41 LIGHT SCATTERING FROM SURFACE 87

In Fig. 4.4 we show the calculated Raman spectra for a 100 km film of GaP for scattering angles of 1.2", 1.6", 2.0" and 4.0" (measured from the normal, outside the crystal). This is a relatively thick film compared to the wavelength of the incident light, as well as to that of the transverse optic mode. Hence, the degree of non-conservation of wavevector, Ak, =

Ik~ ' -k~ ' -QLJ-2.rr /L is very small. The frequency of the guided wave polariton with QII = k$O)-kf) and QL= ky)- kf) is indistinguishable from that of the bulk polariton mode with wavevector Q = (Qi+ 0:);. Thus, in Fig. 4.4 one sees a peak corresponding to the lower branch bulk polariton (labeled .rr) and a peak corresponding to the bulk LO phonon mode (labeled L) in these spectra. It should be noted that when conservation of wavevector is very nearly valid, the upper branch of the bulk polariton is not accessible to the light scattering probe (HENRY and HOPFIELD [1965]). The dispersion of the bulk polariton peak (i.e., its up-shift as the scattering angle increases) is evident from these spectra. A dispersionless small peak (labeled T) at a frequency corresponding to the volume TO mode (i.e., the asymptotic frequency of the bulk polariton mode for large wavevectors) is also present in the spectra. This peak has its origin in backscattering events (of the type shown in Fig. 4.2b) where the back scattered light enters the spectrometer after a reflection at the

GEOMETRY A L = 100 g m L

w/wTO

Fig. 4.4. Calculated Raman spectrum of a 100 pm GaP film in geometry A for various scattering angles.


back surface of the crystal. Since our theory includes the contributions of all such events, this weak TO peak appears in the calculated spectra. Indeed, the experimental spectra for thick GaP films in this geometry do show the three peak structure of Fig. 4.4 (see VALDEZ, MATTEI and USHIODA [1978]). The presence of the peak from the backscattered light provides a remarkable illustration of the complete description provided by the present theory.

In Fig. 4.5 we show the Raman spectra for films of three different thicknesses (100 pm, 50 pm and 20 pm) at a fixed scattering angle of 1.6". In going from the 100 pm film to the 50 pm film the bulk polariton peak (T) undergoes a large distortion, and for the 20 pm film this evolves into a well defined mode structure. This structure is due to scattering from the guided waves. The frequency of each mode varies smoothly with angle, as in bulk polariton spectroscopy. This is shown in Fig. 4.6. The number near each peak is the value of n such that Q, = n.rr/L. No peaks with n < 4 appear in the spectrum since they lie in a frequency region where there is a strong minimum in the bulk Raman tensor 6 [eq. (4.17b)l of GaP (FAUST and HENRY [1966]).

As the thickness decreases two things take place, both of which play a key role in rendering the guided waves visible. As remarked in 5 2 the frequency separation between adjacent modes of fixed Q,, increases (like 1/L) until it is larger than the mode width. Also, the extent of non- conservation of wavevector normal to the surface increases. Thus modes

GEOMETRY A 8,. 1.6" L

L, L.20 ,urn

0.5 I .o 1.5

TO

Fig. 4.5. Thickness dependence of the calculated Raman spectra at fixed scattering angle in geometry A. Three thicknesses (100 pm, 50 pm and 20 pm) are shown.

LIGHT SCATTERING FROM SURFACE 89

GEOMETRY A L = 20 pm n

6: I\ 2.0"

3 'Un' 3 2 ,5 nT

4 1 V l I I

LO I I I I

0.5 I .o 1.5

w ' 9 0

Fig. 4.6. Angle dependence of the theoretical guided wave polariton spectrum for a 20 pm GaP film in geometry A.

far from the bulk polariton frequency are visible. In fact, small peaks due to the guided waves from the upper branch of the bulk polariton are also evident in the spectra of Fig. 4.6. These provide a dramatic illustration of the breakdown of wave vector conservation.

The guided wave modes corresponding to the peaks in Fig. 4.6 are s-polarized (TE) with u and E along 9. As mentioned before, the 2-component of the p-waves (TM waves) can scatter; but the scattering is weak and does not show up in the spectra. For the p-waves the ratio EJE, is of order QII/Q, [see eq. (3.18)], and for near forward scattering QII/Q,<< 1. Furthermore, E, for the p-waves is much smaller than E,, for the s-waves. Hence, we have a "quasi-selection rule" (a term we borrow from DUTHLER and SPARKS [1970]), where both s- and p-waves are allowed to scatter by symmetry, but one set of lines is weak.

The surface polariton peak in between the TO and LO phonon peaks is very weak in this configuration and is not readily visible in Figs. 4.4-4.6. In Fig. 4.7a we show an enlargement of the low frequency side of the LO peak which shows the surface polariton peak clearly.

Finally, in Fig. 4.8a we show the Raman spectra of a film of thickness 2 pm for four different scattering angles. A surface polariton peak (S) is readily seen, while the guided wave modes associated with the lower

a) GEOMETRY A n

I I I

1.06 1.12

h:b, TO

Fig. 4.7. Surface polariton peak tor a 20 &in GaP film in (a) gconietrq A;

I I

a) GEOMETRY A L = 2 p m

1.2"

lU

nu

' b[ 4.0"

11 GEOMETRY B

(h) geometry H

2.0"

j& 4.0"

Fig. 4.8. Calculated polariton spectra for a 2 Frn GaP film in (a) geometry A; (b) geometry B.

11, P 41 LIGHT SCA”ERINC FROM SURFACE 91

branch of the bulk polariton spectrum do not give dramatic structure. The reason is that the frequencies of modes with Q --- n.rrlL with n L 1 now have effective wavevectors so large that they crowd into the volume TO mode frequency. However, a prominent line (U) from a guided wave from the upper branch of the bulk polariton spectrum may be seen. As 8 decreases, this approaches the volume LO mode frequency.

Geometry B. In this geometry (see Fig. 4.3) only the components uxy,.(bxyx) and uzyy(bzyy) of the Raman tensor contribute to the scattering cross section. Forward scattering (6 , = 0) comes only from components of atomic displacement u, and electric field E parallel to f (hence the volume LO phonon cannot scatter), but for non-zero scattering angles, components parallel to 9 scatter weakly. Thus, most of the scattering comes from the p-polarized guided waves in this geometry. With these differences, the spectra in this geometry are similar to those in geometry A just discussed.

In Fig. 4.9 we show the calculated Raman spectra for a 20 pm film in geometry B. Once again a well defined mode structure due to the guided waves can be seen. These are the p-polarized modes as pointed out in the previous paragraph. One can see that the peak due to the LO mode is

GEOMETRY B L = 2 0 p m T

TO

Fig. 4.9. Calculated polariton spectra for a 20 p,m GaP film in geometry B.

92 LIGHT SCAITERING SPECTRA OF SOLIDS [II, P 4

GEOMETRY C L = 30 p m BS = 1.3"

TO

weak, and grows as the scattering angle increases. For this reason the surface polariton mode (peak labeled S) is more easily discerned in Fig. 4.9. This is also shown in the enlargement of the low frequency tail of the LO peak in Fig. 4.7b. In fact, for a thinner film ( L = 2 Fm) peaks due to both the upper and the lower surface polariton branches [as discussed in connection with Fig. 3.2bI can be easily seen in this geometry. This is shown in Fig. 4.8b where the two surface polariton peaks are labeled S, and Sz, respectively.

Geometry C. Here we deal with crystals having (100) surfaces. The incident and scattered fields are perpendicular to each other, but polarized along principal axes of the crystal. A typical Raman spectrum in this configuration is shown in Fig. 4.10. In the forward direction one sees scattering from components of u and E normal to the film. Thus, the s-polarized guided waves and the bulk TO phonon features are absent. Even the contribution from the p-polarized modes is very small since they scatter only by virtue of the small components of u and E normal to the surface present at finite scattering angles. The volume LO phonon contributes the single prominent peak.

These results illustrate the surface and size effects described in the first two sections as they apply to Raman scattering from polaritons in thin films. Our principal conclusions correlate well with the experiments on GaP films by VALDEZ and USHIODA [1977] and by VALDEZ, MATIEI and USHIODA [1978]. Surface polariton peaks such as those in Fig. 4.7 have

- - 1 "

Fig. 4.10. Theoretical Raman spectrum for a 30 k m G a P (100) film in geometry C.

11, § 41 LIGHT SCATERING FROM SURFACE 93

been observed by Valdez and Ushioda who also traced out the surface polariton dispersion curve. For GaP films with (111) surfaces Valdez, Mattei and Ushioda find clear structure from guided waves of both p and s polarization. The measured Raman intensities are indeed a fair fraction of the volume LO phonon peak, as the theoretical calculations indicate. Finally, a film with (100) faces shows only the LO phonon peak, in accord with Fig. 4.10.

There is, however, one major discrepancy between the experimental results and our theory. We have assumed the light couples to the modes via the bulk Raman tensor, which contains two contributions from a,,, and bFVA [see eq. (4.16)]. As mentioned already, these interfere destruc- tively for frequencies less than oTo to produce an experimentally observed broad interference minimum (FAUST and HENRY [1966]), with a near zero at 250 cm-I. This results in the appearance of no guided wave peaks in this frequency region [see the discussion following Fig. 4.61 in the calculated spectra. There is no such minimum in the experimental spectra from GaP films. Furthermore, the raw experimental data from some films with (111) faces show a broad, intense wing to the laser line not seen in films with (100) surfaces. The absence of the interference minimum in the data combined with the presence of the wing suggests other scattering mechanisms not considered by us are present in the films with (111) surfaces. It is clear from our theory that another scattering mechanism that has no such strong minimum can lead to peaks in the 250 cm-’ region. This is illustrated in Fig. 4.1 1 where we compare a

GEOMETRY B d = 2 0 p m e = 1.20 I

0.5 I .o w’wTO

Fig. 4.11. Raman spectrum for a 20 pm film calculated with the coupling tensor a,,, set to zero in order to suppress the minimum in the Raman tensor (see text).

94 LIGHT SCA’ITERING SPECTRA OF SOLIDS HI, § 4

spectrum shown earlier (solid line) with spectrum (dotted line) calculated with electro-optic scattering only (a,,, set to zero). The latter (dotted line) provides a measure of the strength of the electric field fluctuations in the guided wave modes “seen” by the light through a coupling mechanism with no interference minimum.

4.2.2. Surface polaritons in a GaAs film on a sapphire substrate

The first successful experimental study of surface polaritons by Raman scattering was reported by EVANS, USHIODA and MCMULLEN [1973]. These authors examined the Raman spectrum of light scattered from a GaAs film 2500A thick on a sapphire substrate. The laser beam was incident through the transparent substrate, and the experiment examined light scattered in the near forward direction. In contrast, several earlier attempts to find surface polaritons in the Raman spectrum of light back-scattered from semi-infinite materials opaque to the incident radiation were unsuccessful. MILLS, CHEN and BURSTEIN [1976] developed and applied the theory described in this section to the system studied experimentally by EVANS, USHIODA and MCMULLEN [1973].

The theory readily accounts for the large forward-backward asymmetry in the surface polariton intensity just described. The theoretical result is shown in Fig. 4.12, and is easily understood. In the forward scattering geometry the dominant process is the one shown in Fig. 4.2a, whose

GaAs/SAPPHIRE

FORWARD SCATTERING BACKSCATTERING

L I 1 I I I J I I I 1 1 I I

1.0 1.04 1.06 1.0 1.04 1.06

w/wTO fJJ/fJJ TO

Fig. 4.12. Theoretical Raman spectra of a GaAs film on a sapphire substrate: (a) forward scattering; (b) back scattering.

11, § 51 ACOUSTICAL PHONONS AND SPIN WAVES 95

Fig. 4.13. The dominant scattering process for the back-scattering geometry.

contribution to the matrix element involves the difference of normal wavevectors kio’- k q . On the other hand, in the back-scattering geometry the dominant process is the one shown in Fig. 4.13, which involves the sum of normal wavevectors k i o ’ + k f ’ . For typical semiconducting materials at optical frequencies, the real parts of k?’ and k:”‘ are an order of magnitude greater than the imaginary parts (see CHEN, BURSTEIN and MILLS [1975]). Thus, the phase factor exp [i(k$”+ k f ’ ) z ] that enters the back-scattering matrix elements oscillates very rapidly with z, and reduces the back-scattering matrix element strongly compared to the forward scattering matrix element. This accounts for the large forward-backward asymmetry of the surface polariton intensity observed.

This completes our discussion of surface and size effects on Raman scattering from polaritons. In the past few years the formalism described in this section has been applied successfully to the study of Brillouin scattering from ,acoustic phonons in metals and opaque semiconductors, as well as surface and bulk spin waves in ferromagnets. Recent advances in spectroscopic techniques have led to successful experimental investigations of these excitations by Brillouin spectroscopy. We now turn to a summary of these recent developments.

0 5. Surface and Size Effects on Brillouin Scattering from Acoustical Phonons and Spin Waves

5.1. ACOUSTICAL PHONONS IN OPAQUE SOLIDS

The interest in surface and size effects on the Brillouin spectra of solids began after the pioneering experimental innovations by SANDERCOCK [ 1972a, 19781. With his improved spectroscopic techniques Sandercock reported the first observation of opacity broadened acoustical phonon peaks in the Brillouin spectra of Ge and Si in the back scattering configuration (SANDERCOCK [1972a]). He also measured the spectra of

96 LIGHT SCATTERING SPECTRA OF SOLIDS 111, § 5

thin, free-standing slabs and the resultant size effects (SANDERCOCK [ 1972b1). Subsequently DIL and BRODY [1976] investigated light scattering from acoustical phonons in liquid metals, to find the highly asymmetric lines discussed in $ 2. More recently, SANDERCOCK [1978] has observed highly asymmetric Brillouin peaks in metallic solids, while MISHRA and BRAY [1977] have investigated Brillouin scattering in reflection from acoustoelectrically amplified bulk phonons. Thus, we now have in hand data taken on a wide variety of materials. Theoretical developments have kept pace with the experimental developments (SANDERCOCK [1972a], LOUDON [1978a, b, c], SUBBASWAMY and MARADUDIN [1978a, b], DERVISCH and LOUDON [1978], ROWELL and STEGEMAN [1978a, b] and BARTOLANI, NIZZOLI and SANTORO [1978a, b]). All the theoretical approaches are essentially equivalent in principle, but differ in the degree of generality and completeness. The method outlined below (SUBBASWAMY and MARADUDIN [ 1978bl) follows the same Green’s function approach described in § 4, and is particularly complete.

For a perfectly transparent medium of dimensions large compared to the wavelength of the incident light, Brillouin scattering comes mainly from the modulation of the dielectric constant by the bulk acoustic phonons, in a manner analogous to that discussed in $4. For long wavelengths relevant here, the appropriate dynamical variables may be taken to be the displacement gradients

u,p(x, t ) = du,(x, t ) /dx, , (5.1)

where u(x, t ) is the displacement field. In analogy with eq. (4.14) we write

The coefficients kaPvs are called the photoelastic constants, and are simply related to Pockels’ elasto-optic constants [see BORN and HUANG [1954] Ch. VII). Hence, this scattering mechanism is referred to as the photoelastic or elasto-optic coupling. For a cubic crystal, kaPyS is symmetric under interchange of y and 8, so only the symmetrized combination e,, = $(u,, + us?) enters the theory. This is in fact simply the strain tensor. In optically anisotropic media, clearly a local rotation of the lattice modulates the dielectric tensor, as well as strain. In such crystals, kaPvs is not symmetric under interchange of y and 8, so the combination 6.1,~ =

$(du,/dx, -du,/dx,) enters as well; wap is non-zero when a local rotation of the lattice occurs. A complete theory of the coupling of acoustic?’

11,s 51 ACOUSTICAL PHONONS AND SPIN WAVES 91

waves to light, with emphasis on the antisymmetric portion of kupvs has been given by NELSON and LAX [1971] (see also NELSON, LAZAY and LAX [1972]). The effect of finite size, and of surfaces on the scattering from this mechanism may be inferred by substituting eq. (5.2) into our formula for the scattering cross section, eq. (4.9).

We then need to calculate the displacement gradient correlation functions (uapuyS), which in turn depend on the corresponding retarded Green’s functions as in eq. (4.15). It is readily demonstrated (see SUBBAS- WAMY and MARADUDIN [1978b] Appendix A) that the equation of motion of the retarded displacement Green’s functions is isomorphic to the equation determining the classical dynamical Green’s functions of the elastic medium. These Green’s functions incorporate the appropriate acoustic boundary conditions for the given geometry and determine the power spectrum of the long wavelength acoustic phonons. Thus, the problem of determining the Brillouin spectrum is reduced to that of solving for the appropriate classical elastic Green’s functions for the elastic medium. Alternatively, one may view this as a problem of acoustic wave reflection at the boundaries, and determine the appropriate forms of displacement patterns, and hence the necessary response functions (LOUDON [1978a, b]).

The qualitative features to be expected here are evident from our remarks in § 2. For scattering from thin transparent films, the Brillouin peaks break up into a multiplet structure reflecting the discrete values (nr /L ) of the phonon wavevector perpendicular to the film surfaces (see SANDERCOCK [1972b]). For back scattering from opaque substrates (essentially semi-infinite media) the Brillouin peaks are at first broadened, and distorted as the opacity increases until one sees highly asymmetric lineshapes reflecting both the effect of wavevector non-conservation and of the presence of the surface itself.

As remarked earlier, if one ignores the effect of the surface on the phonons, one gets spectral lineshapes with square-root singularities reflecting the one dimensional density of states of acoustic phonons in the direction normal to the surface. However, the presence of the sharp boundary drastically alters the lineshapes, as we shall see. In addition, of course, we shall have scattering from surface waves. Examples of such lineshapes will be presented after we discuss a second important scattering mechanism below.

There is an alternate mechanism for Brillouin scattering, whose importance for back scattering of light from highly opaque materials has

98 LIGHT SCATTERING SPECTRA OF SOLIDS [II, Q 5

been realized only recently (MISHRA and BRAY [1977], LOUDON [1978c]). The presence of surface and bulk elastic waves in a (semi-infinite) crystal necessarily distorts an otherwise flat crystal surface. These surface corrug- ations or ripples act as a dynamical diffraction grating in reflecting the light incident on the surface. This scattering mechanism, called the surface ripple mechanism, plays a very important role in Brillouin back scattering from opaque materials, since the competing photoelastic mechanism described above is confined to a very narrow skin depth. The intensity of the surface ripple scattering is proportional to the mean square amplitude of the ripples ( m k,T), and to the reflectivity. The frequency dependence of the cross section is determined by the power spectrum of the displacement component normal to the surface (as opposed to the displacement gradients that enter the photoelastic mechanism). As pointed out by LOUDON [1978c] the fact that the free surface is a displacement antinode and a node for many components of the displacement gradient explains the dominance of the surface ripple mechanism over the photoelastic coupling in highly opaque materials. Indeed, Sandercock’s data on A1 and Ni (SANDERCOCK [1978]), as well as Dil and Brody’s data on liquid Ga and Hg (DIL and BRODY [1976]) may be successfully interpreted as being due solely to the surface ripple scattering (LOUDON [ 1978~1, DERVISCH and LOUDON [1978]). Of course, both mechanisms are present simultaneously, and a complete theory should include both of them along with the resulting interference terms. The relative importance of the three contributions depends on the opacity of the material. Clearly, in the limit of very large skin depth, the photoelastic mechanism must provide the dominant contribution to the scattered intensity, while for very small skin depths (most particularly in the limit S - O ) , the surface ripple mechanism only contributes. A calculation using the Green’s function approach described in this article has been carried out by SUBBASWAMY and MARADUDIN [1978b]. We outline their theory below. The reflection of light from driven surface acoustic waves has been considered in detail by LEAN [1973], who noted the importance of the surface ripple mechanism in these experiments.

Let us consider a semi-infinite elastic medium occupying the half space 2 > O . In the absence of fluctuations the bounding surface is the plane z = O . However, the presence of elastic fluctuations in the medium distorts the surface (see Fig. 5.1), so the equation that determines the surface profile is now

2 = uz(x,,O, t ) . (5.3)

ACOUSTICAL PHONONS AND SPIN WAVES 99

I Z I

I

Fig. 5.1. The surface ripple and photoelastic mechanisms of Brillouin back scattering from acoustic phonons.

The problem then consists of satisfying the appropriate electromagnetic boundary conditions at the distorted surface. This problem is similar to the one encountered in describing the scattering of electromagnetic waves from a rough surface - a topic that has received much attention in recent years (MARADUDIN and MILLS [1975], ACARWAL [1976], MARVIN, T~oco and CELLI [1975]). In our case the “roughness” is not a static one, but is brought about by the time dependent elastic fluctuations in the medium. In order to keep the algebra simple, we shall assume for the present that we can ignore modulation of the dielectric tensor by the elastic fluctuations in the bulk and describe the optical response of the medium by an isotropic (complex) dielectric constant E ~ . Thus, for the moment, we consider scattering by only the surface ripple mechanism. In the absence of surface ripples the dielectric constant of the elastic medium/vacuum system may be written as

& O ( Z ) = q - 2 ) + & M @ ( Z ) , (5.4)

where O(z) is the unit step function already defined. In the presence of surface ripples, the dielectric constant is modified to

E ( Z ) = 6[U,(X110, t ) -z ]+EM6[Z-U,(X110, t)]. ( 5 . 5 )

With a little algebraic manipulation we may rewrite eq. (5.5) in the form

where

(5.6a)

(5.6b)

100 LIGHT SCAWERING SPECTRA OF SOLIDS [II, 8 5

with

U(x, f ) = f3[z - u,(x,,O, f)]- O ( Z ) . ( 5 . 6 ~ )

Once we cast the dielectric constant of the system in the form of eq. (5.6) we can proceed to calculate the scattering cross section exactly as in § 4, except for a few subtleties associated with the electromagnetic boundary conditions at the corrugated surface. We shall give here only a brief and non-rigorous account of the prescription involved and refer the reader to the original literature for a rigorous justification (AGARWAL [ 19761). Since we are interested only in first order scattering we may approximate eq. (5.6b) to first order in the ripple amplitude,

6&(X, ~)==(1-&M)&(x~\o , f ) 6 ( Z ) . (5.7)

Now, in substituting this expression for 8s into the integral equation for the scattered electric field (eq. (4.6) with L + m), and performing the 6-function integration over z’, one has to take the limit z’ -+ 0 in such a way as to satisfy the electromagnetic boundary conditions to first order in the ripple amplitude. When this is done the resulting expression for the scattered electric field is

Here, 0+, and 0- denote, respectively, whether the limit z’+0 is approached from above or from below. For a more rigorous derivation of eq. (5.8) and for explicit expressions for the Green’s functions we refer the reader to SUBBASWAMY and MARADUDIN [1978b] and the relevant references therein.

The rest of the analysis proceeds as before. The power spectrum of surface ripple scattering is determined by the correlation function (u,(xllO, t)u,(x[O, f ) ) which in turn can be determined from the relevant component of the dynamical elastic Green’s tensor. Finally, we note that the simultaneous effect of both the surface ripple and the photoelastic mechanisms to first order in the phonon amplitudes is included by adding to the scattered electric field in eq. (5.8) a contribution arising from the bulk dielectric fluctuations given in eq. (5.2). Thus, in this approach the interference effects due to the presence of two mechanisms is also included in a consistent manner. The interference terms occur as cross

11, $51 ACOUSTICAL PHONONS AND SPIN WAVES 101

I .o-

0.0- 0.03-

terms between the surface ripple and photoelastic contributions to the cross section when the Poynting vector is formed from the scattered field.

Analytic expressions for the dynamical elastic Green’s functions needed for calculating the above spectral contributions have been derived for an isotropic medium with a stress-free boundary (MARADUDIN and MILLS [1976]), and for a hexagonal crystal with c axis normal to the surface (DOBRZYNSKI and MARADUDIN [1976]). However, most of the explicit calculations of spectra have assumed elastic isotropy. Recently, BARTO- LANI, NIZZOLI and SANTORO [1978a, b], who use a microscopic approach to directly calculate the surface phonon density, have dealt with cubic crystals as well. A numerical determination of the Green’s functions for anisotropic elastic media is quite feasible, but we know of no such study.

We now present several examples of calculated spectra in order to illustrate the effects described above. These spectra are based on the theoretical calculations of SUBBASWAMY and MARADUDIN [ 1978bl and assume the elastic medium to be isotropic. In Fig. 5.2 and Fig. 5.3 we show the calculated total spectra (including surface ripple, photoelastic, and interference contributions) for solids of varying degrees of opacity. These calculations enable one to appreciate the relative importance of the surface ripple and photoelastic contributions to the spectrum, as the skin depth is decreased from a larger value, characteristic of weakly absorbing

TA (Bulk)

b) 6,=0.10

1

O.o--)L I I\ 0.5 5.5 10.5

fi’ctQ11

Fig. 5.2. Calculated total Brillouin spectrum assuming (a) F , = 10.0, ~ ~ = 0 . 0 1 ; (b) E , = 10.0, E 2 = 0.1.


b) €2 ~10.0

I

0.5 5.5 10.5

.Q'C,QII

Fig. 5.3. Same as Fig. 5.2, but with (a) E , = 10.0, F~ = 1.0. and (b) F , = 10.0, E ~ = 10.0

media to a small value, such as found in metals. In each case the wavelength of the incident radiation is assumed to be 5145 A, and polarized perpendicular to the plane of incidence. The angle of incidence is 45", and the back-scattered radiation is gathered along the same direction, with its polarization not being analyzed. The scattering geometry mimics that used by Sandercock in his studies. The isotropic elastic medium is assumed to have Poisson's ratio u = 0.5, which implies the relationship cl = 2c, between the longitudinal and transverse sound velocities in the medium. We assume for the elasto-optic coefficients pI1 = -0.15, pI2 = -0.10 and p44 = -0.05 which are typical of cubic semiconductors. In each case, we also assume the same value of the real part of the dielectric constant, = 10.0, a value typical of semiconductors also.

The four spectra in Figs. 5.2 and 5.3 differ only in the assumed value of the imaginary part of the dielectric constant E ~ . As pointed out in 5 2 breakdown of wavevector conservation normal to the surface allows all bulk modes with frequencies a> ctQII (where c, is the velocity of transverse sound waves in the medium, and QII is the wavevector change parallel to the surface suffered by the radiation) to scatter. Hence, we measure the frequency shifts in the Brillouin spectra in units of ctQll. All intensities are measured relative to that of Fig. 5.2a which corresponds to a relatively transparent solid. Note that the theory does not provide for


the finite lifetime of the excitations and we incorporate such effects phenomenologically by adding a small imaginary part to the frequency shift.

In Fig. 5.2a we show the calculated Brillouin spectrum for a solid with E~ = 0.01. This corresponds to a highly transparent solid, and accordingly the spectrum consists of peaks arising from the bulk T A and the LA phonons at the nearly conserved wavevector QII = kf) - k$’, and Q, = k F ) - k t ) . The scattering from the surface Rayleigh wave, which has a velocity of 0 . 9 3 3 ~ ~ for the elastic constants chosen, is very weak compared to the bulk phonon scattering (-0.1% of the bulk phonon peaks) since the light penetrates rather deeply into the solid. Fig. 5.2b corresponds to .s2=0.1. The spectrum is still dominated by the bulk phonon photoelastic scattering. However, the peak intensity has fallen to about 3% of that in Fig. 5.2a, and the lines have become broadened due to the increase in opacity. The principal effect of the increased absorption is thus primarily simply to decrease the strength of the signal, with a bit of broadening, produced by the breakdown of wave vector conservation normal to the surface. The weak feature due to the surface wave may already be discerned in the spectrum. In going to Fig. 5.3a, where now F ~ = 1.0, the Brillouin spectrum is dominated by the surface ripple scattering. The peak intensity has fallen to about 0.1% of that of the transparent solid in Fig. 5.2a. The most prominent feature is now the surface Rayleigh wave peak at 0 = 0.933ctQll. Minima and ensuing shoulders at 0 = c,QII and O=clQll are characteristic of the surface ripple scattering, as will be described shortly. Peaks corresponding to the bulk modes are now quite weak and considerably broadened. Finally, Fig. 5.3b corresponds to the highly opaque solid, E~ = 10.0, and the spectrum is due essentially entirely to the surface ripple mechanism.

The photoelastic contribution by itself in the absence of surface ripple scattering, corresponding to the spectra of Figs. 5.2a, b and 5.3a, b are shown in Figs. 5.4a, b and 5Sa, b, respectively. In Fig. 5.4a, b the spectrum is almost entirely due to the bulk modes. In Fig. 5.5a the surface Rayleigh mode, and features reminiscent of the square-root singularity alluded to in P 2 begin to develop at 0 = ctQII and 0 = c,QII. Finally, the photoelastic spectrum in Fig. 5.5b bears no resemblance to that of the transparent solid in Fig. 5.4a. Note from the intensity scale that for the case shown in Figs. 5.3b and 5.4b, the surface ripple scattering dominates the Brillouin spectrum.

The surface ripple contribution alone, corresponding to e2 = 10.0 is

104 LIGHT SCATRRINC SPECTRA OF SOLIDS

1.0-

0.0 0.3-

0.0

[II, § 5

a) .+=0.01 PHOTOELASTIC

I , , , , k , ,

b) E ~ = O . I O PHOTOELASTIC

, A 1 L

shown in Fig. 5.6. It is seen that the surface ripple scattering is characterized by zeroes (minima in the presence of finite lifetimes for the phonons) in the spectrum at 0 = c,QII and 0 = cQ. This is a consequence of the free-surface acoustic boundary conditions imposed on the phonons. A close examination of the boundary conditions and the acoustic wave

n nnm, . -.----

I\ a) e2= 1.0 PHOTOELASTIC

"." , I

1 1 0.0002

I I bl c2=10.0 'HOTOE L ASTlC

0.0 1c /" = cJQii

10.5 0.5 5.5

n/c iQ i i

Fig. 5.5. Photoelastic contribution to the calculated spectra in Fig. 5.3a. b, respectively.

11, P 51 ACOUSTICAL PHONONS AND SPIN WAVES 105

SURFACE RAYLEIGH €2 = 10.0

SURFACE RIPPLE / WAVE

Fig. 5.6. Surface ripple contribution to the spectrum of Fig. 5.3b.

reflection coefficients reveals that there are displacement nodes at the surface for waves at grazing incidence, and this produces the zeroes in the spectra. We see from these examples the crucial importance of a full treatment of the scattering problem, with influence of the stress free boundary conditions incorporated fully into the theory. Our simple discussion of 0 2, which ignores this effect entirely, can give only square root singularities at c,Qa and cQ, and may be dramatically in error as a consequence. Minima such as those displayed in Fig. 5.6 have been observed in the Brillouin spectra of highly opaque solids by SANDERCOCK [1978]. A rich variety of boundary and interface effects have been subsequently examined (ROWELL and STEGEMAN [ 1978b1, BARTOLANI, NIZZOLI, SANTORO and SANDERCOCK [1979]). The field is still under active investigation, and shall yield more experimental and theoretical studies in the near future.

5.2. THE SCATTERING OF LIGHT FROM SPIN WAVES ON THE SURFACE OF OPAQUE FERROMAGNETS AND IN THIN FILMS

Up to this point, our attention has been confined to the scattering of light from lattice excitations on the surface of opaque materials, and in

106 LIGHT SCATERING SPECTRA OF SOLIDS [II, II 5

thin films. While the surface and guided wave polaritons discussed in $ 3 and $ 4 were regarded as electromagnetic eigenmodes of the structure in question, they may also be viewed quite equivalently as vibrational modes of optical character in the ionic lattice, which are modified profoundly by the electromagnetic field set up by the spatially varying distribution of electric dipole moment density in the mode.

We now turn to the discussion of light scattering by spin waves on the surface of opaque ferromagnets, and in ferromagnetic films. Spin waves are the elementary excitation of the ordered array of spins in a magnet below its ordering temperature. We shall see that just as in the case of the lattice excitations, we encounter surface modes on the semi-infinite medium, and modes in thin films very similar to the guided wave polaritons examined in 3 3 and E) 4. Recently, a very beautiful series of experiments have appeared which probe these modes by Brillouin scattering techniques. This is a most exciting development since, in our view, the light scattering method is far more flexible than the conventional microwave resonance techniques that until now have been the only means of exciting spin waves in thin films. We discuss the reasons for this below. It is important to note that the experiments discussed in the present section have been made possible by the appearance of the five pass Fabry-Perot spectrometer developed by Sandercock; this device has ushered in a new generation of Brillouin experiments that will provide us with very detailed and quantitative information on the Brillouin spectra of opaque crystals and thin films. In the case of light scattering by spin waves, the method has led to studies of spin wave modes discussed in the theoretical literature, but which have until now eluded experimental study in clean geometries.

Since the reader may be unfamiliar with the properties of spin waves, and most particularly with the striking and unusual properties of the Damon-Eshbach surface spin wave that has been the focus of the recent light scattering studies, we begin with a brief review of the basic properties of these modes. Then we turn to a description of the theory of light scattering from these waves, and a discussion of the recent experimental literature.

A spin in free space, when placed in an external magnetic field of strength H,, executes a Larmor precession at the frequency yH,, where y is the gyromagnetic ratio. With H,-5 X lo3 gauss, and y = 2 x lo7 (gauss/sec)-', the Larmor precession frequency is in the microwave frequency regime accessible also to Brillouin spectroscopy.


If the spin is placed in a ferromagnet rather than in free space, then its precession frequency is modified by its interaction with other spins in the material. For example, a given spin feels a magnetic dipole field from the spins in its near vicinity. The strength of the field produced by a neighboring spin is roughly p/a i , where p is the magnetic moment of the neighbor, and a , the lattice constant. Upon noticing that n,= l /a i is the number of spins per unit volume in the material, and M,= n,F is the magnetic moment per unit volume associated with the aligned spin array, we see that M, provides a measure of the strength of the dipole-dipole interactions between the aligned spins in the ferromagnet. As we shall see, it is actually the combination 47rM, that enters the formulae; in most ferromagnets, M, = 1000 gauss, so 47rMs and typical laboratory Zeeman fields are comparable in magnitude.

In addition, a given spin interacts with those in its near vicinity by exchange interactions of quantum mechanical origin. These interactions, for spins in an insulating material, have their origin in overlap of wave functions of adjacent spins, and thus have a short range, in contrast to the dipole-dipole interaction, which falls off with distance as F3, with r the separation between spins. The exchange interactions are very much stronger than the dipole interactions, and in most ordered magnets are the dominant influence that leads to the spin alignment characteristic of the ferromagnetic state, or ordered configurations of lower symmetry. However, the exchange interactions influence wave-like motions in the spin array (spin waves or magnons) only modestly in the limit of wavelength long compared to the lattice constant, because of the mathematical form assumed by the exchange interaction. If we have two neighboring spins S , and S2, the exchange interaction has the form -JS,-S2. Now suppose the spins are first all aligned parallel to the direction 2, as in the ferromagnetic ground state. Imagine a wave-like excitation of the spin system of wave vector k, where (kl a(,<< 1. In this limit, two neighboring spins that are precisely parallel before imposition of the wave are nearly so afterward; the change in exchange interaction energy between the pair is easily seen to be roughly + J S ( k a J 2 . The exchange couplings thus increase the energy of the wave like motion over that associated with the Zeeman precession and dipole couplings by the amount Dk2 in the limit kao" 1, where the above argument gives the estimate D = JSa&

Thus, while exchange coupling between neighboring spins is very strong, the factor of k2 assures long wavelength excitations in the system are influenced only modestly by exchange. Under conditions of interest to

108 LIGHT SCA'ITERMG SPECTRA OF SOLIDS [II, § 5

us, Dk2 will be typically a few percent of 4.rrMs. While we may ignore exchange in first approximation in the discussion of long wavelength spin waves in ferromagnets, we shall see that its presence has an important qualitative influence on features of the light scattering spectra. Indeed, an important goal of the light scattering studies is to measure the exchange constant D through excitation of spin waves in thin films, where k , = k , =

n.rr/L, n = 1 , 2 , . . . , and DkZ, is large enough to lead to a resolved standing spin wave spectrum.

The discussion of exchange provided above is applicable to insulating materials, strictly speaking. Fundamental conceptual differences occur in discussions of metallic magnets such as Fe or Ni, where the moment- bearing electrons are itinerant in nature. In the end, however, in the limit of long wavelength, as emphasized many years ago by HERRING and KITTEL [1951], both classes of materials are described by the same basic phenomenology.

The discussion above suggests that the principal features in the excitation spectrum of the ferromagnet can, in the long wavelength limit, be described by theory that ignores the influence of exchange. We begin with such a picture, then comment on the influence of exchange.

As the spins precess, they set up a spatially varying magnetic field h ( x , t ) that oscillates with the frequency w of the spin wave. Let m(x, t) be the spatially varying magnetization density associated with the spin wave, that also varies with the frequency w ; this is the analogue of the electric dipole moment density PL(x, t ) [see the paragraph after eq. (3 .2)] . Thus, if we know the relation between b(x, t) = h(x, t ) + 4.rrm(x, t) and h ( x , t) we may, with the help of Maxwell's equations determine the dispersion relation of both surface and bulk spin waves, in the limit when exchange is unimportant. The relation between b(x, t) and h(x , t) may be described by a magnetic permeability tensor c~~~ (0) which depends on the frequency of the motion. We write

where for a ferromagnet with both magnetization M, and Zeeman field H,, aligned along 2, one has (MILLS and BURSTEIN [1974] section 4) c~,,(o) = 1 , pzi = pi, = 0 for i = x or y, and

(5.10a)

11, 9: 51 ACOUSTICAL PHONONS AND SPIN WAVES 109

while

(5.10b)

The principal complication in the discussion of the ferromagnet is the gyrotropic and highly anisotropic nature of the response tensor. This gives a rich structure to the normal mode spectrum, actually. A simplification is that for all practical purposes, retardation effects are totally negligible, so in Maxwell's equations one may take the limit c + safely. We have frequencies w - lo1' sec-' as we have seen, and the kinematics of the light scattering process leads us to examine the response of the system in the wave vector regime of 105-106cm-' (the latter when absorbing substrates are considered). Here, cQll/w - lo5 typically, and retardation effects are very small.

In the absence of retardation, one Maxwell equation reads

V x h = O , (5.11)

so we may write h = -V$. The second is

V - b = O ,

which is equivalent to the anisotropic Laplace equation

(5.12)

(5.13)

In the infinitely extended medium, eq. (5.13) admits solutions in the form $(x, t ) = $, exp [i(Q.x- mot)] where, if 0, is the angle between Q and M,, then eq. (5.13) becomes

pl (wo) sin2 @,+ COS' 0, = 0, (5.14)

which, when solved for mQ gives the bulk spin wave dispersion relation

w, = y[H,(H, + 47rMs sin2 @,)I+. (5.15)

The frequency of the wave, in the long wavelength limit is independent of the magnitude of Q, but does depend on the direction of propagation. This is reminiscent of the behavior of long wavelength, infrared active optical phonons in anisotropic ionic crystals (MILLS and BURSTEIN [1974] section V). The bulk spin wave frequency ranges from a minimum of yH, (0, = 0, propagation parallel to the field) to a maximum of y(H,B)$, where B = H,, + 47rMs. The maximum frequency occurs for 0, = rr/2 and

110 LIGHT SCA7TERING SPECTRA OF SOLIDS [II, 9: s

propagation in a plane perpendicular to the magnetization. One calls the frequency regime y H , 5 w 5 - y (H, ,B)~ the bulk spin wave manifold.

There are surface spin waves with very striking properties that emerge from the present analysis. They occur when the magnetization M, lies parallel to the surface. We turn to a description of them, since they play a key role in the light scattering studies.

Let the surface of a ferromagnet lie in the xz plane with magnetization M, parallel to 2. The material lies in the half space y > O . Then if Qll is a wave vector that lies in the xz plane and x also lies in the xz plane, the search for surface spin waves begins by examining eq. (5.13) for solutions of the form

+(x, t ) = 4, exp * xII- iw,t] exp [-ay]. (5.16)

If 4 is the angle between QII and the i direction, we have

~ , (w , ) [Q~cos2 4-a’ ]+Q;fs in24=0. (5.17)

Further constraints are found by matching the solution in eq. (5.16) to one outside the material ( y <O) , which has the form

+(x, t ) = +< exp [iQll-xll-iw,tl exp (iQ,y). (5.18)

We must require tangential components of h to be conserved, along with the normal component ( 9 component) of b. Tangential components of h are conserved automatically by setting +< = +,, but conservation of by across the surface leads to a second constraint on the relation between a, w, and 01,. This second constraint, combined with eq. (5.17) will lead us to a dispersion relation for the surface wave. The second constraint reads

(5.19)

This may be combined with eq. (5.17) to eliminate a, and we obtain an implicit dispersion relation for the surface mode. One has

[1-Pl(ws)l[1+PI(ws) ~ ~ ~ 2 4 1 + P * ~ ~ , ~ ~ ~ ~ 4 [ ~ + P 2 ~ ~ , ~ ~ ~ ~ 2 4 1 = ~ . (5.20)

When the explicit form of pl(w,) and p2(w,) are inserted into eq. (5.20), a bit of algebra leads to an explicit dispersion relation for the surface wave:

w, = w,(f$) = - Y (-+ Ho B cos 4). 2 c o s 4

(5.21)


This is a most unusual result. The sign of a,(+) is the same as that of cos& Thus, for - 7 r / 2 5 4 5 + 1 ~ / 2 , w,($)>O, while if r r / 2 1 + 5 3 1 r / 2 , us(+) <O. Thus, if QII is changed in sign, by letting 4 + 4 + rr, w,($) does also. This means that in the surface spin wave, no matter what the direction of QII, planes of constant phase always move from left to right across the magnetization. We have a unidirectional wave, with only one sense of propagation! This result will influence the light scattering spectrum in a remarkable way, as we shall see shortly.

Actually, not all values of 4 are allowed for the wave. We must check the behavior of the attenuation constant a, to make certain that a(&)>O. After some algebra, one finds

(5.22)

so in fact our attention is limited to the range - c $ ~ s ~ s + & , where

For + = O , the frequency of the surface spin wave is ws(7r/2)= ( y /2 ) (H0+ B) > y(H,B):, so the surface spin wave lies above the bulk spin wave manifold. As 4+*&, ws(q5)-+y(HoB)f, and the surface wave merges with the bulk spin wave manifold. Note that as 4 + &, a ( 4 ) -+

00, and the surface wave becomes tightly compacted to the surface, and the wave is influenced by the nature of the magnetic environment very near the surface (RAHMAN and MILLS [1979a]).

The remarkable wave described above was discussed many years ago by Damon and Eshbach, and is frequently referred to as the Damon- Eshbach wave in the current literature (DAMON and ESHBACH [1960]). It is the light scattering method that allows it to be studied in detail, in a geometry that approximates the semi-infinite limit considered here. Ear- lier ferromagnetic resonance experiments, influenced strongly by both the finite thickness and width of the magnetic films used there, have been summarized in a review article by WOLFRAM and DE WAMES [1972].

So far, we have ignored the influence of the exchange coupling between spins on the spin wave spectrum. To include this, one proceeds as follows. The permeability tensor pi i (w) that forms the basis of the analysis, was constructed by examining the equation of motion for the precessing magnetization m. This has the form

cos 4c = (H"/B)h.

dm/dt = y(m x HT), (5.23)

where HT is the total magnetic field, the sum of the Zeeman field i H o

112 LIGHT SCATI’ERING SPECTRA OF SOLIDS [II, 0 5

and the field h ( x , t ) generated by the precessional motion of the spins. One ignores non-linear terms of the form mihi on the right hand side of eq. (5.23), then obtains a linear relationship between rn and h.

The influence of exchange may be incorporated into the theory by replacing the Zeeman field iH, by i (H,-D V2), where D is the same parameter that entered our earlier qualitative discussion of the influence of exchange. This parameter plays a role in the theory of ferromagnetism quite analogous to the elastic constant of elasticity theory. Then to discuss the response of the system, or the normal modes of it, one solves the modified eq. (5.23) simultaneously with eq. (5.11) and eq. (5.12).

In the infinitely extended medium, we still have spin waves characterized by well defined wave vector, but the dispersion relation becomes, in place of eq. (5.15)

wQ= y[(H,,+ DQ2)(H,+4.rrMssin2 O,+DQ”)]’. (5.24)

In effect, the Zeeman field Ha is augmented by the exchange field DQ’. In the presence of exchange, yH, remains the lowest possible bulk spin

wave frequency, but the spectrum is no longer bounded from above by y(H,B);. For DQ2 >>Ifo or 4.rrMs, we have wQ = yDQ2, so in fact there is no upper bound to the spin wave spectrum in our continuum theory. (Upon taking due account of the crystal structure, the maximum value assumed by Q is rrla,, where a, is the lattice constant. Then the spin wave spectrum is bounded from above and below. Here we always have Q << .rr/an.)

To discuss the influence of exchange on the Damon-Eshbach wave, we must supplement our mathematical structure with one further statement. At the moment, we are to solve eq. (5.11), eq. (5.12) and the modified form of eq. (5 .23) in a semi-infinite crystal, and we have boundary conditions that require conservation of tangential components of h and normal b. After adding exchange, eq. (5.23) becomes a second order differential equation and, if we attempt to solve the system of equations on a half space, we shall discover that additional boundary conditions are required.

This may be seen as follows, though we omit details. With D=O, eq. (5.19) provides a unique value of a’, once the frequency o and wave vector QII parallel to the surface are specified. If a’ > 0, we have solutions “bound” to the surface, of the form given in eq. (5.16) with a real. With exchange added, eq. (5.17) is replaced by an equation cubic in a2, so we now have six values of a. In our solution, clearly we keep only those real


roots with a > O , and if there are pure imaginary roots for a, we keep those that describe energy flow from the surface to the crystal interior. We shall see there is always at least one pure imaginary root, and we discuss its physical significance later. Thus, we have three possible solutions in the bulk, so $(x, t ) now has the form, in place of eq. (5.16),

3

$(x, t ) = exp [iQll-xll- iwst] 1 +$) exp [-aiy]. (5.25)

Quite clearly, in addition to the boundary conditions imposed before, we require two additional boundary conditions as a supplement before all I,@)

are uniquely determined. The reader familiar with the literature on the optical response of

crystals will recognize that precisely the same issue (that of additional boundary conditions) arises there in the discussion of the optics of crystals that exhibit a non-local response to the optical field, i.e. in situations where the electric dipole moment Pi(x , t ) depends not only on the electric field E,(x, t ) at precisely the same point x, but on an average of E over a small volume which surrounds x. Then the relationship between Pi (x , t ) and Ei(x, t ) takes the form (for a wave of frequency o)

i = l

Pi(x , t ) = 1 d3x’xii(x, x’; w)Ei(x’ , t). i J (5.26)

Most particularly for frequencies near exciton absorption lines in insulators and semiconductors, the non-local nature of the optical response affects the reflectivity of the crystal strongly. The mathematical description of the non-local optical response of insulating crystals, referred to as spatial dispersion theory, is almost identical to that we are encountering in our theory of the influence of exchange on the response of ferromagnets. It is for this reason that the introductory portion of the present section is rather long. We wish the reader familiar with spatial dispersion theory to appreciate the analogy. The theory of spatial dispersion has been discussed in detail in an earlier volume of the present series by AGRANOVICH and GINZBURG [1971]; more recently a brief review of the more current literature can be found elsewhere (MILLS and BURSTEIN [1974] section 6) , and we call the readers’ attention to a recent experiment which bears directly on the form of the additional boundary condition (Yu and EVANGELISTI [1979]). The last cited paper also contains references to the recent theoretical literature.

In spatial dispersion theory, it is difficult to derive the form of the

114 LIGHT SCATI’ERING SPECTRA OF SOLIDS [II, 0 5

additional boundary condition from a complete microscopic theory that properly incorporates the influence of the surface on the optical response. As a consequence, after many years of research activity, its form remains a topic of lively debate. In constrast to this, in the problem of the magnetic response of surface one may derive the form. of the conditions from microscopic models of the surface environment of the spins, and there is general agreement on the mathematical form of the boundary conditions. Thus, we have the possibility of exploring the predictions of the theory in an atmosphere where the underlying mathematical structure is uncontroversial. One frequently uses the form, with the y axis perpendicular to the surface,

wJm,,,laY) + pmx,, = 0, (5.27)

where p is called the pinning parameter, and is influenced by the microscopic details of the surface environment. If p = 0, the picture is that the spins at the surface are free to precess very much as spins in the bulk move. It is as if one sends a wave down a rope with a free end; the displacement is a maximum at the end of the rope. The low symmetry of the surface may lead to local magnetic fields which inhibit the spin motion, to render p f 0. Note that as p -+ m, the boundary condition becomes m,, = 0, corresponding to “pinning fields” so strong that spin precession in the surface is totally inhibited. In practice, p can vary from sample to sample, or in carefully prepared materials, may depend on the method of preparation. Recent experiments on high quality YIG films show pinning levels believed to be intrinsic (RADO [1978]).

Finally, we turn our attention to the influence of exchange on the Damon-Eshbach wave. Before we added exchange to our theory, the Damon-Eshbach wave had frequency that lies outside the bulk spin wave manifold. Now from eq. (5.24) we see, as remarked earlier, that one has bulk spin waves with frequency greater than -y(H,B)f. In fact, if we consider a Damon-Eshbach surface spin wave with wave vector Qa and frequency given by eq. (5.21), there is now necessarily a bulk spin wave with frequency identical to the surface wave, and with wave vector Q that has its projection onto the xz plane (the plane of the surface) identical to QII. If we call Q, the component of the wave vector of this bulk mode normal to the surface, then one of the ai that enters eq. (5.23) must be iQ,. That is, with exchange added to the problem, there is no longer a solution of the equations for which the fields are localized to the surface. With exchange added along with the new boundary condition in eq.

II ,# 51 ACOUSTICAL PHONONS AND SPIN WAVES 115

(5.27), the normal mode that was formerly the pure surface wave now contains a component that carries energy from the surface region into the bulk of the crystal; in the limit D+O, the amplitude of this bulk spin wave component vanishes, so the new solution degenerates into the former one.

A surface wave with the above properties is often referred to as a “leaky surface mode”. It is no longer an infinitely long-lived eigenmode of the system, but acquires a finite lifetime because of the leakage of energy into the bulk. From a mathematical point of view, if we begin with a real wave vector QII, and search for the eigen frequency w of the new mode, the solution to the boundary value problem yields a complex value of w.

In our view, a more satisfactory way of discussing the properties of such a leaky surface wave is to analyze response functions for spins near the surface. There are often ambiguities in the interpretation of the complex frequencies and/or wave vectors that emerge from attempts to characterize such modes with a pseudo-dispersion relation in the presence of such damping (for an example, see BENSON and MILLS [1970]). Recently a complete and quantitative study of the influence of exchange and spin pinning on the Damon-Eshbach wave has been presented by CAMLEY and MILLS [1978a]. These authors base their treatment on an analysis of the response of the surface region to an experimental probe with well-defined frequency w (real) and wave vector QII (real); information about the mode may be extracted from such an analysis in an unambiguous manner.

Earlier we pointed out the analogy between the present discussion of the magnetic response of the surface region, and the theory of spatial dispersion in optics. If we consider the surface between an isotropic dielectric and vacuum, then in frequency regimes where the dielectric constant E ( W ) becomes negative, it is by now well known that surface electromagnetic waves (surface polaritons) can propagate along the interface (see MILLS and BURSTEIN [1974] section X). If spatial dispersion effects are added to the theory, then the surface polariton becomes a “leaky surface wave”, in precise analogy with the Damon-Eshbach wave on the magnetic surface (MARADUDIN and MILLS [1973]).

At this point, after the rather lengthy introduction, we may direct our attention to the question of light scattering from surface spin waves. As we see from remarks earlier in the present subsection, we are interested in the Brillouin method here, and as in earlier portions of the article, we consider first backscattering of light from a semi-infinite material with

116 LIGHT SCA’ITERING SPECTRA OF SOLIDS [II. § 5

skin depth 6 small enough for the signal from the Damon-Eshbach wave and the bulk spin waves to be comparable. As we shall see, this condition is comfortably realized in the experiments discussed below.

We have seen that the Damon-Eshbach wave is a uni-directional mode, and can run only from left to right across the magnetization, if the magnetization lies parallel to the surface and one faces the crystal with M, directed upward. This has the following striking consequence. If one considers a scattering geometry where a surface spin wave is created with the proper sense of propagation in a Stokes process (frequency of scattered light less than incident light) then it is not possible for a surface wave feature to appear on the anti-Stokes side of the line, since the anti-Stokes wave has the reverse sense of propagation. Thus, we have a “one-sided” spectrum, with surface wave on one side of the laser line only, but bulk spin wave features on both sides. Consideration of a fixed scattering geometry with a surface wave Stokes process allowed shows that reversal of the external magnetic field, which also reverses the direction of M,, causes the surface wave to disappear from the Stokes side, to appear as an anti-Stokes feature.

This behavior was observed first by GRUNBERG and METAWE [1977], in their study of backscattering from spin waves on the surface of EuO. The experiment by Grunberg and Metawe is also the first study of the Damon-Eshbach mode under conditions that mimic the semi-infinite geometry that forms the basis of the theoretical discussion above. In light scattering, one excites spin waves with a probe that has wavelength very small compared to the sample size, while the converse is true in typical microwave resonance studies. Subsequent to the work of Grunberg and Metawe on EuO, SANDERCOCK and WETTLING [1978] have explored spin waves on the surface of Fe and Ni, while GRIMSDITCH, MALOZEMOFF and BRUNSCH [ 19791 have examined thin ferromagnetic films. We discuss the experiments in more detail below.

The theory of scattering from spin waves on the surface of semi-infinite ferromagnets has been developed by Camley and Mills, using methods very similar to those outlined in the previous sections of the present article. COTTAM [1978] has also discussed the theory within the framework of a calculation that does not include the influence of exchange. There is thus n o reason to present a detailed discussion of the method, though we sketch the principal features of the approach used in the work of Camley and Mills.

Again the light scatters from thermal fluctuations in the system because

11, (i 51 ACOUSTICAL PHONONS AND SPIN WAVES 117

the fluctuations modulate the dielectric tensor E,, to produce a piece SE,,(X, t ) that varies in space and time. Fleury and Loudon have discussed how this modulation occurs from a microscopic point of view. While their theory is applied explicitly only to anti-ferromagnetic materials, the mechanisms outlined by them may be applied to other ordered configurations of localized spins; it is also not difficult to see how one may extend their treatment to itinerant electron systems as well (FLEURY and LOUDON [ 19681).

We shall proceed in a phenomenological fashion by expanding the fluctuating component of the dielectric tensor SE,,(X, t ) in powers of the spin density S,(x, t ) . One has

~E,,(x, t ) = 1 K,,ASA(x, t ) + 1 GILYATSA (x, t)s, (x. t ) + . . . (5.28) A ATl

where K,,A and G,,,A,, are third and fourth rank tensors to be found upon considering the crystal symmetry combined with the symmetry properties of the spin density SA(x, t ) .

We wish to apply eq. (5.28) to scattering from thermal spin fluctuations (spin waves) well below the Curie temperature T,. Then the magnetization is almost fully aligned along 2, with ( S , (x, t ) ) closely approximated by nos, where no is the number of spins per unit volume. It is the fluctuations in Sx(x, t ) and Sy(x, t ) that modulate SE,~(X, t ) to first order in this circumstance. It is at first glance surprising that we include the second order terms in eq. (5.28). However, in general these second order terms contain contributions of the form S,S, or S,Sy and with the replacement S, = nos, these terms are then first order in S, and S,, respectively. It was first pointed out by WETTLING, COTTAM and SANDERCOCK [1975] that interference between the two terms in eq. (5.28) produces an asymmetry in the Stokeslanti-Stokes ratio for scattering from bulk spin waves in transparent media. This is a reflection of the fact that in the ordered state of a ferromagnet, time reversal is no longer a good symmetry operation, so one has a breakdown of the oft quoted detailed balance argument that gives the Stokeslanti-Stokes ratio as exp [h Ao/k,T], with A o the magnitude of the frequency shift suffered by the light.

Upon noting that the spin density Sh(x, t ) is not left invariant by a time reversal operation, one sees that the symmetry arguments used to deduce which elements of K,"A may be non-vanishing differ fundamentally from those used in the theory of Raman scattering from phonons [see our eq. (4.14)]. For example, in a cubic crystal with unit cell that possesses an


inversion center, the Raman tensor aFVA that couples light to the optical phonon displacement uA (x, t ) necessarily vanishes, while KwYA remains non-zero. In the theory of magnetic scattering phrased in the present phenomenological language, one may proceed by noting that S,(X, t ) has the same properties under rotation and time reversal as a magnetic field HA. Then LANDAU and LIFSHITZ [1960] have provided a complete and careful discussion of the influence of a magnetic field on the dielectric tensor of crystals. In general, KwVA is anti-symmetric under interchange of

the Levi-Civita tensor. The constant K is in general complex, but is real in spectral regions where the crystal is transparent. The non-zero elements of GFuAq are the same as the non-zero elements of the elastic coefficients that relate stress to strain, and only G,, = G,,,, = G,,,, = * - is of interest here.

Once we have the form of SE~,,(X, t ) as outlined above, the Green’s function method outlined in B 4 of the present article proves a convenient means of calculating the light scattering spectrum, to relate it ultimately to the spin correlation functions (SA(x, t’)Sv(x, t ) ) . The discussion proceeds precisely as before, and the final task is to calculate the spin correlation function. The method for doing this, including both the influence of exchange and damping of the spin motion, has been outlined by CAMLEY and MILLS [1978a]. These authors have carried out detailed calculations of the light scattering spectra for the two very different cases of scattering from EuO (here the skin depth is quite large, and exchange is weak), and then from Fe and Ni (small skin depth, exchange effects substantial). We turn to a description of these calculations, and also of the experimental data. We note that the theory has been extended to apply to thin films (CAMLEY, RAHMAN and MILLS [1980]), where standing spin waves (analogous to the guided wave polaritons of 0 3 and 5 4) may be excited in the light scattering experiments (GRIMSDITCH, MALOZEMOFF and BRUNSCH [ 19791).

At the time of this writing, there have been three principal experimental studies of Brillouin scattering from spin waves on the surface of ferromagnets, or in thin films. Each explores a distinctly different physical regime, so it will prove informative to examine them in turn.

As remarked earlier, the first observation of the Damon-Eshbach surface spin wave was reported by Grunberg and Metawe in their study of light scattering from the surface of the ferromagnetic crystal EuO, which has a Curie temperature of 77 K. This work confirms directly the strange uni-directional character of the wave through presence of the mode on

and v, while for a cubic crystal we may write KwVA = iKEpVA with

11, § 51 ACOUSTICAL PHONONS AND SPIN WAVES 119

ANTI - STOKES

1 MODE SURFACE

either the Stokes side of the line or the anti-Stokes, but never on both sides in the same spectrum. As remarked earlier, as far as we know, this is the first experimental study of the Damon-Eshbach wave in a geometry that approximates the simple semi-infinite limit where, as we have seen, a description of its properties follows from a simple theory.

The spectra reported by Grunberg and Metawe are influenced very weakly by exchange, for two reasons. First of all, the Curie temperature of EuO is not high, indicating the exchange interactions there are very much smaller than the effective exchange in Fe and Ni (discussed below), which have ordering temperatures an order of magnitude larger. Also, the optical skin depth in EuO is about 1500& again larger than that in Fe and Ni by about an order of magnitude. The maximum wave vector of a spin wave excited by light is on the order of 8 - ' , as we have seen in § 2, and in EuO the combination D8-2 is very small compared to yH,.

Thus, we have here an example of scattering from a material where exchange effects are very weak. We show in Fig. 5.7a a Rrillouin

STOKES

a) Surface Scattering From Eu 0:

STOKES 20T

ANTI - STOKES

SURFACE MODE

WAVE

-5 -3 -I I 3 5 (wo- w,) / y Ho

b) Surface Scattering From Fe:

Jk,, 1 , ,;k WAVES WAVES

-0 -6 -4 -2 0 2 4 6 0 (a+p~.J/yHo

Fig. 5.7. (a) A theoretical calculation of the spectrum of light inelastically back scattered from the surface of EuO. The incident and scattered light have wave vector which lies in the plane perpendicular to the magnetization, so the Damon-Eshbach wave propagation is perpendicular to the magnetization. (b) A theoretical calculation of the light inelastically back scattered from the surface of Fe. Again the incident and scattered light have wave

vectors which lie in the plane perpendicular to the magnetization.


spectrum calculated theoretically for parameters characteristic of EuO (CAMLEY and MILLS [1978a]). We see the Damon-Eshbach mode on only one side of the laser line, and two sharp bulk spin wave peaks at *ty(H,B)I for this geometry, which has both the incident and scattered photon wave vectors in the plane perpendicular to the magnetization. The Stokes/anti-Stokes ratio is very different from the value very close to unity predicted from detailed balancing arguments. This shows the importance of the quadratic terms in the expansion displayed in eq. (5.28), as discussed earlier. Unfortunately, Grunberg and Metawe explored only geometries where the wave vector of the Damon-Eshbach wave is perpendicular to the magnetization. We refer the reader to the literature for an explicit discussion of the parameters used to generate Fig. 5.7a, and for other examples of theoretical spectra.

Grunberg and Metawe find a rather strong temperature variation of the Damon-Eshbach wave frequency which seems larger than expected from the theory of the intrinsic temperature variation of spin wave frequencies, even when the enhanced degree of thermal disorder near the surface is incorporated into the theory (RAHMAN and MILLS [1979a]). The data is sketchy at present, and a complete, detailed study with emphasis on lower temperatures would be most welcome. Photoemission anomalies have been reported in studies of ELI chalcogenides, and this has led to conjectures that the ferromagnetic spin arrangement may be unstable at the surface (DEMANGEAT and MILLS [1976, 19771, CASTIEL [1976]). Much remains to be learned about the nature of magnetism at the surface of these materials, and light scattering may prove to be a most useful probe.

As remarked earlier, the ferromagnetic metals Fe and Ni take one into a rather different parameter regime. The optical skin depth 6 is now only 200& so the spin waves excited by the laser have a very much larger component of wave vector normal to the surface than was the case for EuO. Also, as we saw, the exchange constant is also very much larger. We can see the contrast between the case of EuO and Fe by comparing the theoretical spectrum displayed in Fig. 5.7b with that in Fig. 5.7a. The calculation, reported in an earlier paper (CAMLEY and MILLS [1978a]), was carried out for parameters characteristic of Fe, with Qll = k r ) - kf) again perpendicular to the magnetization. The strong influence of exchange may be seen by noting that the bulk spin wave portion of the spectrum is no longer a line feature near -y(HoB)f, but is now a broad, asymmetric feature reminiscent of the form displayed in Fig. 2.10. We see scattering from bulk spin waves with frequency very much larger than the maximum

11.5 51 ACOUSTICAL PHONONS AND SPIN WAVES 121

y(H,B)k in the theory which includes only dipole coupling. In fact, the Damon-Eshbach line clearly sits on top of the bulk spin wave "tail"; we see here quite explicitly the bulk spin waves degenerate with the surface mode that, in the presence of exchange, cause the Damon-Eshbach mode to become a "leaky" surface wave. The calculations on Fe are in excellent accord with the data of SANDERCOCK and WETTLING [1978] in that they reproduce the shape of spectrum extremely well, along with the relative intensity of the surface and bulk wave features. Evidently the quadratic terms in eq. (5.28) play a minor role in the transition metals.

Fig. 5.8 shows a series of spectra calculated as the values of QII= kf"- k f ) swing away from the perpendicular to the magnetization. The angle C$ is the angle between QII and the x axis. For the parameters used in the calculation, the critical angle &= 71.8". We see the Damon- Eshbach wave decrease in frequency, to come down to y(H,B)t at and disappear as a well defined elementary excitation beyond &. We have learned that Sandercock has recently obtained spectra for the case where QII is no longer perpendicular to M, (SANDERCOCK [1979]). The results are in remarkable agreement with new calculations carried out by CAMLEY [ 1979al.

ANGLE DEPENDENCE OF LIGHT SCATTERING SPECTRA FROM Fe.

Ho= 2 Kilogauss Ms= 1.46 Kilogauss +c= 71.8'

I (p = 72"

-7 -5 -3 -I I 3 5 7

(w,-w,) /yH,

Fig. 5.8. A series of theoretical spectra for back scattering of light from spin waves on the surface of Fe. The angle 4 is the angle between Ql,= kp'-k(i" and the f direction. We see that as 4 is varied toward the critical angle @ = @c = 71.8". the Damon-Eshbach wave sinks

down toward ?(HOB)$ to disappear as a well defined feature in the spectrum for @ > @=.

122 LIGHT SCAWERING SPECTRA OF SOLIDS [II, § 5

While experiment and theory seem in excellent agreement for the case of Fe, this is not the case for the data reported so far on Ni. The experimental spectra are in qualitative accord with the theory, but the positions of the features do not agree well with the values predicted through use of the bulk value of M,. It may be that the Ni samples used are inhomogeneous near the surface; early spectra taken on Fe also disagreed with the expectations, and this could be correlated with the presence of oxide on the surface. The Brillouin spectra are thus sensitive to the surface preparation techniques and can be used to monitor the magnetic response of the surface region to correlate it with sample fabrication method. We know of no other means of doing this, and the light scattering technique may become a powerful analytic tool if the systematics can be understood and exploited.

In P 5.1, we saw that the Rayleigh surface acoustic wave scatters light very strongly through the surface ripple mechanism. It would be interesting to study the regime where the Rayleigh wave dispersion relation crosses through the Damon-Eshbach frequency. The theory of interactions between Rayleigh waves and spin waves, including the Damon- Eshbach wave has been examined by several authors for the semi-infinite geometry (see references in the paper on this topic by CAMLEY and SCOTT [1978]), and recently a complete study of the coupling between various acoustic normal modes and spin waves in a thin ferromagnetic film on a substrate has appeared (CAMLEY [ 1979bl). If the parameters characteristic of Fe and Ni are examined, in Fe the cross over between the Rayleigh wave and the Damon-Eshbach mode is not accessible to light scattering, simply because the convenient laser sources do not allow a large enough value of QII to be achieved. In Ni, however, if the observed modes were in the expected position, the cross over should be accessible. The spin wave modes as observed do not allow access to the regime where strong interaction between the Rayleigh wave and spins occurs. It may also be that ferromagnetic alloys, for which M, may differ substantially from that in Fe and Ni, will allow the coupling to be studied.

We now turn to light scattering from spin waves in thin ferromagnetic films. Very much as in the case of the guided wave polaritons examined in P 3 and $4, as the thickness L of a film of ferromagnetic material is decreased, the wave vector q1 perpendicular to the surface becomes quantized at the values q:"'==n.rrlL. For L sufficiently small, individual standing spin waves can be resolved in the spectra. As we have seen in P 5.1, one may resolve standing acoustic waves in thin films also. Standing spin waves in films of amorphous ferromagnetic material have been


studied by Grimsditch, Malozemoff and co-workers. A recent publication discusses their data and its interpretation for thin films of thickness from 600 A to 1100 A. Several standing spin wave modes can be resolved in each spectrum, and through use of eq. (5.24) combined with the assumption Q = Q, = nrr/L, these authors extract values for the exchange constant D from the data.

Actually, there is need for a more complete treatment of the theory before reliable values of D may be obtained. We see from eq. (5.25) that in the limit L +a, when exchange is present, the normal modes of the system are a synthesis of three waves, each with its characteristic wave vector Q, normal to the surface. While eq. (5.25) is written in a notation that lends itself to the discussion of the Damon-Eshbach surface wave, a similar form applies to bulk spin waves as well. For the semi-infinite material, the description provided by Camley and Mills includes fully this complex character of the spin wave modes. (This is illustrated dramatically in Fig. 3 and Fig. 4 of CAMLEY and MILLS [1978a]. These figures show effects that arise from the interference between terms in eq. (5.25) with different ai, in a frequency regime where the ai are pure imaginary.) A theory of light scattering from spin waves in thin films which includes a full description of the normal modes is described elsewhere (CWEY, RAHMAN and MILLS [1980]).

There is an extensive literature on microwave excitation of standing spin waves in thin films (see, for example, Yu, TURK and WIGEN [1975]). Thus, it may prove useful to compare the two methods, since the light scattering data may be thought of as ferromagnetic resonance data, with a laser used as the exciting source.

There are several advantages the light scattering method offers. First of all, microwave studies are conventionally carried out in cavity resonators which operate at a single frequency. The spin wave modes must then be swept through the cavity resonance by varying an external magnetic field. Thus, in the end, one has data on the response of the film at only a single frequency, and the amount of information that can be extracted from the data is limited. In the light scattering method, at fixed magnetic field, one obtains information on the frequency response over a wide range of frequencies. The field may then be swept continuously over a wide range, so information on the frequency response of the film over a wide range may be obtained.

In a microwave experiment, the microwave skin depth is frequently large compared to the sample thickness. The rate at which energy is absorbed from the field is then proportional simply to Jkdym,(y), for the


case where the microwaves have magnetic field along the x direction. If Q, = n.rr/L, one sees that oscillator strengths of the modes fall off rapidly as n increases. This may be appreciated by examining the integral for m,(y) - cos (nn-y/L + +,,). I n the light scattering method, in the metallic films examined so far, the skin depth 6 is at most a few hundred Angstroms. The exciting field is, in effect, highly localized to one surface. Under this condition, the intensity of the nth spin wave mode varies slowly with n, until n becomes so large that n - L/6. This feature is evident in the data reported by Grimsditch, Malozemoff and Brunsch. Thus, at least in principle, by the light scattering method, one should be able to excite modes with rather large values of the quantum number n.

It is our understanding that to perform microwave resonance studies of thin films, one must be able to fabricate highly homogeneous films, simply because the microwaves illuminate the whole film. If the thickness varies from point to point, then since the combination D(n.rr/L)2 varies across the film, one cannot resolve individual modes unless the film is very homogeneous. In the light scattering technique, the laser beam may be focused down to a spot of small diameter, and modes can be resolved if t h e film is uniform over this region. It may thus be possible to use light scattering to study spin resonance in materials that are hard to fabricate in films of high enough quality for successful study in a microwave spectrometer.

On the other side of the sheet, the microwave method has resolution far superior to that offered by light scattering measurements with present day spectrometers. The signals are very weak in the light scattering method. The detection technique relies on photon counting, and the statistics can be poor when one has to extract a weak signal from the background. The five pass spectrometers used in the work reported to data are highly sophisticated instruments found presently in only a few laboratories. While the development of the instrument by Sandercock is a major event in Brillouin spectroscopy, we are not yet at the point where these measurements can be carried ou t easily, and with high resolution. Thus, as is usually the case, the light scattering method will prove complementary to the older techniques, which have a substantial number of virtues.

Q 6. Light Scattering as a Microscopic Probe of the Surface Region

The previous three sections examine the scattering of light by a variety of surface waves that may exist at the interface between a crystal and

11, 9; 61 LIGHT SCATTERING AS A MICROSCOPIC PROBE 125

vacuum, or between a film and a substrate upon which the film has been grown. In addition, we have seen that the contribution to the spectrum from bulk waves is affected profoundly by a surface on the bounding planes of a film. All of the waves discussed above may be described by long wavelength theories of the material response characteristics and as a result are macroscopic in nature. As yet, we have had rather little contact with the use of light scattering as a probe of material inhomogeneities near a surface or interface, and as a probe of truly microscopic phenomena there. Before we proceed, we hasten to add that our discussion has not been entirely free of contact with microscopic physics. The pinning parameter p which enters the boundary condition used in the spin wave theory of $5 .2 is in fact such a microscopic parameter which, in principle requires a microscopic theory for its description (see, for example, RADO [ 19781 or DEMANGEAT and MILLS [ 19761). But the basic modes which enter the theory remain macroscopic in nature.

At the moment, the use of light scattering techniques to probe the surface in a microscopic sense is a rapidly evolving and exciting area of current research. We shall review this area here, and much is likely to happen before the present article emerges in print. Also, while the thrust of this article has been to review theories of surface and size effects in light scattering spectra, we are now entering an area which is unde- veloped from a theoretical point of view. Thus, the focus of the present article is on experimental results rather than theory. We begin by discussing experiments which explore inhomogeneities on the length scale of a few hundred ingstroms, then new data which looks on the scale of a few tens of Angstroms, and finally we turn to the study of submonolayer amounts of impurity on the surface.

In doped semiconductors, the carrier concentration can vary substantially in the near vicinity of the surface, by virtue of pinning of t h e Fermi level there by a high density of electronic surface states of extrinsic or intrinsic origin. One may have either a depletion layer, with the carrier concentration at the surface essentially zero or possibly smaller than that in the bulk by some orders of magnitude, or an accumulation layer, where the carrier density increases dramatically. These layers have a thickness which is a few hundred Angstroms, and which decreases as the carrier concentration increases. In essence, the surface perturbation which produces the inhomogeneity is screened from the bulk more efficiently by the carriers, 2s the concentration increases.

Light scattering is a convenient method for probing the physics of the depletion or accumulation layer since, if the sample is illuminated with


light of frequency well beyond the absorption edge, the depletion or accumulation layer extends through a considerable fraction of the skin depth 6. Since 6 varies with frequency, back scattering experiments with different exciting frequencies provide one with a probe of variable depth.

It may be worthwhile to discuss briefly the long wavelength optical vibrations of doped semiconductors before we turn to the experiments.

In the absence of free carriers, in Ei 3 we discussed the long wavelength transverse excitations of an optically isotropic ionic crystal. In the present section, we are concerned only with the response at wave vectors Q such that cQ>>o. Then retardation is unimportant, and in the infinitely extended medium, we have a transverse optical mode with frequency o = oT0. The dielectric constant of the crystal has the form in eq. (3 .9 ,

so the TO phonon frequency coincides with a singularity in E ( w ) .

As remarked in 0 3, we have also a longitudinal mode of oscillation. If D = E(w)E , and V - D = E ( W ) V-E, then a longitudinal oscillation with V - E f 0 of frequency wL0 can occur at the frequency for which E ( W )

vanishes. As remarked in 0 3, the frequency wLo satisfies

WLO= (EsI&m)IWTO, (6.2)

where E,= ~ ,+4 r rne* /po$~ is the static dielectric constant. Since E , > E,, we have wLo > wT0. The LO frequency is higher than the

TO frequency because the longitudinal motion of the ionic lattice generates a macroscopic electric field (V - Ef 0) that stiffens the response of the lattice. For the TO phonon, EGO since V - E = O by symmetry and V x E = 0 when retardation effects are set aside.

Now suppose free carriers, assumed here to be electrons of mass ma for simplicity, are added to the material. The electrons will clearly tend to screen the macroscopic electric field of the LO phonon, and thus lower its frequency. The TO phonon, which fails to generate a macroscopic field when c Q >>w, is left unaffected. Thus the longitudinal phonon is affected strongly by the free carriers and, if the carrier density varies dramatically near the surface as described above, the light scattering signal from a depletion or accumulation layer may differ dramatically from that which originates from the bulk,

We may describe the screening effect on the LO phonon mathematically by adding to the right hand side of eq. (6.1) the free carrier

11, P 61 LIGHT SCATCERING AS A MICROSCOPIC PROBE 127

contribution -4rrn,e2/m*w2, where n, is the free carrier concentration. Thus,

which has zeros at two frequencies w, given by, with w; = 4rrn,e2/m*~, the electron plasma frequency,

Consider first the limit of very small carrier concentrations, where

W: = W t 0 (6.5a)

W? = 4rrn,e21m*~,. (6.5b)

The root W+ describes the LO phonon of the lattice, unaffected by free carriers. In the limit of small carrier concentration, the electrons are unable to screen out the macroscopic field of the LO phonon. The root w- is a collective motion of the electron gas, the plasma oscillation. The expression describes the plasma frequency of a gas of electrons embedded in matrix of dielectric constant E,.

wp<<wL0 or wT0. Then eq. (6.4) becomes

In the opposite limit wp >> oLo, the roots of eq. (6.4) become

w: = w2p= 4 ~ n , e ~ / m * ~ , (6.6a)

and

0: = W+O. (6.6b)

Eq. (6.6a) describes the plasma oscillation of electrons embedded in a medium with dielectric constant E,. The frequency of the motion is too high for the lattice to participate in screening the electric field set up by the electron charge density oscillation. Eq. (6.6b) describes a mode of longitudinal character, but with frequency wT0 rather than wL0. The electrons have now completely screened out the electric field generated by the lattice motion, and the frequency of the oscillation becomes wT0 rather than wLo.

The two modes in eq. (6.4) are referred to in the literature as the L, and the L- modes. A detailed study of these modes by Raman scattering has been described by MOORADIAN and McWnoR-rm [1969] who examine the long wavelength longitudinal normal modes in n-type GaAs, under conditions where the incident laser light penetrates deeply into the bulk.


We are now prepared to discuss spectra taken when the material is opaque to the incident radiation, and the carrier density varies strongly near the surface.

PINCZUK and BURSTEIN [1968, 19691 examined the spectrum of light back scattered from both p and n type InSb, under conditions where 6 = 500 A. In these samples, the carrier concentration was sufficiently high that the condition wp>>wLo was comfortably satisfied. Yet in the spectrum, they observed a clear signal from a mode with the frequency wLo, the unscreened bulk LO phonon frequency. In these samples, a depletion layer some few hundred ingstroms thick is present, and within the depletion layer, the local plasma frequency w:) is in fact small compared to wLo. Thus, the experiment observes a mode which exists only by virtue of the non-uniform electron density near the surface.

It is intriguing that the unscreened LO phonon scatters light in a forbidden geometry, if one presumes the LO mode in the depletion couples to light by means of the same coupling tensors used in the bulk of the material. Thus, not only is it that the mode exists only by virtue of the depletion layer, but the mechanism that couples the light to the mode is also a surface induced coupling. Pinczuk and Burstein note that within the depletion layer, there is necessarily a strong electric field; this must be so if the carriers are repelled from the surface. This strong electric field induces Raman scattering by breaking down the selection rules applicable in the bulk of the material. This mechanism, when examined from symmetry considerations, allows scattering in the forbidden geometry. Furthermore, by invoking it one may understand changes in the scattering intensity with changes in carrier concentration (the electric field increases in strength as the carrier concentration increases), and also with applied D.C. voltage.

Ushioda has examined n-type GaAs under conditions where wp >> wLo, to find a line at wTo and a line at wLo, under conditions where the skin depth and the thickness of the depletion layer are comparable. He also examined the spectrum of the same sample taken with 1.06 pm radiation. This frequency is well below the absorption edge, so the light penetrates deeply into the substrate before scattering. With the 1.06 pm radiation, only a feature at wTo was observed, as expected from eq. (6.6b). This study shows quite explicitly that the mode at wLo is confined to a thin layer near the crystal surface (USHIODA [1970]).

Murase and collaborators have also examined scattering from n-type GaAs, with Ar ion laser radiation with frequency well beyond the

11, D 61 LIGHT SCATERING AS A MICROSCOPIC PROBE 129

absorption edge (MURASE, KATAYAMA, ANDO and KAWAMURA [1974]). Under conditions used in this experiment, the wave vector transfer in the scattering event was sufficiently large that the bulk LO mode (the w-

mode of eq. (6.4) with wp>>wLo) was heavily damped (Landau damped) by virtue of its coupling to the electron gas. The spectrum thus consists of a relatively narrow line at the unscreened LO frequency, and the TO frequency. These features come from the depletion region, where the carrier density is small. In addition, the spectra show a broad, asymmetric background in the vicinity of wTo. This is the heavily Landau damped w-

mode excited by the light that penetrates deeply into the bulk. These spectra were analyzed quantitatively by Murase et al., to demonstrate that the Landau damping mechanism accounts for the position and width of the broad feature that originates in the bulk.

An interesting elaboration of this class of experiment is the study by Buchner et al. of n- and p-type InSb prepared with (111) surfaces. An ideal surface of this zinc blende structure will consist entirely of ions from one sublattice, either In or Sb. A film prepared with (111) surfaces will have one surface that consists entirely of In, and the second will be entirely Sb. This is required to maintain overall electrical neutrality of the structure. In practice, the surfaces are far from ideal, but one can realize an In rich and an Sb rich surface. BUCHNER, CHINC and BURSTEIN [1976] study scattering from LO phonons in the depletion layer that exists at each surface, to find dramatic differences in relative intensity of the modes, and also very different dependences of the scattering intensities on laser frequency, for frequencies in the near vicinity of the El electronic energy gap.

The body of data described above shows that light scattering can be a most useful probe of the physical properties of the depletion layer region of semiconductors. The modes observed in these experiments are localized to the region of interest, and at the same time the mechanism that couples the light to the modes seen in the spectra also differs substantially from that in the bulk. Rather little theoretical attention has been devoted to this general area, however. It would be extremely interesting to see a theory of optical lattice vibrations, when the lattice is immersed in an electron gas with highly non-uniform density. One description of lattice vibrations in this circumstance has been presented (MILLS [1971]), but this description is based on a method which sets aside the non-local character of the electron response. A full and quantitative theory must address this issue. We have here an important area for

130 LIGHT SCATlFRING SPECTRA OF SOLIDS [II, P 6

further study, if maximum information is to be extracted from light scattering data like that described above.

Very recent experiments probe surface inhomogeneities on the scale of a few tens of Angstroms. On this length scale, it is again semiconductor surfaces and interfaces that have been the primary focus of experimental activity in the months that preceded this writing. It is not the slowly varying charge density associated with a depletion or accumulation layer that is explored here, but rather true quantum mechanical bound states of electrons at interfaces.

The body of experimental and theoretical literature that forms the background to the recent light scattering studies is concerned with electrons trapped at the interface between silicon and SiO, in MOS devices. These are multilayer devices with silicon overlaid with 20 or 30A of oxide, followed by a metal film on top of the oxide. Such structures are widely used in contemporary semiconductor devices. In. such a system, imagine the silicon is p-type, and the metal is biased positively with respect to the silicon substrate. The holes are repelled from the surface; in essence the bands are bent in such a way that near the surface, the bottom of the conduction band is driven downward in energy toward the Fermi level. With sufficient voltage (a rather modest amount, actually), the bottom of the conduction band is driven through the Fermi level. One then has a thin layer of electrons trapped on the interface, free to move parallel to it, but with motion normal to the interface inhibited by a strong electric field that traps the carriers there. Behind the electrons is an insulating region, with the p-type region as a third element of the silicon substrate.

We thus have a two dimensional plasma, with electrons free to engage in translational motion parallel to the interface. A simple quantum mechanical description follows upon considering the electron motion to be governed by a Schrodinger equation with mass m*, with the electrons bound to the interface (an infinite potential barrier to first approximation) by a potential -eE,z, with z the spatial coordinate normal to the interface. In this description the electrons form two dimensional energy bands, with the wave function of the nth band given

h ( x ) = $,,(z) exp [ikll-xlll, (6.7)

&,(k,,)= -A, +hZk;f/2m*. (6.8)

and with energy levels of the form

Here kll is a two dimensional wave vector parallel to the interface.

11, § 61 LIGHT SCATTERING AS A MICROSCOPIC PROBE 131

Elementary uncertainty principle arguments show the wave function cCr,(z) extends a distance Az (h2/2rn*eE0)f into the material. If we take rn*=0.05rn and E,=106V/cm as typical parameters, then Az = 30 A. The separation between adjacent subbands, controlled by A,- A, for the two lowest, is typically the order of 30 meV; the voltage may be adjusted so only the lowest subband is occupied.

The two dimensional electron gas on MOS structures has been studied extensively in recent years, by a variety of solid state spectroscopic and transport techniques (see KOCH [1975] for a review, and recent experimental studies are described by KNESCHAUREK, KAMGAR and KOCH [1977], GORNIK and TSUI [1976] and WHEELER and GOLDBERG [1975]). Optical probes include photoconductivity studies, interband transitions (between the subbands described in eq. (6.8)) excited by infrared radiation and cyclotron resonance of the two dimensional electron plasma.

The interband energy of -30meV is well suited to Raman spectroscopy. Thus, light scattering should prove a convenient probe of these systems. However, a laser beam intersects very few electrons in a pass through an MOS device. Thus, the signal is expected to be very weak, unless the experiment can be carried out under extreme resonance Raman conditions. With presently available laser sources, and in an indirect gap material such as silicon, the advantage of signal enhancement through the resonance Raman effect is hard to realize.

However, recently two dimensional electron states very similar to those described above have been detected by light scattering in GaAs/Ga,Al,-,As heterostructures. These structures are alternating layers of each material, with each layer =200A thick. Donors are embedded in the Ga,Al,-,As, and the electrons collect in the GaAs (impurity-free) layers to form a high mobility electron gas. Strong electric fields at the interface trap carriers in states similar to those at the interface of the MOS structure. In a direct gap material such as GaAs, resonance Raman scattering becomes possible. Modern dye lasers allow one to tune in very close to the resonance, with the consequence that the cross section per electron becomes very large. One can detect the interband transitions between the electrons in the lowest subband and the first excited and possibly higher subbands. So far, these experiments have been carried out successfully in two laboratories. ABSTREITER and PLOOG [1979] report the observation of the interband transitions, in a polarization combination that allows spin flip scattering. At Bell Laboratories, inter subband transitions are observed in both a geometry that induces

132 LIGHT SCATIERING SPECTRA OF SOLIDS [II, B 6

spin flip transitions (scattered light polarization perpendicular to incident light polarization), and a geometry that selects spin flip scattering (scattered light polarized parallel to incident light) (PINCZUK, STORMEK, DINGLE, WORLOCK, WIEGMAN and GOSSORD [ 19791).

The physics of the electric field induced subband states in polar materials has interesting features absent in a material such as silicon. Most particularly, in the polar materials, one expects strong coupling between the electronic excitations and the LO phonons in the material. Also, the energy hoLo of an LO phonon quantum is not greatly different than the splitting A , - A , between the two lowest subbands. One sees this clearly from the Bell light scattering spectra, which show the GaAs LO phonons (more precisely, the w- mode of our earlier discussion) and the interband excitations in the same spectrum. Coupling between the LO phonons and the inter subband excitations should shift the transition frequency of these excitations. A simple theoretical description of this phenomenon has appeared recently (BURSTEIN, PINCZUK and MILLS [ 19791). Also, one should have intriguing polaron effects under these circumstances. It has been pointed out, in a different physical context, that electron-phonon coupling leads to off diagonal contributions to the electron proper self energy in such systems (RAHMAN and MILLS [ 19801). Phonon-induced interband mixing should be appreciable in the subbands associated with interfaces between polar materials. Further experimental studies of these systems by light scattering spectroscopy may provide very direct information on these questions.

We now step down in length scale from a few tens of Angstroms, to phenomena associated with the outermost atomic layer. In particular, the study of submonolayer coverages of adsorbed material on surfaces has been a sought after goal for some years now. While the vibrational motions of adsorbates can now be probed in many laboratories through use of high resolution electron spectroscopy, light scattering offers greatly improved resolution. Also, light scattering can be used to explore interfaces between solids and solutions or dense media superimposed on the solid, provided the material over the substrate is transparent. Such a measurement is quite impossible to carry out by means of electron spectroscopy.

However, as soon as one contemplates such an experiment, as in the case with electrons in the inversion layers of silicon based MOS devices, it becomes apparent that the signal should be very weak, if Raman cross sections for adsorbate molecules and their gas phase counterparts are

11, P 61 LIGHT SCATTERING AS A MICROSCOPIC PROBE 133

similar in magnitude. A number of early attempts to detect adsorbate spectra were unsuccessful.

It has been proposed that through use of a prism placed above the sample, with an air gap between the prism and a sample, the magnitude of the electric field in both the incident and scattered radiation may be enhanced at the sample surface very substantially. The prism must be placed approximately one wavelength above the substrate, and both the incident and scattered radiation must be directed to take advantage of certain resonances in the response of the prism/air gap/sample structure. An analysis of this geometry suggests that the Raman signal may be enhanced by two orders of magnitude over the value expected in the absence of the prism (CHEN, CHEN and BURSTEIN [1976]).

While such techniques can and have been employed successfully in surface Raman spectroscopy, in certain systems it has proved possible to detect strong Raman signals from submonolayer coverages of molecules on metal surfaces.

That this can happen became apparent in studies of pyridine adsorbed on Ag, in circumstances where the pyridine is contained in an electrolytic solution above the Ag, and some molecules bond to the substrate (FLEISHMANN, HENDRA and MCQUILLAN [ 19741, JEANMAIRE and VAN

DUYNE [1977]). The adsorbed molecules give rise to Raman lines shifted in frequency and therefore are distinct from those in solution. Despite the fact that the molecules in solution are very much more numerous than those adsorbed on the surface, the two sets of lines are quite comparable in intensity. Evidently the Raman cross section for a pyridine molecule adsorbed on the Ag surface is larger by several orders of magnitude when compared to those in solution. The precise value of the enhancement is a topic of current discussion but the estimates range from lo4 to lo6. Strong signals are observed also from CN on Ag (OTTO [1978], BILLMANN, KOVACS and OTTO [1980]), and from C O on Ag (WOOD and KLEIN [ 19791).

As of this time, despite a proliferation of theoretical papers that in total cover most conceivable explanations, the origin of the phenomenon does not appear to be well understood. The reason for this is that the experiments are carried out in circumstances where the surface and adsorbate geometry are ill defined. Many have an electrolytic solution above the surface, and observe the enhanced signal only after the surface has been subjected to one or more electro-chemical cycles. It is unclear precisely what the cycle does to the surface. The consensus is that the


surface is roughened by this procedure, but the nature of the resulting surface has not been characterized precisely. The work of Wood and Klein has been carried out in high vacuum, but they use evaporated films surely rough on a microscopic scale.

In some theoretical pictures, roughness enters in a crucial manner, and in others the enhancement mechanism is operative on a smooth surface. The data suggests strongly that resonance Raman scattering with a sharp electronic level as an intermediate state is not responsible for the enhancement. All the data shows the Raman intensity varies smoothly with frequency, though the precise power law is laboratory dependent at the moment. It seems as if the very large signals are observed for molecules adsorbed only on Ag substrates, and it will prove disappointing if the phenomenon is operative on only one or possibly a small number of substrates.

At this time, a number of laboratories are setting up to explore surface Raman spectroscopy and the field is developing rapidly. There is now n o doubt one may see vibrational spectra of adsorbates clearly in some cases, and it remains to see whether this can be done with sufficient flexibility for light scattering spectroscopy to take its place as an analytical tool of major importance in modern surface science.

§ 7. Concluding Remarks

Since the mid-sixties, the development of a variety of laser sources and highly sophisticated spectrometers has enabled Raman spectroscopy to become a tool of major importance in the arsenal of the solid state physics community.

Until the past few years, the light scattering method has been used primarily to study elementary excitations characteristic of the infinitely extended medium. We see from this article that in the past few years, a substantial body of theoretical and experimental literature has extended these methods to the study of waves which propagate on surfaces and along interfaces, to “standing wave” or “guided wave” modes of free standing films and films on substrates, and finally to the study of inhomogeneities or subtle features near the surface with spatial extent very much smaller than the wavelength of light. This is a new era of light scattering spectroscopy, and as in all new and rapidly evolving areas of research, the primary question that remains is whether we have in hand a truly

111 REFERENCES 135

substantive new tool with the flexibility to make a major impact after the first generation of new and very beautiful experiments come forth.

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282.

dam) Ch. 111.



I11

LIGHT SCATTERING SPECTROSCOPY OF SURFACE ELECTROMAGNETIC WAVES IN SOLIDS

BY

S. USHIODA

Department of Physics, University of California, Irvine, California 9271 7, U.S.A.

CONTENTS

PAGE

$ 1 . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . 141

$ 2. SURFACE POLARITONS IN DIFFERENT GEOMETRIES . . . . . . . . . . . . . . . . . . . . . . . 144

$ 3. RAMAN SCATTERING BY SURFACE POLARITONS . . 155

0 4. EXPERIMENTAL RESULTS. . . . . . . . . . . . . . . . 171

§ 5. EFFECTS OF SURFACE ROUGHNESS . . . . . . . . . . 190

§ 6. CONCLUDING REMARKS . . . . . . . . . . . . . . . . 202

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . 202

APPENDIX: DERIVATION OF THE DISPERSION RELA- TION FOR SURFACE POLARITONS AND GUIDED-WAVE POLARITONS IN A DOUBLE INTERFACE GEOMETRY . . 203

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . 208

§ 1. Introduction

In the past several years there has been increasing interest in the phenomena occurring near the surface of solids. Work on solid surfaces can be grouped into two general categories according to the degree of localization of the phenomena of interest. The first type of surface studies are concerned with physical and chemical phenomena that take place within distances of the order of the lattice constant at the surface. These studies focus on chemisorption, physisorption, surface reconstruction, catalysis, etc., on a microscopic or atomic scale. The second type of surface studies focus on macroscopic phenomena that arise from the presence of a surface but do not depend on the microscopic details of the surface atomic structure. The subject of this article belongs to the second category. The surface electromagnetic waves with which we are concerned here are macroscopic normal modes of solid surfaces that owe their existence to the presence of a surface or an interface, but their wavelengths are long enough so that the solid can be treated as a continuous dielectric medium. The macroscopic properties of a dielectric (or metallic) medium are completely specified by its dielectric function E ( k , w ) which contains the information about the optically active elementary excitations of a solid that couple to light. The coupled surface electromagnetic modes of optically active excitations of a solid and photons are called “surface polaritons”. The corresponding bulk normal modes of coupled photon and dipole excitations are “bulk polaritons”. (See, for example, BURSTEIN and DE MARTINI [1974].) The nature of surface polaritons is determined by the equation of motion for the excitation of the solid, Maxwell’s equations, and the boundary conditions at the surface. Corresponding to different dipole excitations of solids to which the surface electromagnetic (EM) wave is coupled, there are different surface polaritons; phonon-surface-polaritons, exciton-surface- polaritons, and surface-plasmons. As we shall see later, the theory of surface polaritons can be developed without specifying which dipole excitations are coupled to surface EM waves, because the specific nature

141

142 LIGHT SCAlTERING SPECTROSCOPY [III, li 1

of the dipole excitations of a solid is contained in the dielectric function E ( k , a). The subject of this article is light scattering studies of phonon- surface-polaritons whose energies lie in the far infrared.

Surface modes are distinguished from bulk modes by the fact that their amplitudes decay exponentially away from the surface in the direction normal to it. This means that the normal component of the wave-vector of a surface mode is purely imaginary and consequently it cannot propagate away from the surface; i.e., it is a non-radiative mode and propagates only along the surface with a real wave-vector. These modes are said to be non-radiative, because they do not couple linearly with bulk electromagnetic modes either inside or outside the surface. For a given frequency the wave-vector of surface polaritons parallel to the surface is either too large or too small for wave-vector matching with bulk electromagnetic waves approaching the surface either from inside or outside the medium. Thus they cannot be observed either by absorption or reflection measurements at the surface.

In order to observe surface electromagnetic waves experimentally, these waves must be coupled to an external probe by some means. Linear coupling between bulk electromagnetic waves and surface polaritons has been accomplished by two methods. One method is called the attenuated total reflection (ATR) method. In this method a coupler prism is placed a small distance away from the solid surface and a light beam is directed so that it is at an' angle for total reflection at the bottom of the prism. In this geometry an evanescent wave whose wave-vector component normal to the prism surface is purely imaginary is generated in the space between the prism and the solid surface. This evanescent wave can couple linearly with the surface polariton of the solid surface draining the energy from the incident bulk electromagnetic radiation. The coupling with surface polaritons is detected as a decrease in the reflected light intensity. The ATR method of surface polariton investigation has been reviewed in detail by OTTO [1974] and by BORSTEL, FALGE and OTTO [1974]. The interested reader should consult these references as well as a book by HARRICK [1967].

Another method of linear coupling between surface polaritons and bulk electromagnetic waves is to draw a grating on the sample surface. The wave-vector matching condition between the bulk radiation and surface polaritons is created by virtue of an additional periodicity due to the grating. This method has been used for measuring the absorption by

III,P 11 INTRODUCTION 143

surface plasmons in metals (see, for example, RITCHIE, ARAKAWA, COWAN and HAMM [ 19681 and references therein) and doped semiconductors (MARSCHALL, FISHER and QUEISSER [1971]).

Scattering experiments measure non-linear coupling between the pro- jectiles (photons, electrons or neutrons) and elementary excitations. So far no neutron scattering result has been reported for surface polaritons, but there are numerous examples of inelastic electron scattering from surface plasmons (RAETHER [1977]) and some from phonon surface polaritons (IBACH [1970, 19711).

In this article we will describe inelastic scattering of photons (Raman scattering) by surface polaritons in various geometries. Scattering experiments involve three particles (waves) in contrast to two in the case of absorption experiments. Thus, there is an additional degree of freedom in the interaction process and wave-vector (momentum) conservation can be achieved easily by adjusting the angle between the directions of the incident and scattered waves (particles).

The purpose of this article is to review the theoretical and experimental aspects of the Raman scattering studies of surface electromagnetic waves. All the experimental work reported so far on Raman scattering from surface electromagnetic waves has been done by the author’s group, and the main emphasis will be on the experimental aspects. However, theories of surface polaritons and their Raman scattering will be reviewed to the extent that they are necessary to understand the experimental results. In § 2 we will review the theory of surface polaritons in various sample geometries. The discussion will center on the mode dispersion and the field pattern of three kinds of surface modes; single interface modes, double interface modes and guided wave polaritons. In 0 3 we will see how these surface modes couple to the light and what information can be obtained by Raman scattering experiments. Since the scattering intensity is extremely weak due to the small scattering volume involved, various special spectroscopic techniques need be applied in order to observe the Raman spectrum. These experimental methods are described in § 3. § 4 deals with the details of the individual experimental results for single interface modes, double interface modes and guided wave polaritons. In li 5 we will discuss the effects of surface roughness on surface polaritons. A short review of theories on surface roughness effect is given, and the results are presented. § 6 contains concluding remarks and prospects for the future.

144 LIGHT SCAlTERTNG SPECTROSCOPY [ In , B 2

0 2. Surface Polaritons in Different Geometries

The purpose of this section is to review the theory of surface polaritons, in particular their dispersion and the field patterns, for different sample geometries which are relevant to the understanding of the experimental results described in D 4. We will define sample geometries, coordinates and symbol conventions so that we can refer back to this section in explaining specific examples in later sections. The theory will be reviewed in a schematic fashion only to the extent that is necessary for a systematic discussion of experimental observations. A convenient way to classify different modes of surface polaritons is by different sample geometries in which the modes are found. First we start with the simplest case where the surface polariton propagates at the flat surface of a semi-infinite dielectric (single interface mode). Next, we consider surface polaritons in a flat dielectric slab having two interfaces in contact with external media (double interface mode). A dielectric slab can support another kind of normal modes called guided wave polaritons; these modes are discussed at the end of this section.

As we have stated earlier, surface polaritons are macroscopic coupled normal modes of the electromagnetic radiation and the surface dipole excitations of the solid. Since surface polaritons are macroscopic normal modes, their behavior is determined completely once the form of the dielectric function ~ ( k , o) and the sample geometry are given. Thus, the following discussion of surface polaritons applies equally to phonon- surface-polaritons, surface plasmons and exciton-surface-polaritons when an appropriate dielectric function e(k , o) is provided. Here we are mainly concerned with phonon-surface-polaritons, and we will assume that the dielectric function of the sample medium is given by

where E ~ ) and E , are the static and optical dielectric constants, respectively, and wTo is the transverse optical phonon frequency. Eq. (2.1) is appropriate for an isotropic dielectric with a single branch of infrared active optical phonons.

Since optical phonons are essentially dispersionless near the center of the Brillouin zone, we can assume that the dielectric function is independent of the wave-vector in the cases of our present interest. When the

111, $21 SURFACE POLARITIONS IN DIFFERENT GEOMETRIES 145

dielectric function has no spacial dispersion, the usual boundary conditions of the electromagnetic theory are sufficient to determine the nature of surface polaritons completely. However, when the dielectric function has a wave-vector dependence, such as in the case of exciton-surface- polaritons, there are still unresolved questions about “additional boundary conditions” (ABC). Although the question of the ABC is a very interesting one, we need not concern ourselves with this problem here, because we assume a wave-vector independent form of the dielectric function given in eq. (2.1). A recent paper by Yu and EVANGELISTI [1979] and the references cited there provide more details on the latest developments on this problem.

2.1. SINGLE INTERFACE SURFACE POLARITONS

We consider surface polaritons propagating at the interface between two semi-infinite media with dielectric constants and E * = E ( W ) given by eq. (2.1). The geometry is illustrated in Fig. 2.1; medium 1 with frequency and wave-vector independent positive dielectric constant E~

fills the upper half-space (z >0) and medium 2 with the dielectric constant E* fills the lower half-space (2 (0).

The surface polaritons are the wave solutions of Maxwell’s equations in a charge and current free space that propagate along the interface at z = 0 with a real wave-vector kll and whose amplitude decays exponentially as z goes to *m. Since both media 1 and 2 are isotropic, there is no loss of generality by assuming k,, to be along the x-direction, i.e. kll = k,. For this case, only transverse magnetic (TM) solutions (E, = H, = H, = 0 in both media) exist, and their field amplitudes are given by the following

Fig. 2.1. Single interface geometry.

146 LIGHT SCATTERING SPECTROSCOPY [III, 9: 2

relations:

(2.2a)

(2.2b)

(2.3)

for z > 0 and similarly,

(2.4a)

(2.4b)

a 2 = (k: - E Z W ~ / C ~ ) ~ (2.5)

for z s 0 . Eb” and Er’ are the arbitrary amplitudes in media 1 and 2, respectively, and the relation between them is determined by the boundary conditions at z = 0. From the requirement that non-trivial solutions satisfying the boundary conditions exist, we arrive at the dispersion relation for surface polaritons:

c2k,2- E ~ E ~

W 2 E , + E 2 ’

The boundary conditions also require that

(2.7) Ehl) = 0

The requirement that the fields in eqs. (2.2a), (2.2b), (2.4a) and (2.4b) are truly localized at the interface between the two media is that both a1 and a2 be real and positive. We note that the conditions a1 = 0 and a2 = 0 correspond to the dispersion relations of the bulk polaritons in media 1 and 2, respectively, namely:

c2kZIw2 = E~ (2.8a) and

c2k:lw2 = E ~ . (2.8b)

111,s 21 SURFACE POLARITIONS IN DIFFERENT GEOMETRIES 147

Thus, the dispersion curve of the surface polaritons lies to the right of the bulk polariton dispersion curves of both media 1 and 2. Therefore, linear coupling between the bulk polaritons and the surface polaritons cannot take place, because of the mismatch in the wave-vector.

The discussion so far has not used any specific properties of the dielectric constants and E ~ . Thus what has been said so far applies generally to any surface polariton. Now let us examine the specific case of phonon-surface-polaritons by setting E~ = ~ ( w ) of eq. (2.1) and E , = 1 for vacuum outside the solid. Then eq. (2.6) becomes:

The dispersion curve of eq. (2.9) is plotted in Fig. 2.2. The curve appears between the TO and LO phonon frequencies where E ( w ) < 0. For large k , t he curve approaches the frequency determined by ~ ( w ) = - 1. Thus the asymptotic frequency for large wave-vectors is given by:

(2.10)

The lower end of the curve stops at the light line in vacuum (a ,=0) beyond which a , becomes imaginary.

From eq. (2.2a) and eq. (2.4a), we see that the polarization of the surface polariton is elliptical in the x-z plane (sagittal plane) and the sense of rotation of the electric field vectors in media 1 and 2 are

Fig. 2.2. Dispersion relation for single interface surface polaritons (SIM).

148 LIGHT SCAlTERlNG SPECTROSCOPY [III, § 2

Fig. 2.3. Electric field pattern for single interface modes. The electric field is elliptically polarized in the sagittal plane (x-z plane).

opposite. The electric field pattern is sketched in Fig. 2.3. The penetration depth of the fields in medium 2 is given by l/a,. A typical value of l / a , encountered for phonon-surface-polaritons is on the order of a few microns. Thus, the behavior of typical surface polaritons in the far infrared frequency region is determined entirely by the surface geometry and the dielectric constant within the first few microns at the surface of a crystal.

2.2. DOUBLE INTERFACE SURFACE POLARITONS

Now we consider surface polaritons that propagate in a double interface geometry illustrated in Fig. 2.4 along with the coordinate system that we will use. The three layers have an infinite extent in the x-y directions. Region 1 ( z > a ) and region 3 ( z < - a ) are semi-infinite in the z -

Fig. 2.4. Double interface geometry.

111, (i 21 SURFACE POLARITIONS IN DlFFEREhT GEOMETRIES 149

direction and are filled by media of dielectric constants and E ~ ,

respectively. Region 2 is filled with a dielectric medium of thickness d = 2a and dielectric constant E,. For simplicity we assume that all three media are isotropic and look for surface EM waves that propagate with a real wave-vector k, in the x-direction parallel to the surface. In region 1 and region 3 we look for solutions of Maxwell's equations having the following form for the electric fields:

Region 1 t ) = Ahe-a,zei(kxx-wO (2.11)

Region 3 ~ : 3 ( ~ , t ) = ~ , ~ a ~ z ~ i ( k ~ x - m O (2.12)

where A, and C, are the A-Cartesian components of the electric field amplitude; and a I and a3 are the decay constants of the amplitudes in regions 1 and 3, respectively. In order for the modes to be localized near the interface, we must require that a, and a3 be real and positive. Since region 2 is bounded at z = +a, we can have both positive and negative exponential dependence of the amplitude on z. Thus, we assume the form of electric field given by:

Ep)(x, t ) = [B;e%'+ B;e-%z]ei(kxx-wt) (2.13)

for region 2. BL and BC are the A-Cartesian components of the field amplitudes to be determined by Maxwell's equations and the boundary conditions at z = f a . The z-component of the wave-vector a, in region 2 can be either real or imaginary as we find out later. Since the derivation of dispersion relations and the relationship among the field amplitudes A,, Bt and C, is quite involved in terms of algebraic complexity, we leave this task to the Appendix, and here we quote only the main results.

As we see in the Appendix, the solutions for this geometry separate into two groups, one with the polarization in the x-z plane (TM modes) and the other polarized along the y-direction (TE modes). The TM modes can have solutions for both a2 real and imaginary. The TM solutions with real a2 are the surface polaritons, and imaginary a, corresponds to guided wave polaritons. The TE solutions allow only imaginary a,; thus, surface polaritons are TM polarized, while there are guided wave polaritons with both TM and TE polarizations.

The dispersion relation for surface polaritons with a, real is given by (A.41):

150 LIGHT SCATTERING SPECTROSCOPY [III, § 2

where a , , az and a3 are given by

ai = (kX- & i o * / c * ) ~ ; i = 1 , 2 , 3 (2.15)

and we require all a’s to be real and positive for surface polaritons. A similar dispersion relation to eq. (2.14) was derived earlier by MILLS and MARADUD~N [1973]. The physical implications of eq. (2.14) are not apparent as it stands, but when the solutions ( k , , w ) for this implicit equation are plotted. the nature of the modes become clear. From eq. (2.14) we obtain two branches labeled UM (upper modes) and LM (lower modes) in Fig. 2.5 where we assumed F ~ > E ~ . As before a1 = 0 and a3 = 0 lines are the bulk polariton dispersion curves in media 1 and 3, respectively. The LM starts at the intersection of a3 = 0 line and w = wTo line and asymptotically approaches the frequency determined by E~ =_

E ( W ) = - c3 for large k , values. The UM “formally” starts at the intersection of w = wTo line and a 1 = 0 line, and goes to the asymptotic frequency determined by the condition that E ~ = E ( ~ ) = - E ~ . However, the dashed part of the UM curve to the left of a3 = 0 l ine does not correspond to true surface polaritons, because a3 is imaginary in this region; that is, the UM is ‘‘leaky’’ into medium 3. The origin of these surface polaritons, UM and LM, becomes clear when we consider the limit of large thickness ( d + a)

for medium 2. In this limit the second term in eq. (2.14) vanishes because

€2 = 0

€2: -€ ,

€2 = - €3

€ 2 = + O D

Fig. 2.5. Upper mode (UM) and lower mode (LM) of surface polaritons in the double interface geometry. 1-2 mode and 2-3 mode are obtained in the limit of a large slab

thickness d.

111, 5 21 SURFACE POLAREIONS IN DIFFERENT GEOMETRIES

of the exp { - 2a2d} factor, and eq. (2.14) becomes:

+ a I E 2 / a 2 & 1 ) ( 1 + a 2 & 3 / a 3 & 2 ) = 0

which is equivalent to:

c 2 k : -- -

w 2 E l + E 2

and

151

(2.16)

(2.17)

(2.18)

Thus, we see that eq. (2.14) reduces to two separate single interface surface polariton dispersion relations at the two interfaces between media 1 and 2, and media 2 and 3. The two dispersion curves corresponding to eq. (2.17) and eq. (2.18) are shown in Fig. 2.5 labeled by “1-2 mode” and “2-3 mode,” respectively. The 2-3 mode has lower frequencies than the 1-2 mode because of our choice, E ~ > E ~ . Now the physical origin of the UM and LM can be easily understood. When the thickness d of medium 2 is sufficiently small so that the second term in eq. (2.14) is not negligible, the two single interface modes 1-2 and 2-3 interact and repel each other to form the UM and the LM. When d is large the interaction between the surface polaritons at the opposite faces of medium 2 di- minishes and single interface surface polaritons appropriate to the two separate interfaces appear.

Next we consider a special case where E~ = E~ = E , (symmetric slab geometry). Physically, this geometry is realized when a dielectric slab of thickness d and dielectric constant E~ is placed in a medium of dielectric constant F , . Then eq. (2.14) becomes:

(2.19) E ~ / E , = -(a2/a,) tanh a2u

and

E J E , = -(a2/a,) cotanh a2u. (2.20)

a, = c Y ~ = ( ~ $ - P , w ~ / c ~ ) ~ . (2.21)

Here we have used the fact that d = 2u and the definition:

Fig. 2.6 illustrates the dispersion relations given by eq. (2.19) (UM) and eq. (2.20) (LM) for the case when F, = 1. This situation corresponds to a dielectric slab placed in vacuum. Both UM and LM start at the intersection of w = w T o and the vacuum light line, a,=0 or w = c k , and

152 LIGHT SCATTERING SPECTROSCOPY [III, 5 2

w = ck,(al = a 3 = O ) !

/ /I// Fig. 2.6. Surface polaritons in a symmetric slab geometry. UM and LM converge to SIM

(single interface mode) in the limit of a large slab thickness d.

approaches the asymptotic frequency given by eq. (2.10). When the thickness of the slab is large (aZd + a), U M and LM become degenerate and the mode is identical to the single interface surface polaritons of medium 2 in vacuum (labeled SIM in Fig. 2.6) whose dispersion relation is given by eq. (2.9). These symmetric slab modes were described in detail by KLIEWER and FUCHS [1966].

2.3. GUIDED WAVE POLARITONS (GWP)

As we see in the Appendix, guided wave polaritons with imaginary a2 can have either TM or TE polarization. For TM polarization the dispersion relation is given by eq. (2.14) (or (A.41)). The guided wave polaritons with the TE polarization (y-polarized) has the dispersion relation given by (A.39):

Since both eq. (2.14) and eq. (2.22) are too complex to see the underlying physics, we will consider a simplified case of a dielectric slab in vacuum. Then E,,, = E , = E~ = 1 and a , = a3 3 a,,,, where a, is given by eq. (2.21). In discussing experimental results, we will encounter only this simple situation. Since we are interested in guided wave polaritons for

111, § 21 SURFACE POLARITIONS IN DIFFERENT GEOMETRIES 153

which a2 is imaginary, we set a2 = ip2 as is done in the Appendix. Then eq. (2.14) and eq. (2.22) for TM and TE guided wave polaritons, respectively, reduce to:

e2= 4 ~ ) = ( P 2 / q , , ) tan P2a (2.23a)

and E* = & ( W ) = - (P2/a,) cot P2a (2.23 b)

(2.24a)

(2.24b)

for the TE modes. These results for a symmetric slab geometry were earlier derived by KLIEWER and FUCHS [1966]. A graphic study of eq. (2.23) and eq. (2.24) shows that the allowed values of P2 for eq. (2.23a) and eq. (2.24a) are given by

p2 = ( m + 6)(71./2a) (2.25)

where m is an even integer and 6 is a small positive number less than unity. For eq. (2.23b) and eq. (2.24b), the allowed values of P2 are given by eq. (2.25) with odd integers m. Physically, guided wave polaritons have a standing wave pattern across the thickness of the slab as shown in Fig. 2.7. If the boundaries at z = *a were made of a metal, there would be

Z = - a 2 = + a

Fig. 2.7. Standing wave amplitude pattern for GWP for m = 1, 2 and 3 .

154 LIGHT SCATERING SPECTROSCOPY [III, 0 2

exactly an integral number of half-waves across the thickness and 6 in eq. (2.25) would be zero. However, at dielectric boundaries the electric field does not vanish and as a consequence p2 does not correspond to the reciprocal of an exact integral number of half-wavelengths. 6 approaches zero as rn increases.

The dispersion curves for the TM and T E guided wave polaritons obtained from eq. (2.23a, b) and eq. (2.24a, b) are plotted in Fig. 2.8a and Fig. 2.8b, respectively. Both TM modes and TE modes appear between t h e light line (a,,, = 0) and the bulk polariton dispersion curve (az = 0). The apparent difference between the TM and TE guided wave polariton dispersion is that the TE mode dispersion curve departs from the light line sharply while the TM dispersion curve moves along the light line before it deviates from it.

Our attention in the above discussion was focused on GWP’s that appear in the far infrared frequencies where the dielectric function is strongly dispersive due to the presence of optical phonons. The GWP’s in the far infrared do not seem to be very familiar waves, but the GWP’s in the visible have been studied in detail by workers in the field of integrated

kT 2 k T

Fig. 2.8. Dispersion curves of GWP. (a) Transverse magnetic modes (TM); (b) Transverse electric modes (TE). The wave-vector is scaled by k , = W,/C.

111,831 RAMAN SCA’ITERING BY SURFACE POLAFUTONS 155

optics and dielectric wave-guides. A recent review article by TIEN [1977] and books by MARCUSE [1974] and by KAPANY and BURKE [1972] treat GWP’s from the viewpoint of optical wave-guide modes. The interested reader should consult these references to learn about the visible counterparts of the GWP’s that we have discussed.

Throughout our discussion in the present section we have assumed that all the dielectric constants involved are real; in particular, we assumed the dielectric function of medium 2 to have the form given by eq. (2.1) without any imaginary part. Both E , and E~ can be assumed to be real in the cases we encounter experimentally in this article. However, for medium 2 the form given by Fig. 2.1 is an over-simplification; the optical phonon has a finite life-time and E ( W ) has an appreciable imaginary part. Thus, in real crystals all the surface EM modes have a finite damping reflecting the lifetime of the optical phonon (SCHOENWALD, BURSTEIN and ELSON [1973] and MCMULLEN [1975]) to which they are coupled. Addi- tional damping peculiar to surface modes occurs when the surface is not perfectly smooth. The surface roughness induced damping and frequency shift of surface polaritons will be discussed in 0 5.

This concludes a short review of the theory of surface electromagnetic waves. The materials presented here are sufficient for understanding the experimental results described later in this article. However, there are many interesting related effects that we could not cover in the limited scope of this article. For instance, when a semiconductor is doped with free carriers, many new phenomena can be expected due to the coupling of LO phonons to plasmons near the surface (TAJIMA and USHIODA [1978]). Furthermore when a magnetic field is applied to such samples, various magnetoplasma effects modified by the presence of a surface take place (WALLIS, BRION, BURSTEIN and HARTSTEIN [ 19741). More related references can be found in the above papers; also a very extensive list is found in a review article by O n 0 [1976].

0 3. Raman Scattering by Surface Polaritons

In this section we will summarize the essential features of light scattering spectroscopy that are important in understanding the experiments described in later sections. For the reader who wishes to learn further details of the theory and experimental methods of light scattering spectroscopy we list several general references on this subject (LOUDON


[1964], HAYES and LOUDON [1978], BERKE and PECORA [1976], CARDONA [ 19751). For historical reasons inelastic scattering of light by optical phonons is referred to as Raman scattering and scattering by acoustic phonons is called Brillouin scattering. Since we are interested in light scattering by surface polaritons which are coupled modes of optical phonons and infrared photons, we will use the words Raman scattering.

3.1. BASIC CONCEPTS OF RAMAN SCATTERING

In Raman scattering experiments, one sends a beam of monochromatic light into a sample and analyzes the energy of the scattered light emerging at some scattering angle 0 from the direction of the incident light as shown in Fig. 3.1. Let the wave-vector and the frequency of the incident and the scattered light be (ki, mi) and (ks, oJ, respectively. If the incident photons of momentum Aki and energy hi interact with quanta of elementary excitations in the sample (e.g. phonons, polaritons), the spectrum of the scattered photons (ks, w,) contains inelastic components whose energies and momenta are shifted from those of the incident photons. The momentum (Ak,) and the energy (Aw,) of the elementary excitations that scatter the photon can be found from the conservation laws:

k, = ki * k , (3.1)

W s = O i f O , . (3 .2)

The plus sign in eqs. (3.1) and (3.2) corresponds to annihilation of the elementary excitations that are present in the sample due to thermal excitations. In this process the scattered photon has a higher frequency (shorter wavelength) than the incident photon; this process is called anti-Stokes scattering. This process vanishes when the sample temperature is zero because there is no thermally excited quantum in the sample.

SAMPLE INCIDENT BEAM

SCATTERED LIGHT

Fig. 3.1, Conceptual scheme for light scattering.

111,531 RAMAN SCA'TTERING BY SURFACE POLARlTONS 157

FR EOU E NCY

Fig. 3.2. Conceptual spectrum of the scattered light.

The minus sign corresponds to creation of an elementary excitation and consequent loss of energy for the incident photon; this is Stokes scattering. Thus, when there is a single species of elementary excitations that scatter light (Raman active) the spectrum of the scattered light appears as shown schematically in Fig. 3.2. Thus, knowing (ki, mi) and finding the scattered light spectrum (ks, us), one can determine the dispersion (ke, we)

of the elementary excitations that scatter light. One important fact that we should note in considering light scattering is

that the wave-vectors involved have a very small magnitude compared to the size of the Brillouin zone of ordinary crystals. The wave-vector k Z s = T / U (a = lattice constant) at the zone boundary is on the order of 10' cm-I, while the wave-vector of light is on the order of lo5 cm-' in the visible. Thus, the elementary excitations that can be investigated by light scattering exist near the center of the Brillouin zone. (This is true only for first order scattering processes indicated by eqs. (3.1) and (3 .2 ) . ) As we have seen in the preceding section, the most dispersive part of the surface polariton branches lie close to the light line near the Brillouin zone center. Thus, light scattering is a convenient method of studying the dispersion of surface polaritons.

Earlier we remarked that surface polaritons cannot be observed by optical absorption or reflection, because surface polaritons do not couple directly with bulk EM modes either inside or outside a dielectric medium for kinematic reasons; i.e. both wave-vector and energy cannot be conserved simultaneously in a linear coupling process involving one photon and one surface polariton. Raman scattering is a non-linear process that involves three waves, the incident and scattered waves and an elementary

158 LIGHT SCATTERING SPECTROSCOPY [III, 5 3

excitation of the sample. Thus, we have energy and wave-vector conservation among the three waves, (ki, wi), (ks, 0,) and (ke, we) allowing one to probe a region of (ke, w,)-space by varying the angle 8 between the incident and scattered waves. In order to appreciate what happens to the energy-wave-vector conservation conditions in the Raman scattering process by surface polaritons, we must digress a little and review how the conservation rules, eq. (3.1) and eq. (3.2), arose for the usual Raman scattering by bulk excitations in a large sample.

Light scattering takes place because of the fluctuations ~ E ( x , t) in the dielectric constant at position x and time t caused by excitations of the sample in the path of the incident light. Then it can be shown that the intensity of light scattered with wave-vector k, and frequency w, is given by the space-time Fourier transform of the thermally averaged correlation function of the dielectric fluctuation SE(X, t ) (see, for example, BERNE and PECORA [1976])

x exp {i(ki- k,).(x-x’))(ij~*(x’, 0 ) 6 ~ ( r , t ) ) (3.3)

where I(, is the intensity of the incident light; R is the distance between the sample and the detector; and n is the refractive index of the sample. The angular bracket ( a 1 .) indicates a thermally averaged time correlation function. The space integration limit V is the scattering volume which is essentially infinite for a transparent bulk sample. Now if the fluctuation SE(X, t ) is caused by a modulation of the dielectric constant due to a well-defined elementary excitation with wave-vector k, and frequency we,

the correlation function has the form:

(~E*(x ’ , 0) ~ E ( x , t )> = B[1 f n ( 4 l exp {-irke - (x- x’) - wet]) (3.4)

for Stokes scattering, where B is a constant determined by the kind of elementary excitation, and n ( w ) is the Bose-Einstein factor for scattering by bosons. When we combine eq. (3.3) and eq. (3.4), we have:

m

Z(ks, w,) = A[1+ n(w) l dt exp {i(ws- wi + wJt) I, X JV d’x lv d3x’ exp{i(k,-k,-k,).(x-x’)} (3.5)

where A is a weakly frequency dependent constant. The first integral

111, § 31 RAMAN SCATTERING BY SURFACE POI-ARITONS 159

gives a delta function 6(w,- wi + we) which results in energy conservation, eq. (3.2). In arriving at eq. (3.4) an assumption was made that the sample is large enough so that the correlation function depends on x and x’ only through the difference (x-x’). If we invoke the same assumption in eq. ( 3 . 9 , the space integral gives a factor proportional to the scattering volume V and a delta-function 8(ki - k,- ke). The wave-vector conservation condition, eq. (3. I ) , results from this delta-function.

We have tacitly assumed, as is done in the usual theory of Raman scattering, that all the wave-vectors and the frequencies are real. How- ever, if the sample is absorbing, ki and k, become complex. Also all excitations have a finite life-time resulting in a complex we with a small imaginary part corresponding to damping. In the case of Raman scattering from surface polaritons in an opaque crystal, both ki and k, are complex and the component of k,=kSp normal to the surface (icu,) is imaginary. Moreover, the scattering volume V is determined either by the sample thickness or by the skin-depth of the incident and scattered light, i.e. by the imaginary part of ki and k,. Thus, we see immediately that the first integral of eq. (3.5) corresponding to energy conservation remains intact, but the second integral is subject to modifications for Raman scattering by surface polaritons.

Very crudely speaking, the correlation function for surface polaritons corresponding to eq. (3.4) takes the form:

(~E*(x’ , 0) 8~ (x, t ) ) [I + n(osp)] exp {-i[ksp,l - (xll- xi) - wSptl) x

(3.6)

where k,,, is the wave-vector of surface polaritons parallel to the surface, and XII and xi are two dimensional position vectors in the plane of the surface. In addition the space integration limit of eq. (3.5) in the z- direction becomes either --oo to 0 for the geometry of Fig. 2.1 or - a to + a for the geometry of Fig. 2.4. Then in a very rough sense, instead of having delta-functions in the expression for the scattering intensity, we have the form:

(3.7)

Thus, the energy is conserved, and the components of the wave-vectors parallel to the surface, kill, ksIl and ksPII, are conserved, but the components perpendicular to the surface, kil, k,, and ia, are not conserved exactly.

160 LIGHT SCA7TERING SPECTROSCOPY [III, § 3

Indeed, it is the last factor of eq. (3.7) that makes it possible to observe surface polariton scattering in a near-forward geometry with thin films, but impossible to observe backward scattering from the surface of semi- infinite crystals, as we shall see in the next section.

Now consider an experimental geometry shown in Fig. 3.3 where the incident light is directed normal to the surface of the sample. In this geometry kill = 0 and the wave-vector of the surface polariton that scatter light by angle 0 is given by:

kSPll = f ksll (3.8)

(3.9)

In arriving at eq. (3.9) we used the fact that wsp<< wi and consequently oi= us, and thus ki 2- k,. So one can trace the dispersion curve of surface polaritons by Raman scattering, if one measures the frequency shift wsp as a function of the scattering angle 0. The degree of wave-vector conservation in the direction normal to the surface is measured by the third factor in eq. (3.7) and will affect the intensity of the scattered light. In anticipation for the experimental results, we note that if ki, and k,, point in opposite directions (backward scattering), the scattering intensity is smaller than in the forward scattering, because (ki,- ksL) is larger for the backward scattering than for the forward scattering. In actual experimental situations a*<< k, , = k,,, and the forward scattering is lo2- lo3 times more intense than the backward scattering.

In the preceding paragraphs and in what follows we assume that the modulation in dielectric constant &(x, t ) due to elementary excitations is a known quantity. Once this quantity is known the Raman scattering intensity can be deduced by classical electrodynamics straightforwardly even though algebraic complexities are involved due to particular geomet-

and the magnitude of kSPIl is:

ksN1 = k , sin 8 = ki sin 0.

SAMPLE

Fig. 3.3. Experimental geometry for surface polariton scattering with the incident beam normal to the sample surface.

I I I , § 31 RAMAN SCATCERING BY SURFACE POLARITONS 161

ries. What determines the magnitude of &(x, t ) is the interaction strength between a particular elementary excitation and light in the medium. In the case of optical phonons and polaritons, it has been shown that the dominant interaction between the incident photon and the phonons or polaritons occurs via the electrons (LOUDON [ 19631). Thus, the factors that determine the magnitude of the modulation of the dielectric constant and consequently the Raman scattering intensity are the electron-photon interaction and the electron-phonon (polariton) interaction. The phenomenological coefficients upYs and bpYs which are introduced in the following discussion summarize both of these interactions via the atomic displacement and via the electric field of the polariton, respectively.

When the incident or scattered photon energy is close to electronic transition energies of the crystal, the electron-photon interaction is large and the Raman scattering intensity is enhanced; this phenomenon is called resonance Raman effect or enhancement. (See, for example, MAR- TIN and FALICOV [1975], RICHTER [1976] and BENDOW [1978] for recent reviews.) When the energy of the incident light coincides with one of the electronic transition energies there is strong absorption of the light. Thus, strong electron-photon interaction implies strong absorption as well, and therefore resonance enhancement of Raman intensities is necessarily accompanied by strong absorption. Thus, it may seem that the gain in the Raman scattering intensity is cancelled by the loss due to absorption of the incident and/or scattered light. It is found, however, that under certain experimental conditions the gain due to resonance enhanced Raman scattering can be arranged to outweigh the loss due to absorption. Then one can gain Raman intensity under resonance conditions where the incident photon energy is strongly absorbed.

In Raman spectroscopy of surface EM waves, the scattering volume is always very small compared to the bulk mode scattering situations. Thus, it becomes essential to take advantage of resonance enhancement in order just to observe Raman scattering by surface polaritons. All the experiments described here were done under resonance conditions to gain the intensity.

3.2. RAMAN SCATTERING INTENSITY AND SELECTION RULE FOR SURFACE POLARITONS

As we have suggested by crude arguments in the preceding section, the theory of Raman scattering from surface polaritons must take a careful

162 LIGHT SCATI'ERING SPECTROSCOPY [III, 5 3

account of the effects of the finite and small scattering volume resulting from the sample geometry and also the absorption of the incident and scattered light. These effects not only relax the wave-vector conservation condition normal to the surface, but also severely affect the intensity of scattering. Comprehensive theories addressing these effects have been developed recently. In this section we will review these theories and summarize essential predictions that are relevant in understanding the experimental results.

Early theories of Raman scattering by surface polaritons had been presented by RUPPIN and ENGLMAN [1969] and by AGRANOVICH and GINZBURC [1972] before the experimental data became available (EVANS, USHIODA and MCMULLEN [1973]). These papers pointed out basic differences between Raman scattering by surface modes and bulk modes, but did not focus on the factors that are important in determining the feasibility of experimental observation. The first theory that focused upon and accounted for the factors that are responsible for successful observation of surface polaritons by Raman scattering was reported by CHEN, BURSTEIN and MILLS [1975]. They examined the phase factor (i.e. a factor essentially equivalent to the third factor in eq. (3.7)) that enters the expression for the Raman scattering intensity by surface polaritons, and clearly explained the reasons why early attempts at observing surface polaritons by backward scattering from opaque crystal surfaces had been unsuccessful. They attributed the success of the forward scattering experiment from a thin film (EVANS, USHIODA and MCMULLEN [1973]) to the asymmetry in the scattering intensity for forward and backward directions; they showed that the intensity in the forward direction is stronger by a factor of 102-103 than in the backward direction, making it impossible to observe backward scattering. Also they could correctly account for the observed relative intensity of the surface polariton scattering to that of the bulk LO and TO phonon scattering. The detailed derivation of the theory just mentioned above as well as other interesting aspects of Raman scattering from thin films was later presented by MILLS, CHEN and BURSTEIN [1976]. Later in this section, we will outline their theory and exhibit certain important results in it.

At about the same time the above mentioned theory was developed, NKOMA and LOUDON [1975] took a different but equivalent approach and formulated a parallel theory of Raman scattering by surface polaritons. Their theory uses the response function approach which was described earlier by BARKER and LOUDON [1972]. NKOMA and LOUDON [1975]

111, § 31 RAMAN SCATERMG BY SURFACE POLARlTONS 163

consider backward Raman scattering from surface polaritons in a semi- infinite opaque crystal. A later work by NKOMA [1975] formulated a theory for forward scattering from thin films, reaching the same conclusion that was obtained by MILLS, CHEN and BURSTEIN [1976]. The theories by the two groups agree completely in their main conclusions, although they used different degrees of approximations at various stages of calculation. The equivalence of the electromagnetic Green’s function approach by Mills et al. and the response function approach by NKOMA and LOUDON [1975] was demonstrated earlier by MILLS and BURSTEIN [1974] for the case of scattering from bulk polaritons. In what follows we will outline the theoretical approach taken by MILLS, CHEN and BURSTEIN [1976] and attempt to extract the essential physics involved; the reader should, however, keep in mind the full equivalence of the theory by Nkoma and Loudon.

Mills, Chen and Burstein consider the geometry of Fig. 2.4 and write the pv-component of the dielectric tensor of this entire structure in the form:

E&, t ) = S f i ” E ( Z ) + SE,”(X, t ) (3.10)

where E ( Z ) takes on the appropriate values in vacuum above, the sample and the substrate. ~ E , , ( x , t ) is the fluctuation created by surface polaritons. Then the electric field E‘”’(x, t ) of the scattered light can be written in terms of the Green’s function Dwu(x, x‘; t - t’) of Maxwell’s equations with the appropriate boundary conditions for the present geometry shown in Fig. 2.4. The scattered power (P‘”’) measured at position x is given by:

(3.11)

Since the integral for E‘”’(x, t ) contains SE,,(X, t ) as the source term as well as the Green’s function D,,(x, x’; t - t ’ ) , eq. (3.11) contains a correlation function ( ~ E , , ( x , t)8&,&’, 0)) which is related to the spectral density function of surface polaritons. To proceed, one needs the explicit form of this correlation function. Mills et al. assume the form:

(3.12) S

where 8,(x, t ) and Us(x, t ) are the electric field and atomic displacement amplitudes of surface polaritons, respectively; boys is the electro-optic coefficient and aPvs is the atomic displacement susceptibility defined


earlier by BURSTEIN, USHIODA and PINCZUK [1968] in describing the Raman scattering intensity of bulk polaritons. By using the relation between the displacement and the accompanying polarization due to surface polaritons, the two terms in eq. (3.12) can be combined to write:

where

with O = q-0, and e* is the transverse effective charge of the optical phonon and N is the number of unit cells per unit volume. Now the correlation function of tkBY(x, t ) can be transformed into the correlation function of the electric field fluctuations of surface polaritons:

At this point Mills et al. show that this correlation function can be related to the original Green’s function D,,(x, x’; t - t ’ ) which describes the scattered light field. This remarkable fact is essentially based on the following physics. D,,(x, x’; t - t ’ ) satisfies Maxwell’s equations with the boundary conditions for the three-layered dielectric structure of Fig. 2.4; that is D,,,(x, x’; t - t ’ ) describes the normal modes of the electromagnetic radiation coupled to the excitations of the dielectrics through E~,,(x, t ) of eq. (3.10). Now, surface polaritons are also normal modes of the electromagnetic waves of the same layered structure satisfying Maxwell’s equations and the same boundary conditions that D+,(x, x’; t - t’) satisfies. Thus, information on the field 8,(x, t ) of the surface polaritons is also contained within D,,(x, x’; t - t’), although the frequency associated with 8,(x, t ) is in the far infrared and the scattered light is in the visible. Using this fact Mills et al. construct the spectral density function corresponding to the correlation function of 8,(x, t ) from the two-dimensional Fourier transform of the Green’s function D,,,(x, x’; t - t’). The general expression for the scattered power per unit frequency range per unit solid angle dZP(‘)/dwsdO that they obtain is quite complex and contains the scattering from bulk LO and TO phonons, surface polaritons as well as guided wave polaritons. Under certain simplifying assumptions and ap-

HI,§ 31 RAMAN SCATERING BY SURFACE POLARITONS 165

proximations, they arrive at a tractable form that we exhibit below:

x exp [ - 2 (i + t ) d ] (i) Im [S, , (k!? - k 9 1 (3.14)

for forward scattering with the incident light wave-vector k, directed normal to the interface from region 3 to region 2 in Fig. 2.4. For backward scattering with the incident beam directed in the same way, they obtain a similar equation:

x exp [ - 2 (i + : ) d ] (i) Im [S, , ( - k y ) - k f))] (3.15)

where we have altered the notations to conform to ours defined in § 2. Now we need to define some of the symbols appearing in eq. (3.14) and eq. (3.15): b(o) is one of the elements of 6fip,,(fl) defined by eq. (3.13) and Tg3') are the transmission coefficients of the incident (superscript i) and scattered (s) light from medium i to j (subscripts). li and 1, are the attenuation lengths of the incident and scattered light, respectively. S,, is essentially the spectral density function for the film (medium 2). The portion of S,, arising from surface polaritons can be isolated when the damping is small, and can be written as:

(3.16)

where new symbols a , and b, were introduced as a shorthand for:

(3.17)

(3.18)

in the present notation, and QII= kill- ksIl. d(QI1, 0) is equal to the left hand side of eq. (2.14); thus, the poles of d-'(QIl, a ) trace the dispersion curve of surface polaritons in the (QII,f2) plane. The square of the absolute value in eq. (3.16) corresponds to the last factor of eq. (3.7). For

166 LIGHT SCAIITERING SPEmROSCOPY [III, § 3

forward scattering Ak is given by:

A k = ki, - k,, (3.19)

and

A k = - k i l - k , , (3.20)

for backward scattering. Thus, IAk( is much smaller for forward scattering than for backward scattering. When ( A k J is large the integrand in eq. (3.16) oscillates rapidly, producing a small value for the integral, while it becomes large for small values of IAkl corresponding to forward scattering. Thus, the forward scattering intensity given by eq. (3.14) is much greater than the backward scattering intensity given by eq. (3.15). This is one of the main conclusions of the theory by Mills et al. The numerical calculation of the spectrum of the film geometry by these authors using the general expression (not exhibited here) can reproduce the complete spectrum of surface polaritons as well as bulk LO and TO phonons. Later SUBBASWAMY and MILLS [1978] found that the general expression contains the spectrum of guided wave polaritons also as it should, and provided a motivation for a Raman scattering study of guided wave polaritons by VALDEZ, MATTEI and USHIODA [1978].

Now we need to comment on the polarization selection rules for surface polaritons. Because surface polaritions are macroscopic waves, we could write the modulation of the dielectric constant ikpy(x, t ) in terms of the electric and atomic displacement fields, 8,(x, t ) and Us(x, t), of the surface polariton in the form given in eq. (3.12). This equation involves the same coefficients, bpT8 and uPys, which enter the corresponding expression for bulk polar excitations (BURSTEIN, USHIODA and PINCZUK [1968]). Thus, the same Raman tensor (LOUDON [1963]) used for bulk polar modes can be applied to surface polaritons, as long as the polarization of the surface polariton is properly taken into account.

3.3. EXPERIMENTAL METHOD

As we have seen in the first part of this section, the basic conceptual aspects of Raman scattering experiments are quite straightforward. The apparatus consists of a source of monochromatic light, optics to define the direction and the polarization of the incident and scattered light, a spectrometer that analyzes the spectrum of the scattered light, and a detection

111, 8 31 RAMAN SCATTERING BY SURFACE POLARITONS 167

system that detects and records the spectrum of the scattered light. Fig. 3.4 shows a conceptual scheme of the Raman scattering spectroscopy apparatus.

The monochromatic light sources used these days are almost always lasers of various wavelengths. Lasers are ideal for Raman spectroscopy because of the high power and monochromaticity of the output beam as well as the high degree of beam collimation. Availability of high monochromatic power from lasers has been the single dominant cause for the rapid progress of light scattering spectroscopy in the last two decades. Since the spectral linewidths of most of the visible lasers are so much narrower ( 5 0.1 cm-') than the linewidth of elementary excitations of interest, a laser beam can be considered completely monochromatic in Raman scattering applications. This is not the case for Brillouin spectroscopy, because both the frequency shift and the linewidth of acoustic phonons are much smaller, and single moding of a laser is usually required for Brillouin scattering applications (SANDERCOCK [ 19751).

The optics before and after the sample consists of polarizers, filters and lenses. The output beam of gas and dye lasers are usually linearly polarized by Brewster windows inside the resonator cavity, and any desired polarization can be obtained by a combination of a half-wave plate, a quarter-wave plate and a linear polarizing element such as a glan-air prism. A lens is used to concentrate the incident beam into a small scattering volume whose image can be focused onto the entrance slit of a spectrometer. The scattered light is collected by the input optics

SAMPLE

INPUT OPTICS POLARIZER

I I DETECTOR

l-lJ RECORDER

Fig. 3.4. Conceptual scheme of a Raman scattering experiment.

168 LIGHT SCATTERING SPECTROSCOPY [HI, P 3

of the spectrometer that includes a collecting lens, a polarization analyzer and often another lens. One of the important factors in designing input optics is matching of the f-numbers of this system and the spectrometer. A matching f-number will ensure that all the light that passes through the input optics will just fill the grating surface resulting in the optimum efficiency and resolution of the spectrometer. The design of the input optics that was used in the present work is depicted in Fig. 3.5. The space between lens 2 and lens 3 where the input beam is parallel and narrow is used to place a polarization analyzer or sometimes a dove prism to rotate the sample image. The solid angle of acceptance and the distance between the sample and the first collecting lens L, can be varied according to need by changing lenses L, and L,. The f-number of the entire input optics is fixed by the beam diameter & and the focal length of L3 (f = f3/&) and this ratio is matched to that of the spectrometer. The f-number for our spectrometer is approximately 7.

When one wishes to investigate Raman scattering from dispersionless (k independent) excitations, one wants to maximize the solid angle of acceptance because the spectrum does not depend on the scattering angle. On the other hand, when one wishes to investigate the dispersion of a mode (such as surface polaritons) one must define the scattering angle by restricting the acceptance solid angle of the input optics so that the scattering wave-vector is well defined. If the dispersion of the mode does not depend on the azimuthal angle about the incident beam direction ki, then an annular aperture in front of the collection lens L, defines the scattering angle by sin 0 = r / l where r is the mean radius of the annular opening and 1 is the distance between the sample and the aperture. The

LASER

SAMPLE ANNULAR A PPE R TURE

SPECTROMETER ENTRANCE

SLIT

Fig. 3.5. Input optics used in surface polariton experiments.

111, 8 31 RAMAN SCATTERING BY SURFACE POLARlTONS 169

uncertainty A 0 in the scattering angle 0 is determined by the width b of the opening according to A 0 = b/21.

The heart of Raman spectroscopy is the spectrometer and the detector system which must meet many difficult requirements. The main challenge arises from the extremely low scattering cross-section for Raman scattering. To appreciate the necessary conditions for Raman measurements, it is useful to consider the following typical experimental parameters. In surface polariton scattering experiments the number of scattered photons detected at the peak is usually 1 - 10 photon counts/sec when the incident laser power is on the order of a half watt ( - loL8 photons/sec). Moreover, the spectral distance between the very strong elastic scattering peak (due to surface and bulk crystal imperfections) is often less than lOOA ( - 400 cm-' at 5000 A). Thus, one must be able to detect a very weak peak in the scattered light very close to a high elastic peak which in the above example is on the order of lo6- 10' photons/sec. Therefore the spectrometer must have a high resolution, a high stray light rejection ratio and a high throughput (low loss). Several models of commercially .produced spectrometers which satisfy these requirements are available. Most of them use a Czerny-Turner type double grating system. The rejection ratio of stray light and the resolution are improved over a single grating spectrometer by use of two gratings in tandem.

For the work described here we used a matched pair of ruled gratings with 600 grooves/mm. The efficiency of these gratings is maximum at 1 km; therefore, we use the second order diffraction for the analysis of spectra around 5000A where argon ion laser lines lie. When these gratings are used in our spectrometer (focal length 75 cm), the linear dispersion is 118,lmm of the entrance and exit slits for second order diffraction. Thus, the slit width required to achieve 1 cm-' resolution around 5000 8, is 50 k. The middle slit between the two halves prevents stray light from entering the second stage of the spectrometer. This slit is usually kept open at twice the widths of the entrance and exit slits.

For detection of the analyzed spectrum, photomultipliers have become a standard device, and the light intensity is measured by counting pulses in the photoelectric current in low signal applications. Some of the photomultipliers are especially suited for photon counting applications characterized by the low dark count requirement. By cooling and appropriate adjustment of the high voltage, the amplifier gain and the pulse height discriminator level, one can achieve a dark count level of 1-2 counts/sec. Accomplishment of a low dark count level is essential when

170 LIGHT SCATTERING SPECTROSCOPY [III, P 3

one wishes to detect a few counts/sec of signal. In order to minimize the dark count, we keep a photomultiplier cooled to - 40°C in the dark with a high voltage applied at all times.

The photon pulses emerging from the anode of a photomultiplier is amplified, converted into pulses of a standard width (1 Fsec), and finally counted either by an analog rate-meter or a digital scalar. We use a standard pulse counting circuitry developed for nuclear counting applications. The heights of true photon pulses are distributed around a higher average than the dark count pulses. We use this difference in pulse height distributions for discriminating against the dark counts.

Since the scattering process is a random process, the signal-to-noise ratio is N/& where N is the total photon counts proportional to the data accumulation time t. Thus, the SIN ratio improves with accumulation time as h. In order to be able to count photons for an extended period with minimum effort, we built a spectrometer system interfaced to a minicomputer which controls the wavelength setting of the spectrometer and records the accumulated photon counts. This system is depicted schematically in Fig. 3.6. A PDP-11 minicomputer is programmed to bring the spectrometer to any desired wavelength setting by controlling a stepping motor and measure the scattered light intensity for a pre-set

CRT PDP-II

FLPY DISC TERM

I CLK INTERFACE HRLY I

XI, WPM REC AMP

Fig. 3.6. Schematic diagram of a computer-controlled Raman spectroscopy system. FLPY DISC: floppy disc system, TTY: teletype, CLK: clock, RLY: bank of relays, MOT: stepping motor, ENC: shaft encoder, PM: photomultiplier, AMP: amplifier. Arrows indicate the flow

of control and signal pulses.

111, B 41 EXPERIMENTAL RESULTS 171

length of accumulation time. The photon count is accumulated in a scalar and this digital information is stored in the computer memory as the spectrometer is stepped along a sequence of points in the wavelength. The computer also controls a bank of relays which are used to insert, take out or rotate polarizers, mirrors and filters. This capability is used in taking a difference spectrum for different polarizations and scattering angles as we see later. If the laser output is not stable over a long term, it is necessary to measure the entire spectrum in a relatively short time. To do this and yet accumulate enough signal, this system can be scanned rapidly over a desired spectral range and repeat the scans many times.

Although no sophisticated computer controlled Raman spectroscopy system was available when our system was built initially, nowadays there are several commercial systems with similar capabilities. The details of our system in its initial stage are described in our earlier report (USHIODA, VALDEZ, WARD and EVANS [1974]).

Q 4. Experimental Results

There have been many reports of the observation of surface polaritons (optical phonon-coupled ones as well as plasmon-coupled ones) by the ATR method as we indicated in D 1. However, so far it appears that the author’s group is the only one reporting light scattering measurements of surface polaritons. Before the first observation of surface polaritons by Raman scattering by EVANS, USHIODA and MCMULLEN [1973], many attempts had been made to detect surface polariton Raman scattering from the surfaces of opaque bulk samples in a backward scattering geometry. We tried such experiments ourselves. In samples opaque to the incident light, the scattering takes place within the skin depth of the incident radiation which is on the order of a few thousand ingstrom in the case of GaAs with the visible incident light. Bulk LO and TO phonon scattering in such a backward scattering geometry can be observed routinely. Since surface polariton amplitudes are localized in the same skin depth region of the sample, it seemed reasonable to expect Raman scattering from surface polaritons with an intensity comparable to that of the bulk modes. Opaque samples appeared better than transparent samples for two reasons: First, the scattering takes place only near the surface and makes the relative intensities between the bulk mode scattering and the surface mode scattering comparable; second, opaqueness implies the


absorption of the incident light which is necessary for resonance enhancement of the scattering process. In retrospect, however, the crucial condition for observation turned out to be the combination of a forward scattering geometry and resonance enhancement as we saw in 5 3.2. These conditions were met in experiments by Evans et al.

Chronologically, the UM of the double interface geometry of § 2.2 was the first surface polariton branch to be observed by light scattering; however, in this review we will follow the logical order given in § 2 and describe the experimental results starting with the single interface modes (SIM) .

4. I . SINGLE INTERFACE MODES (SIM) AND THE SELECTION RULE

If one wants to combine the intensity gain due to resonance enhancement which is accompanied by strong absorption and the gain due to a near-forward scattering geometry, a semi-infinite sample cannot be used. To keep both of these conditions and still have the SIM, thin single crystal samples of GaP were prepared. The thickness d was chosen so that exp (- 2 ~ x 4 ) << 1 to satisfy eq. (2.16). Then although the sample is in the form of a thin slab, the surface modes at the opposite faces do not couple, and the resulting modes are identical to the SIM of a single interface geometry as we saw in § 2.2.

GaP was chosen so that the strong resonance enhancement (SCOTT, DAMEN, LEI= and SILFVAST [1969], BELL, TYTE and CARDONA [1973], WEINSTEIN and CARDONA [1973]) with the blue and green light of an argon ion laser could be used to gain the scattering intensity. This resonance is due to the coupling of light at the Eo transition, and the complex index of refraction in this energy range is well known (SERAPHIN and BENNEP [1967]). Also the properties of the optical phonons in GaP have been studied by many methods, so the infrared and Raman scattering properties of the bulk phonons (HOBDEN and RUSSELL [1964]) and bulk polaritons (HENRY and HOPFIELD [1965]) are very well characterized in this material. In fact the surface polaritons (SIM) in GaP were the first phonon-surface-polaritons whose dispersion was measured by the ATR method (MARSCHALL and FISCHER [1972]). Thus, the objectives of the experiment were to see the surface polaritons by light scattering and to determine the selection rule in a single crystal with known orientations (VALDEZ and USHIODA [1977]).

The samples were prepared from oriented rectangular slabs of dimen-

111, § 41 EXPERIMENTAL RESULTS 173

sions - 2.5 x 2.5 x 1 mm3 cut out from a single crystal bode. These slabs were then ground and polished, using various polishing techniques (VAL- DEZ [1978]), to a final thickness of 15 to 25 microns. The final polish was done on a silk cloth using 0.05 micron alumina powder. Raman measurements described here were made on a sample having a thickness of 20 microns and large (111) faces. The directions of [ilO] and [Ti21 crystal axes were known in the (111) plane from natural cleavage of the (il0) face as well as from the Laue X-ray pattern. We will use the notation for the three crystal axes: P' = [lll], 3' = [ 1701 and 2' = [iT2]. Note that 2' is normal to the surface in this sample. The bulk Raman tensors for this axis system is given by:

2 0 0

0 - 1 R ( X ' ) = - $ [ : -1 01;

R(z')=-$[-Y 0 0 -1 0 1

0 -Jz

R(y')=- - 1 -; -{I; (4.1)

in LOUDON'S notation [ 19641. These Raman tensors correspond to 6fib,s(0) of Q 3 with

Rfiy(xg) = 6p,s(o). (4.2)

Relevant crystal parameters of GaP are summarized in Table 4.1. Raman scattering measurements were made at room temperature with

the 5145 A line of an argon ion laser as the exciting source. The incident beam of about 400mW was directed normal to the sample surface and the scattered light was collected using the input optics illustrated in Fig. 3.5. The scattering angle in the range of I" to 4" was obtained by adjusting the lens to sample distance 1 and the radius r of the annular

TABLE 4.1 Crystal parameters of GaP

Crystal symmetry Td (zincblende) No. of atoms/unit cell 2 Phonon symmetry F2 WL.0 403.0 cm-'

W . r o 367.3 cm-l

E , 9.09 I

Ell 10.94

4, 2.78 eV

174 LIGHT SCATTERING SPECTROSCOPY [III, (i 4

opening of the masks. The value of 1 was between 27 and 29 cm, while r ranged between 4.5 and 16mm with the width of the opening b fixed at 2 mm. The scattered light intensity was measured by photon counting with the integration time per position of 60 to 240 seconds. The step size between data points was typically 0.5 cm-l, and the instrumental resolution was set at -0.25 cm-'.

The surface polariton spectra for several scattering angle 8 are depicted in Fig. 4.1. These spectra were taken with the incident beam linearly polarized and the scattered light was not analyzed; i.e., all the scattered light emerging at angle 8 from the incident beam direction was collected without a polarization analyzer. Because annular apertures were used, k,, direction in the (1 11) plane for surface polaritons are not specified. When we needed to know the direction of kll in the (1 11) plane, we used a small segment of the annular aperture with openings at specific positions. The large peak at 403 cm-' in Fig. 4.1 is the bulk LO phonon scattering and the small feature pointed by an arrow is the surface polariton peak. The small peaks were identified as due to surface polaritons by plotting the peak position as a function of the scattering angle 8 and comparing it with the theoretical prediction of the surface polariton dispersion given by eq. (2.9) or eqs. (2.10) and (2.20). The comparison is illustrated in Fig. 4.2.

8 = 3.6*

e = 3.00

8 = 2 . 5 O

- e 2.00

e 1.60

1 1 I I I I 390 395 400 405 410

Frequency Shift (cm-')

Fig. 4.1. Spectra of SIM surface polaritons in a GaP slab of thickness 20 krn as a function of the scattering angle 0. The large peak is the bulk LO phonon peak.

111, J 41 EXPERIMENTAL. RESULTS 175

Fig.

. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - d - -__ - - 400 - - -

I - 5 398- - % - 0

al g 396- 2 LL -

394 - I I

10 2 O 30 40 Scatter ing Angle

4.2. Measured dispersion of the SIM surface polaritons (dots) in GaP and the theoretical dispersion curve (solid curve).

The solid curve is the plot of eq. (2.9) and the circles are the plot of the peak positions obtained from Fig. 4.1. From this comparison, we see that the small peak on the low frequency side of the LO phonon peak is indeed due to surface polaritons. The angles I9 indicated in Figs. 4.1 and 4.2 are the scattering angles measured outside the sample which are related to the wave-vector k,, of surface polaritons by:

(4.3) IkllJ = (2n/Ai) sin I9

where A i is the wavelength of the exciting light (5145 A in this experiment). In calculating the theoretical dispersion curve of Fig. 4.2, the values of wLo, wTo, E() and E, given in Table 4.1 were used. We see that the observed dispersion of surface polaritons follows the theoretical prediction quite closely, although there is a noticeable systematic deviation of the data points downward from the theoretical curve. We will comment on this point in § 5.

Now we need an explanation on how the peak positions (arrows) in Fig. 4.1 were determined. As one can see in Fig. 4.1, it is difficult to specify where the exact position of the surface polariton peak is located especially at small scattering angles where the peak broadens. The method we used in pinpointing the peak position can be understood most easily by referring to Fig. 4.3. First, the analytic form of the spectrum near the LO phonon and the surface polariton peaks was assumed to be:

(4.4)

where DLo(w) and D,,(w) are given by the damped harmonic oscillator

F(w) = u + bw + DLo(o) + Dsp(w)

176 LIGHT SCATIERING SPECTROSCOPY [III, (i 4

. 0..

390 400 410 FREQUENCY (cm-')

Fig. 4.3. Detail of the digital spectrum fitting procedure. (a) Original data points and the calculated background curve. (b) Curve fitting for the surface polariton peak after the

background is subtracted.

function:

Dso,sp(w) = ALO,SP[(~2,P,S, - w2Y + w2Eo.sPl-1. (4.5)

ALO,SP is the peak intensity of the LO phonon (surface polariton-SP) and the rLo,sp is the damping of these modes. In eq. (4.4) a + bw is the background contribution. First eq. (4.4) without the surface polariton contribution Dsp(w) was fitted to the digital data spectrum ignoring the fit in the region of the surface polariton peak (the region between tall bars in Fig. 4.3b). In this process A,,, rL0 and wLo are fixed. Next this background spectrum indicated by the solid curve in Fig. 4.3a was subtracted from the data spectrum producing the pure surface polariton contribution shown in Fig. 4.3b. Then this surface polariton contribution is fitted by Dsp(w). This process fixes the parameters wsp. Asp and rsp which are the desired quantities. The solid curve of Fig. 4.1 and Fig. 4.3a is the background contribution mentioned above. The arrows in Fig. 4.1 point to the surface polariton peak positions wsp determined by the above procedure. All the digital data fitting was performed visually using the APL graphics software developed by A. Bork at the University of California, Irvine. The optimum set of the fitting parameters was deter-

EXPERIMENTAL RESULTS 177 111, § 41

mined by using the X2-test (BEVINGTON [1969]) and minimizing the x2 with the grid least square method.

Having determined that the small peaks of Fig. 4.1 are due to surface polaritons of the single interface geometry (SIM), the next step was to determine the polarization selection rule. Polarization experiments were performed at 8 = 2.5” (outside crystal) where the surface polariton peak is clearly resolved from the LO phonon peak. Again the incident beam was directed normal to the surface so that kill = 0, and the scattered light with kSl, along 3’ and 2’ was analyzed. When k,,, is parallel to q’(2’) the surface polariton wave-vector kll is also parallel to Q ’ ( i ’ ) . The polarization configurations that were used can be expressed in the standard notation used in light scattering: x’(y’y’)x’ and x’(y’z’)x’. The first and the last symbols in this notation signify the incident and scattered light directions, respectively, and the symbols in the parentheses are the polarization directions of the incident (first) and scattered (second) light. By combining the two directions of kll and the two polarization combinations, the Raman scattering intensities for four configurations were obtained.

Fig: 4.4 depicts the spectra for two of these four configurations. Evidently Raman scattering from surface polaritons is allowed for spectrum (a) in Fig. 4.4, while it is not allowed for spectrum (b). In spectrum

1 1 I I I 390 395 4 0 0 405

Frequency Shift (cm-’)

Fig. 4.4. Polarization selection rule: (a) k,, )I 2’. x‘(y’, y’jx’; (b) k,, (1 i’, x’(y’, z’jx’

178 LIGHT SCAlTERING SPECTROSCOPY [III, P 4

(b) the configuration is unfavorable for the LO phonon scattering also. Since the two spectra are normalized to make the height of the LO phonon peak appear equal, the scale for spectrum (b) is expanded by a factor of about 10 relative to that of spectrum (a). The surface polariton scattering intensity for each configuration was found by the curve fitting procedure described before, using eq. (4.4). Then the values of A,, were compared for the four configurations; the relative intensities are shown in Table 4.2. These intensity values are estimated to be accurate to *40°/o.

Theoretical predictions for the intensity can be calculated by using the Raman tensors given in eq. (4.1) and the formula (LOUDON [1963]):

where p, a , T = x', y', 2'; A is a constant, and 8, is the electric field associated with surface polaritons that appear in eq. (3.12). In writing eq. (4.6) we have combined the atomic displacement and the electro-optic contributions to the scattering intensity as we have done in eq. (3.13) and eq. (4.2). When kll(I $', the non-zero components of the surface polariton field 8 are 8,. and EY,, because surface polaritons are elliptically polarized in the sagittal plane (see § 2). Then using eq. (4.1) and eq. (4.6), we find the scattering intensities for (y'y') and (y'z') polarization combinations:

and

(4.7)

When kll I( i', 8= (gX,, 0, 8=,) and we obtain the corresponding expressions

TABLE 4.2

Polarization kl, direction x'(y'y')x' x'(y'2')x'

(0.2

IILO 41 EXPERIMENTAL RESULTS 179

for this case:

and

Iz,(y’z’) = 0. (4.10)

In eqs. (4.7)-(4.10) the subscript on I indicates the direction of kll. These expressions are shown in parentheses in the corresponding places in Table 4.2.

We see that the theoretical prediction and the observed relative intensities agree if we assume that 8,. =gY, for kll 11 9’, and that the very small observed intensity for kll 11 2’ and x’(y’z‘)x’ matches with the prediction of zero intensity for this configuration. The case for kll 11 2’ and x’(y’y‘)x’ requires a detailed knowledge of 8%, and g2,. Now we consider the relative magnitudes and phases of the three components of 8 by referring back to 0 2. Let us first consider the case for kll (1 9’. In the present geometry and coordinate system, 8,. is the component of 8 normal to the surface (Z1) and gYt is the parallel component (Ell). Thus, referring to eq. (2.4a), we see that the ratio between 8,. and ZY, is given by:

8,,/8,, = - az/ikll. (4.11)

Similarly, we find

8,,/8,, = - a,/ikll. (4.12)

Now we need the ratio a,/kll for 0=2.5” where the intensity measurements were made. Using the parameter values in Table 4.1 and eq. (4.3) for hi = 5145 A, we find a2/kll= 1.14. Thus, we expect the ratio between IJy’y’) and IJy’z’) to be given by:

I,,,(y’z’)/Iy,(y’y’) = 2 18yj/8x.12 = 2.6. (4.13)

The observed ratio is 2*40%, so the agreement is reasonable. By using eq. (4.12) with the above value of a,/kll, we predict:

Z24y’y’)/I,,(y’y’) = I&(- a,/ikll) - 11’ = 3.6 (4.14)

to compare with the observed value of 3*40%. Here again the agreement is within the estimated uncertainty of the measurement. Thus, we conclude that the Raman scattering selection rule for surface polaritons is indeed determined by the Raman tensors of the bulk crystal. However, in

180 LIGHT SCATIERING SPECTROSCOPY [III, !? 4

the case of surface polaritons, the elliptical polarization of the modes must be carefully taken into account as we have done in arriving at eqs. (4.13)-(4.14).

4.2. DOUBLE INTERFACE MODES (DIM)

Double interface modes are found when the sample is sufficiently thin so that the two SIM's on the opposite faces of the sample interact. Mathematically this corresponds to the case where exp { - 2a2d} in eq. (2.14) is appreciable, and the resulting modes have two branches UM and LM. The LM was the first surface polaritons observed by Raman scattering (EVANS, USHIODA and MCMULLEN [1973]) and subsequently the UM was also found after some effort (PRIEUR and USHIODA [1975]). We will review the findings of these two experiments in this subsection. Since the experimental setup is essentially identical to the one described in the preceding subsection, we will point out only the notable differences.

The sample used in these experiments was a thin (- 2500 A thick) film of GaAs deposited on a sapphire substrate by chemical vapor deposition (MANASEVIT and THORSEN [1970]). The carrier density level was nearly intrinsic showing no sign of the presence of plasmons. The sapphire substrate had the (0001) face where the interatomic distances are close to that of the (1 11) surface of GaAs. Thus, the film is supposed to have the (111) face of GaAs, but the observed Raman selection rule indicated that the film is polycrystalline. Table 4.3 summarizes the relevant parameters of GaAs.

The measurements on the LM were made with the 4880 8, line of an argon ion laser as the exciting source; the power level was approximately 400 mW. Sapphire is completely transparent at this wavelength, but

TABLE 4.3 Crystal parameters of GaAs

Crystal symmetry Td (zincblende) No. of atomslunit cell 2 Phonon symmetry F2 W L O 292 cm-l W T O 270 cm-' E , 1 1 . 1

13.1 E" 1.43 eV

111,s 41 EXPERIMENTAL RESULTS 181

GaAs is opaque with the refractive index, n = 4 . 4 and the extinction constant, K = 0.4 (SERAPHIN and BENNET [1967]). The skin depth of the 4880A radiation is approximately 900 8, which is comparable to the thickness of the film 2500 A. Thus, it was possible to detect the scattered light in the near-forward direction. The exciting beam was incident from the sapphire side normal to the surface; the incident angle was precisely kept at normal by insuring that the reflected beam coincide with the incoming beam. The polarization of the incident beam was linear and the scattered light of all polarization was collected.

The spectra measured for various near-forward directions are depicted in Fig. 4.5. Note that the scattering angle + is measured inside GaAs, so B(outside) = n+ with n = 4.4. The large peaks at 270 cm-' and 292 cm-' are the bulk TO and LO phonons, respectively. The small peak between the two large peaks pointed by an arrow is the LM of the double interface geometry. The dependence of the peak position on the scattering wave- vector (qr in the figure) and + is shown in Fig. 4.6 along with the uncertainties in frequency and wave-vector.

The solid curve is the theoretical dispersion curve of LM for the

I I I I 1 ,

255 270 205 300

frequency shift (crn-'1

Fig. 4.5. Raman spectra of DIM surface polaritons in a GaAs film on a sapphire substrate as a function of the scattering angle I/I inside GaAs. The arrows indicate the surface

polariton peak positions.

182 LIGHT SCATIFRING SPECTROSCOPY [III, P 4

105: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - - - - - - I - - - - 4. I -& lo" do 12" e

130 150 170 190 210 3 ' a n ' ' ' ' a '

91 w m 7 c

Fig. 4.6. Measured dispersion of the DIM surface polaritons as a function of the scattering angle $ and the theoretical dispersion curve.

present geometry. In terms of the symbols used in B 2.2, = 1 for air and c2= E ( W ) for GaAs given by eq. (2.1) with the GaAs parameters collected in Table 4.3. Because sapphire is uniaxial and the substrate has the c-axis normal to its surface, the value of c3 in 0 2.2 takes on two values, ql for polarization normal to the surface (parallel to c-axis), and E, for polarization in the surface plane (perpendicular to c-axis). The implicit dispersion relation that takes into account the anisotropy of the substrate was given by MILLS and MARADUDIN [1973]. For the present geometry the corresponding equation to eq. (2.14) is:

(4.15)

where a, and a2 are defined by eq. (2 .19, but a3 is defined differently to take account of the anisotropy of medium 3 (sapphire) by:

a3 = [(&J&li)(kf- &p2/c2)?. (4.16)

The low frequency branch (LM) of eq. (4.15) is plotted in Fig. 4.6 with d = 2500 A. We see that the agreement between the data and theory is excellent; in fact, the observed peaks were identified as LM by comparison of its dispersion with the theoretical prediction.

According to the theory presented in Q2 .2 there should be another branch closer to the LO phonon line (UM), but no peak corresponding to UM could be found in the spectra of Fig. 4.5. However, upon closer inspection it was noticed that the bulk LO phonon peak in Fig. 4.5 has a slight asymmetry with higher intensity on the low frequency side than on the high frequency side. This asymmetry is particularly evident at 4 = 3.5" in Fig. 4.5. Furthermore, calculation of the UM dispersion curve using eq.

111, P 41 EXPERIMENTAL RESULTS 183

(4.15) shows that the UM branch lies within the line-width of the LO phonon peak. Thus, it was supposed that the peak due to UM is causing the asymmetry of the LO phonon peak.

In order to measure the dispersion of UM, it was necessary to extract the position of the UM peak from the asymmetric shape of the LO peak. To accomplish this, two schemes were used. One was to place a dielectric liquid in contact with the GaAs film replacing the air. The liquid increases the dielectric constant in region 1, and a computer study of eq. (4.15) shows that the frequency of the UM branch is lowered by an increase in E , . Then the UM should appear more prominently, increasing the observed asymmetry of the LO peak. The other scheme depends on the asymmetry of surface polariton scattering intensities in the forward and backward directions discussed in B 3.2. According to the theory the backward scattering by surface polaritons is weaker by a factor of lo2- lo3 than the forward scattering. Since the observed forward scattering intensity is on the order of 2 - 3 photons/sec at the peak, the backward scattering intensity is essentially zero for surface polaritons, while the bulk phonons scatter light in the backward direction as strongly as in the forward direction. To exploit this forward-backward asymmetry, the difference spectrum between the forward and backward scattering was obtained by using a scheme illustrated in Fig. 4.7. Then the difference spectrum contains only contributions from surface polaritons, both LM and UM. When the moving mirror intercepts the laser beam, the light follows the dashed path, and the backward scattering geometry results, while the forward scattering geometry is obtained when the mirror is rotated out of the laser beam. The moving mirror is controlled by one of the relays which in turn is controlled by the minicomputer (see Fig. 3.6). Thus, both forward and backward scattering spectra are recorded at a given position of the grating; then the difference is calculated by digital subtraction of the recorded data at the end of a run. One might think that the same result could be obtained by taking the forward and backward spectra separately. However, it was found that because the difference is very small compared with the background, slight shift in the grating positions in two separate runs introduce unacceptably large errors. Also it was very important to take the difference spectra of the same spot on the sample, because of the non-uniformity of the sample from spot to spot. Fig. 4.7 also depicts a liquid cell which was filled with benzene as a dielectric liquid. Benzene was selected because its dielectric properties in the far infrared are well characterized (ZELANO and KING [1970]).

184 LIGHT WATERING SPECTROSCOPY [III, § 4

CELL 4, I ',SAMPLE I . . + I

I I I

Fig. 4.7. Experimental scheme for obtaining forward and backward scattering spectra in a single run using a computer-controlled moving mirror.

The spectra were measured with the 5145 A line of an argon ion laser as the exciting source; the incident power was approximately 400 mW. A typical spectrum obtained for the benzene-GaAs-sapphire structure is shown in Fig. 4.8. The difference spectrum between the forward and backward directions indeed reveals the UM which emerges from the low frequency wing of the LO peak. The data obtained on the dispersion of the UM as well as the LM with and without benzene in contact with the GaAs film surface are plotted in Fig. 4.9. We see that the change of E ,

due to a change from air to benzene affects the frequency of the UM significantly, but not the LM frequency. Referring back to 0 2.2 we recall that when < E~ the UM is more strongly localized at the 1-2 interface than at the 2-3 interface. With benzene and sapphire as medium 1 and medium 3 , respectively, this condition E , < E~ is satisfied. Thus, the UM amplitude is high at the GaAs-air interface, so the UM dispersion is strongly affected by replacing of air by benzene as medium 1. The solid theoretical curves were plotted using the full frequency dependent dielectric functions of sapphire and benzene (PRIEUR and USHIODA [1975]). The frequency shift caused by changing of medium 1 from air to

111, fr 41 EXPERIMENTAL RESULTS

1.02

1.00-

185

- I /.---~cl..

I I I I I I I I I 1 -

FREQUENCY (cm-'1

Fig. 4.8. Forward and backward scattering spectra and the difference spectrum. Note the asymmetry of the peak near oLo in the forward scattering spectrum.

benzene is clearly seen in the two theoretical curves, and the amount of the shift agrees quite well with the observation.

4.3. GUIDED WAVE POLARITONS (GWP)

The motivation for an experimental search for Raman scattering from GWP was provided by the results of a theoretical calculation of the

0 EXTERNAL MEDIUM AIR

0 EXTERNAL MEDIUM BENZENE 1,061

. - Fig. 4.9. Measured dispersion for UM and LM, with air (empty circles) and benzene (solid circles) in contact with the GaAs film. The solid curves are the theoretical dispersion curves.

186 LIGHT SCA"ER1NG SPECIXOSCOPY [III, D 4

spectrum of thin films by SUBBASWAMY and MILLS [1978]. They performed a computer simulation of the Raman spectrum of a thin dielectric film using the full theoretical expression of the Raman intensity given by MILLS, CHEN and BURSTEIN [1976], and found that the peaks due to GWP have comparable strengths to those of surface polaritons and bulk phonons. The GWP peaks lie in the frequency region (below wTo and above wLo) which had not been explored closely in the studies of surface polaritons before.

The experiments on GWP were performed on GaP samples of thickness ranging from 5 microns to 125 microns. The thinning procedure for these samples was identical to that used for the SIM experiment in P 4.1. The sample thickness was determined by a mechanical gauge as well as optical density measurements at 4880 A and 5145 A, both of which gave the thickness within f 10% of each other. Crystals with both (100) and (111) faces were used with the incident laser beam normal to the surface as before. The incident light was at 5145A with the power level of

c

v)

z 3

t a

c_

a a c_ m a a

k

1

t

v) z w k

f

50 300 350 400 FREQUENCY (cm-1)

Fig. 4.10. Raman spectra of GWP in a GaP sample of thickness 30 p,m for different scattering angles 0. The peak labeled by "B" is not a GWP peak.

III ,§ 41 EXPERIMENTAL RESULTS 187

approximately 100 mW. Typical counting rates for the individual GWP peaks were -0.1 to 1 cps.

Fig. 4.10 presents the Raman spectra of GWP for a GaP sample with a (111) face and thickness 30 microns. The large peak at 403 cm-’ is the bulk LO phonon, and the bulk TO phonon peak appears at 367 cm-’. A group of small peaks on the low frequency side of the TO phonon peak is the GWP. The vertical bars indicate the theoretically expected locations of GWP peaks for a 30 micron thick GaP crystal at each scattering angle. We see that the peaks in the spectra correspond closely to the expected GWP frequencies. The peak with a label “B” appears in the backward scattering spectra also; thus, this peak was identified not to be due to GWP.

In Fig. 4.1 1 the frequencies and wave-vectors of the GWP peaks in Fig. 4.10 are plotted along with the theoretical dispersion curves of GWP. Each theoretical curve is labeled by the mode number rn of eq. (2.27). The .nearly vertical lines labeled by the scattering angles 8 are the kinematic trajectories on which both k,, and w are conserved in the scattering process; these trajectories are given by:

k,, = ( ki - wlc) sin 8. (4.17)

__-. -_-- _-. 6 325

I - . .- * * S -POLARIZED I

I ( T E I .; 2751 ’ I I I 0.5 1.0 1.5 2.0 2.5 3.0

kll ’kt

Fig. 4.11. Measured dispersion of the GWP peaks (solid circles). The measured peak positions fall on the crossing points of the kinematic trajectories (labeled by the scattering angle) and the theoretical dispersion curves. The mode numbers rn for each dispersion curve

are indicated on the right margin. The wave-vector is measured in the unit of kT=wTO/c,

188 LIGHT SCA'ITERING SPEmROSCOPY [III, !i 4

The GWP peaks are expected to appear in the spectrum at the crossings of the kinematic trajectory and the dispersion curves. We see that the peak positions of the spectra in Fig. 4.10 indeed fall on the crossing points. Since the polarization of the scattered light was not analyzed, the separation into TE and TM modes could not be made on the basis of the polarization selection rule. Both TE and TM modes appear in Fig. 4.10. The sorting of the peaks indicated in Fig. 4.11a and 4.11b was done on the basis of frequency matching with the theory. The solid portion of the theoretical dispersion curves indicates the region in which the surface normal component ( k , ) of the wave-vector is conserved within 27r/d; i.e., A k , = k , - k , < 27rld. In the dashed region Akl 2 27rld, resulting in weak scattering according eq. (3.16).

The calculated spectra according to SUBBASWAMY and MILLS [ 19781 corresponding to the spectra of Fig. 4.10 are shown in Fig. 4.12 for comparison. The experimental spectra show a modulation of the intensity not seen in the theoretical simulation. Also it was found that the selection rules do not hold for GWP on the (1 11) face. Another interesting finding was that the intensity of GWP scattering depends on the surface quality, producing a higher intensity for rougher surfaces than for smooth ones.

GEOMETRY A L = 20 p m

I , LO

I 0.5 1.0 1.5

w'+l

Fig. 4.12. Theoretical Rarnan spectra of GWP in GaP (after SURRASWAMY and MILLS [1978]).

111, § 41 EXPENMENTAL RESULTS 189

These discrepancies between theory and experimental results suggest that there are scattering mechanisms not included in the present theory.

In order to exclude the possibility that the observed spectra contain second order Raman scattering features, the spectra were measured at liquid nitrogen temperature also. The relative intensity of the GWP peaks to that of the LO phonon peak stayed constant at the value observed at room temperature; thus, the GWP peaks are due to first order Raman scattering processes. Another check on the identity of the GWP peaks was made by measuring the backward scattering spectra. These spectra did not show any of the features identified as GWP peaks in the forward scattering spectra of Fig. 4.10, as expected on the basis of the forward- backward asymmetry discussed in Q 3 .

Fig. 4.13 illustrates an assortment of spectra for different sample thicknesses and orientations. Spectrum (a) is for a sample with the same thickness (30 pm) as before but with (100) faces. The incident beam was linearly polarized in the (100) plane and the scattered light was not analyzed. In this geometry Raman scattering occurs from the GWP fluctuations normal to the surface, but the amplitude of the fluctuations in this direction is small according to theory (SUBBASWAMY and MILLS [1978]). Thus, n o scattering from GWP could be observed in spectrum (a). Spectrum (b) is for a thick sample (125 pm) with (111) faces. Here the mode separation between different rn modes has

GaP A c

v)

t z 3

> a a a I- - m a a

I\ ( a ) (100) 3 0 p m 0 = 1 . 3 O

\ I- I

250 300 350 4 0 0

I I I

6pm 0; 1 . 6 O

I

300 3 50 4 0 0 4 5 0 FREQUENCY (cm- ' )

Fig. 4.13. Raman spectra of GaP slabs of different thicknesses and crystalline orientations.

190 LIGHT S C A m R I N G SPECTROSCOPY [III, 8 5

decreased and all the GWP peaks have converged to the bulk polariton peak indicated by 7~ in spectrum (b). So we see that a 125 km sample behaves like a true bulk sample. Spectrum (c) depicts the case of a thin sample limit (6 km). The GWP peaks are weaker, but better resolved from each other than in the 30 krn sample, because the separation between the branches ( T / d ) is greater in this sample. We can also see peaks above wLo corresponding to the upper branches of GWP.

This concludes the discussion of the experimental results on Raman scattering by various surface electromagnetic (EM) waves described in § 2. All the modes discussed in § 2, surface polaritons in single (SIM) and double (DIM) interface geometries as well as guided wave polaritons (GWP), have been observed by light scattering. However, we note that the experiments were done by choosing crystals which are particularly strong scatterers of light and whose dielectric properties are already known. Thus far no truly new information in terms of light scattering mechanisms or materials properties near surfaces has been obtained by this experimental method. Now that the light scattering by surface EM waves can be measured, the next step is to utilize this capability in characterizing surfaces and the excitations near the surface. We discuss a first step in this direction in the next section.

Q 5. Effects of Surface Roughness

In the preceding sections we have considered surface polaritons and guided wave polaritons under the assumption that the surfaces are perfectly flat and smooth. However, real surfaces, even optically flat ones, are not smooth on the scale of a few hundred or thousand Sngstroms. Since the decay length of the surface polariton amplitude (1/aZ) in the far infrared is on the order of a few microns the surface roughness of the order of 1 - 10% of the decay length may have a significant effect on the behavior of surface polaritons. Also if the distance between peaks and valleys of the surface roughness along the surface plane is comparable to the surface polariton wavelength (1/k,J, it may suffer a diffraction effect from the roughness. The surface polaritons discussed in § 2 are the normal modes localized at a smooth interface. So when roughness is introduced on the surface, they are no longer the true eigenmodes of the geometry. Then one expects that these modes will interact with each other as well as the bulk modes on either side of the surface. Thus, the

111, 8 51 EFFECTS OF SURFACE ROUGHNESS 191

surface polariton should shift its frequency and change its mean free path upon the introduction of roughness. This kind of consideration leads to the theoretical work and experiments discussed in this section.

5.1 . THEORETICAL CONSIDERATIONS

Theories on the interaction of the electromagnetic waves with rough solid surfaces address themselves to basically two questions; one is the scattering and absorption of an incident wave induced by surface roughness and the other is the effects of roughness on the dispersion of surface electromagnetic waves. These two apparently different problems are intimately connected, because both of these effects are caused by the perturbation of the boundary conditions at the surface. (See the discussion on the reen’s function and the spectral density in g3.2.) Many theoretical an 1 experimental papers have appeared recently dealing with the effects of surface roughness (WILLIAMS and ASPNES [1978] and references cited therein). Although the subject matter discussed in these papers is closely connected with what we treat in this subsection, it is not possible to review all of them and clarify the connections among them in the limited space here. Thus, we simply call the reader’s attention to the existence of the above references on this subject. Here we focus on three theories by MILLS [1975], MARADUDIN and ZIERAU [1976] (MZ), and KROGER and KRETSCHMANN [1976] (KK) which deal directly with the damping and frequency shift of surface polaritons induced by roughness on the surface.

In all three theories above Im E ( W ) = 0 is assumed so that the surface polariton has no damping (infinite life-time and mean free path) in the absence of surface roughness. Then Mills calculated the mean free path of surface polaritons with surface roughness, while M Z calculated both the damping and the shift in frequency due to roughening of the surface. Both of these theories are based on the use of electromagnetic Green’s functions for a semi-infinite crystal geometry perturbed by the introduction of surface roughness (MARADUDIN and MILLS [1975]). The unperturbed Green’s function in the absence of roughness was derived earlier by MILLS, MARADUDIN and BURSTEIN [1970]. Mills calculates the energy loss from a surface polariton through radiation into bulk modes and into another surface polariton. The energy loss rate is then converted into a mean free path which corresponds to damping. MZ start with the same

192 LIGHT SCAlTElUNG SPECTROSCOPY [III, P 5

unperturbed Green’s function to calculate the proper self-energy of the spectral density function for inelastic light scattering. Thus, their theory basically follows the line of argument described in § 3.2 except for the fact that the Green’s function now contains the effect of surface roughness. Finally from the poles of the perturbed spectral density function they find the frequency shift and damping due to surface roughness.

KK’s approach to the problem is quite different from the above two. Their starting point is the “transformed boundary conditions” at a rough surface which were derived in their earlier work (KROGER and KRETSCHMANN [1970]). Instead of the usual continuity conditions for tangential and normal components of the fields, the transformed boundary conditions prescribe certain discontinuities that account for the induced currents due to roughness at the surface. Then they apply the perturbation due to the boundary condition change to the dispersion relation of surface polaritons at a single interface. The result is given in terms of a complex shift hk in the surface polariton wave-vector parallel to the surface. It can be seen quite easily that the imaginary part of hk is directly related to the mean free path calculated by Mills and the results agree exactly. Thus, the theoretical result on the imaginary part of bk is considered well established. However, with respect to the real part of hk, the results by MZ and KK are in slight disagreement, and the KK result appears to be correct (MARADUDIN [ 19791).

In all three theories the effect of surface roughness is taken into account by a perturbation approach, and the leading term is proportional to the mean square height of roughness (S2). For later use we quote the explicit result according to KK’s theory:

= ( S 2 ) P y P d2k g(k - kll)A(k, kll) C

where

cos2 cp + ( - &)f

(5.2)

111, Q 51

and

EFFECTS OF SURFACE ROUGHNESS 193

with

(5.3)

-ik, = (k2- .mgp/c2)f, (5 .5)

k 2 = (w&/c~- k2)i; (5.6)

P j d 2 k means the Cauchy principal integral taken over the two- dimensional space in k. wSp and kll are the frequency and wave-vector of the surface polaritons satisfying the unperturbed dispersion relation given by eq. (2.9). g(k-kll) is the Fourier transform of the autocorrelation function of the surface roughness profile z = S(x, y) given by:

g ( k - kll)=,- d2x exp{i(k-kll)-x)(S(x, y )S(O,o ) ) (5.7) ( S ‘ I )

where the integration is over the two-dimensional surface scanned by x = (x, y). A physically reasonable and mathematically manageable form for g ( k - kll) can be obtained if we assume that the roughness profile is stochastic with the average height (S) = 0. Then g ( k - kll) has the Gaussian form given by:

(5.8)

where a is the transverse correlation distance between peaks and valleys of z = S(x, y). In what follows we will assume the form of g ( k - kll) to be given by eq. (5.8), but real surfaces may not be represented well by this assumption, depending on the surface preparation methods. For instance, if a surface is prepared by polishing with long strokes in one direction, g(k-kIl) will not be isotropic in k, but rather have a structure similar to a grating. On the other hand, etching of certain crystal surfaces is known to produce structures peculiar to them with characteristic textures.

With the Gaussian form for g(k-kll), one can perform the angular part of the integration in eq. (5.1) and obtain the result in terms of modified Bessel functions, but the radial part of the integration must be performed numerically. When the angular part of the integration is performed, one sees that the second integral corresponds to l/l‘sp’ given by Mills, where 1(”) is the mean free path determined by decay of a given mode of

g(k - kll) = (a/2.rrI2 exp { - (a/2)2(k- k11)2)


surface polariton with wave-vector kll into all other surface polaritons. The first integral in eq. (5.1) contains both real and imaginary parts. The real part corresponds to the frequency shift and the imaginary part corresponds to the damping due to radiative energy loss to bulk modes; this part corresponds to l/l(R) in Mills’ theory. Upon numerical integration one finds that the imaginary part corresponding to the radiative loss to bulk modes (l/l(R)) is negligible compared to the loss to other surface polaritons represented by the second integral in eq. (5.1). We will compare the numerical results based on eq. (5.1) with experimental results in the following subsection.

5.2. EXPERIMENTAL RESULTS A N D COMPARISON WITH THEORY

The experimental work on this subject was initially motivated by two observations that have already been mentioned in §4. In the dispersion relation of SIM in GaP it was seen that the observed frequencies are systematically lower than the theoretical values based on the bulk dielectric function. Then it was found that the scattering intensity of GWP depends on the quality of the surface. Both of these effects appeared to be connected to the roughness of the sample surface. Thus, a systematic study was conducted on the effects of surface roughness on the Raman scattering intensity, the surface polariton frequency and the linewidth by using samples whose surface roughness is controlled by preparation procedures (USHIODA, AZIZA, VALDEZ and MATTEI [ 19791).

The experimental method and procedure used in this work were identical to those described in § 4.1. The only difference was that the GaP samples used in the present experiment were prepared under varied and controlled conditions so as to obtain different roughness. The five kinds of samples used in this experiment are listed in Table 5.1. The surface

TABLE 5.1 Samples prepared by different procedures

Surface preparation Thickness (pm)

Sample A 0.3 pm polish 15 Sample B 0.3 pm polish + annealing 15 Sample C 0.05 pm polish 15 Sample D 0.05 pm polish+annealing 20 Sample E 0.05 pm polish fetching 15

111, § 51 EFFECTS OF SURFACE ROUGHNESS 195

roughness of these samples was controlled by the grit size of the final polishing powder. The samples were first thinned to about 250 pm by grinding with a slurry of 3 pm alumina in deionized water, and then they were further polished on a silk cloth with 1 pm alumina powder. After this preliminary polishing process, sample A was polished with 0.3 pm alumina. Sample B was obtained by annealing a piece from a batch of sample A in lop6 Torr vacuum for 3 hours at 500°C. Some pieces of sample A were further polished with 0.05 pm alumina powder to obtain a smoother surface; these are sample C. Sample D was prepared by annealing a piece of sample C under the same condition that was used in preparing sample B. Sample E was prepared from sample C by etching its surface in a water solution of 0.5M KOH+ 1.OM K,Fe(CN),. Thus, samples A and B have the same roughness, but after annealing sample B has less residual surface strain due to polishing. The relationship between samples C and D is similar. The etched sample E presumably had the strained region of the surface removed. The effect of surface strain as well as the geometrical roughness of the surface was considered, because it had been learned earlier that the residual surface strain plays an important role in determining the bulk phonon linewidth observed in backward scattering from opaque surfaces (EVANS and USHIODA [1974]).

Fig. 5.1 illustrates the difference in the Raman spectra of surface polaritons (SIM) for a “rough” surface (sample A) and a “smooth” surface (sample C). Evidently a “rough” surface produces a stronger surface polariton scattering than a “smooth” surface, but it is not easy to see if there is any difference in the peak position (wsp) and the damping

I I I I 1

>

cn z W I-

k

z w

I-

-J w a

I I I I I

385 390 395 400 405 (cm-1)

(-4). Fig. 5.1. Surface polariton (SIM) spectra of a “smooth” surface (C) and a “rough” surface

196 LIGHT SCAWERtNG SPECTROSCOPY [III, d 5

10 20 3" 40

e. Fig. 5.2. Dispersion of surface polaritom (SIM) for different surfaces. Theoretical dispersion

curves correspond to different values of E,: (a) E,= 9.091, (b) E,= 7.8, (c) E,= 7.0.

width (rsp). In order to obtain quantitative data, the digital spectral decomposition technique described in Q 4.1 using eqs. (4.4) and (4.5) was applied. By this method the desired parameters, asp, r,, and A,, of eq. (4.5) were determined as the best fitting parameters for a given spectrum. In determining r,, the spectrometer resolution width of - 1 cm-' was not taken into account. It is difficult to compare the raw magnitudes of the surface polariton scattering intensity A,, for different samples, because the absolute intensity changes from run to run depending on the slightest details of the set-up. Therefore, the ratio of the integrated intensity of surface polaritons to that of the LO phonons, Isp/ILo, was chosen for the measure of intensity. Figs. 5.2, 5.3 and 5.4 show the frequency (asp), the damping (rs,) and the intensity ratio (ISP/ZLO), respectively, as a function of the scattering angle ( 6 ) for the five different samples of Table 5.1. The estimated uncertainties in these data are as follows: o,,f 1 cm-', rs,* 0.5 cm-' and Isp/IL0f0.05.

Let us first focus on the dispersion data shown in Fig. 5.2. As we observed in Fig. 4.2, the measured frequencies of surface polaritons are systematically lower than the values expected on the basis of the bulk dielectric constant (E,= 9.091); the theoretical curve for this value of E ,

is labeled (a) in Fig. 5.2. The solid triangles are the "old" data points transferred from Fig. 4.2. The data points for different surface roughness are mixed and there appears to be no systematic trend that depends on

111, B 51

5 -

4 -

3 -

5 - a v)

L. 2 -

I -

0

EFFECTS OF SURFACE ROUGHNESS

I I I

0

0 ROUGH ( -

A 0

-

0 0 0

X

0 - m . 0

0 . m o -

t -

I I I I I

197

0.3

s I 4

-

I0 20 39 40

Fig. 5.3. Surface polariton damping r,, for different roughness and scattering angles.

the surface roughness; however, most of the data points fall 0.5 to l.Ocm-' below the theoretical curve (a). Thus, we must conclude that within the range of roughness examined here there is no clear-cut dependence of wsp on the roughness, although all the data points are depressed below the theoretical curve.

0.4 1 I I I I I I

0 0

A 0 ROUGH ( -

0

Fig. 5.4. Relative surface polariton scattering intensity I,,/I,, for different roughness and scattering angles.

198 LIGHT SCATTERING SPECTROSCOPY

Fig. 5.5. Profile of a rough surface.

One way of physically interpreting the present data is the following. When the surface is not perfectly smooth, there is a transition region between the true bulk and the air above where the dielectric material fills only a portion of the volume as illustrated in Fig. 5.5. In this transition (or roughness) region the dielectric constant is lower than in the bulk, because there is less material per unit volume than in the interior of the bulk. To check the validity of this idea, the value of E, was varied and the surface dispersion curves were plotted as indicated by labels (b) and (c) for E,= 7.80 and ~ , = 7 . 0 0 , respectively. From Fig. 5.2 it is seen that the appropriate average value of E, near the surface is close to E,= 7.80 which is about 15% lower than the bulk value.

In the case of surface polariton damping rsp, there is a clear trend that depends on the surface quality as seen in Fig. 5.3. Smooth surfaces show smaller damping than rough surfaces, and annealing has the effect of decreasing the damping. This is the same trend observed earlier in the bulk LO and TO phonon damping (EVANS and USHIODA [1974]). The change in going from sample A to sample D is on the order of 1 - 2 cm-'. The scatter in the data points comes from the lack of uniformity on the surface from spot to spot.

We can see in Fig. 5.4 that there is a definite correlation between the surface roughness and the Raman scattering intensity; rough surfaces scatter more strongly than smooth surfaces. The intensity range between sample A and sample D is a factor of 2 to 4. To the author's knowledge, there is n o theoretical work dealing with the effect of surface roughness on the Raman scattering intensity. This is an area that is worthy of theoretical exploration in the future, especially in view of recent developments on Raman scattering from adsorbed molecules on a rough silver surface (see, for example, VAN DUYNE [1977] and BURSTEIN, CHEN, CHEN, LUNDQUIST and TOSSATTI [1979]).

Now we will compare the above experimental observations with the theoretical predictions of eq. (5.1). The frequency shift Au and the

EFFECTS OF SURFACE ROUGHNESS 199 111, Q 51

damping rs, can be obtained from bk by the relations:

A o = V, Re (ak) (5.9)

(5.10)

where V, is the group velocity of the surface polariton. Ao and r,, were calculated by numerical integration of eq. (5.1). The only adjustable parameters of the theory are the transverse correlation length a and the mean square height of roughness (S') . The calculation was done for S = 5 0 O A which is a realistic value for samples prepared with the final polishing powder size of 0.05 pm (500 A). As one can see in eq. (5.1) both Aw and r,, are simply proportional to ( S 2 ) , while their dependence on a is not obvious because a appears in the integral. The plots of Aw and r,, were obtained as functions of both a and kll; the experimental value of k,, for 8 = 4" is k,, = 8000 cm-'.

Fig. 5.6 illustrates the dependence of Aw on k,, for different values of a whose values in the range of 1-10 ~ r n were chosen to approximate the real conditions of the sample surface. For small values of a = 1 pm and 2 pm, A o goes through a zero and changes its sign in the relevant range of kll values. When a becomes large the zero of Aw moves to a smaller value of kl, and Aw is negative in the entire range of k,, depicted in Fig. 5.6. Thus, a large transverse correlation length a has the effect of depressing the

Fig. 5.6. Frequency shift \Aw\ as a function of wave-vector kll and transverse correlation length a.

200 LIGHT S C A m R I N G SPECTROSCOPY [III, 8 5

dispersion curve in the region of k,, observed in the present experiment; this correlates well with the observation. The change of sign in Aw means that the roughness puts a small wiggle in the dispersion curve. This effect is similar to Bragg diffraction, but because the roughness has a distributed periodicity with its average equal to a, the diffraction effect is weak and a real energy gap does not appear. When the effect of roughness is strong enough, the dispersion curve can have a double bend and it becomes possible to observe a double peak in the spectrum (KRETSCHMANN, FER- RELL and ASHLEY [1979]). The maximum size of Aw in the experimental range of k , ( 5 1 0 000 cm-') is about 0.1 cm-' for S = 500 A. Thus, for S = 3 000 A we expect A w = 3.6 cm-'. This is comparable to the observed range of Aw = 0.5 - 1.0 cm-'.

r,, as a function of kll and a is plotted in Fig. 5.7. It is seen that I',, increases monotonically with kIl. The magnitude of rsp for realistic values of a (1 - 2 pm) and the experimental range of kll (I 10 000 cm-') is about 0.1 cm-' for S = 500 A; consequently rsp= 3.6 cm-' for S = 3 000 A. Thus, the change in r,, in going from sample A to C on the order of 1 - 2 cm-' is consistent with these theoretical values.

Fig. 5.8 is a plot of Aw and rs, as a function of a for a fixed value of kl l=8000cm- ' (h,,=7.85 pm). We see that Aw and rs, have the Kramers-Kronig-type relation as a function of a. It is interesting to note that the maximum effect of roughness is felt by surface polaritons not

lo-'- - - I

5

fl - a

0 2 4 6 8 10 12 14 16x103 k II (crn-l)

Fig. 5.7. Surface polariton damping r,, versus kll for different values of a.

111, § 51 EFFEcrS OF SURFACE ROUGHNESS 201

a ( p m )

Fig. 5.8. (Awl and r,, versus a at a fixed value of k , l = 8 x lo3 cm-'. A w is negative on the left side of the dip at a = 2 gm, and positive on the right side.

when the transverse correlation length a is equal to its wavelength (ASP), but when a is considerably smaller than Asp.

The above comparisons show that the theory and experiment are in agreement within the semi-quantitative accuracy of the present data. In order to refine the experimental data, the surface roughness must be characterized in quantitative terms. Recent work by WILLIAMS and ASPNES [ 19781 suggests a way to improve the surface characterization, and their method can be used in conjunction with a surface polariton experiment, if one wishes to gain more quantitative information on surface roughness. It was assumed that g(k , k,,) has a Gaussian form in calculating Ao and rsp. According to Williams and Aspnes, there are two types of roughness (one Gaussian plus another component) coexisting on their surface of Si. A similar situation may exist on GaP samples also. Then the theoretical calculation requires some revisions.

The question of the effects of surface roughness on surface polaritons, surface scattering and absorption is an interesting problem from a purely intellectual viewpoint as well as the viewpoint of device applications. When a surface EM mode is used to carry signals in integrated optics devices the attenuation length of the mode is one of the most important parameters. The Raman scattering method can provide quantitative information neccessary in such applications when appropriate refinements are added as we have suggested above.

202 LIGHT SCATERING SPECTROSCOPY [111, $6

0 6. Concluding Remarks

In this review we have covered only the essential part of the theories and experiments on light scattering by surface electromagnetic waves. The main purpose was to organize an understandable account of the subject matter into one article. Thus, in § 2, for example, the simplest and most expedient way was chosen to arrive at the necessary results. In order to linearize the discussion, a conscious effort was made to leave out related subjects not absolutely essential to the understanding of the material in 0 4 and § 5. For instance, various surface related modes reviewed by BURSTEIN, HARTSTEIN, SCHOENWALD, MARADUDIN, MILLS and WALLIS [1974] were not included in our discussion. The same is true with regard to references, and no effort was made to present references on related areas or to give a comprehensive list for the areas covered by this article.

The light scattering method of surface polariton studies has not been used as commonly as the ATR method, principally because of the difficulty of detecting the weak signals. However, the light scattering probe has the advantage over the ATR method and the grating method in that it allows direct measurements of the surface modes undisturbed by the probing techniques. In the ATR method the true mode that causes absorption is a double interface mode of a three-layered structure com- posed of a prism, an air gap and a sample, and not the true single interface mode of the sample surface. In the case of the grating method, the sample surface must be disturbed by drawing a grating. Thus, analysis of the light scattering spectra is much more straightforward than the ATR spectra or the absorption spectra of the grating method. Now that the necessary conditions for successful observation are well known, it is hoped that light scattering will be used more frequently in studies of surface electromagnetic waves.

Studies on the effects of surface roughness in the present context have just begun. It is clear that much more careful and extensive studies are needed on different samples and geometries. As a humorous colleague remarked recently, “We have only scratched the surface of the problem”, so far.

Acknowledgments

The experimental work discussed here would not have been possible without the collaboration of my students and colleagues. The experiments

111, App.1 DERIVATION OF THE DISPERSION RELATION 203

were done in collaboration with D. J. Evans, J. D. McMullen, J. B. Valdez, J.-Y. Prieur, G. Mattei and A. Aziza at different stages. Interac- tions with theorists, D. L. Mills, K. R. Subbaswamy, A. A. Maradudin and R. F. Wallis have been most valuable, motivated new experiments in some cases, and overall provided a stimulating environment to work in. I would like to thank them for the important part they played in this research. Also I would like to express my thanks to E. Burstein who stimulated new ideas and provided much valuable advice throughout the course of this research.

The experimental research was supported in part by a grant from the Air Force Office of Scientific Research.

The main portion of this article was written while I was on sabbatical leave at the Institute for Solid State Physics (ISSP) of the University of Tokyo. I would like to thank my colleagues at ISSP, in particular Professor A. Ikushima, for their hospitality and stimulating discussions. I would also like to thank the Yamada Science Foundation, Osaka, Japan, for financial support during my stay in Japan.

Finally, I would like to thank M. Doi for expert typing, text editing and assembly of the reference list.

Appendix: Derivation of the Dispersion Relation for Surface Polaritons and Guided-wave Polaritons in a Double Interface Geometry

The Maxwell’s equations in a charge and current free non-magnetic ( k = 1) medium are:

V * D = O

V*H=O

1 aH V X E = - - - c at

1 aD VXH=--

c at

where

D = EE.

(A.4)

(A.5)

Now we look for surface waves that propagate in the x-direction in the geometry illustrated in Fig. 2.4. Since we assume the three media 1, 2 and

204 LIGHT SCATTERING SPECTROSCOPY [IK App.

3 to be isotropic, we can specialize on the waves that propagate in the x-direction with wave-vector k, without loss of generality. Let us assume the A-Cartesian components of the electric field amplitudes in the three regions of Fig. 2.4 to be as given in eqs. (2.11), (2.12) and (2.13):

EY)(x, t ) = Ahe-a,zei(kxx-w*) (A.6)

EF)(x, t ) = [Bh+eaaZ + B;e-a2z]ei(k=xpW') (A.7)

E? ) (~ , t ) = CAea3zei(kxX-w*). (A.8)

Here, a l , a2 and a j are the z-components of the wave-vectors in the three regions. The main task now is to determine the relationship among a l , a2, a3 and k, (dispersion relation) and the relationship among Ah, €3,' and C, (field amplitudes) by Maxwell's equations and the boundary conditions, and further to elucidate the nature of the surface waves that propagate in the geometry illustrated in Fig. 2.4. In order that the modes propagate along the x-direction and be localized near the two interfaces, we must require that k, be real and a1 and a3 be real and positive; a2 may be either real or imaginary.

By using (A.3) we first obtain the magnetic field amplitudes in the three regions:

ia,c (A.9)

(A. 10)

H =--A w Y

H, =- (a lA, +ik,A,)

H, =-A

ic 0

(A.l l ) ck, l x w y

in region 1 and similarly in region 3:

[ H , = 2 A,.

(A.12)

(A.13)

(A. 14)

111, App.] DERIVATION OF THE DISPERSION RELATION 205

In region 2 we have:

(A. 15)

I - ik, (B:e"2' + B;e-".')] (A.16)

In (A.9) through (A. 17) we suppressed a common multiplication factor exp {i(kxx - ot)} which appears on all the fields. These equations determine the magnetic fields when the relationship among A,, €3: and C, are fixed. Now we look for the relationship among these coefficients.

From (A.l), using the fact that q's are isotropic and uniform in each region, we obtain:

When (A.3) and (A.4) are combined, we have:

1 a 2 0 c a t

V(V. E) - V E + ~ - = 0.

(A.18)

(A.19)

(A.20)

(A.21)

Now we substitute the explicit forms of E f ) in the three regions into (A.21) and use (A.18) and (A.20). Then we have:

(a;- kZ+ &lw2/c2 )A , = 0

(a:- kZ+ &1w2/c2)A, = 0

(a:- k:+ e3w2/c2)Cx = O

(a: - k: + e3w2/c2)Cy = 0

(0;- kZ+ E ~ w ~ / c ~ ) E : ~ ) ( z ) = 0

( a ; - k ; + E ~ o ~ / c ' ) E ~ z ) = 0

(A.22)

(A.23)

(A.24)

(A.25)

(A.26)

(A.27)

206 LIGHT SCA'ITERING SPELTROSCOPY UII, AQQ.

where we have used abbreviations defined by:

Ei2'(z) =B:e"z"+ B,e-"2'. (A.28)

In order to have non-trivial solutions satisfying (A.22) through (A.27) as well as (A.18) through (A.20), we must have:

( ~ i = (k:- E ~ O ~ / C * ) ~ ; i = 1,2, 3. (A.29)

Now we apply the boundary conditions at z = + a. From the condition that the components of the E and H fields parallel to the surface be continuous at z = + a, we obtain the relations:

Axe-"," = B:e"2"+ B;e-a2a (A.30)

Aye-"]" = B:e'za+ B;e-"z" (A.31)

a 1 x A e-"la+ikxA,e-ula= -[a2B:e"2"--a 2 x B-e-"2"-ik,E~"(a)]. (A.33)

The continuity of 0, at the boundary gives:

(A.34)

The continuity requirement for €3, at the surface gives the same relation as (A.31). Next, elimination of A,'s among (A. 18), (A.30)-(A.34) results in three equations for six undetermined coefficients B:'s:

B:e"2"[(al - kZ/al) + a,]+ B;e-"2"[(al - kZ/a,) - a2]

+ B:e"z"( - ik,) + B;e-"."( -ik,) = 0, (A.35)

B:e"z"( 1 + a2/a1) + B;e-"2"(1 - a2/a1) = 0, (A.36)

B:e"za(ik2/a2) + B;e-"2"(ikx/a,) + B:e"Za- B;e-" 2 a = O . (A.37)

The corresponding equations from the boundary conditions at z = - u can be obtained by changing a to - a and a1 to -a3. Thus, we have six undetermined coefficients B: and six linear homogeneous equations. We note here that the y-components are decoupled from the x- and z - components. Thus, we obtain two kinds of modes separated by polarization properties; i.e. y -polarized transverse electric (TE) modes and x-z- polarized transverse magnetic (TM) modes.

The condition for the existence of non-trivial TE (y-polarized) solu-

111, App.1 DERIVATION OF THE DISPERSION RELATION 207

tions is that the secular determinant for the coefficients B: be zero, that is:

(A.38)

In obtaining (A.38), we used (A.36) and the corresponding relation at z =-a. By solving (A.38) we obtain the dispersion relation for TE modes:

(1 + a2/a1)(1 + a2/a3) -eP2-zd(l - az/a,)(l - a2/a3) = 0 (A.39)

e Y 1 + a,/al) e--q 1 - a2/a3)

e--Za( 1 - a2/a,)

e"2"(1+ a2/a3) l = O .

where we used d =2a. Similarly the condition for the existence of non-trivial TM solutions (x-

z-polarized) is obtained from (A.35), (A.37) and the corresponding relations at z = - a in the form of a 4 x 4 determinant:

=O. (A.40)

After some algebra (A.40) reduces to an implicit dispersion relation given by:

(1 +=) a I E 2 (1 + G ) + e - Z + ( a2E3 1 -=) a1E2 (1 -=) a2E3 = 0 ( ~ . 4 1 )

for TM modes. As we have mentioned earlier, the boundary conditions allow both real

and imaginary values for a2. Now we will investigate what conditions are imposed on a2. Let us first examine the dispersion relation for TE modes given by (A.39). This equation can be put into the form:

(a:+a,a,) tanh a,d= -(a,+a3)a2. (A.42)

Let us first assume that a2 is real. Since both a1 and a3 are real and positive, the left-hand side of (A.42) is positive for a2>0 and negative for a,<O. The right-hand side is negative for a2>0 and positive for a2 < 0. Thus, the only real solution for a2 is the trivial one at a2 = 0, and therefore there is n o surface polariton with TE polarization.

208 LIGHT SCAlTERING SPECTROSCOPY [111

Next we assume a2 to be imaginary. Then we can let a2 = ip2 for real

( -p$+aIa3) tan P 2 d = - ( a l + a 3 ) P 2 . (A.43)

A graphic inspection of (A.43) shows that there is a series of allowed values for p2 approaching p2d=fm(7r/2) for m =odd integers as increases. Each allowed value of p2 corresponds to a branch of guided wave polaritons whose amplitude oscillates as exp { *ip2z} inside medium 2.

p2, and (A.42) becomes:

For the TM modes, (A.41) can be transformed into the form:

(A.44)

If e, > 0 in (A.44), the same analysis as in the TE case applies, and there is no real solution for a2 except for a2 = 0. Thus, n o TM polarized surfacepol- ariton solution exists for E~ > 0. However, if E~ < 0 then E ~ ( ~ ~ / E ~ + CYJE~)

is negative and there are solutions with real a2 corresponding to surface polaritons. If a2 is assumed to be imaginary a2 = ip2 again with p2 real, (A.44) becomes:

(A.45) (-P:+----)tan f f I 4 p 2 d = - ~ 2 ( $ + 2 ) p 2 . & I F 2

This equation gives discrete allowed values of p2, each of which corresponds to a branch of TM polarized guided wave polaritons. The allowed values of p2 occur close to p2d = *m(7r/2) for odd integers rn.

Now we can summarize the normal modes of a three layer double interface geometry as follows: Surface polaritons for which a l , a2 and as are all real are TM polarized and the dispersion curves lie in the region where e , = e ( w ) is negative, i.e. between the TO and LO phonon frequencies. When a I and a3 are real and a2 is imaginary (a2=ip2) , the propagation modes are guided waves whose amplitude along the z - direction is oscillatory inside medium 2. Guided wave polaritons can have either TM or TE polarization.

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SCHOENWALD, J., E. BURSTEIN and J. ELSON, 1973, Solid State Comm. 12, 185. SCOTT, J. F., T. C. DAMEN, R. C. C. LEITE and W. T. SILFVAST, 1969, Solid State Comm. 7,

953. SERAPHIN, B. 0. and H. E. BENNETT, 1967, Semiconductors and Semimetals, eds. R. K.

Williardson and A. C. Beer (Academic Press, New York) Vol. 3, p. 509. SUBBASWAMY. K. R. and D. L. MILLS, 1978. Solid State Comm. 27, 1085. TAJIMA, T. and S. USHIODA. 1978, Phys. Rev. B18, 1892. TIEN, P. K.. 1977, Rev. Mod. Phys. 49, 361. USHIODA, S., A. AZIZA, J. B. VALDEZ and G. MAWEI, 1979, Phys. Rev. B19, 4012. USHIODA, S., J. B. VALDEZ, W. H. WARD and A. R. EVANS, 1974, Rev. Sci. Instrum. 45,

VALDEZ, J. B., 1978, Ph.D. Dissertation (University of California, Irvine). VALDEZ. J . B., G. MATTEI and S. USHIODA, 1978. Solid State Comm. 27, 1089. VALDEZ, J. B. and s. USHIODA. 1977, Phys. Rev. Letters 38, 1098. VAN DUYNE, R. P., 1977, J. Phys. (Paris) 38, C5-239. WALLIS, R. F., J. J. BRION, E. BURSTEIN and A. HARTSTEIN, 1974, Phys. Rev. B9, 3424. WEINSTEIN, B. A. and M. CARDONA, 1973, Phys. Rev. B8, 2795. WILLIAMS. M. D. and D. E. ASPNES, 1978, Phys. Rev. Letters 41, 1667. Yu, P. Y. and F. EVANGELISTI, 1979, Phys. Rev. Letters 42, 1642. ZELANO, A. J. and W. T. KING, 1970, J. Chem. Phys. 53, 4444.

(North-Holland, Amsterdam) p. 678.

21, 1530.

(Springer, Berlin) p. 157.

p. 183.

419.


IV

PRINCIPLES OF OPTICAL DATA-PROCESSING

BY

H. J. BUTTERWECK

Eindhouen University of Technology, Eindhouen, The Netherlands

CONTENTS

PAGE

$ 1. INTRODUCTION. . . . . . . . . . . . . . . . , . . . . 213

0 2. FIELD THEORY OF OPTICAL SYSTEMS. . . , . . . . 216

§ 3. SYSTEM-THEORETICAL APPROACH TO COHERENT OPTICAL SIGNAL PROCESSORS . . . . . . . . . . . . 222

§ 4. PARTIALLY COHERENT ILLUMINATION. . . . . . . 227

§ 5. BASIC SYSTEM CONSTRAINTS. . . . . . . . . . . . . 232

9 6. EXAMPLES OF PHYSICAL AND ABSTRACT SYSTEMS 245

5 7. OPERATIONAL NOTATION OF OPTICAL SYSTEMS AND BASIC CASCADE EQUIVALENCES . . . . . . . 252

$ 8. OPERATIONAL ANALYSIS OF OPTICAL SYSTEMS 256

$9. SYSTEMS COMPOUNDED OF LENSES AND SEC- TIONS OF FREE SPACE (5%-SYSTEMS) . . , . . . . . 263

$ 10. SHIFT-INVARIANT SYSTEMS: COHERENT VERSUS INCOHERENT ILLUMINATION. . . . . . . . . . . . . 268

§ 11. RELATED TOPICS. . . . . . . . . . . . . . . . . . . . 275

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . 279

§ 1. Introduction

In communication theory, a transmission system denotes a physical arrangement which, with or without distortion, transmits a signal from a source (transmitter) to a receiver, thereby communicating a certain amount of information. Occasionally one defines a system as the set of its constituents and their mutual arrangement (the “interior”), but in a more common approach a system is viewed as a “black box”, of which only the behaviour at the input and output terminals is studied. In earlier treatments (KUPFMULLER [1948], BAGHDADY [1961], PAPOULIS [1962]) the usually electrical signals are throughout considered as functions of time t , which implies that a system is mathematically defined as an operator

a t ) = T{f ( t ) ) (1.1)

which transforms the input signal f ( t ) into the associated output signal g ( t ) . It is easily recognized that physically realizable systems have to satisfy the fundamental constraints of causality and realness which state that g ( t ) is specified only by the past history of f ( t ) and that any real function f ( t ) is transformed into a real function g ( t ) .

An important class of systems moreover satisfies the requirements of linearity and time invariance which implies that the operator T has the mathematical form of a convolution integral (PAPOULIS [ 19621)

def - g ( t ) = h(f-T)f(T)dT= h(T)f(t-T)dT = h(t)*f(t), (1.2)

where the real weighting function & ( t ) denotes the impulse response of the system. If a linear, time-invariant system is excited by a harmonic signal with circular frequency w, the output is likewise harmonic with frequency w. In mathematical terms, f(t) = exp ( - i d ) is an eigenfunction of a linear, time-invariant system, as appears from

L-

g ( t ) = J h ( ~ ) exp ( - i d + iwT) dT = h ( w ) exp ( - id) , (1.3) 0

213

214 PRINCIPLES OF ORICAL DATA-PROCESSING [IV, § 1

where the complex “system function” h(w) is the Fourier transform of the impulse response:*

h ( w ) = [ h ( ~ ) exp ( i w ) dT = h*(-w). (1.4)

In the present paper, the constraints of linearity and time invariance are assumed to be satisfied throughout. This assumption excludes all effects in the realm of nonlinear optics as well as large-signal nonlinearities in electronic picture processors. Also randomly fluctuating media and fast- modulating electro-optic devices are thus left out of consideration.

If a linear, time-invariant system is excited by a non-monochromatic signal the Fourier transform method applies. With f(o) and g ( w ) denoting the Fourier transforms of f(t) and g ( t ) , we then obtain the simple product relation

= h(o)f(w). (1.5)

However, most non-monochromatic signals in optics have a random character and do not admit a Fourier representation. In such cases the theory of partially coherent light applies (cf. 0 4).

Thus far, the system concept was concerned only with the transformation of time signals. Mainly through the advent of two-dimensional image processing, this concept has been extended in the past decades (O’NEILL [1963], GOODMAN [1968]). Signals are, in addition to their time dependence, also considered as functions of the spatial coordinates and, as such, are processed through electronic or optical systems. From a black box point of view, an image- (or data-) processing system is then defined as any arrangement of electronic scanning devices (T.V. circuitry, two- dimensional digital filters) and/or optical components (lenses, masks, gratings, holograms) which is operated between two suitably chosen reference planes. A two-dimensional light distribution in the “input plane” (the “object”) excites the system and is transformed into another light distribution in the “output plane” (the “image”).

Obviously, the behaviour of electronic scanning systems differs strongly from that of purely optical systems. In an electronic system the outgoing light emerges from a built-in source (coherent or incoherent) and, as such, exhibits no correlation with the incoming light. Unlike an optical system with a strong correlation between the light disturbances (coherent or incoherent) in the two end-planes, no interference phenomena can occur

*The asterisk denotes the complex conjugate. ,

IV, 8 11 INTRODUCI‘ION 215

between the input and output light distributions. In addition, electronic systems which commonly “start” with a video camera and end up with a cathode-ray tube display, process optical data only in one direction with well-defined input and output planes, whereas optical systems are inherently bidirectional processors.

The significance of a general theory of optical systems reaches farther than might be expected from its primary objectives. Since any number of fictitious intermediate planes can be inserted between the input and output plane, the system under consideration can not only be split up into a number of possibly more elementary subsystems, but also can the optical field at each interior point in the system be considered by inserting the plane through that point”. This potentiality explains that even geometric-optical approximations can be elegantly derived from optical system theory (3 11). Further it has been conjectured (MENZEL, MIRANDE and WEINGARTNER [ 19731) that also the human perception of light can be adequately described with the tools of linear system analysis which then provide suitable methods for the experimental determination of the pertinent system properties.

When comparing the signal transformation (“filtering”) in two- dimensional space with that in time, we observe a number of significant differences (O’NEILL [1963]). Since “left”, “right”, “above” and “below” are not preferred by nature, causality has no meaningful counterpart in spatial filters. Likewise, realness is no longer a fundamental constraintt. Linearity will remain an important restriction, but “shift invariance” (the counterpart of time invariance) which will be discussed in § 5.1, has to be considered as a special property, albeit of utmost significance.

To avoid ambiguities in the presentation, the main article is concernea only with optical systems excited by light of strictly harmonic time dependence. Due to linearity and time invariance, all field quantities inside the optical system are then time-harmonic, too. Partially coherent illumination will be discussed in § 4. There we shall show that much of the formalism developed thus far also applies to the extreme case of incoherent illumination. Furthermore it appears that electronic scanning systems

* In principle, a theory of optical systems can also be set up for curved input, output, or intermediate surfaces. Difficulties with respect to the choice of suitable coordinate systems have hitherto prohibited a practical elaboration of that idea (with the incidental exception of spherical surfaces).

t This is true only for coherent processors. For incoherent and electronic processors t h e input and output signals are not only real, but also nonnegative (cf. P 10).

216 PRINCIPLES OF OPTlCAL DATA-PROCESSING [IV, I 2

can be described in much the same way as incoherently illuminated systems.

0 2. Field Theory of Optical Systems

2.1. THE DATA-PROCESSING MODE

An optical system contains linear, time-invariant, source-free matter. As such, the electromagnetic field inside the system obeys Maxwell’s equations which, for harmonic time dependence proportional to exp(- id) , read as

curl H = -ioe - E , (2.1 a)

curl E = i w p - H. (2.lb)

Like the complex electric and magnetic field vectors E, H, the material properties as reflected by the tensor functions E and p are dependent on position (x, y, 2). For isotropic media E and p degenerate into scalar functions E(X, y. z ) and p(x, y, z ) which, due to potential dispersion, may also be complex functions of frequency.

As indicated in Fig. 2.1 (shaded part), the interior of the optical system is characterized by a certain distribution of matter (E, p). contiguous to a vacuum region (E = E ~ , p = wo) in the vicinity of the end-planes. The latter assumption, which fairly corresponds to actual realizations of optical

\ vacuum / Fig. 2.1. Geometry of a general optical system.

IV, I21 MELD THEORY OF OPTICAL SYSTEMS 217

systems (with “vacuum” replaced by “air”) yields a considerable simplification of the further analysis. The two end-planes carry two parallel coordinate systems xl, y1 and x2, y2. A point in either plane i ( i = 1,2) will be indicated by a two-dimensional position vector r i = ( x i , y,) and a surface element dxi dy, will be shortly denoted by dr,. If a three- dimensional coordinate system is required, we choose an x and y axis coinciding with the x2 and y2 axis and a z-axis pointing to the exterior of the optical system.

In either reference plane we have four two-dimensional distributions of electromagnetic field quantities, viz. Ex, E,, H,, H,. The normal components E,, H, can be left out of consideration, because they are directly related to the tangential components, due to Maxwell’s equations: E, -

Concerning the tangential components a well-known theorem of resonator theory (SLATER [1954], GOUBAU [1961], BORCNIS and PAPAS [1955], KUPRADSE [1965]) states that the electromagnetic field in a cavity is uniquely determined if the tangential component of E or H is prescribed on the boundary surface. The same is true, when the tangential component E,,,, is prescribed on part of the boundary surface and Iftang is prescribed on the complementary part. Finally, on part of the boundary also an impedance boundary condition of the type Etang = Zw(Htang X n) can be imposed, where Z, and R denote the complex wall impedance and the outward normal vector, respectively.

For our optical system the two end-planes plus a cylindrical surface (cf. Fig. 2.2) with an infinite radius h constitute the boundary surface. The radiation field on the infinitely remote cylindrical surface locally resembles a plane wave* with a boundary impedance Z, = G. Hence we can conclude that in either reference plane Etang or H,,,, can be prescribed ad libitum and that the total electromagnetic field in the optical system including the remaining quantities in the reference planes is then uniquely determined?.

dHx/8y -dH,ldx; H, -dE,/dy -dEJdx.

* It is tacitly assumed that the material part (E f q,, p f go) of the optical system has bounded axial and transverse dimensions.

t It can be easily recognized that the electromagnetic field must be unique for a given E,,,, or ITtang in the reference planes. ,Assume that there are two solutions. Then the difference solution has to satisfy Maxwell’s equation with a vanishing E,, , , or H,,,,, in the reference planes. This implies that there also the normal component of Poynting’s vector vanishes. Since then no power is fed into the system, which has to account for energy dissipation in the material and radiation losses, the total difference solution vanishes, too, Q.E.D. We remark, however, that this simple proof does not guarantee the existence of a field solution under the prescribed boundary conditions.

218 PRlNCIPLES OF OPTICAL DATA-PROCESSING [IV, P 2

/ plane 1

\ plane 2

Fig. 2.2. Optical system with supplementary cylindrical surface.

This view seems to be in contradiction with the common idea that in an optical data-processing system the field is uniquely determined everywhere, when the tangential electric or magnetic field is prescribed in the input plane only. To resolve this contradiction we have to realize that in the processor “mode” the output plane is assumed to be contiguous to a source-free half-space with E = E ~ , p = po, into which electromagnetic waves are radiated. This implies that the tangential electric and magnetic fields in the output plane are linked to each other by an impedance boundary condition. However, unlike the impedance discussed above, this boundary condition is non-local: At a certain point P, Etang(P) is not only determined by Htang(P), but also by neighbouring values of Htang. As discussed further down, this generalized impedance becomes local only for fields E,;,, and H,,,, with sufficiently slow spatial variations.

In an electric network analogy we can compare an optical system with a two-port (two-terminal pair network) whose electric state is described by two voltages and two currents (VAN VALKENBURG [1965]). The two-port properties find expression in two equations which implies that at either port one quantity (voltage or current) can be chosen ad libitum, thereby determining the remaining quantities. When one port is terminated with a certain impedance the voltage (or current) at the other port completely determines the electric state of the network; thus, like the situation in an optical processor, an input-output relation is established.

In the data-processing mode an optical system radiates at the output plane into free space. In other words, there are no external reflections. In

IV, B 21 FIELD THEORY OF O!TICAL SYSTEMS 219

a common approximation one further assumes that free space can be replaced by another optical system without disturbing the transmission properties of the original system. Then the undisturbed output of the first system forms the excitation of the second, and a simple formalism can be developed for the cascades of optical systems. The restriction under consideration can be referred to as “absence of internal reflections”; it will be presumed to be satisfied throughout*.

The tangential electric or magnetic field in the input plane forms the excitation of an optical processor. Likewise we can look at the tangential electric or magnetic field in the output plane. Since we suppose that human light perception and photographic registration is intimately associated with the intensity of the electric field, we can henceforth focus our attention on E alone. In this perspective Eta,,, in the input plane “causes” Eta,, in the output plane. With the observation that an optical system is inherently bidirectional, we can formulatet the input-output relations (VAN WEERT [1978])

c

(2.2a)

(2.2b)

where (2.2a) and (2.2b) pertain to the two directions of transmission 1 + 2 and 2- 1. These relations which form the basis for all theory of optical data-processing systems. are a direct consequence of the principle of superposition, valid for any linear system. The tensor functions g2, and g 12 completely reflect the data-processing properties of the system under consideration.

In the following we further adopt the “scalar” approximation, in which the “cross-polarization’’ between orthogonal field components is neglected. In this approximation the x-component of E in the input plane does not excite a y-component in the output plane and vice versa. In other words, the tensors gZ, and g,, become diagonal and the optical processor corresponds to two independent systems with only x-x and y-y couplings. If we then focus our attention on one of these systems, we can

*In the special case of an aperture in an otherwise opaque screen the assumptions discussed here also form part of Kirchhoff-Huygens’ principle (STRATTON [1941]). In fact, they are the basis of the Fresnel-Kirchhoff diffraction formula (BORN and WOLF [1965]).

T If not stated otherwise, all integrations extend over the entire plane.

220 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, B 2

replace (2.2) by

(2.3a)

&(rJ = 1 g12(r1, r2)42(r2) dr2, (2.3b)

where the scalar function + stands for the x or y component of the electric field vector, and the scalar functions g,,, g,, describe the transmission properties in the two directions 1 + 2 and 2 + 1. In the following, the conditions for the validity of the scalar approximations are assumed to hold throughout. Apart from birefringent media which form inherently “vectorial” systems the approximately scalar character of most optical systems is related to the fact that the characteristic physical dimensions of the usual components considerably exceed the wavelength of light. Even in grounded glass with a very fine structure cross- polarization can hardly be observed (BASTIAANS [ 1979a1). It should be mentioned, however, that the scalar treatment of optical systems basically violates the principle of reciprocity (VAN WEERT [1978]) and, apart from a few exceptional cases, constitutes not more than an approximation.

2.2. THE RECIPROCITY THEOREM

Two different solutions of Maxwell’s equations (2.1) E‘”, H‘”, E(*’, H‘,’ satisfy the reciprocity theorem (STRATTON [ 19411)

valid for any closed surface A provided that the tensors p and E are symmetrical (which includes the degenerate case of scalar functions I*. and E ) . Physically, this condition is violated only in the presence of a static magnetic field (CASIMIR [1963]), e.g. in a Faraday rotator. As in other branches of applied physics (e.g. radio communication, electrical networks) we expect that reciprocity imposes certain restrictions upon the behaviour of an optical system when operated as a signal processor.

Suppose that for the system under consideration the scalar approximation applies. Moreover the scalar input and output functions are assumed to vary so slowly that in a spatial Fourier representation (cf. § 3) the

IV, P 21 FIELD THEORY OF OWICAL SYSTEMS 22 1

highest spatial frequencies “contained” in the signal are small compared with the wave number k = 2r/A = “6. VAN WEERT [1978] has shown that this “paraxial” approximation just implies the validity of the former scalar approximation.

The fields E, H with superscripts 1 will now be associated with the transmission 1 + 2 of the optical processor. In either plane, @& is assumed to have only an x-component. Due to the slow transverse variations of E y ) ef +(’) a locally plane wave is excited in the output plane with HL1’=O and H I ” = G E P ’ . As the system is free from internal reflections the same relation holds for the input plane. If the system is operated in the opposite direction 2 -+ 1, the associated fields are indicated by a superscript 2. All plane waves travel in the opposite direction and we have a remarkable sign change: HY) = - G E F ) .

With the geometry of Fig. 2.2, the two end-planes supplemented by a cylindrical surface with radius h + 00 form the closed surface in the reciprocity integral (*). Due to the asymptotic behaviour of E and H for great distances one easily estimates (SOMMERFELD [ 19541) that the infinitely remote cylindrical surface does not yield a contribution to (*).

There remains

(E(”H(2) - E(2)H‘’)) dr + (EYJH(2) - E(2)H(1)) dr - 0 -1 lane 1 lx l y Ix l y 1 L I a n e 2 2 Y 2x 2 y 2 - .

With E , = 4 and the above relations between E, and Hy, and after division by the common factor 2=, we obtain

I +y)+y) dr, = I +k1)+k2’ dr,. (2.4) plane I plane 2

This important reciprocity relation valid in the scalar and paraxial approximation can be readily interpreted in terms of the characteristic functions g,, and g12 of the optical system. Assume point excitations 4:”= 6(rl - a) in plane 1 and 4:”=6(r2-b) in plane 2 . Then (2.4) yields

+:“’(a) = +Y)(b). (2.5)

On the other hand (2.3) states that $?)= 6(rl -a) causes +Y)= g21(r2, a) and &) = 6(r2- b) causes 4y) = g12(rl, b) so that (2.5) implies

g12(a, 6) = gzi(b, a).

Since a and b are arbitrary, we can finally conclude that

222 PRINCIPLES OF OFTICAL DATA-PROCESSING [IV, 5 3

Hence, the transmission from a point r l in plane 1 to a point r2 in plane 2 equals that in the opposite direction.

This simple result is only valid in the paraxial approximation. If the input and output signals contain high spatial frequencies, the scalar treatment in general fails; but even if the scalar assumption is taken for granted (as in acoustical systems), (2.6) has to be properly modified (BUTIERWECK [1978]). Then it turns out that only with respect to their macroscopic structure, g,, and g,, are equal. When “viewed” through an instrument with a spatial resolution of a few wavelengths or less, one observes a considerable difference in fine structure.

0 3. System-theoretical Approach to Coherent Optical Signal Processors

3.1. INPUT-OUTPUT RELATIONS IN SPACE AND FREQUENCY DOMAIN

In this section we consider an optical signal processor from a pure “black box” point of view. In this approach the light vibrations in the two reference planes are assumed to be describable by two scalar, complex- valued functions +l(x, , y l ) and &(x,, y,) of which one plays the role of excitation and the other that of response*. Again the system is assumed to behave reflexion-free but, on the other hand, no a priori assumptions are introduced with respect to reciprocity and paraxial approximations.

Henceforth, all two-dimensional signals are equivalently described in the space and frequency domain. For any signal, a Fourier transform pair +(x, y), @(X, Y ) is defined according to

@ ( X Y) = 4(x, y ) exp [WXx + Yy)] dx dy, (3.la)

Again we mark a “space point” by a position vector r = (x, y), whereas a spatial “frequency point” is marked by a vector R = (X, Y). Then (3.1)

* As discussed in B 2, this scalar approach applies to many optical systems, exactly or approximately. On the other hand, a theory of acoustic systems is inherently scalar, with identifiable as the sound pressure.

IV, P 31 SYSTEM-THEORETICAL APPROACH 223

can be written more compactly:

@(R) = J 4 ( r ) exp ( G R - r ) dr, (3.2a)

(3.2b)

We agree that the upper signs in the Fourier transformations refer to the transmission 1 + 2, and the lower signs to the transmission 2 + 1. This double sign convention (BUTTERWECK [1977]) will yield a number of formal advantages in the further course of this section. The ultimate reason is, however, of physical nature: we want to identify spatial frequencies with directions in three-dimensional space. Let us consider, for instance, the illumination exp [i(X,x + Y,y)] and inquire, how this two-dimensional plane wave has to be continued in three-dimensional free space. This continuation satisfies Helmholtz’s equation A 4 + k2+ =

0, with k = w 6 , and obviously represents the three-dimensional plane wave (assume Xi + Yo2 s k2)

(*) exp [i(X,x + Y,y + Z,z)]

with

ZO=Jk2-X:- Y;.

When we now require that the wave propagates in +z-direction the positive Z, value and the upper signs in (3.1) have to be chosen, corresponding to a 1 + 2 operation. The spectral description 47r2 S(X- X,, Y - Yo) of our illumination then follows from (3.la). In the 2 4 1 operation the same function corresponds to exp [-i(X,x + Y,y)] with exp [-i(X,x + Yoy + Zoz)] as three-dimensional continuation, propagating in (-2) direction. Obviously this is the conjugate of the former wave (*), and as such it has the same equiphase planes but opposite direction of propagation. If the same signs in (3.1) had been used for either operation 1 + 2 and 2 + 1, the last result would have become exp [i(X,,x + Yoy - Zoz)] which corresponds to a completely different direction of propagation in three-dimensional space. In our notation the frequency pair X,, Y,) corresponds to a direction uniquely described by the “wave vector” k = (X,, Yo, +dk2 - Xz - Y;) with the understanding that in the two modes 1 -+ 2 and 2-+ 1 the wave propagates in the directions k and (-k), respectively.

224 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, D 3

In many practical problems functions +(x, y ) with rotational symmetry occur. With r = m we then have $(x, y ) = +,(r). Then also the Fourier transform exhibits rotation symmetry: @ ( X , Y ) = @,(I?) with R = m. The Fourier transformation (3.1) then degenerates into a Hankel (or Fourier-Bessel) transformation (GOODMAN [ 19681) as given by

(3.3b)

where J o ( . ) denotes the zero-order Bessel function. Note that (3.3) holds for either sign in (3.1).

As we have learned in 9 2, the signal transformation in a linear optical processor is governed by a superposition integral. Due to (2.3) we have

+2(r2) = j g21(r2, r1)41(r1) drl ,

+l(rl) = j g12(rI9 r2)42(r2) dr2,

1 - 4&) = 271. I, @,(R)J,(Rr)R dR,

(3.4a)

(3.4b)

for the two directions 1 + 2 and 2 + 1 . Of course, there are similar relations in the frequency domain, viz.

@z(&) = 5 G2,(R2, Rl)@,(R1) dR1, (3.5a)

W R J = G12(Rl, R2)@2(R2) dRz, (3.5b)

where the G-functions are coupled to the g-functions due to (3.2). We obtain

with similar relations (only i is replaced by -i) for the second pair g,,, GI,. The highly symmetrical transformations (3.6) (which have to be carefully distinguished from ordinary Fourier transformations) will henceforth be referred to as “mixed” transformations.

IV, § 31 SYSTEM-THEORETICAL APPROACH 225

Each of the pairs g,,, g,, and GI,, G,, provides a complete characterization of the optical system. These complex-valued weighting (or Green’s) functions have simple physical interpretations: the g’s are point spreads (the counterparts of impulse responses in time domain and thus responsible for the blur in an optical system) and the G’s are wave spreads (spectral representations of responses to plane-wave excitations).

In the special case

(3.7a)

(3.7b)

for which +1=+2 , we have a “through connection” or an “identity system”. Notice that g,, and G,, form a mixed pair according to (3.6) and not a Fourier transformation pair!

3.2. CASCADES AND INVERSE SYSTEMS

If two systems are placed in cascade, the output of the first system forms the input of the second (cf. Fig. 3.1). Considering only one direction of transmission we then have

and, after elimination of the intermediate signal +,(r2),

1 2 3

Fig. 3.1. Cascade of two optical systems

226 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, P 3

with

g,,(r,, rl) = j g32(r3, r2)g21(r2, r l ) dr2. (3.11)

We notice that, apart from some important exceptions to be discussed below, a change of the order of the individual systems in a cascade also changes the properties of the overall system. Further, the cascade formulas can be extended to more than two subsystems. For n systems (3.11) then becomes an (n- 1)-fold integral in which integrations are carried out over all intermediate planes with surface elements dr,, dr,, . . . dr,.

Finally, we consider the question whether for a given output signal an associated input signal can be found. Mathematically, this amounts to the solution of an integral equation. If and only if it can be solved (in D 5 a number of explicitly solvable cases is compiled), the inverse signal transformation can be written in the form

Although these equations must not be read as cause - effect - relations (the transmission remains in the direction 1 +. 2) we can associate with them a fictitious inverse optical system whose weighting functions are gil(rl, r2) and Gi1(R1, R2) and which operates in the 2 -+ 1 direction. The question whether the inverse system can be realized belongs to the domain of system synthesis and can, for the moment being, remain unanswered.

The cascade of the original and the (fictitious) inverse system yields, of course, the identity system. Since this is one of the few cascades where the order of the subsystems may be interchanged, we obtain with (3.7) and (3.11) the two equivalent relations

(3.13b)

Similar relations can be derived for the pair G,,, Gil and also for the

IV, I41 PARTIALLY COHERENT ILLUMINATION 221

functions corresponding to the 2 + 1 direction. Unlike the relations (3.12a) where also the signals &, &, occur, (3.13) should be viewed as the definition* of the point spread of the inverse system and as the starting point for its evaluation.

§ 4. Partially Coherent Illumination

4.1. SPECTRAL TREATMENT OF PARTIAL COHERENCE

In the preceding sections time-harmonically illuminated optical systems have been the objects of field-theoretical (Q 2) and system-theoretical (Q 3) investigations. In this section we temporarily exchange time- harmonic signals for signals associated with stationary random processes. In optical terms, we deal with partially coherent light including the extreme cases of complete incoherence and complete coherence. In our treatment we adopt the modern frequency-domain approach developed by MANDEL and WOLF [1976] and BASTIAANS [1977]. This approach saves a number of necessary quasi-monochromaticity assumptions of former theories (BERAN and PERRANT [1964]).

Resuming the lines of 9 1, we consider a single-input-single-output, linear, time-invariant system with impulse response h( t ) and system function h(o). If a random signal f ( t ) excites such a system, the convolution integral (1.2) remains valid for the determination of the output signal g ( t ) . Since, however, a random signal does not possess an ordinary Fourier transform, the simple product relation (1.5) in the frequency domain becomes meaningless. On the other hand we are, in general, interested in statistical averages rather than in the detailed structures of the pertinent time functions (this is certainly true for optical signals of which, due to the high frequencies involved, not more than the mean- square values can be measured). Such a statistical average is the autocorrelation function

Ef(T) = ( f ( t ) f ( t - T ) ) ,

where ( ) denotes ensemble or (due to ergodicity) time averaging. From (1.2) we can easily derive the autocorrelation function of the output signal

*The two equations (3.13) are analogous to the two definitions of the inverse of a matrix A, viz. AA-' = 1 and A-'A = 1 with 1 the unit matrix. These relations are equivalent, to be sure, but their numerical elaboration leads to a different set of equations.

228 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, 0 4

g ( t ) from that of the input signal f( t ) as

c, (7) = h(7) * Sy(7) * h( -T ) ,

where * denotes convolution with respect to the time shift variable T. Notice that this relation involves only deterministic functions and easily admits Fourier transformation. With the power spectra S,(w) and S f ( w ) defined as the Fourier transforms of the pertinent autocorrelation functions we simply obtain

S , ( O ) = lh(w)12 S f ( W ) , (4.1)

i.e. the power spectrum of the output signal is obtained from that of the input signal through multiplication by the squared magnitude of the system function.

The necessary generalization to multiple-input-multiple-output systems is plain sailing. Assume that a system with N inputs is excited by functions $ ? ( t ) (j = 1,2, . . . N) and that these are transformed into N output functions &?‘(t) ( I = 1,2 , . . . N) such that through a generalization of (1.2)

N

&‘“‘(t)= 1 hii(t)*4?(t), ( 1 = 1, 2 , . . . N). j - 1

Then we obtain

where

. . .

is called the (cross-)correlation function” which, for j = k , degenerates into the autocorrelation function. In optics, where the signals 4 are functions of position r, i j k ( 7 ) = (&rj, t ) & * ( r k , t - 7)) is usually referred to as the (mutual) coherence function. The frequency-domain equivalent of (4.3) reads as

j = l k = l

* Notice that (4.4) also applies in the case of complex time functions like the “analytic signals’’ (BORN and WOLF [1965]).

IV, § 41 PARTIAFLY COHERENT ILLUMINATION 229

with

the (mutual) power spectrum which for j = k degenerates into the (real, non-negative) auto-power spectrum.

An optical signal processor is a system with an infinite number of inputs and an infinite number of outputs. The discrete variables 1, j in (4.2) then become the continuous variables r,, rl in (3.4), hij(w) becomes gZ1(r2, r l ) (which is also a function of frequency w ) , and the sums pass into integrals.

The continuous analogue of (4.5) can then be written as

S(r& r;, w ) = II g2!(r$, r;, w)S(r;, ry, o )gz l ( r i , ry, w ) dr; dry. (4.7)

A number of conclusions can be drawn from this fundamental relation. First we recognize the importance of second-order statistics: the output mutual power spectrum can be uniquely determined from the input mutual power spectrum provided that the system properties are known. The measurable total intensity at a certain point r2 in the output plane then follows from (4.7) by consideration of r2 = r; = r;. With the inverse of (4.6) we obtain:

- intensity at r2 = mean-square value of 4(r2, t)

= S(r2, r2, 0 ) = - S(r2, r2, w ) dw. (4.8) 27T ‘ I

Further, we notice that (4.7) is linear in the mutual power spectrum. As such, it is the generalization of (3.4) for the case of partially coherent illumination. It is important to recognize that the time-harmonic analysis of an optical system is entirely sufficient for the prediction of its behaviour under partially coherent illumination. With the exception of narrow-band excitation (for which S ( r ; , fl, w ) resembles a Dirac pulse around w=w,, ) , we need, however, have disposal of the complete frequency characteristics of the system, as given by gZ1(r2, r l , w ) . Anyway, due to the “universality” of the optical system functions g2,(r2, r l , w ) we can conclude that the system properties are inherently independent of the light coherence and that, more particularly, two systems equivalent for coherent illumination are equivalent for any degree of partial coherence.

230 PRINCIPLES OF OPTICAL DATA-PROCESSING DV, 5 4

4 . 2 . INCOHERENT ILLUMINATION

The important degeneracy of a vanishing mutual power spectrum for r\ # ry corresponds to incoherent illumination. We then have

S(r\, r;, o) = p(ri, w ) 6(r; -r;) (4.9)

for the input power spectrum, where p(rl, o) denotes the (auto-) power spectrum* at the point rl. Insertion into (4.7) yields

which relation allows the conclusion that incoherent light in general does not remain incoherent: the light vibrations at two points in the output plane receive contributions from all points in the input plane and hence exhibit a certain degree of correlation.

Often, one is merely interested in the auto-power spectrum in the output plane:

This famous relation is usually referred to as “superposition of power” for incoherent illumination. We note, however, the difference of symbols: S(r2, r2, w ) is a true auto-power spectrum, whilst p(rl, w ) was only shortly denoted as such. Strictly speaking, S(rl, r l , w ) is infinite with a finite S(r2, r2, w ) , which puts the poor efficiency of incoherently illuminated systems into evidence. Actually, a finite “correlation area” is required in the input plane, in order to produce a nonvanishing response in the output plane?.

Formally (4.11) exhibits a certain resemblance with the superposition integral (3.4) for the time-harmonic case. The main difference is that the input and output signals as well as the weighting function lg21(2 are real

* Strictly speaking, the auto-power spectrum of incoherent light according to (4 .9 ) becomes infinite. However, we do not wish to introduce a new name for p which has most properties in common with the power spectrum.

? The necessity for a finite correlation area is also revealed by dimensional considerations. With ( 3 . 4 ) the dimension of g is (area)-’ and that of lgI2 is (area)-2. If, as is done in many textbooks, p in (4. t1) is replaced by S, we obviously need an additional constant multiplier with the dimension of an area. This is just the correlation area as mentioned above.

IV, 5 41 PARTlALLY COHERENT ILLUMINATION 231

and positive”. Since the positiveness is difficult to translate into the frequency domain the spectral counterpart (3.5) of (3.4) is seldom constructed in the incoherent case. An important exception is the shift- invariant system discussed in § 5. The “modulation transfer function” introduced for these systems will be treated in § 10.2.

For two systems arranged in cascade, caution has to be used when applying (4.11). After the light has passed the first system, incoherent light has become partially coherent, and the general formula (4.7) applies. If this transformation incoherence + partial coherence would not take place, no imaging with incoherent light would be possible. This implies that in the relation (3.11) for the cascade of two systems g must not be replaced by 1gI2 for incoherent illumination?. Rather, one has first to employ (3.11) with the complex g-functions and take the squared magnitude after that.

4.3. COHERENT ILLUMINATION

An interesting special situation occurs when an optical system is illuminated by a point source located at r l = a. Then we have to insert

S ( r i , r;l, w ) = 6(r{ - a, r ; - a)q(w) (with q(o) 2 0 ) (4.12)

into (4.7) and thereby obtain

(4.13)

with

t ( r , 0) = g2,(r, a, w)JiGJ. (4.14)

Light, whose mutual power spectrum can be factorized according to (4.13) is referred to as coherent, although not necessarily monochromatic. It has the remarkable property that it remains coherent after passage through linear systems. Indeed, if the input power spectrum can be

* The present theory applies also to electronic scanning systems if they provide a linear intensity mapping. Of course, the transformation from incoherent into partially coherent light, as arises in pure optical systems, does not occur here. A possible combination of blur and nonlinear behaviour leads to complications which are beyond the scope of this article.

t The situation is different when in the intermediate plane a rotating diffuser is inserted which removes the partial coherence.

232 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, B 5

factorized according to (4.13) we obtain from (4.7)

and the output power spectrum can be likewise factorized:

S(r;, r;, w ) = t2(r;, w)tT(r’l, o)

t2(r2, w ) = 1 gZ1(r2, r l r w)tl(rl, a) drl .

(4.15)

with

(4.16)

Hence, the function t(r, w ) transforms like our former function &(r) according to (3.4) so that we are justified to state that coherent light propagates in much the same way as strictly time-harmonic light or, more rigorously, as deterministic, Fourier-transformable light pulses. For (4.16) would exactly apply in the latter case with t(r, o) denoting the Fourier transforms of the signals under consideration.

In this section we have abandoned the consistent space-frequency approach of the previous time-harmonic treatment. The reason is two- fold: First, partial coherence is only a collateral subject within the framework of optical system theory, and second, the dual of incoherent light, the spatially stationary light, is of minor practical significance. However, in connection with the somewhat more specialized case of transmission of incoherent light through shift-invariant systems we shall resume the space-frequency dualism in FI 10.

§ 5. Basic System Constraints

5.1. SINGLE CONSTRAINTS

After this “stochastic” intermezzo we return to strictly harmonic time dependence and resume the line of P 3. In this connection we consider a number of basic restrictions which can be imposed upon the system behaviour and which can be expressed in terms of conditions for the weighting functions g,,, g12. As these are linked to the corresponding frequency functions GZ1, GI, via the mixed transformations (3.6) each condition applying to the g’s has a frequency pendant applying to the G’s. The pertinent checks for the correctness of these various interrela- tions are left to the reader. At the end of this section some physical

IV, § 51 RASIC SYSTEM CONSTRAINTS 233

systems are considered which satisfy the constraints in question with a certain degree of accuracy.

The reciprocity condition forms a link between the two directions of transmission, as formulated by

gZl(r2, rd = g d r 1 , r2Ir

G21(R2, RI) = Gi,(Ri, R2).

(5.la)

(5.lb)

In 5 2 we have shown that systems filled with isotropic material do approximately satisfy this condition. Notice that we profit by the double sign convention introduced in (3.2). With a uniform sign (5.la) would have transformed into the asymmetrical condition GZ1(R2, R,) =

Gi2(-Ri7 -RJ. The remaining conditions discussed in this section apply to a single

direction of transmission. which henceforth will be chosen in the 1 + 2 mode. This does not exclude that the same condition holds for the 2 + 1 direction, too. In that case the condition under consideration will be said to be satisfied completely.

Next we consider losslessness” which expresses equality of input and output signal “energies”:

(5.2a)

(5.2b)

When we insert (3.4a) in (5.2a) and require that the resulting identity holds for all input signals 41(rl) , we obtain (with the same steps in the frequency domain)

g21(r2, r d g ? ~ h , r;) dr2 = 6(r l - r 3 (5.3a)

(5.3b)

as necessary and sufficient condition for the validity of (5.2). We note, in passing, that if (5.3) is satisfied for all temporal frequencies w, even the

* This signal-theoretical definition has not an immediate physical meaning. Physical losslessness involves the time average of Poynting’s vector EX H, but it can be shown that in the scalar, paraxial approximation its normal component is proportional to 1&1*. See also VAN WEERT [1980].

I G21(R2, RI)G?I(Rz, Ri)dR,= W - R ; ) ,


time-space integrals of the squared input and output signals are equal (in the case of light pulses), or the space integrals of the temporal mean- square values are equal (in the case of “power” signals including partially coherent signals). For partially coherent illumination* the pertinent proof can be given with the aid of (4.7).

A concomitant property of utmost importance is easily found for lossless systems: they admit an explicit determination of their inverse systems. Comparison of (5.3) and (3.13) reveals that

The next property to be discussed is symmetry. Owing to the mixed transformation formulas (3.6) we have to distinguish two types. Spatial symmetry is defined by

whereas spectral symmetry is defined by

Spatial and spectral symmetry are not mutually excluding properties. Indeed, a number of important systems are spatially and spectrally symmetric (cf. B 6). In order to give an idea about the physical meaning of spatial (spectral) symmetry we can state that the transmission from a space (frequency) point A in the input plane to a point B in the output plane equals that from the projection of B on the input plane to the projection of A on the output plane.

A subclass of spatially symmetric systems is formed by the spreadless systems, whereas the shift-invariant systems are a subclass of the spec-

* This result holds also for incoherent illumination. This seems to be in contradiction with the poor efficiency observed in connection with (4.11). The paradox is resolved by the fact that incoherent light has an enormous blur in comparison with coherent light so that the integral of the autopower spectrum ultimately remains the same. Also note in this connection that, due to (5.3) the integral of lg2 , ( r2 , r,)12 over r z becomes infinite, which effect compensates the seemingly poor efficiency due to (4.1 1).

IV, § 51 BASIC SYSTEM CONSTRAINTS 235

trally symmetric systems. Their characteristic functions are given by

(5.7a)

(5.7b)

M2,(R) = 1 mZ1(r) exp (-iR * r) dr (5.7c)

for spreadlessness, and

= hZl(r) exp (-iR r) dr (5 .8~)

for shift invariance. Similar relations hold for the 2 + 1 mode provided that (4) is replaced by (+i) in (5 .7~) and (5.8~).

We first discuss the behaviour of these important systems in the space domain. Inserting (5.7a) in (3.4a) yields

42(r) = m21(r)41(r) (5.9)

for a spreadless system, i.e. a local relation between the input and output signal, without blurring effects. Such a system therefore acts as a (amplitude and/or phase) modulator. For a shift-invariant system we obtain the spatial convolution

42b) = h21(r) * 41(r),

i.e.

4 2 ( 4 = 1 ~ ( r 2 - r 1 ) 4 1 ( r ~ drl (5.10)

from (5.8a) and (3.4a) which reflects the inherent property that if &(rl) is replaced by &(rl -a) ( a is a constant vector), the corresponding output signal changes from &(rJ to 42(r2- a). Hence, the image is shifted over the same (vectorial) distance a as the object, without change of its form. This is in contrast with shift-variant image formation (LOHMANN and PARIS [1965]), where the structure of the image depends upon the location of the object.


If we compare the spatial behaviour of these systems with their spectral behaviour, as given by (5.7b) and (5.8b), we have

(5.11)

for the spreadless and shift-invariant system, i.e. the signal transformation for one system in the space domain has the same character as that for the other in the frequency domain. The two types of systems are said to be dual in a wide sense (PAPOULIS [1968b]), whereas we speak about strict- sense duality if moreover the functions m2,(r ) and H21(R) (and herewith Mzl(R) and hZ1(r)) are similar with respect to their mathematical structure. Due to this definition, free space and a lens are, within their respective approximations, dual in the strict sense (cf. Q 6). Besides the inherent spatial symmetry of spreadless systems and the inherent spectral symmetry of shift-invariant systems the complementary symmetry condition can also be satisfied. Obviously this occurs, when h21(r) or Mzl(R) are even functions.

When a shift-invariant system is illuminated with a plane wave exp (iR * r ) insertion into (5.10) yields H21(R) exp (iR - r ) for the output signal which allows the conclusion that exp (iR - r ) is an “eigenfunction” of a shift-invariant system with the “system function” H2,(R) as proportionality factor (eigenvalue). Indeed, this property forms the background for the simple product relation (5.12) in the frequency domain, which states that there n o frequency “mixing” takes place. Dual statements hold for the spreadless system.

Either type of systems is of practical importance: spreadless systems are always associated with small axial dimensions (transparencies and thin lenses belong to this category), whereas shift-invariant systems are often required for optical filtering (deblurring, matched filtering, differentiation, contrast improvement etc.), but also, albeit in special forms, readily provided by nature (free space and, with some restriction, diffraction- limited imaging). Numerous examples, including the problems encountered with incoherent illumination will be treated in forthcoming sections.

The last restriction to be discussed in this section is that of rotation invariance. Unlike the former restrictions it applies only to genuinely two-dimensional systems. In analogy to shift invariance, it means that with an object rotation round the origin r = O an equal-angle image

IV, 9: 51 BASIC SYSTEM CONSTRAINTS 231

rotation is associated, with the image structure maintained. First considering a general system and introducing polar coordinates x = r cos a, y =

r sin a, we rewrite (3.4a) as

(5.13)

If now rotation invariance is required, the response to a point source at r I = r,, a,=O has to be rotated by an angle a, if the source moves to rl = ro, a t = a,, i.e. gz1Jr2, r,, a2, 0) becomes g21.p(r2, r,, a2- al, 0 ) =

g21.p(r2, r,, az, al ) . This implies that

g21.p = q21(r2r r 1 7 f f2- a,) (5.14)

is a function of a2-a1 only so that the integration over a1 in (5.13) degenerates into a convolution.

Assume now that the rotated object is proportional to the original object, then, on account of linearity and rotation invariance, the same statement holds for the image with an equal proportionality factor. Functions with this property have the general shape

exp (irna)f(r), rn = 0, *l, &2, . . . .

Hence, an input function with an angular dependence according to exp(irna) is transformed into a similar function at the output. Note, however, that the radial functions need not be equal so that in general the total function is no eigenfunction. With rn = 0 we find that functions 4 with rotational symmetry retain this property after passage through a rotation-invariant system.

Since a rotation in the space domain r corresponds to an equal-angle rotation in the frequency domain R, there is no sense to seek for the dual system: a rotation-invariant system is its own dual.

Reciprocity, losslessness, and the various forms of symmetry have hitherto been defined in a rather abstract manner. We shall now link these constraints to certain physical or geometrical properties of an optical system. Moreover we present illustrating examples and counterex- amples, for which the conditions defining the above constraints are satisfied or violated.

As shown in 8 2.2, most optical systems satisfy the reciprocity condition (2.6) in the paraxial approximation. Beyond that approximation, only special systems, like free space, remain reciprocal in the sense of (2.6). On the other hand, non-reciprocity in the paraxial region has to be sought

238 PRINCIPLES OF OITICAL DATA-PROCESSING [IV, § 5

among the devices operating with static magnetic fields. So, the cascade of a 90"-Faraday rotator and a 90" reciprocal rotator (consisting of an optically active material) constitutes a scalar, nonreciprocal system. Due to 0" and 180" rotations in the two opposite transmission modes the pertinent system functions have different signs and, as such, describe a nonreciprocal 180" phase shifter.

In general, losslessness can be shown to be equivalent to absence of energy dissipation within the optical system (VAN WEERT [1980]). In mathematical terms, losslessness then implies realness of E(X, y, 2). If, however, the spatial frequencies involved are so high that nonuniform plane waves are excited, attenuation need not be associated with energy dissipation. This is the case, e.g., when free space is excited with spatial frequencies exceeding the wave number k . According to P 6.1, we thus obtain the lowpass characteristic of free space. In the low-frequency region, where the above dissipation mechanism applies, typical lossless representatives are formed by the dissipation-free phase modulators, whereas dissipative films act as lossy amplitude modulators with a modulation function smaller than unity.

The formal conditions for symmetry can always be satisfied by suitable structural symmetries. Let a fictitious intermediate plane be inserted midway between the input and output plane and let, for the moment being, the x-y-plane of the coordinate system be shifted to this intermediate plane. Let furthermore the system be filled with an isotropic medium with a scalar dielectric constant E(X, y, z ) so that reciprocity applies. Then we have spatial symmetry if E ( X , y, z ) = E ( X , y, -2) (i.e. mirror symmetry with respect to the intermediate plane) and we have spectral symmetry if E ( X , y, z ) = E ( - x , -y, -z) (i.e. symmetry with respect to the origin). Finally, rotation symmetry is obtained if E ( X , y, z ) is a rotationally symmetric function, depending on (x"+ y") and z. The proof for the spatially symmetrical case follows from the simple observation that, due to the special distribution of E, the transmission from a point A in plane 1 to a point B in plane 2 equals that from the projection A' of A onto plane 2 to the projection B' of B onto plane 1 which, due to reciprocity, again equals that from B' to A'. So, the definition for spatial symmetry is fulfilled. A similar reasoning applies to the realization of spectral symmetry. Attention should be paid to the fact that reasoning of this sort does not admit the conclusion that "black-box-symmetry" necessarily implies structural symmetry. There is some evidence, that this need not be true!

IV, 9: 51 BASIC SYSTEM CONSTRAINTS 239

5.2. CONSERVATION LAWS

The constraints discussed above are fundamental in the sense that most of them are preserved in cascade combinations and under system inversion. (In fact, only symmetry is not per se preserved in cascades.)

This finds expression in the following theorems.

Theorem 1. In a cascade of optical systems, each of which satisfies the condition of reciprocity, losslessness, spreadlessness, shift invariance, or rotation invariance, the overall system satisfies that condition, too.

Theorem 2 . If an optical system admits system inversion and if it satisfies the condition of reciprocity, losslessness, symmetry, spreadlessness, shift invariance, or rotation invariance, the inverse system satisfies that condition, too. (In the case of reciprocity system inversion involves both directions (1 + 2 and 2 -+ I).)

The proofs of the various aspects of these theorems can be readily furnished through combination of the pertinent system constraints with the cascade and inversion relations (3.11), (3.13); they are left to the reader. Rather, we want to discuss a special consequence of the conservation of reciprocity in cascades. If two reciprocal systems which are realized as cascades of reciprocal components and which differ in their structure and their components have been proven to be equivalent (i.e. to have equal weighting functions g, G ) in one direction, they are also equivalent in the opposite direction. Therefore, from the equivalence proofs for the two directions one (in general the easiest) may be chosen ad libitum. For example, if a certain cascade of free space, lens, free space (which are reciprocal components*) appears to be equivalent to an abstract, ideally imaging system (which occurs, when the “lens law” is satisfied) in one direction, the same is automatically true for the opposite direction. For further, less trivial applications of this principle cf. 0 7.

With respect to cascades a special result holds for spreadless and shift-invariant systems. Not only are the two properties preserved but also may the order in which the individual systems are arranged be interchanged without any influence upon the overall system properties. We

* Note that in our approach a section of free space is always considered as a separate system.

240 PRINCIPLES OF OWICAL DATA-PROCESSING [IV, 5 5

then have with (5.9) and (5.12)

m,,(r)= mn.,-i(r) . . . m32(r)m21(r), (5.15a)

Hn,,(R) = Hn,n-l(R) . . H32(R)H21(R) (5.1%)

for the cascade of (n - 1) spreadless or shift-invariant systems. In the complementary domains these ordinary products are transformed into convolution products. Due to the commutativity of both types of products the order of arrangement of the individual systems is arbitrary.

5.3. MULTIPLE CONSTRAINTS

Many practical systems exactly or approximately satisfy a number of the foregoing constraints simultaneously. The resulting properties, some of which interesting and surprising, are now discussed.

First we combine reciprocity and (complete) losslessness. Then (5.1) and (5.4) combine to form the relations

gbI(r1, r2) = gTAr1, r2), (5.16a)

GLI(R1, R2) = GTZ(R1, R2). (5.1 6b)

Thus, the transmission of the “inverse” system is the complex conjugate of the reverse system.

As can be concluded from Fig. 5.1 this amounts to the following property: if an excitation 41(r l ) in plane 1 causes a response 42(r2) in plane 2, then 4:(r2) in plane 2 causes 4T(r,) in plane 1 (cf. Fig. 5.2).

Anticipating a result of the next section we mention that free space is approximately lossless and reciprocal. Then the above property can be visualized by a point source in plane 1 exciting a divergent wave in plane 2, while its conjugate there produces a wave now converging towards plane 1 (Fig. 5.3).

Since, however, the losslessness condition is not exactly fulfilled, the wave does not converge to an exact point, but only to a “focal” region with dimensions in the order of a wavelength*.

If, for the moment being, harmonic time dependence is abandoned and

* In holography one encounters a practical application of the principle illustrated in Fig. 5.3. After illumination of a developed hologram with the reference wave one obtains on the observer’s side besides the wanted virtual image (representing the original wave front) also an unwanted real image (due to the complex conjugate wavefront).

BASIC SYSTEM CONSTRAINTS 241

2 ’ original system

inverse system

Fig. 5.1. Relation between the inverse and reverse transmission in a lossless, reciprocal system.

Fig. 5.2. Two compatible signal pairs for the opposite transmission directions in a lossless, reciprocal system.

Fig. 5.3. Divergent and convergent waves in free space. The illuminations in the right-hand planes are each other’s complex conjugates.

real polychromatic signals i i ( r i , t ) ( i = 1, 2) with spectral components in a certain frequency band are considered and if, moreover, the system is lossless and reciprocal in that frequency band, we can draw the following conclusion from Fig. 5.2: when &(rl, t) excites i2(r2, t ) , then i 2 ( r2 , -t) excites &(rl, - t ) . This result originates in the simple fact that under time reversal the Fourier transform of a real function of time changes into its


complex conjugate. Hence we can conclude that joint fulfilment of losslessness and reciprocity implies time reuersibility.

Combination of reciprocity with other constraints yields some formal simplification of the signal-processing description. Jointly symmetrical, reciprocal systems are characterized by (cf. (5.1), ( 5 . 5 ) , (5.6))

g2,(r2, rJ = g12(r2, r l ) for spatial symmetry, (5.17a)

G21(R2, Rl) = G12(R2, R,) for spectral symmetry, (5.17b)

which implies that for these functions the indices 12 ,21 may be dropped (this is not true for the complementary functions). A spatially symmetrical, reciprocal system is characterized by one symmetrical space transmission function g(r2, r l ) = g(r l , r2) , whilst for a spectrally symmetrical, reciprocal system we have G ( R 2 , R,) = G(Rl, R2) applying in either direction of transmission.

For the special cases of spreadless and shift-invariant systems we can conclude that the transmission properties are completely described by the bidirectional functions m(r) = m21(r) = m12(r) and H(R) = =

H12(R) . On the other hand, we have in the complementary domains M2,(R) = M12(-R) and h21(r) = h12(-r) so that, except for even functions, there the specification of a transmission function must always be accompanied by a reference direction of transmission.

Combination of reciprocity and rotation invariance yields with (5.14)

q 2 , ( r 2 , r17 a 2 - = q, , (r , , r2 , a1 - a 2 )

or

q21(r2, r l , a) = q 1 2 ( r l , r2, -a). (5.18a)

If the functions are even with respect to a, we have the simpler result

q21(r27 rl , a) = q12(r1, r2 , a). (5.18b)

Even functions of a imply that the response to a point source at r l = r,, a1 = a, is angularly symmetric with respect to the source, and the system can be said to be rotation-free. This condition can be expected to be met in most rotation-invariant systems of practical interest.

The remainder of this section is concerned with one direction (1 -+ 2) only; so reciprocity is no longer involved. First, we consider the combination of shift and rotation invariance. Shift invariance implies validity of (5.8a) :

g21(r2, r l ) = h 2 1 b - r l ) .

IV, § 51 BASIC SYSTEM CONSTRAINTS 243

Rotation invariance implies that rotation symmetry of a signal is preserved. As the point source 6 ( r , ) has to be reckoned among the rotationally symmetrical signals its response hZ1(r2) must belong to that class, too:

hz1(r) = hZl,&) = h21,p(J=7). (5.19)

and the condition (5.14) for rotation invariance is obviously met. With the knowledge that every motion of a two-dimensional figure (a

“rigid body” in kinematics) can be considered as a succession of a rotation round a fixed point 0 and a translation (cf. Fig. 5.4), we realize that shift- and rotation-invariant systems are invariant with respect to any figure motion. What has to be kept constant during the motion are merely the mutual distances between all points of the figure: distortions and magnifications are thus excluded.

From Fig. 5.4 we can further conclude that any motion can also be interpreted as a rotation round some point 0’. In this view, shift invariance plus rotation invariance round a fixed point is equivalent to

Fig. 5.4. General motion of a two-dimensional figure interpreted (i) as a rotation B-B‘ round 0 followed by a translation B’-B”, (ii) as a pure rotation B-B” round 0’.

244 PRINCIPLES OF OVTICAL DATA-PROCESSING [IV, § 5

rotation invariance round all points. Apparently this again implies shift invariance, since a shift can be viewed as a rotation round an infinitely remote point*.

For a shift- and rotation-invariant system the relation between the point spread h21(r) and the system function H21(R), both rotationally symmetric, is given by a Hankel transformation (3 .3) . When also the illumination is rotationally symmetric, the image has this property, too, and with (RI = R (5.12) passes into

(5.21)

If now @,.,(R) is given by a 6-function at R = A, the same holds true for @2,p(R). Such a 6-function corresponds to a Bessel function J,(Ar) in the space domain; hence J,(Ar) is an eigenfunction of a shift- and rotation- invariant system with H21,p(R) as proportionality factor (eigenvalue).

The dual of the above shift- and rotation-invariant system is a spreadless system with rotationally symmetric modulation function rn(r). This seems, however, to have less practical significance and therefore deserves no further consideration.

On the other hand, the combination of losslessness with spreadlessness or shift invariance occurs frequently; it is characterized by the condition

(rn2,(r)1 = 1 for spreadlessness, (5.22a)

IH21(R)I = 1 for shift invariance, (5.22 b)

which follows from (5.2) combined with (5.9) or (5.12). Apparently a lossless and spreadless system constitutes a pure phase modulator, while the corresponding shift-invariant system represents what is called an “all-pass’’ in electrical engineering. If IR( does not exceed the wave number k = 27r/A, free space belongs to this class (cf. P 6). It is important to note that the conditions (5.22) are hardly to translate into the complementary domains, i.e. there are no simple conditions for MZ1(R) and

h21(r).

* In either point of view, a two-dimensional rigid body is characterized by three degrees of freedom. Starting from a certain initial position, the pure rotation is given by the x-y-coordinates of the rotation center plus the rotation angle, and in the combined rotation-translation the rotation angle plus the displacements in the x and y directions characterize the motion quantitatively.

IV, B 61 EXAMPLES OF PHYSICAL AND ABSTRACT SYSTEMS 245

P 6. Examples of Physical and Abstract Systems

In this section we first analyze some basic structures which are in common use as building blocks of optical systems. In the second part we compile several abstract systems with desirable properties without as yet investigating their realizability.

6.1. PHYSICAL SYSTEMS

The probably most basic optical “component” is a section of free space with length d . For the sake of convenience, we assume that the index of refraction n equals unity (vacuum), otherwise the wave number k =

06 occurring in forthcoming formulas has to be multiplied by n. The fact that free space is included in the catalogue of optical components (a strange idea in classical optics) is typical of the system-theoretical approach of optics.

Free space is reciprocal, shift- and rotation-invariant and, as such, can be described by a rotationally symmetric system function H(R) = HJR) with a signal transformation obeying (5.12). On the other hand, H ( R ) can also be interpreted as the eigenvalue (proportionality factor) pertaining to the eigenfunction exp (iR r ) = exp [i(Xx + Yy)] . This two-dimensional plane wave propagates as

exp [i(Xx + Yy + Zz)]

in three-dimensional space, where

Z = J k 2 - X 2 - y2, k = 2r/A = 06, and z denotes the distance of a field point from the input plane. For X 2 + Y 2 < k 2 this plane wave propagates in +z-direction, whereas it decays exponentially for X 2 + Y 2 > k2 . For this to be true the square root for Z has obviously to be taken in the first quadrant.

In the output plane z = d we then obtain the field distribution

exp (iZd) exp [i(Xx + Yy)]

which yields the system function

H(R) = H,(R) = exp (iZd)

=exp ( i d J k 2 - X 2 - Y2) = exp (id-). (6.1)

246 PRINCIPLES OF OITICAL DATA-PROCESSING [IV, 8 6

The corresponding point spread follows from (3.3) as

1 ” h,(r) = I, H,(R)J~WW d~

a ad

= -- {exp [ik-]/(2~-)}, (6.2a)

which integral was first evaluated by Sommerfeld (cf. WATSON [ 19661). This h,(r) is conveniently approximated in two steps. First, differentiating with respect to d and neglecting a term small for d > > 2 ~ / k = A, one obtains

k d exp[ikdr2+d2] h,(r )=- -

2 7 r i & G F J7T-Z 9 (6.2b)

whereupon the small-angle (“paraxial”) approximation r << d yields

k h,(r) == exp (ikr2/2d). (6 .2~)

Notice that a constant phase factor exp(ikd) (which only might be interesting in interferometric applications, where the absolute phase shift of free space is involved) has been dropped in (6 .2~) . The widely used result (6 .2~) is usually referred to as the Fresnel approximation of free-space propagation; its validity has been the object of various investigations (PAPOULIS [ 1968a1, GOODMAN [ 19681, MAITHIJSSE and HAMMER [1975]). If not stated otherwise it is henceforth taken for granted.

The system function pertinent to the Fresnel approximation is found by Hankel transformation of (6 .2~) . We obtain

d 2k

H,(R) = exp (-i - R2), (6.3)

which (curiously enough) is the low-frequency approximation of (6.1). We observe that, due to [H,(R)I= 1, free space is lossless in the Fresnel approximation, whereas the exact result (6.1) reflects a low-pass transmission characteristic, in which all frequencies R > k are attenuated and, in fact, completely lost if kd >> 1.

For large distances d, but finite extent of the input illumination, the Fresnel diffraction passes into Fraunhofer diffraction. Inserting (6 .2~) into

IV, 8 61 EXAMPLES OF PHYSICAL AND ABSTRACl SYSTEMS 247

(5.10) then yields k

2md b2(r2) = - [exp [ik Ir2- r l~z /2dldJ l~rd dr,

k 2rrid

=- exp (ikr:/2d) exp (-ikr2 - rl/d)dJl(rl) dr,. (6.4)

Apart from a quadratic phase factor (which can be avoided by looking on an output sphere instead of an output plane) one “sees” the Fourier transform of the input signal dJ1(rl).

An important class of optical components which, moreover, is relatively easy to manufacture, is formed by the modulators (spreadless systems). Its simplest representative is an aperture in an otherwise opaque screen, with a modulating function

1 in the aperture, 0 elsewhere.

m(r) =

It can be easily extended to a multiple-aperture modulator which, for instance, is met in the half-tone realization of a transparency with continuous gray-shades.

An approximate representative of the phase modulators which, moreover, is in widespread use, is provided by the lens. With reference to Fig. 6.1, we assume that between the two reference planes 1-2, a distance d o apart, a dielectric material with constant refractive index n, but variable thickness d ( r ) 5 do is inserted. Under the condition that the input signal and the thickness function d(r) do not vary too rapidly (i.e. that they contain only low frequencies) we obtain a local input-output relation

1 ,vacuum

’1 refractive index

n (constant)

Fig. 6.1. Geometry of a phase modulator.


Fig. 6.2 . A thin, plano-convex lens.

that is determined as if a plane wave was normally incident. Neglecting reflections at the boundary surfaces we then obtain the phase delay

knd(r)+ k [ d , - d ( r ) ] = k [ ( n - l ) d ( r ) f d , l

and the modulating function

m ( r ) = exp [ik(n - l ) d ( r ) ] ,

where we have dropped the (uninteresting) constant phase factor exp (ikd,). Apart from the proportionality factor k ( n - 1 ) the thickness function d ( r ) then determines the local phase delay. For a spherical lens d ( r ) = dp(r) is rotationally symmetric and follows a quadratic law in the paraxial approximation ( r < rmax << p in Fig. 6.2). Simple geometrical considerations reveal that for a plano-convex* lens with curvature radius p the local thickness becomes (apart from an additive constant)

dp( r ) = -r2/2p

which leads to the two characteristic functions

m(r) = m,(r) = exp (-ikr2/2f), (6.6a)

(6.6b)

with

f = p/ (n - 1 )

in that approximation. In § 8 it will turn out that f equals the focal distance of the lens. For r = r,,, (cf. Fig. 6.2) the phase function attains

* For other types (double-convex, positive meniscus, double- and plano-concave, negative meniscus) of lenses cf. GOODMAN [1968]. Notice that for an overall concave lens f becomes negative.

IV, 8 61 EXAMPLES OF PHYSICAL AND ABSTRACT SYSTEMS 249

the maximum absolute value kr$,,/2f which remains constant for r > r,,,. The pertaining mathematically inconvenient modulation function is usually replaced by m = 0 for r > r,,,, corresponding to a window modulator cascaded with the idealized lens according to (6.6). This rough approximation is guided by the geometric-optical idea that “rays” travelling past the lens are definitely leaving the optical system and do no longer contribute to the output signal.

In a final approximation (the “thin-lens” model) one completely neg- lects the finite lateral lens dimensions. This model is mathematically described by (6.6) for all values of r and R. We have to keep in mind, however, that (6.6) is the result of a number of rather simplifying assumptions (spreadlessness, normal plane wave incidence, weakly curved lens surfaces, infinite lateral dimensions, absence of reflections) so that in certain optical systems this model possibly predicts the lens behaviour with insufficient accuracy.

6.2. ABSTRACT SYSTEMS

Next we treat the Fourier transformer, which has, in fact, furnished the name “Fourier optics” for the modern field of optical signal processing. It is characterized by the pair of functions

k exp (-i- r2 rl),

d k

g21(r2, r1) =

d d G21(R2, R,) = ~ 27rik exp (-i R2 - R1),

which leads to the input-output relation

(6.7a)

(6.7b)

in the space domain and a corresponding relation in the frequency domain.

In rough terms, the output signal indeed represents the Fourier transform of the input signal. However, a scaling constant with the dimension (length)-2 is required to provide a mapping of the frequency domain of the usual Fourier transformation into the output space domain. This constant equals k/d in (6.8), where the length d is a characteristic

250 PRINCIPLES OF ORICAL DATA-PROCESSING [IV, P 6

constant of the Fourier transformer under consideration and the wave number k occurs in connection with its actual realization (cf. 0 8.1). The proportionality factor kl(2rrid) in front of the integral is SO chosen that the Fourier transformer becomes lossless.

We further assume that g,, = g,, and G,, = G,, which implies reciprocity. Moreover, the Fourier transformer is (spatially and spectrally) symmetrical, and rotation-invariant, but neither shift-invariant nor spreadless. Notice that a shift of the input signal merely introduces a linear phase factor in the output signal which remains invisible in case only the output intensify is observed*. This property manages to detect certain frequencies in undetermined regions of the input plane.

In 0 8 it will be shown that a Fourier transformer can be simply realized as a cascade of a lens and two sections of free space. While the real-time realization of a time-domain Fourier transformation is prohibited due to the required knowledge of the future values of the input signal, no causality requirement impedes a spatial Fourier transformation.

A system of equal theoretical and practical importance is formed by the magnifier. Its characteristic functions are given by

g21(r2, rJ = t S(rl - tr2L (6.9a)

(6.9b)

which implies

M r ) = t&(tr), (6.10)

and a similar relation in the frequency domain. Obviously the magnifier is a generalization of the identity system (through connection) into which it degenerates for t = 1. For It( < 1 it furnishes a size magnification and for ltl>1 a size reduction. Furthermore, for t < O the scene is inverted.

The occurrence of the proportionality factor t in (6.10) guarantees the losslessness of the magnifier. Reduction of the linear dimensions by a factor t reduces the surface dimensions by t2, but simultaneously increases the intensity (=square of the amplitude) by t2.

Again we assume that reciprocity applies, i.e. g,, = g,, and GI, = Gzl. This implies that (6.10) is also valid in the direction 2- 1. Then upscal- ing in one direction means downscaling in the opposite direction and vice

* Note that the output intensity can also be interpreted as the Fourier transform of the autocorrelation function of the input signal.

IV, 8 61 EXAMPLES OF PHYSICAL AND ABSTRACT SYSTEMS 251

versa so that a specification of the magnification t has to be associated with a reference direction. How this is accomplished formally, will be discussed in § 7. Although it constitutes a device with a simple mathematical description, a magnifier with I t \ # 1 does not satisfy the constraints of spatial and spectral symmetry (and herewith those of shift invariance and spreadlessness). However, it is rotation-invariant.

6.3. CASCADES, INVERSIONS, AND DUALITIES OF ELEMENTARY SYSTEMS

When two or more sections of free space, lenses or magnifiers are arranged in cascade the resulting system is again equivalent to a section of free space, a lens or a magnifier. Only in the case of Fourier transformers another type of system can be (but need not be) created after cascading (cf. § 7.2).

For two sections of free space with lengths d , and d , the resulting free space has the length d , + d , . Notice that this trivial result is also found formally by applying the Fresnel approximation (6.3)!

For two lenses in cascade the reciprocal focal distances l/fl and l/f2 (the “powers”) have to be added, while for two cascaded magnifiers the magnifications have to be multiplied.

On account of the losslessness of the four basic optical systems, inversion according to (5.4) can be easily accomplished. For free space with length d inversion again leads to free space, now with length ( - d ) . Thus, retrieval of the input signal of a free-space section amounts to transmission of the output signal through a fictitious free space with corresponding negative length. Notice, however, that this is true only in the Fresnel approximation. If this is abandoned, system inversion for d sufficiently large is completely impossible because the high frequencies IR\ > k are lost.

For a lens with focal distance f system inversion leads to a focal distance (-f). Hence, a convex lens is transformed into a concave lens (and vice versa) and their cascade provides an identity system. Inversion of a Fourier transformer changes its characteristic length into its negative value ( d + - d ) , while in the case of a magnifier t is transformed into l / t .

Finally we remark that free space and lens are dual in the strict sense (compare (6.6a) with (6.3) and (6.6b) with (6.2~)) . The dual of a magnifier with magnification t is a magnifier with magnification l l t (which is also its inverse). The Fourier transformer is its own dual.

252 PRINCIPLES OF OFITCAL DATA-PROCESSING [IV, 9: 7

P 7. Operational Notation of Optical Systems and Basic Cascade Equivalences

7.1. AN OPERATIONAL NOTATION

Most optical systems can be viewed as cascades of more or less elementary systems, among which the four basic components of the last section. In order to facilitate the analysis of such cascades we now introduce an operational notation*.

In this notation, a system is represented by a Gothic symbol with the pertaining transmission functions ad libitum added between brackets. A general system is denoted by

Wgzl(r2, r J r g12(r,, 4 1 or

WGzi(Rz, RI), G12(R1, R2)I,

where the first notation applies to the space domain and the second to the frequency domain. The inverse system is represented by W and the identity equation G a s @ , means equivalence of the systems Ga and 8, in either direction.

Two systems Ga[gzi(rZ7 r l ) , glArl, rZ)l and ab[g32(r3, 4 , g2,(r2, 4 1 , which are cascaded so that ab is to the right of @a, form the new system aC =(By,@$ with

@c[g3l(r3, rl), g13(rl? r3)1*

Clearly, plane no. 1 is at the left, plane no. 3 at the right, while the intermediate plane 2 disappears when only the overall system GC is viewed. The new system function g3, is determined according to (3.11) with a corresponding formula for g13.

Special systems admit simplifications. Reciprocal systems require only one g-function, as in

W&, rA1,

where g obviously applies to either direction. Integration over r l is required for the 1 4 2 transmission, and integration over r2 for the 2 4 1 transmission. Henceforth, all systems are assumed to be reciprocal.

*This operational notation should be viewed as a continuation and refinement of a method proposed by VAN DER LUGT [1966]. The present analysis follows BUTTERWECK [1977].

IV, § 71 OPERATIONAL NOTATION 253

Subclasses of reciprocal systems are indicated by special symbols. So, @[H(R)] and m[m(r)] denote shift-invariant and spreadless systems, and G(d) , L?(d), 3 ( d ) , X ( t : 1) represent the four systems of the preceding section, viz. free space, lens*, Fourier transformer and magnifier. The characteristic quantities d (length of free space, focal distance [formerly denoted by f], and scaling constant of Fourier transformation, respectively) and t (magnification) are added between brackets.

For some reciprocal systems it has to be explicitly stated to which direction the quantity inside brackets refer. This is the case, for instance, with the magnifier, where the notation t : 1 indicates that the physical dimensions of the left-hand scene are It1 times those of the right-hand scene so that (6.10) applies. The same is true if shift-invariant systems are to be described in the space domain and spreadless systems in the frequency domain. Then, according to h2*(r) = hI2(-r) and M21(R) =

MI2(-R) , we can write

@[h,,(r)I or @[G(r)I,

YJ2[M2,(R)] or YJl[fi(R)],

with the understanding that the pertinent functions have to be mirrored for the (t) direction.

From the above considerations we would expect that shift-invariant and spreadless systems are advantageously described by the functions H ( R ) and rn ( r ) , respectively. Indeed, the modulation function rn (r) completely characterizes a spreadless system, independent of the transmission direction. The situation is, however, more complicated for a shift-invariant system. With the double sign convention in the Fourier relations (3.2) we have established a one-to-one correspondence between R and a direction in three-dimensional space (cf. B 3.1). As is illustrated in Fig. 7.1, where the hatching indicates the pass-band of a spatial narrow-band filter, the reversed system passes a different wave direction. Thus, for a shift- invariant system, the specification of the system function H(R) has to be accompanied by a labelling (1,2) of the two reference planes. This way the spatial orientation of the system is fixed. When the system has to be reversed or, what amounts to the same result, the system is viewed from “behind the paper”, the reference planes are interchanged and H(R) is replaced by H(-R) .

* In % ( d ) the finite,dimensions of the lens pupil are not taken into account. If desired a window modulator can be cascaded with 2 ( d ) .


‘passed “ray”

Fig. 7.1. Change of the pass-band of a narrow-band spatial filter due to reversal.

7.2. CASCADE EQUIVALENCES

We are now prepared to discuss a number of basic equivalences of system cascades, which facilitate the analysis of complicated systems. How these tools are applied to actual systems, will be treated in the next section. In the present section we confine ourselves to a statement of the various equivalences and a brief discussion of their implications. The pertinent proofs, which are throughout easy to construct, have been given elsewhere ( B ~ R W E C K [1977]) and are omitted here. We only want to recall that, due to the reciprocity of all building blocks, all equivalence proofs need to be given only for one (the “easiest”) direction of transmission.

All equivalences to be discussed in this section can be traced back to five identities:

(7.la)

(7.2a)

(7.3a)

(7.4a)

(7.5)

The first three relations are concerned with the interaction of a magnifier X with a Fourier transformer 8, a general shift-invariant (9) or a spreadless (!LR) system. We observe that 5Z is “absorbed” by a Fourier transformer (thereby changing its scaling constant from d to rd), whereas

IV, § 71 OPERATIONAL NOTATION 255

it is “pushed” through a shift-invariant and a spreadless system (thereby again changing the scaling of their characteristic functions). The next result (7.4a) states that a Fourier transformer can be pushed through a shift-invariant system thereby transforming it into the dual spreadless system, or vice versa. It can be traced back to the well-known “convolution theorem” which states that the Fourier transform of a convolution of two functions equals the product of their Fourier transforms. In this interpretation, @ corresponds to (spatial) convolution and YJI corresponds to (spatial) multiplication. Finally, (7.5) is a special result for the family 2,8,G, which does not admit a generalization towards more general systems. It is easily found by working out the first integral in (6.4) and properly identifying the resulting individual terms. It states that a free- space section G can be replaced by two lenses (which are concave for d > O ) and a Fourier transformer in cascade.

These results admit some modifications and specializations. First, from (7.la) we have

8(4)%d2) = % ( - ( 4 / d 2 ) : I), (7.lb)

Xdi)8(d2)8(d3) X-didJdJ. (7 .1~)

A cascade of two (or, more generally, an even number of) Fourier transformers thus constitutes a magnifier, whereas three (or, more generally, an odd number of) Fourier transformers again equals a Fourier transformer. We remark that (7.lb) is easily found from (7.la) through “post-multiplying’’ (7. l a ) by the inverse system f’J(-d) which cancels f’J(d). Finally (7 .1~) is found through application of (7.la) and (7.lb) to its left- hand side.

Next, we apply (7.2a) and (7.3a) to the special shift-invariant and spreadless systems G and 2. We then find

E(t: l)G(d) =G(t2d)5( t : l), (7.2b)

X ( t : 1)2(d)=2(r2d)Z(t: 1). (7.3b)

Again, the magnifier re-scales the adjacent systems by transforming d into t2d.

By viewing (7.4a) from “behind the paper”, H(R) is transformed into H ( - R ) , while 8 and 9.X as spatially symmetric systems remain unchanged. We thus are led to:

??(d)@[H(-R)l=m[ H(: r)]%d). (7.4b)

256 PRINCPLES OF OPTICAL DATA-PROCESSING [ N , 9: 8

Some minor notational changes further yield

Xd)lDl[m(r)I=@[ m(i R)]5(d), (7.4c)

DmCm (r)B:(d) = 5(d)@[ m (- $ R)]. (7.4d)

Application of (7.4) to I! and 6 leads to

tY(dJWd2) E I!(d?/dJ%di), (7.4e)

Wd2)5(dJ %di)Wd?/dd. (7.4f)

Hitherto, 6 was described in terms of R and Vl in terms of r. In rare cases also a description with interchanged roles is required. For instance, with

@[H(R)I = HL(r)I

one obtains

in (7.2a), or with

mn[m (r)I = lDln[fi(~>I

one obtains

in (7.3a). Finally (7.4a) reads as

Q[K(r)]%(d) -S(d)W[4r2 $6(-$ R ) ] .

All results follow from well-known scaling theorems of Fourier theory.

§ 8. Operational Analysis of Optical Systems

8.1. ACTUAL REALIZATIONS OF FOURIER TRANSFORMER AND MAGNIFIER

First we apply the general results of the previous section to analyze two well-known realizations of a Fourier transformer. Observe that the operational notation throughout obviates the need for illustrations.

IV, 0 81 OPERATIONAL ANALYSIS OF OPTICAL SYSTEMS 257

In the cascade z(d)G(d)Z(d) we have two lenses, each of which is placed in the focal plane of the other. With (7.5) and the observation that 2 ( d ) and its inverse 2 ( - d ) cancel each other, we have

E(d)G(d)2( d ) 3 2(d )2 ( -d )%(d)2 ( - d ) 2 ( d ) g(d). (8.1)

The dual of this realization is the more familiar cascade G(d)2(d)G(d), i.e. one lens operated between its two focal planes. As g(d) is its own dual, we expect again a Fourier transformer %(d) as equivalent.

In a formal proof, we cascade G(d)g(d)G(d) from the left with the identity system %(-d)%(d) and then push % ( d ) through the cascade. With (7.4e) and (8.1) we obtain:

Wd)Wd)Wd) %:(-d)%(d)G(d)2(d)G(d)

-%:(-d)2(d)G(d)2(d)g(d)

27- d ) % . ( d ) X d ) ~ % ( d ) . (8.2)

Note that the realizations according to (8.1) and (8.2) provide a new proof for the losslessness and rotation invariance of a Fourier transformer: it is based upon the knowledge that these constraints are preserved in cascades and that they are satisfied by the individual components G and 2.

The “classical” optical system is a lens providing a sharp imaging. In our notation this reads as G(u)C(f)G(b), where the two lengths a, b are related to the focal distance f according to the lens law

With (7.5) we write

G(aP(f)G(b) 2(- a)%(a)W- a)C(fR(- b)%(b)W-- b)

and state that, due to (8.3) and the additivity of the lens powers, the three central lenses cancel out. Then the two adjacent Fourier transformers are combined into a magnifier (cf. (7.lb)) and finally the lens is pushed through the magnifier (cf. (7.3b)):

G(a)2!Cf)G(b) = 2(-a)g(a)g(b)2( - b ) - B ( - a ) 2 ( -: : l)2(- b )

=B(-; a : 1)2(--)2(-b)=X(-- b2 U : 1)2(f- b). a b

(8.4a)

[IV, 5 8 258 PRINCIPLES OF OFIXXI.. DATA-PROCESSING

In the last step, we have combined the two lenses at the right, making use of the lens law (8.3). This way we have derived the well-known result that the cascade G2G satisfying (8.3) provides a sharp imaging with a quadratic phase error. Hence, an ideal magnifier (with a negative magnification) is constructed by placing a correcting convex lens with focal distance b - f > O at the right end:

6 ( a ) 8 ( f ) Q ( b ) ~ ( b - n ~ ~ ( - ~ : 1). (8.4b)

Note that the left-hand focal plane of the correcting lens and the right-hand focal plane of the main lens coincide.

8.2. FOURIER FILTERING

A shift-invariant system is conveniently synthesized as a modulator inserted between two Fourier transformers (thereby transforming the former into its dual). With (7 .4~) and the fact that g ( d ) and g ( - d ) cancel each other, we then have:

This relation is the basis of Fourier filtering. In the intermediate plane where the modulator 2R is inserted, the Fourier transform of the left-hand input signal is manipulated in some way or another. Inverse Fourier transformation then yields a filtered version of the input signal with the properly rescaled modulation function rn acting as system function of the resulting shift-invariant system.

In the ultimate realization of (8.5) the Fourier transformers are synthesized according to (8.2). A minor flaw occurs due to the unrealizability of g ( - d ) in (8.5) which would require an G ( - d ) . Instead we replace g ( - d ) by g ( d ) and so obtain

i.e. a shift-invariant system followed by an uninteresting inversion (Fig. 8.1).

IV, Is 81 OPERATIONAL. ANALYSIS OF OPTICAL SYSTEMS 259

Imagine now that the modulator is not carefully adjusted in the central “Fourier plane”, but undergoes an axial displacement 6 to the right (cf. Fig. 8.1). Then we can advantageously utilize the notion of a negative length of free space.

We first go to the right of the original Fourier plane, insert YJJ1 there, and then go back to the left. The resulting cascade can then be described as

where (7.4e) has been applied. Clearly, the effect of the displacement 6 is reflected by two fictitious lenses arranged on either side of the wanted 6. Thereby the shift-invariance property is lost. The right-hand lens is not disturbing, when only the output intensity is measured, but the left-hand lens introduces errors for illuminations with sufficiently large linear dimensions. Since, like all equivalences discussed in this and the previous section, the identity (8.7) applies to all degrees of coherence, we can state an interesting result for incoherent illumination: then also the left-hand lens does not exert any influence upon the system properties, with the result that the modulator can be placed ad libitum between the two lenses in Fig. 8.1!

8.3. INSERTION OF A MODULATOR IN FRONT OF A FOCUS; ABERRATION ERRORS

Assume now that a convergent beam is focused on the axis ( r = 0) and that a modulator \%R[m(r)] is inserted in front of the focus, at a distance d. We can then apply the notion of a negative-length free-space section, moving backwards over a distance ( -d ) , inserting Yh!, and going forward to the former focus. With (7.5) we obtain:

The three central components commute so that 2 ( d ) and %-d) cancel each other. Further, with (7 .4~) and the eventual annihilation of the pair 3( -d) , g(d) we have

(8.8) d

G(-d>n[rn ( r ) ]G(d) = 2 ( d ) @ [ m(- R ) ] 2 ( - d ) ,

PRINCIPLES OF OF’TICAL DATA-PROCESSING [IV, 9: 8 260

input plane

. . -. -. - c ---- -.- I output plane

Fig. 8.1. Standard realization of a shift-invariant system with inversion. Included is an unwanted shift 6 of the modulator away from the exact “Fourier plane”.

from which we can, incidentally, conclude that this configuration is its own dual.

Our above assumption that without ?m the light was focused on the axis, amounts to a left-hand illumination of our system with a fictitious point source S(r ) which, of course, remains undisturbed by the first lens 13(d). If also the phase distortion due to 5 2 - d ) can be left out of consideration (which is allowed with pure intensity measurements), we “view” the point spread of @ at the system output. This is determined as the inverse Fourier transform of the system function H(R) = m(-(d/k)R), yielding

Hence, apart from the quadratic phase factor due to 13(-d) we “see” the properly scaled Fourier transform of the modulation function m(r) . We conclude from (8.9) that the pattern h(r) is enlarged with increasing distance d.

The most obvious application of this result is found, when m ( r ) represents the exit pupil of an otherwise ideally imaging system. For a circular aperture with radius a we have

(8.10)

IV, 8 81 OPERATIONAL ANALYSIS OF OFTICAL SYSTEMS 261

and (8.9) becomes the “Airy” pattern

ka J,(kar/d) h ( r ) = h,( r ) = -

27rd r (8.1 1)

In D 9.2 we shall show that the result (8.9) and its application (8.11) also holds true if a modulator is inserted in any intermediate plane of a perfectly imaging system even if such a plane is separated from the focus by one or more lenses.

An actual optical system does not form an exact focus in the hypotheti- cal absence of the window modulator YJl. Due to spherical aberrations the illumination in the modulator plane exhibits phase errors that can be represented by an additional phase modulator with modulation function exp [ib(r)] in tandem with the pupil modulation function (8.10). Notice that such a fictitious phase modulator can be elegantly compensated along holographic way by recording exp[ib(r)] on an off-axis hologram and using its “conjugate” wave exp [-ib(r)] for compensation (LEITH [1977]).

Without compensation, the real function m ( r ) of (8.10) is converted into a complex function, whose absolute value is now given by (8.10). This implies that the “energy” of m ( r ) (i.e. the integral of its squared modulus) remains unchanged and that, due to Parseval’s theorem, the same is true for the point spread h(r ) . But there is some indication that h ( r ) becomes “broader” after addition of the phase distortion. To see that we determine the value of Ih(r)l at the origin r = 0, which according to (8.9), is found as

Among all functions m ( r ) with equal absolute value the constant-phase function yields the maximum value of the last integral and, hence, of Ih(0)l. (Particularly when the phase variations of m(r) exceed 27r, the different contributions to the integral will tend to interfere in a destructive way.) Since the integral of Ih(r)JZ has been found to be independent of the phase function, the constant-phase m ( r ) thus leads to the strongest concentration of lh(r)l around the origin.

The beneficial effect of suitable amplitude modulators in the exit pupil (apodization) is discussed in D 10.1.

262 PRINCIPLES OF OWICAL DATA-PROCESSING [IV. Q 8

8.4. SOME PHENOMENA IN FREE-SPACE PROPAGATION

Assume that a free-space section of length d is subsequently illuminated by an arbitrary function f ( r ) and its complex conjugate f* ( r ) transmitted through a Fourier transformer %(-d) . We maintain that the observed moduli at the system output are equal in both cases. With g(r ) denoting the output signal for the first illumination, as illustrated by - G ( d ) -

f(r) K ( r )

we have the situation - % ( - d ) S ( d ) E- G ( - d ) C ( - d ) G ( - d ) G ( d ) f*(r) i* (r )

for the second illumination, where use was made of (8.2). Observe that in the last identity G ( - d ) and 6 ( d ) cancel each other and that at the output of the left-hand G ( - d ) the function g * ( r ) appears. For, if f ( r ) is transformed into g ( r ) by G ( d ) , then f * ( r ) is transformed into g*(r) by the (conjugate) system G ( - d ) . Since lg*(r)l= Ig(r)l and 2 ( - d ) merely introduces an additional phase factor, we have proved the above assertion.

Application of this result to various functions f ( r ) reveals that the function pairs

exp [ - a ( x 2 + y')]

sin ( k a x l d ) sin ( k a y l d ) 2

kyld

lead to the same output intensities ( p , (x) denotes the rectangular pulse with p , ( x ) = 1 for Ixl<a and zero outside, and S(x, y ) denotes the two-dimensional Dirac function). Thus, square-aperture illumination and its %(-d) transform produce the same output intensity. The same is true for two certain Gaussian functions with different spreads and also for a periodic function and the associated line spectrum.

The last example belongs to the class of periodic functions. It is for this

263 IV, P 91 LENSES AND SECTIONS OF FREE SPACE

class, that under a certain condition self-imaging can take place, for which the output signal equals the input signal (“Talbot effect”). With the free-space transmission function

we see that for all spatial frequencies

R = { p m d , v-4 p, v=O, 1,2, . . . ,

IRl2 = ( p 2 + v2)47rk/d

we obtain H(R) = 1. Thus, for all periodic functions whose fundamental frequency equals w d (or a multiple of that) all harmonics are transmitted the same way and n o (linear) signal distortion occurs. The reader is invited to verify that the difference between the distances from an image point to the corresponding object point and to its “neighbour” point (a period distant) then becomes a quarter wavelength (or a submultiple of that). Only under this somewhat surprising condition constructive and destructive interference take care for the fact that an image point “receives” light only from the corresponding source point.

0 9. Systems Compounded of Lenses and Sections of Free Space (53.5-systems)

In this section we pay special attention to those important systems which are built up of only lenses and sections of free space (“5%-systems”). Since we have shown ( 5 8.1) that Fourier transformers and magnifiers can be realized as 2G-systems (i.e. 3 and 5Z form subsets of the set of %-systems), we can as well speak about EgE-systems. This class of systems has a number of interesting properties.

9.1. EQUIVALENT “CIRCUITS”

We first show that every $%-system can be replaced by an equivalent system of at most three components from the catalogue 2,G,g,E. Borrowing an appropriate term of the electrical engineers community, we will henceforth speak about an equivalent “circuit”.

First let us prove that a general 5%-system containing a total of N

264 PRINCIPLES OF OFTICAL DATA-PROCESSING “, 8 9

lenses+sections of free space and to be denoted by GN, can always be replaced by a cascade 282 containing not more than two lenses and a Fourier transformer. We assert then:

@N 2(d 1)8(d2)2(d3) > (9.1)

where the d, ( i = 1,2 ,3) have to be properly adapted to @”. We prove (9.1) by induction and assume that it is valid* for a certain N . Then we add from the right-hand side a further component and prove that the new @)N+l again can be represented in the form (9.1). Addition of a lens 2 ( d 4 ) preserves the general form (9.1) in a trivial way: 2(d4) and 2(d3) can be directly combined into a new lens.

Addition of a free-space section G(d4) and reduction to the form (9.1) is somewhat more laborious. For convenience, we omit the characteristic constants and only write 2,G,%, 2. Then we have

@N+ 1 = @NG = 2826

from (9.1). Subsequently we apply 2=- (the dual of (7.5)) obtaining @N+1=2WWG=2@@Z and then push the right-hand 8 to the left and combine it with the left-hand f’j to a magnifier E:

@N+1 = 222G.

With G=282 (cf. (7.5)) we further obtain @N+l=2E2@2 and finally push X to the right and absorb it in 8:

@,+I = 22X$j%? = ,282, Q.E.D.

Likewise we can show that any 2G-system possesses also the following equivalents:

@N = (9.2)

@jN = G2G, (9.3)

@N = Gi%, (9.4)

@ j N = 2G8, (9.5)

@ j N = G28, (9.6)

@N = 2Gc. (9.7)

The equivalence (9.3) which states that a general 2G-system can always be replaced by a single lens operated between two appropriate “reference

*With (7.5) it is valid for N = 1, if (31, =G. With d,= -d, and d 2 A 0 it is also valid if = 9.

IV, 9: 91 LENSES AND SECTlONS OF FREE SPACE 265

planes” is well known from geometrical optics (O’NEILL [1963]). On the other hand, (9.1) is most suitable to derive the general expression for the point spread of an 26-system. With (6.6) and (6.7) and the cascade formula (3.11) we easily obtain

for the point spread of GN according to (9.1). We conclude that a general 26-system has an exponential point spread with a quadratic form in r l and r 2 as exponent. The factor in front of the exponential function guarantees the losslessness of the overall system. From the dual equivalent circuit (9.4) it follows that an expression similar to (9.8) also applies in the frequency domain. Finally we note that the proofs for the validity of the various equivalent circuits need be given for only one circuit, since all these cascades can be easily transformed into each other. Like their electric counterparts, the different equivalent circuits exist, however, only for almost every 26-system in the sense that one or more of the characteristic constants might degenerate (become zero or infinite) in special cases.

9.2. MODULATORS IN BG-SYSTEMS

With the notation of the inverse system and the equivalent circuits of the preceding paragraph we are able to generalize a result of 0 8.3 which stated that, apart from some quadratic phase factor, insertion of a modulator into a converging beam produces the Fourier transform of the modulating function in the focal plane. We now prove that this result also holds if a modulator is inserted somewhere in. a perfectly imaging system illuminated by a point source.

A perfectly imaging 26-system transforms a point illumination into a point image. Like the simple lens, obeying the “lens law” (8.3), we then have the equivalent circuit (8.4) consisting of a magnifier E followed by a lens 2 representing a quadratic phase aberration. This system to be denoted by @ is now conceptually split up into two parts G1 and G2 with the understanding that the modulator 2R is inserted between them:

G& = G + G 1 r n 2 .

Instead of the last notation, we can describe the system with modulator in

266 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, 9: 9

another way: We first go through the whole system @, then go back to the modulator plane (which involves system inversion a;), insert the modulator 9Jl and finally move through a2 to the output. In symbols, the whole system then reads as

@W2rn2 . (9.9)

@ and a2 are next replaced by equivalent circuits:

(9.10)

according to (9.1). Inversion of CS2 implies inversion of the individual components of the equivalent circuit and reversal of order. This implies

@lrn@2 = @@$Jm2 X( t : 1)2(d1)2(-d4)8(-d3)2(-~2)rn2(~*)%(d3)2(d~).

Now 2 ( d 2 ) absorbs 2 ( - d 2 ) , like %(d3) is absorbed into 8 ( - d 3 ) after having transformed 2X into a shift-invariant system 6. Omitting all characteristic constants we then obtain

@,rn@, = Z2@2 (9.11)

which - apart from the magnifier E - equals the former result (8.8). Hence a modulator inserted in an arbitrary intermediate plane of a perfectly imaging system produces the properly scaled Fourier transform of its modulation function in the output plane. If the modulator represents a circular aperture (e.g. due to finite lens dimensions), we get an Airy pattern. Such an aperture can always be transformed to another plane whereby, due to rescaling, its diameter in general is changed. Especially it can be transformed to the conceptual beginning or end* of the system there forming the entrance or exit pupil.

If two modulators YX1, n2 are inserted into a perfectly imaging system, similar considerations as those which led to (9.11) now yield

@Irn1@2rn2@3 = X52 &2&2 (9.12)

for the overall system, where @,@2@3 represents the original system and YJll and YJ, are transformed into and Q2. If the intermediate lens between 6, and 6, were absent, 6, and 6, would be in cascade which

* The “beginning” and the “end” of a system are formed by arbitrary boundary planes (e.g. the tangential planes of the “first” and “last” lens) and have to be carefully distinguished from the input and output planes.

IV, § 91 LENSES AND SECTIONS OF FREE SPACE 267

amounts to multiplication of their system functions Hl(R) and H2(R) which, apart from scaling factors, equal the modulation functions of YJll and n2. If n1 and n2 then represent apertures one of them will completely cover the other (reckoning with the pertaining scaling constants) and thus completely determines the system’s behaviour. This is what geometrical optics predicts: The finite extent of the entrance or exit pupil is found by geometrically projecting the smallest aperture of the system onto the entrance or exit plane (GOODMAN [1968]). The presence of the intermediate lens in (9.12) makes this result, however, not perfectly correct. Only when the apertures are rather large compared to the wavelength, 6, and 6, are low-pass filters with rather high cut-off frequencies or, what amounts to the same, with narrow point spreads. Point illumination of (9.12) from the left then causes an almost-point illumination of the intermediate lens, which gives rise to a negligible phase distortion. A similar reasoning applies when more than two apertures form part of the optical system.

9.3. SYSTEMS CONTAINING CYLINDRICAL LENSES

A component of practical importance that renders a number of interesting signal transformations possible and that was not treated hitherto, is the cylindrical lens. If the axis of the cylinder is oriented parallel to the y-axis, the modulation function becomes y-independent and is given by

m ( r ) = exp (-ikx2/2f). (9.13a)

In the frequency domain we obtain

(9.13b)

Systems containing sections of free space, spherical lenses and cylindrical lenses of arbitrary orientations are no longer rotation-invariant and, as such, not describable by the simple expression (9.8) for the point spread of a general EG-system. In that expression the occurrence of )r112, r l r2, and lr2)2 reflects the rotation invariance of the overall system, with the pleasant result that not more than three constants dl , dZ, d3 describe the system behaviour completely. When cylindrical lenses are added to the catalogue of components, the quadratic form (9.8) is maintained, to be sure, but also mixed terms as xIy1,x2y1 occur and the total system is


described by 10 coefficients. Our operational notation seems to offer n o longer specific advantages and the analysis of systems with cylindrical lenses will in general be carried out with the aid of the cascade formulas (3.11). However, it is our strong feeling that a systematic theory of such generalized 5%-systems which would yield more insight into their behaviour has still to be developed. Probably, van der Lugt’s formalism (VAN

DER LUGT [1966]) might render a suitable starting point in this direction.

5 10. Shift-invariant Systems: Coherent Versus Incoherent Illumination

On account of their practical usefulness, the shift-inuariant systems deserve special consideration. Only shift-invariant systems handle all “portions” of the spatial input signal in an equal manner, thus making possible numerous types of signal processing: low-pass, high-pass, band-pass and band-suppressing filtering, spatial differentiation (GORLITZ and LANZL [1975], BUTTERWECK and WIERSMA [1977]), contrast improvement, phase- contrast methods (ZERNIKE [ 1935]), matched filtering, pattern multiplication, image deblurring (GOODMAN [1968]). In contrast with their shift- variant counterparts, such systems are also amenable to a strikingly simple realization. According to (8.5), any modulator inserted between two Fourier transformers constitutes a shift-invariant system.

In this section we first review the coherent behaviour of shift-invariant systems. In the further course a spectral analysis of incoherent illumination is developed with special emphasis upon low-pass filtering.

10.1. COHERENT ILLUMINATION

The behaviour of a shift-invariant system with strictly time-harmonic illumination was studied in P 5.1. With the shorthand notations

+l(rl) = +in(rl),

+2(r*) = +out(r*),

hz1(r) = h(r),

&(R) = H(R) = I h(r) exp (-iR * r) dr,

269 IV, 0 101 SHIFT-INVARIANT SYSTEMS

the main results (5.10) and (5.12) can be rewritten as

(10. la )

( 10.1 b)

Hence, in space domain the signal transformation in a shift-invariant system is governed by a (two-dimensional) convolution of the input signal with the point spread h(r ) , whereas in the frequency domain an ordinary multiplication with the system function H ( R ) is involved.

Next, (8.5) reveals that a modulator with modulation function m ( r ) is transformed by two adjacent Fourier transformers ’fJ(d), ’fJ(-d) into a shift-invariant system with a system function

H(R) = m ( t R). (10.2)

If n o constraints are imposed upon m ( r ) , any desirable system function H ( R ) can obviously be realized, and herewith any point spread h(r) .

The situation changes when e.g. only amplitude modulation can be performed in the “Fourier plane”. Then m ( r ) is positive real and so is H(R). As the inverse Fourier transform of a positive function, the point spread h(r ) is then subject to severe restrictions. We note that this formal problem is well known from communication theory where h(r ) is replaced by the autocorrelation function, whose Fourier transform is the positive power spectrum. For the practical implications of the constraints under consideration we refer to the next section, where this problem is reconsidered, albeit in a different physical context.

Another restriction of practical interest appears when m(r) = 0 for Irl> r,. Such a restriction is associated with diffraction-limited imaging, where the exit pupil with aperture radius r, plays the role of the modulator. With (10.2), H(R) then vanishes for IR( > kr,/d and we can speak about a low-pass filtering. The cases m(r) = 1 for Irl< ro as well as m ( r ) = exp (ib(r)) were extensively discussed in 0 8.3, where we reached the conclusion that a varying b(r) due to spherical aberration errors in general broadens the point spread h(r) and so deteriorates the imaging properties. On the other hand, manipulation of ( m ( r ) ( for lr(<ro can improve h(r ) . Such an “apodization” can be performed e.g. by coating the (last) lens of an imaging system. As an example, the originally rectangular one-dimensional modulation function ml(x) of Fig. 10.1 can better be replaced by the triangular m2(x) that possesses a positive

270 PRINCIPLES OF OPTICAL DATA-PROCESSING “, 5 10

Fig. 10.1. About apodization of a diffraction-limited system.

inverse Fourier transform (corresponding to the “line spread”) and therefore exhibits no overshoot after transmission of a black-white jump. Also the width of the pertaining h,(x) (if properly defined) becomes smaller than that of h , ( x ) . For optimization of apodization procedures we further refer to PAPOULIS [1968a].

10.2. INCOHERENT ILLUMINATION

Like coherent light, there is another extreme case within the framework of partially coherent light, that is only realizable in an approximate manner: the incoherent light. What it distinguishes from its coherent counterpart is the property that the idealization degenerates after passage through linear systems, i.e. incoherent light becomes partially coherent after linear transformations.

For incoherent illumination, we have a linear mapping of intensities. It is governed by (4.1 1) and will now be specialized for the shift-invariant system. With the new notations*

p(rJ = lin(ri),

S(rz, r2) = lout(r2),

gzl(rz, r l ) = h(r2-r1),

we obtain

I O u h ) = lh(r)12*Mr)y (10.3)

i.e. the output “intensity” is found as the convolution of the input “intensity” with the squared modulus of the point spread h(r). In a

* The reader is warned against the different dimensions of I , , and I,,,,, as was discussed in 54.2.

SHIFT-INVARIANT SYSTEMS 27 1 IV, 8 101

comparison with (10.1) for coherent illumination, we immediately observe that only positive, real quantities are involved in the incoherent signal transformation. This fact has an important implication for the manner in which the frequency counterpart of (10.3) is usually formulated. As the Fourier transforms of positive functions assume their maximum values at the origin (GOODMAN [1968]), they are advantageously normalized with respect to that value. In our case, this yields

f ( R ) = I ( r ) exp (-iR * r) d r / j I ( r ) dr, 5 (10.4)

(h(r)I2 exp (-iR * r) dr/]lh(r)12 dr, (10.5)

with the indices “in” and “out” to be added to f(R) and I ( r ) . Clearly, f ( R ) and R(R) assume unity values for R = 0, which cannot be exceeded for any R # 0. With these normalized spectral functions (10.3) is transformed into

i0”,(R) = H(R)iin(R). (10.6)

While fin(R) and fout(R) are commonly known as the normalized input and output spectra, R(R) is usually referred to as the complex optical transfer function (OTF). Its squared modulus Ik(R)12 is called the modulation transfer function (MTF). Like the system functions of time-domain filters, f i ( R ) can be experimentally determined through excitation by harmonic functions (“eigenfunctions”). For that purpose, sinusoidal test patterns” are in use for which we can easily derive that the ratio of the relative fluctuations (I,,,=- I~”)/(ImaX + I ~ , ) at the output and the input directly equals Ifi(R)I, while the shift between the location of the maxima determines arg f i (R). The spatial frequency of the sine pattern equals the modulus of R and the normal to the equiphase lines determines its direction.

Since H(R) is the normalized Fourier transform of lh(r)I2 = h(r)h*(r) it can also be written as

E.T(R) = {WR) * H*(-R))I{H(R) * H*(-R)),=,, (10.7)

which is the normalized autocorrelation of H(R) or (except for the scaling constant) the normalized autocorrelation of the modulation function m(r)

* On account of 1 2 0 , these test patterns contain a constant bias term besides the desired sinusoidal function.

272 PRINCPLES OF OPTICAL DATA-PROCESSING [IV, 8 10

in the “Fourier plane”. From (10.7) we also note the Hermitian property

From the foregoing we conclude that not every (Hermitian) function of R is realizable as an OTF f i (R) . If and only if it has a positive Fourier transform or (what amounts to the same) if it can be written as an autocorrelation, such a function is permitted. From the viewpoint of filter synthesis this rules out a great number of interesting frequency responses.

If a function f i ( R ) is permitted, it can be written in the form (10.7) in an infinite number of ways. This can be easily seen in the space domain where the transition lhI2+ h is associated with an arbitrary phase factor”. In certain (but not all) cases this degree of freedom can be exploited to make H(R) positive real. If that succeeds, Fourier filtering can be performed with a pure amplitude modulator.

An important “natural” shift-invariant system is formed by diffraction- limited imaging. As discussed in previous sections, we then have m ( r ) = 0 for Irl> r, or, what amounts to the same, H ( R ) = 0 for (RI >R,. Autocorre- lation according to (10.7) then broadens the frequency characteristics and makes that the low-pass cut-off frequency pertaining to the incoherent system function f i ( R ) equals 2R,. At a first glance this seems to improve the imaging quality and this the more, as the effects of autocorrelation resemble that of apodization (cf. Fig. 10.1; the autocorrelation of a rectangle becomes a triangle). Moreover, the positiveness of the incoherent point spread prevents overshoot. A thorough investigation (GOOD- MAN [1968]) reveals, however, that the belief that incoherent imaging in general is superior to coherent imaging, has insufficient foundation.

A(-R) = fi*(R).

10.3. LOW-PASS FILTERS

The problems around diffraction-limited imaging will now be generalized to more general low-pass filtering. On the one hand this problem is important enough to deserve a separate discussion and on the other hand it illustrates general principles of incoherent filtering. Our treatment follows the ingenious lines developed by LUKOSZ [1962]. For sake of brevity, we confine ourselves to one-dimensional low-pass filters, where the vectors r , R are replaced by x (position) and X (spatial frequency). The cut-off frequency of the filter is denoted by X , so that

* WALTHER [1963] has shown that this arbitrariness is strongly restricted if h is band- limited. This occurs when the modulation function m ( r ) in the Fourier plane has finite “dimensions”.

IV, § 101 SHUT-INVARIANT SYSTEMS 273

fi(X)=O for IXl>X,. Moreover, Ih(x)l' is assumed to be even which implies realness of fi(X). With these premises in mind we now derive an upper bound for fi(X) in the pass-band (XI < X,.

One of the features of incoherent filtering is that with any positive input signal a positive output signal is associated. We shall choose a particularly simple input signal which, in fact, contains not more than one free parameter. The requirement of a positive output signal then yields a necessary condition for R(X).

As illustrated in Fig. 10.2 (left) the input signal lin(x) is given by the function 1 + C cos X,x which is sampled at the equidistant points xk = 2rk/NXo with k = 0, *l, +2, . . . and N an integer to be specified in due course. Without sampling, the spectrum of the above signal would consist of three lines at the frequencies 0, *X,, but due to sampling we obtain the periodic repetition of this line triplet, with a period NX, in the frequency domain (PAPOULIS [1962]).

Assume now that the cut-off frequency X, is below (N-l)Xo. Then only the central triplet is transmitted by the filter, attended by a modification of the line intensities. While the ratio of the line heights at X = *Xo and at X = 0 originally equals C/2, this becomes after passage through the filter fi(X0)C/2. In time domain we obtain at the output lout(x) = IDC,out X

(1 + Cfi(X,) cos X,x) which remains positive if IR(X,)l 5 1/C. Under which condition is the input signal positive? We consider the

marginal condition, viz. l i n ? O where the equality sign holds for some x. This amounts to the requirement that the smallest sampling value be zero. For N even, this yields C = 1, but for N odd we obtain C cos (.rrlN) = 1. The latter case (N even yields a trivial result) leads after combination with the above condition for Iout 2 0 to

Ifi(xo)l 5 cos (dNL (10.8)

which is valid for Xc5(N- l)Xo or

X,ZX,/(N- 1). (10.9)

At a first glance (10.8) and (10.9) seem to apply only for N odd. That this result is also valid for N even, can, however, be easily concluded when the sample points are shifted by half a sampling distance (from 2.rrk/NXo to 2 4 k +i)/NX,) so that the minimum sampling value again becomes zero for C cos (r/N) = 1.

Thus, for N = 3,4,5, . . . corresponding to X,, 2 XJ2, X, 2 X,/3, X, 2

X,/4, . . . we have ~ f i ~ ~ c o s ( r / 3 ) = ~ , ) f i ~ ~ c o s ( ~ / 4 ) = ~ ~ , ~ f i ~ ~

274 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV, § 10

Fig. 10.2. A suitable input signal li,(x) and its Fourier transform for N = 7.

cos (.rr/5) = 0.81, . . . , respectively. Combining these inequalities then yields the “Lukosz bound’’ as depicted in Fig. 10.3.

Only transmission characteristics Ifi(X)) which are below this bound are permitted. Of course, this condition is only necessary, i.e. not every function satisfying it can be realized. In fact, realizability can only be checked via the sign of the inverse Fourier transform of fi(X). From a synthesis viewpoint, permitted functions fi(X) can, moreover, always be constructed through autocorrelation of an arbitrary function H(X), according to (10.7).

A fundamental, permitted function fi(X), which is obtained by autocorrelation of a rectangular H(X) is the triangular characteristic fitri(X), as inserted in Fig. 10.3. Apart from scaling constants, its inverse Fourier transform equals (sin xlx)’ which becomes zero in an infinite number of points, viz. x = n r (n# 0). Since any deviation from the triangular characteristic threatens to destroy the positiveness of the inverse Fourier transform, we are justified to call the triangular function marginally permitted.

It is now logical to ask which functions Afi(X) superimposed upon the triangular function fitri(X) are absolutely forbidden. The answer is that the inverse Fourier transform of Afi(X) must not be negative, where that of &(X) vanishes; otherwise the sum would be locally negative. So we have to solve a nice problem of sampling theory, viz. to look for those functions Afi(X) whose inverse Fourier transforms have nonnegative

I

Fig. 10.3. Lukosz bound for the system function of a low-pass filter.

IV, 5 111 RELATED TOPICS 215

sampling values at the equidistant zeros of (sin x/x)*. As we demonstrate below one readily reaches the conclusion that AR(X) + AR(XC- X), i.e. the OTF plus its mirrored version, must be negative for all X. In other words, if at some frequency X, the total function R(X) exceeds fit"(X), it has to remain below fitri at the mirror frequency (X,-X) by at least the same amount! In the mean, G(X) has to be smaller than fitri(X); excesses in some regions have to be (over-) compensated in others. This implies that the average value of R(X) cannot exceed 4 (= the average value of Rtri(X)). Notice, however, that all these restrictions are only necessary; fulfilment does not guarantee realizability !

We sketch the proof of the above statement. First we notice that sampling in the x-domain corresponds to periodic repetition in the X-domain; the period turns out to become X,. As Afi(X) extends from -X, to +X,, we get an overlapping due to this repetition process such that Afi(X,- X) has to be superimposed upon Afi(X). Positive sampling values now imply that the Fourier series expansion

m

Afi(X)+Afi(X,-X)=c,+ 1 C, cos(2n.rrXIXc) n = l

contains positive coefficients c,, cz, c 3 . . . (c, may become negative because at x = 0, (sin XIX)' does not vanish). This again implies that the above function attains its maximum values at X = 0 and X = X, , where it becomes c,+ c, + c2 + c3 + * * . On the other hand it must vanish there, because we must not deviate from &(X) at these end points. Hence it is negative in the interior O < X < x , Q.E.D.

0 11. Related Topics

In the present article, general optical data-processing systems have been studied under various aspects. In retrospect, however, we are aware that our description is far from complete. As an exhaustive discussion of the missing aspects would fill at least another article, we cannot do more than mention their existence.

Optical systems were hitherto described in wave-theoretical terms only. Another possible description, albeit of restricted validity, involves geometric-optical principles. This applies particularly to 2%-systems, where our three characteristic parameters (viz. the elements of one of the equivalent circuits) find their counterpart in the elements of the 2 x 2 ray

276 PRINCIPLES OF OPTICAL DATA-PROCESSING “, § 11

transformation matrix (DESCHAMPS [1972], O’NEILL [1963]). As its determinant equals unity, this matrix likewise contains three free parameters. An agreeable property of that description is that system cascading amounts to simple matrix multiplication. The fourth system parameter is the price to be paid for this computational advantage.

Recently the Wigner distribution has proved to be a useful tool in optical system analysis (BASTIAANS [ 1978a1, WOLF [ 19781). This simultaneous space-frequency formalism* is based upon wave theory, to be sure, but its results are most conveniently interpreted in geometric-optical terms. In fact, the Wigner distribution constitutes a modern approach to derive geometrical optics from wave optics in the limit of extremely small wavelengths (BASTIAANS [ 1979b1).

Quantum electrodynamics and light propagation in terms of photons likewise belong to the vocabulary of modern optics. These descriptions have found widespread application in the theory of partial coherence (PERINA [1972]), with the remarkable result that the final conclusions are in formal agreement with those of classical wave theory.

While the present article deals with deterministic signals and systems, stochastic aspects can also be taken into account. Noise, like the granularity of a photographic film, has to be considered as a stochastic signal, but also test patterns can be treated as such (O’NEILL [1963]). Speckle is due to unpredictable system fluctuations (dust particles etc.). Finally, grounded glass is an example of a special optical system (a modulator) that has a meaningful description only on a stochastic basis. Rotating grounded glass is a convenient means to convert coherent or partially coherent light into almost incoherent light.

In our treatment optical signals were represented throughout by continuous, two-dimensional functions, i.e. to every position r = ( x , y) a signal value was assigned. For several problems it is more appropriate to use a discrete-space description, in which the signals are represented by two-dimensional arrays of numbers. Then the fundamental superposition integral (3.4) is converted into a sum. If, for the moment being, only one-dimensional signals are taken into account, we have a transformation

* Also without the Wigner formalism and occasionally without proper awareness combined space-frequency argumentations are in common use. As an example, consider a diffraction grating of finite dimensions illuminated by a plane wave. The field behind the grating is then usually decomposed into diffracted “bundles” or “beams”, which e.g. are “lost” if they travel past the following lens. The notion of a beam with a direction (frequency) and a position (space) is such a mixed concept alien to rigorous Fourier theory. Also the well-known space-bandwidth product (LOHMANN [1967]) belongs to this category.

IV,0111 RELATED TOPICS 277

of an input “vector” into an output “vector”, for which the system properties can be represented by a matrix. Cascading then finds expres- sing in matrix multiplication, system inversion means matrix inversion. Losslessness involves a unitary matrix, and symmetry a symmetrical matrix. The conservation theorems of 0 5.2 are then interpretable in terms of well-known matrix properties.

The practical significance of a discrete-space representation is two-fold: First, halftone realizations of continuous-tone objects are inherently discrete. When, moreover, the measurement of the output signal is performed only at the corresponding array points, we have, in fact, an entirely discrete signal-and-system representation. In a second case dis- creteness is less tangible. We mean the possible representation of a band-limited signal by its sampling values. If the system (e.g., as in the shift-invariant case) does not produce additional higher frequencies, the input and output signal can be likewise sampled, and we obtain a fictitious discrete system”.

A complete optical system theory has to reckon with the vectorial nature of light. This implies that the pertinent catalogue of optical components needs to comprise polarizers, quarter-wavelength plates etc., which manipulate the state of polarization of the incoming light. At present, it seems that only components without lateral parameter variations have been studied, which transform normally incident plane waves in normally outgoing plane waves. Fundamental work in this direction has been done by JONES [1956].

An important aspect of optical system theory is that of synthesis. Whereas we have extensively discussed the realization of shift-invariant systems (cf. 0 lo), that of shift-variant systems was hardly touched upon. In fact, at present no general synthesis procedure is available?, and the question arises whether it would be very useful (FRANCOIS and CARLSON [1979]). The variety of these systems is so vast, that every subclass asks for a specific treatment. We only mention systems which involve a geometric distortion and which have been studied by BRYNGDAHL [1974]. In this context we like to mention that not all practical systems fit into our approach which is essentially based upon cascade connections of simple

*Also series expansions of the input and output signal in terms of given sets of (orthogonal) functions lead to discretizations of optical systems.

t This assertion applies only to two-dimensional systems with point spreads g2, (r2 , r l ) . For one-dimensional systems the pertinent point spread g,,(x,, xl ) can be relatively simply realized with two-dimensional means (GOODMAN [1977]).


components. This approach excludes e.g. volume holograms, but also systems containing mirrors and beam-splitters (LOHMANN and RHODES [ 19781). Arrangements using these components add new possibilities to the solution of a number of synthesis problems.

Concerning synthesis, we have to deal with two basic questions. First, are there any fundamental restrictions to be imposed upon the point spread of an optical system? Certainly, the familiar causality constraint of time-domain filtering has no spatial counterpart. For incoherent systems, we have the restriction that all space functions including the point spread, are real and nonnegatiue. On the other hand, there is sufficient evidence for the belief, that coherent processing is unrestricted in the sense that any well-behaving complex point spread function can, in principle, be realized.

Secondly, which systems (i.e. which point spreads) are actually required? It seems that the main interest is directed towards the easily realizable shift-invariant systems. Besides the systems producing geometric distortions as discussed above and some very special systems (like the Mellin transformer required for pattern recognition (GOODMAN [1977])) the class of the weakly shift-variant (i.e. the almost shift-invariant) systems will probably attract the most attention. For further information cf. GOODMAN [1977], CASASENT and PSALTIS [1978].

A comprehensive study of optical systems has also to include a number of (semi-) technological aspects. We only mention the practical realization of a modulator. For m ( r ) = Im(r)l< 1 we have an amplitude modulator with a positive real modulation function which, due to passivity, cannot exceed unity. A photographic realization of such a modulator with continuous shades of gray meets considerable difficulties due to the nonlinear characteristics of the photographic emulsion. With a high- resolution film one can simulate “gray” by a “half-tone’’ technique that utilizes only the two levels m = 0 (black) and m = 1 (white). The pertinent pattern which can be drawn and reproduced with high precision can be optimized in the sense that after proper low-pass filtering exactly the required continuous function appears (BASTIAANS [ 1978bl)). For circularly symmetric functions another two-level simulation applies making use of concentric rings (WIERSMA [1978]).

More severe problems are envisaged if a phase modulator has to be realized. In the present state of the art, bleaching and etching processes have not yet led to satisfactory, well-reproducible results (VAN DER LUGT [1974]). Due to the ingenious method of VAN DER LUGT [1964] phase

IVI REFERENCES 279

variations can, however, be modulated upon a high-frequency carrier thereby transforming the desired phase modulator into an amplitude modulator. Using holographic methods, complex light distributions* can be directly recorded on photographic film (provided that the linear part of its characteristics is used) and thereby transformed into modulation functions m ( r ) . Another way which ultimately leads to two-level modulators, is that of computer holography as proposed by LOHMANN and PARIS [1967]. We finally note that a lens as a special phase modulator can be simulated by a Fresnel-zone plate which consists of concentric transparent rings (PAPOULIS [1968a]).

References

BAGHDADY, E. J., 1961, Lectures on Communications system theory (McGraw-Hill, New

BASTIAANS, M. J., 1977, Opt. Acta 24, 261. BASTIAANS, M. J., 1978a, Opt. Comm. 25, 26. BASTIAANS, M. J. , 1978b, J . Opt. SOC. Amer. 68, 1658. BASTIAANS, M. J., 1979a, private communication. BASTIAANS, M. J., 1979b. Opt. Comm. 30, 321. BERAN, M. J. and G. B. PERRANT, 1964, Theory of Partial Coherence (Prentice-Hall,

BORGNIS, F. E. and C. H. PAPAS, 1955, Randwertprobleme der Mikrowellenphysik (Springer,

BORN, M. and E. WOLF, 1965, Principles of Optics, Fifth ed. (Pergamon Press, Oxford)

BRYNGDAHL, O., 1974. J . Opt. SOC. Amer. 64, 1092. BUTERWECK, H. J., 1977, J. Opt. SOC. Amer. 67, 60. BU~TERWECK, H. J., 1978, Arch. f . Elektronik u. Ubertragungstechnik 31, 335. BUITERWECK, H. J . and P. WIERSMA, 1977, J . Appl. Sc. and Eng. 2, 213. CASASENT, D. and D. PSALTIS, 1978, in: Progress in Optics, ed. E. Wolf (North-Holland,

CASIMIR, H. B. G., 1963, Proc. IEEE 51, 1570. DESCHAMPS, G. A., 1972, Proc. IEEE 60, 1022. FRANCOIS, R. E. and F. P. CARLSON, 1979, Appl. Opt. 16, 2775. GOODMAN, J . W., 1968, Introduction to Fourier Optics (McCraw-Hill Book Co., New

GOODMAN, J. W., 1977, Proc. IEEE 65, 29. GORLITZ, D. and F. LANZL, 1975, Japan J. Appl. Physics, Suppl. 14-1, 223. GOUBAU, G., 1961, Electromagnetic Waveguides and Cavities (Pergamon, Oxford) Sect. 21.

*If the complex light distribution is the output signal of a Fourier transformer, the Fourier transform of some required function can be recorded. This yields a method to record any wanted point spread of a shift-invariant system with optical means. The developed film is then inserted in the “Fourier plane” of the well-known 82R8 arrangement (8.5).

York).

Englewoods Cliffs, New Jersey).

Berlin) Section 2.4.

Sect. 8.3, 10.2.

Amsterdam) Vol. 16, p. 291.

York) Sect. 2, 4.1, 5.1, 6.1, 6.3, 6.5.

280 PRINCIPLES OF OPTICAL DATA-PROCESSING [IV

JONES. R. C., 1956. J . Opt. SOC. Amer. 46, 126. KUPFMULLER, K., 1948, Die Systemtheorie der elektrischen Nachrichtenubertragung (Hir-

KUPRADSE, W. D., 1965, Randwertaufgaben der Schwingungstheorie und Integral-

LEITH. E. N., 1977, Proc. IEEE 65, 18. LOHMANN, A., 1967, IBM Research Paper RJ-438. LOHMANN, A. and D. P. PARIS, 1965. J . Opt. Soc. Amer. 55, 1007. LOHMANN, A. and D. P. PARIS. 1967, Appl. Opt. 6, 1739. LOHMANN, A. and W. T. RHODES, 197X, Appl. Opt. 17, 1141. LUKOSZ, W., 1962, Opt. Acta 9, 335. MANDEL. L. and E. WOI.F, 1076. J . Opt. Soc. Amer. 66. 529. MATTHIJSSE. P. and P. J. G. HAMMER, 1975. J . Opt. Soc. Amer. 65, 188. MENZEL, E., W. MIRANDE and I. WEINGARTNER, 1973. Fourier-Optik und Holographie

O'NEILL, E. L., 1963, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass.)

PAPOULIS, A., 1962, The Fourier Integral and its Applications (McGrdw-Hill, New York)

PAPOULIS, A., lY68a. Systems and Transforms with Applications in Optics (McGraw-Hill.

PAPOULIS, A.. 1968b. J . Opt. SOC. Amer. 58, 653. PERINA, J., 1972, Coherence of Light (Van Nostrand, London). SLATER, J. C., 1954, Microwave Electronics (Van Nostrand, New York) Sect. 4.2. SOMMERFELD, A., 1954, Partielle Differentialgleichunyen der Physik (Akad. Verlags-

STRATTON, J . A., 1941, Electromagnetic Theory (McGraw-Hill Book Co.. New York) Sect.

VAN DER LUGT, A.. 1964, IEEE Trans. Inf. Theory IT-10, 139. VAN DER LUGT, A., 1966, Proc. IEEE 54, 10.55. VAN DER LUGT, A,, 1974, Proc. IEEE 62, 1300. VAN VALKENBURG, M. E., 1965, Network Analysis (Prentice Hall. Englewoods Cliffs) Sect.

VAN WEERT, M. J., 1978. J. Opt. Soc. Amer. 68, 1775. VAN WEERT, M. J., 1980, J. Opt. SOC. Amer. 70, 565. WALTHER, A,, 1963. Opt. Acta 10, 41. WATSON, G. N., 1966, A Treatise on the Theory of Bessel Functions (University Press,

WIERSMA, P.. 1978. Opt. Acta 25, 917. WOLF, E., 1978, J . Opt. SOC. Amer. 68, 6. ZERNKE, F., 1935. Z. Techn. Physik 16, 354.

zel, Stuttgart).

gleichungen (Deutscher Verlag der Wissenschaften, Berlin) Sect. 4.

(Springer, Wien) Sect. 2.9d.

Sect. 2.4., 3.4.. 3.7.. 7.1.

Sect. 1.3., 3.4.

New York) Sect. 9.3, 11.3. 11.5.

gesellschaft, Leipzig) Sect. 28.

8.13.. Problem 13 of Sect. 8.

11.1.

Cambridge) p. 416.

E. WOLF, PROGRESS IN OPTICS XIX @I NORTH-HOLLAND 1981

V

THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY

BY

F. RODDIER

DCpurtemenf d'Astrophysique de 1'1. M.S.P., Equipe de recherche ussociie au C.N.R.S. no 669, Uniuersitt! de Nice, Parc Valrose, 06034 Nice Cedex, France

CONTENTS

§ 1 .

§ 2 .

§ 3 .

§ 4 .

§ 5 .

§ 6 .

9 7 .

§ 8 .

§ 9 .

PAGE

INTRODUCTION . . . . . . . . . . . . . . . 283

STATISTICAL PROPERTIES OF ATMOSPHERIC TUR- BULENCE . . . . . . . . . . . . . . . . . . 284

STATISTICAL PROPERTIES OF THE PERTURBED COMPLEX FIELD . . . . . . . . . . . . . . . 291

LONG-EXPOSURE IMAGES . . . . . . . . . . . 297

SHORT-EXPOSURE IMAGES . . . . . . . . . . 309

EXPOSURE-TIME AND NON-ISOPLANICITU' EF- FECTS . . . . . . . . . . . . . . . . . . . 319

OPTICAL PATH FLUCTUATIONS . . . . . . . . . 328

STELLAR SCINTILLATION . . . . . . . . . . . 341

APPLICATIONS TO HIGH RESOLUTION IMAGING . 350

8 10 . SEEING MONITORS AND SITE TESTING . . . . . . 360

5 11 . CONCLUSION . . . . . . . . . . . . . . . . 367

REFERENCES . . . . . . . . . . . . . . . . . . 368

Q 1. Introduction

Atmospheric turbulence is a major problem in optical astronomy as it drastically reduces the angular resolution of telescopes. The diameter of the image of a star, also called the seeing disk, varies approximately from 0.3 arcsecond to 10 arcseconds or more, according to weather conditions. A typical diameter is 2 arcseconds, which is the resolution limit of a 6 cm aperture in the visible.

When a stellar image is observed through a telescope with high magnification, the observed image structure, even with perfect optics, is usually far from the theoretical diffraction pattern and changes rapidly with time. The appearance of the image depends strongly on the aperture of the telescope. With small apertures a random motion of the image is often the main effect. With large apertures spreading and blurring of the image occur. A speckle structure is often observed, somewhat similar in appearance to a bunch of grapes, as noted by ROSCH [1958b].

In the early sixties, the physics of image degradation by atmospheric turbulence was still little understood. Several techniques were worked out on an empirical basis in order to make quantitative estimates of the amount of degradation. They are reviewed by STOCK and KELLER [1960], MEINEL [1960], and in the Proceedings of the I.A.U. Symposium no 19 (ROSCH, COURTES and DOMMANGET [1963]). The influence of the telescope location on image quality was recognised and these techniques were extensively used during the site testing campaigns leading up to the construction of most of our modern observatories. An I.A.U. working group on site testing was created and a Symposium on this subject held in Rome in October 1962. Several questions arose: is it possible to describe image degradation with a single atmospheric parameter? Is it possible to predict image behaviour through a large telescope from measurements through a small one? No answer could be given due to the absence of any theory of image formation through turbulence.

Since that time, the situation has evolved considerably. Observation sites once having been chosen, astronomers became less and less

283

284 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [v, 5 2

interested in measuring image degradation. At the same time, due to the discovery of the laser, optical physicists became more and more interested in optical propagation through turbulence, leading to considerable advances in this field. Unfortunately, the work of optical physicists has often remained unknown to astronomers (for example LINFOOT and WITCOMB [1972] and GRIFFIN [1973]). At the same time, the application by A. LABEYRIE [ 19701 of speckle interferometry techniques to stellar images gave a new impetus to high resolution imaging techniques as well as to the theory of image formation through turbulence. This theory is now well established on a solid experimental basis but its importance is not yet fully appreciated by the majority of astronomers.

The purpose of this paper to to summarise the present state of the theory, review the experimental checks that have been made and discuss the implications in the domain of astronomical observations. We hope that this review will help to acquaint the astronomical community with recent advances in this field.

0 2 briefly summarises the theory of energy cascades due to KOL- MOGOROV [1941] and the related spectral properties of temperature and refractive index fluctuations in the atmosphere derived by OBUKHOV [1949], YAGLOM [1949] and CORRSIN [1951]. § 3 summarises the spectral properties of the complex amplitude fluctuations of a perturbed plane wave, derived by CHERNOV [1955] and TATARSKI [1956]. In order to minimise mathematical calculations we have adopted a phase screen approach similar to that of LEE and HARP [1969]. § 4, § 5 and § 6 deal with the statistical properties of a point source image derived mainly by HUEVAGEL and STANLEY [1964], FRIED [1966] and KORFF [1973]. Applica- tion to Michelson stellar interferometry and aperture synthesis is also discussed. In 0 7 and 08, the phase screen approach is used again to derive the statistical properties of amplitude and phase fluctuations, i.e. of stellar scintillation and of optical path or angle-of-arrival fluctuations. Applications to high resolution imaging are discussed in § 9. Applications to site testing and the measurement of image quality are discussed in 8 10.

0 2. Statistical Properties of Atmospheric Turbulence

2.1. STRUCTURE OF TURBULENCE

Mathematical descriptions of turbulence are presented in the books by BATCHELOR [1970] and by HINZE [1959]. A didactic presentation, with

v, 5 21 STATISTICAL PROPERTIES OF ATMOSPHERIC TURBULENCE 285

particular emphasis on atmospheric turbulence, can be found in the book by TENNEKES and LUMLEY [ 19721. Properties of atmospheric turbulence are described in detail in the book by LUMLEY and PANOFSKY [ 19641. Here we shall briefly summarise the results pertinent to optical propagation, as reviewed by TATARSKI [1961].

A flow becomes turbulent when the Reynolds number R e = VoLo/vo exceeds a critical value which depends only upon the geometrical structure of the flow. Here, V, is a characteristic velocity and Lo a characteristic size of the flow; v, is the kinematic viscosity of the fluid. Atmospheric air flow is nearly always turbulent. Since the kinematic viscosity of air is of the order of uo= 15x 10-6m2s-', taking V,= 1 m/s and Lo= 15 m leads to R e = lo6 which, in general, corresponds to fully developed turbulence.

KOLMOGOROV [ 19411 suggested that, in fully developed turbulence, the kinetic energy of large scale motions is transferred to smaller and smaller scale motions. Motions on a small scale are statistically isotropic. Motions at scale L have a characteristic velocity V. When the Reynolds number VL/u, becomes small enough, the break up process stops and the kinetic energy is dissipated into heat by viscous friction. In a stationary state, the rate E , of viscous dissipation must be equal to the rate of production of turbulent energy. It is therefore reasonable to assume that the velocity V of motions at scale L depends only upon L and upon the rate E~ of energy production and dissipation. A dimensional reasoning then easily shows that

v 0: EdLt . (2.1) In a spectral analysis of the kinetic energy as a function of the modulus

K of the wave vector K, the energy E ( K ) dK between K and K + d ~ is proportional to V'(K) . Taking (2.1) into account, with L 0: 1 / ~ , leads to:

which expresses the Kolmogorov law, valid only in the inertial range L;'<< K << I ; ' , where Lo is the outer scale (generally the scale of the motions which give rise to turbulence) and 1, is the inner scale at which viscous dissipation arises.

In the troposphere, I , ranges from a few millimetres near the ground to about l c m near the tropopause. Lo is of the order of the thickness of turbulent layers, that is about 100 m. Near the ground, it is of the order of the height above the ground. We shall see that the size of the wavefront perturbations, degrading astronomical images, ranges between a few


centimetres and the size of the telescope aperture. Kolmogorov’s law therefore entirely applies and is indeed found consistent with observations. However in the case of long baseline interferometry it may become questionable.

2.2. TEMPERATURE AND HUMIDITY FLUCTUATIONS

Temperature and humidity are both functions of height in the atmosphere. Turbulent mixing therefore creates inhomogeneities of temperature and humidity at scales comparable to eddy sizes. OBUKHOV [1949] and YAGLOM [1949] have shown that, in a turbulent flow, the concentration of an additive which is passive (i.e. does not affect the dynamics of turbulence) and which is conservative (i.e. does not disappear by chemical reaction of the other), also follows Kolmogorov’s law. The inner and outer scales differ, in principle, from 1, and Lo, the inner scale being related to molecular diffusion. However, as far as we are concerned, they happen to be of the same order of magnitude. This result applies, to a good approximation, to the mixing of air with water vapour or the mixing of cool air with warm air. The power spectrum Q T ( ~ ) of temperature fluctuations and the power spectrum @,-(K) of humidity fluctuations are therefore also described by

@ T ( K ) CC K-? , @ c ( K ) CI: K d . (2.3)

In the following, we shall deal with three-dimensional spectra @ ( K ) =

@( K,, K ~ , K ~ ) . One-dimensional spectra are related to three-dimensional spectra by integration over all directions so that, in the isotropic case

@ ( K ) = 4TK2@(K),

theref ore

GT(~) K-? and @-(K) CI: ~-4’. (2.4)

BT ( P ) = ( @ ( r ) @ ( r + p ) ) . (2.5)

The covariance of the temperature fluctuations 0 = T - ( T ) is

According to the Wiener-Khinchine theorem, it is the three-dimensional Fourier transform of Q T ( ~ ) defined as

v, 021 STATISTICAL PROPERTIES OF ATMOSPHEFUC TURBULENCE 287

However it cannot be derived from (2.4) since integral (2.6) would diverge at the origin (where (2.4) is not valid). Following TATARSKI [1961] and others, it is convenient to consider the structure function

DT(p) = (I@(r+p)- @(r)12) (2.7)

D, ( P ) = 2[B, (0) - Blr @)I (2.8)

related to the covariance by

which remains finite as long as IpI is finite. Again, an expression for D T ( p ) , valid in the inertial range, can easily be derived by dimensional reasoning, assuming that it depends only upon ( P I = p, upon the rate c0 of production of turbulent energy and upon the rate qo of production of temperature fluctuation (which is also the rate of molecular dissipation). Then necessarily Da ( p ) is proportional to

D, ( p ) Lx qo&gfpf (2.9)

D,(p) = c2,pl (2.10)

as derived by OBUKHOV [1949]. Eq. (2.9) is usually written

defining C: as the structure constant of temperature fluctuations. With such a definition, TATARSKI [1961] has shown that (2.4) becomes

(2.11)

Similar expressions can be derived for humidity fluctuations c = C- ( C ) .

2.3. REFRACTIVE INDEX FLUCTUATIONS

The refractive index N of air is a function N(U , C) of the temperature U and of the concentration C of water vapour. Its fluctuations n = N-(N) are therefore given by

aN dN =- @+- c = A@ +Bc au ac (2.12)

with variance

(n’)=A2(@2)+2AB(@c)+B2(c2). (2.13)

In optical propagation the last term is always negligible. FRIEHE and LARUE [1974], ANTONIA, CHAMBERS and FRIEHE [1978] have shown that

288 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN ORICAL ASTRONOMY [v, 5 2

the second term can be a significant correction factor, either positive or negative, in a marine boundary layer. However, it seems to be negligible in the case of most astronomical observations.

An expression for the coefficient A can easily be derived from Glad- stone’s relation. Assuming pressure equilibrium leads to:

A = 80 x lop6 P/U ’, (2.14)

where the air pressure P is expressed in millibars and the air temperature U in Kelvin degrees. Since n =A@, the structure function of n also follows Obukhov’s law

DN(p) = c&pa (2.15)

where CN is the index structure constant. It is related to the temperature structure constant C, by

CN=ACT. (2.16)

Similarly, the power spectrum of the index fluctuations is given by

QN(tc) = 0.033C&~-y. (2.17)

The parameter CL expresses the contribution of turbulence to optical propagation. Measuring its dependence upon height has been the aim of many investigations, the results of which will be now reviewed.

2.4. DEPENDENCE OF C i WITH HEIGHT AND TIME

An old, but good description of the origin of astronomical seeing can be found in the book by TEXEREAU [1961]. The sources of image degradation can be roughly divided into four origins: turbulence associated with the telescope and the dome, turbulence in the surface boundary layer or due to ground convection, turbulence in the planetary boundary layer or associated with orographic disturbances, and turbulence in the tropopause or above.

Turbulence inside the telescope and the dome are due to temperature gradients and can therefore be avoided. Turbulence due to velocity gradients at the interface between still air inside the dome and wind driven air outside the dome cannot be avoided and is a major source of image degradation. However the associated temperature or refractivity fluctuations are entirely due to temperature gradients. They can therefore

v, 6 21 STATISTICAL PROPERTIES OF ATMOSPHERIC TURBULENCE 289

be considerably reduced by carefully equalising inside and outside air temperatures. Difficulties arise during daytime observations because of the solar heating of the instrument.

Turbulence in the surface boundary layer and its diurnal cycle has long been known by astronomical observers. During the last decade, it has been extensively studied by people interested in laser beam propagation in the atmosphere. A review of the results can be found in the monograph by CLIFFORD [1978]. This turbulence layer extends roughly up to several tens of meters above the ground. C: values have been derived from high-speed temperature sensors either ground based (LAWRENCE, OCHS and CLIFFORD [ 19701) or tower mounted (KALLISTRATOVA and TIMANOVS- KIY [1971], OCHS and LAWRENCE [1972], NEFF [1975]). They can also be derived from heat flux estimates (WESELY and ALCARAZ [1973]). Acoustic soundings allow direct visualisation of refractivity fluctuations up to about 1 km. This new technique, reviewed by SINGAL [1974] and AUBRY [1975] also leads to C: estimates (NEFF [1975]). Radar soundings (METCALF [ 19751) and aircraft soundings (TWANG [ 19691, OCHS and LAWRENCE [ 19721) have also been used to study refractivity fluctuations within the first kilometre above the ground.

The diurnal cycle, well known to solar observers, is due to the solar heating of the ground (HESS [ 19591). Typically, turbulence reaches a minimum just after sunrise and steeply increases until early afternoon. Growing thermal plumes are seen on acoustic sounding records. Then, turbulence decreases to a secondary minimum after sunset. It slightly increases again during nighttime. 12 m above the ground, KALLISTRATOVA and TIMANOVSKIY [1971] find typical C’, values of the order of m-3 during daytime and m-3 during nighttime. A h-‘ height dependence has been predicted by WYNGAARD, IZUMI and COLLINS [1971], under unstable daytime conditions on flat land, and has been found to agree with observations (TWANG [1969], NEFF [1975]). They also predict a h-; dependence under neutral conditions and a slower decrease under the stable conditions that occur during nighttime.

The behaviour of turbulence above 1 km has been little known until recently. An exponential decrease of C’, with height was often assumed (REIGER [1962, 19631, YOUNG [1969]). A more realistic model was derived by HUFNAGEL [1966] taking into account both atmospheric data and stellar scintillation data. The use of balloon-borne thermal probes initiated by COULMAN [1973] and BUFTON [1973a, b] greatly improved our knowledge. The same technique has since been extensively used by


h lo-’ 0.1 1 10 kn

Lll I 1 I l l I I I l l

Fig. 1. Average CL profile (HUFNAGEL. r197-11) extended towards low altitudes according a h d law (neutral nighttime conditions). Dotted line: extension according a h-f law (unstable

daytime conditions).

BARLETIT, CEPPATELLI, MORODER, PATERNO and RIGHINI [1974] and BAR- LETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [ 19771 during the J.O.S.O. site testing campaign. Turbulence appears to be concentrated into thin layers with a typical thickness of 100-200 m where C$ increases by more than one order of magnitude above its background level. Below 4 km, orographic disturbances certainly play an important role, but above 4 km, BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [1976] conclude that the behaviour of turbulence is almost independent of the location. Indeed, when the layered structure is smoothed out, and the average of many profiles is taken, a typical behaviour is observed. It reaches a minimum of the order of m-f around 6-9 km, slightly increases to a secondary maximum near the tropopause and decreases again in the stratosphere. Low resolution C$ profiles showing this general behaviour have also been obtained from stellar scintillation analysis by

V, § 31 STATISTICAL PROPERTIES OF THE PERTURBED COMPLEX FELD 291

OCHS, TIN-I-WANG, LAWRENCE and CLIFFORD [1976] and by VERNIN, BARLETTI, CEPPATELLI, PATERNO, RIGHIN~ and SPERONI [ 19791 who found a good agreement with simultaneous thermal soundings. Radar measurements of C’, profiles have also been reported by VANZANDT, GREEN, GAGE and CLARK [1977, 19781. Turbulence near the tropopause is due to strong wind shears frequently occurring in this region. Its time evolution can be followed by radar soundings (BROWNING [1971]) or by stellar scintillation analysis as recently shown by AZOUIT and VERNIN [ 19801.

An improved model for the average C’, profile and its fluctuations has been proposed by HUFWAGEL [1974] and found to agree well with the observations of BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [1976] who also derived a lucky observer model by plotting the smallest C’, value ever observed at each altitude. Hufnagel’s average C’, profile is presented in Fig. 1 with an extension towards lower altitudes according to the predicted power laws. Although individual atmospheric measurements show considerable departures from the model, the order of magnitude of the predicted effects on seeing are well reproduced, as discussed in the following sections.

0 3. Statistical Properties of the Perturbed Complex Field

Because the atmosphere determines the ultimate limitations of optical telescopes, astronomers were among the first to be interested in optical propagation through turbulence (see LITTLE [ 195 11, CHANDRASEKHAR [1952], KELLER [1953, 19551). VAN ISACKER [1954] attempted to derive the spectral distribution of thermal fluctuations in the atmosphere from scintillation measurements. However CHERNOV [ 19551 and TATARSKI [1956] were the first to introduce Kolmogorov’s law into their theory. The English translation of their monographs (CHERNOV [ 19601, TATARSKI [1961, 19711) made available the first complete treatment of the problem of wave propagation in turbulent media. Applications to optical propagation in the atmosphere and discussions on the limitations of the theory can be found in several articles by STROHBEHN [1968, 1970a, 1971, 19731 and CLIFFORD [1978]. A simple physical approach has been derived by LEE and HARP [1969]. Modern theories include the effect of strong fluctuations and multiple scattering. They are discussed by USCINSKI [1977], ISHIMARU [1978] and STROHBEHN [1978]. In this section, we shall


deal only with the complex amplitude. We shall briefly derive its properties relevant to image formation (coherence functions) using a phase- screen approach similar to that of LEE and HARP [1969] and the formalism of Fourier optics as presented by GOODMAN [1968]. Fluctuations of amplitude (scintillation) and of angle-of-arrival, will be similarly examined in § 7 and § 8, using the small perturbation approximation.

For the sake of simplicity, we shall consider only horizontal monochromatic plane waves, at wavelength A, propagating downward from a star at the zenith, towards a ground-based observer. Each point of the atmosphere will be designated by a horizontal coordinate vector x and an altitude h above the ground. The scalar vibration located at coordinates (x, h ) will be described by its complex amplitude

q h (x) = I q h (x)l exp [ i q h (XI]. (3.1)

At each altitude h, the phase (ph(x) will be referred to its average value so that, for any h, ((Ph(X))=O. Furthermore, we shall normalise to unity the unperturbed complex amplitude outside the atmosphere, so that TJx) = 1. Finally we shall consider the atmosphere as non-absorbing and hori- zontally stratified (its statistical properties depend only upon h) .

3.1. OUTPUT OF A THIN TURBULENCE LAYER

Let us now assume that the earth atmosphere is still and homogeneous everywhere except inside a thin horizontal layer, between altitude h and h + 6h. The layer thickness is chosen to be large compared to the correlation scale of the inhomogeneities but small enough for diffraction effects to be negligible over the distance 6 h (thin screen approximation). Since, at the layer input, qh+ah(x)= 1, at the output

* h (4 = exp [icp ( 4 1 (3.2)

where cp(x) is the phase shift introduced by index fluctuations n(x , h ) inside the layer

h+Sh

q ( x ) = k l dz - n(x, z ) h

(3.3)

where k = 2r/A is the wavenumber of the vibration. The second order moment of the complex random field qh(x) at the

STATISTICAL PROPERTIES OF THE PERTURBED COMPLEX HELD 293 V, 531

layer output is the coherence function

Bh(&) = (ph(x)p?(x+ 8). (3.4)

(3.5)

Putting eq. (3.2) into eq. (3.4) leads to

Bh (5) = (exp i[q (x) - cp (x + &>I>. Since q ( x ) is the sum of a great number of independent variables (eq. (3.3)), it has Gaussian statistics. The expression between square brackets in (3.5) is therefore also Gaussian and has a zero mean. Since &(&) is its characteristic function (Fourier transform of the probability density function) at unit frequency, it is equal to

Bh (8 = exp -%lq (x) - q ( w + &)Iz) (3.6)

or, introducing the two-dimensional (horizontal) structure function 0, (5) of the phase cp(x),

Bh (6) = exp (8- (3.7)

We must now relate 0,(&) to the statistics of the index fluctuations. Let B,(&) be the covariance of cp(w)

B , ( & ) = ( d x ) c p ( x + & ) ) . (3.8)

Putting (3.3) into (3.8) leads to

h+Sh h+Sh

B,(&)= k ’ j dzIh dz’(n(x, z ) n ( x + & , 2’)).

Introducing C = z’ - z and the three-dimensional covariance ElN(&, 1;) of the index fluctuations, gives

h

h+Sh-z

B , ( & ) = k Z I h i B h d ~ j dc-BN(f; 1;). (3.9) h h-z

Since 6h has been assumed to be much larger than the correlation scale of the index fluctuations, the integration over C can be taken from --co to +-co

so that

B,(S) = k 2 d5 M 5 , I ) . (3.10)

The phase structure function is related to its covariance by

0, ( f ) = 2[B, (0) - B, (511.


or

Dq(&)= k 2 6hld5[DN(&? 5)-DN(o, 511

where

DN(&, 5) = 2[BN(0, O ) - BN(6, 511 is the index structure function. Assuming Obukhov's law (2.15)

a(&, 5) = c3t2 + f2)' where 6 = \&I, so that (3.11) leads to

or after integration

D,(&) = 2.91k2C$ ah@.

(3.11)

(3.12)

Putting (3.12) into (3.7) gives the second order moment of the complex field ?Ph(x) at the layer output

I?,,(&) = exp -$(2.91kZC$ ah@). (3.13)

The complex field ?Po(x) at ground level is the field diffracted by the layer. Since optical wavelengths are much smaller than the scale of the observed wavefront perturbations, the Fresnel approximation can be used safely. Therefore (GOODMAN [1968])

(3.14)

where the symbol * denotes a two-dimensional convolution with respect to the variable x. A remarkable property is that coherence functions &(&) are invariant by Fresnel diffraction. Indeed, the coherence function Bo(&) at ground level is

V, Q 31 STATISTICAL PROPERTIES OF THE PERTURBED COMPLEX FIELD 295

Now, as easily shown by taking the Fourier transform,

ihh

where 8(x) is Dirac’s impulse symbol. Therefore

B O ( 5 ) = Bh (8

or, taking (3.7) into account,

(3.15)

BO(5) = exp -50, (5). (3.16)

For high altitude layers, the complex field P,,(x) at ground level will fluctuate both in amplitude (scintillation) and in phase. Therefore, in (3.16), 0, must not be taken as the structure function of the phase at ground level. However, as discussed in § 7.3, the correction remains small in the conditions of astronomical observations. Taking 0, as the phase structure function at ground level is called the near-field approximation.

3.2. MULTIPLE LAYERS AND THICK LAYERS

Let us now assume (as is often true) that turbulence is located in a number of thin layers between altitudes h, and h, +ah,. The complex amplitude q h , at the output of layer j is related to the complex amplitude *,,+Ah, at the input by

*hj (x) = *h,+Sh, (x) * exp [iqj (XI] (3.17)

where q,(x) is the phase fluctuation introduced by layer j . Since ‘p, is statistically independent of *h,+*h,, the coherence at the output is related to the coherence at the input by

( w h , ( x ) * *ff(x+&))= (*h,+*h,(x) * *$+*h, (X+ 8) (exp i[qJ - qJ (x + &>I)* (3.18)

Following (3 .9 , and (3.13), we obtain

(exp i[ql (x) - q, (x + &)I) = exp -&[2.91 k2C2,(h,) Sh,@]. (3.19)

Through each layer, the coherence function is multiplied by expression (3.19). Between layers, it remains unaffected. Its value on the ground is

296 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OFI'ICAL ASTRONOMY [v, 8 3

therefore

Bo(&) = fl exp -i[2.91k2C;(hj) 6hj53] i

1 2.91 k2@ C$(hj) ahj i

(3.20)

which easily generalises, for a continuous distribution of turbulence, into

El,,(&) = exp -- 2.91k2$ dh - C;(h)] 2 " I (3.21)

the integral being extended all over the earth's atmosphere. When observing at an angular distance y from the zenith, the thickness

6h of each layer is multiplied by (cos y)-' and, to a good approximation, the coherence function Bo(&) on the telescope aperture plane is

Bo(&) = exp -5 2.91k2(cos y)- '@ I dh C:(h)]. (3.22) " It will be shown, in § 4, that the properties of long exposure images are entirely determined by the coherence of the complex field Vo. Expression (3.22) is therefore of fundamental importance. It has been criticised by LUTOMIRSKI and YURA [1971] who demonstrated the influence of a finite outer scale. The influence of both finite inner and outer scales has recently been investigated by VALLEY [ 19791. Departures from Kol- mogorov's spectrum at small scales have also been reported and their influence on optical propagation have been examined by HILL and CLIF- FORD [ 19781. Experimental evidence for departures from the 5/3 power law in (3.22) has been reported by BOURICIUS and CLIFFORD [1970], by CLIFFORD, BOURICIUS, OCHS and ACKLEY [1971] and by BUSER [1971] in the case of horizontal propagation near the ground. However, no depar- ture has been observed in the experiments, described in § 4 and § 7, made under the conditions of astronomical observations. Expression (3.22) can therefore be considered as a good approximation for describing astronomical seeing.

3.3. FOURTH ORDER MOMENTS

In order to describe the statistical behaviour of short exposure images, an expression will be needed for the fourth order moment

M~(&, & ' ) = { q o ( x ) G ( x + &)'J',*(x+C)'J'dx+ &+ &')>. (3.23)

V, B 41 LONG-EXPOSURE IMAGES 291

Such an expression cannot be derived without any additional assumption about the statistics of complex field To. When turbulence is located near the ground, expression (3.2) holds with a Gaussian phase cp. To is said to be log-normal. As shown in § 7.3, (3.2) still holds to a good approximation in the case of astronomical observations, although scintillation occurs.


MO(575’) = (exp i[cp(x) - cp (x + 5) - cp (x + 5’) + cp (x + 5 + 5‘11). (3.24)

Since the quantity between square brackets has Gaussian statistics, the same arguments used to derive (3.6) lead to

MO(& 5’) = exp -4([cp (x) - cp (x + 5) - cp (x + 5’) + cp (x + 5 + 4”). (3.25)

Introducing the phase structure function D,, we find after some manipu- lations

M,(&,&’) = exp -[D, (8 + 0, (6’) - lo, (t + 5’) - lo, (5 - 5’13. (3.26)

Since we assume turbulence to be located near the ground, (3.12) holds and leads to

(3.27)

Statistics of stellar speckle patterns described in 0 5 are found to agree with this expression.

Q 4. Long-Exposure Images

The appearance of a turbulence degraded image, as seen through a telescope, has been briefly discussed in the introduction. The structure of the image undergoes random changes related to the motion of atmospheric inhomogeneities in front of the telescope. Exposure times as short as a few milliseconds are necessary in order to freeze the image. The eye is therefore unable to follow the most rapid changes. In a conventional astronomical photograph, the exposure time easily exceeds a few seconds,


in which case the recorded image is no longer random. It is an average and we shall refer to it as a “long exposure image”. Properties of long exposure images were first derived by HUFNAGEL and STANLEY [1964] and FRIED [1965, 19661. Applications to astronomy have been discussed by YOUNG [1974].

4.1. RELATION BETWEEN THE OBJECT AND THE IMAGE

Let us denote O ( a ) the irradiance distribution from the object as a function of the direction (Y on the sky. I(@) will be the observed irradiance distribution, in the instantaneous image, as a function of the same variable (Y. A long exposure image will be considered as the ensemble average ( I ( (Y)) . Since astronomical objects are entirely incoherent, the relation between (I(@)) and O((Y) is linear. We shall moreover assume that it is shift invariant, i.e. the telescope is isoplanatic and the average effect of turbulence is the same all over. the telescope field of view. In such a case, ( I (&) ) is related to O ( a ) by a convolution relation

( I ( d = O(a) * ( S ( d (4.1)

the point spread function ( S ( ( Y ) ) being the average image of a point source.

We shall define the two-dimensional complex Fourier transform I(/) of I ( a ) as

(4.2)

with similar relations for the Fourier transform 0 and of 0 and S. In these expressions the spatial frequency vector up has the dimension of the inverse of the angle (Y and must therefore be expressed in radian-’. With such a definition, (4.1) becomes, in the Fourier space

(f(f)> = 0(f) * (WN (4.3)

where &up)) is the optical transfer function of the whole system, telescope and atmosphere.

4.2. EXPRESSION FOR THE OPTICAL TRANSFER FUNCTION

In order to relate this transfer function to atmospheric properties, we shall assume that we are observing, through the atmosphere, a mono-

v, 541 LONG-EXPOSURE IMAGES 299

chromatic point source, of wavelength A. Again, we shall denote To(x) as the complex amplitude at the telescope aperture. The complex amplitude &(a) diffracted at an angle a in the telescope focal plane is proportional to

(4.4) &(a) cc I dx * To(x)Po(r) exp ( - 2 i r a * x/A)

where Po(x) is the transmission function of the telescope aperture. For an ideal diffraction-limited telescope,

1 inside the aperture

0 outside the aperture. P*(X) = (4.5)

In the case of aberrated optics, wavefront errors are introduced as an argument of the complex transmission Po(r).

In the following, we shall make extensive use of the non-dimensional reduced variable

u = x/A. (4.6)

Let us call

T(u) = To(Au) and P(u) = P,,(Au). (4.7)

With such notation (4.4) becomes

.@J;a(a> CC 9 r w u ) P(u)l (4.8)

where 9 is the complex Fourier transform defined by (4.2). The point spread function is the irradiance diffracted in the direction (Y

S ( a ) = I.Na)I2 cc l 9 [ ~ ( u ) ~ ( 4 l I Z . (4.9)

Its Fourier transform is given by the autocorrelation function of P(u)P(u)

(4.10)

In the absence of any turbulence, we assume that T(u) = 1 ( 5 3) so that, normalising g(/) to unity at the origin,

S(/) = Y-l J du * P(u)P*(u+ /) = T ( / ) (4.11)

where Y is the pupil area (in wavelength squared units). Eq. (4.11) is the classical expression for the optical transfer function T ( / ) of a telescope.


In the presence of turbulence (4.11) becomes

s'(/) = 9-' du - ?P(u)!P*(u+ / )P(u)P*(u+ /) (4.12)

and the optical transfer function for long exposures is

In (4.13) appears the second order moment

B (8) = (W) * T * ( U + 9)) = BOO /) (4.14)

the properties of which have been studied in § 3. Since B ( / ) depends only upon /, (4.13) can be written, taking (4.11) into account,

(s'(/)> = B ( / ) * T ( / ) (4.15)

showing the fundamental result that, for long exposures, the optical transfer function of the whole system, telescope and atmosphere, is the product of the transfer function of the telescope with an atmospheric transfer function equal to the coherence function B(/).

4.3. RESOLVING POWER

Following FRIED [1966] we shall use Strehl's criterion (see for instance O'NEILL [1963]), defining the resolving power '2 as the integral of the optical transfer function, which is analogous to the bandwidth in electronics. According to (4.15) the resolution of a telescope through turbulence is

It is limited either relative width of

by the telescope or the atmosphere according the two functions B ( / ) and T ( / ) . For a

(4.16)

to the small,

diffraction-limited telescope of diameter D, turbulence effects are negligible and, assuming a free circular aperture,

'2 -- Bed = d / * T ( / ) = $7r(D/A)*. (4.17)

For a large good telescope, the resolving power depends only upon

I

V, 141 LONG-EXPOSURE IMAGES 30 1

turbulence and

%=9L= d/.B(/) . I (4.18)

According to eq. (3.22), B ( / ) can be written

B(/)=B,(AJ)=exp-KJ; (4.19)

where K describes the seeing conditions and p = 141. With such a notation (4.18) becomes, after integration,

3- = (6.rr/5)~-W(6/5) (4.20)

where r is the usual gamma function.

critical diameter r, of the telescope for which A very convenient measure of seeing introduced by FRIED [1966] is the

I d/ - B ( / ) = I d / * W). (4.21)

Putting D= ro in (4.17) and equating it with (4.20) leads to

K = [(24/5)r(6/5)]~(ro/A)-: = 3.44(r0/h)-%

so that expression (4.19) can be written

B(/) = exp- 3.44(A{/ro): (4.22)

B,(&) = exp- 3.44(5/ro)2. (4.23)

Expressions (4.22) and (4.23) will be used in all that follows. The resolving power is limited by the telescope when its diameter D is smaller than ro. It is limited by the atmosphere when D is greater than r,. Large r,, values mean good seeing, small values mean bad seeing.

The relation between Fried’s parameter r, and the profile C& with height is obtained by equating (4.23) and (3.22). We get

or

-2

0.423k2(cos y ) - l I dh * C&(h)] ’ (4.24)

which shows that r,, also depends upon the zenith angle y and the wavelength h = 2.rr/k. The wavelength dependence is given by

r, cx ( ~ - 2 ) - : = ~4 (4.25)

as confirmed experimentally by BOYD [1978] and by SELBY, WADE and

302 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [v, fi 4

SANCHEZ MAGRO [1979]. In the following, whenever y or A are not specified, we assume y = 0 (observations at the zenith) and A = 0.5 km. Table 1 gives r, values at three different wavelengths as a function of

Measurements of r, are described in detail in the last section. Typical values for astronomical observations in the visible lie between 2cm and 20cm, which shows that the telescope resolving power is always limited by the atmosphere. However, under favourable circumstances, it can be diffraction limited in the infrared. When limited by the atmosphere, the seeing angle w,, also given in Table 1, is of the order of A/rO (D7.6). According to (4.25), it varies as A - i , which shows that seeing slowly improves at increasing wavelengths as already noticed by astronomers observing in the infrared.

dh * CZ,(h).

4.4. APPLICATION TO MICHELSON’S STELLAR INTERFEROMETRY

Expression (4.15) applies whatever the shape of the telescope entrance pupil. It therefore applies to Michelson’s stellar interferometry, if we assume that the entrance pupil is made of two small apertures at some distance A{,) apart. In such a case, the pupil transmission function can be written

P(u) = P(U) + P b - 90) (4.26)

where p(u) is the transmission function of each aperture. Assuming no turbulence occurs, the optical transfer function is obtained by putting (4.26) into (4.1 1) leading to

T(B) = t(9) + 4w + 90) + - 90) (4.27) where

f(/) = s - 1 1 du * P(U)P(U+ 4) (4.28)

is the optical transfer function of each aperture of area s. In the limit of very small apertures (4.27) can be written as a sum of Dirac’s 6 distributions, or after renormalisation,

T(/) = W) + $(9 + 90) + 4w - 90) (4.29)

and the image Fourier transform is

focn = a/) * T ( 9 ) = m) +30(-Bo) 90) + t w o ) S(9 - 90)- (4.30)

TABLE 1

(4.24), (5.13) and (7.70”). Fried’s parameter ro. coherence area A’o and seeing angle w, at three different wavelengths. for typical values of dh . CZ,(h), according to eqs.

A =0.5 p.m A = 2.2 fim h = l O w m

J d h . C $ ( h ) w, w, 0,

(m!) r,, (cm) A 2 0 (an2) (arc second) ro (cm) A’o (cm’) (arc second) r,, (cm) A ’ o (cm’) (arc second)

21.1 16.5 13.9 10.9 9.19 8.04 6.30 5.30 4.16 3.50 2.14 2.02

152 93.1 66.3 40.8 28.9 22.1 13.6 9.62 5.91 4.19 2.57 1.39

0.62 0.79 0.94 1.20 1.43 1.63 2.08 2.41 3.15 3.74 4.77 6.49

125 97.9 82.4 64.6 54.5 47.6 37.3 31.4 24.6 20.7 16.2 11.9

5.34x lo3

2.32 x 10’

1.01 x 10’

3.28 x 103

1 . 4 3 ~ 103

174 476 331 207 147 90.1 48.8

0.46 0.59 0.70 0.89 1.06 1.21 1.55 1.84 2.34 2.78 3.55 4.82

769 603 507 398 335 293 229 193 151 127 100 73.5

202 x 10’

8 7 . 9 ~ lo3 54.1 x lo3 38.3 x lo3 29.3 x lo3 18.0 x 103 1 2 . 8 ~ lo3 7.84 x 103 5.55 x 103 3.41 x 103 1.85 x 10’

1 2 4 ~ 103 0.34 0.43 0.52 0.66 0.78 0.89 1.14 1.36 1.73 2.06 2.62 3.56

Ic) 0 W


Since O(g) is hermitian, the irradiance distribution in the image plane is

I o ( a ) = O(0) + Re [O(go) exp ( 2 h a - go)] (4.31)

or, introducing the argument O0 of the complex quantity O(g0)

= O(o) + 10(go)l cos ( 2 ~ - go+ eo). (4.32)

Expression (4.32) describes the interference fringe pattern observed in the telescope focal plane. The fringe visibility

v = l ~ ( / o ) I / O ( O > (4.33)

is related to the modulus 10(go)l, whereas the fringe position is related to the argument O0 of O(go).

When turbulence occurs, according to (4.15), the Fourier transform of the long exposure image is obtained by multiplying T ( g ) and B ( g ) so that (4.30) becomes

(I(/)> = O(f) * T ( A * B ( A

= O(0) ~(p)+lO(-40)B(-go) w+ Bo)+mgo)B(40) W-Bo) (4.34)

and the long exposure fringe pattern is described by

Ma)) = O(0) + R e [OYo)W,J exp ( - 2 i ~ a * go)] = O(0) + (O(go)( B ( f 0 ) cos ( 2 r a * go + 00). (4.35)

The related fringe visibility is

v = B(f0) - lO(go)l/O(o). (4.36)

When observing a stellar point source, b ( g ) = 1 and the fringe visibility becomes

v = B(B0). (4.37)

This is the essence of the interferometric measurements of the atmospheric transfer function B ( g ) , described in 0 4.5. Expression (4.34) shows that, when long exposures are taken through a Michelson stellar interferometer, the attenuation of spectral components of the object is exactly the same as in normal imaging. As we shall see, this attenuation is mainly due to the motion of the fringes during the exposure. In the actual Michelson experiment (MICHELSON [ 19201) fringes were observed visually. The success of the experiment comes from the ability of the eye to follow fringe motions.

v, 041 LONG-EXPOSURE IMAGES 305

Let us call TI and T2 the instantaneous complex field at each aperture. The instantaneous transfer function is obtained by multiplying p(u) with TI and ~ ( u - 9 ~ ) with T2 so that (4.29) becomes

s(/) = $(I Ti12 + I a(/) + ;TI 'J'? a(/ + 90) + iTTT2 a(/- /a) (4.38)

and (4.30) becomes

f(/) = a/) * S ( / ) = 5(1TIl2+ 1T2I2)m a(/) + $o(-/" 1 Wl T? a(/ + 90) + iW0) 9 7 9 2 a(/ - 40). (4.39)

The instantaneous irradiance distribution in the image plane is therefore

I ( a ) =t(lT'l12+(T2(2)O(0)+Re [O(g0)TTT2 exp ( - 2 i ~ a (4.40)

Since turbulence introduces mainly phase disturbances

TI = exp icp, and T2 = exp icp,

so that

I(@) = O(0) + Re [O(/,) exp- i (27~a - f 0 + cp, - q2)] (4.41)

showing that fringes are mainly shifted by a random amount (cp l - cp,)/2~ lfol. When the r.m.s. shift is greater or equal to about one fringe spacing, fringes will disappear in a long exposure. However they are still visible to the eye as long as their motion can be followed (see § 7.4).

An expression for the long exposure fringe pattern is obtained by taking the average of (4.40). Since the average irradiance is not affected by turbulence,

(I Td2) = (I V212) = 1 (4.42)

and (4.40) leads to

( I ( a ) ) = O ( O ) + Re [O(g0)(T7T2) exp (-2i7ra BO)] (4.43)

which is equivalent to (4.35) since (TTT2) = B ( f 0 ) . Instead of making a long exposure, we can envisage recording a sequence of short exposures or rapid photoelectric scans in order to freeze the instantaneous fringe pattern. In each recorded pattern, the variance of the modulation is, according to (4.40),

u;=; )O(g0))2 * pP7T2\2. (4.44)

Averaging over a large number of independent scans, the variance

306 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OITICAL ASTRONOMY [v, 5 4

becomes

The quantity

(4.45)

(4.46)

is the second order moment of the intensity fluctuations on each aperture. As shown in 0 8 (Fig. 14) these fluctuations become uncorrelated when the baseline exceeds about 10 cm, in which case

and (4.45) becomes

(4.47)

which is exactly the variance that we would obtain by scanning an undisturbed fringe pattern as described by (4.32). This is the principle of photoelectric interferometry as developed by CURRIE, KNAPP and LIEWER [ 19741. The object energy spectrum (6(g)(2 is entirely recovered through turbulence but the phase of 6(/) is lost. It will be shown in 0 5 that the object energy spectrum can similarly be recovered by a second-order statistical analysis of a sequence of short-exposure images taken with the full telescope aperture (the so-called speckle interferometry technique).

4.5. EXPERIMENTAL MEASUREMENTS OF THE LONG-EXPOSURE TRANSFER FUNCTION

Astronomers have long tried to measure the photometric profile of star images on long-exposure photographs. Because of the non-linearity and small dynamic range of photographic emulsions, images of several stars of different magnitudes must be analysed on a single carefully calibrated plate (see for instance KING [1971]). The observed profile is always nearly Gaussian so astronomers are used to fitting their stellar profiles with Gaussian curves. According to the theory presented here, stellar profiles are given by the two-dimensional Fourier transform of the atmospheric transfer function B ( / ) described by (4.22). Since 5/3 is not far from 6/3 = 2, B(/) and therefore its Fourier transform, are nearly, but not exactly, Gaussian functions. Numerical computations of the predicted stellar profile are presented in Table 2. The core is indeed nearly Gaussian but the fall off in the wings is not as steep as that of a Gaussian

TABLE 2 Theoretical profile of a long exposure stellar image, through a large telescope, obtained by taking the Hankel transform of Fried's transfer

function (eq. (4.22)), after KADIRI [1979]. For ro = 10 cm, A a = 0.16 arcsecond.

a/Aa 0 1 2 3 4 5 6 7 8 5

I (a) / I (O) 1.00000 0.92893 0.74724 0.52615 0.33039 0.19005 0.10364 0.05567 0.03052 0

a/Aa 9 10 11 12 13 14 15 16 17

I (a) /r(o) 0.01751 0.01062 0.00681 0.00459 0.00322 0.00234 0.00175 0.00134 0.00104

a / A a 18 19 20 21 22 23 24 25 26

I(a)lI(o) 0.00083 0.00067 0.00055 0.00045 0.00038 0.00032 0.00027 0.00023 0.00020

a/Aa 27 28 29 30 31 32 33 34 35

2

8

I (a) l I (O) 0.00018 0.00016 0.00014 0.00012 0.00011 0.00010 0.00009 0.00008 0.00007

W 0 4

308 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OWICAI. ASTRONOMY [v, 5 4

a i arc second 1

I I I I I I

1/100 lll0 1

Fig. 2 . Open and full circles: profile of a stellar image after KING [1971]. Solid line: theoretical profile from Table 2 assuming r,, = 5.5 cm. Dashed line: Gaussian profile. Dotted

line: expected inner scale cut-off.

curve. Fig. 2 shows a stellar profile published by KING [1971] together with a fit of the central core with a Gaussian curve and a fit with the theoretical profile of Table 2. Experimental points clearly agree well with the extended wing predicted by Table 2, at least up to 6 arcseconds. At larger angular distances an inverse-square slope aureole is observed. Since a turbulence inner scale of the order of a few millimeters implies a steep fall-off of the profile in this region, the aureole has necessarily a different origin. It is probably due to scattering by small particles or scratches in the telescope, since Fraunhofer diffraction by a sharp edge leads to an inverse square law.

The atmospheric transfer function B ( / ) is best measured interferometrically. The principle of such measurements has been given in § 4.4. Compared to stellar profile measurements, it eliminates both the emulsion dynamic range problem and the effects of telescope aberrations or focusing errors. Interferometric measurements of B(f) had initially been made by optical physicists in the case of horizontal laser beam propagation (WESSELY and BOLSTAD [1970], BERTOLOTTI, Muzir and SETTE [1970], BOZEC, CACNET and ROGER [1971]). They provided the first experimental check of expression (4.22). Similar measurements were later performed on stellar sources by C. RODDIER and F. RODDIER [1973], KELSALL [1973], DAINTY and SCADDAN [1974, 19751, C. RODDIER [1976]

V, 8 51

B35)

4 . . i 'Y

lk., .

-1 * ' \

1

5 0- T I - - - r - - -

SHORT-EXPOSURE IMAGES 309

and BROWN and SCADDAN [1979]. Fig. 3 shows experimental data obtained with a rotation interferometer by C. RODDIER [1976], together with a best fit with expression (4.23). A check of the 513 law is obtained by plotting experimental values of log [-log El(/)] as a function of logfas shown on Fig. 4. Experimental results are found entirely consistent with the expected 513 slope. More accurate checks of the 513 law will be discussed in P 7.

0 5. Short-Exposure Images

Because of turbulence, the instantaneous illumination in the focal plane of a telescope must be considered as a random function. In the last section we studied its first moment or average value. Here, we shall be interested in its second order statistics or energy spectrum.

5.1 . THE IMAGE ENERGY SPECTRUM

We will again assume that the image I ( a ) is related to the object O ( a ) by a convolution relation

[(a) = O(a)*S(a) . (5.1)

Fig. 3. Square of the coherence function BJt) . Full circles: experimental measurements made with a rotation interferometer (C. RODDIER [1976]). Solid line: theoretical curve

assuming r0 = 6 cm (eq. (4.23)).

310

0-

-a5-

.

- 1

- 1.5-

THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [v, 8 5

c

- ma m - I

Ln -9

-0.7

-0.9

log F I r ' ~ I I ~ ~ ' * I

The relation is indeed linear but shift invariance is now a crude assumption. Here I ( a ) and S ( a ) are both random functions and (5.1) means that, at any given instant, the distorted point spread function is the same for all points of the image, which implies that the instant wavefront perturbations are identical for all wavefront directions. As easily seen in Fig. 5 this can only be true if turbulence is entirely located near the telescope aperture. Since it is not the case, our assumption is an approximation valid only in a limited field of view called the isoplanatic patch.

* - --------- - -

I #'' -&-= T turbulent pubil

layer plane image plane

Fig. 5. If the observed object has sufficiently small angular dimensions (I, or if the turbulence is localised near the telescope pupil, beams originating from any point o n the object and arriving on the pupil can be considered to have encountered almost identical

regions of the perturbing atmosphere. The related image aberrations are isoplanatic.

Fig. 4. Full circles: experimental data from C. RODDIER [1976]. Solid line: least square fit with a 5/3 slope allowing an accurate estimation of r,,. Here rO = 3.9 cm.

V, § 51 SHORT-EXPOSURE IMAGES 311

Observations and theoretical estimations show that, in the case of astronomical images, the size of the isoplanatic patch is of the order of a few arcseconds. The isoplanicity approximation will be discussed in details in the next section.

From (5.1), the image energy spectrum is given by

(Im”) = IO(8)l” (ls(f)l’) (5.2)

where 16(/)1” is the object energy spectrum and (ls(/)l’) is the energy spectrum of a point source image. (Is(g)l’) describes the transmission of the spectral components in the image energy spectrum. Its square root is sometimes called the speckle modulation transfer function. It will be shown that, in contrast with the long exposure transfer function, it has a high frequency component, extending up to the telescope cut-off frequency, corresponding to a speckle structure observed in monochromatic short exposures. This high frequency component allows retrieval of high resolution information by the so-called speckle interferometry technique (see 0 9.4).

Assuming monochromaticity, the random instantaneous transfer function s(g) is given by expression (4.10) derived in 04.2. Its average squared modulus is therefore

(Is(S)l’)= Y-” 1 du du’(ly(u)ly*(u+g)ly*(u’)ly(,’+ 4))

x P(u)P*(u+ g)P*(u’)P(u’+ 4)

which depends upon the fourth order moment

M(f, 8’) = ( ly(u) ly ’”(u+8) ly*(u+g’) ly(u+g+g’) )

where f ’=u’ -u . Putting (5.4) into (5.3) leads to

describes the telescope contribution. In contrast with the long exposure transfer function, ((s(g)l’) is no longer the product of a telescope function with an atmospheric function, because of the integral in (5.5).

Let us first derive the general asymptotic behaviour of (Is(g)l”) for a large telescope (03 >> ro), at high spatial frequencies (f>> ro/A). In this case,


T(u) ?P*(u + f’) is uncorrelated with T*(u + f) ?P(u + / + 8’) except when f’=-f, so that

w/, f ’ k ( T ( U ) T * ( U + / ’ ) ) ( P * ( U + / ) T ( U + f + 9’)) (5.7)

everywhere in the 8’ plane, except near f ’= -8, the contribution of which is negligible in the integral (5 .5) . Therefore, when { >>r,/X,

(5.8)


(I$f)(2)=Y-2/ duP(u)P*(u+/) df’ - B 2 ( f ’ ) P * ( u + / ’ ) P ( u + / + / ’ ) .

Since B 2 ( / ’ ) falls off very rapidly on a distance of the order ro/A, whereas the pupil transmission P has slow variations,

(5.9) I

where

c = / df’ * B2(/’)

is a measure of the coherence area of the wavefront perturbations and

(5.11)

(5.12)

is the transfer function of an ideal diffraction limited telescope of the same aperture. Eq. (5.10) shows that extends up to the telescope diffraction cut off f c and that, to a first approximation, telescope aberrations and focusing errors have a negligible influence. Eq. (5.10) also shows that, in the image, the typical speckle size is of the order of f;’ = X/D. Since the size of the seeing disk is of the order of A/ro, the number of speckles in the image is of the order of (D/ro)2. Putting (4.22) into (5.11) leads, after integration, to

cr = 0.342(r,/h)2. (5.13)

Values of h2cr are given in Table 1 (p. 303) in cm2 units. Since Y-

V, 8 51 SHORT-EXPOSURE IMAGES 313

(D/A)2, the number of speckles is also of the order of Wcr, so that the attenuation factor in (5.10) is approximately the inverse of the number of speckles in the image.

A complete analytical expression for (lg(f)12) can be derived if “(u) is assumed to follow circular Gaussian statistics (DAINTY [1975]). This assumption is widely used in optics when dealing with ground-glass diffusers. In this case

w 9 , 9’) = B 2 ( / ) + B2(9’). (5.14)

Putting (5.14) into (5.5) and still assuming D >> r,, leads to

(Is(/>12>= B 2 ( 9 ) IT(/))*+ ( d Y ) T o ( f )


(IW)l2) = (W)>”+ (dWTo(9) (5.15)

which describes the speckle energy spectrum (\s(f)12) as the sum of the low frequency long-exposure energy spectrum and the high frequency component previously derived.

In the case of atmospheric fluctuations, W(u) is better described by log-normal statistics and the fourth-order moment is given by (3.26). Comparison between (3.16) and (4.23) shows that, at the near-field approximation,

&(O = 6 . ~ W r 0 ) : . (5.16)


M ( 9 , 9’) = Mo(A9, A/’)

= exp{-6.88(A/r0): * [ 1 f ( z + 1 f f 1 2 - 4 \f+f’l:-i I/-f’l+]}.

(5.17)

In such a case, has no simple analytical expression. Integration of (5 .5) was performed numerically for the first time by KORFF [1973] using (5.17). The general shape of (Is(g)l’> remains the same. The asymptotic behaviour at high frequencies is still described by (5.10). However, the low frequency part is slightly wider than the long exposure energy spectrum. As shown by KORFF [1973], it fits the energy spectrum of a long exposure made with a fast automatic guider removing image motion, i.e. wavefront tilts, on the telescope aperture. Such a spectrum has been studied by FRIED [1966] who derived the approximate expression, in the


near-field case (see § 7.6)

(Is(g)l’) = exp {-6.88(AP/d2[l - (APE@Il). (5.18)

Its square root, called the “short exposure” transfer function by Fried, must be distinguished from the speckle transfer function.

Accurate experimental checks of Korff’s calculations are difficult since short exposure stellar images are always noisy. However, since the two-dimensional speckle energy spectrum is expected to have a circular symmetry, its values along any axis can be found by taking the average squared Fourier transform of the speckle image integrated along a direction perpendicular to that axis. In the experiment by KARO and SCHNEIDERMAN [ 1976b], the integration was performed electronically along each line scanned by a T.V. camera. In a more recent experiment by AIME, KADIRI, RICORT, RODDIER and VERNIN [1979] a photomultiplier is used in association with a scanning slit. One of their typical results is presented in Fig. 6 showing a good agreement with Korff’s theory. Excellent agreement has also been obtained in the infrared by CHELLI, LENA, C. RODDIER, F. RODDIER and SIBILLE [1979] again using a single detector and a scanning slit.

Energy Spectrum

Fig. 6. Spatial energy spectrum of the image of an unresolved star. Dots: experimental data obtained by AIME, KADIRI, RICORT, RODDIER and VERNIN [1979], on star Vega, with the 193 cm telescope of the Haute Provence Observatory (France). Solid lines: theoretical curves, according to Korff‘s theory, for r,, = 4.3 cm, 5.6 cm and 6.9 cm. The telescope central

obscuration has been taken into account in the calculations.

V, S 51 SHORT-EXPOSURE IMAGES 315

5.2. THE APERTURE-SYNTHESIS APPROACH

In 0 4.4, we investigated the influence of atmospheric turbulence on a Michelson stellar interferometer. Let us generalise the results to an array of No small identical apertures. The pupil transmission function will be

(5.19)

Assuming no turbulence, the related optical transfer function is

1 No Nu

T ( / ) = - C C t ( / + / i - / j ) (5.20)

where t ( / ) is the optical transfer function of each aperture. In the limit of very small apertures, (5.20) can be written after renormalisation

No i - 1 j = 1

Several pairs of apertures may have the same separation

/j - / i = /m.

(5.21)

(5.22)

Let Nm be the number of aperture pairs separated by /,. Nm is called the redundancy coefficient of the array at frequency f m . Grouping these pairs together in (5.21) and assuming f O = O , leads to

with c, = Nm/No.

The image Fourier transform is

(5.23)

(5.24)

(5.25)

Let us now consider the entrance pupil of a large diffraction limited telescope as an array of small adjacent apertures. As seen in Fig. 7, within such a pupil, the number Nm of small-aperture pairs separated by f m is proportional to the overlap area of two pupil images shifted by f m . The ratio NmJNo is therefore given by the integral (5.12) so that

316 THE EFFECTS OF ATMOSPHERJC TURBULENCE IN OPTICAL ASTRONOMY [v, 8 5

Fig. I . within vector

The number of pairs of points (open and full circles) separated by a vector f (arrow) a given pupil is proportional to the overlap area of two such pupils separated by the 9 This number, as a function of f, determines the telescope optical transfer

function.

The optical transfer function of a diffraction limited telescope is a measure of the redundancy of its pupil. Putting (5.26) into (5.25) leads to

or, in the limit of infinitely small apertures,

(5.28)

thus recovering the usual expression for a diffraction limited telescope. The effect of atmospheric turbulence on a two-aperture stellar inter-

ferometer has been described in P 4.4. Following a similar approach, we shall now investigate the effect of turbulence on an array of No small apertures. Let !Pi be the instantaneous complex field at aperture i . The instantaneous transfer function is obtained by multiplying p(u - f i ) with ?Pi in (5.19) so that (5.20) becomes

(5.29)

Let us group again aperture pairs having the same separation

f j - 9i = 9 m (5.22)

and let Nm be the number of such pairs. Renumbering the apertures, we shall call ?P,,,q and !Ph,q the two values of the complex field associated with the qth pair separated by fm. Hence (5.29) becomes, in the limit of

V, 8 51 SHORT-EXPOSURE IMAGES

very small apertures,

N", 1 s(/)=-C C q m , q q C q a ( / - / m ) . NO m q = t

The instantaneous transfer function still has the form

with

317

(5.30)

(5.31)

(5.32)

The long exposure transfer function (s(f)) is obtained by taking the average of (5.31). The related coefficients are

(5.33)

in agreement with (4.15). In the following, we shall assume that, for m f 0, the separation l f m l between the apertures is always larger than Fried's parameter r, so that

( q r n , q ~ ~ q > = B ( / m ) -0. (5.34)

According to (5.31), the energy spectrum of a point source image is a discrete spectrum with components

(5.35)

Since the quantities 'P,,,,,qCq -have been assumed statistically independent with zero mean (5.34), the variance of their sum in (5.35) is the sum of their variances and

Nm

q = l = C ( I 9 m . q 12)(1 K . q 1 2 ) = N m (5.36)

the average irradiances (1 Fm,q 1') and (\Tk,q\2) being normalised to unity. Hence, for m # 0, (5.35) becomes

< c i > = Nm/%* (5.37)


This is to be compared with the energy spectrum of a point-source image without turbulence. According to (5.23) and (5.24), it has components

c’, = Pm/%. (5.38)

Turbulence therefore introduces the attenuation factor l/Nm. For a non-redundant pupil such as the two-aperture pupil of a Michelson interferometer, there is no attenuation as seen in 0 4.4. The significance of pupil redundancy in the degradation of images by aberrations was first underlined by RUSSELL and GOODMAN [1971] and by RHODES and GOOD- MAN [1973].

Let us again consider the entrance pupil of a large telescope as an array of adjacent apertures with size of the order of r,, so that the wavefront perturbations on each aperture are uncorrelated. According to (5.37), the energy spectrum of a point-source image, for frr ,JA, can be approximated as


(5.40)

and, approximating the sum as an integral,

= N,’T(/). (5.41)

Thus recovering expression (5.10) where No= Y/w is the number of coherence areas on the pupil, or the number of speckles in the image. This heuristic derivation of (5.10) is due to KORFF, DRYDEN and MILLER [1972]. Application to aperture synthesis was considered by F. RODDIER { 19741.

5.3. THE PROBABILITY DENSITY FUNCTIONS OF STELLAR SPECKLES

A laser speckle pattern is generally considered as an interference pattern formed by the combination of light beams issued from a large number of independent scatterers. The resulting complex amplitude, being the sum of a large number of independent random fields, tends to a

V, 8 61 EXPOSURE TIME AND NON-ISOPLANTCITY EFFECTS 3 19

complex Gaussian process, as a result of the central limit theorem (GOODMAN [1975], DAINTY [1976]). According to Q 5.2, the same result applies to stellar speckles provided that the telescope diameter D is much larger than the correlation scale of the wavefront perturbations, that is D >> r,,. In such a case the complex amplitude &(a) in the image plane is a circular Gaussian process and the related illumination S ( a ) = (&(a)J’ has an exponential distribution

P(S) = (S)-’ exp ( - S / ( S ) ) (5.42)

as suggested by LABEYRIE [1975]. Accurate experimental verification of (5.42) is difficult because the speckle pattern is always blurred by the finite width of the scanning aperture, the finite optical bandwidth or the finite “exposure time”. GOODMAN [1975] and DAINTY [1976] have given an approximate expression for the probability density function of such blurred speckles which has been found to agree well with observations through a 91 cm telescope (SCADDAN and WALKER [1978]).

The probability density function of the illumination in the image plane of a small telescope cannot be derived from the central limit theorem and will depend upon the statistics of the wavefront perturbations. However no simple analytical expression can be derived unless we assume that the wavefront perturbations have Gaussian statistics in which case &(a) is also Gaussian.

6 6. Exposure -Time and Non-Isoplanicity Effects

In the last section, we assumed that images were recorded with an exposure time short enough to freeze the instantaneous wavefront perturbations. Here, we shall investigate the effect of a longer exposure time, i.e. the effect of the evolution of wavefront perturbations during the exposure time. We shall also investigate the effect of non-isoplanicity, i.e. the effect of the evolution of wavefront perturbations with the direction of observation. The two theories are very similar. Both involve the calculation of speckle cross-spectra.

6.1. SPECKLE CROSS-SPECTRA

Let ?Pl(u) and ?P2(u) describe two different wavefronts and let &(a) and S,( a) be the associated speckle patterns. Their Fourier transforms

320 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [V, $ 6

gl(f) and $(/) are given by (4.12) and the cross-spectrum is

where A(/, 4’) is defined by (5.6) and

M12(9,9’) = ( ? P l ( U ) ? P ~ ( U + / ) ~ ( U + / ’ ) ? P 2 ( U + If+/’)). (6.2)

In the following, we shall assume that ?PI and ?Pz obey the same Gaussian statistics. Although log-normal statistics would be more realistic, we know from § 5 that Gaussian statistics give a fairly good qualitative account of the observations. Moreover, the high frequency tail of the image energy- spectrum (which is of interest in speckle interferometry experiments) does not depend upon the statistics of ?P. Assuming Gaussian statistics,

6.2. EFFECT OF NON-ISOPLANICITY

Let us drop the assumption of isoplanicity. The relation between the image I and the object 0 takes the more general form

(6.6)

where the point-spread function S now depends upon the direction of observation. Taking the Fourier transform of (6.6) leads to

f (9) = I d P * O(P)s(9, P ) exp (-2i.rrP - 9). (6.7)

The image energy spectrum therefore becomes

(If(9)l’) = I I d P * dP‘ - O(P)O(P ’ ) (W, P ) S * ( / . P”

x exp [-2i.rr(P - P’) * /I (6.8)

V, 161 EXPOSURE TIME AND NON-ISOPLANICITY EFFECTS 321

or, with 8 = p - @’,

x exp ( - 2 i d /)

= J d 8 - Co(8)(s(p, p)s*(g, p - 0)) exp ( -2 i r8 - p)

(6.9)

where CO(8) is the object autocorrelation function. Putting (6.5) into (6.9) leads to

(lf(pe)I2) = B 2 ( 8 ) IT(S)l’ IW)l”

(6.10) where

Be(/’) = (T ‘ (u , P)T*(U+pe’, P’ ) ) (6.11)

describes the covariance between wavefront perturbations associated with directions p and p’.

Let us first assume a single thin turbulent layer at altitude h. The related perturbations T(u, p ) and T(u, p’) will be identical but shifted one from the other by an amount Oh so that

Be(/’) =(T(u+@h/A, O)T*(u+/’+p’h/A, 0))

= B ( / ’ - 8h/A). (6.12)

For a moderate to large size telescope A(f, 8’) does not vary appreciably over the width of the function B Z ( J ‘ - 8 h / A ) . In (6.10) we shall therefore approximate this function with a Dirac distribution

B:(/’) = B2(/’- 8h/A) C T ~ ( / ’ - 0h/A) (6.13)

where cr is defined by (5.11). Putting (6.13) into (6.10) leads to

(ImI’) = B 2 ( / ) lT(A1’ IO(8)l’

+cTY-’I d 8 CO(8)A(g, 8hlA) exp (-2i7~8 * pe).

(6.14)

When both \ / I and (8h/A( are much smaller than the telescope frequency

322 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [V, 0 6

cut off pc = D/A, A(/, 8 h / h ) can be approximated with

A(/, 8hlA) = fl(/)TT(Bh/A). (6.15)

Putting (6.15) into (6.14) shows that, within a large frequency range r,/A < I/( << D/A, the image energy spectrum is approximately described by

In this frequency range, the object spectrum is smoothed by convolution with the Fourier transform of T(Bh/A). The related features in the object autocorrelation are attenuated by the factor T(Bh/A). Half attenuation occurs when (8) =D/2h. With D = 1 m and h = lo4 m, the isoplanatic patch size is of the order of 10 arcseconds.

Observed values are smaller than that, of the order of 4 arcseconds (NISENSON and STACHNIK [1978], SCHNEIDERMAN and KARO [1978a], POL- LAINE, BUFFINGTON and CRAWFORD [1979]). This is not surprising since turbulence can hardly be assumed to be concentrated into a single layer. According to (4.22) and (6.12), the single layer model leads to

B;(/’) = exp {-6.88(A/r0): 14’- 8h/AIg}. (6.17)

A reasoning similar to that of 0 3.2 shows that, in the case of several layers at altitudes h,, (6.17) becomes

B2,(/’) = n exp {-6.88(A/ro,,): I/’ - 8h,/AIz) (6.18)

where rO,] is Fried’s parameter that would produce layer j alone. Eq. (6.18) can also be written

I

(6.19)

or, assuming layers of thickness 6h, with a uniform turbulence of stfuc- ture constant C&( h,),

1 c (ro.,)F I/‘- @h,/AF B;(/’) = exp -6.88(A/rO)z ’ ( r($

(6.20)

I

c C&(h1) ah, I/’- 8h, /h( :

c C&(h,) 6 4

V, § a EXPOSURE TIME AND NON-ISOPLANICITY EFFECTS 323

since ro,, and Ch(hi) ahi are related by (4.24). For a continuous distribution of turbulence, (6.20) easily generalises into

. (6.21)

(6.22)

1 J,-dh Cg(h) If’ - &/A12

L,md h * Ch(h 1 B i ( f ’ ) = exp -6.88(A/r,,)s 1

In such a case, approximation (6.13) becomes

B;(f’) 21 a(f3) ti(/’- Sh/A)

where 6 is an average turbulence altitude and

a(@) = df’ - Bi(/’). (6.23)

The quantity a(@) can be computed numerically from (6.21). It is clear from (6.18) that a(@) will fall to zero as soon as 161 21 r,/Ah where Ah is some measure of the height dispersion of turbulence layers. FRIED [1976, 19791 suggests that

Ah=[\: dh * Ck(h) ] .

5

m 3

dh * hW&(h) (6.24)

Under different assumptions,

A h =

SHAPIRO [1976] gives

(6.25)

Recently Loos and HOGGE [1979] have quoted the similarity between these integrals and eqs. (8.11) and (8.12) giving the scintillation index (see § 8). They suggested that scintillation measurements through an aperture of appropriate size (about 10 cm) would give the parameter Ah, as defined by (6.24), thus permitting r,/Ah to be evaluated, giving the order of magnitude of the isoplanatic patch size.

Putting (6.22) into (6.10) and using approximation (6.15) again shows that, in the frequency range r , /A < lfl<< D/A the image energy spectrum is


approximately given by

(6.26)

Since for a large telescope a(@) decreases steeper than T(BK/A) , in this frequency range, the image spectrum is mainly smoothed by convolution with the Fourier transform of a ( O ) / a ( O ) . The related features in the object autocorrelation are attenuated by the factor a(C))/a(O). Using Hufnagel’s model (§ 2.4) in (6.21) and (6.23) shows that half attenuation should occur when 101 =- 2 arcsecond which is smaller than observed values (see FRIED [1979]). This is not surprising since turbulence is never continuously distributed as in the model, which must be considered as a measure of the probability of observing a turbulence layer at a given altitude. As shown in $ 2 , any true distribution of turbulence with height consists of discrete layers so that the observed isoplanatic patch size will lie between the average model value and the single layer value.

Detailed calculations of the cross correlation (s(f, p ) - g*(p, p‘) ) have been made by KORFF, DRYDEN and LEAVITT [1975] using log-normal statistics. A discussion of isoplanicity can also be found in WANG [1975] under the assumption of Gaussian statistics.

6.3. THE TIME EVOLUTION OF SPECKLES

Let Tu,(u) = T(u, t ) and TJu) = T(u, t - T ) describe the wavefront perturbations at time t and t - T. According to (6.5) the cross-spectrum of the associated speckle patterns is

describes the spatio-temporal covariance of T. The temporal correlation function of speckles C ( T ) is obtained by integrating (6.27) over all

V, S61 EXPOSURE TIME AND NON-ISOPLANICITY EFFECTS 325

frequencies, leading to

C(T) = j df * B2(f) IT(f)12+ J df’ - B:(f’) lT(f’)12. (6.29)

As yet, little is known about the function B : ( f ’ ) . It is often assumed that a wind, uniform throughout the turbulence, merely drives the wavefront perturbations without deformation over the telescope pupil plane (Taylor’s approximation). In such a case, u being the wind velocity,

B , ( f ’ ) = (Yr(u, O)Yr*(u+ f’- UT/A, 0))

= B ( / ’ - U T / ~ ) , (6.30)

an expression similar to (6.12). According to (4.22)

B : ( f ’ ) = exp {-6.88(A/r0): If’- U T / A ) $ } . (6.31)

Measurements of the spatio-temporal covariance of irradiance fluctuations (TI2 indeed show evidence for propagation. They also show a steep decay of the correlation with time delay 7 (Fig. 16, p. 348). A similar behaviour is therefore expected for B : ( f ‘ ) . Assuming several turbulent layers with wind velocities ui leads to an expression similar to (6.18)

B : ( f ’ ) = n exp {-6.88(A/r0,,)z If’ - u,~/h(:}

where rO,, is Fried’s parameter produced by layer j alone. ous distribution of turbulence, eq. (6.32) generalises into

I

(6.32)

For a continu-

. (6.33) [dh . C2,(h) lf‘-u(h)T/Al:

B : ( f ’ ) = exp -6.88(A/r0): i JOadh * C2,(h)

Expressions (6.32) and (6.33) are both able to describe the decay of €I:(/’’) with T. For a moderate to large size telescope, lT(f)I2 does not vary appreciably over the width of the functions B 2 ( f ) and B : ( f ) . In eq. (6.29), we shall therefore approximate these functions with Dirac distributions

B2(f) = US(/) (6.34)

B : ( f ’ ) = U ( T ) a(/’- l j ~ / h ) (6.35)

where l j is an average wind velocity,

V ( T ) = df’ - B : ( f ’ ) , (6.36) J

326 THE EFFECTS OF ATMOSPHERIC' TURRULENCE I N OWICAL ASTRONOMY [V, 5 6

and c = ~ ( 0 ) is given by (5.13). The quantity U ( T ) can be computed numerically from (6.33), knowing both the atmospheric wind and turbulence profiles. It is clear from (6.32) that C ( T ) will fall to zero as soon as T = r,/Av where A v is some measure of the velocity dispersion of turbulence layers.

Putting (6.34) and (6.35) into (6.29) leads to

C(T) = U{ 1 -k [C(T)/(+(O)] ) T ( f i T / A ) I 2 } . (6.37)

Measurements of C(T) by SCADDAN and WALKER [1978] and PARRY, WALKER and SCADDAN [1979] reveal a steep decay due to speckle boiling and a slow decay due to image motion (neglected in our approximations). According to (6.37), the normalised covariance of the speckle boiling is given by

(6.38)

Assuming a single wind velocity implies that u ( T ) = u ( O ) so that the speckle lifetime becomes equal to the transit time of perturbations over the telescope pupil. The measurements quoted above clearly show a smaller lifetime, typically 10 ms which must be attributed to the decay of ~ ( 7 ) . It is therefore of the order of ro/Au. This leads to a velocity dispersion Au = 5 m/s for rn = 5 cm. The lifetime is also found to increase with wavelength as expected, since r, increases as AS (eq (4.25)).

6.4. EFFECT OF THE EXPOSURE TIME ON THE IMAGE SPECTRUM

For an exposure time T, the optical transfer function is given by

I +TI2

S T ( / ) = T-' dt . S(/, t ) = T-' dt U(r/T)S(g, t ) (6.39)

where U(t/T) is a rectangle function of width T. The energy spectrum of a point-source image is therefore

I-,,

dT * n ( ~ / T ) ( s ( / , t)S*(/, t - 7 ) ) (6.40)

V, $61 EXPOSURE TIME AND NON-ISOPLANICITY EFFECTS 327

where TA(T/T) is the autocorrelation function of IZ(t/T). Putting (6.27) into (6.40) leads to

(6.41)

or, using approximation (6.35),

(ls,(,/)l”> = B 2 ( / ) IT(,/)12+ Y 2 T - ’ d7 * A(T/T)u(T)A(~P, i h / A ) .

(6.42)

When both I,/\ and 1 i h / A I are much smaller than the telescope frequency cut-off fC = D/A, A(,/, f i ~ / A ) can again be approximated as

I

A(,/, k / A ) = YT(gP)T(ik/A) (6.43)

so that (6.42) becomes

+ &-‘T-’T(,/) d7 . A ( T / T ) [ ~ ( ~ ) / u ( O ) ] T ( ~ T / A ) . (6.44) I Since the equivalent width of a(~) /u(O) is of the order of ro/Au, expression (6.44) shows that the high frequency wing of the image spectrum is uniformly depressed by a factor

T-’ d7 . A ( ~ / T ) [ ~ ( ~ ) / u ( O ) ] T ( ~ T / A ) = T-’ min [TI ro/Au, D/lijl]

(6.45)

where “min” stands for “the smallest value among”. As seen in § 6.3, for a large telescope, ro/Av is smaller than D/lijl so that, within a large frequency range r o / A < (,/I<< D/A the attenuation factor is uniform and equal to r, /TAv, as soon as T r r o / A u . Such a uniform depression of the energy spectrum has been observed by KARO and SCHNEIDERMAN [ 19781. The observed amount of depression is consistent with r,,/Av values of the order of 20 ms. This is also consistent with the exposure time of speckle interferometers based on standard T.V., systems. It has been shown (see § 9) that the signal-to-noise ratio in speckle interferometry is, for a single exposure, proportional to the number N, of photons per speckle, as long as Nscc 1. Since N, is proportional to T, whereas the signal decreases as

I

328 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OFTICAL ASTRONOMY [v, 5.7

T-', the signal-to-noise ratio for a single exposure remains constant independent of T, whenever Trr, Av and N,<< 1.

Detailed numerical computations of the theoretical shape of the image energy spectrum as a function of the exposure time have been done by C. RODDIER and F. RODDIER [1975], assuming log-normal statistics. Unfortu- nately, a single wind velocity is also assumed which now appears to be unrealistic.

0 7. Optical Path Fluctuations

Up to now, we have considered the statistical properties of the complex field T&) only. We shall now examine the statistical properties of its modulus ITo(x)l and of its argument or phase &i). Fluctuations of the squared modulus (W, (X )~~ describe the random illumination at the telescope aperture or the stellar shadow pattern easily seen when directly viewing the telescope objective mirror. The motion of the shadows produces stellar scintillation as seen with the naked eye. Fluctuations of the phase cp,(x) or of the optical path hqO(x)/2.rr describe the shape of the wavefront surface. They are related to the fluctuations of the angle of arrival of light, that is to the image motion as seen through a very small aperture.

When dealing with imaging, scintillation or angle-of-arrival fluctuations need not to be considered separately, as we have seen. They are, however, to be considered in some other cases. For instance, scintillation interferes directly with photometric observations, especially in occultation measurements. Angle-of-arrival fluctuations interfere with astrometric measurements. In 0 10, we shall see that scintillation and angle-of-arrival measurements are useful tools for probing atmospheric turbulence and testing astronomical sites.

As we did in 0 3, we shall again use the phase screen approach and the formalism of Fourier optics in order to derive briefly the fundamental relations due to CHERNOV [1955] and TATARSKI [1956] and presented in detail in the books quoted in 0 3.

7.1. EFFECT OF A THIN TURBULENT LAYER

Let us consider again horizontal monochromatic plane waves propagating downward, towards a ground based observer. As in P3.1, we shall

v, 0 71 OPTICAL PATH FLUCTUATIONS 329

first assume that the atmosphere is homogeneous everywhere except inside a thin horizontal layer between altitudes h and h + ah, introducing a phase shift cp(x) described by (3.3).

We shall now make a new and very restrictive assumption that we did not make in D 3. We shall assume that

cp(x)<< 1. (7.1)

This assumption, called the small perturbation approximation, is not valid for most of the experiments made on propagation along horizontal paths. Many efforts have been made during the last decade to account for the saturation of scintillation which occurs when this approximation is not valid (see for instance STROHBEHN [1978]). In the case of astronomical observations, saturation effects have been considered by YOUNG [1970]. Saturation regimes usually occur when the distance from the zenith exceeds 60". Near the zenith, they are exceptional and (7.1) can be taken as a good approximation, so that the field at the layer output given by (3.2) can be written

q h ( x ) = l+iCp(x). (7.2) The complex field W,(x) at ground level is given by the Fresnel

convolution described by eq. (3.14) of 0 3.1, so that

Since, as is easily seen from its Fourier transform,

(7.3)

(7.4)

eq. (7.3) becomes

where the complex quantity

Ah (7.6)

describes the relative fluctuations of the complex amplitude at ground level.

Its real part

(7.7)

describes the relative fluctuations of the modulus I!Po(x)l.


Its imaginary part

Ah

describes the fluctuations of the phase. Since q ( x ) has Gaussian statistics (see § 3), the linearly related quan-

tities E(x), x(x) and cpo(x) also have Gaussian statistics. Let us define the two-dimensional power spectrum W,( f ) of cp(x) as?

w,( f ) = j d5 * ~ ~ ( 5 ) exp (-2i.rrf

where B,(5) is the covariance of cp(x). With that squared modulus of the Fourier transform of the (7.6), (7.7) and (7.8), leads to the following power and cpo(x):

W,(f) = W,(f)

W , ( f ) = W,(f)(sin ,rrAhf2)2

W,&f) = w,(f)(CoSTAhf2)*.

- 5) (7.9)

definition, taking the convolution factor in spectra for E(x), x(x)

(7.10)

(7.11)

(7.12)

Taking the Fourier transform of (3.10) according to (7.9) gives

W,(f) = k 2 Sh WhU, 0) (7.13)

where

is the three-dimensional power spectrum of index fluctuations. Assuming Kolmogorov's law (2.17) to be valid, with f = I f \

W,(fO) = ( 2 ~ ) ~ x 0.033C$(h)(2~f)-y

= 9.7 x 1 0 - ~ c $ ( h ) f p (7.15)

so that (7.13) becomes

W , ( f ) = 9 . 7 ~ 10-3kZf-yC$(h) 6h (7.16)

t Although this definition is not the most usual, it is consistent with the definition (4.2) of the Fourier transform given in B 4. Care must be taken that the frequency vector f now has the dimension of an inverse length.

v, 871 OPTICAL PATH FLUCTUATIONS 331

and (7.10) to (7.12) can be written, since k = 2.rr/A,

W, ( f ) = 0.38A-’f-yC$(h) Sh

W,(f) = 0.38A-’f-~C~(h) Sh(sin mAhf2)’

W,,(f) = 0.38A-’f-YC$(h) Gh(cos ~Ahf’)”.

(7.17)

(7.18)

(7.19)

7.2. MULTIPLE LAYERS AND THICK LAYERS

Let us first consider two thin layers of turbulence at altitudes h l and h, (where h l > h,) and with thicknesses Sh, and Sh,. Let cpl(x) and cpz(x) be the phase shifts introduced by each layers. According to (7.6) the relative fluctuations of the complex amplitude at the input of the lower layer are

exp i r xz 1. (7.20) 1

E (x, h, + ah,) = icpl(x) * iA(h, - h,) [ A ( h l - h J

They are, at the output,

and, at ground level

*-exp IT- iAh, ( * A:,)

(7.22)

As easily shown by taking its Fourier transform,

1 exp ia?’-]*-exp 1 (i.rr$)

ih(h, - h,) [ h(hl-h,) ihh,

-- - ihh,

Therefore

showing that, at the small perturbations approximation, fluctuations produced at ground level by several turbulent layers add linearly. Since they are statistically independent, their power spectra also add linearly and

332 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OWICAL ASTRONOMY [v, 8 7

expressions (7.17) to (7.19) easily generalise, for any distribution of turbulence, into

W,(f) = 0.38A-’f-Y I dh * Ck(h)

W x ( f ) = 0.38h-2f-5! I d h * Ck(h)(sin ~Ahf’)’

W,,,(f) = 0.38A-’fp dh C$(h)(cos rhhf2)’. (7.27)

(7.25)

(7.26)

We note that

W , ( f ) = W,( f )+ W,,,(f) (7.28)

showing that the cross-spectrum of x and ‘po is zero. Since x and ‘po are Gaussian variables, this implies that they are statistically independent.

7.3. THE NEAR-FIELD APPROXIMATION

The near field approximation consists of neglecting W,(f) in (7.28) so that

W,(f)== W,,,(f). (7.29)

The validity of this approximation for astronomical observations has been discussed by YOUNG [1974]. From (7.26) and (7.27), it is clear that W x ( f ) is negligible at low frequencies. At high frequencies, both spectra take similar but very small values due to their decrease as the -9 power. Therefore, the near-field approximation holds in many applications. For instance, the variance (x’) of the log-amplitude x is completely negligible compared to the variance (&) of the phase

(7.30)

We already know from (3.21) that the coherence function at ground level has the form

B”(5) = exp - mo. According to FRIED [1966], the wave structure function can be written

D(5) = Dx(5) + D,<)(&)

(7.31)

(7.32)

V, 5 71 OPTICAL PATH FLUCTUATIONS 333

where Ox(&) and Ow,,(&) are, respectively, the structure functions of the log-amplitude and of the phase at ground level. Eq. (7.32) is the analog of (7.28) for the structure functions. Comparing (7.31) with (4.23) yields

D(&) = 6.88(5/r0);. (7.33)

Fig. 8 shows a plot of D(&) for r,= 11 cm, together with an estimation of Ox(&) from the relation

Ox(&> = 2[Bx(0)- B,(&)I (7.34)

where B,(&) is the covariance of the log-amplitude taken, according to (8.1), as 0.25 times the covariance of the scintillation plotted in Fig. 14 (8 8), assuming a typical scintillation index of 20% produced by a turbulent layer at 10 km. The phase structure function Ow,,(&) is obtained by subtracting Ox(&) from D(&) according to (7.32). Clearly, the near-field approximation

Ow,,(&) = WS) (7.35)

holds with a maximum error of the order of 8% occurring when 5 5 2.5 cm.

Fig. 8. Wave structure function (solid line), log amplitude structure function (dotted line) and phase structure function (dashed line) assuming ro = 11 cm and scintillation entirely

produced by the tropopause layer with a: = 0.2.

3 34 THE EFI;ECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL. ASTRONOMY [v, 8 7

7.4. PHASE FLUCTUATIONS

From (7.33), the near-field approximation (7.35) leads to

D,,,(S) = (Ivo(x) - qo(x + 511') = 6.88(6/ro)' (7.36)

which shows that the standard deviation of the phase difference between point x and point x + S

(7.37)

increases as the 5 power of the distance 6. Interferometric measurements provide an experimental check of (7.36) and (7.37). Such experiments have been done by BUSER [ 197 13 and by CLIFFORD, BOURICIUS, OCHS and ACKLEY [1971] o n horizontal laser beam propagation. The first test on stellar sources was done by BRECKINRIDCE [1976] and agreed w'ell with the 5/6 power law, for displacement values 6 ranging from 0.2 m to 1.5 m. Departures from the 5/6 power are expected at large distances when 6 becomes comparable with the outer scale of turbulence. However, the distance up to which (7.37) is valid is still unknown.

Since r, varies as A;, u,~, varies as A- ' and the standard deviation of the optical path difference

u, = (A/27r)a,,, = (2.62/27r)AO(6/ro)2 (7.38)

is wavelength independent. In (7.38), rO is the value of Fried's parameter at wavelength A,. Long baseline optical interferometers are now operating in astronomy (LABEYRIE [1975]). Assuming (7.38) is still valid on a 20 m baseline and r, = 11 cm (at A, = 0.5 ym) leads to optical path differences of the order of 16 ym, that is typical fringe excursions of the order of 30 fringes in the visible. Assuming frozen turbulence driven by the wind velocity u, the characteristic evolution time is of the order of u / [ . Taking u = 2 m/s and a 20 m baseline leads to a typical fringe motion of the order of 3 fringes per second, which explains that fringes are visible to the eye whenever the seeing is good and the wind speed is low. More rigorous calculations of the fringe visibility as a function of the baseline and the exposure time, assuming frozen wavefronts with log-normal statistics and Kolmogorov's law, can be found in C. RODDIER [1976].

a,,,([) = [D,,,(S)I; = 2.62(6/rO)2

7.5. ANGLE-OF-ARRIVAL FLUCTUATIONS

The deviation z(x, y ) of the wave surface from the average plane at ground level is

Z(X, Y ) = ( A / ~ T ) ~ x , Y ) . (7.39)

V, §71 OPTICAL PATH FLUCTUATIONS 335

I n terms of geometrical optics, the light rays are normal to the wave surface (see STROHBEHN and CLIFFORD [ 19671). Their angle-of-arrival therefore fluctuates. The fluctuations are

in the x direction and

a a P ( X , y ) = -- z(x, y ) = -(A/27r) - d x , y ) (7.41)

in the y direction. They obey Gaussian statistics. The power spectra of a and p are related to the power spectrum of cpo by

a Y aY

W, ( f ) = A2c W,<,C f )

Wo ( f 1 = A2f: WqC,C f 1

(7.43)

(7.44)

where fx and f, are the x and y components of the frequency vector f. The variance of a and p are

( a z ) = A 2 J d f f:W,(,(f) (7.45)

(7.46)

The standard deviation cr, of the angle-of-arrival is therefore given by

ofn = ( a 2 ) + ( p 2 ) = A 2 J df * f'W,<,( f ) . (7.47)

At the near-field approximation (7.29), W,,,( f ) is given by (7.25). Putting (7.25) into (7.47) and integrating over all directions in the frequency plane leads to

(7.48)

The integration over f cannot be taken from 0 to s. A more realistic expression is

af,a [)=dh . Ck(h) 1" 'df . f-:. (7.49)

where L,, is the turbulence outer scale and D-' is the high frequency cut-off due to averaging over the observing aperture of diameter D.

,


Integrating (7.49) leads to m

a: 0~ [W- L-:] [ dh C$(h) (7.50)

or, neglecting L-: compared to D-t (which can be a crude assumption), c m

(7.51)

Expression (7.51) has been established for observations at the zenith. It remains valid at any zenith angle y by taking h as a distance along the line of sight. Replacing h by h/cos y, (7.5 I ) takes the more general form

(7.52)

TATARSKI [ 197 11 has indeed shown more rigorously that (7.52) describes the motion of the centre of gravity of a star image as seen through a telescope of diameter D, when scintillation effects are neglected. The (cos y)- ' law was first established by KRASILNIKOV [1949]. Observations of the quivering of stellar images were reviewed by KOLCHINSKI [1952, 19571 and found consistent with (7.52) (see also IRWIN [1966]). More recent observations by BUFTON and GENATT [1971] agree both with the D-: law and the (cos y)-' law. It is worthwhile noting that u$ is independent of wavelength. Introducing Fried's parameter r, as expressed by (4.24) leads to

I ui 0~ D-~(cos y)-l dh . Ch(h).

ufn 0~ AZD-:r(;:. (7.53)

More rigorous estimations by TATARSKI .[1971] and FRIED [1965, 19751 give the following coefficient of proportionality in radians squared

c r i = (3.44/7r2)A2D-:r-:

(7.54)

The covariance of angle-of-arrival fluctuations, defined by

B a ( ~ , q ) = ( 4 x , y ) a ( x + p , y + q ) ) (7.55)

is obtained by taking the Fourier transform of their power spectrum (7.43). Therefore

(7.56)

V, §71 OPTICAL PATH FLUCTUATIONS 337

or, introducing the phase structure function

(7.57)

At the near-field approximation (7.36),

D,,)(p, q ) = 6.88r$(p2+ q2)g (7.58)

and

Putting (7.59) into (7.57) gives the longitudinal covariance in radians squared

B,(p, 0) = 0.097(A/ro)t(A/p)t (7.60)

and the lateral covariance

B,(O, q ) = 0.145(A/ro);(A/q)f. (7.61)

It must be remembered that these expressions are only valid in the inertial range. The divergence at the origin is not physical. In any practical situation, the value at the origin is limited by aperture averaging and given by (7.54). STROHBEHN [1970b] has pointed out the sensitivity of these covariance functions to the shape of the turbulence temperature spectrum. Experimental measurements by BORGNINO and MARTIN [ 19771 and by BORCNINO and VERNIN [1978] on the solar limb motion are in excellent agreement with (7.60) and (7.61) for displacement values p and q ranging from 5 cm to 30 cm. The agreement extends up to 1.1 m in similar measurements made by AZOUIT, BORGNINO and VERNIN [1978] on lunar limb motion. These measurements provide one of the best proofs of the validity of the inertial model in the case of astronomical observations.

7.6. IMAGE MOTION AND BLURRING

Astronomers are used to describing image degradation in terms of image motion and blurring. Image motion describes the wandering of the image centre of “gravity”. As seen in B 7.5, the deviation a’ of the image from its average position has a Gaussian probability density

(7.62)


with standard deviation urn given by eq. (7.54). Image motion does not degrade short exposures. Its effect on long exposures can be removed by using a fast automatic guider or, after detection, by adding short exposures properly centred (see § 9.1). The remaining degradation is called blurring.

Let &(a) = S ( a + a‘) be the instantaneous illumination in the centred image of a point source. The classical long exposure ( S ( a ) ) is related to the improved long exposure ( S , ( a)) by

( S ( a ) ) = ( I d a ’ -So(a-af)9(a’))

= ( S o ( a ) ) * %a). (7.63)

The associated modulation transfer functions are therefore related to each other by

(a#)> = (W)) - @(PI (7.64)

where

@(#I = exp - .rr2uip2 (7.65)

is the Fourier transform of 9(a ) and f = I+’\. Putting (7.54) into (7.65) leads to

@(#) = exp- 3 . 4 4 ( A ~ ~ ) ~ ( h { / r o ) ~ . (7.66)

Now, putting (4.15), (4.22) and (7.66) into (7.64) gives

(%#I) = T(#) exp- 3.44(A{/ro)’

= (s,,({)) exp- 3.44(h{/D)i(hf/ro)2

so that

(so(+’)) = T ( J ) exp- 3.44(h{/ro)z[1 - (h{D)f]. (7.67)

Expression (7.67) was first derived by FRIED [1966] who defines the associated improved resolution 9?o as (see § 4.3):

(7.68)

B0 is the resolution obtained after the removal of image motions. It is degraded only by blurring. Astronomers define blurring as a spread angle. Here we shall define it as the diameter oo of a disk, with uniform illumination equal to &(O)), having the same integral as (&,(a)). oo is the

v, 571 OPTICAL PATH FLUCTUATIONS 339

equivalent width of ( S o ( a ) ) . With such a definition

(.rr/4)wi = dcx - [(s0(~~)/(s0(0))1= 1 / 3 0

(7.69)

Similarly, we shall define the spread angle for normal imaging as

0 = (4/&)f (7.70)

where 3 is the resolution for normal imaging given by (4.16). For a small diffraction limited telescope of diameter D

w = wd = (4/7r)(A/D) = 1.27(A/D). (7.70’)

For an infinitely large telescope limited by turbulence

w = w, = 1.27(A/r0). (7.70”)

Numerical values of 9 and iB0 have been computed by FRIED [1966] as a function of D/ro. The associated spread angles w and wo, expressed in A/ro unit, are plotted in Fig. 9, together with a spread angle w, for image motion similarly defined as

w, = (4/&,): (7.71)

where

With such a definitiont

0, = 2am (7.73)

Fig. 9 shows that blurring reaches a minimum for D/ro= 3.7. For smaller Dlr, values, it is mainly due to diffraction by the telescope aperture. For larger D/ro values it is mainly due to turbulence. Fig. 9 also shows that the maximum improvement in resolution, due to image motion removal, is of the order of a factor 2 and occurs when D/ro= 3 ( 5 9.1). For large D/ro values, image motion is certainly overestimated because the finite outer scale of turbulence has not been taken into account. The effect of the outer scale has been studied by VALLEY [1979]. It must be pointed out

is only *,,,/A that is w,/2&, t Care must be taken that the standard deviation of image motion along a given direction


10 1\

4 I I I

0.1+--- - 7 I I , I I I I ;’... I , , >

01 1 10 100

Fig. 9. Image motion w,, blurring wn and total spread angle w, expressed in h/r, unit, as a function of D/ro. The asymptotic value 4/7-r is indicated with an arrow. Dotted line: blurring

due to diffraction by the telescope aperture alone.

that w2# w:+w; (7.74)

although such an equality has often been assumed by astronomers in the past. The equality would hold if o, oo and om were defined as standard deviations. Unfortunately this is not possible since the variance of the spatial irradiance distribution in a long exposure point source image is infinite as it is in an Airy pattern.

The first measurements of blurring as a function of the telescope aperture are due to ROSCH [1958a]. They were made at the Pic du Midi Observatory on the star a Lyrae. Rosch defines a blurring factor equal to the ratio of the width of the unperturbed Airy pattern to the measured

TABLE 3 Comparison between measurements of blurring by ROSCH [1958a] and theory. A good fit is

obtained when assuming ro = 7.6 cm.

Telescope 9cm 18cm 27cm 36cm 45cm 54cm diameter D

measured 0.96 0.70 0.51 0.37 0.30 0.23 ( R ~ S C H

Rosch’s [ 1958aI) blurring factor theoretical 0.90 0.74 0.54 0.40 0.29 0.23

(rn = 7.6 cm)

V, § 81 STELLAR SCINTILLATION 341

width of the instantaneous image. Rosch’s blurring factor can be easily read on Fig. 9 by measuring the distance between the curve of blurring and the dotted line. As shown on Table 3, a good fit is obtained with Rosch’s measurements by assuming ro = 7.6 cm. ROSCH [ 1958al also measured image motion. His data agrees very well with a D-f law. Measurements of blurring and of image motion are widely used by astronomers to estimate image degradation by atmospheric turbulence (§ 10).

0 8. Stellar Scintillation

A review of the properties of stellar scintillation has been done recently by JAKEMAN, PARRY, PIKE and PUSEY [1978]. Statistical properties of stellar scintillation have been experimentally investigated by many observers, mainly GAVIOLA [ 19491, NEITLEBALD [ 195 11, MIKESELL, HOAG and HALL [1951], ELLISON and SEDDON [1952], ELLISON [1954], BUTLER [1952, 19541, MIKESELL [1955], Z H U K O V ~ [ 19581, BARNHART, KELLER and MITCH- ELL [1959], PROTHEROE [1955a, b; 1961a, b; 19641, PROTHEROE and CHEN [1960] and more recently by BURON and GENATT [1971], VERNIN and RODDIER [1973], ROCCA, RODDIER and VERNIN [1974], PATERNO [1976], JAKEMAN, PIKE and PUSEY [1976]. Comparisons with simultaneous in situ atmospheric soundings were made by BUFTON [ 1973b], BARLEITI, CEP- PATELLI, PATERNO, RIGHINI and SPERONI [1977] and by VERNIN, BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [ 19791. Experiments using stellar scintillation analysis as a means for remote sensing of atmospheric turbulence have been developed by ROCCA, RODDIER and VERNIN [ 19741, OCHS, TING-I-WANG, LAWRENCE and CLIFFORD [1976] and AZOUIT and VERNIN [ 19801. The effects of scintillation on astronomical observations have been discussed by YOUNG [1967, 19691, SEDMAK [1973] and BELVE- DERE and PATERNO [1976].

8.1. FIRST ORDER STATISTICS

The easiest quantity to measure is the “amount” of scintillation, or scintillation index a:, defined as the variance of the relative irradiance fluctuations. It is related to the variance u: of the relative amplitude fluctuations ,y by

u: = 4 4 (8.1)

342 THE EFFE(JTS OF ATMOSPHERIC TURBULENCE IN OlTICAL ASTRONOMY [v, 5 8

Since a: is the integral of the power spectrum of x r

J

putting (7.26) into (8.2) and (8.1) and integrating over all directions in the frequency plane leads to

a: = 8.rr x 0.3%-2 dh * C$(h) df * f-y(sin ~ A h f ~ ) ~ . (8.3) 6 Lm a:= 12.5A-ZL dh 9 h;C&(h)[) dw w-'k(sin w ) ~

L=

Introducing the dimensionless variable w = TAhf gives m m

(8.4)

or, after integration,

a:= 19.12A-2 dh . hzC&(h). (8.5)

Fig. 10 shows a histogram of a: values obtained by VERNIN [1979] in the visible, near the zenith over fifty-two nights. Typical values of a: are of the order of 20%. a: may be as low as a few percent under good seeing conditions. Eq. (8.5) shows that scintillation decreases with increasing wavelength and becomes very small in the infrared. Eq. (8.5) also shows that the contribution of a turbulence layer to scintillation increases with height as the 5/6 power. Fig. l l a illustrates the average contribution of atmospheric turbulence to scintillation as derived from Hufnagel's model, described in § 2.4 (Fig. 1) for nighttime. There are two maxima, one at 1.2 km and the other at 10.8 km, separated by a minimum at 5 km, showing that scintillation comes mainly from two different parts of the atmosphere: 25% comes from layers lower than 5 km, mainly in the 1-2 km range, while 75% comes from layers above 5 km, mainly in the tropopause and lower stratosphere.

0 5LYln 0 0.5 an.,. 1

. -7-

1.5

Fig. 10. Histogram of scintillation indexes observed over fifty-two nights by VERNIN [1979].

V, 0 81 STELLAR SCINTILLATION 343

Fig. 11. Average contribution of atmospheric turbulence to stellar scintillation, according to Hufnagel's model; (a) as seen through a small aperture, (b) as seen through a large aperture.

It must be remembered that the results stated above are valid only when irradiance fluctuations are measured through an aperture smaller than their typical scale (in actual practice the aperture diameter must be smaller than or equal to about 3cm). When a larger telescope is used, high spatial frequency components are smoothed out. Aperture filtering has been described by TATARSKI [1961], YOUNG [1970] and FRIED [1973] and experimentally investigated by YOUNG [1967], MINOTT [1972], BUF- TON and GENAW [1971] and IYER and BUTON [1977]. The telescope filtering function G(f/fc) = [p0(f)12 must be introduced in (8.3). When the aperture frequency cut-off f, (of the order of the inverse D-' of the aperture diameter D) is small enough so that

rrhhfz<< 1 (8.6)

then the approximation

(sin rrAhf2)*= rr2h2h2f4 (8.7)

can be used in (8.3) which becomes m m

CT: = 95 I dh . h2C2,(h) 1 df ?G(f/f,) (8.8) 0 0

or, introducing the dimensionless parameter a = f/fc

a: = 9Sfj [:dh * h2C2,(h) (8.9)

344 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL

or

ASTRONOMY [V, $8

(8.10)

The approximation (8.7) is called the geometrical approximation since it can be derived from geometrical optics (REIGER [1962, 19631). From (8.6), it is valid whenever D>>(.rrhh):, that is, taking A =0.5 km and h = 10 km, D >> 13 cm. Eq. (8.10) shows that, in such a case, the amount of scintillation is independent of wavelength and decreases when the aperture diameter D increases, according to a - f power law. Eq. (8.10) also show that, for large apertures, the contribution of turbulent layers to scintillation increases with height h as h2 instead of hz. Fig. l l b shows the average contribution of atmospheric turbulence, in this case, as derived from Hufnagel’s model. The effect of low altitude layers is clearly filtered out. Experimental checks of the h 2 and hz laws have been made by BUFTON [1973b] and by BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [1977].

Expressions (8.5) and (8.10) have been established for stellar scintillation at the zenith. It remains valid at any zenith angle y by taking h as a distance along the line of sight. Replacing h by wcos y, eqs. (8.5) and (8.10) take the more general form

a:= 19.2A-a(~0s y)-y dh * h:C$(h) (8.11) r L- U: 0~ D-~(cos y)p3 dh . h2C’,(h). (8.12)

Such a dependence on zenith angle was first derived by TATARSKI [1961] and found in reasonable agreement with observations up to y = 60”. At greater zenith angles U: increases less rapidly and even saturates. As already quoted, the phenomenon of saturation of scintillation has been extensively studied on horizontal laser beam propagation (see for instance CLIFFORD, OCHS and LAWRENCE [1974]). In the case of astronomical observations it has been studied by YOUNG [1970] and more recently by PARRY, WALKER and SCADDAN [1979]. To my knowledge, modern theories on saturation (STROHBEHN [1978]) have not yet been applied to the study of stellar scintillation.

The small perturbation theory predicts log-normal statistics for the irradiance fluctuations. The probability density function (p.d.f.) of stellar scintillation has been experimentally investigated by JAKEMAN, PIKE and

V, 8 81 STELLAR SCINTILLATION 345

PUSEY [1976], JAKEMAN, PARRY, PIKE and PUSEY [1978] and VERNIN [1979]. Results are found to agree well with a log-normal law far from the saturation regime. The Rice distribution is clearly excluded. At large zenith distances, where saturation occurs, data approximately fit a K distribution (JAKEMAN and PUSEY [ 19761).

8.2. SECOND ORDER STATISTICS

According to (7.26) the spatial power spectrum of the relative irradiance fluctuations is given by

W ( f ) = 4 W X ( f ) = 3 . 9 x 10- k f dh - C$,(h)(sin rAhf2)*.

(8.13)

Fig. 12 shows theoretical spatial spectra that would produce a single turbulent layer at several altitudes. The increase of the fluctuation energy

'-T

Fig. 12. Theoretical spatial power spectra of stellar shadow patterns, in arbitrary scale, assuming a thin turbulent layer at the latitude indicated in km on each curve (2.5, 5 , 7.5 and 10 km). The integral J dh . C2,(h) over the layer thickness is assumed to be the same in each

case.


Fig. 13. Typical spatial power spectrum of stellar shadow patterns. Full circles: data from VERNIN and RODDIER [ 19731. Solid line: theoretical spectrum assuming two turbulent layers

at altitudes 2.5 km and 10 km. Dotted lines: assumed contribution of each layer.

with the altitude is clearly visible. Fig. 12 also shows that the frequency of the energy maximum decreases when the altitude increases. There are few experimental measurements of the spatial spectrum of scintillation. The first measurements were made by PROTHEROE [1961a, b; 19641. Fig. 13 shows experimental data obtained by Vernin using an improved version of Protheroe's technique described in VERNIN and RODDIER [1973]. A good fit with theory is obtained by assuming that the scintillation is produced by two turbulent layers at approximate altitudes of 2.5 km and 10 km in agreement with the average behaviour described in Fig. l l a (see also TOWNSEND [1965]).

The spatial covariance of irradiance fluctuations is given by the two- dimensional Fourier transform of (8.13). Fig. 14 shows the result of numerical computations for a single turbulent layer at several altitudes. The characteristic size of the shadow pattern clearly increases with the altitude of the layer. Visual observation of shadow patterns often reveals several patterns propagating in different directions. Patterns displaying large shadows can clearly be attributed to turbulence at the tropopause

V, §81 STELLAR SCINTILLATION 347

Fig. 14. Theoretical spatial covariance of stellar shadow patterns in arbitrary scale, assuming a thin turbulent layer at the altitude indicated in km on each curve (2.5, 5, 7.5 and 10 km). The integral dh . Ck(h) over the layer thickness is assumed to be the same in each

case.

level, whereas patterns displaying small shadows are produced by low altitude turbulence. Experimental measurements of the spatial covariance were first made by PROTHEROE [1955b] and by BARNHART, KELLER and MITCHELL [ 19591. Again experimental results can be interpreted in terms of an appropriate C’, profile.

The problem of inverting the integral in (8.13) in order to derive C& profiles from experimental spatial spectra or spatial covariance has been discussed by PESKOFF [1968] and FRIED [1969]. The poor accuracy of the method has been underlined by STROHBEHN [ 1970~1. However, by using combinations of appropriate spatial filters, OCHS, TING-I-WANG, LAW- RENCE and CLIFFORD [1976] were able to obtain C’, profiles with about 4 degrees of freedom. Systematic measurements have been done by Loos and HOGGE [ 19791. Spatio-temporal and spatio-angular analysis provide additional useful information for remote sensing of atmospheric turbulence. To a good approximation, the time behaviour of shadow patterns is entirely described by the motion of the wind driven atmospheric inhomogeneities. Fig. 15 shows a two-dimensional spatio-temporal spectrum obtained by VERNIN and RODDIER [1973]. It reveals two different contributions. The associated wind speed and altitudes deduced from the spectrum are in good agreement with meteorological soundings. Fig. 16


Fig. IS. Two-dimensional spatio-temporal power spectrum of stellar shadow patterns obtained by VERNIN and RODDIER [1973]. Lines of equal power are displayed in a temporal-frequency versus spatial-frequency plot. The spectrum shows evidence for two structures propagating at different speeds. Their velocities projected along iX are given by the slope of the dashed line as indicated. They agree with wind velocities at altitudes of 5 and 10 km, as observed in meteorological soundings. Similar altitudes are deduced from the

shape of the spatial spectrum associated with each structure.

IN C-L 0 10 20 30

sl Fig. 16. Two-dimensional spatio-temporal correlation of stellar shadow patterns obtained by AZOUIT and VERNIN [1980]. Thin lines: curves of equal correlation for a given time delay T = 4 ms. Thick line: wind hodograph from meteorological soundings with altitudes indicated in km. In this case, scintillation originates partly from turbulence around 2 km and partly

from turbulence near the tropopause.

V, 581 STELLAR SCINTILLATION 349

shows a spatio-temporal covariance, for a given delay 7 = 4 ms, obtained by AZOUIT and VERNIN [1980]. It also reveals two contributions and their associated wind speed. Altitudes are deduced from meteorological wind hodographs. Spatio-angular analysis on double stars seems highly promising. This technique, first described by ROCCA, RODDIER and VERNIN [1974], allowed AZOUIT and VERNIN [1980] to obtain a C; profile every 1 0 s with a resolution of 2 km. A typical result, displayed as a time sequence, is presented in Fig. 17.

Since scintillation is a noise source in high speed stellar photometry, purely temporal power spectra are of interest to astronomers. Such measurements have been performed mainly by MIKESELL, HOAG and HALL [1951], MIKESELL [1955], YOUNG [1967] and more recently by PATERNO [ 19761. The behaviour of scintillation spectra with the telescope aperture has been theoretically investigated by TATARSKI [ 19611, REIGER [1962, 19631 and by YOUNG [1967, 19691. Assuming a single wind velocity 2) in the x direction, the photoelectric signal is given by

Q(r) = F(vr, 0) (8.14)

where

F ( x , y ) = $ ( x , Y)*P(X, Y ) (8.15)

is the convolution of the irradiance $(x, y) on the aperture plane with the aperture transmission function P(x, y). The power spectrum W,(v) of

10 5 0 ' t ime cmnr

Fig. 17. Time sequence showing the evolution of turbulent layers producing stellar scintillation, from AZOUIT and VERNIN [1980]. A CL profile is obtained every 10 s, with a vertical resolution of about 2 krn, from a statistical analysis of the shadow pattern produced by a

double star.


Q ( t ) is therefore

W d v ) = u-1 j df, . W(v/u, f,) I&/u, fJ2 (8.16)

where W(f.., f,) is the spatial power spectrum described by (8.13). In the case of a full circular aperture of diameter D, without a central obscuration, the aperture filter function is given by

Im, f,)l” = (D2/4) IJ , (m)/f l* . (8.17)

It decreases as the -4 power of the spatial frequency f = (fi+@. When the aperture is large, high spatial frequencies are filtered out. According to (8.13), W ( f ) varies as f’ in its low frequency part. The product W ( f ) IP(f)l’ therefore decreases as f-‘ and its integral over f, decreases as f-’. Scintillation amplitude spectra [ W(v)$ are therefore expected to decrease as up:. Spectral indexes ranging from - 1.5 to -2.5 reported by PATERNO [1976] with 15 cm and 30cm apertures probably indicate a significant contribution of stratospheric turbulence.

Detailed numerical computations assuming non-uniform wind profiles, and taking into account the telescope central obscuration, have been performed by YOUNG [1969] who also extended the theory to planetary scintillation.

§ 9. Applications to High Resolution Imaging

Many methods have been attempted in order to improve the angular resolution of astronomical observations through atmospheric turbulence. In this section, we shall review these methods in the light of the theory developed in the preceding sections. It will be shown how the parameters describing image degradation limit the possibilities of image restoration.

9.1. CLASSICAL METHODS

A discussion on classical techniques can be found in ROSCH [ 19721 and YOUNG [ 19741. Expression (4.1) shows that a long exposure image can be restored by deconvolution or Wiener filtering. This has been achieved by HARRIS [1966], MUELLER and REYNOLDS [ 19671 and MCGLAMERY [1967], with images degraded by laboratory-generated turbulence. It is now also

V, 891 APPLICATIONS TO HIGH RESOLUTION IMAGING 35 1

used for imaging in astronomy (COUPINOT [1973], COUPINOT and HECQUET [1979], HAWKINS [1979]). The point spread function can either be obtained from a reference point source (MUELLER and REYNOLDS [1967]) or be estimated from a priori knowledge of the object (COUPINOT [1973]). It may also be obtained by empirically adjusting a parameter in a theoretical model (HARRIS [ 19661). The two-dimensional Fourier transform of the point spread function is the optical transfer function. Its shape is generally assumed to be Gaussian. Fried’s theoretical expression (4.22) has - to my knowledge - never been used in a deconvolution procedure.

From the preceding sections, we draw the following conclusions. Fried’s theoretical transfer function has been accurately confirmed by experiment. It should therefore be employed in any deconvolution procedure. In certain cases considerable differences might arise if a deconvolved image were obtained using Fried’s model instead of that obtained with an assumed Gaussian transfer function. Fried’s parameter r, can be estimated either from a priori knowledge of the object or from simultaneous measurements of a reference point source. The importance of simul- taneity must be underlined because of the great variability of this parameter (KARO and SCHNEIDERMAN [1976a], MILLER and ZIESKE [1977], RICORT and Arm [1979]). Techniques for measuring ro are reviewed in 0 10. The exponential decrease of the modulation transfer function described by (4.22) implies that the high frequency components of the image become rapidly buried by detector noise. A very low noise level and a good linearity are both essential to an efficient deconvolution. In the case of ultimate photon noise limitation, GOODMAN and BELSHER [1976a, b] have derived an expression for the maximum restorable frequency. In the case of a high signal-to-noise ratio, improved resolution could be obtained by using non-linear techniques, taking into account a priori knowledge of the object (positivity, finite extent) as shown by radio-astronomers (BIRAUD [ 19691).

Expression (5.1) shows that - within the limits of isoplanicity - short exposure images can also be restored by deconvolution. In this case, however, the point-spread function has a complicated speckle structure and a reference point source is needed within the isoplanatic patch. Such a deconvolution has been performed by MCGLAMERY [ 19671 with images degraded by laboratory-generated turbulence. He obtained better results from short exposures than from long exposures, confirming that higher spatial frequencies are recorded in short exposures. Unfortunately astronomical sources are too faint and the isoplanatic patch is too small for

352 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY w, 5 9

the method to be practicable in astronomy. It becomes practicable when sequences of many short exposures are used, as shown by WEIGELT

A method commonly used in astronomy, at least for solar observations (MULLER [1973]), is frame selection. It consists in selecting the best frames in a sequence of many short exposures. The probability of getting a good picture has been theoretically investigated by FRIED [1978]. He defines a good image as one for which the squared wave-front distortion over the aperture is 1 Tad2 or less, and finds that, if the aperture diameter D is greater than 3.5r,, the probability of getting such an image is approximately 5.6 exp [-0.1557(D/r,)*]. For example, with a 40 cm aperture in very good daytime seeing conditions ( r , = 8 cm), one image out of ten is good, in reasonable agreement with observations. With r, = 4 cm (average daytime conditions at a good site), the probability falls to The same probability holds for a larger aperture D = 80 cm in very good seeing conditions (r, = 8 cm). This explains why solar astronomers tend to reduce their aperture in order to obtain better images. Selection of the best frames in a sequence of short exposures is effective for bright objects, when D / r , ~ 6 . Owing to the recent advances in I.R. imaging devices, it will probably be an effective method for imaging bright I.R. sources through large telescopes. Indeed, r, = 13 cm at A = 0.5 km corresponds to r, = 77 cm at 2.2 krn so that, for a 3.60 m aperture, D/r, = 4.7.

Planetary surfaces are too faint to be properly recorded by a single short exposure. However, composite pictures can be obtained by adding many frames, each being recentred in order to cancel the effect of image wander. The associated transfer function, derived by FRIED [ 19661, is given by (5.18). Photon noise limitation has also been discussed for this case by GOODMAN and BELSHER [1976a, b]. The maximum increase in resolution is of the order of 2 and occurs when D/r,= 3 . Combined with frame selection, the technique should be effective up to D/r, = 6. Image recentring can be obtained automatically, using a high speed servomechanism. This is the simplest form of active or adaptive optics.

[ 1978cl.

9.2. ADAPTIVE OPTICS

Adaptive optical systems are those in which real-time control over optical wavefronts is employed to maximise the angular resolving power of a telescope viewing through atmospheric turbulence. Images are not

v, 8 91 APPLICATIONS TO HIGH RESOLUTION IMAGING 353

only centred but also sharpened. This is achieved by real-time modification of the shape of an optical component, which cancels the wavefront aberrations introduced by turbulence. The beginning of adaptive optics dates from the work of BABCOCK [1953, 19581. The present state of the art has been reviewed by HARDY [1978a, b]. A special issue of the Journal of the Optical Society of America (Vol. 67, no 3 (1977)) and a meeting of the Society of Photo-optical Instrumentation Engineers (1978) have been dedicated to this topic. Applications to solar observations are discussed by HARVEY [1978].

The necessary performances of an adaptive system are entirely dictated by the statistical properties of wavefront perturbations described in the previous sections. Fried’s parameter r, again plays a central role. Since r,, is the diameter of a coherent cell, the number of independently controlla- ble elements required to correct a filled aperture of diameter D is about (D/rJ2. According to eq. (7.38) in 97.4, the standard deviation of the optical path difference (T, over a baseline D is independent of wavelength and equal to (2 .62/2~)A~(D/r~)”. The peak-to-peak amplitude of the corrections to be made is at least three times larger, that is about ten wavelengths for D/r, = 12. The perturbations of wavefront amplitudes are negligible and can be ignored as shown in 0 7.3.

The response time of the servomechanism is dictated by the “Eulerian” evolution time of wavefront perturbations, or transit time of perturbations over a coherence cell. It is of the order of ro/v, where v is the typical wind speed of turbulent layers, and may vary from 1 to 10ms or more, according to weather conditions. It is a little shorter than the speckle lifetime which is related to the “Lagrangian” evolution time of wavefront perturbations and is approximately r,/Av, Au being the velocity dispersion, as seen in 9 6. Corrections made in the pupil plane are valid only within an isoplanatic patch. The effect of non-isoplanicity for adaptive optics has been studied by FRIED [1974, 19751. The isoplanatic patch size has been found to be essentially the same as that for speckle interferometry (FRIED [ 19791). However, the isoplanatic patch can be much wider if several corrections are made along planes conjugate to the locations of turbulence layers. Photon-noise limitations are discussed by GOODMAN and BELSHER [1976a].

Many wavefront correction devices have been developed in order to meet all these requirements. The most successful are active mirrors including segmented mirrors, continuous thin-plate mirrors, monolithic mirrors and membrane or pellicle mirrors. The corrections to be applied

354 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OFTICAL ASTRONOMY [v, 8 9

can be determined either by using a wavefront sensing device or by trial and error using an image-sharpening algorithm. For astronomical applications only white-light wavefront sensors, such as Hartmann tests or achromatic interferometers, can be used. Their sensitivity decreases as the inverse of the object size so that, for large objects, image-sharpening algorithms are necessary. Using such techniques, Buffington and his co-workers (MULLER and BUFFINGTON [ 19741, BUFFINGTON, CRAWFORD, MULLER, SCHWEMIN and SMITS [1977], BUFFINGTON, CRAWFORD, MULLER and ORTH [1977], POLLAINE, BUFFINGTON and CRAWFORD [1979]) have been able to sharpen, in real time, images of single stars (Sirius, Arcturus) and of double stars (a Gem, y Leo).

9.3. MICHELSON INTERFEROMETRY

Interferometric techniques have already been reviewed by LABEYRIE [1976, 19781. Applications to solar astronomy are reviewed by F. ROD- DIER [1978a]. Recent advances are presented in the proceedings of the I.A.U. Colloquium no 50 (DAVIS and TANGO [1979]). Here, we shall mainly discuss the limitations due to atmospheric turbulence. The principle of Michelson interferometry has been described in details in § 4.4. In Michelson’s original experiment (MICHELSON [ 19201, ANDERSON [ 19201, MICHELSON and PEASE [1921]) the fringe visibility was estimated visually. More recently HARVEY [1972], using the same technique, qualitatively demonstrated the existence of solar features with dimensions of about 100 km or smaller. In 9 4.4 we have shown how atmospheric effects can be eliminated in order to make quantitative astronomical measurements. Such measurements had not been attempted until recently, namely by CURRIE, KNAPP and LIEWER [ 19741 who measured stellar diameters, by KINAHAN [1976] and AIME, RICORT and GREC [1975, 19771 who derived the power spectrum of solar granulation, and by MCCARTHY and Low [1975] and MCCARTHY. Low and HOWELL [1977] who measured stellar envelopes in the infrared. Corrections for atmospheric turbulence can be avoided only if the diameter of the two apertures is much smaller than ro, in which case the signal-to-noise ratio is low.

Photon noise limitation has been discussed by GOODMAN and BELSHER [ 1976al. At low light level the signal-to-noise ratio is proportional to the number of photons received per “exposure” (see also AIME and RODDIER [1977]). For a given optical bandwidth and a given exposure time, it is

v, P 91 APPLICATIONS TO HIGH RESOLUTION IMAGING 355

proportional to r: since the area of the apertures should be comparable in size with rO for optimal results. The “exposure time”, or maximum integration time allowing fringes to be frozen, is dictated by the transit time of perturbations over the aperture, as for adaptive optics. .It is therefore of the order of r,/u ( u being the velocity of the perturbations). When fringes are scanned, it is proportional to ri as shown by AIME [1978]. For small objects, the optical bandwidth AA is restricted by the condition that the beams from the two apertures must interfere, in spite of the optical path difference vz introduced by turbulence. It is therefore proportional tot A2v;’ or, taking (7.38) into account, for a baseline L,

AA 0~ (rob);. (9.1)

Since the number of photons per “exposure” is proportional to the aperture area, the exposure time and the bandwidth, the signal-to-noise ratio is proportional to

S/N 0~ ~(ro /u ) (ro / iL)~ = r$u-’K:. (9.2)

It increases almost as the fourth power of r,. The signal-to-noise ratio is also proportional to the square root of the observation time. In order to obtain a given signal-to-noise ratio, the necessary observation time will therefore increase almost as the 8th power of r,. As we shall see, the same result stands for speckle interferometry, thus underlining the fundamental importance of good seeing conditions. The size of the isoplanatic patch for Michelson interferometry does not yet seem to have been considered in the literature. Since Michelson interferometry is a special case of speckle interferometry, it is very likely to be of the same order of magnitude.

An important drawback to Michelson’s interferometry is ihat the Fourier space is explored sequentially. An alternative method envisaged by RUSSELL and GOODMAN [1971] consists of using multiple small apertures. As shown in § 5.2, there is no attenuation of the object Fourier components, as long as the array of apertures is non-redundant. How- ever, the number of Fourier components which can be measured at the same time remains limited. This drawback can be overcome by using shearing interferometers. Such interferometers have already been used for astronomical observation (SAUNDERS [ 1964, 19671, CURRIE, KNAPP and LIEWER [1974]). They have been suggested as an alternative for speckle interferometry by KENKNIGHT [ 1972, 19751 and BRECKINRIDGE [ 1972,

t Note the steep improvement of tolerances with increasing wavelength.


19741 as they are equivalent to a large number of Michelson interferometers working together. F. RODDIER and C. RODDIER [1978] have underlined the advantages of a variable rotational shear, and have shown that fringes can be obtained with a wide optical bandwidth by means of a chromatic corrector (see also F. RODDIER [1978b]). The limitations due to atmospheric turbulence are exactly the same as for classical Michelson interferometry .

The main advantage of the two-aperture scheme is that the baseline can be considerably extended as was done by MICHELSON and PEASE [1921] or more recently by LAREYRIE [1975]. Mechanical and atmospheric instabilities are the two main difficulties to overcome. Heterodyne detection solves the problem of mechanical stability at the cost of a drastically reduced bandwidth. The effects of atmospheric turbulence on heterodyne detection are discussed by FRIED [1967] and RABBIA [1978]. The technique has been used in the infrared (SUTTON [1978], Assus, CHOPLIN, CORTEGGIANI, CUOT, CAY, JOURNET, MERLIN and RABBIA [ 19791). Homodyne detection (or intensity interferometry) solves both the mechanical and the atmospheric problems at the cost of very low sensitivity (HANBURY BROWN [1977, 19781). The best signal-to-noise ratios are obtained by direct optical interference. The recent success of Labeyrie’s two-telescope experiment has opened new hopes and initiated new projects in the visible (University of Maryland, University of Sydney) as well as in the infrared (C.E.R.G.A., Berkeley, Imperial College). Due to the optical bandwidth limitation, the signal-to-noise ratio should decrease as L-2 (eq. (9.2)) as long as the baseline is much smaller than the outer scale of atmospheric turbulence. The baseline at which the outer scale begins to play a role is still unknown.

LABEYRIE [ 19741 suggested applying speckle interferometric techniques to long-baseline interferometry with large apertures. The associated theory has been developed by C. RODDIER and F. RODDIER [1976a, b]. They have shown that if the wavefront perturbations on the two apertures are uncorrelated, atmospheric effects are eliminated by defining the fringe contrast as the ratio of high spatial frequency energy of short exposures to the low frequency energy of long exposures. GREENAWAY and DAINTY [1978] have investigated the use of an interferometer working in the pupil space. The signal-to-noise ratios are essentially the same in both cases (DAINTY and GREENAWAY [1978]).

The concept of a telescope array as the basis for a “Next Generation Telescope” has recently attracted attention (PACINI, RICHTER and WILSON [ 19781) and the construction of coherent arrays has been envisaged

V, § 91 APPLICATIONS TO HIGH RESOLUTION IMAGING 357

(LABEYRIE [1977], GUSH [1979]). The effect of atmospheric turbulence on a telescope array has been investigated by F. RODDIER [1974] in the case of small apertures and by AIME and RODDIER [ 19761 in the case of large apertures. For non-redundant arrays the highest observing efficiency is achieved by arrays containing only a few elements. For highly-redundant arrays, the efficiency is maximised by maximising the number of elements (GREENAWAY [ 19791).

9.4. SPECKLE INTERFEROMETRY

The principle of stellar speckle interferometry is due to LABEYRIE [ 19701 who first recognised the similarity between short exposure stellar images and laser speckles. The method consists of a second order statistical analysis of the image speckle pattern. Short exposures are usually recorded photographically (GEZARI, LABEYRIE and STACHNIK [ 19721, BRECKINRIDGE, MCALISTER and ROBINSON [ 19791, BEDDOES, DAINTY, MOR- GAN and SCADDAN [1976], WEIGELT [1978a, b], BALEGA and TIKHONOV [1977]). The image energy spectrum is then obtained by optical processing. The use of a television camera has been initiated by LABEYRIE [1974] and by KARO and SCHNEIDERMAN [1976b]. Electronic image processors have been built in order to compute in real time the energy spectrum of the autocorrelation function of speckled images. On-line digital correlation of photon counting T.V. images, as described by BLAZIT, KOECHLIN and ONETO [ 19751, seems to be the most promising technique for faint objects. Better photometric accuracy is obtained by photoelectric scans, as pointed out by AIME and RODDIER [1977]. They suggest one-dimensional scans as a good compromise between Michelson interferometry and two-dimensional speckle interferometry. This technique, which has been found valuable for the study of solar granulation (RICORT and AIME [1979]), may provide the best accuracy for the observation of bright objects (see also AIME, KADIRI, RICORT, VERNIN and RODDIER [1979] and KADIRI [1979]). One dimensional scans have also been used for infrared speckle interferometry (SIBILLE, CHELLI and LENA [1979], WADE and SELBY [1978], SELBY, WADE and SANCHEZ MAGRO [1979]). Recent advances in I.R. diode arrays make I.R. speckle interferometry very promising.

Signal-to-noise ratio in speckle interferometry has been studied by many authors. Assuming photon-noise limitation, the uncertainty o n the autocorrelation function has been derived more or less heuristically by


LABEYRIE [1974] and DAINTY [1974]. The uncertainty on the energy spectrum has been derived by F. RODDIER [1975] and by GOODMAN and BELSHER [1976a]. Since the autocorrelation function is an integral over all spatial frequencies, the S/N ratio in this quantity exceeds that in the energy spectrum by a factor equal to the square root of the number of speckles. The equivalence of the two expressions have recently been demonstrated by GREENAWAY and DAINTY [ 19781. Consequences have been drawn by BARNETT and PARRY [1977] and by WALKER [1978, 19791 for the autocorrelation, by MILLER [1977], and by DAINTY and GREENA- WAY [1978, 19791 for the energy spectrum. The signal-to-noise ratio for infrared speckle interferometry has been derived by SIBILLE, CHELLI and LENA [1979] (see also CHELLI [1979]). Whereas, in the visible, photon noise limitations imply that Michelson and speckle interferometry have a similar signal-to-noise ratio, background noise limitations afford the multiplex advantage to I.R. speckle interferometry.

Assuming photon noise only, the signal-to-noise ratio on the energy spectrum, at a very low light level, is proportional to the number of photons per speckle. For a given optical bandwidth and a given exposure time, it is therefore proportional to 6. The optical bandwidth limitation is similar to that for Michelson interferometry (KARO and SCHNEIDERMAN [1978]) as given by eq. (9.1). It increases as ri. As shown in Q 6.4, the exposure time must be of the order of r,/Av, where A v is the dispersion of wind velocities in turbulent layers. The signal-to-noise ratio therefore depends upon the atmospheric parameters r, and A v as

S/N cz rfr: Av-’. (9.3)

For a given S/N ratio, the observation time increases almost as the 8th power of r,, as for Michelson interferometry (Q 9.3).

Eq. (5.2) in Q 5 implies that the object energy spectrum IO(f)(” is obtained by dividing the image energy spectrum by the energy spectrum of a point source image (ls(g)l’). The function (\s(f)[’) describes the atmospheric attenuation of the object spectral components. Its determination remains a major experimental problem. In Labeyrie’s initial experiment, calibration of the atmospheric attenuation was obtained by observing unresolved stars between each measurement. Unfor- tunately, atmospheric effects are often strongly time-dependent. They also depend upon the location of the observed star in the sky, or at least upon its distance from the zenith. Accurate calibration is therefore difficult.

v, 491 APPLICATIONS TO HIGH RESOLUTION IMAGING 359

WELTER and WORDEN [1978] have suggested taking the difference between the average autocorrelation of speckle patterns and the average cross-correlation of successive patterns as the autocorrelation for the image effectively uninfluenced by the atmosphere. Indeed, assuming that successive patterns are uncorrelated, their average cross-correlation is equal to the autocorrelation of the long exposure image. In the Fourier space, Welter and Worden’s procedure is therefore equivalent to taking the difference between the short exposure spectrum and the long exposure spectrum. According to (5.15) it is effectively proportional to the unperturbed object spectrum multiplied by the diffraction limited transfer function of the telescope. However eq. (5.15) assumes Gaussian statistics which are known to be unrealistic for wavefront perturbations. The low-frequency part of the short exposure spectrum is better represented by (5.18) such that an improvement of Welter and Worden’s procedure would consist in subtracting the cross-correlation of successive recentred patterns. Moreover, the short exposure energy spectrum (\s(i)12) is often wrongly normalised to unity at the origin, as the long exposure spectrum (s(#))2. In the subtraction, Iow frequencies are thus incorrectly eliminated. The difference ( ~ ~ ( O ) ~ ” ) - ( s ( O ) ) ’ represents the variance of the illumination integrated over the telescope pupil. I n the Gaussian approximation (5.15) it is equal to v/9’ and cannot be neglected. Similar criticisms of Welter and Worden’s procedure have been made by FANTE [1979] and by BRUCK and SODIN [1980]. Bruck and Sodin suggest a procedure minimising the observed width of the object autocorrelation.

The good agreement recently obtained between observed and theoretical transfer functions, assuming log-normal statistics (AIME, KADIRI, RICORT, RODDIER and VERNIN [1979], CHELLI, LENA, F. RODDIER, C. RODDIER and SIBILLE [ 19791) suggests that these theoretical transfer functions could be used to restore information. Only Fried’s seeing parameter ro needs to be measured. It can be determined simultaneously and independently of the observations or it can be derived from the observations themselves using a known part of the object spectrum, such as the low frequency part, or using a value measured with a Michelson interferometer. A self calibration method has been worked out by RICORT and AIME [1979], based on the consistency of the results obtained from different seeing conditions.

However, theoretical transfer functions apply only to ideal conditions. Corrections for finite optical bandwidths and finite exposure time may be necessary. Focusing errors and telescope aberrations also play a role

360 THE EFFEOS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY [v, 8 10

when the seeing disk becomes of the order of the transverse aberration (DAINTY [1974], KARO and SCHNEIDERMAN [1977], F. RODDIER, RICORT and C . RODDIER [1978]). The effects of non-isoplanicity as well as finite exposure time have been discussed in detail in § 6.

9.5. IMAGE RECONSTRUCTION

Classical interferometry or speckle interferometry give only the modulus of the Fourier transform of the intensity distribution in the observed object. Although images can be derived from such limited information (FIENUP [1978]), the solution is not in general unique (FIDDY and GREENA- WAY [1978]). It may however be unique in many cases (GREENAWAY [1977], BATES, MILNER, LUND and SEAGAR [1978], BRUCK and SODIN [1979]). Techniques of phase recovery have long been developed by radioastronomers (VAN SCHOONEVELD [ 19791) and similar techniques are now envisaged in optics. Many proposals have been made to extract the phase information from speckle images. They are reviewed by WORDEN [1977] and by NISENSON and STACHNIK [1979]. Some of them have already been applied to astronomy with some success (LYNDS, WORDEN and HARVEY [1976], WEICELT [1978c]). The most general method, due to KNOX and THOMPSON [1974], has been applied to the restoration of sunspot images by STACHNIK, NISENSON, EHN, HUDGIN and SCHIRF [1977]. No expression for the signal-to-noise ratio in the Knox and Thompson procedure has yet been given in the literature. However its dependence upon atmospheric parameters is very likely to be the same as for speckle interferometry (eq. (9.3)).

0 10. Seeing Monitors and Site Testing

Astronomers have long tried to estimate image degradation by atmospheric turbulence for various reasons, such as the choice of an observatory site, comparison between existing sites, selecting observations to be made under given seeing conditions, the decision as to whether an observation is possible, choosing the best instant for a measurement and the calibration of atmospheric effects.

Most of the classical methods of measuring image quality were developed in the early sixties with the intention of selecting optimal sites for the construction of new telescopes. These methods are reviewed by STOCK

v, I101 SEEING MONITORS AND SITE TESTING 361

and KELLER [1960], MEINEL [1960] and in the proceedings of the I.A.U. Symposium no 19 (ROSCH, COURTES and DOMMANGET [1963]). The work of Soviet scientists is reviewed in KUCHEROV [1965]. Unfortunately, at that time, the physics of image degradation by atmospheric turbulence was still little understood and experimental efforts were essentially empirical. Questions which arose were: what should be measured? And, more importantly, how many parameters are needed to define the quality of an image? Most of the experimental attempts were based on image motion and blurring measurements. We now know that image motion and blurring are strictly related (0 7.6) and that image quality can be almost entirely determined by a single parameter such as Fried’s parameter r,. Other parameters of lesser importance are the size of the isoplanatic patch and the characteristic evolution time of the image structure. Fried’s parameter is now universally adopted as a convenient measure of image quality. It can be deduced either directly, from optical measurements (seeing monitors), or indirectly from atmospheric soundings. Both types of measurements will be reviewed in this section.

10.1. SEEING MONITORS

Estimations of image quality are often made visually. Unfortunately, the appearance of the image depends strongly upon the aperture of the telescope. Through a large telescope the size w of the seeing disk is often estimated by comparison with the known angular separation of a double star. The result slightly depends upon the stellar magnitude. Assuming negligible image motion, r, is known to be of the order of h l o with A = 0.5 pm. A calibration by comparison with more quantitative estimates is highly recommended. Slow image motion is difficult to perceive. Inspection of Fig. 9 (p. 340) shows that it may not be negligible. Indeed visual estimations of image quality are known to be often optimistic. Corrections for image motion can be envisaged from the theoretical curves of Fig. 9. However, for large telescopes, care must be taken. In this case the theoretical value of image motion may be overestimated due to the finite outer scale of turbulence being neglected.

Through a small telescope, visual estimations of image quality are difficult and subject to errors. Experience shows that the quality of seeing is always overestimated. A method based on the appearance of stellar diffraction patterns has been proposed by Danjon. Unfortunately, this appearance is not only very sensitive to the telescope diameter but also to


the evolution time of the image structure which is of the order of the response time of the eye. Danjon’s scale cannot therefore be calibrated purely in terms of ro. Solar observers are used to estimating image quality from the contrast of solar features, such as granulation or sunspot umbrae and penumbrae, and from the appearance of the solar limb. A scale has been proposed by Kiepenheuer. It is subject to the same criticism as Danjon’s scale. Visual measurements of image motion with a micrometer probably give the most reliable results. An estimation of ro is obtained by means of eq. (7.54). Unfortunately this method is very sensitive to telescope vibrations due to wind shake. A differential method has been developed with a two-aperture telescope (STOCK and KELLER [1960]). It can be considered as a simplified Hartman test and will be discussed below.

Photographic methods are expected to give more quantitative results. In principle the size of the seeing disk can be measured on stellar photographs taken with a large telescope. Exposure times of at least 10 s take image motion into account and can be considered as long exposures. Unfortunately one encounters a considerable magnitude effect. In practice, the image size grows almost indefinitely with the increasing brightness of the stars. As quoted in § 4.5, the size of the seeing disk can be obtained only by a combination of measurements made on a carefully calibrated plate. On the other hand, exposures as short as a few milliseconds taken with an image intensifier reveal the speckle structure of the image. The area of the telescope aperture divided by the estimated number of speckles gives the wavefront coherence area o, from which r, can be deduced by means of expression (5.13).

With small telescopes, the most widely used method consists in photo- graphing star trails. The polar star is generally observed with a fixed telescope mounting. A systematic comparison with visual estimations of the size of the seeing disk through a large telescope has been reported by HARLAN and WALKER [1965]. Star trails provide a continuous record of image motion. Rapid motions are however filtered out. MORODER and RIGHINI [1973] state that the standard deviation measured on photographic plates is about half of that measured visually. After corrections, they derive estimations of ro by means of eq. (7.54). A more reliable method seems to derive the line spread function, and hence ro, as reported by WALTERS, FAVIER and HINES [1979].

Only photoelectric devices permit continuous monitoring of image quality. Their high sensitivity and small response time allow the instantaneous image structure to be frozen. With large telescopes, rapid scans

v, § 101 SEEING MONITORS AND SITE TESTING 363

through a bright stellar image with a narrow slit give one-dimensional instantaneous intensity profiles. As shown by AIME, KADIRI, RICORT, RODDIER and VERNIN [ 19791, reliable speckle energy spectra are obtained by processing the signal through a spectrum analyser. Accurate values are obtained by fitting theoretical curves to observations. SCHNEIDERMAN and KARO [1978b] use a similar procedure with a T.V. camera. MLLER and ZIESKE [ 19771 have developed a seeing monitor in which spectrum analysis is obtained by scanning the stellar image with a variable spatial frequency spinning reticle. The spatial frequency at which the spectrum drops to one half is taken as a measure of blurring. Rapid photoelectric scans are also used to estimate the quality of the solar image through a large telescope. The observed contrast of the solar granulation is taken as a measure of the quality of the image. Theoretical computations have been done by RICORT, AIME, RODDIER and BORCNINO [1980] in order to calibrate the observed contrast in terms of r,,. Fig. 18 illustrates the result of their computations. Unfortunately the method is sensitive to scattered light. Spectrum analysis seems preferable to contrast measurements. Rapid image scans apply only to bright objects. When dealing with stars fainter than magnitude 3 or 4, the simplest approach seems to estimate the size of the long exposure seeing disk by measuring the stellar light through a series of diaphragms of different sizes. An iris diaphragm could

10 -

5 -

......... *.. ...... -...?a ......................... ........ ._---ds-e=- .....

,/

3cm _----- 01 I I I I

0 0.5 1 1.5 Wavelength ( f m )

Fig. 18. Observed r.m.s. brightness fluctuations of the solar granulation in O/O of mean brightness, as a function of wavelength, for 3 telescope diameters and 4 values of ro

(measured at A = 0.5 pm), according RICORT, AIME, RODDIER and BORGNINO [1980].

364 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY p, 5 10

be automatically driven for continuous monitoring with a response time of the order of 1 0 s or more. Accurate tracking of the telescope is essential.

With a small telescope, most of the seeing effects appear as image motion. In the astronomical seeing monitor (A.S.M.) of BABCOCK [1963], the motion of a stellar image is measured and recorded with a photoelectric image follower. Babcock’s A.S.M. was extensively used in Chile for site testing (IRWIN [1966]). Photoelectric recording of the motion of the solar limb was performed by KALLISTRATOVA [1966]. For the J.O.S.O. site testing campaign, BRANDT [ 1969, 19701 developed a similar technique allowing motion and blurring of the solar limb to be measured simultaneously. BRANDT [1969] also derived time frequency spectra of the image motion displaying considerable energy at very low frequencies, as could be expected from Kolmogorov’s law. A maximum is expected at a frequency of the order of the ratio vlL, of the wind speed to the turbulence outer scale. It can easily be as low as 0.1 Hz, showing that integration times of at least 10 s are necessary in order to obtain reliable values for the standard deviation a,. BORGNINO, VERNIN, AIME and RICORT [1979] have recently brought attention to the spatial and angular filtering associated with the use of a slit on the solar limb. Such filtering must be taken into account in the interpretation of the results.

Up until now, we have reviewed only seeing monitors working in the image plane and measuring atmospheric effects on the image structure. In fact, the best results have been obtained by directly measuring wavefront perturbations in the pupil plane. Hartmann tests have been used by ROSCH [ 1954a, b]. However, quantitative statistical analysis of Hartmann photographs is time consuming. A simplified Hartmann test with two apertures can be made with a small, interferometer-like instrument (STOCK and KELLER [1960]). According to (7.54), (7.60) and (7.61), the mean squared amplitude of the relative longitudinal motion of the two stellar images is

2[B,(O, 0) - Bm(k, O)] = 2(A/rO);[0. 18(A/D); - 0.097(A/p)t] (10.1)

and the mean squared amplitude of the relative transverse motion is

2[B,(O, 0) -B@(k, O)] = 2(A/r,)Z[O.l8(A/D)f- 0.145(A/p)f] (10.2)

where D is the diameter of each aperture, assumed to be small and p the distance between the two apertures (see MILLER and KELLEN [1975]). More reliable estimations of r, would be obtained by using three or four

v, P 101 SEEING MONITORS AND SITE TESTING 365

apertures and by following the procedure of BoRGNlNO and VERNIN [ 19781. The Foucault test, also called occultation, schlieren or strioscopic technique, consists in focusing a star image onto a knife edge. Irradiance fluctuations proportional to the slope of the wavefront surface are observed in the pupil plane. The use of Foucault tests was also suggested by ROSCH [1957]. For solar observations, a similar test can be done by observing the telescope pupil through a diaphragm or a slit at the solar limb as shown by KOZHEVNIKOV [1961] and BECKERS [1966]. This method allowed BORGNINO and VERNIN [ 19781 to make quantitative estimations of the spatial covariance of angle-of-arrival fluctuations, providing evidence for the inertial structure of atmospheric turbulence. As a by-product, accurate r, values are obtained at the near field approximation. They can be used as a standard to calibrate image motion measurements made with a simpler instrument as shown by BORGNINO, VERNIN, AIME and RICORT [1979]. The results have recently been found to be in excellent agreement with in situ thermal soundings (BORGNINO, CEPPATELLI, RICORT and RIGHINI [ 19801). When dealing with stellar sources, interferometric techniques are by far the most satisfying. The motion of fringes due to atmospheric turbulence was first investigated by DANJON [ 19551. The last report is due to BRECKINRIDGE [1976]. Interferometric measurements of the atmospheric transfer function are reviewed in 04.5. They are direct measurements of the coherence scale of wavefront perturbations according to the definition (4.23) of r,, The results must therefore be considered as primary standards from which any seeing monitor should be calibrated.

10.2. ATMOSPHERIC SOUNDINGS

Since the seeing parameter r, is related to the integral Jdh CZ,(h) of the structure constant of refractive index fluctuations by eq. (4.24), it can be deduced from atmospheric measurements of C; as a function of altitude. From (5 2, we know that C; is essentially related to the structure constant C; of temperature fluctuations (eq. (2.16)). Measurements of Ct are made with thin wire thermometers. Such microthermal sensors have been widely used in site testing campaigns. Lynds (in ROSCH, COURTES and DOMMANGET [ 19631) investigated temperature fluctuations in the surface boundary layer at Kitt Peak using seven sensors arrayed along the side of a vertical tower up to 22.5m. COULMAN [1974] made microthermal measurements in the immediate vicinity of telescopes.


KIEPENHEUER [ 1972, 19731 made similar measurements from aircraft, for the J.O.S.O. site testing campaigns. For the same campaigns, BARLETTI, LEMMETI and PATERNO [ 19741 designed a balloon-borne radiosonde. Their results are summarised in a paper by BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [1977]. Since radars cannot be easily moved, they are of little use in site testing campaigns. Sodars were used during the J.O.S.O. campaigns. They have been found too sensitive to wind which blows out the return signal. Scintillation measurements have now been proved capable of yielding quantitative results. A comparison between Fig. 1 l a and 1 l b (p. 343) shows that scintillation measurements through a 30 cm aperture give the contribution of the tropopause, whereas simultaneous measurements with a 3 cm aperture would give the contribution of the planetary boundary layer by taking the difference. Although more sophisticated, the system developed by OCHS, TIN-I-WANG, LAWRENCE and CLIFFORD [1976] and extensively used by Loos and HOGGE [1979] would certainly be extremely useful in a site testing campaign. The method developed by ROCCA, RODDIER and VERNIN [1974] also gives wind velocities. It has been found to agree with in situ measurements (VERNIN, BARLETTI, CEPPATELLI, PATERNO, RIGHINI and SPERONI [ 19791).

10.3. DISCUSSION

We shall now discuss the most appropriate method to be chosen according to the goals briefly outlined at the beginning of this section. For site testing, an approximate knowledge of the origin of atmospheric disturbances is needed and these disturbances must be continuously monitored. Therefore, scintillation measurements seem the most appropriate for turbulence above 1 km. Near the ground, turbulence is best measured with thermal sensors. A gap remains between 1 km and 20 or 30 meters. It is very likely that a carefully calibrated seeing monitor, measuring the integral of (2; over the whole atmosphere, would give that contribution by taking the difference between the results of other measurements. Such a monitor could be a rotation interferometer, as developed by C. RODDIER [1976], mounted on a 20cm telescope or an interferometrically calibrated image motion monitor. For solar site testing, an image motion monitor calibrated as described by BORGNINO, VERNIN, AIME and RICORT [ 19791 seems appropriate. Considerable efforts have been made in the past to put seeing monitors on top of towers

v, J 111 CONCLUSION 367

carefully protected from the wind. There is now no doubt that the measurements of a ground based monitor could be corrected for the effect of lower layer turbulence, deduced from simultaneous and independent measurements with thermal sensors. The approximate knowledge of the distribution of turbulence with height enables one to estimate the size of the isoplanatic patch for high angular resolution imaging (§6). A knowledge of the distribution of wind velocities in turbulence layers is also useful in order to estimate the time scale of speckle boiling (§ 6), which limits the exposure time in speckle interferometry.

Seeing monitors have also been developed for real time selection of the best images (ROSCH [1958b, 19601). As we have seen ($9) this is only worthwhile with telescopes of moderate size (D/ro -- 3 to 5). Post- detection frame selection is now considered a much easier solution (MULLER [1973]). Calibration of atmospheric effects for a given observation remains a difficult problem. With high signal-to-noise ratios calibration is better made on the image itself, using some a priori knowledge of the object or using a self-consistent method, as reported by RICORT and A i m [1979]. For speckle work on faint objects, simultaneous offset measurements on a bright star could be envisaged.

0 11. Conclusion

During the last decade, considerable advances have been achieved in high angular resolution optical imaging in astronomy, as attested by the I.A.U. Symposium n o 50 held in August 1978 at the University of Maryland (DAVIS and TANGO [ 19791). Diffraction limited resolution has been obtained with large telescopes up to magnitude 13 by means of speckle interferometry. A Michelson interferometer is already working in the visible on a 20 m baseline up to magnitude 4 and many projects are nearing completion. However, such technological progress will be fully useful only if accurate quantitative measurements prove to be feasible through atmospheric turbulence. As pointed out by Hanbury Brown (in DAVIS and TANGO [1979]), the accuracy needed on stellar diameters is of the order of 2%. Furthermore, most of the physical information is in the centre-to-limb brightness distribution as a function of wavelength. The considerable progress made recently in the understanding of atmospheric effects gives strong hopes that such a goal will be achieved in the near future. There will be strong competition between spatial and ground

368 THE EFFECTS OF ATMOSPHERIC TURBULENCE IN OpTlCAL ASTRONOMY [V

based techniques. However, spatial techniques are extremely expensive and a telescope larger than 2.4 m is very unlikely to be launched before decades. The next step might be to launch a long baseline interferometer in space. Better support will be given to this project once ground based techniques are proven to be scientifically productive.

As pointed out in § 9, the quality of seeing is of fundamental importance for achieving high angular resolution from the ground. International efforts should be made to find the most appropriate site for such observations, and at existing sites, priority should be given to these observations whenever the seeing is exceptionally good. Considerable progress could also be made by the proper modification of domes and their environment, as carried out at the Pic du Midi observatory.

Much work remains to be done before achieving a full understanding of atmospheric effects. The relationship between isoplanicity or speckle boiling and the structure of the atmosphere is still not clear. The statistics of the fluctuations of seeing with time remain unknown. The effects of the finite outer scale of turbulence have never been experimentally investigated. The low frequency part of the spectrum of image motion is almost entirely unknown, although long term fluctuations are known to exist from astrometric measurements. Such a lack of knowledge has dramatic consequences in the selection of the astrometric satellite Hypparcos, since there still remains considerable controversy over the fundamental limitations imposed by the atmosphere on astrometric measurements. A whole area of investigation remains open and results are urgently needed. As a by-product, astronomical instruments may become useful for atmospheric remote sensing, as is now the case in studies of stellar scintillation.

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WALKER, J. G., 1979, Opt. Commun. 29, 273. WALTERS, D. L., D. L. FAVIER and J. R. HINES, 1979, J . Opt. SOC. Am. 69, 828. WANG, C. P., 1975, Opt. Commun. 14, 200. WEIGELT, G. P., 1978a, Astron. Astrophys. 67, L 11. WEIGELT, G. P., 1978b. Astron. Astrophys. 68, L 5. WEIGELT, G. P., 1978~. Appl. Opt. 17, 2660. WELTER, G. L. and S. P. WORDEN, 1978, J. Opt. SOC. Am. 68, 1271. WESELY, M. L. and E. C. ALCARAZ, 1973, J. Geophys. Res. 78, 6224. WESSELY, H. W. and J. 0. BOLSTAD, 1970, J. Opt. SOC. Am. 60, 678. WORDEN, S. P., 1977, Vistas in Astronomy 20, 301. WYNGAARD, J. C., Y. IZUMI and S. A. COLLINS JR. , 1971, J. Opt. SOC. Am. 61, 1646. YAGLOM, A. M., 1949, Dan. S.S.S.R. 69 (6). 743. YOUNG, A. T., 1967, Astron. J. 72, 747. YOUNG, A. T., 1969, Appl. Opt. 8, 869. YOUNG, A. T., 1970, J. Opt. SOC. Am. 60, 248. YOUNG, A. T., 1974, Astron. J. 189, 587. ZHUKOVA, L. N., 1958, Izv. Glav. Astron. Observ., Akad. Nauk S.S.S.R. 21, n o 162.

AUTHOR INDEX

A ABELLA, I. D. 5, 40 ASRIKOSOV, A. A. 83, 84, 135 ABSTREITER, G. 131, 135 ACKERHALT, J. R. 23, 24, 40 ACKLEY, M. H. 296, 334, 370 AGARWAL, G. S. 24, 30, 40, 99, 100, 135 AGRANOVICH, V. M. 113, 135, 162, 208 AHMAD, F. 23, 24, 42 AIME, C. 314,351,354,355,357,359,363,

ALCARAZ, E. C. 289, 376 ALLEN, L. 23, 40 ANASTASAKIS, E. 60, 135 ANDERSON, J. A. 354, 368 ANDO, Y. 129, 136 ANTONIA, R. A. 287, 368 APANASEVICH, P. A. 6, 7, 8, 40 ARAKAWA, E. T. 143, 210 ASHKIN, M. 52, 135, 136 ASHLEY, J. C. 200, 209 ASPNES, D. E. 191, 201, 210 Assus, P. 356, 368 AUBRY, M. 289, 369 AVAN, P. 28, 30, 40 AZIZA, A. 194, 210 Azourr, M. 291, 337, 341, 348, 349, 369

364, 365, 366, 367, 368, 369, 374

B BABCOCK, H. W. 353, 364, 369 BAGHDADY, E. J. 213, 279 BAKLANOV, E. V. 20, 40 BALEGA, Y. Y. 357, 369 BALLAGH, R. J. 38, 41 BARKER, A. S. 83, 135, 162, 208 BARLETTI, R. 290, 291, 341, 344, 366, 369,

BARNETT, M. E. 358, 369 BARNHART, P. E. 341, 347, 369 BARTOLANI, V. 96, 101, 105, 135 BASTIAANS, M. J. 220, 227, 276, 278, 279 BATCHELOR, G. K. 284, 369

375

BATES, R. H. T. 360, 369 BECKERS, J. M. 365, 369 BEDDOES, D. R. 357, 369 BELL, M. I. 172, 209 BELSHER, J. F. 351,352,353,354,358,371 BELVEDERE, G. 341, 369 BENDOW, B. 161, 209 BENNETT, B. I. 61, 135 BENNETT, H. E. 172, 181, 210 BENSON, H. J. 115, 135 BERAN, M. J. 227, 279 BERGMANN, S. M. 11, 40 BERNE, B. J. 156, 158, 209 BERTOLOITI, M. 308, 369 BEVINGTON, P. R. 177, 209 BILLMAN, J. 133, 135 BIRAUD, Y. G. 351, 369 BLAZIT, A. 357, 369 BLOCH, F. 16, 40 BOILEAU, E. 29, 42 BOLSTAD, J. 0. 308, 376 BORGNINO, J. 337, 363, 364, 365, 366, 369,

BORGNIS, F. E. 217, 279 BORN, M. 96, 135, 219, 228, 279 BORSTEL, G. 142, 209 BOURICIUS, G. M. B. 296, 334, 369, 370 BOYD, R. W. 301, 369 BOZEC, P. 308, 369 BRANDT, P. N. 364, 369 BRAY, R. 96, 98, 136 BRECKINRIDGE, J. B. 334, 355, 357, 365,

BRION, J. J. 155, 210 BRODY, E. 61, 96, 98, 135 BROWN, D. S. 309, 369 BROWNING, G. K. A. 291, 369 BRUCK, Y. M. 359, 360, 369 BRUNSCH, A. 116, 118, 136 BRYNGDAHL, 0. 277, 279 BUCHNER, S. 129, 135 BUFFINGTON, A. 322, 354, 370, 373

374

369

377

378 AUTHOR INDEX

BUFTON, J. L. 289, 336, 341, 343, 344, 370,

BLJLLOUGH, R. K. 23, 24, 40, 41, 42 BURKE, J. J. 155, 209 BURSHTEIN, A. I. 7, 8, 9, 30, 31, 32.40, 43 BURSTEIN, E. 56, 58, 60, 64, 67, 70, 78, 80,

81, 84, 85, 94, 95, 108, 109, 113, 115, 128, 129, 132, 133, 135, 136, 141, 155, 162, 163, 164, 166, 186, 191, 198, 202, 209, 210

372

BUSER, R. G. 296, 334, 370 BUTLER, H. E. 341, 370 BUTTERWECK, H. J. 222, 223, 252, 254,

268, 279

C CAGNET, M. 308, 369 CAMLEY, R. E. 115, 118, 120, 121, 122,

CARDONA, M. 156, 172, 209, 210 CARLSON, F. P. 277, 279 CARLSTEN, J. L. 11, 33, 40, 41 CARMICHAEL, H. J. 21, 27, 41 CASASENT, D. 278, 279 CASIMIR, H. B. G. 220, 279 CASTIEL, D. 120, 135 CELLI, V. 99, 136 CEPPATELLI, G. 290, 291, 341, 344, 365,

366, 369, 375 CHAMBERS, A. J. 287, 368 CHANDRASEKHAR, S. 291, 370 CHELLI, A. 314, 357, 358, 359, 370, 375 CHEN, C. Y. 162, 198, 209 CHEN, K. Y. 341, 374 CHEN, L. 198, 209 CHEN, W. 133, 135

123, 135

CHEN, Y. J. 78, 80, 81, 84, 85, 94, 95, 133, 13.5, 136, 162, 163, 186, 209

CHERNOV, L. A. 284, 291, 328, 370 CHING, L. Y. 129, 135 CHOPLIN, H. 356, 368 CITRON, M. L. 21, 41 CLARK, W. L. 291, 315 CLIFFORD, S. F. 289, 291, 296, 334, 335,

341, 344, 347, 366, 369, 370, 371, 372, 373, 375

COHEN-TANNOUDJI, C. 21, 22, 23, 28, 30, 34, 40, 41

COLLINS Jr., S. A. 289, 376 COOPER, J. 31, 32, 38, 39, 41, 42

CORRSIN, S. 284, 370 CORTEGGIANI, J. P. 356, 368 COWAM, M. G. 116, 117, 135, 137 COULMAN, C. E. 289, 365, 370 COUPINOT, G. 351, 370 COURTENS, E. 34, 41 COURTES, G. 283, 361, 365, 374 COWAN, J. J. 143, 210 CRAWFORD, F. S. 322, 354, 370, 373 CUOT, E. 356, 368 CURRIE. D. G. 306. 354. 355. 370

D DAGENAIS, M. 27, 41 DAINTY, J. C. 308,313,319,356,357,358,

360, 369, 370, 371 DAMEN, T. C. 172, 210 DAMON, R. W. 111, 135 DANJON, A. 365, 370 DAVIS, J. 354, 367, 370 DEMANGEAT, C. 120, 125, 135 DE MARTINI, F. 141, 209 DERVISCH, A. 96, 98, 135 DESCHAMPS, G. A. 276, 279 DEWAMES, R. 111, 137 DIL, J. 61, 96, 98, 135 DILLARD, M. 4, 41 DINGLE, R. 132, 137 DOBRZYNSKI, L. 101, 135 DOMMANCET, J. 283, 361, 365, 374 DRYDEN, G. 318, 324, 372 DUTHLER, C. J. 89, 135 DZYALOSHINSKI, I. E. 83. 84, 135

E EBERLY, J. H. 23, 24, 30, 40, 41 EHLOTZKY, F. 30, 43 EHN, D. C. 360, 375 ELLISON, M. A. 341, 370 ELSON, J. 155, 210 ENGLMAN, R. 162, 210 ESHBACH, J. R. 111, 135 EVANGELISTI, F. 113, 137, 145, 210 EVANS, A. R. 171, 210 EVANS, D. J. 94, 135, 162, 171, 180, 195,

EZEKIEL, S. 17, 20, 21, 30, 41, 43

F FALGE, H. J. 142, 209 FALICOV, L. M. 161, 209

198, 209

AUTHOR INDEX 379

FANTE, R. L. 359, 370 FAUST, W. L. 88, 93, 135 FAVIER, D. L. 362, 376

FEOKTISTOV, A. A. 7, 9, 10, 37, 38, 39,42 FERRELL, T. L. 200, 209 FIDDY, M. A. 360, 370 FIENUP, J. R. 360, 370 FISCHER, B. 143, 172, 209 FLEISCHMANN, M. 133, 135 FLEURY, P. 117, 136 FRANCOIS, R. E. 277, 279 FREEDHOFF, H. S. 27, 42 FRIED, D. L. 284, 298, 300, 301, 313, 323,

324, 332, 336, 338, 339, 343, 347, 352, 353, 356, 370, 371

FELDMAN, D. W. 52, 135, 136

FRIEHE, C. A. 287, 368, 371 FUCHS, R. 73, 75, 136, 152, 153, 209

G GABEL, C. W. 21, 41 GAGE, K. S. 291, 375 GAVIOLA, E. 341, 371 GAY, J. 356, 368 G E N A ~ , S. H. 336, 341, 343, 370 GEORGE, E. V. 33, 42 GEZARI, D. Y. 357, 371 GIBBS, H. M. 17, 41 GINZBURG, V. L. 113, 135, 162, 208 GLAUBER, R. J. 13, 14, 28, 29, 41 GOLDBERG, H. S. 131, 137 GOODMAN, M. F. 38, 41 GOODMAN, J. W. 214, 224, 246, 248, 267,

268, 271, 272, 277, 278, 279, 292, 294, 318, 319, 351, 352, 353, 354, 355, 358, 371, 374

GORKOV, L. P. 83, 84, 135 GORLITZ, D. 268, 279 GORNtK, E. 131, 136 Gossom, A. C. 132, 137 GOUBAU, G. 217, 279 GRAY, H. R. 21, 41 GREC, G. 354, 368 GREEN, J. L. 291, 375 GREENAWAY, A. H. 356, 357, 358, 360,

GRIFFIN, R. F. 284, 371 GRIMSDITCH, M. 116, 118, 136 GROVE, R. E. 17, 20, 21, 30, 41, 43 GRUNBERG, P. 116, 136

370, 371

GUSH, H. P. 16, 34, 41, 357, 371 GUSH, R. 16, 34, 41

H HAHN, E. H. 5, 42 HALL, J. S. 341, 349, 373 HAMM, R. N. 143, 210 HAMMER, P. J. G. 246, 280 HANBURY BROWN, R. 356, 371 HARDY, J. W. 353, 371 HARLAN, E. A. 362, 371 HARP, J. C. 284, 291, 292, 373 HARRICK, N. J. 142, 209 HARRIS, J. L. 350, 351, 371 HARTIG, W. 20, 21, 41 HARTMA", S. R. 5, 40 HARTSTEIN, A. 155, 202, 209, 210 HARVEY, J. W. 353, 354, 360, 371, 373 HASSAN, S. S. 23, 24, 41 HAWKINS, M. R. S. 351, 371 HAYES, W. 47, 58, 83, 86, 136, 156, 209 HECQUET, J. 351, 370 HEITLER, W. 28, 38, 41 HENDRA, P. 133, 135 HENRY, C. H. 87, 88, 93, 135, 136, 172,

HERCHER, M. 20,42 HERRING, C. 108, 136 HESS, S. L. 289, 371 HILL, R. J. 296, 371 HINES, J. R. 362, 376 HINZE, J. 0. 284, 371 HOAG, A. A. 341, 349, 373 HOBDEN, M. V. 172, 209 HOGGE, C. B. 323, 347, 366, 373 HOPFIELD, J. J. 65, 67, 87, 136, 172, 209 HOWELL, R. 354, 373 HUANG, K. 96, 135 HUBER, D. L. 31, 32, 39, 41 HUDGIN, R. H. 360, 375 HUFNAGEL, R. E. 284, 289, 290, 291, 298,

209

371. 372

I IBACH, H. 54, 136, 143, 209 IPATOVA, I. P. 64, 136

ISHIMARU, A. 291, 372 IYER, S . 343, 372 IZUMI, Y. 289, 376

IRWIN, J. W. 336, 364, 372

380 AUTHOR INDEX

J JAKEMAN, A. 345, 372 JAKEMAN, E. 341, 344, 345, 372 JEANMAIRE, D. L. 133, 136 JONES, R. C. 277, 280 JOURNET, A. 356, 368

K WIN, S. 307, 314, 357, 359, 363, 368,

KALLISTRATOVA, M. A. 289, 364, 372 KAMGAR, A. 131, 136 KAPANY, N. S. 155, 209 -0, D. P. 314, 322, 327, 351, 357, 358,

360, 363, 372, 374 KARPLus, R. 4, 10, 11, 32, 37, 41 KATAYAMA, S. 129, 136 KAWAMURA, H. 129, 136 KAZANTSEV, A. P. 20, 41 KELLEN, P. F. 364, 373 KELLER, G. 283, 291, 341, 347, 360, 362,

364, 369, 372, 375 KELSALL, D. 308, 372 KENKNIGHT., C. E. 355, 372 KIEPENHEUER, K. 0. 366, 372 KIMBLE, H. J. 23, 24, 25, 27, 30, 41, 42 KINAHAN, B. F. 354, 372 KING, J. R. 306, 308, 372 KING, W. T. 183, 210 KITTEL, C. 48, 59, 66, 77, 108, 136 KLEIN, M. V. 133, 137 KLIEWER, K. L. 73,75, 136, 152, 153,209 KNAPP, S. L. 306, 354, 355, 370 KNESCHAUREK, P. 131, 136

KNOX, K. T. 360, 372 KOCH, J. F. 131, 136 KOECHLIN, L. 357, 369 KOLCHINSKI, I. G. 336, 372 KOLMOGOROV, A. N. 284, 285, 372 KORFF, D. 284, 313, 318, 324, 372 KOVACS, G. 133, 135 KOZHEVNIKOV, N. I. 365, 372 KRASILNIKOV, V. A. 336, 372 KRETSCHMANN, E. 191, 192, 200, 209 KROGER, E. 191, 192, 209 KUCHEROV, N. I. 361, 372 KUPMLLER, K. 213, 280 KUPRADSE, W. D. 217, 280 KURNIT, N. A. 5, 40

372

KNIGHT, P. L. 23, 40

L LABEYRIE, A. 284, 319, 334, 354, 356, 357,

358, 371, 372 LANDAU, L. D. 53, 54, 77, 118, 136 LANZL, F. 268, 279 L A R ~ , J. C. 287, 371 LAWRENCE, R. S. 289, 291, 341, 344, 347,

366, 370, 372, 373 LA& M. 17, 41, 97, 136 LAZAY, P. D. 97, 136 LEAN, E. G. 98, 136 LEAVIIT, R. P. 324, 372 LEE, R. W. 284, 291, 292, 373 LEHMBERG, R. H. 16, 35, 41 LEITE, R. C. C. 172, 210 LEITH, E. N. 261, 280 LEMMETI, P. 366, 369 LENA, P. 314, 357, 358, 359, 370, 375 LEWER, K. M. 306, 354, 355, 370 LIFSHITZ, E. M. 53, 54, 77, 118, 136 LINFOOT, E. H. 284, 373 LISITSA, V. S. 36, 41 L m , C. G. 291, 373 LOHMANN, A. 235, 276, 278, 279, 280 Loos, G. C. 323, 347, 366, 373 LOUDON, R. 47, 58, 83, 84, 86, 96, 97, 98,

117, 135, 136, 155, 156, 161, 162, 163, 166, 173, 178, 208, 209

Low, F. J. 354, 373 LUKOSZ, W. 272, 280 LUMLEY, J. L. 285, 373, 375 LUND, G. I. 360, 369 LUNDQUIST, S. 198, 209 LUTOMIRSKI, R. F. 296, 373 LYNDS, C. R. 360, 373

M ~ L O Z E M O F F , A. 116, 118, 136 MANASEVIT, H. M. 180, 209 MANDEL, L. 23, 24, 25, 27, 30,41, 42, 227,

280 MARADUDIN, A. A. 56, 58, 60, 61, 64, 78,

96, 97, 98, 99, 100, 101, 115, 135, 136, 137, 150, 182, 191, 192, 202, 209

MARCUSE, D. 155, 209 MARSCHALL, N. 143, 172, 209 MARTIN, F. 337, 369 MARTIN, R. M. 60, 136, 161, 209 MARVIN, A. 99, 136 ~ T T E I , G. 88, 92, 137, 166, 194, 210

AUTHOR INDEX 381

MATTHIWE, P. 246, 280 MCALISTER, H. A. 357, 369 MCCALL, J. L. 5, 42 MCCARTHY, D. W. 354, 373 MCGLAMERY, B. L. 350, 351, 373 MCMULLEN, J. D. 94, 135, 155, 162, 171,

MCQUILLAN, A. J. 133, 135 MCWHORTER, A. L. 127, 136 MEINEL, A. B. 283, 361, 373 MENZEL, E. 215, 280 MERLIN, G. 356, 368 METAWE, F. 116, 136 METCALF, J. I. 289, 373 MICHELSON, A. A. 304, 354, 356, 373 MIKESELL, A. H. 341, 349, 373 MILLER, M. G. 318, 351, 358, 363, 364,

MILLER, M. M. 15, 16, 28, 31, 42 MILLS, D. L. 56, 58, 60, 64, 67, 70, 78, 80,

81, 84, 85, 94, 95, 99, 101, 108, 109, 111, 113, 115, 118, 120, 123, 125, 129, 132, 135, 136, 137, 150, 162, 163, 166, 182, 186, 188, 189, 191, 202, 209, 210

180, 209

372, 373

MILNER, M. 0. 360, 369 MINOIT, P. 0. 343, 373 MIRANDE, W. 215, 280 MISHRA, S. 96, 98, 136 MITCHELL, W. E. 341, 347, 369 MOLLOW, B. R. 5, 7, 8, 9, 10, 11, 13, 14,

15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27,28, 31,32, 33,34,37,38, 39,40,42

MONTROLL, E. W. 64, 136 MOORADIAN, A. 127, 136 MORGAN, B. L. 357, 369 MORODER, E. 290, 362, 369, 373 MUELLER, P. F. 350, 351, 373 MULLER, R. 352, 367, 373 MULLER, R. A. 354, 370, 373 MURASE, K. 129, 136 Mum, L. 308, 369

N NEW, W. D. 289, 373 NELSON, D. F. 97, 136 NETCLEBALD, F. 341, 373 NEWSTEIN, M. C. 7, 10, 11, 42 NISENSON, P. 322, 360, 373, 375 NIZZOLI, F. 96, 101, 105, 135 NKOMA, J. S. 82, 84, 136, 162, 163, 209

NOTKIN, G. E. 7, 9, 10, 37, 38, 39, 42

0 OBUKHOV, A. M. 284, 286, 287, 373 OCHS, G. R. 289, 291, 296, 334, 341, 344,

347, 366, 370, 372, 373

OMOW, A. 31, 32, 38, 39, 42 OLIVER, G. 21, 26, 42

O’NEILL, E. L. 214, 215, 265, 276, 280, 300, 373

ONETO, J. L. 357, 369 ORTH, c. D. 354, 370 07~0, A. 133,135,136,142,155,209,210

P PACINI, F. 356, 373 PANOFSKY, H. A. 285, 373 PAPAS, C. H. 217, 279 PAPOLILIS, A. 213, 236,246,270, 273, 279,

280 PARIS, D. P. 235, 279, 280 PARKER, J. H. 52, 135, 136 PARRY, G. 326, 341, 344, 345, 358, 369,

372, 373

366, 369, 373, 375 PATERNO, L. 290, 291, 341, 344, 349, 350,

PATTERSON, G. D. 40, 42 PEASE, F. G. 354, 356, 373

P E ~ I N A , J. 276, 280 PECORA, R. 156, 158, 209

PERRANT, G. B. 227, 279 PESKOFF, A. 347, 373 PIKE, E. R. 341, 344, 345, 372 PINCINBONO, B. 29, 42 PNCZUK, A. 128, 132, 135, 136, 137, 164,

166, 209 PLOOC, K. 131, 135 POLDER, D. 23, 42 POLLAINE, S. 322, 354, 373 PRIEUR, J.-Y. 180, 184, 210

PROTHEROE, W. M. 341,346,347,373,374 PSALTIS, D. 278, 279

PROSNITZ, D. 33, 42

PUSEY, P. N. 341, 344, 345, 372

Q QUEISSER, H. J. 143, 209

R RABBIA, Y. 356, 368, 374 RAIJO, G. T. 115, 125, 137

382 AUTHOR INDEX

RAETHER, H. 143, 210 RAHMAN, T. S. 111, 118, 120, 123, 132,

RASMUSSEN, W. 20, 21, 41 RAUTIAN, S. G. 6, 7, 8, 9, 10, 37, 38, 39, 42 RAYMER, M. G. 11, 33, 41 REIGER, S. H. 289, 344, 349, 374 RENAUD, B. 24, 42 RESSAYRE, E. 21, 26, 42 REYNAUD, S. 21, 22, 23, 34, 41 REYNOLDS, G. 0. 350, 351, 373 RHODES, W. T. 278, 280, 318, 374 RICHTER, W. 161, 210, 356, 373 RICORT, G. 314, 351, 354, 357, 359, 360,

363, 364, 365, 366, 367, 368, 369, 374 RIGHINI, A. 290, 291, 341, 344, 362, 365,

366, 369, 373, 375 RITCHIE, R. H. 143, 210 ROBINSON, W. G. 357, 369 ROBL, H. R. 4, 41 ROCCA, A. 341, 349, 366, 314 RODDIER, C. 308, 309, 310, 314, 328, 334,

135, 137

356, 357, 359, 360, 363, 366, 368, 370, 374

RODDIER, F. 308, 314, 318, 328, 341, 346, 347, 348, 349, 354, 356, 357, 358, 359, 360, 366, 368, 370, 374, 375

ROGER, G. 308, 369 ROSCH, J. 283, 340, 341, 350, 361, 364,

365, 367, 374 ROUSSEAU, D. L. 40, 42 ROWELL, N. 96, 105, 137 RUPPIN, R. 162, 210 RUSSELL, F. D. 318, 355, RUSSELL, J. P. 172, 209

374

S SANCHEZ MAGRO, C. 301, 357, 375 SANDERCOCK, J. R. 61, 95, 96, 97, 98, 105,

SANTORO, G. 96, 101, 105, 135 SAUNDERS, J . B. 355, 374 SAUNDERS, R. 23, 24, 42 SCADDAN, R. J. 308, 309, 319, 326, 344,

SCHIEDER, R. 20, 21, 41 SCHIRF, V. E. 360, 375 SCHNEIDERMAN, A. M. 314, 322, 327, 351,

SCHOENWALD, J. 155, 202, 209, 210

116, 117, 121, 135, 137, 167, 210

357, 369, 370, 373, 374

357, 358, 360, 363, 372, 374

SCHUDA, F. 20, 42 SCHUURMANS, M. F. H. 23, 42 SCHWEMIN, A. J. 354, 370 SCHWINGER, J. 4, 10, 11, 32, 37, 41 SCOTT, J. F. 172, 210 SCOTT, R. Q. 122, 135 SEAGAR, A. D. 360, 369 SEDDON, H. 341, 370 SEDMARK, G. 341, 375 SELBY, M. J. 301, 357, 375 SERAPHIN, B. 0. 172, 181, 210 SETTE, D. 308, 369 SHAPIRO, J. H. 323, 375 SIBLLE, F. 314, 357, 358, 359, 370, 375 SIEGERT, A. 16, 40 SILFVAST, W. T. 172, 210 SINGAL, S. P. 289, 375 SLATER, .I. C. 217, 280 SMITH, E. W. 31, 32, 38, 39, 42 SMITHERS, M. E. 27, 42 SMITS, R. G. 354, 370 SOBEL’MAN, I. I. 6, 7, 8, 9, 10, 37, 38, 42 SODIN, L. G. 359, 360, 369 SOKOLOVSKII, R. I. 13, 18, 42 SOMMERFELD, A. 221, 280 SPARKS, M. 89, 135 SPERONI, N. 290, 291, 341, 344, 366, 369,

STACHNIK, R. V. 322, 357, 360, 371, 373,

STANLEY, N. R. 284, 298, 372 STEGNAN, G . I. 96, 105, 137 STOCK, J . 283, 360, 362, 364, 375 STORMER, H. 132, 137 STRATTON, J. A. 219, 220, 280 STROHBEHN, J. W. 291,329,335,337, 344,

STROUD Jr., C. R. 20, 21, 24, 25, 41, 42 SUBBASWAMY, K. R. 84,85,96,97,98, 100,

101, 137, 166, 186, 188, 189, 210 SUDARSHAN, E. C. G. 14, 28, 42 S ~ N , E. C. 356, 375 SWAIN, S. 27, 42

SZIKLAS, E. 29, 35, 36, 42 SZOKE, A,, 11, 33, 34, 40, 41

T TAJIMA, T. 155, 210 TALLET, A. 21, 26, 42 TANGO, W. J. 354, 367, 370

375

375

347, 375

SWANSON, L. R. 61, 135

AUTHOR INDEX 383

TATARSKI, V. I. 284, 285, 287, 291, 328,

TENNEKES, H. 285, 375 TEXEREAU, J. 288, 375 THIELE, E. 38, 41

THORSEN, A. C. 180, 209 TIEN, P. K. 74, 137, 155, 210 TIKHONOV, N. A. 357, 369 TIMANOVSKIY, D. F. 289, 372 TING-I-WANG, 291, 341, 347, 366, 373 TIOGO, F. 99, 136 TOSSAITI, E. 198, 209 TOWNSEND, A. A. 346, 375 TSUI, D. C. 131, 136 TSVANG, L. R. 289, 375 TURK, R. A. 123, 137 TWE, R. N. 172, 209

U USCINSKI, B. J. 291, 375 USHIODA, S. 52, 85, 86, 88, 92, 94, 128,

135, 137, 155, 162, 164, 166, 171, 172, 180, 184, 194, 195, 198, 209, 210

336, 343, 344, 349, 375

THOMPSON, B. J. 360, 372

V VALDEZ, J. B. 73, 75,88,92, 137, 166, 171,

VALLEY, G. C. 296, 339, 375 VAN DER LUGT, A. 252, 268, 278, 280 VAN DWNE, R. P. 133, 136, 198, 210 VAN ISACKER, J. 291, 375 VAN SCHOONEVELD, C. 360, 375 VAN VALKENBURG, M. E. 218, 280 VAN VLECK, J. H. 31, 42 VAN WEERT, M. J. 219,220,221,233,238,

VANZANDT, T. E. 291, 375 VENKATESAN, T. N. C. 17, 41 VERNIN, J. 291, 314, 337, 341, 342, 345,

346, 347, 348, 349, 357, 359, 363, 364, 365, 366, 368, 369, 374, 375

172, 173, 194, 210

280

W WADE, R. 301, 357, 375 WALKER, J. G. 319, 326, 344, 358, 373,

WALKER, M. F. 362, 371 WALLIS, R. F. 53, 54, 137, 155, 202, 209,

WALLS, D. F. 21, 27, 41

374, 376

210

WALTERS, D. L. 362, 376 WALTHER, A. 272, 280 WALTHER, H. 20, 21, 41, 42 WANG, C. P. 324, 376 WARD, W. H. 171, 210 WATSON, G. N. 246, 280 WEIGELT, G. P. 352, 357, 360, 376 WEING~TNER, I. 215, 280 WEINSTEIN, B. A. 172, 210 WEISS, G. H. 64, 136 WEISSKOPF, V. 17, 28, 31, 42

WESELY, M. L. 289, 376 WESSELY, H. W. 308, 376 WETTLING, W. 116, 117, 121, 137 WHEELER, R. G. 131, 137 WHITLEY, R. M. 24, 42 WIEGMAN, W. 132, 137 WIERSMA, P. 268, 278, 279, 280 WIGEN, P. E. 123, 137 WIGNER, E. 17, 42 WILDMAN, D. W. 33, 42 WILLIAMS, M. D. 191, 201, 210 WILLIAMS, P. F. 40, 42 WILSON, R. N. 356, 373 WITCOMB, R. C. 284, 373 WOLF, E. 219, 227, 228, 276, 279, 280 WOLFRAM, T. 111, 137 WOOD, T. H. 133, 137 WORDEN, S. P. 359, 360, 373, 376 WORLOCK, J. M. 132, 137 Wu, F. Y. 17, 20, 21, 30, 41, 43 WYNGAARD, J. C. 289, 376

Y YAGLOM, A. M. 284, 286, 376 YAKOVLENKO, S. I. 36, 41 YOUNG, A. T. 289,298,329,332,341,343,

344, 349, 350, 376 Yu, J. T. 113, 123, 137 Yu, P. Y. 145, 210 YURA, H. T. 296, 373

Z ZELANO, A. J. 183, 210 ZERNIKE, F. 268, 280 ZHUKOVA, L. N. 341, 376 ZIERAU, W. 191, 209 ZIESKE, P. L. 351, 363, 373 ZOLLER, P. 30, 43 ZUSMAN, L. D. 31, 43

WELTER, G. L. 359, 376


SUBJECT INDEX

A E

adaptive optics, 352, 353 Airy pattern, 266 antibunching, photon, 27 apodization, 269, 270 autocorrelation function, 227, 269, 299,

321. 357

B

benzene, 184 Bessel function, 244 Bloch equations, optical, 19, 21, 22, 24-26,

Born approximation, 80 Bose-Einstein factor, 158 Brillouin scattering, 50, 59, 61, 95-97, 106,

- spectroscopy, 106, 167 - spectrum, 102-106, 119 - zone, 48, 49, 60, 157

29, 35-37, 39

118

C

coherence function, 228, 293-295 coherent state, 13, 14, 28 collisional relaxation, 31 communication theory, 213 convolution theorem, 255 Curie temperature, 117, 119

D

Einstein A-coefficient, 7, 15, 20, 22, 23, 31 - B-coefficient, 9, 28 elasticity theory, 112 electro-optic device, 214 evanescent wave, 142

F

Fabry-Perot interferometer, 49, 50 spectrometer, 106

Faraday rotator, 220 Fraunhofer diffraction, 246, 308 Fresnel approximation, 246, 294 - diffraction, 246, 294 - -Kirchhoff diffraction, 219 --zone plate, 279 fringe visibility, 304, 354

G

Gaussian random process, 30 - statistics, 319, 320, 324, 330, 359 Green’s function, 79 grounded glass, 276 guided waves, TM, 74, 86

_-_

H

Hankel transformation, 224, 244 Hartman test, 362, 364 Heisenberg picture, 23, 24 Helmholtz equation, 223 holography, computer, 279 homodyne detection, 356 Hufnagel,s model, 342, 344 Damon-Eshbach wave, 106, 11 1, 112, 114-

116, 118-120, 122, 123

I

illumination, coherent, 231, 268 -, incoherent, 230, 231, 234, 268 imaging, diffraction limited, 269, 272 impulse function, 213

Danjon’s scale, 362 data-processing, optical, 21 1. 216, 218 dielectric tensor, 79 dispersion relation, 69, 70, 72, 73, 150 Doppler broadening, 20 dressed atom method, 21

385

386 SUBJECT INDEX

interferometry, Michelson, 302, 304, 354.

-, speckle, 284, 327, 355, 357 irradiance distribution, 298, 305

355, 357, 358, 367

K

Kirchhoff-Huygens’ principle, 219 Kolmogorov law, 285, 286, 291, 330 Kramers-Kronig relation, 200

L

Larmor precession, 106 laser, 284 log-normal statistics, 324, 328, 344 Lorentzian, 60 Lukosz bound, 274 Lyddane-Sachs-Teller relation, 66

M

Maxwell’s equations, 65, 67, 68, 71, 72, 77, 78, 108, 109, 141, 145, 149, 164, 204, 216, 217, 220

multiphoton emission, 9

N

non-linear optics, 62

0

Obukhov’s law, 294 optical astronomy, 281, 283 - signal processors, 222

P

paraxial approximation, 221, 222, 237 Parseval’s theorem, 261 partial coherence, 227, 231, 276 phonons, 47, 49, 51, 53, 55 -, acoustical, 49, 59, 61, 95, 99 - branch, longitudinal, 48, 89, 93, 127-

129, 164, 166, 174, 175-178, 182, 183, 198

198 - branch, transverse, 48, 89, 92, 164, 166,

-, guided wave, 76, 85 -, optical, 49, 156, 161

-, surface, 54, 85, 92, 165 photoelastic constants, 96 photon echoes, 5 Pic du Midi observatory, 368 Pockels’ elasto-optic constants, 96 Poisson formula, 26 polariton, 47, 65

-,guided wave, 64,73, 152, 153-155, 185, 190, 194

-, surface, 64, 68-71, 89, 141, 142, 144- 148, 151, 155, 159-161, 171, 180, 195

power-broadening, 21 - spectrum, 228, 229, 23 1, 269, 286, 330,

Poynting vector, 101, 233 P-representation (see coherent state)

-, bulk, 141, 146, 147

350

R

Rabi modulation, 8 - nutation frequency, 3, 6 radiation-reaction theory, 23 Raman effect, 131, 161 - scattering, 50, 60, 92, 93, 95, 134, 143,

- spectroscopy, 134, 167, 169, 171 - spectrum, 85, 94 randomly fluctuating media, 214 Rayleigh scattering, 6 - wave, 54, 56, 61, 122 reciprocity theorem, 220 refractive index fluctuations, 287 regresion theorem, fluctuation, 17, 24-26,

resonance fluorescence, 1, 7 - light scattering, 1 , 4, 6, 20 Reynolds number, 285 Rice distribution, 345 rotating wave approximation, 3 rotation invariance, 242

155-162, 168, 179, 198, 201

32

S

sapphire, 180, 182, 184 Schrodinger equation, 130 - picture, 15, 24, 26 self-induced transparency, 5 size effects, 75, 95 skin depth, optical, 56, 62, 123, 159 speckle, 297. 311, 312, 313, 318, 324

SUBJECT INDEX 387

spectral density, 9, 10 T spectroscopy, light scattering, 139 spectrum, emission, 8, 10 -, scattering, 8, 50, 76 spin waves, 47, 50, 95, 105, 106 Stark effect, dynamical, 6, 19, 25 - splitting, 8, 11, 29 steller scintillation, 341, 344 structure function, 287, 288, 332, 333 surface effects, 75, 95 symmetry, rotation, 238 -, spatial, 234, 238, 242 -, spectral, 234, 238, 242 system function, 214

thin films, 47, 51, 62.64, 75, 98, 105, 118 transfer function, modulation, 271

, optical, 271, 298-300 turbulence, 288, 289 -, atmosphere, 281, 283, 284, 342, 343

W

Wiener-Khinchine theorem, 286 Wigner distribution function, 276

--

2

Zeeman field, 111, 112


CUMULATIVE INDEX - VOLUMES I-XM

A B E L ~ , F., Methods for Determining Optical Parameters of Thin Films ABELLA, I. D., Echoes at Optical Frequencies

ABITBOL, C. I., see J. J. Clair

AGARWAL, G. S., Master Equation Methods in Quantum Optics

AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion

ALLEN, L., D. G. C. JONES, Mode Locking in Gas Lasers

AMMANN, E. O., Synthesis of Optical Birefringent Networks

ARMSTRONG, J. A., A. W. SMITH, Experimental Studies of Intensity Fluctua-

ARNAUD, J. A,, Hamiltonian Theory of Beam Mode Propagation BALTES, H. P., On the Validity of Kirchhoffs Law of Heat Radiation for a

BARAKAT, R., The Intensity Distribution and Total Illumination of Aberration-

BASHKIN, S., Beam-Foil Spectroscopy

BECKMANN, P., Scattering of Light by Rough Surfaces

BERRY, M. V., C. UPSTILL, Catastrophe Optics: Morphologies of Caustics and

tions in Lasers

Body in a Nonequilibrium Environment

Free Diffraction Images

11,249

VII, 139

XVI, 71 XI, 1

IX, 235

IX, 179

IX, 123

VI. 21 1 XI, 247

XII, 1

I, 67 XII, 287

VI. 53

their Diffraction Patterns XVIII, 259 BEVERLY 111, R. E., Light Emission from High-Current Surface-Spark Dis-

BLOOM, A. L., Gas Lasers and their Application to Precise Length Measure-

BOUSOUET, P., see P. Rouard

BRUNNER, W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation

BRYNGDAHL, O., Applications of Shearing Interferometry BRYNGDAHL, O., Evanescent Waves in Optical Imaging

BURCH, J. M., The Metrological Applications of Diffraction Gratings

BLITTERWECK, H. J., Principles of Optical Data-Processing CAGNAC, B., see E. Giacobino CASASENT, D., D. PSALTIS, Deformation Invariant, Space-Variant Optical Pat-

CHRISTENSEN, J. L., see W. M. Rosenblum CLAIR, J. J., C. I. ABITBOL. Recent Advances in Phase Profiles Generation

charges

ments

tern Recognition

389

XVI, 357

IX, 1

IV, 145

xv, 1

IV, 37

XI, 167 11, 73

XIX, 211 XVII, 85

XVI, 289

XIII, 69 XVI, 71

390 CUMULATIVE INDEX

CLARRICOATS, P. J. B., Optical Fibre Waveguides-A Review

COHEN-TANNOUDJI, C., A. KASTLER, Optical Pumping COLE, T. W., Quasi-Optical Techniques of Radio Astronomy CREWE, A. V., Production of Electron Probes Using a Field Emission Source

CUMMINS, H. Z., H. L. SWINNEY, Light Beating Spectroscopy

DAINTY, J. C., The Statistics of Speckle Patterns

D ~ D L I K E R , R., Heterodyne Holographic Interferometry DECKER Jr., J. A,, see M. Hanvit

DELANO, E., R. J. PEGIS, Methods of Synthesis for Dielectric Multilayer Filters

DEMARIA, A. J., Picosecond Laser Pulses

DEXTER, D. L., see D. Y. Smith

DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers DUGUAY, M. A., The Ultrafast Optical Kerr Shutter EBERLY, J. H., Interaction of Very Intense Light with Free Electrons

ENNOS, A. E., Speckle Interferometry

FIORENTINI, A., Dynamic Characteristics of Visual Process FOCKE, J., Higher Order Aberration Theory

FRANCON, M., S. MALLICK, Measurement of the Second Order Degree of

FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical

Signals, Based on Use of the Prolate Functions

FRY, G. A., The Optical Performance of the Human Eye GABOR, D., Light and Information GAMO, H., Matrix Treatment of Partial Coherence

GHATAK, A. K., see M. S. Sodha GHATAK, A., K. THYAGARAJAN, Graded Index Optical Waveguides: A Review

GIACOBINO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy

GINZBURG, V. L., see V. M. Agranovich

GIOVANELLI, R. G., Diffusion Through Non-Uniform Media

GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the Diffrac-

GOODMAN, J. W., Synthetic-Aperture Optics GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emis-

HARWIT, M., J. A. DECKER Jr., Modulation Techniques in Spectrometry HELSTROM, C. W., Quantum Detection Theory

HERRIOTT, D. R., Some Applications of Lasers to Interferometry

HUANG, T. S., Bandwidth Compression of Optical Images

JACOBSSON, R., Light Reflection from Films of Continuously Varying Refrac-

Coherence

tion Theory of Elastic Waves

sion

tive Index JACQUINOT, P., B. ROIZEN-DOSSIER, Apodisation

XIV, 327

v , 1 XV, 187

XI, 223 VIII, 133

XIV, 1 XVII, 1

XII, 101

VII, 67

IX, 31

X, 165 XII, 163

XIV, 161 VII, 359

XVI, 233 I, 253

IV, 1

VI, 71

IX, 311

VIII, 51 I, 109

111, 187 XIII, 169

XVIII, 1

XVII, 85

IX, 235

11,109

IX, 281

VIII, 1

XII, 233 XII, 101

X, 289

VI, 171

x , 1

V, 247 111, 29

JONES, D. G. C., see L. Allen IX, 179

CUMULATIVE INDEX

KASTLER, A,, see C. Cohen-Tannoudji KINOSITA, K., Surface Deterioration of Optical Glasses

KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open

KOTTLER, F., The Elements of Radiative Transfer

KOITLER, F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory

KOITLER, F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory

KUBOTA, H., Interference Color LABEYRIE, A,, High-Resolution Techniques in Optical Astronomy LEAN, E. G., Interaction of Light and Acoustic Surface Waves

LEE, W.-H., Computer-Generated Holograms: Techniques and Applications LEITH, E. N., J. UPATNIEKS, Recent Advances in Holography LETOKHOV, V. S., Laser Selective Photophysics and Photochemistry

LEVI, L., Vision in Communication

LIPSON, H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch

MALLICK, S., see M. Franeon MANDEL, L., Fluctuations of Light Beams

MANDEL, L., The Case for and against Semiclassical Radiation Theory MARCHAND, E. W., Gradient Index Lenses

MEESSEN, A,, see P. Rouard

MEHTA, C. L., Theroy of Photoelectron Counting MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Quasi-Classical Theory of Laser

MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction

MILL% D. L., K. R. SUBBASWAMY, Surface and Size Effects on the Light

MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design

MOLLOW, B. R., Theory of Intensity Dependent Resonance Light Scattering

MURATA, K., Instruments for the Measuring of Optical Transfer Functions

MUSSET, A,, A. THELEN, Multilayer Antireflection Coatings

OKOSHI, T., Projection-Type Holography OOUE, S., The Photographic Image

PAUL, H., see W. Brunner PEGIS, R. J., The Modern Development of Hamiltonian Optics

PEGIS, R. J., see E. Delano

PERINA, J., Photocount Statistics of Radiation Propagating through Random

Resonators

of Physical Optics

Radiation

Scattering Spectra of Solids

and Resonance Fluorescence

and Nonlinear Media PERSHAN, P. S., Non-Linear Optics PETYKIEWICZ, J., see K. Gniadek

PICHT, J., The Wave of a Moving Classical Electron

391

v , 1 IV, 85

VII, 1 111, 1

IV, 281

VI, 331

1,211 XIV, 47

XI, 123 XVI, 119

VI, 1

XVI, 1 VIII, 343

V, 287

VI, 71

11, 181 XIII. 27

XI, 305

x v , 77

VIII, 373

VII, 231

XVII, 279

XIX, 43 I, 31

XIX, 1

V, 199

VIII, 201 XV, 139

VII, 299 x v , 1

1, 1 VII, 67

XVIII, 129 V, 83

IX, 281 V, 351

392 CUMULATIVE INDEX

PSALTIS, D., see D. Casasent RISEBERG, L. A., M. J. WEBER, Relaxation Phenomena in Rare-Earth

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

ROUARD, P., P. BOUSQUET, Optical Constants of Thin Films

ROUARD, P., A. MEESSEN, Optical Properties of Thin Metal Films

RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave

RUDOLPH, D., see G. Schmahl SAKAI, H., see G. A. Vanasse

SCHMAHL, G., D. RUDOLPH, Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence between Coherence

Properties of Light and Nonlinear Optical Processes SCHULZ, G., J. SCHWTDER, Interferometric Testing of Smooth Surfaces

SCHWIDER, J., see G. Schulz

SCULLY, M. O., K. G. WHITNEY, Tools of Theoretical Quantum Optics SENITZKY, I. R., Semiclassical Radiation Theory within a Quantum-Mechanical

SIPE, J. E., see J. Van Kranendonk S ~ I G , E. K., Elastooptic Light Modulation and Deflection

SLUSHER, R. E., Self-Induced Transparency SMVIITH, A. W., see J. A. Armstrong

SMITH, D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in

S M ~ , R. W., The Use of Image Tubes as Shutters SODHA, M. S., A. K. GHATAK, V. K. TRIPATHI, Self Focusing of Laser Beams in

STEEL, W. H., Two-Beam Interferometry STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere

STROKE, G. W., Ruling, Testing and Use of Optical Gratings for High-

SUBBASWAMY, K. R., see D. L. MILLS

SVELTO, O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser

SWINNEY, H. H., see H. 2. Cummins TANGO, W. J., R. Q. TWISS, Michelson Stellar Interferometry TATARSKII, V. I., V. U. ZAVOROTNYI, Strong Fluctuations in Light Propagation

Luminescence

Aberration Measurements of the Human Eye

Framework

Insulators

Plasmas and Semiconductors

Resolution Spectroscopy

Beams

XVI, 289

XIV, 89 VIII, 239

XIX, 281

111, 29

XIII, 69 IV, 145

XV, 77

IV, 199 XIV, 195

VI, 259

XIV, 195

XVII, 163

XIII, 93

XIII, 93 X, 89

XVI, 413 XV, 245

X, 229 XII, 53

VI, 211

X, 165

x , 45

XIII, 169

V, 145 IX, 73

11, 1 XIX, 43

XII, 1

VIII, 133 XVII, 239

in a Randomly Inhomogeneous Medium

TAYLOR, C. A., see H. Lipson XVIII, 207

V, 287

CUMULATIVE INDEX 393

TER-MIKAELIAN, M. L., see A. L. Mikaelian

THELEN, A., see A. Musset

THOMPSON, B. J., Image Formation with Partially Coherent Light THYAGARAJAN, K., see A. Ghatak

TRIPATHI, V. K., see M. S. Sodha

TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations

TWISS, R. Q., see W. J. Tango

UPATNIEKS, J., see E. N. Leith UPSTILL, C., see M. V. Berry USHIODA, S., Light Scattering Spectroscopy of Surface Electromagnetic Waves

VANASSE, G. A,, H. SAKAI, Fourier Spectroscopy

VAN HEEL, A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE, Foundations of the Macroscopic Elec-

tromagnetic 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

WOLTER, H., On Basic Analogies and Principal Differences between Optical

WYNNE. C. G., Field Correctors for Astronomical Telescopes

YAMAJI, K., Design of Zoom Lenses

YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interfer-

YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Tech-

ZAVOROTNYI, V. U., see V. I. Tatarskii

and by Spatial Frequency Filtering

in Solids

and Electronic Information

ence Microscopy

niques

VII, 231

VIII, 20 1

VII, 169

XVIII, 1

XII1, 169

11, 131

XVII,239 VI, 1

XVIII, 259

XIX, 139

VI, 259

I, 289

XV, 245 XIV, 245

XIV, 89 IV, 241

XIII, 267 XVII, 163

X, 89

I, 155 X, 137

VI, 105

VIII, 295

XI, 77 XVIII, 207


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