Progress in Optics, Vol. 20

PROGRESS IN OPTICS VOLUME X X EDITORIAL ADVISORY BOARD L. ALLEN, Brighton, England M. FRANCON, Paris, France E. I...

Author: Emil Wolf

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PROGRESS IN OPTICS VOLUME X X

EDITORIAL ADVISORY BOARD L. ALLEN,

Brighton, England

M. FRANCON,

Paris, France

E. INGELSTAM,

Stockholm, Sweden

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, Fed. Rep. Germany

M. MOVSESSIAN,

Armenia, U.S.S.R.

G.

Berlin, D.D.R.

SCHnZ,

W. H. STEEL,

Sydney, Australia

J . TSUJIUCHI,

Tokyo, Japan

w. T. WEWORD,

London, England

PROGRESS I N OPTICS VOLUME XX

EDITED BY

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

Contributors G. COURTES, P. CRUVELLIER. M. DETAILLE, M. SAPSSE, C. FROEHLY, B. COLOMBEAU, M. VAMPOUILLE, S. KIELICH, P. HARIHARAN, W. JAMROZ, B. P. STOICHEFF

1983

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM * NEW YORK . OXFORD

NORTH-HOLLANDPUBLISHING COMPANY-I 983

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 61-19297

PUBLISHERS:

NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM . OXFORD

SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

CONTENTS OF VOLUME I (1961) 1-29 THE MODERNDEVELOPMENTOF HAMILTONIAN Omcs. R . J . PEGIS . . . I. I1 . WAVE Omcs AND GEOMETRICALOmcs IN OFTICAL DESIGN. K . 3 1-66 MIYAMOM . . . . . . . . . . . . . . . . . . . . . . . . . . . AND TOTAL ILLUMINATION OF ABERRATIONI11. THEINTENSITYDISTRIBLITION FREEDIFFRACTION IMAGES,R . BARAKAT. . . . . . . . . . . . . . 67-108 IV . LIGHTAND INFORMATION. D . GABOR . . . . . . . . . . . . . . . . 109-153 AND PRINCIPAL DIFFERENCESBETWEEN OPTICAL V . ON BASICANALOGIES 155-210 AND EtECl'RONIC INFORMATION H . WOLTER . . . . . . . . . . . . 211-251 COLOR.H . KUBOTA . . . . . . . . . . . . . . . . VI . INTERFERENCE . . . 253-288 CHARACEIUSTICSOF VISUAL PROCESSES. A . RORE"I VII . DYNAMIC DEVICES.A . C. S.VAN HEEL . . . . . . . . . . 289-329 VIII . MODERNALIGNMENT

.

CONTENTS O F VOLUME I 1 (1963) I.

RULING.TESTINGAND USEOF O ~ C A GRATINGS L FOR HIGH-RESOLUTION 1-72 SPECTROSCOPY. G . w. STROKE . . . . . . . . . . . . . . . . . . APPLICATIONS OF DIFFRACTION GRATINGS.J . M. I1. THE METROLOGICAL 73-108 BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I11. DIFFUSIONTHROUGH NON-UNIFORM MEDIA.R . G . GIOVANELLI . . . . 109-1 29 IV . CORRECTION OF O ~ C A IMAGES L BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING.J . TWIIUCHI . . . . . . . . . . . 131-180 OF LIGHT BEAMS. L. -EL . . . . . . . . . . . . 181-248 V . FLUCTLIATIONS OPTICAL PARAMETERS OF THINFILMS.F. VI . METHODS FOR DETERMINING ABELES

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

249-288

CONTENTS O F VOLUME I11 (1964) I. I1. 111.

I. I1.

I11.

IV . V. VI . VII .

THEELEMENTS OF RADIATIVE TRANSFER. F. K ~ . .R . . . . . . APODISATION. P . JACQUINOT. B . ROIZEN-DOSSIER . . . . . . . . . . MATRIXTREATMENT OF PARTIAL COHERENCE. H. GAMO . . . . . . .

CONTENTS OF V O L U M E IV (1965) HIGHERORDERABERRATION THEORY. J . FOCKE . . . . . . . . . .

APPLICATIONS OF SHEARINGINTERFEROMETRY. 0. BRYNGDAHL . . . . OF OPTICALGLASSES.K . KINOSITA . . . . . SURFACEDETERIORATION OF THINFILMS. P.ROUARD. P . BOUSQUET . . . . O ~ C A CONSTANTS L THE MIYAMOM-WOLF DIFFFWXON WAVE.A . RUBINOWICZ . . . . . ABERRATION THEORY OF GRATINGSAND GRATINGM0U"GS. W . T. WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . . DIFFRACTION AT A BLACK SCREEN.PART I: KIRCHHOFF'S THEORY. F. KOTIZER . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CONTENTS O F VOLUME V I. OPTICALPUMPING. C . COHEN.TANNOUDJI. A . KASTLER. Omcs. P . S. PEFISHAN. . . . . . . . . I1 . NON-LINEAR 111. TWO-BEAM INTERFEROMETRY. W . H . STEEL . . . . . . V

(1966) . . . . . . . . . . . . . . . . . . . . .

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 .

INSTRUMENTS FOR THE MEASURING OF O ~ C ATRANSFER L FUNCTIONS. K. 199-245 MURATA . . . . . . . . . . . . . . . . . . . . . . . . . . . . v . LIGHTREFLECTION FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE 247-286 INDEX. R . JACOBSSON . . . . . . . . . . . . . . . . . . . . . . DETERMINATION AS A BRANCH OF PHYSICAL VI . X-RAY CRYSTAL-STRUCTURE Oprrcs. H . LIPSON.C. A . TAYLQR. . . . . . . . . . . . . . . . . 287-350 CLASSICAL ELECTRON. J . PICHT . . . . . . . 351-370 VII . THEWAVEOF A MOVING

CONTENTS OF V O L U M E V I (1967) RECENTADVANCES IN HOLOGRAPHY. E . N . LEITH.J . UPATNIEKS 1-52 . . . . I. 53-69 I1. S C A m G OF LIGHTBY ROUGHSURFACES. P. BECKMA" . . . . . . OF THE SECOND ORDER DEGREEOF COHERENCE.M. 111. MEASUREMENT FRANC ON,^. MALLICK. . . . . . . . . . . . . . . . . . . . . . 7 1-104 OF ZOOMLENSES.K . YAMAJI . . . . . . . . . . . . . . . . 105-170 IV . DESIGN V . SOMEAPPLICATIONS OF LASERSTO INTERFEROMETRY. D . R . HERRIOTT . 171-209 STUDIES OF INTENSITYFLUCIUATIONS IN LASERS.J . A . VI . EXPERIMENTAL W T R O N G . A . w . SMITH 211-257 .................... G . A . VANASSEAND H . SAKAI . . . . . . . . 259-330 VII. FOUIUERSPECTROSCOPY. AT A BLACK SCREEN. PART 11: ELECTROMAGNETIC THEORY. VIII . DIFFRACTION F.KOTIZER . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1-377

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

MULTIPLE-BEAM INTERFERENCE AND NATURALMODESIN OPEN RES1-66 ONATORS. G . KOPPELMAN ..................... MULTILAYER FILTERS.E . DEL11. METHODS OF SYNTHESISFOR DIELECTRIC 67-137 ANO. R . J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . . . L I . D . ABELLA . . . . . . . . . . 139-168 111. ECHOES AT O ~ C AFREQUENCIES. 169-230 W~THPARTIALLY COHERENT LIGHT.B . J . THOMPSON IV . IMGE FORMATION A . L. MIKAELIAN. V. QUASI-CLASSICALTHEORY OF LASERRADIATION. 231-297 M. L.TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . 299-358 VI . THEPHOTOGRAPHICIMAGE. S. OOUE . . . . . . . . . . . . . . . J . H. VII . INTERACTION OF VERY INTENSELIGHTw r r ~FREE ELECTRONS. 359-415 EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 Omrcs. J . W. GOODMAN . . . . . . . . . THE OFTICAL PERFORMANCE OF THE HUMAN EYE.G. A . FRY . . . . .

1-50 I. 51-131 I1. H . L . SWINNEY . . . . 133-200 SPECTROSCOPY. H . Z . CUMMINS. 111. LIGHTBEATING COATINGS. A . MUSSET.A . THELEN. . . 201-237 IV . MULTILAYERANTIREFLECTION PROPERTIES OF LASERLIGHT.H . RISKEN . . . . . . . . 239-294 V . STATISTICAL THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE VI . COHERENCE MICROSCOPY.T. YAMAMOTO. . . . . . . . . . . . . . . . . . . 295-341 L . LEVI . . . . . . . . . . . . . . . . 343-372 VII . VISIONIN COMMUNICATION. OF PHOTOELECTRON COUNTING. C. L. MEHTA . . . . . . . . 373-440 VIII. THEORY

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

GASLASERSAND THEIR APPLICATION TO PRECISELENGTHMEASUREMENTS.

A . L . BLOOM .

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

1-30

.

31-71 PICOSECOND LASERPULSES,A. J. DEMARIA . . . . . . . . . . . 111. OPTICAL PROPAGATION %ROUGH THE TURBULENT ATMOSPHERE,1. 73-122 STROHBEHN . . . . . . . . . . . . . . . . . . . . . . . IV. SYNTHESIS OF OPTICAL BIREFRINGENT NETWORKS, E. 0.AMMA" . . . 123-177 V. MODELOCmw IN GASLASERS,L. ALLEN,D. G . C. JONES . . . . . . 179-234 VI. CRYSTALOPTICS WITH SPATIALDISPERSION, v. M. AGRANOVICH,v. L.GINZBURG . . . . . . . . . . . . . . . . . . . . . . . 235-280 VII. APPLICATIONS OF OFTICAL METHODSIN THE DIFFRACTIONTHEORY OF ELASTICWAVES,K. GNIADEK, J. PETYKIEWIU. . . . . . . . 281-310 VIII. EVALUATION,DESIGNAND EXTRAPOLATIONMETHODSFOR O m c ~ SIGNALS,BASEDO N USE OF THE PROLATEFUNCTIONS, B. R. FRIEDEN . 3 11-407 11.

..

w. .

.

. ..

..

. .

.

C O N T E N T S OF V O L U M E X (1972) BANDWIDTH COMPRFSSIONOF OPTICAL IMAGES, T. S. HUANG . . . . . THE USEOF IMAGETusm AS SHU~TERS, R. W. SMITH . . . . . . . . TOOLS OF THEORETICALQ U A N T UOPTICS, M M. 0. SCULLY, K. G. WHITNEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. FIELD C O R R E W R S FOR ASTRONOMICAL TELESCOPES,c . G. WY"E . . O ~ C A ABSORPTION L STRENGTHOF DEFECKS IN INSULATORS, D. Y. V. SIVIITH, D. L. DEXTER.. . . . . . . . . . . . . . . . . . . , . . VI. ELASTOOPTIC LIGHTMODULATION AND DEFLECTION, E. K. S ~ G ... VII. QUANTUM DETECTIONTHEORY, C. W. HELSTROM . . . . . . . . . . I. 11. 111.

1-44 45-87 89-135 137-1 64 165-228 229-288 289-369

C O N T E N T S OF V O L U M E X I (1973) MASTEREQUATION METHODS IN QUANTUM OPTICS, G . S. AGARWAL . . 1-76 RECENT DEVELOPMENTS IN FARINFRARED SPECTROSCOPIC ~ C H N I Q U E S , H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . 77-122 111. INTERACTION OF LIGHTAND ACOUSTIC SURFACEWAVES,E. G. LEAN . 123-166 IV. EVANESCENT WAVESIN O m c h IMAGING, 0. BRYNGDAHL . . . . . . 167-22 1 V. PRODUCTION OF ELECTRON PROBES USINGA FIELDEMISSION SOURCE,A. v.cREwE . . . . . . . . . . . . . . . . . . . . . . . . . . . 223-246 VI. HAMILTONIAN THEORY OF BEAMMODEPROPAGATION, J. A. ARNAUD . 247-304 VII. GRADIENT INDEXLENSES,E. W. MARCHAND . . . . . . . . . . . . 305-337 I. 11.

.

.

C O N T E N T S O F V O L U M E XI1 (1974) I. 11. 111.

SELF-FOCUSING,SELF-TRAPPING, AND SELF-PHASE MODULATION OF LASERBEAM,0. SVELW . . . . . . . . . . . . . . . . . . . . SELF-INDUCED TRANSPARENCY, R. E. SLUSHER . . . . . . . . . . . MODULATION TECHNIQUES IN SPECTROMETRY, M. HARWIT, J. A. DECKER

JR. IV. V.

VI.

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

INTERACITON OF

1-51 53-100 101- 162

LIGHT WITH MONOMOLECULAR DYE LAYERS, K. H.

163-232 DREXHAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . % PHASE TRANSITION CONCEPTAND COHERENCEIN ATOMICEMISSION, 233-286 R . G w . . . . . . . . . . . . . . . . . . . . . . . . . . BEAM-Fon SPECTROSCOPY, s. BASHKIN. . . . . . . . . . . . . . . 287-344

CONTENTS OF VOLUME XI11 (1976) I.

ONTHE VALIDWOF KIRCHHOFFSLAWOF HEATRADIATION FOR A BODY IN A NONEQUILIBNUM ENVIRONMENT, H. P. BALTES . . . . . . . . .

1-25

11.

THE CASE FORAND AGAINST SEMICLASSICAL RADIATION THEORY, L. 27-68 MANDEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . OF 111. OBJECTIVE AND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS 69-91 THE HUMAN EYE,W. M. ROSENBLUM, J. L. CHRISTENSEN. . . . . . . OF SMOOTH SURFACES, G. SCHULZ, J. IV. IN’IERFEROMETRIC TESTING 93-1 67 SCHWIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . V. SELF FOCUSING OF LASERBEAMS IN PLASMAS AND SEMICONOUCTORS, M. S. SODHA,A. K. GHATAK, V. K. TRIPATHI . . . . . . . . . . . 169-265 v1. APLANATISM AND ISOPLANATISM, w . T. WELFORD . . . . . . . . . . 267-292

C O N T E N T S OF V O L U M E XIV (1977) I. 11. 111.

.D W . . . . . . . . . OPTICAL ASTRONOMY, A. LABEYRIE . RARE-EARTHLUMINESCENCE, L. A.

THE STATISTICS OF SPECKLEPATERNS, J. c

HIGH-RESOLUTION TECHNIOLJES RELAXATION

PHENOMENA

IN

IN

RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . IV. THE ULTRAFAST OPTICALKERRSHU~TER, M. A. DUGUAY . . . . . . . V. HOLOGRAPHIC DIFFRACTIONGRATINGS, G. SCHMAHL, D. RUDOLPH . . VI. PHOTOEMISSION, P. J. VERNIER . . . , . . . . . . . . . . . . . . REVIEW,P. J. B. CLARRICOATS . . . VII. OPTICALFIBREWAVEGUIDES-A

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

C O N T E N T S OF V O L U M E XV (1977) I.

u. 111. IV. V.

THEORY OF OPTICAL PARAMETRICAMF-LEICATION AND OSCILLATION, W. BRU”ER,H.PAUL . . . . . . . . . . . . . . . . . . . . . . OPTICALPROPERTIES OF THINMETALFILMS, P. ROUARD,A. MEESSEN . PROJECI’ION-mEHOLOGRAPHY, T. OKOSHl . . . . . . . . . . . . QUASI-OITTCAL TECHNIQUES OF RADIOASTRONOMY, T. W. COLE . . . FOUNDATTONS OF THE MACROSCOPIC ELECTROMAGNETIC THEORYOF DIELECIWCMEDIA, J. VAN KRANENDONK,J. E. SmE . . . . . . . .

.

.

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

C O N T E N T S OF V O L U M E X V I ( 1 9 7 8 ) I.

11.

LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY. V. S. LETOKHOV . . . . . . . . . . . . . . . . . . . . . . . . . . . RECENT ADVANCES IN PHASE P R o m ~ sGENERATION,J. J. CLAIR,C. I. ABrnOL

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

1-69 71-117

COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS, W.-H. LEE . . . . . . . . . . . . . . . . . . . . . . . . . . . 119-232 A. E. ENNOS . . . . . . . . . . . . . . 233-288 IV. SPECKLEINTERFEROMETRY, DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATTERN RECOGNIV. TION, D. CASASENT, D. PSALTIS . . . . . . . . . . . . . . . . . . 289-356 VI. LIGHT EMISSION FROM HIGH-CURRENT SURFACE-SPARK DISCHARGES, R. E. BEVERLY111 . . . . . . . . . . . . . . . . . . . . . . . . 357-411 VII. SEMICLASSICAL RADIATION THEORY WITHIN A QUANTUM-MECHANICAL 413-448 FRAMEWORK. I. R. SENITZKY . . . . . . . . . . . . . . . . . . 111.

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C O N T E N T S OF V O L U M E X V I I (1980) 1.

11.

HETERODYNE HOLOGRAPHICINTERFEROMETRY, R. DANDLIKER . . . . DOPPLER-FREEMULTIPHOTON SPECTROSCOPY, E. GIACOBNO,B. CAGNAC

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

1-84 85-161

111. IV. V.

THE MUTUAL DEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR O ~ C APROCESSES, L M. SCHUBERT, B. WILHELMI . . . 163-238 MICHELSON STELLAR INTERFEROMETRY, W. J. TANGO, R. Q. Twss . . . 239-277 SELF-FOCUSING MEDIAWITH VARIABLEINDEXOF REFRACTION, A. L. MIKAEL~AN. . . . . . . . . . . . . . . . . . . . . . . . . . . 219-345

CO N T E N T S OF V O L U M E XVIII (1 9 8 0 ) I. 11. 111.

IV.

GRADEDINDEXOWICALWAVEGUIDES: A REVIEW,A. GHATAK,K. THYAGARAJAN .......................... 1-126 PHOTOCOUNT STATISTICS OF RADIATION PROPAGATING THROUGH RANDOM AND NONLINEARMEDIA, J. P E ~ I N A.. . . . . . . . . . . . . . 127-203 IN LIGHT PROPAGATION IN A RANDOMLY STRONGFLUCTUATIONS INHOMOGENEOUS MEDIUM,V. I. TATARSKII, V. U. ZAVOROTNYI . . . . 205-256 CATASTROPHE ~ P ~ I cMORPHOLOGIES s: OF CAUSTICS AND THEIR D i m c TIONPATIERNS, M. V. BERRY,C. UPSTILL.. . . . . . . . . . . . . 251-346

CO N T E N T S OF V O L U M E XIX (1981) INTENSITY DEPENDENT RESONANCE LIGHTSCATI'ERING AND RESONANCE FLUORESCENCE, B. R. MOLLOW . . . . . . . . . . . . . 1-43 11. SURFACE AND SIZE EFFECTS ON THE LIGHTSCATTERING SPECTRA OF SOLIDS, D. L. MILLS,K. R. SUBBASWAMY . . . . . . . . . . . . . . 45-137 OF SURFACE ELECTROMAGNETIC 111. LIGHT SCATTERINGSPECTROSCOPY WAVESIN SOLIDS,S. USHIODA. . . . . . . . . . . . . . . . . . . 139-210 OF OPTICAL DATA-PROCESSING, H. J. BUTIERWECK . . . . . 211-280 Iv. PRINCIPLES OF ATMOSPHERIC TURBULENCE IN OPTICAL ASTRONOMY, V. THE EFFECTS F. RODDLER. . . . . . . . . . . . . . . . . . . . . . . . . . . 281-376 I.

THEORY OF

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PREFACE When the first volume of PROGRESS IN OPTICS was published in 1961 it would not have been easy to predict that optics would soon become one of the most dynamic of sciences. To-day we are happy to be presenting the twentieth volume of this series with the knowledge that these volumes reflect some of the exciting developments that have taken place in optics in recent years. The credit for the success of this undertaking must primarily go to the authors of the more than 120 articles that have appesred in these volumes to-date. To them, as well as to all the past and present members of the Editorial Advisory Board who rendered much helpful advice I wish to express, on this special occasion, my warmest thanks. I also wish to acknowledge my indebtedness to h4rs. Ruth F. Andrus for valuable editorial assistance that she has very ably provided for more than fifteen years. My thanks go also to Dr. M. S. Zubairy for preparing, with much patience, the subject indices for many of the volumes. The quality of production of this series reflects the high standards that one associates with the North-Holland Publishing Company, whose fine cooperation, over many years, I gratefully acknowledge. I am particularly indebted to Drs. Catharina Korswagen of their editorial staff for her unfailing help. Finally, I wish to express my appreciation to the many reviewers of these volumes for the fine reception that they accorded this series. EMILWOLF

Department of Physics and Astronomy University of Rochester Rochester, N Y 14627 February 1983

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CONTENTS I . SOME NEW OPTICAL DESIGNS FOR ULTRA-VIOLET BIDIMENSIONAL DETECTION OF ASTRONOMICAL OBJECTS by G . Corn*.

P . CRUVELLIER.M. DETAILLE and M. SAISSE (MARSEILLE. FRANCE)

1. INTRODUCTION .................... ......... 1.1 Spectral analysis of the sources . . . . . . . . . . . . . . . . . . . . 1.2 Bidimensional information . . . . . . . . . . . . . . . . . . . . . . 1.3 First approaches to bidimensional information . . . . . . . . . . . . . . . . . . .. . . . . . . . . 1.4 Direct images of the sky in W radiation 2. W PHOTOGRAPHIC SURVEYS USINGWIDEF r u ~ CAMERAS . . . . . . . . . 2.1 Early wide field cameras . . . . . . . . . . . . . . . . . . . . . . . 2.2 The very wide field camera (1 ES 022 in Spacelab-1) . . . . . . . . . . 2.2.1 Scientificprogram . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The detector performance . . . . . . . . . . . . . . . . . . . 2.2.3 Compromises to accomplish the scientificprogram . . . . . . . . . 2.2.4 Geometrical conception of the camera . . . . . . . . . . . . . . 2.2.5 Photometric properties of the imagery mode of the VWFC . . . . . 2.2.6 Main characteristicsof the VWFC . . . . . . . . . . . . . . . . 2.3 The nebular spectrograph (NS) of the VWFC . . . . . . . . . . . . . 2.3.1 Scientificobjectives . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Limitation of the spectral range . . . . . . . . . . . . . . . . . 2.3.3 Optical design . . . . . . . . . . . . . . . . . . . . . . . . 3. AVERAGEFIELDTELESCOPES . . . . . . . . . . . . . . . . . . . . . . 3.1 Early telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The FAUST telescope on board Spacelab-1 (1 NS 05) . . . . . . . . . . 3.3 Three-mirror anastigmat 40 cm diameter telescope solution (TMA-1000) . 3.4 Geneva-Marseille W balloon program . . . . . . . . . . . . . . . . 3.4.1 Balloon experiment . . . . . . . . . . . . . . . . . . . . . . 3.4.2 The SCAP telescope . . . . . . . . . . . . . . . . . . . . . 3.4.3 The siderostat . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Optical design of FOCA-1000: astronomical telescope for balloon observations . . . . . . . . . . . . . . . . . . . . . . . . . 4 . W SPACE TELESCOPES OF THE FUTURE . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The space Schmidt telescope (SST) . . . . . . . . . . . . . . . . . . 4.2.1 Description of the optical design . . . . . . . . . . . . . . . . 4.3 The space telescope (ST) ...................... 4.3.1 The Wide Field and Planetary Camera (WF/PC) of the ST: The radial-bay instrument . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Faint Object Camera (FOC) . . . . . . . . . . . . . . . .

3 3 4 4 5 6 6 7 7 9 10 10 11 12 16 16 17 17 19 19 22 25 28 30 31 33 37 39 39 43 43 47 48 50

XIV

CONTENTS

4.3.2.1 The detector . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 Description of the optical design . . . . . . . . . . . . . 4.3.2.3 Verification of the optical performance of the FOC and its coronograph mode . . . . . . . . . . . . . . . . . . . 5 . CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

ACKNOWLEDGE MEN^ REpeRENcEs

51 53 57 58 59 59

I1. SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES by C. FROEHLY.B . ~OLOM~EAU and M. VMOUILLE (LIMOGES. FRANCE)

INTRODUCXTON. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. FRAMEWORKFOR SCALAR DE~CRIP~ON OF O m c a Prns~s. . . . . . . . . . 1.1 Complex analytic representation of space-time pulses . . . . . . . . . . 1.2 Sampling of optical pulses and number of their space-time modes . . . . . 1.3 Conditions leading either to deterministic analysis or to statistical analysis of optical pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Three examples of optical pulses exhibiting different coherence . . . . . . 1.4.1 Deterministic temporal analysis of purely temporal pulses . . . . . 1.4.2 Purely temporal analysis of space-time pulses . . . . . . . . . . . 1.4.3 Partially coherent temporal analysis of purely temporal pulses ... 2. SPATIAL AND TEMPORAL PULSE FILTEXING ON PROPAGATION AND DIFFRACTION 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Diffraction and propagation of quasi-monochromaticpulses . . . . . . . 2.2.1 Definition of quasi-monochromaticpulses . . . . . . . . . . . . 2.2.2 Spatial phase filtering on monochromatic pulses by free space propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Linear dispersion of quasi-single space frequency pulses . . . . . . . . . 2.3.1 Definition of single space frequency pulses . . . . . . . . . . . . 2.3.2 Linear dispersion of single space frequency pulses on free space propagation . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Definition of quasi-single space frequency pulses; finesse of the spatial frequency spectrum of a pulse . . . . . . . . . . . . . . . . . 2.3.4 Experiments on the dispersion of quasi-single space frequency pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Two pulse interference: temporal Young’s experiment . . . . . . . 2.3.6 Temporal Fourier analysis by “far field dispersion” of single space frequency pulses . . . . . . . . . . . . . . . . . . . . . . . 2.4 Temporal filtering of pulses by transmission through time independent opticalpupils . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Time impulse response of Young’s slits . . . . . . . . . . . . . 2.4.2 Time impulse response of a periodic grating . . . . . . . . . . . 2.4.3 Time impulse response of other time independent apertures . . . . 3. TrrvlE SHAPINGOF PICOSECONDOPllcAL hnsm . . . . . . . . . . . . . . 3.1 Pulse shaping by optical filtering of time frequencies (spectral modulation) . 3.1.1 General principles and limitations . . . . . . . . . . . . . . . . 3.1.2 Examples of typical shapes produced by amplitude or phase filtering 3.1.3 Filtering experiments . . . . . . . . . . . . . . . . . . . . . 3.2 Pulse shaping by temporal modulation . . . . . . . . . . . . . . . .

.

.

65 66 66 68 70 74 74 75 76 77 77 78 78 80 84 84

85 90 93 94 96 97 98

99 100 102 103 103 103 105 121

CONTENTS

3.2.1 Pulse shortening by self-amplitude modulation in saturable absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Frequency modulation by self-induced refractive index variation of transparent materials . . . . . . . . . . . . . . . . . . . . . 3.3 Shaping by a combination of temporal and spectral modulations . . . . . 3.3.1 Self-phase modulation of pulses after temporal shaping of their intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Combination of linear dispersion and self-phase modulation of Gaussianpulses: compressionandother pulsedistortions . . . . . . . . 3.3.3 A few other examples of combined modulation and filtering . . . . 4. O ~ C AANALYSIS L OF PICOSECOND LIGHT PUISES ............. 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Measurements of pulse intensity profiles . . . . . . . . . . . . . . . . 4.3 Coherent optical analysis of the temporal structure of picosecond pulses . . 4.3.1 Measurements of the number of temporal modes (samples) of a pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Measurements of phase (frequency) modulation by pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Measurements of “instantaneous frequencies” . . . . . . . . . . 4.3.4 Temporal phase measurements by optical beating . . . . . . . . . 4.3.5 Coherent pulse imaging by amplitude correlations or spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS ........................... REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

122 123 128 128 131 133 134 134 135 137 138 138 139 140 141 149 150

III. MULTI-PHOTON SCA’ITERING MOLECULAR SPECI’ROSCOPY by S. KIELICH (Porn&. POLAND)

1. HIWORICALDEVELOPMENTSANDOUTLINEOFTHEPRESENTREVIEW . . . . . 157 1.1 The definition of spontaneous multi-photon scattering . . . . . . . . . . 157 161 1.2 Spontaneous hyper-Rayleigh light scattering studies . . . . . . . . . . . 1.3 Spontaneous hyper-Raman scattering studies . . . . . . . . . . . . . . 163 165 1.4 The purpose of this paper . . . . . . . . . . . . . . . . . . . . . . 2. NONLINEAR MOLECULAR RAMAN POLARIZABUIES . . . . . . . . . . . . . 166 167 2.1 The multipole interaction Hamiltonian . . . . . . . . . . . . . . . . 2.2 The equation of motion for the vector of state . . . . . . . . . . . . . 169 2.3 Nonlinear polarizabilities in the electric-dipole approximation . . . . . . 172 2.4 Multipole electric and magnetic polarkabilities . . . . . . . . . . . . . 175 3. INCOHERENT AND NONREBONANT MULTI-PHOTON SCATTERING BY MOLECULES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 178 3.1 The electric and magnetic fields of the scattered wave . . . . . . . . . . 3.2 Harmonic electricdipole elastic scattering processes . . . . . . . . . . 179 182 3.3 Multi-photon vibrational Raman scattering (classical approach) . . . . . . 3.4 Rotational, vibrational and rotational-vibrational multi-photon scattering processes (semi-classical approach) . . . . . . . . . . . . . . . . . . 183 187 3.4.1 Three-photon Raman scattering . . . . . . . . . . . . . . . . 198 3.4.2 Four-photon scattering . . . . . . . . . . . . . . . . . . . . 4. LINEWIDTH BROADENING IN QUASI-Ew\snC MULTI-PHOTON ~CA~TERINGBY 201 CORREL~TED MOLECULES. . . . . . . . . . . . . . . . . . . . . . . . 4.1 The electric field and correlation tensor of scattered light . . . . . . . . 201

XVI

CONTENTS

4.2 Linear scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Isotropic incoherent and coherent scattering . . . . . . . . . . . 4.2.2 Anisotropic incoherent and coherent scattering . . . . . . . . . . 4.3 Three-photon scattering . . . . . . . . . . . . . . . . . . . . . . . 4.4 Four-photon scattering . . . . . . . . . . . . . . . . . . . . . . . 5. COOPERATIVE THREE-PHOMN !kATTEZUNG . . . . . . . . . . . . . . . . 5.1 Fluctuational variations of the nonlinear molecular polarizabilities . . . . 5.2 The time-comelation function for interacting atoms and centrosymmetric molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Many-body atomic multipole interaction . . . . . . . . . . . . . 5.2.2 Molecules with centre of inversion destroyed by the field of electric multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . 6. RAMAN LINE BROADENING IN MULTI-PHOTONSCATTERING(CLASSICALTREATMENT)

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

6.1 Three-photon Raman scattering . . . . . . . . . . . . . . . . . . . 6.2 Four-photon Raman scattering . . . . . . . . . . . . . . . . . . . . 7 . A N G DISTRIBUTION ~ AND POLARIZATION STATESOF MULTI-PHOTON SCATTERED LIGHT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The scattering tensors in terms of Stokes parameters . . . . . . . . . . 7.2 Natural incident light . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Linearly polarized incident light . . . . . . . . . . . . . . . . . . . 7.4 Circularly polarized incident light . . . . . . . . . . . . . . . . . . . 7.5 Four-photon light scattering . . . . . . . . . . . . . . . . . . . . . 7.6 Reciprocity relations . . . . . . . . . . . . . . . . . . . . . . . . 8. CONCLUDING REMARKS, AND OUTLOOK. . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS ........................... APPENDIX A . k R E D U C I s t E CARTESIAN TENSORS ............... APPENDIX B . ISOTROPIC AVERAGING OF CARTESIAN TENSORS .......... REFERENCES

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

204 205 207 210 215 216 216 219 220 222 226 230 232 233 233 236 238 239 241 244 246 249 250 251 254

IV. CULOUR HOLOGRAPHY by P.

HAFXHARAN (SYDNEY. AUSTRALIA)

1. INTRODUCTION ............................. 1.1 Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The cross-talk problem . . . . . . . . . . . . . . . . . . . . . . . 2. EARLY llnmaves FOR C~LOUR HOLOGRAPHY . . . . . . . . . . . . . . 2.1 Thin holograms . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Frequency multiplexing . . . . . . . . . . . . . . . . . . . . 2.1.2 Spatial multiplexing . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Coded reference beams . . . . . . . . . . . . . . . . . . . . 2.1.4 Division of the aperture field . . . . . . . . . . . . . . . . . . 2.1.5 Separation of spectra in image holograms . . . . . . . . . . . . . 2.2 Volume holograms . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Volume transmission holograms . . . . . . . . . . . . . . . . 2.2.2 Volume reflection holograms . . . . . . . . . . . . . . . . . . 2.3 Problems with early techniques . . . . . . . . . . . . . . . . . . . . 2.3.1 Diffraction efficiency . . . . . . . . . . . . . . . . . . . . . 2.3.2 Emulsion shrinkage . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Colour rendering . . . . . . . . . . . . . . . . . . . . . . .

265 265 266 268 268 268 269 271 272 274 275 276 278 279 279 279 280

XVII

CONTENTS

3. M~LTICOLOUR "BOW HOLOGRAMS ................... 3.1 The rainbow hologram . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multicolour images with rainbow holograms . . . . . . . . . . . . . . 3.3 One-step multicolour rainbow holograms . . . . . . . . . . . . . . . 3.4 Image blur ............................ 3.4.1 Wavelength spread . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Source size . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Recording materials . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The sandwich technique . . . . . . . . . . . . . . . . . . . . 3.5.2 Gain in image luminance with the sandwich technique . . . . . . . 4 . VOLUMEREFLECTIONHOLOGRAMS: NEWTECHNIQUES ............ 4.1 Alternative recording materials . . . . . . . . . . . . . . . . . . . . 4.2 Bleached reflection holograms . . . . . . . . . . . . . . . . . . . . 4.3 Sandwich technique . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Concentration of the diffracted light ................. 5 . PSEuDoCOLoUR IMAGES . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Colourcoding . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rainbow holograms . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Volume reflection holograms . . . . . . . . . . . . . . . . . . . . . 6. ACHROMATICIMAGES .......................... 6.1 Dispersion compensation . . . . . . . . . . . . . . . . . . . . . . 6.2 Rainbow holograms ........................ 7. APPUCATIONS OF COLOUR HOLOGRAPHY . . . . . . . . . . . . . . . . . 7.1 Storage of colour images . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Systems using image holograms . . . . . . . . . . . . . . . . . 7.1.2 Systems using spatial filtration . . . . . . . . . . . . . . . . . 7.1.3 Systems using rainbow holograms . . . . . . . . . . . . . . . . 7.2 Colour holographic stereograms . . . . . . . . . . . . . . . . . . . 7.2.1 White-light holographic stereograms . . . . . . . . . . . . . . . 7.2.2 Achromatic holographic stereograms . . . . . . . . . . . . . . 7.3 Computer-generated colour holograms . . . . . . . . . . . . . . . . 7.3.1 Technique using multilayer colour lilm . . . . . . . . . . . . . . 7.3.2 Techniques using holographic stereograms . . . . . . . . . . . . 7.4 Holographic cinematography . . . . . . . . . . . . . . . . . . . . . 8 CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS ........................... REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

283 284 285 288 290 290 291 292 292 292 294 295 296 296 297 299 300 300 301 302 303 303 305 307 307 308 309 310 312 313 316 316 317 318 318 321 321 321

.

V GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION by W . JAMROZ and B . P . STOICHEFF (TORONTO. CANADA)

1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nonlinear susceptibilities . . . . . . . . . . . . . . . . . . . . . . 2.2 Resonant enhancement and tunability in gases . . . . . . . . . . . 2.3 Conversion efficiency . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Saturation effects and other limiting processes . . . . . . . . . . . 2.5 Higher order nonlinear effects . . . . . . . . . . . . . . . . . . . .

..

..

327 328 328 331 337 341 346

XVIII

CONTENTS

3 . &PERIME.NTAL&SULTS ........................ 3.1 General techniques of frequency conversion . . . . . . . . . . . . . . 3.2 Tunable generation in rare gases . . . . . . . . . . . . . . . . . . . 3.3 Tunable generation in metal vapors . . . . . . . . . . . . . . . . . . 3.3.1 Strontium . . . . . . . . . . . . . . . . . . . . . . . . . . ......................... 3.3.2 Magnesium 3.3.3Zinc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4Mercury . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Beryllium,calcium . . . . . . . . . . . . . . . . . . . . . . 3.4 Tunable generation in molecular gases . . . . . . . . . . . . . . . . 3.5 W and X W generation by higher order processes . . . . . . . . . . 3.6 Generation of tunable XUV radiation by anti-Stokes Raman scattering . . 4 . CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . &ZERJ?,NCF.S

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

AUTHORINDEX . . . . . . . . . . . . SUBJECTINDEX . . . . . . . . . . . . VOLUMES I-XX CUMULATIW INDEX .

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

349 349 353 361 363 364 367 369 371 372 374 376 377 377 381 391 395

E. WOLF, PROGRESS IN OPTICS XX @ NORTH-HOLLAND 1983

I

SOME NEW OPTICAL DESIGNS FOR ULTRA-VIOLET BIDIMENSIONAL DETECTION OF ASTRONOMICAL OBJECTS? BY

G. COURTES", P. CRUVELLIER*, M. DETAILLE and M. SAISSE Laboratoire d'Asrronomie Spatiale du C.N.R.S.,Marseille, France

t Manuscript completed April 1981.

* Also with

Observatoire de Marseille.

CONTENTS PAGE

§

1. INTRODUCTION

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

3

0 2. UV PHOTOGRAPHIC SURVEYS USING WIDE FIELD CAMERAS

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

6

. . . . . . . . . . $ 4 . WSPACETELESCOPESOFTHEFUTURE . . . . .

38

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

58

§

3. AVERAGE FIELD TELESCOPES

0 5. CONCLUSION REFERENCES

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

19

59

0 1. Introduction The application of space optics to the observation of the W astronomical objects has undergone important developments since the first use of space vehicles. The large set of objects of very different appearance (from stellar point sources to the most extended ones like the Milky Way or the Zodiacal Light) brought about many imaginative solutions. The first period of space astronomy instrumentation was devoted chiefly to stellar spectroscopy with the famous series of O A O telescopes (HALLOCK [1962]; DAVISand RUSTZI[1962]), but there has been a very limited interest in direct or indirect (scanning) imagery. The subject of the present paper is to describe these new approaches of space astronomy from the small pioneering cameras of a few centimeters aperture up to the 2.4 meter Space Telescope.

1.1. SPECTRAL ANALYSIS OF THE SOURCES

Among the most important information is, of course, the spectrophotometry of the celestial objects. For the reader not familiar with astronomical observations, it is first interesting to recall the general methods used by astrophysicists to obtain accurate, indirect, spectroscopic observations of more and more distant objects, hence fainter and fainter to sight. For example, Morgan’s U-B-V photometry consists in using three large bandwidth filters (U = ultraviolet, B = blue, V =visible), giving the possibility of reaching the faintest objects at the limit magnitude of a given telescope. The “colour indices” U-B and B-V are calibrated with great accuracy as a function of stellar spectral classification, with observations of bright stars for which it is easy to obtain both good spectral classification with relatively high dispersion spectrographs and excellent determination of the photometric magnitudes U, B and V. Thus, for the faintest objects, for which it is impossible to obtain spectra, the relation between 3

4

DETECTION OF ASTRONOMCAL OBJECIS

[I, § 1

the spectrum and the U-B-V parameters will give an indirect stellar spectral classification. The transposition to the UV range follows the same method. One can consider now that the W spectral classification of stars has reached an excellent quality since the spectrographic surveys of the OAO-11, TD1, Copernicus and, more recently, the IUE (International W Explorer). These surveys, made on relatively bright stars (V<8), are now the basis from which we can extrapolate the classifications of distant stars and galaxies.

1.2. BIDIMENSIONAL INFORMATION

Another obvious need is the bidimensional knowledge leading to the morphology, necessary as an important contribution to the interpretation of all extended objects like galaxies, HI1 regions (gaseous nebulae), diffuse nebulae (dust clouds), galactic and globular clusters and large stellar clouds.

1.3. FIRST APPROACHES TO BIDIMENSIONAL INFORMATION

Before there were any space telescopes capable of forming images of celestial objects, observers used the natural spin of the rockets and satellites to scan the sky with a collector equipped with photo-electric devices (KUPPERIAN, BOGGESSand MILLICAN [1958]) from the Goddard Space Flight Center (GSFC). A scanning mirror on a 4" x 4" field has been used by the Laboratoire d'Astronomie Spatiale (LAS) of the CNRS, for the Persee rocket experiment (CRUVELLER, Roussm and VALERIO[19701 and COURT&[1971]) on the galaxy M31, giving evidence of a UV excess in this galaxy. The best results of the scanning method were obtained with the following astronomical satellites on the Magellanic clouds: the TD1 (CARNOCHAN, DWORETSKY, TODD,WILLISand WILSON[1975]); the ANS (BORGMAN, DUINEN and KOORNNEEF[1975]) and the D2B-AURA (MAUCKERAT-JOUBERT, CRUVELLIER and DEHARVENG [1979]), Isophotes of the W radiation of the Magellanic Clouds and of the Milky Way have been obtained with these satellites. The angular width of the slit on the sky and the scanning velocity limit the spatial resolution of this type of

I, I11

INTRODUCTION

5

observation. In the case of the D2B, there is no slit, but a baffle limiting CRUVELLIER and DEHARVENG [1979], the field (MAUCHERAT-JOUBERT, COURT~S [1971]). Other baffles have been developed also (MCGRATH and MAR^ [19691). Fortunately, the Magellanic Clouds and the Milky Way are objects of very large apparent diameter, but there is no question of extending this method of scanning to small apparent diameter objects because the width of the slit is always too large if one wants to integrate a significant quantity of light. Even in the case of large objects, the time devoted to the bidimensional scan is very long for reasons of dimension and faintness (this requires integration over many orbits). This evidence leads to real imagery solutions with bidimensional detectors (photographic film, image intensifier +photographic film and, more recently, image intensifier with microchannel array + TV camera). 1.4. DIRECT IMAGES OF THE SKY IN UV RADIATION

A survey which covered 10% of the sky was achieved by the Smithsonian Telescope experiment on board the OAO-I1 astronomical satellite (CODE[1969]). The detector was an image dissector, and the collector a small reflecting telescope. The first photographs were obtained with small photographic (anastigmatic lens) cameras designed by HENIZE,WACKERLING and O'CALLAGHAN [1967] of Northwestern University (USA) and used by the astronauts (Gemini-I1 flights). Another similar instrument, but with a Maksutov telescope, was used by cosmonauts during early Soyuz flights by GURZADYAN [1974] from the Burakan Observatory (USSR). AII these instruments have, because of the photographic emulsion, a large spectral range comprising the visible and the near U V > 2000 A. MORTON and SPITZER [19661 (Princeton University) used small Schmidt telescopes with successful results on bright star spectra, thanks to objective-grating techniques down to 1100 A (with a LiF corrector) and down to 1000 A with all-reflective cameras. These Schmidt telescopes were in the payload of rockets (Morton) and on the manned platform Skylab as S-019 camera (~'CALLAGHAN, HENIZEand WRAY[1977]) and S 183 (COURT~S [1971]) from LAS, as well as on the best stabilized platform, the Moon's surface itself, during the Apollo-16 Mission (CARRUTHERS [1973]). All-reflective cameras (MORTON,JENKINS and BOHLIN [19681) were developed by Princeton University.

6

DETECTION OF ASTRONOMICAL. OBJECTS

[I, 5 2

More recently, Smith, from the Goddard Space Flight Center, used an all-reflective camera equipped with an objective grating. The detector was an image intensifier.

0 2. UV Photographic Surveys Using Wide Field Cameras 2.1. EARLY WIDE FIELD CAMERAS

Several other instruments have been flown on rockets by the LAS. Their scientific program was set up to obtain images of very wide, faint brightness areas of the sky owing to the faint UV radiation of the Milky Way* and of the Zodiacal Light?. The field of view was of the order of 100" and the focal ratio had to be as large as possible because of the faintness of these extended sources and the short duration of the flight (<5 minutes). These conditions, combined with dimensional constraints, led to very short focal lengths, and resulted in poor stabilization of the rockets. The compromises between stabilization, focal length, focal ratio and field were discussed in some previous publications (SIVAN and VITON [1970], COURTI3S [1971] and COURTES, V r r o ~and SWAN[1967]). The optician has to resolve paradoxical conditions: a large field of about loo", a focal ratio of about F/1, a high transmission in the UV-hence a small number of optical elements-and a gradient of vignetting as small as possible in this large field. The general optical solution, shown in Fig. 2.1, was the use of a small Schmidt telescope designed to work in diverging beams (at a short distance) and playing the role of relay optics from a wide field image of the sky given by a convex hyperbolic mirror. The use of new techniques developed at the Laboratoire d'Astronomie Spatiale (LAS) and at the Observatoire de Marseille (BARANNE, DETAILLE and LEMAITRE [1969]) had allowed for very fast UV Schmidt telescope plates, to be obtained. Several rockets have been equipped with such an optical system *The Milky Way concerns the appearance of the Galaxy viewed from the Sun, as a belt of 30"X 360". Several problems of stellar composition, dust absorption and extension of the galactic system, scattering of interstellar and interplanetary dust can be studied owing to this imagery method. 7 The Zodiacal Light extends to the whole sky. Due to the sunlight dust scattering in the solar system its maxima of light distribution are on the ecliptic.

1, § 21

7

WIDE FIELD CAMERAS

Hyperbolic m i r r o r

pseudo-flat aspherical mirror

Fig. 2.1. Basic optical layout of the wide field all reflection camera.

(COURTES [1971]), but the latest instrument of this type is the Very Wide Field Camera (VWFC) of the Laboratoire d'Astronomie Spatiale selected for the first SPACELAB mission, ESA payload. The VWFC is an enlargement of the first rocket experiment. 2.2. THE VERY WIDE FIELD CAMERA (1 ES 022 in Spacelab-I)

2.2.1. Scientific program A short description of the scientific program will show how the optician can define the best compromise between the general properties of this design. The VWFC concerns: (i) the stellar statistics of blue dwarf stars of high galactic latitude. The main information about these stars, at the end of their evolution, has been the TD1 satellite survey (spectrometer scanning the sky) reaching magnitude 8 (CARNOCHAN, DWORETSKY, TODD, WILLISand WILSON [1975]). It was necessary to make the entrance pupil of the VWFC large enough to detect fainter stars than TD1. (ii) detection of gaseous clouds (HI1 regions). Extended objects up to 10" apparent diameter are very faint, but their emission is mainly monochromatic. In the spectral range of the VWFC, the carbon line C 1111 (1909 A), will be one of the best indicators of gaseous emission of high excitation. The geometry of the beams of the VWFC will give the possibility of using interference filters, in spite of the high focal ratio and the wide field. The chosen design (Fig. 2.2) shows the small incidence

8

DETECTION OF ASTRONOMICAL OBJEcrS

SPECTRO -

SPHERICAL MIRROR

\

PLANE MIRROR

/

Fig. 2.2a. Optical design of the VWFC.

I, §21

WIDE FIELD CAMERAS

9

angle of the beams between the hyperbolic mirror and the Schmidt camera, where it is easy to put the interference filters. The interference filter for monochromatic detection has to be as narrow as possible. The state of the art for good transmission (faint objects) leads to a bandwidth of AA = 300 A; in fact, the transmission T of the W metal-dielectric interference filter corresponds roughly to the empirical formula: T I AA/lOOO A. (iii) detection of very extended sources (scattering by the interplanetary and interstellar dust). This type of source concerns the entire Milky Way and needs a field at least wide enough to reach the extragalactic sky background on both sides of the Milky Way. Considering the angular width of the Milky Way (about 30"), one chooses 60" as a minimum field in order to keep the maximum scale for the image. After their correct rejection (owing to their morphology), the large angular distribution of the zodiacal light and the Gegenschein will permit an accurate determination of the faint extragalactic sky background - about 200 ph. (cm*.s.A.ster)-' -which is one of the most exciting cosmological problems. High galactic latitude dust clouds discovered in some directions by SANDAGE [1976] with the 6" field, 48-inch Palomar Schmidt telescope could contribute significantly to this background. The 60" field of the VWFC will provide a complete verification of the real extension of these clouds and the best "window" to measure the extragalactic background. Items (ii) and (iii) concerning extended sources have been almost entirely neglected to date. A special effort has been made to obtain the best performance. As nebular space spectrography of extended clouds does not yet exist, the VWFC is equipped with a wide angle (14" high) slit spectrograph.

2.2.2. The detector performance All optical parameters depend on the detector, an ITT 40 mm diameter multichannel array (proximity focus intensifier tube (PFCIT)). The TABLE2.1 VWFC PFClT detector performance Diameter of the microchannel element: -13 pm Expected photometric resolution verified at the laboratory: -60 pm Sensitivity: 2 x erg. cm-* at 250 nm Wavelength range limited by CsTe photocathode (A < 300 nm) * Dynamic range limited by the film * Gain = 1000. *

*

-

10

DETECTION OF ASTRONOMICAL OBJECTS

[I,

§2

performance of this detector, associated with Kodak film, is given in Table 2.1.

2.2.3. Compromises to accomplish the scientific program The diameter of the detector and the need for a 60"field leads to a final focal length of 30-40 mm. Because of its plane-parallel 5 mm thick MgFz window, good optical behaviour of the detector requires a focal ratio
2.2.4. Geometrical conception of the camera The pure geometrical concept of Fig. 2.1 has been adapted to the unavoidable dimensional constraints of the Spacelab by DETAILLE [19811 who has designed the following optical system as a result of a very complex compromise. This solution optimizes the number of optical elements (three in principle) and solves the field curvature and the astigmatism problems. Indeed, the hyperbolic mirror diaphragmed at the geometrical focus and used for an object at infinity is free of astigmatism in all orders. This results from the calculation of the sagittal and the tangential focal distances fs and ft, respectively. If R, and R, are the values of the radii of curvature at the point M, for an incident point M on the hyperbolic mirror, we have:

fs=ft=~(R,R,)"2. Hence, the field curvature given by the third order Petzval curvature will be flat, if the paraxial radius of curvature of the hyperbolic and the spherical mirrors are equal and opposite (because the Schmidt camera is free of astigmatism).

I, $21

WIDE FlELD CAMERAS

11

If the pupil magnification of the hyperbolic mirror is y, the field of view on the sky is p, and the final focal number is N , the hyperbolic mirror works with an N / y focal number and the Schmidt telescope has an equivalent field of pfy. Thus the aperture aberrations of the hyperbolic mirror are slight and can be corrected by the Schmidt telescope (the spherical aberration by an additional aspherisation of the pseudo-flat mirror of the Schmidt camera, and the coma by a shift of the pseudo-flat mirror from the center of curvature of the spherical mirror). Finally, the focal number N and the field p are limited by the performance (mainly astigmatism) that we can expect from an all-reflexion Schmidt telescope and by the length of the whole optical system of the camera, which increases with y for a given focal distance. This solution, solving the field curvature problems, leads to a flat field which is very convenient for most of the detectors. In addition, this system has the merit of having no entrance pupil limitation for large fields*.

2.2.5. Photometric properties of the imagery mode of the VWFC Unlike some wide field cameras which have a vignetting effect, this type of telescope has a pupil surface crp which increases with the field angle (for the VWFC, crp increases by 50% from p = O " up to p = 7 0 ° ) . Therefore, the stellar sensitivity is better at the edge of the field. O n the other hand, the background sensitivity varies as K ( p ) = a,(sinp/r)ap/dr, where r is the axial distance of the point in the focal plane (in the VWFC, K ( p ) is constant to within less than *lo% over the full field) (Fig. 2.3). This condition is very interesting because it provides a uniform sensitivity to the sky background and hence the best signal/noise ratio evaluation over all the detector. As we pointed out before, it is also the only way to obtain representative images of faint, extended sources like the Zodiacal Light, the Gegenschein, the Milky Way or the high galactic latitude dust clouds. This condition of uniform

* Some wide field cameras have been constructed in the past following similar prinicples, but in visible light. In that case, an anastigmatic lens is combined with a concave mirror for correction of the field curvature. The first ones were the lS0°F/2 camera of the Yerkes and OSTERBROCK Observatory designed by L. C. Heyney and J. L. Greenstein (SHARPLESS [1952]) and the 150"F/1.8camera of the Haute Provence Observatory (COURT~S [1952; 19601).

12

DETECTION OF ASTRONOMICAL OBJECX

[I, 9: 2

Fig. 2.3. VWFC field photometric function.

sensitivity in the field is very rarely fullfilled in conventional wide angle objectives and is the reason for the bad morphology usually obtained, e.g. in most photographs of the Zodiacal Light. This uniform response to extended sources is obviously indispensable for defining the real extension of this galactic and interplanetary diffuse light, in order to observe the intergalactic medium (IGM) in the clearest directions.

2.2.6. Main characteristics of the VWFC The main optical characteristics of the VWFC are shown in Table 2.2. The spot diagrams obtained with the ACCOS Program at the center and at the edges of the field are shown in Fig. 2.4. Fig. 2.5 shows an example TABLE 2.2 Main optical characteristics of the VWFC *

Paraxial focal length: F>29.3 mm

- Field in the symmetric plane: 58"

- Total solid angle: 0.9 steradian or 2950 0" Geometric F number: F/I - Obstruction ratio: 0.18 - energy Maximum spot diameter: 0.08 mm or 8 arc/minute for 100% of *

.8

I, Ei 21

13

WIDE FIELD CAMERAS

I I

Mean field

I

Paraxial focus for t h e mean field

Chooser1 nominal f o c u s ( + 0.05 1

Fig. 2.4. Spot diagrams of the photometric mode of the VWFC. The circle diameter corresponds to 0.1 mm.

14

DETECTION OF ASTRONOMICAL OBJECTS

Fig. 2.5. A stellar field obtained with the mock-up of the VWFC. Circle corresponds to the 58" nominal field.

of a 58" star field obtained with the mock-up of the VWFC, which is very consistent with the computation. We can comment as follows: (i) The field was limited to 58", and the beams between the hyperbolic mirror and the Schmidt camera were folded by a flat mirror, because of the Spacelab's dimensional constraints. (See Fig. 2.2.) (ii) The aspherical mirror follows the concepts of an all-reflection Schmidt systems (EPSTEIN [1967]; LEMA~TRE [1979]) in order to be free of the chromatism of the conventional Schmidt correcting plate in this UV spectral range, where the dispersion is steeply wavelength dependent (Fig. 2.6). (iii) Interference filter (AA = 300 A) define three passbands of astrophysical interest: A = 2500 A (stellar continua); A = 1909 A, C 1111 (gaseous nebulae emission) and A = 1550 A (stellar continua). This last

WIDE FIELD CAMERAS

15

Fig. 2.6. The aspherical mirror interference pattern (Michelson test) of the VWFC Schmidt all-reflection camera. This deformed surface is a plastic copy of a controlled flexure mirror following the Lemaitre methode (LEMAITRE[1979]).

Fig. 2.7. VWFC - interference filters.

16

DETECTION OF ASTRONOMICAL OBJECTS

[I, 6 2

passband is carefully adapted to a complete rejection of the strong Lyman a emission of the geocorona (2X lo3 Rayleigh) thanks to a calcium fluoride support of the interference filter. The CaFz is heated to 20°C to maintain the cutoff above Lyman a. The interference filter transmission curves are shown in Fig. 2.7. (iv) The Schmidt camera is of the CoudC type to allow easy access to the detector.

2.3. THE NEBULAR SPECTROGRAPH (NS) OF THE VWFC

2.3.1. Scientific objectives

This spectrograph was conceived to obtain UV spectra of the gaseous clouds (HI1 regions) of the Milky Way. These HI1 regions have an apparent diameter of a few degrees; some extend to 20". The length (14") of the slit is defined by the mean apparent diameter of the gas clouds; this length depends on the focal length of the small telescope and must be optimized with the aberrations of the grating, perpendicular to the mean plane of dispersion (Fig. 2.2). The maximum angular resolution is given by the 0.4" slit width. The spectral resolution, because of the small number of emission lines expected from the interstellar gas, does not need to be high. The chosen resolution is 5 A, sufficient to separate the monochromatic emission lines and small enough to detect the continuum. This small resolution corresponds, however, to an interesting monochromaticlcontinuum ratio and will allow elimination of all risk of misinterpretation between monochromatic gas emission and dust continuum scattering in the direct image of the VWFC. In fact, the coefficient of monochromatic discrimination r of the spectrograph (COURT~S [1973]) is: - A , (bandwidth of the filter) r=Ahh l(width of the slit spectrograph) * Hence, the spectrograph is, for example, about 40 times more efficient than the interference filter. As we saw above, the nebular spectrograph (NS) of the VWFC has two main functions: (1) Spectral analysis of extended sources (astronomical objects as well as residual high atmospheric emission) ;

I, 421

WIDE FIELD CAMERAS

17

(2) Detection of the monochromatic emission (higher monochromatic/continuum contrast than interference filters - and hence, no doubt, greater monochromatic detection). 2.3.2. Limitation of the spectral range At the shortest wavelength, the main limitation is due to the Lyman a hydrogen emission of the geocorona. The brightness of this emission can reach 2 x lo3 Rayleigh and requires absolute elimination if one wants to accomplish the scientific program on faint extended objects. Even for the spectrograph equipped with a very low diffusion holographic grating, there is no question of allowing the Lyman a radiation to enter directly into the slit. The CaF, plano-concave lens I," is needed to put the pupil at the right position. This precaution should be sufficient to cut Lyman (Y but some light could come directly into the Schmidt camera of the VWFC through the window of the spectrograph. A convenient baffle and a CaF, planeparallel plate shut this window only in the direction of the imagery mode box of the VWFC. Both plates are heated (20°C) in order to keep the cutoff of the transmission far enough from Lyman a. At the largest wavelengths, no important emission line is expected above 2000 A.

2.3.3. Optical design For reasons of operation (easy pointing) the optical axis of the small telescope of the nebular spectrograph is parallel to the pointing direction of the field center of the VWFC. For reasons of mechanism simplicity, the grating is operated by the filter exchange system. The grating disperses the light coming from a slit independant of the VWFC itself. This slit is in the focal plane of a small off-axis telescope. The use of the same Schmidt camera for both the VWFC and the NS is an obvious weight and dimensional optimisation, but leads to some complications in the design of the grating. With the NS optical system, the flat field of the detector corresponds in fact to the strongly curved virtual image given by the hyperbolic mirror. An ideal spectrum would have to match the same curved field. There is no solution for a grating, ruled or holographic, giving, in these conditions, * I t is in fact a slit-lens.

18

[I, 8 2

DETECTION OF ASTRONOMICAL OBJECTS

Fig. 2.8. Spot diagrams of the nebular spectrograph of the VWFC. T: on the tangential focal plane of the grating; H: on the focal plane of the hyperbolic mirror (DUBAN[1978]).

the same curvature sign for the tangential focal surface, but a good compromise has been found with a torroidal holographic grating (DUBAN [1978]), for the minimisation of this curvature as well as for reduction of the other aberrations. The spot diagrams are shown on Fig. 2.8. The spectral resolution and the slit image height in the plane of symmetry are shown in Fig. 2.9. One sees that the spectral range (from 1450A to 2000A) can be correctly achieved with Duban’s compromise. Image Height

A

(wn)

I

300

/

Resolution( )

/ /

1500

2000

xr8,

Fig. 2.9. (a) Variation of the resolution with wavelength. (b) Variation of the slit image height with wavelength (DUBAN [19783.

I, $31

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AVERAGE FIELD TELESCOPE

TABLE2.3 Characteristicsof the NS of the VWFC (1 ES 022 - SPACELAB) ~~

-

Focal length: 45 mm * F number: F/1.8 (the same as V W C ) * Dispersion: 30 A/mm * Axial spectral resolution: -3 A from 1450 A to 2000 A *

A classical ruled concave grating has only one parameter, its radius. Only one wavelength should be coincident with this particular focal surface. The holographic grating, owing to the additional parameters due to the conditions of recording, permits us to obtain a spectrum that crosses the focal surface at two different wavelengths (1550A and 1920A) and remains free of coma in this spectral range. Thus, the spectrum is good enough over the whole observed spectral range, the maximum defocusing corresponding to less than 1A of loss of resolution (DUBAN 119781). This gives an exceptionally good compromise for such difficult initial conditions. The general performance shown in Table 2.3 gives a very good solution to the scientific problem. The grating resolution is better than 2 A in the most interesting part of the spectrum (Fig. 2.9), but the space resolution of the detector limits this spectral resolution to about 3 A.

B

3. Average Field Telescopes

3.1. EARLY TELESCOPES

As soon as the spacecraft stabilization was improved, longer focal lengths were possible; consequently, the entrance pupil was increased step by step. Several small telescope designs have been used; there are of course many relatively simple solutions for this problem (MONNET[1970], COURT^ [1971J, CARRUTHERS, HECKATHORN and OPAL119781). We recall that Morton, of Princeton University, successfully flew such a small Schmidt telescope, both with and without a plane objective grating (MORTONand S P ~ E [1966], R MORTON,JENKINSand BOHLIN [1968]). Smith, of the Goddard Space Flight Center, designed a similar system [19781). Carruthers, of the Naval Research Laboratory, (CARRUTHERS built a remarkable Schmidt system equipped with an electronic camera

20

DETECITON OF ASTRONOMICAL OBJECTS

[I,

P3

PRIMARY

CORRECTOR

*------

Fig. 3.1. The optical system of the field camera of the Laboratoire d’Astronornie Spatiale. S 183 experiment of Skylab.

for his famous observation from the Moon during the Apollo-16 Mission. The Carruthers camera used a CsI photocathode of very high quantum efficiency in the far U V , which was solar blind to the visible and near UV. Henize, of Northwestern University, designed a flat field Schmidt Cassegrain equipped with an objective prism (S 019 experiment) for the Skylab Mission; the receptor was Schumann UV emulsion on Kodak film

Fig. 3.2. A photograph of the compact optics S 183 Schmidt Cassegrain telescope.

I, 5 31

21

AVERAGE FIELD TELESCOPE

TABLE 3.1 S 183 Schmidt Cassegrain telescope characteristics Components

Characteristics

Corrector plates

a=1.11; diam 46 mm thickness 1mm

Fused silica

Primary

R , = 65.06 mm

B 1664

-

E,

Secondary

= O ; diam 5 2 m m

R , =69.24mm E~ = 0;

diam 29 mm

Materials

Coating

A/4multilayer

(Sovirel)

B 1664

A/4 multilayer

(Sovirel)

(O’CALLAGHAN, HENIZEand WRAY[1977]). All these American instruments have been described in a former review paper (CARRUTHERS [19781). For the Skylab Mission, another Schmidt Cassegrain using 16 mm 103 a 0 Kodak film in the S 183 experiment of the Laboratoire d’Astronomie Spatiale of CNRS was designed by M. Saisse (COUR& [1971]) (Figs. 3.1, 3.2 and Table 3.1). In this design, the flat field was directly obtained by the two mirrors with a chosen zero Petzval sum of their radii of curvature (DETAILLE and SA~SSE [1971], LAGETand SA~SSE [1974]). The filtering was obtained by the reflexion coatings of the two mirrors (Fig. 3.3). This instrument detected 4000 stars, the Magellanic Clouds and the central parts of the M 3 1 galaxy and numerous galactic fields (Fig. 3.4)

WAVELENQTH

la)

Fig. 3.3. This figure shows the very small change of the bandwidth before and after the flight. (Multilayer coatings on the two mirrors.)

22

DETECTION OF ASTRONOMICAL OBJECTS

[I, 9: 3

Fig. 3.4. A galactic field in the Eta Carinae area, 7" x 9" field of the S 183 LAS Experiment aboard Skylab. See also other fields in COURTES,SIVAN,VITON,VUILLEMIN and ATKINS [19751).

(COURTES, LAGET, SWAN,VITON,VUILLEMIN and ATKINS [19751, DEHARMONNETand VUILLEMIN [1976]) in UV light on fields of 7" x 9" from Skylab in the 2500 A wavelength range. This S 183 experiment aboard Skylab was equipped, in addition to this small Schmidt camera, with 1878 A and 2970 A simultaneous bandwidths of 600& thanks to the BPM filter (field grating techniques) (COURT& [1971]). [1964]) and small MgF, lens arrays (COURT& VENG, LAGET,

3.2. THE FAUST TELESCOPE ON BOARD SPACELAB-1 (1 NS05)

The FAUST telescope was developed by LAS with special technical support of CNES", to achieve a new step in the detection of faint extragalactic objects in the far UV owing to rocket flights. The entrance pupil had to be much larger than the entrance pupil in the VWFC. The need for a high focal ratio, flat field (zero Petzval sum of the mirror radii), compactness and a minimum of reflections (only two) to optimize the transmission, led to the use of Wynne's two-aspherical-mirror telescope * CNES = Centre National d'Etudes Spatiales (France space agency).

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AVERAGE FIELD TELESCOPE

23

system (WYNNE[1969]) which has the great advantage, for rocket use, of being axisyrnrnetric, thus giving possibilities of steady, mechanical, elegant solutions. After success in the detection of the central parts of the galaxy M 31 in Andromeda (DEHARVENG, LAGET,MONNETand VUILLEMIN [1976]), this

FAUST INSTRUMENT

Fig. 3.5. Cut of the LAS 1-NS-05Spacelab FAUST telescope. One sees the two mirror 180 mm diameter Wynne telescope giving a flat field and a high focal ratio of H1.12.

24

[I, § 3

DETECTION OF ASTRONOMICAL OBJECTS

instrument has now been selected by NASA to fly on the first Spacelab Mission with a common scientific program of Courtks and Deharveng from the Laboratoire d'Astronomie Spatiale of the CNRS Marseilles, and Bowyer and Malina, from the Space Sciences Laboratory of the University of California (Berkeley). On the Spacelab, the stabilization is expected to be about 0.1" with a possible drift of *0.01" s-'. This gives a focal length of 180 mm. The final solution defined by Monnet, Deharveng, Deshayes and Rivibre (DEHARVENG, R I V ~ R E MONNET, , MOUTONNET, C O U R ~DESHAYES , and BERGES [19793> was designed by Detaille and Cohendet, as is shown in Fig. 3.5, with some new improvements with respect to the first rocket telescope. The resolution of 1' is compatible with the short duration exposures authorized by the high sensitivity of the micro-channel arrays, and the expected drift of the Spacelab stabilization. The physical characteristics of the telescope are given in Table 3.2. TABLE 3.2 Optical parameters of the Faust experiment (Spacelab - 1 NS 05) (Parameters of the mirrors with profile

Z

= Cy '[

1+ JI - (x + 1)C2y*I-'

+ dy + ey + fy'.)

Focal length: 180 m m .F number: F/1.12 * Field of view: 7"5 Spectral range 145-245 nm * Angular resolution: 1arclmin * Detector: 25 m m Csl PFCIT *

-

convex primary concave secondary Distance between primary and secondary (mm) Overall diameter (mm) Useful diameter (mm) l/C: Radius of curvature at the vertex (mm) x: Conic constant d[mm-3) e(mm+) aspheric deformation -7 coefficients

I

f(mm 1

Maximum departure from the osculating sphere at the vertex of the surface (pm) Maximum departure from the nearest sphere (pm) Maximum departure from the osculating quadric (pm)

385.2 180 161 573.96 7.24 0.4265 X lo-' -0.1842~ -0.2086 x lo-''

385.2 437 413 517.37 0.178 -0.1187 x lo-'' 0.6628X lo-" 0.1433 X

220

270

35

55

18

20

I, 5 31

AVERAGE FIELD TELESCOPE

25

Fig. 3.6. Computer-generated spot diagram of the Wynne camera ( F = 180 mm, Fil. 12) at 1600 A: chief ray height (mm) (a) 0.0, (b) 7.18, (c) 0.0, (d) 7.18. Focus shift (mm)(a) 0.0, (b) 0.0, (c) 0.01, (d) 0.01. Half circle diameter: 10 arc sec.

Wynne shows that in the third order theory, there is a unique solution with two mirrors to obtain an anastigmat telescope with a flat focal plane. Schmidt, Bowen and Couder designed systems of the same family but without a flat focal plane. All these telescopes allow a large field (5" or 7" and 10" for the Schmidt) with a very good sensitivity given by the small number of optical surfaces and the very high focal number (extended sources). From the third order, the FAUST telescope was optimized with a 2.5 mm MgF, window for the CEMA and gave the spot diagram of Fig. 3.6. Because of the very strong aspherisation of the mirrors, the optical image quality of the telescope is limited to 1 arc min diameter. This quality is not spoiled by a 5 mm thick MgF, window if we put a CaF, lens in front of the telescope on the aperture of the telescope; this lens also cuts the Lyman a of the geocorona. We do not know yet if we shall use a CsI-CEMA detector without selective coating or a CsTe-CEMA detector with a selective coating on the CaF, lens. 3.3. THREE-MIRROR ANASTIGMAT 40 CM DIAMETER TELESCOPE SOLUTION (TMA-1000)

Three mirror systems have been proposed in the past by PAUL[1935] and by MEINELand SHACK[1966], and more recently by Tim (in CARRUTHERS [1978]). The main advantage of those systems is that they are

26

[I, § 3

DETECTION OF ASTRONOMICAL OBJECTS

'

5.4"

Fig. 3.7. Layout of NASA 6-inch F/1.8 wide field far ultra-violet camera for APOLLOAAP astronomy use by the University of Arizona ( ~ ~ E I N Eand L SHACK [1966]).

t 0.002"

0.000"

Q

fi

6

& .....

.,

*?

..

:

ENTRANCE PUPIL

00

1.890

1.35'

2.70'

FIELD ANGLE

Fig. 3.8. Spot diagrams for NASA 6-inch camera for optimum vignetting of image flare with aperture stop at the tertiary mirror.

I, § 31

AVERAGE FIELD TELESCOPE

27

made with reflective elements only. The Meinel and Shack telescope (Fig. 3.7)is an F/1.8,5'4 field, 150 mm diameter telescope, using UV photographic film. The image quality is quite good for such a complex combination (Fig. 3.8).The central obstruction is not exaggerated (0.58).Our purpose was to design a larger size telescope of 40cm diameter capable of reaching extragalactic objects. A 40 cm F/2.5 three-mirror anastigmat telescope is under study by LAS. Such an all-reflective solution would offer the possibility of being used on a Spacelab flight with far UV wavelength bandpasses. It would have dimensions similar to the FAUST Wynne telescope, but with the obvious advantage of an entrance pupil three times larger. The general design of SAISSE[1981]is composed of three hyperboloidal mirrors (Fig. 3.9,Table 3.3). The solution has been optimized in order to balance spherical aberration, coma and astigmatism over the whole 2.5" field of view and to minimize the central obstruction (0.3).The only remaining aberration is the field curvature which is removed by the use of a simple lens corrector located near the focus. Assuming the possibility of introducing fourth and sixth order asphericity in the mirror's figure, the theoretical image diameter is smaller than 2 arc seconds over the entire 2.5" field (Fig. 3.10). The problem not yet solved is the mirror manufacturing; the solution may be found in the use of new techniques either from Lemaitre or from optical figuring under computer control. Success of such an instrument could provide a deep probe of UV extragalactic objects up to the magnitude V = 20.

Fig. 3.9. The LAS three mirror 40 cm diameter telescope (TMA-1000).

28

[I, 5 3

DETECTION OF ASTRONOMICAL OBJECTS

TABLE3.3 TMA 1000 optical characteristics TMA 1000 - First order parameters: Focal length: 975 mm F number: 2.4 Field of view: 2.5" Image quality: 2 arclsec blur circle diameter Spectral range: 1500 A to 1850 A with filters TMA optical prescription (partially optimized design data) E d ThickRadius conic fourth e ness (mm) (mm) constant* order* six order*

-2.89

-1600

0.8532X lo-''

0.5473X 400

-1600

-44.63

0.1335 x lo-"

0.5707 x 300

-1642 -259.83

-46.23

0.1284X

0.8975x -300 -5

ffi

* Figuring depth

of mirror profile x

= h2/{R(1 +J1- ( E

Material Air Mirror Air Mirror Air Mirror Air MgF,

Diameter (mm) 400 200 144

+ l)h2/R2)}+dh4+eh6.

A

B

C

D

0 DEG

0.5 D€G

0.8 DEG

t,2 DEG

Fig. 3.10. Spot diagrams of the 1 m focal length three mirrors anastigmat telescope studied by the LAS for a future Spacelab mission. The circle diameter is 2 arc sec.

3.4. GENEVA-MARSEILLE UV BALLOON PROGRAM

Two-dimensional detection of celestial objects has been limited to the most sophisticated, stabilized space platform facilities: rockets (Morton, Carruthers, Smith, Viton, Court&, Deharveng) or the large, manned, stabilized Skylab spacecraft (Henize and Wray ; Court&, Laget and Vuillemin) and the Moon itself (Carruthers), during the Apollo-16 Mission as was shown earlier. Some ballon observations of nuclei of galaxies

I, 831

AVERAGE FIELD TELESCOPE

29

Fig. 3.11. (a) Photograph of the Large Magellanic Cloud in visible light. One sees the internal elliptic central stellar bulge of evolved stars. (b) Photograph of the same field in 2500 A ultraviolet light thanks to the S 183 experiment of the Laboratoire d’htronomie Spatiale, aboard Skylab. Only the hot star associations are detected; the central bulge disappears.

30

DETECTION OF ASTRONOMICAL OBJECTS

[I,

B3

have been made, thanks to the 12-inch diameter telescope of Stratoscope 1 and the 36-inch Stratoscope I1 by SCHWARZSCHILD [1973], but in the visible spectral range. Considering the UV scientific program of the Laboratoire d’Astronomie Spatiale, one of the most interesting possibilities of extraatmospheric observation is to select the very hot objects related to the fundamental problem of star formation and evolution. The population of high temperature stars, usually young in evolution, are linked to the interstellar gas and dust corresponding to the stellar birth place criteria. These hot objects have to be recognized among a much larger number of objects of lower temperature (for example, the evolved population of stars, similar to the central parts of spiral galaxies or elliptical galaxies and globular clusters). The maximum of the Planck blackbody spectral energy distribution of young population objects is situated in the far UV, since this maximum is in the visible and the IR for the evolved cooler population. The intensity of the average Planck curve of the low temperature population is practically negligible below 2500 A;thus, observations around 2000 A, for the study of evolutionary and morphological problems, give results very similar to those obtained, with the more severe technical difficulties (rockets and satellites) in the far UV. This has been verified, especially by comparison of the Large Magellanic Cloud images obtained in the 1500 A range by Carruthers (Apollo-16 Mission on the Moon) with the S 183 Skylab results (COURT&,LAGET,SWAN,VrroN, VUILLEMIN and ATKINS [1975]). The central bulge of the Large Magellanic Cloud is one of the best examples of evolved stars, very strong in visible light; on the LAS Skylab S 183 photograph it was not detected at 2500 A (Fig. 3.11).

3.4.1. Balloon experiment The transmission of the Earth’s atmosphere above the first ozone layer (above 40km) shows a window of 130A at half intensity peaked at 2020 A (Fig. 3.12) (ACKERMAN, BIAUMEand KOCKARTS[1970]), usuable for observing hot blue objects from a balloon. The emissions of the upper atmosphere are faint enough in these bandwidths to not compromise the astronomical observations, as has been shown by COHEN-SABBAN and VUILLEMIN [1973] and COURTES[19713. Indeed, this atmospheric bandwidth at 2020 A gives the certainty of selecting only high temperature objects since this bandwidth is at a shorter wavelength (2000 A)than that

I,

P 31

AVERAGE FIELD TELESCOPE

31

k Two mirrors bandwidth

Combination of the two mirrors bandwidth and lha atmosuherir

I I I I

I I I

/

Fig. 3.12. Bandpass given by the two mirror coatings of the SCAP Schmidt Cassegrain telescope. The dotted part of the atmospheric handpass corresponds to the envelope of the bottoms of the narrow molecular 0, absorption bands. The C 1111 interstellaremission line is by chance transmitted by coincidence with a transmission window near 1909 A.

of the S 183 Schmidt camera (2500 A) of Skylab (Fig. 3.11). Another advantage is that of being able to reach one of the best high excitation indicators of the interstellar gas, the C 1111 emission line at 1909 A. All these considerations led Viton and S a k e to design a new flat field anastigmat Schmidt Cassegrain telescope (DETAILLE and S ~ S S[E 19751) to be flown on a balloon gondola of the Geneva Observatory and the Laboratoire d’Astronomie Spatiale of the CNRS (Fig. 3.13). The balloons and their logistics were under the responsibility of the CNES (Centre National d’Etudes Spatiales).

3.4.2. The SCAP telescope The detector is an I l T multichannel array of 25 mm diameter combined with conventional Kodak IIaO 35mm photographic film. The

32

[I, 8 3

DETECTION OF ASTRONOMICAL OBJECTS

Fig. 3.13. The Geneva-Marseille gondola with its Schmidt Cassegrain telescope and its siderostat monitored by a screw jack tilting system.

maximum resolution obtained is about 100 p,m; the accuracy of the pointing system is k30”. The need for easy photometric reduction of the results lead us to have, as nearly as possible, a purely Gaussian distribution of the light for each stellar image in the focal plane. For this reason, the focal length has been

A 0 DEG

B 1 Dft

C 1.I

DfC

D 2.5 DEC

Fig. 3.14. SCAP telescope for balloon experiments. Image quality versus the field angle. The circle diameter is 1.5 arc min.

I, § 31

AVERAGE FIELD TELESCOPE

33

TABLE3.4 The SCAP telescope: Main characteristics

chosen smaller in order that the maximum diameter of the limit cycle be smaller than the 100 pm resolution of the detector-optical combination. The choice was a Schmidt Cassegrain flat field (zero Petzval sum) telescope (Fig. 3.14, Table 3.4).

3.4.3. The siderostat The SCAP telescope with its startracker must not be too difficult to point towards different directions of the sky. The adopted solution is a light-weight siderostate (Fig. 3.15), thanks to the startracker (HUGUENIN and MACNAN [1978]). The main difiiculty is the rapid change of temperature, requiring heating and, consequently, a small expansion coefficient for the Pyrex of the lightened mirror. The mirror remains flat enough for the image quality defined above. This type of azimuthal astronomical mounting gives rise, during the exposures, to a field rotation; but the high sensitivity of the ITT multichannel array allows exposure times short enough to reach the noise and background before any deterioration of the image quality occurs. One of the best successes of the SCAP has been to provide the first good-quality UV images of the nearby galaxies M 31, M 33, M 81 (Figs.

34

DETECTION OF ASTRONOMICAL. OBIECTS

[I, § 3

Fig. 3.15. Light-weight Pyrex siderostat mirror.

3.16, 3.17) and M 101, and to detect, as point sources, nearly 300 more distant galaxies during two flights (COURTES, GOLAY,VITON,BENTZ, DEHARVENG, LAGET, DONAS, SWANand MILLARD [19801). Detection of UV objects follows the predictions and reveals the most active places of stellar formation, especially the spiral arms. In the case of M 31 (the Andromeda Galaxy), the central bulge is detected in the UV. Previous experiments (CODE[ 19691, CRUVELLIER, ROLJSSIN and VALERIO [1970], COLJRTES, LAGET,SIVAN, VITON,VU~LLEMIN and ATKINS [ 19751, DEHARVENG, LAGET, MONNETand VUILLEMIN [19761, CARRUTHERS [19781) have shown this unexpected UV radiation in the central parts. The SCAP experiment provides, for the first time, a well defined morphology. A possible interpretation of this result could be in a large amount of dwarf blue stars at the end of their evolution (DEHARVENG, LAGET, MONNET and VUILLEMIN [19761, DEHARVENG, JAKOBSEN, MILLIARD and LAGET[19801). These results are extremely interesting since they are the beginning of

1, 5 31

AVERAGE FIELD TELESCOPE

35

Fig. 3.16. (a) M 31 at 2000 A. Bright hot population of the spiral arms of the galaxy M 31. One notes the bright UV central parts in the middle of the evolved populations of the bulge (Marseille-Geneva balloon program). (b) M 31 in the “visible” radiation. One notes the strong brightness of the bulge in visible light. (From the Palomar Sky Survey.) (COURTI~, GOLAY,VITON,BENTZ,DEHAFWENG, LAGET,DONAS,SIVANand MILLIARD [1980].)

36

DETECTION OF ASTRONOMICAL OBJECTS

Fig. 3.17. ( a ) M81 in visible light (from the Palomar Sky Survey). (b) M 81 at 2000 A. Only the spiral arms populations are detected. (Marseille-Geneva balloon program.) (COURT&, GOLAY,VITON,BENTZ,DEHARVENG, LAGET,DONAS, WAN and MILLIARD [1980].)

1, §31

AVERAGE F’IELD TELESCOPE

37

imagery of the most important “unit” of the Universe, the galaxies; but many other observations (diffuse nebulae, blue stars of the halo, etc.) have been made, up to the magnitude V = 13, for point-like sources. Continuation of this program leads to a large telescope (40 cm diameter) (FOCA-1000) capable of reaching magnitude V = 16 and to approach more distant galaxies for which no other observing program is foreseen before the Space Telescope. 3.4.4. Optical design of FOCA- 1000: astronomical telescope for balloon observations The good behaviour of the SCAP telescope and its pointing system suggests imagining a larger instrument without changing the most technically difficult parts of the SCAP. The siderostat, in particular, is maintained with a small increase in its dimensions.* Improvement of the pointing system allows reaching the resolving power of a one meter focal length telescope combined with the ITT multichannel arrays, say 100 pm or 20”. In any case, the unavoidable limitation in diameter of the ITT detector (40mm) limits the field to a value compatible with the field rotation. The need for compactness and a relatively high focal ratio (sensitivity to extended sources) leads to a Cassegrain telescope with a two-silica-lens corrector near the focal plane. The limited bandwidth given by the

Fig. 3.18. The 40 cm diameter FOCA-1000 telescope for balloon observations. *Note added in proof: During the publishing of this paper we decided to point this telescope directly to the sky without siderostat.

38

[I, 5 4

DETECTION OF ASTRONOMICAL OBJECIS

A

B

D

C

a* D ~ G

0 , ) DEG

0 DEG

t , Z DEG

Fig. 3.19. Image quality in functions of the field angle. The circle diameter is 20 arc sec.

TABLE 3.5 The FOCA- 1000 telescope: Main optical characteristics

- Focal length: 1 meter F number: 2.5 Field of view: 2.5" * Star image quality: 10 arc-second blur circle diameter Spectral range 185-250 nm * *

-

TABLE 3.6 The FOCA-1000 optical parameters (partially optimized design data of the two-lens corrector solution)

Surface

Radius (mm)

1

-1166.6 (parabola)

2

-1166.6

Thickness or seperation (mm)

-342 -202.15 3

Diameter (mm)

Zerodur mirror

400

-1 Zerodur mirror

170

-I

288.45 9.87

4

Material

UV grade fused silica

100

UV grade fused silica

100

139.12 20

5

-849.80 9.87

6

-200.57 186.63'

' Manufactured by Schott. To the image plan.

I, 841

SPACE TELESCOPES OF THE FUTURE

39

atmospheric window allows this lens corrector without excessive chromatism. The general design of Saisse and Grange is composed of a concave parabolic main mirror and a convex spherical secondary mirror. This combination, with the addition of a two lens corrector, leads to a spatial resolution of 10” within the whole spectral band (Figs. 3.18, 3.19; Tables 3.5, 3.6).

6 4. U V Space Telescopes of the Future 4.1.INTRODUCTION

All the preceding descriptions concern relatively small instruments, very different from conventional telescopes for ground based observations. Two large instruments are now studied: the Space Schmidt Telescope (SST) of 75 cm diameter and the Space Telescope (ST) of 2.4 m diameter. This corresponds to the beginning of a new period of Space Astronomy, thanks to these first really big telescopes. The SST provides a wide field (5”). Keeping in mind the present remarkable performance of the big ground based Schmidt telescopes, it is apparent that the SST will give tremendous possibility of discovering new faint UV and “visible” objects up to magnitude V = 26. For example, the Anglo-Australian 48” Schmidt recently detected a large number of faint objects of magnitude V = 23 of very great cosmological interest, within the scope of the future ST observations. The SST is a “survey” instrument giving a high probability of discoveries, thanks to its large field. On the other hand, the ST will be the absolute tool of deep probes for future astronomy. It will be the first instrument in which the two main qualities of space observation will be used simultaneously: (i) access to the UV and the near IR, without the background of the upper atmospheric emission and (ii) high spatial resolution up to the diffraction limit - performance impossible for a 2.4 m ground based telescope because of the atmospheric turbulence. But this second quality, combined with the diameter of the electronographic detectors, leads to a very small field; thus only previously identified objects will be observed with the ST. The Space Schmidt, SST, instrument of discovery of new faint objects, and the Space Telescope, ST, for precise studies, are very complementary, as was already the case for the simultaneous use of the 48” Schmidt telescope and the 200” Hale telescope of Mount Palomar. Concerning high spatial resolution, we can note that even for the smaller Space Telescope, very few experiments have used the possibility

FOCAL P L A N E AT F I L M

FIELD MIRROR/

-INDARY M I R R O R ELLIPSOID

2-POWER L E N S FOR PHOTOGRAPHY

RETRODIVIDER &/SO FOCAL P L A N E (OTHER RETRODIVIOER PMOTOYllLT I P L IE RS NOT S H O W N )

:AL P L A N E SERVOCONTROLLED TRANSFER LENS FINE TELEVISION

FIELD LENSES F / 2 0 FOCAL P L A N E

PRIMARY M I R R O R FOCAL L E N G T H 144”

W

Fig. 4.1. Layout of the Stratoscope I1 telescope (Courtesy,Sky and Telescope).

I. $41

SPACE TELESCOPES OF THE FUTURE

41

Fig. 4.2. Seyfert galaxy NGC 4151 at the diffraction limit of the Stratoscope I1 balloon telescope. One sees the first difiaction ring of the point like nucleus (Courtesy, Sky and Telescope).

of the diffraction limit, chiefly because of stabilization problems; one can and DANIELSON [19701 recall the balloon experiments of SCHWARZSCHILD of Princeton University, on the Sun and on some local galaxies, thanks to Stratoscope I1 equipped with a 90 cm diameter telescope giving a resolution of 0.2” (SCHWARZSCHILD [1973]). This telescope was a gregorian system (Fig. 4.1) with a very large focal length of 90m. It gave the diffraction limit on the nucleus of the Seyfert galaxy NGC 4151 (Fig. 4.2). The short exposures used on the Sun sometimes allow with success the use of small telescopes at the diffraction limit on rockets (Fig. 4.3) and balloons (COURTESand BONNET[1962]*, BONNET,LEMAIRE,VIAL, JOUCHOUX, LEIRACHER, SKURMANICH and VIDALARTZNER, GOUTTEBROZE, [1978], HERSE[1979], BONNET,BRUNER,ACTON,BROWN and MADJAR DECAUDIN [1980]); but the problem is much more difficult for faint stars and nebulae which need at least several minutes exposure time, even with the aid of image intensifiers. *This space experiment was the first to use field grating filter techniques (BPM filter) (COURT&[1962, 1964, 19711). This technique is interesting when one wants to use, simultaneously, several bandpasses with a much better transmission than that given by far UV interference filters. See also BLAMONT and BONNET[1965, 19661.

42

DETECTION OF ASTRONOMCAL OBJECTS

Fig. 4.3a. The first Ly a photograph of the sun obtained at 1 arc sec resolution by BONNET, BRUNER, ACTON,BROWNand DECAUDIN [1980] (Laboratoire de Physique Stellaire et PlanCtaire). One sees the extremely sharp magnetic structures, the spicules and the prominences on the sun limb. NASA rocket launched from the White Sands Base (New Mexico) with the cooperation of Lockheed Palo Alto Research Laboratory. The instrument was a 10.6cm diameter Cassegrainian telescope of 2.5 m focal length. Because of the pinholes of the interference filters, the Ly a filter is in fact made of two separate filters which gives a 50-100 A bandpass.

Fig. 4.3b. Photograph on 103 a 0 UV Kodak emulsion of the Sun obtained by Herst (Service d’Aironomie du C.N.R.S.) on October 5 , 1970 at 2000 A (150 8, bandwidth). The field is 2 . 5 ~ 3 . 7 5arc min. One sees the solar limb and details in the faculae and the granulation up to 0.5 arc sec resolution. Telescope: 20 cm diameter-equivalent focal length 16 m. Astrolabe Gondola of the C.N.E.S.

1, $41

SPACE TELESCOPES OF THE FUTURE

43

4.2. THE SPACE SCHMIDT TELESCOPE (SST)

The large spectral range of the SST excludes the use of a classical aspherical Schmidt plate for correction of the spherical aberration, because of the obvious reason of the strong variation of the refractive index with wavelength in the UV, as well as because of the difficulty of polishing such a large plate in CaF, or MgF,, either as one piece or as a mosaic. Some all-reflexion Schmidt telescopes have been built, especially by EPSTEIN [1967], MORTON,JENKINS and BOHLIN[1968] and O’CAL LAGHAN, HENIZEand WRAY[19771 of Northwestern University, Evanston, as well as by LEMA~TRE $19791 of the Marseille Observatory.

4.2.1. Description of the optical design The detector is a Carruthers magnetically focused electronic camera (CARRUTHERS [1978]) of field 125 mm X 125 mm which is under development but has already been tested at the NRL. The detector will have a curvature coincident with the natural field curvature of the Schmidt. The geocoronal Ly (Y emission is eliminated by a heated* CaF, window. The bandwidth of the CaF, window combined with the sensitivity curve of the CsI photocathode will be 300 A wide. The expected quantum efficiency is 0.15. The resolution of this detector will be 30 pm, corresponding to 3” on the sky, far from the diffraction limit of a 75 cm diameter telescope; but resolution is not the main goal of this telescope, which is devoted to large-field detection of faint objects. Nevertheless, it is essential to obtain excellent image quality all over the field of 5”. A pre-study has been made by the NASA-SST Scientific Group and SCHROEDER [1979] (Table 4.1); according to this study, 0.93 of the light will be in a 1.5” diameter circle for a point source. One sees in Fig. 4.4 and Fig. 4.5 the schematic view of this all-reflexion Schmidt telescope and its location on the Spacelab Pallet. The flat mirror is used only to fold the beams and to provide easier access to the focal surface. In this type of telescope, the center of the pseudo-flat correcting mirror is situated at the center of curvature of the spherical mirror. If 0 is the focal ratio and p the half field of the telescope, and if one requires zero vignetting for the whole 2 p field, the correcting mirror must be tilted at an angle a > p +$a. *See p. 16, 5 2.2.6.

44

DETECTION OF ASTRONOMICAL OBJECTS

TABLE 4.1 First-order parameters of the Perkin-Elmer study for the SST telescope Effective focal length Field of view Image diameter Aperture area

F number ' Back focal length Unfolded length Obstructed area Plate scale Image radius Image size

1.9672 meter 4.667" 160.0 mm 0.865 meter (equivalent diam.) (0.98dia. X 0.6) Fl2.00 X Fl3.28 0.243 meter 3.915 meter 10% 0.572 mmlarc-min 1.9626 meter 1.0 arc-sec (toleranced)

The conventional all-reflection Schmidt uses an axisymmetric aspherical mirror (Fig. 4.6). According to a recent study by L E M A ~ R [1979], E zero power of this mirror should be at a radius of r, = ($)"'rm ( r , = radius of the pseudo-flat mirror aperture) instead of the Kerber or Schroeder classical value or r, = $rm. This value r,, which comes from the third order theory, has the zero power zone outside of the contour of the entrance pupil of the Schmidt combination. The new method of polishing under mechanical deformation ( L E M A ~ R [19791) E allows the use of elliptical instead of axisymmetric deformation (Fig. 4.7). This elliptical deformation is the projection of the axisymmetric deformation on the plane perpendicular to the axis of the spherical mirror XX'. (This solution has been applied to the Very Wide Field Camera Schmidt system, see Fig. 2.6.) In

Fig. 4.4. The optical layout of the Space Schmidt telescope from the Schroeder study.

I, 541

45

SPACE TELESCOPES OF THE FUTURE

APERTURE B A F n E SYSTEM

,

NRL ELECTPRONOGRAPHIC CAMERA

TELESCOPE RDUSING

Fig. 4.5. General mounting of the Space Schmidt telescope on the pallet of the Spacelab.

Fig. 4.6. Conventional axisymmetric aspherical deformation (dotted circle) of an allreflection Schmidt telescope.

46

[I, 5 4

DETECTION OF ASTRONOMICAL OBJECTS

Fig. 4.7. Lemaitre solution with an elliptical aspherisation of the correcting plate.

this solution, proposed for the SST by Lemaitre, the radius of the zero power zone r, is different; in fact, a mechanical figuring solution appears for simply r, = r,. Spot diagrams show the image quality improvement of this new design (LEMA~RE [1979]) (Fig. 4.8). the main dficulties are certainly the field curvature (R/2=2000mm), and the need for the relatively thick face plate in a fast focal ratio. In order to obtain a 1”image diameter, the thickness of the face plate must be less than 1mm.

. ....,..

:

- ..... )*i : ;. . ’., +?JC , ’’

I....

,

-

<.:

.’?

\.

.;

.;g.,

-

- 0 - -

.!!.

-

’. .w:,: .

c:.3

*

3

-low*

- 5 fl

...._ .... : ...;,...’...:::::.. . ... :........ ...... ..,..t

1

I

rol’m

Fig. 4.8. Spot diagrams show the deformation by the residual aberrations of the pupil mask (bottom left) for different values of the radius of the neutral zone. Curves give the variation of the resolution for extreme fields 2 and 3 for the central parts of the field.

1, 441

SPACE TELESCOPES OF THE FUTURE

41

4.3. THE SPACE TELESCOPE (ST)

The 2.4 m Space Telescope has been designed, as we said before, for a difIraction limited mode at A = 6000 A. The use of several auxiliary instruments at the focus of the ST without a difficult mechanical exchanger system leads to an original optical solution never used on ground based telescopes (Fig. 4.9). The optical telescope assembly (OTA) is an F/24 Ritchey-ChrCtien, providing a large available field of sharp images. (The only remaining aberration is the astigmatism which increases with the square of the field angle.) The field of view of the telescope is used by five scientific instruments (SI’s). The radial SI uses the central part of the field (2.67 x 2.67 arc. min); this is the Wide Field and Planetary Camera (WF/PC). Four axial SI’s (faint object camera, faint object spectrograph, high resolution spectrograph, high speed spectrometer (LECKRONE [19801)) use the outer part of the ST field. Ninety-five percent of the surface of the field is not used. In the beginning of the project, proposals to collect this image surface (with a field mirror or a field grating) were rejected by NASA because of excessive telemetry data.

Fig. 4.9. Configuration of the Space Telescope showing locations of four axial-bay scientific instruments and one radial-bay instrument.

48

DETECTION OF ASTRONOMICAL OBJECTS

[I, § 4

In keeping with the bidimensional theme of this paper, we will describe the W / P C for information, and the FOC because of the LAS participation in the optical design of this instrument. It is the only instrument of its kind prepared in Europe, according to an agreement between ESA and NASA. 4.3.1. The Wide Field and Planetary Camera ( WF/PC) of the ST: The radial - bay instrument The problem was to record, on CCD receptors from 1160A to 11500 A, the widest field with the highest quality images from the OTA, compatible with the data storage and the telemetry rate. This condition leads to a surface of detection much larger than that of CCD detectors at the present state of the art. The resolution is as high as possible, but this is not the main accomplishment of the W / P C ; it suffices that the FOC fulfills these high resolution requirements. The solution found by the designer (WESTPHAL [1979]) consists in using a focal reducer (F/24 to l712.88) similar in principle to those used in ground based telescopes (COURT&[1952a], MEINEL[1956], COURTES 1119641) and in some space telescopes like that of the UTEX project (DETAILLE, SCANDONEand BENVENUTI [1976]). The field lens of the conventional reducer is replaced by an off-axis concave field mirror which forms the image of the entrance pupil on a small H12.88 Ritchey-ChrCtien telescope system. The dimensions of the linear field increase by a factor of two, hence of four in surface, thanks to this reduction technique. A complementary way of increasing the field again by another factor of two is to put together four

Fig. 4.10. Optical system principle of the WideField Camera. Only one optical channel the field surface) of four is represented here.

(a of

1, $41

49

SPACE TELESCOPES OF THE FUTURE

square field mirrors close to the focal surface of the RC telescope. Figure 4.10 shows the final arrangement. This optical design provides a field 16 times larger in surface than that of the detector alone. The field is of 2.67x2.67 arc min or 6 0 m m x 60mm in the focal surface of the RC telescope. Obviously the only possible place for four such symmetrical cameras is the very centre of the focal surface. However, because of dimensional constraints, the WF/PC is located in a radial bay; the light is reflected, thanks to a plane pick-off mirror on the optical axis of the telescope. The complex solution of four joined optical systems was unavoidable because the detector was so limited in size. In any case, it is advantageous because of the necessary redundancy of such an exceptional telescope. The increase in focal ratio corresponds to an illumination, at the detector surface, four times that at the direct RC focus. This larger focal ratio favours the detection of extended faint sources, extensions of galaxies, intergalactic matter, etc. A series of 53 interference and coloured filters, transmission grating and polarizers supplements the multipurpose imagery of this instrument. This instrument will reach the 17th magnitude with an exposure of 0.1 sec, and the 28th in a 300 sec exposure. In the planetary camera mode, PC, the focal ratio is F/30 in order to obtain a longer focal distance for a better definition of planetary images (Table 4.2). TMLE4.2 Wide Field Planetary Camera “WF/FC” This instrument can operate at two different focal ratios: F/12.88 or F/30. In the first mode, the instrument is referred to as the Wide Field Camera (WFC) and, in the second mode, as the Planetary Camera (PC). Pictures can be taken in either mode, with any one of a wide variety of spectral filters or transmission gratings. Characteristics Field of view Angular resolution (1 pixel) Linear resolution (1 pixel) Bandwidth (quantum efficiency z=1“/o) Photometric accuracy Dynamic range (SIN 3 3)

WFC 2.67 x 2.67 (arc min)’ 0.1 xO.1 (arcsec12

0.043 x 0.43 (arc sec)’

15pmxlSp.m

15pmx15pm

1.15 x lo3A to 1.1 p 1% 9.5 s m , c 2 8

1.1x103A to 1 . 1 ~ 1% 8.5 S m, =z28

50

DETECTION OF ASTRONOMICAL OBJECTS

[I, § 4

4.3.2. The Faint Object Camera (FOC) Like the WF/PC, the FOC reflects the great variety of observing programs of the worldwide astronomical community. Here, the main purpose is both sensitivity and diffraction-limited resolution, combined with the possibility of observing very faint objects close to bright ones (coronograph mode), and the possibility of obtaining low resolution spectra of very faint objects using two objective prisms or a long slit 20"x 0.1" spectrographic facility composed of a fixed reflection grating and a removable mirror which can be inserted into one of the imaging mode beams. In addition, we note: (i) a high spatial resolution mode (F/288) that will allow a complete analysis of the diffraction pattern of a point source given by the ST, and (ii) a mode using a UV optically contacted magnesium fluoride Rochon polarizer in order to measure polarization of stars and nebulae. Note first that for the ST, in the difh-action-limited mode, sensitivity and spatial resolution go together. In a ground based telescope, the best images obtained with good seeing and during the minimum exposure time needed for faint objects are of the order of 1"; they can be 0.5" during exceptional nights. The ST will give 0.1" at worst; so one sees that, for the same flux collected by the large mirror, the stellar image will be ten times smaller, and hence the illumination of the detector will be a hundred times stronger. This authorizes a jump of 5 magnitudes for the same sky background brightness if the signalhoke ratio is large enough for the point source to be considered as the limiting magnitude. But, in fact, above the atmosphere, the sky brightness is fainter by about a factor of two in the middle of the visible spectral range (5000& and much fainter in the UV and IR (Fig. 4.11). The gain, in comparison with a ground based telescope, will of course be limited by the necessary time of integration, For ten hours cumulative exposure, the ST will reach magnitude V = 28 with a signal/noise ratio of 4 and will give probes of the Universe ten times deeper in distance. A volume of the Universe a thousand times larger will be evaluated, certainly changing much of our cosmological knowledge. In terms of imagery, a distant galaxy, giving a Gaussian light distribution of 1" diameter with a ground based telescope, will show a hundred discrete image elements when it is observed by the Space Telescope. The morphological class and the state of evolution of this distant galaxy will be easily recognized, thanks to these one hundred image elements.

51

SPACE ELESCOPES OF THE FUTURE

magnitudelarc sec’

I

i

L

L

, L L

3000

5000

-

7000

AM

L

.

.

d

SO00

Fig. 4.11. Brightness of the sky background showing the advantage of the extraatmospheric observation even in the “visible” range, chiefly in the red and near infra-red. Dotted line: sky background at Kitt Peak observatory (Arizona). Dark line: sky background from above the atmosphere.

4.3.2.1. The detector (Fig. 4.12). The detector is derived from a Boksenberg Photon Counting detector (BOKSENBERG and COLEMAN [19791); this system consists of a series of three magnetically focused image intensifiers of very high gain (-lo6), followed by a television tube capable of detecting and recording all the bright spots (events) of the phosphor screen due to the accelerated electron impacts. A pixel can be 25 p.m or 50 p.m, thanks to an electronic “zoom”, and its dimension leads to the PERMANENT MAGNET

5-10 PHOTOCATHODE

,,I EBS TARGET

PHOSPHOR

(TOTAL W N 10’)

ASSEMBLY

-

TO * SCIENTIFIC ~ DATA STORE

VIDEO PROCESSING UNIT

Fig. 4.12. Design concept for the faint object camera’s photon-counting detectors. Two [ 19791). identical units are used (BOKSENBERG

TABLE 4.3 The Faint-Object Camera “FOC”

lJl

N

The faint-object camera consists of two independent camera systems that operate, respectively, at F/96 and F/48. The F/96 system contains a coronagraphic facility that can be used to mask the light from bright objects. The F/48 system also provides for long-slit ( 1 0 ~ 0 . (arcsec)’) 1 spectroscopy with a fixed grating F/48 22 x 22 (arc sec)2 44x44 (arcsec)’ at slightly degraded resolution TV (zoom)

F/96 1 111~(arcsec)’ 22 x 22 (arc sec)2 slightly degraded resolution TV (zoom)

Fl288 4 x 4 (arc sec)’ 8 x 8 (arc sec)’

Pixel sue

0.045 x 0.045 (arc s e ~ ) ~

0.022 x 0.022 (arc sec)‘

0.066 x 0.066 (arc sec)’

8

Wavelength range quantum efficiency >1%

1200 A4000 A

1200 A6000 A

1200 A4000 A

54

Dynamic range (cumulative 10 hours observations without attenuating filters or combining pixels; SIN = 4)

point sources: 21 m, to 28 m, extended sources: 15 m,/(arc sec)’ to 22 m,/(arc sec)’

point sources: 21 m, to 28 rn, extended sources: 15 m,/(arc sec)’ to 22 m,/(arc sec)2

Photometric accuracy (when not photon-noise limited)

at least 2%

at least 2%

Spectrographic mode long slit

0.1” x 20“

Characteristics Field of view (see text)

Wavelength ranges

1200-1800A 1800-2700 A 3600-5400

a

Enlargement in one direction (TV-zoom)

resolution 1.5 A 2.2 A 5.4 A

250 x 1000 pixels

frequency 270 4% 35%

3

0

3 z

1, 041

SPACE TELESCOPES OF THE FUTURE

53

focal length needed to achieve the diffraction-limited resolution of 0.1”; in fact, a pixel corresponds to 0.022” for F/96 and 0.044” for F/48. The direct Cassegrain RC focal length is insufficient and must be increased by at least a factor two. This is the reason for this F/48 mode. Better redundancy is obtained with a second optical channel at F/96 and, finally, an F/288 mode will provide a kind of “speckle” interferometry for some very specific objects, as well as photometric evaluation of bright sources (distribution of the light on a larger number of pixels, in order to keep the illumination below the saturation of the individual pixels). It is interesting to remember that non-linear response of the detector appears as soon as a one “event”/pixel-sec flux is reached; in terms of stars, this corresponds to a 21st magnitude star (Table 4.3) in the F/48 and F/96 mode. We have already pointed out the position of the flat pick-off mirror of the WF/PC, right in the centre of the Ritchey-ChrCtien focal surface. Consequently, the FOC cannot be at the centre; thus we have the paradoxical situation of using an astigmatic part of the image to obtain the highest space resolution! The enlarging image system must be designed not only to increase the final focal length, but also to correct this primary astigmatism (1.8 mm length); the toroidal folding mirror resolves this problem. This leads to a slight tilt of the off-axis field because of the field curvature of the Cassegrain system. The enlargement of the FOC increases this difficulty. The detector surface is consequently nonperpendicular to the incident beams. We shall see that the astigmatism also complicates the concept of the occulting mask in the coronograph mode.

4.3.2.2. Description of the optical design. The optical design is the result of the general conclusions of the NASA and the ESA Committees and of continuous cooperation between the LAS group and the principal investigator of the FOC, Dr. Macchetto. Thanks to a permanent feed back with the Committees of the FOC-STC, the ST-WG of the ESA Agency Sai‘sse, Deharveng, Mauron and Court& (SAME [198 11) have adapted the required specifications to the realistic possibilities (MACCHE~O and LAURENCE [19771). The adopted design is the following (Fig. 4.13); the off-axis beams are reflected by a concave spherical mirror and a convex aspherical mirror, giving an aplanatic Cassegrain combination. A pseudo-flat (toroidal)

54

DETECTION OF ASTRONOMICAL OBJECTS

[I,

P4

Fig. 4.13. The optical combinations of the Faint Object Camera (FOC) of the Space Telescope.

mirror folds the beams and corrects the astigmatism of the RitcheyChrCtien system of the ST (Figs. 4.14, 4.15). For dimensional reasons, and because of the large size of the two Boksenberg type detectors, the incident angles of the beams on the concave-convex mirror combination are exceptionaly large. In spite of

I, 941

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Fig. 4.14. Spot diagrams of the whole optical system (OTA+FOC) for F/48 and F/96. Mode and three angular position. [(A) field center, (B) 5.5 arc sec field for F/96, 11 arc sec field for F/48,(C) 11 arc sec field for F/96, 22 arc sec field for F/48.]

Fig. 4.15. Spot diagrams of the whole optical system (OTA+FOC) for the F/48 spectro mode. Comparison between the pixel sue and Airy circle with the geometric blurred function. (B: slit center, A and C: slit edges.)

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this difficulty, one can see the excellent image quality on the spot diagram. This quality is not compromised by practical production problems. The maximum wave-front deviation, with respect to the design, does not exceed A/15 at A = 633 nm, on the flight mirrors made by Stigma-Optique and Matra firms in Paris. There are two Cassegrain relay systems in order to obtain, from the ST focal ratio F/24, one combination at F/48 and another at F/96. The two imaging modes (F/96 and F/48) are located in the meridional plane of the scientific instrument. The F/288 system consists of another small removable Cassegrain, situated at the exit pupil of the ST+F/96 relay system combination (Fig. 4.16).

Fig. 4.16. The F/96-F/288 optical relay and its mock-up in Zerodur (Schott), polishing by Stigma-Optique.

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4.3.2.3. Verification of the optical performance of the FOC and its coronograph mode. A simulation experiment has been performed at LAS [1979]) to verify the quality of the diffraction limited optical (MAURON combination by photographing the enlarged image of an artificial star. One sees, in Fig. 4.17, the diffraction pattern of the final image in the detector. Resolving power has been checked by using an artificial double star of V = 22 and progressively increasing the distance. MAURON[1980] has shown that this criteria of duplicity, corresponding to the diffraction-limit, appears when a large redundancy is given to the image size in comparison to the pixel size. This experiment justifies the F/96 mode. The coronograph has also been verified by Mauron; the result (Fig. 4.18) shows that in a satellite at lo-' the intensity of the central star is easily detected (directed image, without image processing) at only 1.3 arc sec from this central star. Even if the coronograph does not reach the nearest planetary systems, it will be a very powerful instrument for observation of faint extensions around very distant starlike galaxies, interstellar gas close to the bright galactic nuclei, gaseous shells around neighbouring stars, etc. Wavelength selection is obtained for all optical modes of the FOC with a filter on a set of 3 wheels; most of the filters are interference filters. They are situated close to the exit pupil (a better position in the focal plane was impossible because of the pinhole diffraction of the filters). A wheel device of 45 filters is a complex design. Another solution would have been a filter grating system (B.P.M. in French) (COURT&[1962,

Fig. 4.17. Control interferograms of the F/96 relay mirrors and image given by the aplanetic combination.

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(i 5

Fig. 4.18. Images at the coronograph focus obtained with an occulter (cross shape width 0.57") and a classical Lyot apodizer. The intensity of the simulated star on the right is 2X that of the main star, and 2 x lo-' for the first ring. The distance between the two stars is 1.3"(MAURON [1979]).

19641); transmission and bandwidth definition are much better for a grating than for an interference filter. This non-conventional solution has not been adopted.

CJ 5. Conclusion A subject of research completely neglected ten years ago by the principal space astronomy programs, the bidimensional study of extended astronomical objects in UV light, has provoked strong interest for new instrumentation, from the Very Wide Field Camera with a 3 mm entrance pupil of the first generation (V, = 10th magnitude), to the 2400 mm ST

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telescope (V, = 28th magnitude). Since the beginning of this instrumental approach, the very large range of intensities of celestial bodies has permitted astronomers to obtain important new results in spite of the very modest aperture of the first space cameras. The methods of ground based astronomy have always followed the procedure which consists of using progressively large wide field telescopes, like the 18 inch and then the 48 inch Schmidt telescopes of the Mount Palomar Observatory, in order to discover new faint objects which are later observed by the powerful (but limited field) 200 inch Hale telescope. In space astronomy, the philosophy was quite different; the first projects were devoted to stellar spectrography of known bright objects. In recent years, the previous method of observational astrophysics, which was so efficient and successful in the past for ground based astronomy, has been rediscovered with the instruments described in this paper. Undoubtedly, this new path in space research, thanks to the constant progress of space optics, leading to the ingenious use of a large telescope above the atmosphere - the 2.4 m Space Telescope - will drastically change many of our concepts about the origin and the evolution of the Universe. For example, the present most remote Q.S.O. (2= 3.53)* corresponds to the first lo9 years of the Universe’s life. The time of the ST after 1984 will represent a much more abrupt transition in our knowledge than the first observations with the 5 m Hale telescope at Mount Palomar thirty years ago.

Acknowledgements We thank J. Caplan for the revision of the manuscript, Mrs. Reforzo, Mrs. Leclerc and J. P. Gouda1 for the typewriting, the composition and the photographs.

References ACKERMAN,M., F. BIAUMEand G. KOCKARTS,1970, Planetary and 1639-1651. BARANNE,A,, M. DETAILLEand G. LEMAITRE, 1969, Optical Instruments (Oriel Press Ed.). BLAMONT,J. E. and R. M. BONNET,1965, Solar limb darkening in (2000-3000 A), Proc. VIth Intern. Space Science Symp., Mar del Plata,

Space Sci. 18, and Techniques UV continuum Sp. Res. VI, 8.

* Note added in proof: During the publishing of this paper a new quasar, PKS 2000-330, has been discovered with a red shift of 3.78 (0.92 of the speed of the light) at a distance of 16 billions light years from our Galaxy.

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BLAMONT, J. E. and R. M. BONNET, 1966, Mesure de I’assombrissement du centre au bord du disque solaire entre 2000 A et 3000 A, C.R. Acad. Sci. B262, 152. BLAMONT, J. E. and R. M. BONNET,1967, Mesure de I’assombrissement centre-bord du soleil entre 2000 A et 3000 A, C.R. Acad. Sci. B264, 1158. BOKSENBERG, A. and C. I. COLEMAN, 1979, Photon electronic image devices. Advances in Electronics and electron physics, Vol. 52, eds. B. L. Morgan and D. Mc Mullan, p. 255. BONNET, R. M., E. C. BRUNER, L. W. ACTON, W. A. BROWNand M. DECAUDIN, 1980, Astrophys. J. 237, no. 2. BONNET, R. M., P. LEMAIRE, J. C. VIAL, G. ARTZNER, P. G O U ~ B R O ZA. E ,JOUCHOUX, J. W. LEIBACHER, A. SKURMAMCH and A. VIDAL-MADJAR, 1978, Astrophys. J. 221, no. 3, Part 1, 1032. BORGMAN, J., R. J. DUNENand J. KOORNNEEF, 1975, Astron. Astrophys. 40, 461. CARNOCHAN, D. J., M. M. DWORETSKY, J. J. TODD,A. J. WILLISand R. WILSON,1975, Phil. Trans. R. SOC.Lond. A219, 479-485. CARRUTHERS, G. R., 1973, Applied Optics 12, 2501 CARRUTHERS, G. R., 1978, Space Science Instrumentation 4, no. 1, 3. CARRUTHERS, G. R., H. M. HECKATHORN and Chet. B. OPAL, 1978, Astrophys. Journal 225, 346-356. CODE,A. D., 1969, Publ. Astron. SOC.Pacific. 81, 475. COHEN-SABBAN, J. and A. VUILLEMIN, 1973, Astrophys. and Space Sci. 24, 127-132. COURT&,G.,1952a, L‘Astronomie 66,261. ’ COURT&,G., 1952b, C.R. Acad. Sci., Paris 234, 506. COURT&G., 1960, Annales d’Astrophys. 23, 115. COURT&,,G.,1962, C.R. Acad. Sci. Paris 254, 1738. COURT&,G., 1964, Astron. J. 69, 325. COURT&,G., 1971, New Techniques in Space Astronomy, in: Labuhn and Lust (Ed. D. Reidel Publishing Company, Dordrecht, Holland) pp. 273-301. CoUR*, G., 1973, Vistas in Astronomy Vol. 14, ed. A. Bur (Pergamon Press) 81-161. COURT&,G. and R. M. BONNET,1962, Annales d’Astrophys. 25, 36. COLJRTI?~, G., M. GOLAY,M. VITON,W.BENTZ,J. M. DEHARVENG, M. LAGET,J. DONAS, J. P. SWANand B. MILLIARD, 1980, Extragalactic and galactic W observations owing to the balloon borne SCAP telescope at the wavelength 2000 A, COSPAR, Space Research XXIII. COURT^, G., M. LAGET,J. P. SWAN,M. VITON,A. VUILLEMIN and H. ATKINS, 1975, Phil. Trans. R. SOC.Lond. A279, 401-404. COURTSG., M. VRoN and J. P. SWAN,1967, Space Research VIII (North-Holland Publ. Company, Amsterdam) p. 43. CRWELLIER, P., A. ROUSSIN and Y. VALERIO,1970, Ultraviolet stellar spectra and related ground-based observations, IAU Symp. 36, eds. L. Houziaux and H. E. Butler, p. 130. DAVIS,R. J. and 0. M. RIJSTZI,1962, Applied Optics 1, 131. DEHARVENG, J. M., P. JAKOBSEN, B. MILLIARD and M. LAGET,1980, Astron. Astrophys. (in press). DEHARVENG, J. M., M. LAGET,G. MONNET and A. VUILLEMIN, 1976, Astron. Astrophys. 50, 371-375. DEHARVENG, J. M., G. P. RIVIERE,G. MONNET,J. MOUTONNET,G. COURT&,J. P. DESHAYES and J. C. BERGES,1979, Space Sci. Instr. 5, 21. DETAILLE, M., 1981, Camera B trts grand champ, Moscow SF 81 Symp. (USSR Academy of Sciences, Ed.). DETAILLE,M. and M. SALSSE,1971, IIItmes Journees d’Optique, Utilisation de lames de Schmidt B profil de Kerber dans diffkrentes combinaisons optiques (CNES, Ed.) pp. 203-230.

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DETAILLE,M. and M. S ~ S S E1975, , Vkmes Journtes d’optique, Camtra de Schmidt Anastigmat (CNES, Ed.) pp. 207-222. DETAILLE,M., F. SCANDONE and P. BEWENUTI,1976, UTEX Phase A, Optical StudiesESA-Contract SC 2604 HQ. DUBAN,M., 1978, J. Optics, Paris 9, no. 3. EPSTEIN,L., 1967, Sky and Telescope 33, 203. G~RZADYAN, G. A., 1974, Sky and Telescope 48, 213. HAL.LQCK, H.B., 1962, Applied Optics 1, 155. HENIZE,K. G., L. R. WACKERLING and F. G. O’CALLAGHAN, 1967, Science 155, 1407. I-IERSE,M., 1979, Solar Physics 63 (Ed. D. Reidel Publ. Company, Dordrecht, Holland) pp. 35-60. 1978, Proc. Esrange Symp., Ajaccio, ESA-SP, 135. HUGUENIN, D. and A. MAGNAN, KUPPMAN, J. E., A. BOGGESSand T. E. ~ ~ I L L I G1958, A N , Astrophys. J. 128, 453. LAGET,M. and M. S A ~ S E1974, , Applied Optics 16, 4. LECKRONE, D. S., 1980, Publ. Astr. Sac. Pac. 92, no. 545, 5. LEMAITRE,G., 1979, C.R. Acad. Sci. Paris 288B, 297-299. MACCHE-ITO, F. and R. J. LAURENCE, 1977, The Faint Object Camera-ESA-SN-126. MCGRATH,J. F, and M. MARWI-IT,1969, Applied Optics 8, 837. MAUCHERAT-JOUBERT, M., P. CRUVELLIER and J. M. DEHARVENG, 1978, Astron. Astrophys. 70, 467. MAUCHERAT-JOUBERT, M., P. CRUVELLIERand J. M. DEHARVENG,1979, Astron. Astrophys. 74, 218. N., 1979, ESA/ESO Workshop Astronomical Uses of the Space Telescope, 363. MAURON, MAURON, N., 1980, Thesis “Haute rksolution angulaire et coronographie en astronomie spatiale - ttude thCorique et experimentale du mode coronographique de la FOC” (UniversitC de Provence - UER). MEINJX, A. B., 1956, Astrophys. J. 124,652. MEINEL,A. B. and R. V. SHACK,1966, Optical Sciences, Tech. Report no. 6 “A wide angle all-mirror UV camera” (The University of Arizona, Ed.). MONNET,G., 1970, Intern. Summer School of Space Optics, eds. G. Court&s and A. Martchal (C.I.O.). MORTON,D. C., E. B. JENKINS and R. C. BOHLIN,1968, Astrophys. J. 154,661. MORTON,D. C. and L. SPITZER.1966. Astrophys. J. 144, 1. OCALLAGHAN, F. G., K. G. HENIZEand J. WRAY,1977, Skylab S-019 Ultraviolet Stellar Astronomy Experiment, Applied Optics 16, 973. PAUL,F. W., 1935, Revue d’optique, Systbmes correcteurs pour rtflecteurs astronomiques, Vol. XIV, 169. S ~ S S EM., , 1981a, Telescope anastigmat a trois miroirs, Moscow SF8l Symp. (USSR Academy of Sciences, Ed.) SAME, M., 1981b, La faint object camera, Moscow SF81 Symp. (USSR Academy of Sciences, Ed.). SANDAGE, A., 1976, Astrophys. J. 81, IV.11, 954. SHARPLESS,S. and D. OWEMROCK,1952, Astrophys. J. 115, 89. SCHROEDER, D. J., 1979, in Perkin-Elmer report No. 14281. SCHWARZSCHILD, M., 1973, Astrophys. J. 182, 357-361. SCHWARZSCHILD, M. and R. DANIELSON,1970, Sky and Telescope, 39, 365. SWAN, J. P. and M. VITON,1970, IAU Symp. no. 36, eds. L. Houziaux and H. E. Butler, p. 120. WESTPHAL,J. A., 1979, WF/PC GSFC-Preliminary Design Review Package (CM-04) (NASA Doc.). WYNNE, c., 1969, J. Opt. SOC.Am. 59, 572.

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E. WOLF, PROGRESS IN OPTICS XX @ NORTH-HOLLAND 1983

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SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES BY

C. FROEHLY, B. COLOMBEAU and M. VAMPOUILLE U.E.R. des Sciences, Laboraroire d’Optique, E.R.A. CNRS 535, 123 Rue Albert Thomas, 87060 Limoges-Cedex, France

CONTENTS PAGE

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61 . FRAMEWORK FOR SCALAR DESCRIPTION OF OPTICAL PULSES . . . . . . . . . . . . . . . . . .

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6 2 . SPATIAL AND TEMPORAL PULSE FILTERING ON PROPAGATION AND DIFFRACTION. . . . . . . . .

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63 . TIME SHAPING OF PICOSECOND PULSES.. . . . . . . . . . . . . . . . .

OPTICAL

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64. OPTICAL ANALYSIS OF PICOSECOND LIGHT PULSES.. . . . . . . . . . . . . . . . . . . . . . R EF ER EN C E S . .

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Introduction The production of light pulses exhibiting selected temporal shapes and the analysis of optical structures on the picosecond time scale raise some problems that have not been entirely solved. These questions play an important part in various fields, such as the design and building of mode locked lasers, the generation of plasma or the nuclear fusion of pellets with the help of laser sources, optical communications and ranging, and picosecond chemistry. A number of excellent reviews already cover these topics. This new article about the same subject places particular emphasis on the optical peculiarities of the processing techniques. Indeed, it appears that short pulse optics offers new developments to coherent optics: temporal operations on these pulses have to be considered in the familiar framework of diffraction, holography, interferometry and frequency filtering. The translation of these notions in the field of time variables enables us to make use of powerful tools of coherent optics, namely the Huygens principle and the Fourier analysis in the framework of a two dimensional temporal space. We have chosen this unusual point of view, as it allows for deeper understanding of the action and performances of systems designed for pulse shaping and analysis. Concepts of coherent optics in the temporal field will be introduced in this paper in a rather detailed way: the scope of this review will be restricted mainly to coherent techniques, that is techniques performing deterministic operations on the complex amplitude of the temporal optical signal. We will not consider electronic or electrooptical devices, although they reach a few tens of picoseconds; we have also left aside the new developments of optical energy compression by stimulated effects, such as stimulated Raman scattering, although they give rise to active research connected with work on laser fusion. Optical bistability and phase conjugation also raise problems of pulse shaping and analysis in the sub-nanosecond range, but now they are so far developed that they constitute separate fields which we shall not discuss here. 65

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This article is in four sections. The first one deals with the space-time coherence effects due to the new situation arising in optics; namely circumstances when large amounts of energy are carried by every one of the space-time modes of the radiation, a situation realized for example with picosecond laser pulses. The second section presents basic pulse types frequently involved in operations of shaping or analysis: “quasi-monochromatic” pulses, having time frequency well defined enough for them to obey simple scalar diffraction laws, and the corresponding notion of “quasi single space frequency” pulses, with well defined space frequency, obeying simple dispersion laws. Then we discuss the temporal shape distortions encountered by optical pulses on diffraction through apertures. The third section discusses deterministic pulse shaping by interaction with filters and modulators in terms of temporally coherent optics. The last section starts with a review of well-known methods for measurements of pulse durations, then devotes its main development to observations of complex amplitude distributions at the picosecond o r sub-picosecond time scale.

5 1. Framework for Scalar Description of Optical Pulses 1.1. COMPLEX ANALYTIC REPRESENTATION OF SPACE-TIME PULSES

In the following, light signals depending on both spatial and temporal variables are referred to as “space-time pulses”. At any given time, these signals are confined inside finite geometrical volumes; at any given point, they spread out over limited durations. A space-time pulse, propagating in an homogeneous time Isdependent medium, can be considered as resulting from the linear superposition of monochromatic plane waves (plane wave spectrum). The complex amplitude a(M, t) of a plane component at a point M(xyz) in coordinates (0,xyz) is written as

with r =ol\.i;IF(N, v)I2,intensity of this component; q(N,v), initial phase at r = 0, t = 0; u = time frequency of the component ( = 0 / 2 ~if ,o denotes the angular frequency); N = k / 2 ~ k, being the wave vector of the

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component; and the projections N,, N,,, N, of N are the spatial frequencies of general use in Fourier optics. Rigorously, the superposition would be a vectorial sum of polarization states. The scope of the analysis presented here is restricted to the scalar wave approximation. Then, the superposition is fully characterized by the four-dimensional complex spectral distribution F(N, v ) = 1FN.J exp CidN v)). By Fourier transformation of F(N, v), a space-time complex distribution f(r, t ) will be defined: - at a given frequency v, this distribution is monochromatic, written as f(r, t ) = g(r) exp W w + ) ,

where g(r) is the usual “complex amplitude” of coherent optics in the geometrical space, - at a given point Mo this time dependent distribution becomes identical to the “analytical signal” (BORNand WOLF[1965a]) if the positive time frequencies only are considered. Again we will call “complex amplitude” the field distribution in the more general space-time pulse described by the functions f(r, t) or F(N, v). The scalar approximation enables us to study wave groups, the spectrum of which is narrow around the average wave vector. In homogeneous, time independent propagation space, the distribution f(r, t ) = f(x, y, z, t ) is in fact a function of three independent variables only, for instance x, y, z, or x, y, t. The evolution of the radiation field along the fourth dimension obeys a propagation equation; a pulse propagating in the half space z > O is then fully specified by a three variable scalar complex function fo( x, y, t), which describes the temporal evolution of the amplitude distribution in the plane z = 0 , or equivalently by another scalar complex function g,(x, y, z ) , which describes the spatial amplitude distribution at the initial instant t = 0. Further successive states of f i ( x , y, t ) at z > O o r g,(x, y, z ) at t > O can be unambiguously deduced from f o ( x , y, t ) and g,(x, y, z) with the help of diffraction calculations. In the following, optical propagation and interactions are discussed in a three dimensional space-time (x,z, t). Suppressing the y variable lightens the writing of formulas without loss of generality of the results. Of course, the description in the geometrical and temporal space is strictly equivalent to what is given in Fourier space by the spectral

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distributions F,(N,., u ) or G,(N,., N,) defined by the Fourier transformations: fa.

FdN,., u ) = j j f o ( x , t ) exp {-j2rr(NXx+ vt)) dx dt, -m

+m

GdN,.,N,) =

jj

g,(x, z) exp {-j2rr(NXx+N,z)} dx dz.

--m

1.2. SAMPLING OF OPTICAL PULSES AND NUMBER OF THEIR SPACE-TIME MODES

Although space-time pulses are spatially and temporally limited events, their frequency spectra are not unlimited, in contradiction with the mathematical necessities of Fourier analysis. Indeed, the frequency domain where the spectrum power is larger than the noise power - including all the fundamental and practical noise sources - has a limited bandwidth. There are various conventional ways for space-time pulse widths to be defined (BRACEWELL [1965]). Let Ax, Az, At, AN,.,ANz and Au be these widths in the spaces x, z and t, respectively. After the sampling theorem (BRACEWELL [19653, such a pulse is entirely characterized by the complex values of a finite number K of independent samples in the representation space; K = A x ~ A z ~ A N ; A N , = A x ~ & t ~ A N ,~ A u . The sampling decomposition can be performed as well in the geometrical spaces (x, z ) or (x, t) as in the spectral spaces (N,., Nzl or (N,.,v). The number K is invariant with respect to the sampling space. The sampling periodicities (see Fig. 1.1) are equal to l/AN,. and 1/AN, along the coordinates x, z , l/AN, and l/Au along the coordinates x, t, llAx and llAz along the coordinates N,, N,, l/Ax and l / A t along the coordinates N,., u. To one single sample in spaces (N,.,u ) or (N,.,N,) corresponds, in spaces (x, 2) or (x, t), one single monochromatic plane wave, the wave vector k of which has a modulus (k( = 2rr(N: + NZ)1’2= 2rru/u ( u is the phase velocity at frequency u ) and is tilted by an angle 0 with respect to the axis 0 z : c o s 8 =2rrNz/\k\,sin 8 =2rrN,./lkl (Fig. 1.2). Propagation invariant structures are called “modes” of the radiation (see ARNAUD [1976]). Consequently, any monochromatic plane wave is a

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sv "' sv

Fig. 1.1. Sampling of a function f ( t ) = a ( t )exp fi2.nvot} and of its spectrum F ( v ) . Only real parts of the functions are represented. 6u = l / A t , spectral sampling periodicity; St = U A V , temporal sampling periodicity. The number K = At/6t = A U / ~ oVf the samples is the same in hoth t and v spaces.

mode of the time-independent homogeneous free space, and each spectral sample of a light pulse is a mode of this propagation space. A pulse characterized by K samples admits a decomposition in K independent modes. The single mode structures (K = 1) are the simplest ones. They are the best physical approximations of monochromatic plane waves in limited space-time domains. For instance, that is the case for Gaussian

Fig. 1.2. To one sample located at S in the frequency space (h) corresponds the wave vector k of a plane monochromatic wave in the geometrical space (a).

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pulses emitted in a Gaussian beam by mode-locked lasers; that is not the case for permanent light vibrations, even if emitted by very monochromatic sources, such as carefully stabilized gas lasers. The next subsection (9 1.3) reconsiders this problem. Now, let’s point out that a continuous laser radiation, exhibiting very high spectral finesse - say a bandwidth A v = 1MHz- already contains millions of independent temporal samples after emission over a few seconds; a deterministic temporal analysis of the light signal should account for each of these samples, which rapidly becomes impossible. Although such a laser emission exhibits a “coherence length” (BORNand WOLF[1965b]) three hundred meters long, it gives rise to few time coherent optical experiments (mainly frequency beating experiments) because of the large number of samples to be processed. On the contrary, in the picosecond range of duration, poorly monochromatic pulses hold enough small numbers of modes - or samples - for coherent analysis or processing by limited capacity optical systems, as shown in the next subsection.

1.3. CONDITIONS LEADING EITHER TO DETERMINISTIC ANALYSIS OR TO STATISTICAL ANALYSIS OF OPTICAL PULSES

It was previously noted that any light signal can be represented with the help of a well defined complex function. The modulus and argument of this function are known unambiguously at each point of the representation space; the function is necessarily deterministic, whatever its spatial and temporal structure may be. The space-time amplitude distribution in a light pulse can always be considered as fully coherent, whatever its monochromaticity and its angular divergence are, to the extent where the idea of coherence refers to deterministic knowledge of the amplitude and phase of the optical vibration. For instance, let’s consider a single sample pulse, of duration A t , = O.lps, emitted around the instant tl by the point PI in the geometrical space (x, z) (Fig. 1.3). After transmission through any aperture (D) and propagation in space z > 0, the pulse acquires a more complicated spacetime structure, the mode structure of which depends on the geometry of the screen (D), and on the location of the observation point P,. The temporal shape of the amplitude distribution fp,(t) at any point Pzcan be deduced from the diffraction laws, as discussed in § 2.4.3. Obviously, the well defined modulus and argument of the complex amplitude fp,(f)

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71

Fig. 1.3. Deterministic amplitude perturbation of a single mode pulse fp,(t) emitted hy P, and detected as fP2(f) at P, after transmission through the screen (D). Fp,(v): initial pulse spectrum; FP2(u):filtered pulse spectrum.

characterize a deterministic distribution of radiation in the (x, z) space at the time t. Nevertheless, experiments exist where the radiation should be described as a stochastic process, o r considered as “partially coherent”. Two beam interference experiments performed on the optical field available at the point P2would show very low contrast fringes. Such a radiation seems poorly coherent from the point of view of classical “coherent optics”. This apparent contradiction comes from the fact that the coherence properties are not intrinsic features of the complex amplitude itself, provided it carries enough energy; they strongly depend on interactions between the signal and the observer. Either coherent or incoherent optical operations can be performed on the same light signal, according to whether the processing capacity of the operator is large o r small with respect to the complexity of the signal. In other words, we will see now that the result of observations on a space-time light pulse is related to its complex coherent amplitude, either in a deterministic way o r by statistical energy averaging, according to whether the capacity of the detection

12

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II,

§ 1

device is large or small, with respect to the number of samples of the pulse. The light quantification is the first fundamental irreducible cause of uncertainty. After GAROR[1964], let’s calculate the ratio A W = W/K’ of the whole signal energy

W=

(7

I f ( x , t)I2 dx dt =

[IF(N,, v)12 dNXdv

-m

to the number K ’ of the samples of the energy distribution submitted to measurement. If it happens that the energy per sample AW is smaller than the photon energy hvo (v”, average frequency of the signal), we will fundamentally never be able to measure each of the K’ samples independently, as the detector has to add the energies of many successive samples before emitting an observable response. That is the general situation if the radiation comes from thermal sources, where the number of photons per sample is much smaller than 1; but it is also the case if a laser produced radiation suffered energy losses large enough to make the single sample detection impossible. Such a situation cannot lead to coherent analysis of the complex amplitude. It only remains possible to perform spatial and temporal autocorrelation of the vibration, as these operations act by energy integration over numerous samples. Correlation analysis only provides statistical information about the space-time distribution of the complex amplitude across the representation space of the pulse. No other approach is possible in studying spontaneous emissions of light by “thermal sources”. A second source of uncertainty has a purely instrumental cause; it is the possible deficiency of the capacity of the observation channel, as compared with the information content of the pulse to be observed. Let us recall that the capacity of a channel is quantitatively measured by the number C = R log, (1 + sln) (Hartley, Tuller and Shannon famous relation for instance, see SHANNON [1949]), function of the sample number R and signal to noise ratio sln of the output signal delivered by the channel. The capacity equals the number of samples at the limit where sln = 1. Examples: - a photodiode with 200 ps time resolution has a 5 GHz bandwidth. This detection channel has a single spatial sample (or “degree of freedom”) but can record purely temporal information flows up to rates of 5 x 10’ bits * s-’,

11,

5 11

SCALAR DESCRIPTION OF OFTICAL PULSES

73

- two identical contiguous photodiodes form a two channel space-time detector conveying 10” space-time samples per second, - a photographic emulsion resolving lo3 lines mm-’ is a purely spatial memory with one single temporal degree of freedom. It is able to display lo6 independent spatial samples per square millimeter of photosensitive material. If the capacity of the detector is higher than the number of samples of the space-time energy distribution impinging on it, suitable scaling or coding allows settling injective correspondence between the signal samples to be detected and the available degrees of freedom of the detector. Any superposition of samples on the same degree of freedom can be prevented. From the signal emitted by the detector, we are able to go back to the intensity of each sample of the space-time energy distribution in the pulse. Coherent optics studies are possible on these deterministic conditions. On the contrary, too small a detector capacity prevents the independent recording of each sample of the light pulse. The energy of the K‘ samples has to be distributed among the R < K’ degrees of freedom of the detection channel, each degree of freedom recording an energy averaged over n = R / K ’ > 1 samples. The result of the pulse detection is a set of R numbers, each of them representing the sum of the intensities of n samples. Now, the knowledge of these R numbers is no longer sufficient to go back to the individual samples of the pulse energetic distribution. This detection can be considered as an ensemble averaging over n sequences of R samples only. Each of these sequences is a deterministic function describing a part of the pulse. But the ensemble of these n deterministic functions defines a stochastic process (BLANCand FORTET [19531) which cannot be analyzed deterministically. LAPIERRE The observed radiation exhibits “partial coherence”, as its complex amplitude cannot be unambiguously recovered from the measurements; there are at least n independent amplitude distributions yielding the same detected signals in the geometrical, temporal and spectral domains. When the number of radiation modes is enormous with respect to the capacity of the measurement channel (n +.a),the optical field becomes a Gaussian complex variable X, the statistical properties of which are fully described with the help of the first and second order moments ((X) and (X,X:)) only. The optical coherence theory (BORNand WOLF[1965c]) was developed in this situation, in order to account for observable features of radiation emitted by thermal sources. It no longer accounts for observations involving short pulses with small values of the ratio n. In this case,

-

74

[II, 5 1

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

deterministic working conditions can always be fulfilled by using detectors of suitable performance. Examples of coherence effects on pulse detection are now given for illustration.

1.4. THREE EXAMPLES OF OPTICAL PULSES EXHIBITING DIFFERENT COHERENCE

1.4.1. Deterministic temporal analysis of purely temporal pulses (Fig. 1.4) We consider a short pulse produced by the point source 0 around the time t=-L/c, with L = ( I , + 1 2 ) / 2 , l,=OA+AM,+M,P and 12= OM,+M,A+AP. It is split into two subpulses as it reaches the detector (D) at the point P around the time t = 0 . If the amplitude at 0 was

only real parts

of

complex signal5 are drawn.

Fig. 1.4. Deterministic temporal observation of a pair of short pulses by broadband photoelectric detection.

11, 3: 11

SCALAR DESCRIPTION OF OWICAL PULSES

75

expressed for instance as fo(t) = exp { - . r r ( ( l + L / c ) / T )exp ~ } {-j2.rrvo(t + L/c)), the amplitude distribution fP(t) at P would follow immediately: fp(t)

=exp{-.rr((t-APt)/~)~}exp {-j2.rrv0(t-At)} +exp {-r((t+At)/T)2}exp {-j2.rrvo(t +At)}. where At = ( l2 - l l ) / c .

The photodetector (D) yields a current, the intensity i(t) of which results from convolution between Ifp( t)l’ and the time impulse response h ( t ) of the detector. If the detector bandwidth AvD is large enough, the spectra Sp(v) of IfP(t)12 and Ip(v) of i(t) are porportional to each other. The number K ’ = 2 A t / ~of the samples of the intensity distribution Ifp( t)I2 is smaller than the number R =At .AvD of the degrees of freedom available for detection over a time interval At. The observed signal i(t) allows the deterministic recovering of the square modulus Ifp(t)I2 of the complex amplitude. This is a situation of full temporal coherence although the wide-band pulses are not quasimonochromatic wave groups. Various operations of time coherent optics will be possible in such a situation, as reported in PP3 and 4.

1.4.2. Purely temporal analysis of space-time pulses (Fig. 1.5)

If one mirror of the Michelson interferometer is slightly tilted by an angle a,the pulse emitted by 0 assumes a structure dependent on both time t and the geometrical variable x in the plane of detection. One subpulse reaches the point P, at time tl, whereas the second one reaches the point P2 at time r2 = tl + At: I f ( x , t)12=exp{-.rr(t/T)2}(8(S(t- tl, x - x , ) + S ( t - t 2 , x-x2)) the symbols S and (8 respectively denoting the Dirac distribution and the convolution operation. Let’s suppose that the spatial separation x 2 - x1 = A x is just equal to the resolution limit of the lenses (LJ, (L,,). Then the detected space-time pulse has just two spatial samples, the number of its temporal samples remaining equal to 2 A t / ~ . Even if the bandwidth of the detector (D) is larger than the bandwidth 2/7 of the pulse couple, its single spatial degree of freedom is not suited to the spatial features of the signal; obviously, the observation of the

76

[II, I 1

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

0

Fig. 1.

It is not possible to recover the deterministic law I f ( x , ?)I2 i ( t ) : this is a situation of partial coherence.

from c-;ervations on

current i(t) does not allow deterministic recovering of the square modulus lf(x, t)l' of the complex amplitude. This is a situation of partial coherence, where many different distributions Ifa(x,t)I2, If&, t)]" etc. lead to the same detected signal i(t)

If&

t)12=exp{-n(t/~)2}~(S(t-cl, x-xx,)+6(t-tz, x-x,))

)fb(x, t ) 1 2 = e x p ( - ~ ( t / ~ ) 2 } ~ ( 6 ( t - xr -lx, 2 ) + 6 ( t - t 2 ,

x-x,)).

This problem always arises when observing spatially multimodal optical structures, especially at the time of pulse propagation studies in multimodal optical fibers (experiments in § 4).

1.4.3. Partially coherent temporal analysis of purely temporal pulses Coming back to the initial arrangement sketched in Fig. 1.4, let's now suppose that the bandwidth AvD of the detector is narrower than the bandwidth 2/7 of the spectrum S J v ) and the ratio n = K ' / R increases beyond the unity. The photocurrent i(t) can be considered as resulting

11, $21

SPATIAL AND TEMPORAL PULSE FILTERING

77

from averaging over n energy distributions Ifi(t)12, each of them including R degrees of freedom. Also in this case, deterministic recovering of the initial distribution If,(t)l* from i ( t ) is impossible. Coherent information about the complex amplitude of the pulse is not accessible. This situation of partial time coherence has been limiting experimental analysis of intensity fluctuations in light beams (-EL [ 19631, HANBURY BROWN and Twrss [1956]), and should be prevented before performing coherent shaping or spectral analysis of spatially unimodal optical pulses as described in O § 3 and 4. We can point out the full analogy of these problems related to time analysis with the spatial coherence problems raised in holographic or speckle interferometry, by the superposition of a finite number of individually coherent, but mutually incoherent quasi-monochromatic wave surfaces. Only spatial degrees of freedom are available from the photographic detector, so that the space coherence of the time averaged wave surface and the interferogram contrast decrease at each temporal fluctuation of the wave shape. We can summarize the above discussion as follows: the complex amplitude distribution of a spatially and temporally multimodal light pulse can be defined in a deterministic way, on the condition that the number of photons per mode is large enough to allow the individual detection of each mode. If the number of degrees of freedom of the detection channel is larger than the number of the modes to be detected, entirely coherent analysis of the pulse is always possible. If not, only partially coherent statistical analysis can be performed. Shaping and analysis operations considered in the next sections are chiefly space-time coherent operations on various classes of multimodal optical structures.

15 2. Spatial and Temporal Pulse Filtering on Propagation and Difhraction

2.1. INTRODUCTION

The spatial and temporal amplitude distributions defined in the previous section as “space-time pulses” do not exhibit structural invariance on propagation, as would be the case in one dimensional, non-dispersive lines (e.g. vibrating string). The scope of this section is a coherent

78

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, I 2

description of temporal distortions occurring in spatially, temporally and spectrally limited pulses traveling in free space o r through optical apertures. This will give a background to the further discussions about coherent pulse shaping (§ 3) and analysis ( 5 4) of picosecond optical signals. At first we focus our interest on elementary pulse shapes, the propagation of which obeys particularly simple laws. These are “quasimonochromatic” pulses of narrow bandwidth Au and, in an analogous way, pulses that we will call “quasi-single space frequency”, where the variables u and N, play symmetrical parts with respect to the quasimonochromatic situation. The deep analogy between scalar diffraction of quasi-monochromatic signals and linear dispersion of “quasi-single space frequency” temporal structures will be shown and illustrated by a few typical “temporal coherent optics” experiments on the picosecond scale. Finally, the linear filtering of the time frequency spectrum of light pulses through optical pupils will be discussed, in terms of time impulse response, and time frequency transfer of these pupils. 2.2. DIFFRACTION AND PROPAGATION OF QUASI-MONOCHROMATIC PULSES

2.2.1. Definition of quasi-monochromatic pulses When a space-time optical pulse is traveling in the geometrical space

{x, z}, the spatial distribution of the field at any time t may be deduced without ambiguity, from the sole knowledge of one of its space-time amplitude distributions f z ( x , t ) at a propagation distance z, as pointed out in 8 1.1. However, the problem of analytically describing the one to one correspondence between two successive vibration states fi,,(x, t ) and f z , ( x , t ) , at two successive propagation distances zo, zlr is not a trivial one, even on linear propagation in the free space. One of the situations where simple correspondence is well known is provided by monochromatic radiation: fz(x, t ) = X , ( x ) exp fi2.rruot)}.There is a separation of the geometrical X , ( x ) and temporal expCj2wot} variations allowing for a purely geometrical, time independent relationship between the spatial wave structures X , , ( x ) and X,,,(x) at the distances z, and zO: this is “diffraction” of monochromatic light (cf. § 2.2.2).

11, § 21

SPATIAL AND TEMPORAL PULSE FILTERING

79

Monochromatic radiations do not exist physically because of instabilities of light sources or, at the ultimate Fourier limit, because of their finite emission time. Every physical linear pulse propagation may be interpreted as the temporal beating between the various diffraction phenomena occurring at each of the frequencies carried by the pulse. An important practical question is to know how monochromatic a space-time pulse should be for its spatial structure to evolve, according to a single frequency diffraction law. The answer should be more detailed than the classical one, which characterizes the radiation ability to give rise to diffraction or interference phenomena through optical devices depending on spatial variables only. “Quasi-monochromatic” radiation is usually defined (BORNand WOLF[1965b, c]) as exhibiting a “coherence length” larger than the optical path differences involved by the diffracting aperture or interferometer. In the more general case of optical pulses depending on both spatial and temporal variables, quasi-monochromaticity has to be defined in the following way: Let the form rn,(x, t ) exp Cj2rrvot}describe the pulse amplitude fib,t ) , with emphasis put on its average frequency v,. Quasi-monochromatic propagation will take place only if the space-time modulation rn,(x, t ) degenerates into the product of a spatial term X , ( x ) by a temporal term Y,(t); then the field f , ( x , t ) = X , ( x ) s,(t)exp fi27rvot} is a temporal wave train T,(t) exp fi2rrvot} modulated by the spatial distribution X , ( x ) . This radiation is similar to polychromatic radiation spatially modulated by transmission through time independent optical devices. Its spatial structure X , ( x ) will be kept independent on time t at any distance from the origin of the propagation space, on the condition (Fig. 2.1) that the spectral bandwidth Av of 3, (t) satisfies the “quasi-monochromaticity” requirement Av<(c/S max); 6 max being the maximum optical path difference between outermost rays AP and BP of the beam at their most oblique diffraction angle 8,,. It may be related to the spatial width Ax and to the highest spatial frequency N , of the structure X , ( x ) by the equations: 6 max = Ax * sin 8” = ( A X N , ) (clv0). Then the condition for “quasi-monochromatic’’ pulse diffraction to arise is written as 9 ’ = ( v o / A v ) > ( N 1* A x ) ;and the ratio 9=vo/Av is called the “spectral finesse” of the pulse; N , is the upper limit of the space frequency spectrum Fz(Nx)of X , ( x ) : +a

X , ( x ) exp (-j27rNxx) dx.

F,(Nx)= -a

80

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

PI, § 2

Fig. 2.1. Quasi-monochromatic diffraction of a pulse: the “coherence length” Ar is much larger than 6max. (Ic,), (Il): the same pulse at two successive times t, and t , . Dotted line: geometrical beam shape.

Quasi-monochromatic scalar diffraction of low divergence beams is now considered in both plane wave spectrum and Fourier integral formulations, in situations and under forms that allow easy comparisons (§ 2.3) with linear dispersion laws. This subsection will be of little interest to readers familiar with Fourier optics of spatially coherent quasimonochromatic radiation.

2.2.2. Spatial phase filtering of monochromatic pulses by free space propagation The evolution of the spatial modulation term X , ( x ) of a quasimonochromatic pulse, on propagation along the geometrical coordinate 2, only consists of a frequency dependent phase shifting of its spatial frequency spectrum z z ( N x =I’m” ) X , ( x ) exp (-j27rNxx) dx (ARSAC[1961], GOODMAN [1968]). This classical result of paraxial optics will again be proved in the more general case of the oblique diffraction of narrow light pencils. At the well defined wavenumber cr0 of the optical field, a single frequency component A,,Nxexp ( j 2 d X x ) of A z ( N , ) is the plane wave A(x,z ) of wavevector k given by ( 1 / 2 ~Ikl= ) o,,kf 4 k: = \k\*,(1/27r)k, = N, ; k,, k, being the projections of k on the coordinates x, z . Indeed, the amplitude distribution of this plane wave in the space {x, z } may be written as A(x,z ) = A exp Q(k,x + k,z + 4J). Its spatial distribution along the coordinate x at any given z has the single frequency

11, rj 21

81

SPATIAL AND TEMPORAL PULSE FlLTERlNG

expression B,(x) = B exp G(k,x + +,)} - with obvious notations - which is quite identical to A,,,,, exp Q27rNxx} after identification of B exp Qd,}to Az,Nx.

The spatial frequencies of the amplitude distribution X,(x), propagating as plane waves, keep constant intensities in the whole space. Only their phase k,x + k,z varies, with linear dependence versus x and z. The progression of the phase &(N,) of the spatial frequency N,, along the propagation direction 2, is described by the relations

c#~,(N,)=k,* ~ = ( I k l ~ - k : ) ’ / ~ ~ = 27r(a;: - Nf)”*Z.

(2.2.1)

The spatial frequency spectrum %,(N,) of the pulse, at any distance z from the coordinates origin, can be deduced from its expression %(N,) at z = 0 by the following frequency dependent phase shifting: X z ( ~ x ) = X o ( ~ x expQ4,(N,)}=Xo(NX) ) expCj27r(ai-N,2)1/2z).

(2.2.2) For describing the diffraction of narrow divergence beams around an average direction 80 (Fig. 2.2), this expression will be approximated by a second order limited polynomial expansion of d,((N,) in powers of N, -No, where NO= (1/27r) lkl sin 00 = uo sin 80

(2.2.3)

Fig. 2.2. Oblique diffraction of narrow aperture beams; the propagation distance of the Huygens wavelets to be considered is L, not 2. The diffracting aperture size is to be measured transversely to OP, along x‘, not along x : OM’ = OM * cos B0.

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SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, P 2

Hence the spatial frequency transfer function H ( N x )= exp G4z(N,)} may be written as:

Inverse Fourier transformation (F.T.(-'))performed on the phase shifted frequency spectrum k ( N X )provides an integral formulation of the correspondence between the two successive spatial amplitude distributions X O ( X and ) X,(x):

x,(x) = x~(x)@F.T.(-')(H(N,)}, where F.T?{EZ(N)}

= h(x)

[ t-+ 2.rrz(ao

= (a221-1'2 exp j

I

- Neal +i(No)'a~}]

x S[x+(a~-Noaz)z]@exp(-j.rr")}.

2

a2z

(2.2.6)

The symbol @ denotes the convolution operation along the variable x, 6(x) is the Dirac distribution, and the following result has been assumed: F.T.'-"[exp (j.rrNz)J= exp (-j?rxz) exp (j?r/4). The convolution by S[x +(a1-Noa2)z] causes translation of XO(X) parallel to the x direction. If we are concerned only with the alterations of spatial structure of the propagating beam, without regard for its lateral geometrical position, we describe the wave distortion on diffraction from

11, § 21

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SPATIAL AND TEMPORAL PULSE FILTERING

the origin 0 to z # O by the convolution: x,(x>

= K,

*

Xo(x)Wexp (-jm2/a22)),

(2.2.7)

with

K , =(-jazz)-“’

expCj2.rrz(ao-Noal+4(~O)~a2))

or, after introducing the length L = z/cm Oo, we get the final form:

(2.2.8) where

d ( L )= 2.rrz(a0- NOUI +$(No)’a2),

l/uo= ho

(radiation wavelength).

This is nothing but a convolution formulation of the classical one dimensional Fresnel-Kirchhoff diffraction integral. Without going into detailed calculations, let us retain two useful particular cases: = 6(x - XO) + 6(x + XO) (Fig. 2.3(a)), this is the Young’s slit if XO(X) experiment; the energy diffracted at distance z from the origin is the sine law modulated with periodicity p

= ( h o ~ ) / 2 xCOS’ o

eo;

(2.2.9)

Fig. 2.3. Oblique diffraction of (a) a couple of Young’s slits, (b) a Gaussian beam modulated at spatial frequency No.

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SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, § 2

if X O ( x = ) exp [-T(X/&,)*] exp {-j2?riV0x} (Fig. 2.3(b)), this is the oblique diffraction of a Gaussian beam, which diverges at an angle A 8 = Ao/&o cos 80 around its average propagation direction 80. Now we will consider another special class of pulses for which the above arguments and results are to be transposed, in the domain of the geometrical frame of wave diffraction and interference in a suitable two dimensional temporal space, where a Huygens principle can be defined.

2.3. LINEAR DISPERSION OF QUASI-SINGLE SPACE FREQUENCY PULSES

2.3.1. Definition of single space frequency pulses These are spatially single mode pulses, the space-time amplitude distribution of which has the form fz(x, t)=Tz(t)expCj2dV~x}.

(2.3.1)

This notion corresponds to straightforward transposition of the relation fi(x, t ) = X z ( x )- exp Cj27rvot) defining monochromatic radiation; the parts of the temporal and spatial variables are exchanged. Experimental production of single space frequency radiation is not possible rigorously; it would, for instance (Fig. 2.4), require sending a temporal pulse T(t), carried by a transversely unlimited parallel beam, perpendicularly onto an unlimited diffraction grating of spatial frequency NO.But first we will discuss this idealized situation. If the only existing

Fig. 2.4. Generation of a single space frequency pulse through the sinusoidal transmission distribution (G) of groove spacing l/&.

11, § 21

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SPATIAL AND TEMPORAL PULSE FILTERING

diffraction orders were k = +1 and k = - 1, the amplitude at any point of the space z > O would keep the spatial cosine law form

(2.3.2)

f z ( x , t ) = % , ( t ) cos 27TNox.

If only one of the orders + l or -1 existed (due to blazing effects), then

f , ( x , t ) =T,(t) exp{*j2dV0x}.

(2.3.3)

Anyway, the periodical structure of period p = NO along the variable x would remain rigorously invariant on propagation along z, as illustrated in Fig. 2.4. The wave vectors k, = 27r(v/c)i1of each of the frequencies of the temporal pulse T ( t ) ,in the parallel beam on the left of the grating (G), are all parallel to the unit vector i l of the axis Oz; they have no component on the axis Ox, of unit vector i2. Their interaction with the grating wavevectors *27rNoiz then yields new wavevectors k: of modulus 27r(v/c) depending on v, but having independent projections (k$)x= *2?rNo along the direction i2 of Ox. The only effect of the propagation along Oz on the right of the grating is to introduce geometrical path differences between the wave vectors k: of the temporal pulse, as these vectors no longer propagate in the same directions. Thus the various temporal frequencies v of the “single space frequency” structure will only have frequency dependent phase shifts during their propagation, which is known as “frequency dispersion”.

2.3.2. Linear dispersion of single space frequency pulses on free space propagation We have just pointed out that single space frequency structures, having any temporal spectrum, exhibit no spatial distortion, but only phase shifts of their temporal frequencies on propagation in free space. Now, in close analogy with § 2.2.2, first order temporal distortion effects corresponding to “linear frequency dispersion” of ideally single space frequency pulses will be calculated. The phase J/L(v)of the spectral component of frequency v, after propagation over the distance z >O, is expressed as: &(v)=k, ~ = ( l k \ ~ - k f ) ~ ’ ~ * ~ = (27r/C)(u2-N;C2)”2

*

2.

(2.3.4)

The temporal frequency spectrum, gz(v) of the pulse, at any distance z from the coordinates origin, can be deduced from its expression gdv) at

86

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SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

§

2

z = 0 by the following frequency dependent phase shifting:

For describing the dispersion of narrow bandwidth pulses around an average frequency UO, this expression will be approximated by a second order limited polynomial expansion of & ( u ) in powers of u - YO:

where

2T 2 7Tvo & ( ~ o ) = c [ ( ~ o ) ~ - *( cN) o~ ] * ” ~=- C cos 60 *

[dz

&(Y)] u=vo

2,

1

=-[l-(No.c)2(v~)-2]-”2~z=--27T 27T c c cos eo =2

-

.[ =- -I,

m; 211. C

z

c cos eo

I

(2.3.7)

(No * c)2[(v”)2-(No)2c2]-3/2. z

NOC/UO = Noh0 =sin 00. The dispersion vanishes if NO= 0 (then +kz(v) = 2 7 ~u(/ c ) z ) , increasing up to infinity as No reaches the limit UO/C, at which the diffraction orders i 1 propagate at grazing emergence (60= 7r/2rd). Nevertheless, there exists a spectral bandwidth Au around uo, and a range of variation Az > O , inside of which the second order limited expansion of & ( u ) yields a satisfactory agreement with experiments. This approximation, called ‘‘linear dispersion”, takes good account of all the phenomena involved in experiments discussed later on. Then & ( u ) = 2Tboz

+ 27rb1(v- v0)z + 2Trb2(v- uo)2z,

(2.3.8)

with coefficients bo, b~ and bz defined by relations (2.3.7). Inverse Fourier transformation (F.T.(-”) performed on the phase modulated frequency spectrum ??z (v)= $o( v) exp fi Ifbz ( u ) } provides an

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87

integral formulation of the correspondence between the two successive temporal amplitude distributions 3,( t ) and To(t),

T,(t) =To(t)@F.T.(-’){expjlLz(v))

(2.3.9)

(2.3.10) with the same notations and arguments as for the similar spatial relation (2.2.6). The convolution by S [ t + ( b ~ -v0b2)2]causes only the temporal delay of 90(t). If we are only concerned with the temporal structure alterations of the envelope of the propagating pulse, without regard for its “group time of flight”, we describe the pulse distortion, on propagation from the origin 0 to 2 # 0, by the convolution: (2.3.11)

(2.3.12) which is exactly analogous to the spatial convolution (2.2.8) after having changed the notations in the following way: PO= ( v~)-’ sin 80 = N o c ( v o ) - ~ and

7’= T sin 60 = ( d c ) tan 80, (2.3.13)

leading to the form:

T,(t) =expCjJI(T))cos fh@TO(t)@exp POT

{ j r t z ( C poT O S ”)*}

(2.3.14) The similarity between the spatial convolution (2.2.8), representing a difbaction, and the temporal convolution (2.3.14), representing a

88

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

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dispersion, suggests we should consider pulse dispersion as a geometrical diffraction in an appropriate temporal space. For this purpose we have to define (i) this representation space, (ii) the “wavelength” ruling a “temporal diffraction” law in this space, (iii) “temporal wave surfaces” propagated and diffracted at this wavelength. (i) The orthogonal temporal coordinates of the representation space 2 112 = will be {t, s}, with s = T cos 8 0 = (z/c) sin 80 = (z/c)[1- (cos 80) ] (z/c)[l - ( v / c ) ~ ] ” ~ ,where 2) = Z/T is the group velocity of the light pulse along 0 2 . (ii) The quantity playing the part of the wavelength is pa= sin Oo/v0. (iii) By analogy with the monochromatic wave surfaces of classical coherent optics, we now define “temporal wave surfaces” in the space

it, 4. The shape distortions of Tz((t)due to dispersive propagation will be interpreted in the space {t, s} as resulting from the interference of cylindrical wavelets of wavelength PO emitted around the average direction 8 0 by each of the points of the undistorted signal, To(c), distributed on the axis t. The temporal wave surfaces in this space are the envelopes of these wavelets (the Huygens principle). The most simple example of waye surface is the plane wave, representing one component at frequency v = vo+Av of a wave group of average frequency V O . This plane wave, of temporal wavelength P O = sin 8o/vo, propagates along the direction a ( v )= & + A 8 defined by sin a ( v )= sin 80 (vlvo), which can be approximated by the linearized relation ABltan t30=Av/v0<<1. I n the particular case where v = VO, then a(vo)= eo. A group of frequencies of bandwidth Av centered on v = v o is represented by a set of plane waves propagating around the average direction 8 0 with an angular divergence A8 = (Avlvo) tan 80. The analogy between linear pulse dispersion and diffraction of monochromatic waves was especially pointed out by Marburger in the work of SHENand MARBURGER [1975] for defining a “length of stability”, or “temporal Fresnel length” of a pulse, that is the propagation length ZF, over which the temporal structures T,(t) and TO(?)differ from each other by a time delay T = z,/u only, with negligible shape distortion. Example: A representation of a single space frequency pulse whose temporal profile is Gaussian (Fig. 2.5). Let us consider a Gaussian temporal shape of initial duration Ato= s at z = 0, of bandwidth Av = l/Aro = 1OI2Hz around the average

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Fig. 2.5. Representation of pulse dispersion as beam diffraction in the two dimensional temporal space {t, s}.

frequency uo = 6 X l O I 4 Hz (Ao = 0.5 Fm). After diffraction in the first order of an unlimited grating of space frequency No = lo3mrr-’, this pulse has single space frequency structure and suffers temporal distortion by dispersion. In the analogic representation of Fig. 2.5 this is a narrow Gaussian beam of temporal wavelength po = (1/12)x s tilted by an angle Oo==3Oo with respect to the axis s. Its “beam waist” is to= At, cos 8, = (&/2) X lO-”s; after the geometrical results of § 2.2.2 its “temporal divergence” A a is given by A a = po/t0= pO/(At cos 8,) = rd. The pulse duration At increases with z, or T, in an almost linear way if T >> TF= &p0 = 9 X lo-’’ s (temporal Fresnel length), corresponding to z F =(cT,)/tan 8,-470 mm. At a distance z1 = 5 meters (>>zF)from the grating plane, the pulse duration Atl = tb - t, is approximately equal to (TIAa)/cos 8, = (zl/c) sin 8, A a (cos f30)-2==18x lop2s = 18 ps, which is 18 times longer than the initial pulse, still with the same bandwidth. The “instantaneous frequency” of the pulse decreases from v, = Y, sin (a+ Aa/2)/sin fl0 to v b = u, sin (a,- Aa/2)/sin 8, as t increases from t, to tb respectively. The frequency decrease obeys a quasi-linear temporal variation if Aa/Oo<<1, which is the case here (“linear frequency modulation” of a pulse). The Fig. 2.6 represents this evolution along the geometrical coordinate z at three successive group delay times 7,< T~< 9

72.

In fact, as noticed at the beginning of this subsection (§ 2.3.1), it is not possible to produce single space frequency pulses rigorously, as the

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Fig. 2.6. Usual representation of pulse dispersion along Z at three successive times 70,T,,72.

periodic structure along the axis Ox is necessarily limited. The question is to know how large the spatial frequency spectrum can be around its average frequency No with the previous arguments remaining still valid. 2.3.3. Definition of quasi-single space frequency pulses; finesse of the spatial frequency spectrum of a pulse First, let us consider a pulse fo(x,t ) = Y,(t) (exp fi27rN0x}+ exp fi27rN1x}),with Fo(t) = rect (t/Ato) * exp G2.rrvot}.It is a space-time pulse of rectangular temporal shape (duration Ato, average frequency vo), spatially modulated by a beating of two spatial frequencies N,, and N1, with beating spatial periodicity px = l/(Nl- No)= l/AN (AN< - ~ p1 CN = (~ vo)-2cNl, , behind a transparency rect (t/Ato) located at s = 0. In this space, the linear dispersion of this structure will be depicted as a “two wavelength” diffraction problem (Fig. 2.7). The longitudinal amplitude distribution along the axis OP (sin O,=$(No+ N,)(c/v,)) results from the sine law beating between two plane waves of periodicities po and p,, diffracted from the plane s = 0 along two slightly different directions Oo =sin-’ (Noc/vo) and O1 = sin-’ (Nlc/vo).The beating periodicity along OP is AT = plpo/(pl- po);it satisfies the approximate equalities AT/po=AT/kl = N,,/AN= NJAN if 2AN/(N0+N1)<< 1, which we will assume from now on, writing AT= poNo/AN== NiNdAN.

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Fig. 2.7. Dispersion of a pulse modulated at two spatial frequencies No, N,,represented as beam diffraction at two wavelengths F~ and p , .

The amplitude distribution along the axis OP is a periodical wave train of periodicity AT which looks like the spatial distribution (exp fi27rN0x}+ exp fiZ.rrN,x}) after scaling of a factor p o N o ( - p l N , ) . In a more general way, any space-time amplitude distribution of the form fo(x, t ) =To(t) . Xo(x)exp (j27rN0x)in the plane z = 0 may be decomposed in its spatial Fourier components. Let AN<.The evolution of f i ( x , t ) on propagation in the space z > 0 can be described in the space { t , s} as a diffraction of multiple wavelength plane waves through a screen of temporal transmissivity To(t)(Fig. 2.8). The amplitude distribution f , ( x , t ) at any delay time s = ( z / c )sin e0will keep aform fib,t ) = .T,(t) X , ( x ) exp Cj27rN0x},in which temporal and spatial variations are separated from each other, if the bandwidth AN is narrow enough for the wavetrain to keep its frequency equal to No over a length AT> St,,,; St,,,,, = A t sin O0 being the maximum path difference between the outermost rays (“instantaneous frequencies”) AP and BP of the pulse at its two extreme propagation steps: s = o , S+W.

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-___j

Fig. 2.8. Quasi-single space frequency pulse: A T > > A t ‘sin 8,.

This is a straightforward transposition of the “quasi-monochromaticity” requirement, where the parts of the variables t and x, Y and N, and of the representation spaces (x, z} and { t , s} are exchanged. The condition AT>At. sin 8”

(2.3.15)

ATI~O)Y~*A~,

(2.3.16)

may also be written

where the ratio AT/kO=No/AN is the “finesse” N of the spatial frequency spectrum of X , ( x ) exp ci27rN0x}; then the condition enabling “quasi-single space frequency” pulse dispersion to arise is X > v, * At,

(2.3.17)

quite analogous to the previously derived quasi-monochromaticity condition 9 > N o * Ax (§ 2.2.1). An example of quasi-single space frequency requirement: we consider a pulse fo(x, t ) = T,(t) X&) exp Cj2.rrN0x}, of duration At = 100 ps, the temporal and spatial structures of which - respectively T,(t) and X,(x) oscillate at temporal (uo) and spatial (No) average frequencies y o = 6 x 1014Hz and No = lo3mm-’. Its average propagation angle in the { t , s} diagram is O0 = sin-’ (Noc/vo)= sin-’ (0.5); the path difference AT = At * sin 8, in this diagram is equal to 50 ps and the wavelength po= sin 8&, = 2 x lo-’’ s. The condition for the pulse to satisfy the quasisingle space frequency requirement is N>AT/p0 = 6 x lo4. Hence the 9

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spectral bandwidth AN o f X o ( x ) should not exceed AN=No/X= (1/60) mm-'; Ax = l/AN is the minimum geometrical size which has to be covered by the beam along the dimension x for its spatial frequency spectrum to be reduced to a quasi-single component, of frequency No. So the pulse distortions on propagation along z are reduced to pure linear dispersion of its temporal frequency spectrum. It could be shown readily that this condition coincides with the classical condition of spectral resolution, which enables a grating of average spatial frequency No to resolve at least one spectral sample in the temporal frequency spectrum of the pulse.

2.3.4. Experiments on the dispersion of quasi-single space frequency pulses Firstly, it is possible to follow the temporal evolution of the longitudinal energy distribution along the axis z of a pulse (I), modulated at a well defined spatial frequency No by diffraction on a grating (G) (Fig. 2.9). With the help of a telescope (T), of longitudinal magnification g = 1/600, a geometrical space of depth equal to 6 meters will be imaged on the 1cm long entrance slit of a streak camera (S.C.).This device allowed COLOMBEAU and DOHNAIJK [1980] to visualize (Fig. 2.10) the temporal focusing and defocusing, in the {t, s} space, of frequency doubled Nd/YAg single

Fig. 2.9. Observation of chirping of the light pulse (I) behind the strongly dispersive grating (G).(T): telescope; (D.S.):diffusing screen; (S.C.): picosecond streak camera; (G'): image of the grating through the telescope, where no chirping arises.

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Fig. 2.10. Streak camera image of the time focusing and defocusing of a single pulse spatially modulated by a limited grating.

pulses (wavelength 0.53 Fm), after spatial modulation at the frequency No= 1870 mm-’. The minimum pulse duration At at the “beam waist” is near to 30 ps, reciprocal of the laser emission bandwidth; the beam focus is located at z’ = z&, in the image plane (G’) of (G) through (T). At any z’ < z&, there is normal dispersion; the higher temporal frequencies propagate at lower velocity and at any z’>zb., there is anomalous dispersion. The amplitude of the pulse stretching grows almost proportionally to 2’- zb, far enough from the focal region z&, as for diffraction of cylindrical beams (also see Fig. 2.5). TREACY [1968a, b and 1969al was the first to make use of this effect for the compression of frequency modulated Nd/glass pulses (see §§ 3.1.3 and 3.3.2).

2.3.5. Two pulse interference: temporal Young’s experiment If two successive pulses (I1), (I;) are sent on the grating (G), they overlap as soon as their dispersion stretching gives them a duration greater than their spacing At. In the overlapping region, the temporal beatings between different “instantaneous frequencies” have a periodicity at, which may be calculated in a way similar to the Young’s fringe spacing on oblique observation planes (see relation (2.2.9), § 2.2.2): at=-

CLO

p cos eo ’

(2.3.18)

where p, O0, po are respectively angles and “temporal wavelength”

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Fig. 2.11. Temporal Young’s interference of two identical successive pulses (I,) and (I;) after spatial modulation by a grating (GI; (a) usual representation in the geometrical space; (b) representation as two beam interference in the space {t, s}.

defined on Fig. 2.11 in space {t, s}; At p-=cos

60

(2.3.19)

is the angle between the two geometrical rays emitted by the centers of the pulses (I,) and (I:) towards the observation point M. The final form (2.3.20) and is closely analogous to the Young’s formula (2.2.9). COLOMBEAU DOHNALJK [19801 performed the experiment by illuminating the grating (G) of Fig. 2.9 with the pulse couple transmitted from a Michelson interferometer, adjusted at a path difference A L = c * At. Photographs of the interference field in the { t , s } space are shown in Fig. 2.12. At the

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Fig. 2.12. Streak camera images of two different patterns of Young’s fringes in the space It, sl.

“beam waists” the pulses have At’- 100 ps temporal separation; the fringe spacing increases quasi-linearly with the distance between the “sources” and the observation plane, as in the classical Young’s slits experiment .

2.3.6. Temporal Fourier analysis by “far field dispersion” of single space frequency pulses It is well known that the dispersion of light at an infinite distance from a grating (G), that is in the back focal plane of a lens (0)(Fig. 2.13), gives access to the frequency spectrum of the light. The analogy between dispersion and diffraction suggests a way for imaging the temporal frequency spectrum of a pulse in the space of the temporal variable t itself; on the pulse propagating behind the grating, a quadratically time dependent phase shifting will be performed by a convenient frequency modulating device (0,)(Fig. 2.14) ( Q 3.2). After a delay time sf,,, characteristic of

Fig. 2.13. Spectroscopic analysis by dispersion at infinity imaged in the back focal plane ( T ) of a lens (0); f : focal length of (0); (G): diffraction grating.

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Fig. 2.14. Spectroscopic analysis by dispersion in the “temporal back focal plane” i of a linear frequency modulator (0,).The focusing delay s,, is analogous to a focal length.

both frequency modulation and dispersion, the temporal frequency spectrum of the pulse will be displayed sequentially in the space ?, which behaves as a “back focal plane” of the modulator ( O t ) ,This combination of linear dispersion and frequency modulation is the basis of various pulse compression schemes (cf. ri 3.3.2), but the temporal display of frequencies, widely used in electronical spectrum analyzers, is not yet applied at optical frequencies. Such spectral analysis of temporal vibrations may be represented as Fourier diffraction in the space {t. s}. Double spectral analysis in two successive steps allows selective filtering of temporal frequencies with various effects of pulse shaping, as discussed in § 3.1.

2.4. TEMPORAL FILTERING OF PULSES BY TRANSMISSION THROUGH TIME INDEPENDENT OPTICAL PUPILS

If a pulse exhibits neither quasi-monochromatic nor quasi-single space frequency structure, it will be considered as a linear superposition of quasi-monochromatic or quasi-single space frequency components. The distortion of the pulse after propagation or interaction with obstacles might be calculated by the linear superposition of the diffraction and of the dispersion phenomena of each of the temporal and spatial frequencies contained in the pulse. This calculation is generally not straightforward. The notion of “time impulse response” and “time frequency transfer function” of optical [1961] and even by Gouy, as noted by pupils, used early by CONNES SOMMERFELD [ 19671in the theory of spectroscopes, simplifies the temporal analysis of pulse diffraction through optical apertures. After introductory examples of temporal responses of elementary

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transmittance distributions, more general diffracting pupils are considered in connection with the temporal pulse shaping effects they produce.

2.4.1. Time impulse response of Young’s slits

A single short temporal pulse i(t), emitted from a point source So, is split into two identical delayed replicas after transmission by the Young’s slits S1, Sz (Fig. 2.15). At any point P of the space z > O , the output pulse O,(t) will be described by the distribution O,(t) = A [ i ( t - L / c ) + i ( t - ( L + Z ) / c ) ] , =A

i(t)@[s(t - L / c )+ a(t - ( L + I)/c)],

=A

i(t)@ h,(t)

=A

I, +m

i(t‘) h,(t-t’)dt’.

The distribution h,( t ) is the response at the point P of the Young’s slits to a Dirac pulse S(f), emitted by So at t = O ; this is the “time impulse response” of the optical pupil, which behaves as a time independent filter of the temporal frequencies emitted by So. The corresponding frequency transfer function H J u ) is given by the Fourier integral:

1, +m

HJu)

=

h,(t) exp (-j2~ut)dt = 2 cos (mllc) exp (-j2.rrv(L + 1/2)c).

Fig. 2.15. Temporal response O,(t) at a point P of a pair of Young’s slits to the excitation i(t) emitted by So.

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2.4.2. Time impulse response of a periodic grating If the screen consists of n identical, equally spaced apertures (Fig. 2.16), the temporal response observed at any point P generally results from non periodical repetition of the initial pulse i(t):

O,(t) = i(f)@

h,(t),

where i=n

h,(t)

=

C Ai 6 ( t - ( L + L ‘ + + If)/c). Ii

i=l

The “instantaneous periodicity”, or the “instantaneous frequency”, of the response is equal to At = ( li + I;)/c ; it continuously varies over the time interval T = ( I + l’)/c, which is the total duration of the impulse response at the point P. This is the linear frequency variation (chirping) of a pulse behind a periodic grating, already calculated in a quite different way in P 2.3.2. If the points So and P are moved away to infinity in directions il and i2 with respect to the screen, the response h,(t) becomes periodical, of average frequency u = c/(a(sin il -sin i2)), over a limited duration T = na x (sin il -sin i&c, n being the total number of grooves of the grating. The uncertainty Au about the frequency u is of the order of 1/T, corresponding to the bandwidth of a peak of the frequency transfer function H,(v) (Fig. 2.17).

Fig. 2.16. Temporal response O,(t) at a point P of a periodical set of slits to the excitation i(t) emitted by So. ( e ( t ) = i ( t ) . )

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Fig. 2.17. Temporal impulse response h,(t) and frequency transfer function H,(v) at infinite distance from a limited grating of narrow slits.

2.4.3. Time impulse response of other time independent apertures In the general case, the output signal O,(t) at any point P still results from convolution between the signal i(t), emitted from So, and a time impulse response h,(t), depending on the optical shape of the aperture, on condition that the light intensities remain far below the thresholds of occurrence of optical nonlinearities. The temporal shape and duration of h,(t) may sti!l be deduced from LACOURT and VIENOT[19731, BONgeometrical considerations (FROEHLY, NET [1975, 19761) illustrated in Fig. 2.18, a Fourier transformation leading afterwards to H,( v). Firstly, the geometrical locus (2;)of points reached at time to= lo/c by a Dirac pulse 8 ( t ) , emitted at t = O by the point source So, has to be determined. Then a correspondence between the points of (26)and those of a ray OP will be set up by the following operation, analogous to a projection on OP: to each point M of (Z;) corresponds a point R, defined by the intersection of OP with the circle of center P and radius PM= z . This point R carried an amplitude pp(z) dz equal to the length of (ZA), included within the circles of radii z and z +dz. The temporal amplitude distribution h,(t), available around the time tl = ( l o + ll)/c, is shown (GOODMAN [1968]) to be proportional to the

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Fig. 2.18. Graphical approach to the time impulse response of any diffracting screen (D); (a) ideally short impulse S ( t ) emitted by So; (&): impulse at t = &/c, (D) being removed; (2;): impulse transmitted by (D) at t = lo/c; (b) projection p,(z) of (2;)on a ray OP along arcs of circles of center P.; (c) and (d): time impulse response at P versus z or t got by differentiation of p,(z) with respect to z or t (z-ct =const.).

first derivative of p,(ct) with respect to the time t ( z being replaced by (-4):h,(t) = (d/dt)(p,(-ct)). An example of practical importance is given by the temporal response of a free aperture (A) (Fig. 2.19). It illustrates the temporal distortions of a pulse by a “diffraction limited” beam-for instance a single mode TEWobeam. The geometrical locus (2,) is a circle of radius lo, centered on So; the aperture (A) transmits a limited arc (Zo)of the circle (Zh). All the arcs of circles (&), tangent to (So),have centers (P,) located inside the region of larger intensity transmitted by (A). At these points, the time impulse response of (A) is the derivative of a very sharp peak, closely resembling a Dirac pulse; this derivative is approximately the time differentiation operator (e.g. RODDIER [1971]). As the source So emits a signal fo(t), the point PI receives a signal fl(t) = (d/dt)(f,(t)), a delay time (lo+ l,)/c being omitted. The most noticeable effect of this differentiation is a well known (7r/2) phase shifting if the relative signal bandwidth Av/v is narrow. Within this approximation, the temporal structures of fl( t) and fo(t) are similar. But this is only an approximation, and weak distortions of fl(t) are readily observable if the pulses emitted from So are shorter than the total effective duration L,/c of the impulse response at PI. This approximation becomes invalid at points P, out of the region of larger intensity just considered. The centers Pz of circles (&), which

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Fig. 2.19. The temporal response h,(t) of a free aperture (A) to a Dirac impulse 8 0 ) behaves approximately as: (a) the lint derivative (d/dt)[~,,,(r)] of a sharp peak at any point P,, contained inside the geometrical beam or in the central lobe of the far field transmitted by (A); (b) two Dirac impulses of spacing L,/c at any point Pz contained inside the geometrical shadow (near field) or the higher order diffraction lobes of the beam (far field).

cannot be tangent to (&>, lie in lower intensity regions (geometrical shadow), where the filtered temporal structure f2( t ) is generally quite different from the initial one, fo(t). Indeed, the time impulse response is no longer peak shaped; its energy spreads over a time interval LJc, increasing with the size of (A) and the obliqueness of the diffracted rays. This entirely modifies the temporal amplitude distribution in the diffracted pulse, which exhibits a fine structure sharper than LJc, whose temporal frequency spectrum is spread over a bandwidth Av > c/Lz. So, by diffraction of short pulses through apertures of longer time impulse responses, various temporal structures can be yielded; this leads to picosecond pulse shaping processes (§ 3.1).

I 3. TEme Shaping of Picosecond Optical Pulses Because of the strong interaction between the spatial and temporal shapes of a light pulse propagating in free space, acting on the spatial structure modifies the propagation conditions of the time structure and vice versa. The shaping techniques described in this section consider ways of giving time structures to one single spatial sample of a pulse, or to a single

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spatial mode pulse - for instance, propagating as the TEM,, Gaussian mode of a laser. The work is presented from the point of view of Fourier optics in the time domain and related to the filtering and modulation of temporal optical frequencies.

3.1. PULSE SHAPING BY OPTICAL FILTERING OF TIME FREQUENCIES (SPECTRAL MODULATION)

3.1.1. General principles and limitations

In this subsection, the temporal structure of a pulse is modified by frequency dependent attenuation, or phase shifting (amplitude or phase frequency filtering). The shorter the pulse, the easier the filtering, as spectral analysis of wide band short pulses requires less instrumental resolution than analysis of longer pulses which are of narrower bandwidth. Optical filtering techniques are specially powerful in the subpicosecond domain, where no other solution exists at present. But the disadvantage common to all of them lies in the energy loss, being proportional to the number K of the samples introduced by the spectral selection, and the average power loss proportional to K 2 ; filtering of the phase causes no fundamental energy loss, and average power loss proportional to K . The time resolution is limited to At = l/Av by the bandwidth Av of the initial pulse; the duration of the pulse after shaping is limited to 1/Sv by the frequency resolution Sv of the filter. The number K of the samples is equal to Av/ijv.

3.1.2. Examples of typical shapes produced b y amplitude or phase filtering In Fig. 3.1 are shown three types of shaping easily performed by filtering an initially short pulse, the average frequency, bandwidth and duration of which are respectively v,, Avo, At,, with At, . Avo= 1 (temporally unimodal, or single samp1,e pulse): f,(t) = A(sin .rrWt/.rrWt)exp (j2nvOt), (a)

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Fig. 3.1. Three examples of temporal shapes fa(t),fb(t) and f,(t) of an initially Gaussian pulse after frequency filtering of the Gaussian spectrum F J u ) = IF,(u)) exp ( j & ( u ) ) , by frequency transfer functions Ha( u ) , Hb(u) and H,( u ) respectively.

after square band transmission of the spectrum, that is spectral modulation by the rectangular frequency transfer function Ha(v) = rect ( v - vo)/ W (W<
Hb(v) = {rect ( v - vo)/W}@{ S( vo - Av/2)+ S (uo+ A v/2)}

( W << Avo),

fJt) = C * exp Q2.rr(vot- t2/2T2)}, (c) after phase filtering by the complex transfer function Hc(v)= e x p G d v - v0)’T2}. In each of these three situations, as well as in any one imaginable at present, the shaping concerns the envelope modulating a carrier frequency vo >>Avo;a 0.1 ps pulse still includes 50 oscillations of the carrier in the domain of visible radiation (vo=5 x 1014Hz), and this is the order of magnitude of the shortest pulses as yet produced. In these examples, a general effect of the filtering is a stretching proportional to the spectral resolution of the filter; in the literature, shaping systems acting by frequency filtering are often referred to as “pulse stretchers” (see 9 3.1.3).

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3.1.3. Filtering experiments (a) Linear chirping. A simple shaping experiment by phase filtering of the time frequencies consists - as noted earlier by Lord RAYLEIGH [1912] - in transmitting a light pulse through transparent dispersive materials; it only works on the phase of the spectral components, without changing their amplitude. Let n(v) be the refractive index of the material at the frequency v ; after propagation (Fig. 3.2) over a distance z, this frequency has a phase increase cp(v)=(2m/c) * n(v). The transit time, over this distance, of a wave group at average frequency vo is the group delay time t , = z/u,

= (27r)-1(dcp(v)/dv),=, = (z/c)(n

+ v dn/dv),,,,

u, being the group velocity of the pulse in the dispersive material. The energy spreading of the wave group over a time interval At‘ around tg can, in the first approximation, be deduced from the next derivative of q ( v ) with respect to v :

(d2cp(v)/dv2)v=y, =2 ~ ( z / c ) ( 2 dnldv

+ v d2n/dv2),=,.

Fig. 3.2. Linear pulse chirping by propagation over a length z of dispersive material. Representation as a beam diffraction in the two dimensional space {t, s =(z/c)x (l-(ug/c)z)”z}; us: group velocity at average frequency vo of the pulse.

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This term of spectral phase curvature also accounts for the effect of linear “chirping” of the frequency of the vibration. These effects of energy spreading and frequency modulation are interpreted geometrically (Fig. 3.2), as due to the diffraction of a “temporal wave surface” in the two dimensional space referred to the orthonormal temporal coordinates {r, s}, where s = ( z / c )(1 - ( V , / C ) ~ ) ~ ’ ~Similarly . to 0 2.3, the average propagation angle a. will be calculated by giving the spectral phase curvature (d2q(u)/du2),=, the form of a diffraction curvature in the {t, s} space: (d2cp(v)/dv2)= -(27r~0)(c0s

- s,

in close analogy with relation (2.3.7). The “wavelengh” po, characterizing the rate of the wave group spreading, is equal to sin ao/vo,again as in 0 2.3. GIORDMAINE, DUGUAY and HANSEN[19681 demonstrated picosecond pulse chirping by dispersive propagation in laser materials, in order to perform further pulse compression. Unfortunately, the dispersive power of transparent dielectrics is very weak, at frequencies far from the absorption bands. Nevertheless, there now exists the solution of cumulating the dispersion over lengths of low loss single mode fibers; for instance (PUSECKI [1980a, b]; see also 0 4.3) a Ge doped single mode silica fiber yields a spectral phase shift A 2 q = (d2q(u)/dv2)(Av)2of the order of 27r radians each 1.7 meter over a bandwidth Av = 10” Hz, around the average frequency vo = 5 X l O I 4 Hz (Ao = 6000 &, with losses of the order of 20 dB/km. Other workers (GRISCHKOWSKY [1973, 1974a, b], WIGMORE and GRISCHKOWSKY [1978], BJORKHOLM, TURNER and PEARSON [1975]) made systematic use of the strong dispersion of near resonant atomic vapors of alkaline metals (Na, Rb, . . .); the dispersive power can be 1000 times stronger than in previously considered materials. This allowed significant dispersion of nanosecond pulses further allowing efficient pulse compressions (P 3.3). Another solution, due to TREACY [1968al uses the dispersion behind diffraction gratings (Fig. 3.3) discussed above (0 2.3); a parallel beam carrying a pulse f(t) of bandwidth (Av)and centraI frequency vo falls onto the grating (GI) of spatial frequency No. After spatial modulation by the grating transparency, the beam propagating in the diffraction order (+ 1) has a space-time structure described by g(x, t ) =f(t) exp (j27rN0x): it becomes a single spatial frequency beam, at frequency No, and therefore exhibits pure dispersion (ofthe linear anomalous type) as it propagates in

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Fig. 3.3. Anomalous dispersion (TREACY [1968b]) of a Gaussian pulse by a couple of identical gratings ( G , ) ,(CQ. (a) experimental lay-out; (b) diffraction representation in space {t, s}: beam waist w =At cos O,, wavelength F~ = v;' sin Oo; divergence angle of the beam: A6 = k o / w ; pulse duration in (G2)plane: At'= (2T . AO)/cos Oo; frequency sweeping from v, to va. 9

the half space z >O. Another grating ( G J , identical and parellel to (GI), collects the light diffracted from (G,) in its (+1) order at the distance z = zo of the (GI) plane. The amplitude distribution in the pulse reaching the plane of (G,) has the form h(x, t ) =f'(t) exp (j27rN0x), f'(t) being related to f (t) by convolution with the quadratic phase term (j cos O o / G ? ) exp (jd cos' OO/kOT), where T = ( z / c ) tan Oo and po= (v0)-' sin 8" (relations (2.3.13) and (2.3.14)). This convolution is geometrically represented as a diffraction between the two planes (G,) and (G2) in the {t, s} temporal space. The grating (Gz), working in its diffraction order (-l), multiplies the function h(x, t ) by its spatial transmission function m ( x )= exp (-j27rN0x); the transmitted pulse amplitude has the form l ( x , t) =f'(t), which no longer depends on the geometrical variable x, and again propagates in a parallel beam, keeping a steady state of dispersion at any z > zo. This discussion assumed the gratings to be unlimited. In fact, the uncertainty AN on the spatial frequency No, of the waves diffracted by

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(G,) and (G2),is of the order of l/a, a denoting the gratings widths. Thus, the product of the gratings’ amplitude transmittances is no longer a constant along x, but has a narrow spectrum spreading over the finite bandwidth AN = l/a # 0. The temporal signal transmitted by (G2) exhibits negligible x dependence and propagates in a nearly homogeneous parallel beam only if the beam diameter on (G2)slightly differs from a, that is, if the geometrical lateral shift Ax between the extreme frequencies remains much smaller than a at the distance zo. It strongly limits the variation range of this distance, which precludes reaching the highest dispersion theoretically enabled by the intrinsic spectroscopic performance of the gratings. Another disadvantage of this arrangement is its inability to give rise to normal dispersion effects. G I R Eand ~ TOURNOIS [1964] early on described a highly dispersive filter (Fig. 3.4) consisting of a Fabry-Pkrot interferometer working with 100% reflection of its back mirror. Aside from slight absorption and losses of rays at the edge of the Fabry-PCrot plates, the spectral components of a pulse (I) are only dephased on reflection from this mirror pair. The phase shift of the Y frequency is equal to q ( v )=tan-’ [Q tan ( 8 ( v ) / 2 ) ] , where Q = (1+ r ) / ( l - r) and O(v) = ( 2 r u ) ( 2 e / c cos ) i. The second derivative d2q(v)/dv2 responsible for linear chirping exhibits (Fig. 3.5) two maximums of opposite sign over one free spectral range uf= 4 2 e cos i); the first one, around frequency vl, causes anomalous dispersion and the second one, around frequency v2,causes normal dispersion. Linear chirping is obtained in a limited frequency range Av, around v 1 or v2, over which d2q(v)/du2 remains nearly constant. The maximum amount of linear chirping is then proportional to A’q = (d2q(v)/du2)(Av)2.In a further publication about applications of this device to pulse compression

Fig. 3.4. Gires and Tournois interferometer: the total pulse length L has to be larger than 2Qe cos i, (Q = (1+ r)/(l - r ) ) and its “correlation length” larger than 2e cos i.

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Fig. 3.5. Gires and Tournois interferometer: maximum linear dispersion occurs around frequencies v,, v2 at which the curvature of q ( v ) reaches its maximum; vf = c/l = c/(Ze cos i): free spectral range of the interferometer.

(see also Q 3.3) DESBOIS, GIRESand TOURNOIS [1971] underline that the two terms of the product, d2q(u)/du2 and Au have opposite ways of variation; it limits the quadratic phase shift A2q to rather small amounts (of the order of 2.rr radians) although the dispersion rate could be increased at will. (b) Filtering of both amplitude and phase. Using two successive grating filters DESBOIS, GIRESand TOURNOIS [1970, 19731 performed the first experiments of amplitude and phase shaping in the picosecond domain. A grating (GI) diffracts the temporal frequencies of a pulse (I) in frequency dependent angles (Fig. 3.6). A second grating (G2),identical and parallel to (GI), is located at a large distance from (GI);from (G,) emerges an almost parallel beam, the frequencies of which are well separated from each other if the distance e is large enough with respect to the initial laser beam size a. A variable transmittance screen (F) acts as frequency filter in the field diffracted by (G2).A mirror (M) sends back the beams in autocollimation towards the source, after two other diffractions by (G2)and (GI). The narrow light

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Fig. 3.6. Stretching, chirping and shaping of an initially short pulse by the two grating [ ( G , ) (G2)1, , fourfold dispersion filtering device of DESBOIS, GIRESand TOURNOIS [1970, 19731.

pencils corresponding to different frequencies will be recombined on (GJ, after having traveled along different geometrical paths (maximum path difference: Az = c * At’). The nearly parallel narrow beam synthesized by this diffraction on (GI) has the diameter a and carries a pulse (I’), the frequency of which linearly decreases with time. If, for instance, the filter (F) suppresses a particular frequency v,, there is no light at the time to where this frequency should appear in the frequency modulated pulse (1’); in this geometrical framework, the temporal power distribution in the pulse is roughly similar, with the exception of a scaling factor, to the spatial transparency distribution of the filter (F). Highly dispersive gratings ( N = 1872 mm-’), working near to grazing incidence (77”), with e = 400 mm, l2 = 140 mm, enabled the authors to produce nanosecond pulses (At’= 1.7 ns) exhibiting various temporal shapes, 50ps rise time, by the filtering of Nd/glass mode locked laser emission. Some of the performances of the device could not then be tested, as the available detectors were not fast enough for resolving the temporal structure of the filtered pulses. AGOSTINELLJ, HARVEY, STONE and GABEL[1979] recently made accurate observations of the shaping effects with the help of a picosecond streak camera. A detailed difh-action analysis of the rather intricate processes involved by this filter can be given (COLOMBEAU [1980]) in the two-dimensional temporal space {t, s}, where the coordinate s is now defined as s = (z/c) x

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(sin a,+sin al)= zN/vo, the meaning of the geometrical parameters N, 2, al,ao, being explained in Fig. 3.6. Such an analysis will not be outlined here, needing a rather long discussion. It shows that this device does not allow accurate control of amplitude and phase distribution of the optical field, because of intrinsic mixing of temporal and spectral operations. (c) Pulse stacking and stretching. Other shaping techniques exist which can be more readily understood and described by Fourier analysis. J. CONNES pointed out early [ 19611 that a Fabry-Pirot interferometer converts a single short pulse into a train of equally spaced subpulses. The subpulse spacing is equal to At = 2 e / c if the optical thickness of the interferometer is equal to e. The time impulse response h ( t ) and the corresponding frequency transfer function H ( v )= F T ( h ( t ) )give rise to an obvious shaping effect, two parameters of which ( e and 7, see Fig. 3.7) are available for adjustment. If the interferometer works under oblique incidence (Fig. 3.8), it generates a series of parallel beams carrying a sequence of regularly delayed subpulses; each of them can be selectively attenuated by a mask (M). The pulse train, reaching the recombination point P of the parallel beams, is modulated by an envelope which depends on both the reflection coefficient R and on the transmissivity of the mask (M). THOMAS and SIEBERT[ 19761 described "pulse stackers" based on such principles. Other multiple wave interferometers, including a large number of independent beam splitters and mirrors, lead to similar results (HUGHES and

Fig. 3.7. Temporal pulse shaping of a 6t long pulse on transmission through Fabry-Phot plates of spacing e > c . fit.

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Fig. 3.8. Principle of pulse stackers: 2e cos i > c . 6t. The output signal f;(t) consists of a discrete repetition of delayed images of the initial pulse f(t).

DONOHUE [1974], MARTIN and MLAM [1976], MARTIN,JOHNSON, GUINN and LOWDERMILK [1977]). These devices suffer from high losses: a ten beam interferometer transmits about 0.1% of the initial pulse energy. They must satisfy interferometric stability requirements if the initial pulse duration 6t is larger than the interpulse spacing. They then perform “pulse stretching” with the ability to generate continuous smooth shapes, if the interferometric adjustment preserves the optical phase synchronism in pulse overlapping regions (MARTIN and MLAM [1976]). This is a difficult mechanical problem which has not yet been resolved to our knowledge. Such devices are used as discrete “pulse stackers” rather than as continuous “pulse stretchers”. A simple effective first solution for continuous smooth pulse shaping is provided by a slightly modified single grating spectroscope (COLOMBEAU, VAMPOU~LLE and FROEHLY [1976]). A diffraction grating (G), partially blocked by a suitable mask (M), will be illuminated by a point source S of pulses (I) through the collimating lens (L,) (Fig. 3.9). The pulse (I’), collected at the point P of the focal plane of the lens (L), exhibits a temporal shape related to average frequency and bandwidth of the incident pulse, grating resolution, and the transmittance distribution of the mask. The temporal response h,(t), at point P, of the free grating (G) (mask M removed), to a Dirac pulse 6 ( t ) emitted from point S, is a limited sequence of equally spaced subpulses with p/c pulse spacing and llc whole

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Fig. 3.9. Continuous shaping by convolution (@I) of a short pulse f ( t ) by the time impulse grating; (M), mask of transmissivity m ( x ) along x ; response h ( t ) of a masked grating: (C), (LJ, (LJ lenses of focal lengths f,,f2; h,(t): impulse response of (GI, mask removed; h ( t ) : impulse response with mask (M); f’(t): output signal of the shaping device.

duration; p and 1 respectively denote the projections of groove periodicity and grating length along the OP direction. The number of subpulses in h,,(t) is equal to the number of the illuminated grooves in the grating plane. It is clearly seen in Fig. 3.9 - and analytical calculation readily confirms this - that introducing the mask (M) modulates the grating response ho(t) by the function m’(t), which corresponds very closely to the mask transmissivity m ( ( x / c )sin 0) along the projected variable z/c = (x/c) sin 8. The envelope of the time impulse response h ( t ) of the masked grating carefully reproduces the mask shape m ( x ) with time scaling t = (x/c) sin 8; h (t )= h,(t) m ( ( x / c )sin 0). Really, the source S emits a physical pulse rather than a temporal Dirac distribution. This pulse f(t) lasts for the duration At; its frequency spreads over the bandwidth Au around the average frequency vo. The effective

-

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Y

Y I

l

l

Fig. 3.10. Masking of the grating (C) by an opaque screen (M) cut out in a geometrical shape y(x) similar to the amplitude modulation law m(x).

signal f ’ ( t ) , collected at P, results from the convolution between f ( t ) and the time impulse response h ( t ) . It is an amplitude modulated pulse; its envelope modulation resembles the transmissivity distribution m’(t ) as the initial duration At is short with respect to Ilc. Its whole duration is At’= llc + 2 At llc and the points P, where maximum amounts of energy may be collected, correspond to propagation angles 6, at which the projected periodicity p is an integer number k of wavelengths ha = c/va (the integer number k is the working “diffraction order” of the grating). Making the mask (M) can be achieved in quite a simple way by cutting out indentations in an opaque screen along a geometrical curve y = m ( x ) (Fig. 3.10). Then a pinhole, smaller than the diffraction spot of the lens (LJ, should be located at P, rather than the filtering slit erroneously suggested in the publication of COLOMBEAU, VAMPOUILLE and FROEHLY [1976]. In the second order of a plane holographic grating, grooved over 140 mm length at space frequency 1440 mm-’, single mode (At . Av = l), 30ps pulses coming from a frequency doubled Nd/YAg laser were stretched over 700 ps with various temporal shapes, including about 20 independent samples. Further important simplification of the device was achieved (SAUTERET, NOVARO and MARTIN [19Sl]), replacing the three components (LJ, (G) and (LJ by a single concave holographic grating, which can be stigmatic for the two points S and P. These authors studied the resolution, losses and signal-to-noise ratio of this system with the help of a picosecond streak camera, and pointed out the possible combination to ^I

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pulse stackers, for the optical generation of temporally controlled nanosecond pulses. Holographic gratings are nothing but generalized Fresnel zone plates, or “flat lenses”, the shaping abilities of which were noted by CAULFIELD and HIRSCHFELD [1977, 19781, in the light of various pulse processing operations. Pulse stackers and stretchers are well suited to the generation of any real positive law of amplitude modulation. Other pulse shape types, exhibiting phase and amplitude modulation, can be produced more readily by direct selection or phase shifting of the temporal frequencies, as shown later on. (d) Shaping by Fourier analysis. In the recent experiments next discussed, pulse shaping is achieved in two successive steps, involving Fourier analysis and Fourier synthesis respectively in the time domain (COLOMBEAU and DOHNALIK [1980]). These operations are performed in a very simple way by the two stage spectroscope, similar to LACOURTand GOEDGEBUER’S [19771 “spectrographic imaging” device, sketched in Fig. 3.11. The first grating spectroscope, consisting of a telescope (Cl), a plane diffraction grating (GI),and a lens (LJ, displays the frequency spectrum F,(v) of a short mode locked collimated pulse fo(t) in the back focal plane of (LJ. If a mask (F), of variable optical thickness and transmissivity, is located in this plane, it

Fig. 3.1 1. Shaping by double temporal Fourier analysis (double dispersion at infinity). fo(t), F J v ) : single initial pulse (Gaussian pulse); (CJ, (CJ: beam expanders; (GI), (G2): identical diffraction gratings; (L,), (b): collimating lenses; dotted lines: rays representing different temporal frequencies. (F), amplitude and/or phase filter (here, rectangular slit); fl(f’). F,(v): pulse after shaping.

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behaves as a frequency filter, geometrically performing frequency selective attenuation and phase shifting, which can be described with the help of a complex frequency transfer function H ( v ) . The second grating spectroscope (LJ, ( G J , ( C J , reconstructs a parallel beam with new temporal structure f‘(t) from the filtered spectrum Fl(v) = F,(v). H ( v ) ; that is the Fourier synthesis step, where fl(t’)= JT2 F , ( v ) exp (+j2rvt’) dv = f(t)@ h(t), the Fourier transform h(t) of H ( v ) being called the “time impulse response” of the filtering device. The shorter the initial pulse fo(t), the closer the similarity between the shapes of f,(t’) and h(t). In fact, the time variable t’, in the output pulse fl(t’), differs by a constant delay tE from the time variable t in the input pulse f ( t ) , and t ’ - t = t,, the delay time of the wave group, is of no interest in these shaping experiments. In the following, t, is omitted, f l and fo being represented as functions of the variable t only. As shown in Q 2.3, this omission allows for our fruitful interpretation of pulse dispersion as diffraction in the temporal orthogonal coordinates { t, s = ( z / c ) (1( v,/ c 1’)

“”I.

Amplitude filtering effects are caused by variable transparency and constant optical thickness filters (F). Phase filtering effects occur with constant transparency and variable thickness of this screen. The photographs (a), (b), (c) and (d) in Fig. 3.12 are picosecond streak

Fig. 3.12. lf,(t)12, Ifb(t)12, If,(t)12, Ifd(t)lZ: streak camera images of a Gaussian pulse of initial spectrum F,(u) after frequency filtering by H,(u), Hb(u),H,(u) and Hd(v) respectively.

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camera recordings of pulses fa(r), fb(t), fc(t) and fd(f). These are yielded successively by filtering the same initial Gaussian pulse by mask apertures, achieving the frequency transmissions

Ha( v) = rect ( v - vo)/a; Hb(v) = rect ( v - v&a

+ rect ( u - v2)/a;

H,(v)=(rect‘(v-v,)/a)~LU,(~);

&(v)

=a(rect ( v - v,)/a)+P(rect (v- v,)/b);

( P 2 b>> a2a and b << a ) with the following symbol meaning: rect ( v - v o ) / a= 1 if

=O

v € [ v O - ( a / 2 )vo+(a/2)] ,

elsewhere

i=+m

S(v-ip)

W,(v)=

(Dirac comb, periodicity p )

i=-m +m

F ( v ) @ G ( v= )

F(v)* G ( v ‘ - v) dv (spectral convolution).

In the experiments described here, the input Gaussian pulse is a frequency doubled single pulse extracted by electro optical selection from the quasiperiodic emission of a mode-locked YAg/Nd laser: harmonic wavelength A. = 0.53 pm, frequency vo = 5.7 x 1014Hz, duration At 30 X s, and the bandwidth A v = 3.3 X 10” Hz, energy of the order of millijoule. As previously pointed out, the energy losses depend on the diffraction efficiency of the grating - which theoretically could reach 100% - and are proportional to the number of independent samples in the pulse after shaping. The experiment (d), of special interest, will be discussed in a more detailed way. It can be considered as a “temporal holographic” o r a “temporal interferometric” experiment. Indeed, the energy distribution Ifd( t)I2, displayed by the streak camera, results from sinusoidal beatings between the well defined frequency v2 and each one of the frequency spectrum spread over the bandwidth a around vl. The camera recording is nothing but a hologram, in the time domain, of the “object” spectral distribution a(rect (v- v l ) / a ) ,where the narrow line p(rect (v- u2)/b)= S(v - 27,) plays the part of “reference” spectral distribution. This temporal energy recording contains all of the amplitude and phase information o n the “object” frequencies. Comparing photographs (d) and (a) of Ifd(t)12 and Ifa(t)12, tiny “holographic fringes” may be recognized in Ifd( t)l’, the L-

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contrast and phase of which are respectively proportional to the modulus and argument of fa(t) (see for instance the contrast inversion on the side lobes of the function fa(t) = (sin 7rta)/(7rta)). This rather academic example shows that the method can provide images of the temporal evolution of amplitude and phase in picosecond light pulses, on condition that the temporal camera resolution is higher than U A Y (Av is the whole bandwidth covered by the two “object” and “reference” spectra). If the “reference” spectral line is included in the bandwidth of the “object” spectrum itself, the energy distribution lfd(t)I2 looks like an “in HARVEY, STONE line” Gabor type hologram. In such a way AGOSTINELLI, and GABEL [1979] directly measured the frequency modulation of chirped laser pulses, on the output of the two gratings device of DESBOIS, GIRES and TOURNOIS [1970, 19731; they extracted the reference frequency from the pulse spectrum itself with the help of a Fabry-PCrot spectral filter. On the contrary, if the reference frequency lies far enough on the outside of the “object” spectrum, the hologram is of the “off-axis’’ Leith-Upatnieks type; in this case only phase measurements can be carried out without sign or modulo 27r ambiguity. Such a reference signal, synchronized with the pulse under study, can now be generated by the self modulation of amplitude and phase on propagation in nonlinear optical materials (see next § 3.2). Pulse stretching by pure phase modulation is demonstrated in Fig. 3.13. The filter (F) is a converging or a diverging lens which shifts the phase of the spectral components according to the complex transmission function H ( v )= exp {+r ( Y - V ~ > ~ Twith ’ } , sign plus or minus depending on the algebraic convergence of the lens. This phase filtering has either normal or anomalous effects of linear dispersion, and the transmitted pulse exhibits both time stretching and positive or negative linear frequency modulation, ruled by a relation of the type f(t)=exp(-at2) exp(-j27r(v,t*t2/2T2)). The photographic recording of the streak picture showed the magnitude (about 30) of the stretching ratio, but not the chirping effect, which only temporal (see above) or spectral (see § 4.3) interferometry could reveal. Phase filtering effects can be explained in a geometrical picture which represents spectral dispersion as the diffraction of the “temporal wavesurfaces”, previously defined in the two dimensional space (t, s} (Fig. 3.13). In this space, the “instantaneous frequency” v ( t ) , so widely used to

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Fig. 3.13. Shaping by phase filtering of the frequency spectrum: (a) normal, (b) anomalous dispersions after quadratic frequency dependent phase shifting through (a) diverging, and (b) converging achromatic lens (F).

describe frequency modulation, becomes a geometrical ray emitted from the plane s = 0, at the time t, in the direction a ( v )= sin-’ ( ( v / v osin ) O,), according to the results of 0 2.3.2. The dispersion of a pulse propagating from the (Pl) plane to the ( T ) plane resembles cylindral Fourier diffraction of a narrow oblique beam at the “temporal wavelength” p, = ( y o ) - ’ sin do through a lens (L& The phase filter (F), here a converging or a diverging cylindrical lens, changes the curvature of the temporal wave in the ( T ) plane. The Fourier synthesis, by further diffraction from the ( T ) plane to the (P2) plane, yields chirped pulses respectively depicted as converging (Fig. 3.13a) or diverging (Fig. 3.13b) wave surfaces at the output plane (P2). The phase shift A q between the central vo and the extreme frequencies v 1 and v2 is equal to 2mv0Ae/c, where the refractive index n of the filtering lens glass is assumed a constant over the spectral bandwidth v2- vl, Ae denoting the difference of the glass thicknesses crossed by vo

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and v1 (or v,) spectral components. The stretching ratio At’lAt is equal to the number K = A q / 2 n if the spatial frequencies of the gratings (GI) and (G,) are well enough defined for the condition of “quasi-single space frequency” to be satisfied. It leads (COLOMBEAU [19801) to the following requirement as to the gratings size: al sin Bo and a,sin B,> Kc At( = c A t f ) .On this condition, the gratings spectral resolution is larger than (v2- v , ) / K = Au/K (Av 5 spectral bandwidth of the pulses). Filtering by cylindrical converging lenses performs the same operation as free space Fresnel diffraction in the previously described device of DESBOIS, GIFW and TOURNOIS [1970, 19731. But now normal dispersion can be achieved as well using diverging lenses, and many other possibilities also exist. Filtering by a Fresnel biprism causes pulse splitting without frequency modulation (Fig. 3.14a), filtering by a third order aberration gives rise to the compound effects of splitting and frequency modulation (Fig. 3.14b) and “temporal caustics” arise as energy concentration at the pulse edges and pseudo-periodical intensity oscillations. These caustics will be discussed again in 5 3.3.2.

Fig. 3.14. Other shaping effects by various phase filters (F): (a), pulse splitting by Fresnel biprism; (b) caustic effects after filtering by thud order aberration.

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Emphasis should be laid on the theoretically low losses of shaping by phase filtering; using holographic gratings of 80% diffraction efficiency would bring the energy yielded to beyond 60%.

3.2. PULSE SHAPING BY TEMPORAL MODULATION

The devices considered so far generated temporal shapes by direct or indirect modulation of the spectrum of single mode short pulses. This filtering process fundamentally limits the temporal resolution and duration of the filtered signals. For instance, from a single mode pulse of length At = 30 ps, of bandwidth Av l / A t = 3.3 X lo1’ Hz, it is only possible to generate multimode “stretched” pulses longer than 30 ps, including a sample number K proportional to the stretching ratio. In order to perform pulse shortening rather than pulse stretching, fast temporal modulation has to be achieved rather than frequency filtering, whereas frequency filtering narrows the pulse spectrum, the temporal modulation spreads this spectrum over a bandwidth larger than its initial width (before modulation). Temporal modulation of laser pulses on the subnanosecond, or even subpicosecond scale raises more difficult experimental problems than the optical filtering techniques just described. Electro-optical or acousto-optical modulators working at 100 or 200 MHz allowed TREACY [1968a], DUGUAY and HANSEN [1969], DESBOIS, GIRES and TOURNOIS [1970], GRISCHKOWSKY [1974a, b], WIGMORE and GRISCHKOWSKY [1978] to produce the chirping of 500 ps-10 ns long pulses from continuous mode-locked He-Ne or dye lasers. Picosecond electronically driven Pockels (AGOSTINELLI, MOUROUand GABEL [19791) or Kerr cells (STAVOLA, AGOSTINELLI and SCEATS [1979]) even showed rise times of the order of 20 ps. But picosecond or subpicosecond modulations of picosecond optical signals seem, at the present time, to be possible only by wave interaction in nonlinear optical materials. Nonlinear optical experiments, initiated by the use of pulsed laser sources, are rapidly growing as mode-locked pulses become shorter, carrying more peak power at lower energy levels. Many reviews have covered this domain (BLOEMBERGEN [ 19651, ARECCHI and SCHULTZDUBOIS [1972], RABIN and TANG[1975]); picosecond nonlinear optics is thoroughly reviewed by AUSTON [19771. L-

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A common feature of nonlinear wave interactions is the creation of new optical frequencies. Very wide spectral broadening of laser pulses occurs, for instance, in harmonic generation, self-phase modulation and self-focusing, stimulated Raman scattering, and laser induced plasma generation. Unfortunately, the experimental mastership of these various phenomena has not yet reached levels of accuracy high enough to enable deterministic control of the amplitude and phase of each mode of the modulated pulse. This problem of “coherence” remains to be solved before many promising effects can be applied to pulse amplitude shaping, such as fast optical switching by plasma generation (YABLONOVITCH [1973,1974], KWOKand YABLONOVITCH [1977], MCLELLAND and SMITH [1978]), wideband frequency self-modulation by bulk propagation (ALFANO and SHAPIRO [ 197 13, COLLES [197 11, WERNCKE, LAU,PFEIFER, LENZ, and THAC[1972], BUSCH,JONES and RENTZEPIS [1973]) or WEIGMANN guided wave propagation (IPPEN, SHANK and GUSTAFSON [19741, JAIN,LIN, STOLEN and ASHKIN [1977], STOLEN and LIN [1978], BOTINEAU, GIRES, SA~SSY, VANNESTE and AZEMA [1978]). The only optical nonlinear effects currently used for picosecond deterministic amplitude, or phase modulation, are the saturable absorption of organic dyes, self induced variation of the refractive index of dielectrics and, to some extent, self-focusing of single mode pulses. They will now be discussed.

3.2.1. Pulse shortening by self-amplitude modulation in saturable absorbers “Saturable absorbers” increase their transmissivity under strong illumination. This can give rise to the shortening of single mode pulses which are powerful enough for their central part to be less absorbed than their weaker leading and trailing edges. Accurate prevision of the pulse shaping and damping would require processing nonlinear wave equations, the exact solutions of which are neither known analytically nor computable numerically (needing prohibitory computation time). For want of a simple theoretical guideline, a tedious and difficult series of systematic experiments are necessary. PENZKOFER, VON DER LINDE, LAUBEREAU and KAISER [1972] reduced the duration of a gigawatt, Sps, Nd/glass laser pulse to 2 . 6 ~ safter a single pass through a saturable dye exhibiting 9 p s decay time. High energy and intensity losses are reported (typical energy loss

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factor 300 to 400). Sequences of amplifications and absorptions yield further pulse shortening down to one picosecond. These authors provide numerical interpretation of their results after approximations neglecting pulse diffraction and dispersion, assuming two level saturable absorption. NEW[1972, 19741, GARSIDEand LIM[1973], HAUS[1975, 19761, MULLER and NEEF[1977], and MULLER[1979] specially analyzed temporal shapes resulting from a succession of amplifications and saturable absorptions. The main application of these self-modulation effects lies in the widely studied passive mode-locking of lasers, which will not be considered here.

3.2.2. Frequency modulation by self-induced refractive index variation of transparent materials Of great practical importance is the optical nonlinearity which causes the refractive index of materials to increase (occasionally to decrease) to an amount An under illumination of power P, according to the approximated relation An = aP,a being a constant real number. This nonlinearity gives rise to perturbations of spatial and temporal frequency spectra referred to as “self-focusing’’ and “self-phase modulation” phenomena. In a qualitative approach, self-focusing can be considered as selfinduced convergence of wave surfaces in nonlinear dielectrics exhibiting positive a (Fig. 3.15). The available theories do not account for diffraction, which severely limits their range of validity, and only deal with very simple beam and pulse shapes (for instance, Gaussian beams and pulses). Detailed analysis and reviews are given by SHENand MAR~URGER [1975] and SVELTO [1974]. THORNE and LOREE[1972] directly combined simple spatial filtering with geometrical distortion of a Gaussian laser beam weakly self-focused

An (r)= a P(r)

Fig. 3.15. Self-focusing of a Gaussian monochromatic beam through an intensity dependent refractive index (the behaviour at F and after F is not known).

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Fig. 3.16. Power selection by preferential transmission of pulses exactly self-focusing at M (THORNE and LOREE[1972]).

in a CS2 cell (Fig. 3.16), performing relative enhancement of parts of a mode locked train of picosecond pulses. Strong self-focusing of picosecond pulses is very poorly described by this oversimplified model of a converging wave. Pulse distortions by diffraction and dispersion are closely coupled to each other by the nonlinear change of refractive index, making a realistic study of the spatial and temporal phenomena involved very difficult. Nevertheless there exist a range of optical powers where the following elementary argument can work; the optical path l(t), traveled over by a part of a pulse of intensity P, is approximately equal to I ( t ) = A B * n ( t ) = A B ( n o + a P ( t ) ) ,where the length AB is defined in Fig. 3.15, and n(t) is the power dependent refractive index at time t under illumination P(t). The phase q ( t ) of the optical vibration collected at point B exhibits time dependence q ( t ) = - 2 w 0 ( t - l ( t ) / c ) ,u0 denoting the average frequency of the vibration at point A. The first derivative (-drp(t)/dt) is the instantaneous time dependent angular frequency w ( t ) = 2.rr u ( t ) of the radiation at B; this radiation suffered self-frequency modulation, of amplitude Au( t ) = u( t ) - u,,. Thus, a spatially and temporally bell shaped pulse will have its frequency swept around y o , quasi-linearly versus time, in the region of its peak power as explained graphically in Fig. 3.17. Large nonlinear frequency modulation occurs in the picosecond emission of Nd/glass lasers (GLENNand BRIENZA [1967], TKEACY [1969a, 19711, DUGAY, HANSEN and SHAPIRO [1970], SHELTON and SHEN[1971], [1973], ECKARDT, AUSTON [1971], VON DER LINDE[1972], RICHARDSON LEE and BRADFORD [1974], etc.); it is weaker in the Nd/Yag emission (SIEBERT and MONTRY[1978]). It was also produced outside the laser cavity by bulk propagation in CS2 with great experimental care taken to

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Fig. 3.17. Self-modulation of the angular frequency o(t)of a Gaussian pulse by the shifting of its phase ~ ( t proportionally ) to the light intensity P(t).

prevent strong self-focusing (LAUBEREAU [1969], LEHMBERG and MCMAHON [1976]). Guided wave propagation on a few hundred meters of single mode silica fiber enabled STOLEN and LIN [1978] to produce frequency modulated, mode locked argon laser emission at a low power level (a few tens of kilowatts). In spite of the weakness of the nonlinear susceptibility of silica, large phase shifts were reported without observing the usual beam distortions by self-focusing. IPPEN,SHANKand GUSTAFSON [1974] had previously tried to increase the interaction length by guided wave propagation in highly nonlinear liquid core fibers, but the multimode core allowed mode conversion by self-focusing to arise, generating uncontrolled space-time structures. The power spectrum of bell-shaped pulses having suffered self-phase modulation is very characteristic (Fig. 3.18). It exhibits sinelike oscillations of energy, which are usually interpreted with the help of Fig. 3.17. Each of the “instantaneous frequencies” w inside the interval (wo, uof Aw/2) is produced successively at two different times f l , t2. The resulting amplitude at this frequency reaches a minimum if tl - t2 = ( 2 m + 1)/2v (destructive interference) and a maximum if tl - tz = mlv (constructive

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and Fig. 3.18. Typical spectrogram of a self-phase modulated Gaussian pulse (VAMPOUILLE MARTY[1981]).

interference), rn being an integer number. At each of the outermost frequencies of the spectrum (0”+ A0/2, coo - A0/2), there are fringes of maximum brightness corresponding to zero distance tl - t2. Many works reviewed in SVELTO [1974] and SHENand MARBURGER [1975] have related these spectral structures to pulse durations, rate and amplitude of the frequency modulation, by using computer Fourier transformation. Here we give a pictorial interpretation of the spectral features of a self-phase modulated pulse, again by representing spectral analysis as a Fourier diffraction of “temporal waves” in the back focal plane of a lens (L) (Fig. 3.19). The temporal amplitude distribution f ( t ) has modulus If(t)l= (P(t)>i’2 and phase q ( t )= po+2 z v O f- @P(t>, where cpo = 2rrvoAB/c, /3 =2lru0cwAB/c. The curve representing the function q ( t ) in the temporal coordinates (2, s} is a temporal wave surface (2J. In the case of a

*

Fig. 3.19. Diffraction representation of the Fourier correspondence between distributions of temporal amplitude f ( t ) and spectral intensity lF(v)I2 of a self-phase modulated Gaussian pulse. Both “wavelength” ko and “diffraction angle” oo depend on the dispersion performing the spectral analysis.

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bell shaped pulse, this surface also exhibits a bell shape similar to a Schmidt correcting plate, that is similar to a spherical aberration. The wave surface refracted through the lens (L) includes three sheets (&, Z I and X:), the interferences of which are detected in the frequency plane (v). These interferences are nothing but “caustic fringes” observed in the plane of least confusion (BORNand WOLF[1965d]) of a cylindrical converging lens. The three sheets &, Zb and 2; are almost circular, and the variations of the frequency spacing between two consecutive maxima (resp. minima) of the power spectrum obey a p*” law (p being the interference order), similar to the fringe spacing variations on interference between two cylindrical waves of different curvatures. From geometrical calculations or ray tracing through the lens (L), a number of useful results may be inferred: -the number m of the maxima (fringes) in the whole spectrum is equal to qNL/r,qNLdenoting the maximum nonlinear phase shift ( = PI‘,,,,,) reached at the peak power P,,, of the pulse, -there is the same number m/2 of fringes on each side of the initial laser frequency v,, -the spectral width Av of the modulated pulse is 4rn times larger than 6 v = 1/T, the spectral width of the single mode unmodulated pulse, which is also the halfwidth of the narrower fringe centered around the initial frequency v,. This coherent optics approach and these results will be found useful for understanding the mixed operation of frequency modulation and filtering (of $3.3). A little above the power range just discussed, strong self-focusing and self-steepening phenomena open ways to very effective spectral broadening of picosecond pulses-of the order of a hundred. These “catastrophic” phenomena are generally considered irreproducible manifestations of optical instabilities, as they are extremely sensitive to spatial and

Fig. 3.20. Spectroscopic analysis of a sinusoidal fringe pattern self-focused in CS,. Fringe spacing = p along the entrance slit of the spectroscope.

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Fig. 3.21. Spectroscopic analysis of an exponentially decaying pulse sequence self-focused in CS,.

temporal inhomogeneities of the beams. However, they may exhibit unexpected regularities and reproducibility under suitable experimental conditions. Fig. 3.20 (VAMPOUILLE, COLOMBEAU and FROEHLY [19821) shows the reproducibility of the temporal frequency spectra generated by the self-focusing of tight (100 mm-') interference fringes, each of them carrying a 100 kW, 30 ps, frequency doubled Nd/YAg laser pulse in a 5 mm long CS2 cell. The slight differences among the spectra of different fringes seem due to inhomogeneities of the fringe pattern itself rather than to some instability of the self-focusing process. Figure 3.21 (same reference as Fig. 3.20) demonstrates the deterministic relationship linking the peak power of the self-focused beam to the resulting magnitude of the spectral broadening. and LALLEMAND [19661 From the first observations of BLOEMBERCEN and BREWER [1967] extensive work was carried out about these wideband frequency modulations but, as noted above, their experimental and theoretical mastership as yet remains quite unsatisfactory. An application to picosecond shortening of 30-100 ps initially long pulses will be briefly reported in § 3.3.3.

3 . 3 . SHAPING BY A COMBINATION OF TEMPORAL AND SPECTRAL MODULATIONS

3.3.1. Self-phase modulation of pulses after temporal shaping of their intensity (VAMPOUILLE and MARTY[1981]) The device in Fig. 3.22 combines intensity shaping on the masked grating {(G),(M)} with phase modulation through a single mode silica

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Fig. 3.22. Temporally controlled frequency modulation by self-phase modulation of pulses of various intensity profiles through a single mode optical fiber (F). (Io): initially single mode pulse; ( I , ) : intensity shaped pulse after filtering on grating (G) masked by (M);(I2), self-phase modulated pulse (phase shift proportional to P , ( t ) ) .

fiber (F). In the example selected here for illustration, a short pulse (I,) gets the triangular shape of its temporal intensity distribution after spectral filtering by the grating (G) and the fiber (F). This filtered subpulse (I1) then undergoes self-phase modulation proportional to the instantaneous light intensity P( t ) by propagation in the fiber. The temporal evolution of the “instantaneous frequency” is ruled by the geometrical shape of the mask, the peak power of the initial pulse and the fiber length; the frequency modulated pulse (I2)approximately differs from (I1) by translation of its main frequency from vo to v , = v o + A v (Fig. 3.23). The amplitude Av of the frequency translation is nearly equal to m . (26v), where 6 v - l/Tl, T, =duration of pulses (I,) and (I2), rn = q m a x /being r defined above (§ 3.2.2). The power spectrum IF2(v)I2still exhibits “caustic fringes”, this one centered around v1 being the bright “focus” of the tilted plane part AB of the quasi-linearly phase modulated ‘Lwavesurface” X ( t ) . An example of working conditions: (I,) pulse: wavelength 0.53 pm, duration 30 ps, bandwidth 30 GHz, millijoule energy, peak power of a few tens of megawatts, (G) grating: holographic, 1872 mm-’ space frequency, working its second order at 2 x lo5 resolving power,

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Fig. 3.23. Diffraction representation of the spectral structures IF,,(V)~~, IF,(u)12 and IF2(v)I2 of (I”), (I,) and (I2), respectively in case of linear ramp intensity shaping. (L) is a fictitious lens performing the Fourier analysis of the temporal wave surfaces (Xt).

(Il) pulse: quasi-linear intensity profile P(t) over TI = 300 ps; bandwidth of the central frequency peak: 3 GHz, whole bandwidth: 30 GHz; peak power: a few tens of kilowatts, (F) fiber: single mode, silica core 3 p,m size, 10 m length, overall losses including insertion losses: about 10 dB, (I2) pulse: same duration as (I1), peak power of a few kilowatts, submillijoule energy; bandwidth of its frequency peak: 3 GHz; whole bandwidth: 30 GHz; amplitude of the frequency shift A v = 12 GHz. The fiber length is not allowed to increase at will, with the purpose of proportionally increasing the frequency shift Au. The optical path should be short enough in order that the fiber core dispersion does not disturb the intensity shape of the pulse (I1) over its whole bandwidth. Here this limit is equal to about 15 m; this is the “temporal Fresnel length” (§ 2.3.1) of the fiber at the considered frequencies. Various other temporal laws of frequency modulation may be synthesized by changing the geometrical shape of the mask (M).

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3.3.2. Combination of linear dispersion and self-phase modulation of Gaussian pulses: compression and other pulse distortions When propagating in dispersive materials or structures after having suffered self-phase modulation, Gaussian pulses can exhibit two main classes of distortion, according to the relative signs of the dispersion and of the nonlinear phase shift. First situation: The dispersion is normal and the coefficient p of the nonlinear phase shift Q ~ ~ =P (Pt ) is positive. It is the general case for dielectric materials far from their absorption frequencies, under short pulse excitation. The observed effects are the same if both signs of @ and of dispersion change at the same time (anomalous dispersion and negative nonlinear phase shift). Figure 3.24 schematically explains the frequency and intensity distortions of a pulse after having traveled over longer and longer paths sl,s2 and s3 in the dispersive material. The central region of the pulse is represented by a diverging wave surface. On the contrary, the weaker wings exhibit positive concavity; they converge around time s2,

Fig. 3.24. Shape evolution of a self-phase modulated Gaussian pulse as it suffers normal dispersion on its propagation.

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then causing enhancement of the pulse edges with respect to the central part; afterwards they diverge and interfere with the central wave, giving rise to temporal “caustic fringes”, the -umber of which is rn = cpK;l“/.rr. (PNL may be large in the pulses emitted from Nd/glass laser oscillators (DUGUAY, HANSEN and SHAPIRO [1970], VON DER LINDE[1972], RICHARDSON [1973], ECKARDT, LEE and BRADFORD [1974]) or even in glass laser amplifiers (BLISS,SPECKand SIMONS [1974], SUYDAM [1974]); the pulse is said to be “breaking up”. This effect is weaker (rn = 2) in YAg/Nd laser oscillators, where SIEBERT and MONTRY[19781 only reported “pulse splitting”. The edge enhancement occurring between s = 0 and s = s2 was particularly pointed out by LEHMBERG, REINTJESand ECKARDT [ 19771, who called it the “squaring effect”. These authors induced self-phase modulation of Gaussian pulses (wavelength 1.06 pm) by propagation in cesium vapor, exhibiting a negative coefficient 6. Pulse squaring effects were observed after anomalous dispersion by Treacy’s pair of gratings. Second situation: Either the coefficient p is negative and the dispersion normal, or p is positive and the dispersion anomalous. Then the central part of the pulse converges (Fig. 3.25) and its wings diverge on dispersive propagation. Around the group delay time s = sl, the energy content of the converging part will concentrate over a time interval At of the order of 2T/m (T, initial pulse duration), thus multiplying its peak power by q(rn/2)’, q denoting the energy transmission of the whole device. TREACY [1968b; 1969a, b] used the large self-phase modulation of Ndiglass laser picosecond pulses for compressing them by about a factor

Fig. 3.25. Compression of the central part of a self-phase modulated Gaussian pulse at time s, of its dispersive propagation.

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10 by anomalous dispersion through the two gratings delay line (see also § 3.1.3). In a similar way BRADLEY, NEWand CAUGHEY [1970] studied the amplitude of the linear pulse chirp in its central part. In fact, the curvature of the temporal wave surface is not constant; the nonlinearity of the relationship between time and amplitude of the frequency sweep (TREACY[1971], IPPENand SHANK [1975]) reduces the compression efficiency. DREXHAGE and EISENTHAL [19741 reported compression of Nd/glass laser pulses up to 0.5 ps by the insertion of a parallel glass plate inside the laser cavity. The plate works similarly to the Gires and Tournois interferometer (cf. § 3.1.3). The high dispersion of this interferometer was initially applied, by DUGUAY and HANSEN [1969], DESBOIS,GIRESand TOURNOIS [19711, to compressions (factor two) of 0.5 ns electronically chirped He-Ne laser pulses (cf. § 3.2). LAUBEREAU [1969], FISHER, KELLEY and GUSTAFSON [1969], and LEHMBERGand MCMAHON[1976] achieved the compression of initially chirp free pulses by positive selfphase modulation in nonlinear optical materials, followed by anomalous dispersion through Treacy’s grating delay line. Combining frequency modulation and near resonant dispersion of Na vapor, GRISCHKOWSKY [ 1974a, b] performed optical compressions of nanosecond pulses. BJORKHOLM, TURNER and PEARSON [ 19751 even converted continuous 5890A dye laser emission into a train of periodic subpulses with 5 ns spacing, 240ps duration, with peak power six times higher than the initial power of the cw emission and WIGMOREand GRISCHKOWSKY [1978] brought the compression rate to 112 and the power gain to 14, which may be the largest compression effects reported so far.

3.3.3. A few other examples of combined modulation and filtering “Bistable” optical devices, consisting of resonant cavities filled with nonlinear optical material are intensively studied examples of combinations of interferometers and optical modulators. They now cover a wide domain which will not be discussed here. MASSEYand SHANMUGANATHAN [1978] studied the “power limiting” behaviour of various two-beam interferometers, scalar or vectorial, as one of the beams crosses material exhibiting a power dependent refractive index. The power induced phase shift between the two beams causes a

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sine law variation of the interferometer transmission which can compensate exactly for the power fluctuations inducing the phase shift. VAMPOUILLE [1980] selects picosecond temporal substructures in 30 ps-100 ps, millijoule, harmonic pulses of Nd/YAg laser emission, and by combining nearly controlled self-focusing in CS2 (cf. § 3.2.2) and space-time filtering, single mode picosecond pulses are extracted from the initial pulse, unfortunately with a low yield of energy (of the order of

4 4. Optical Analysis of Picosecond Light Pulses 4.1.INTRODUCTION

The conclusion from 0 1 was that optical pulses carrying the energy W distributed over K =K, Kt spatial (K,) and temporal (K,) samples may give rise to deterministic analysis, provided that: -the energy per sample W/K exceeds the threshold of sensitivity of each resolution cell (degree of freedom) of the detector, -the number K does not exceed the detector capacity C. For instance let us single out one of the subpicosecond pulses emitted from a synchronously pumped cw dye laser (argon pumped R6G laser). It is carried by a single mode beam (TEW, Gaussian beam) of average fequency u0= 3 x l O I 4 Hz and its nanojoule energy W is distributed over a time interval At ~ 0 . ps. 5 The maximum number K of its samples is equal to K , = v, At = 150, this also being the number of oscillations of the optical field over At. The average energy of one single oscillation is A W = 7 x lo-’’ J; it exceeds the sensitivity threshold of one resolution cell of usual photographic emulsions (lo-’ J cm-’, 4 x lo4 points resolved each square centimeter). There is no fundamental obstacle to individual detection of each oscillation of the light vibration in this pulse. No experiment has achieved this yet, as no observation device has the required time resolution 8r -- l/uo= 3 X lo-’’ s. Working towards this limit, “picosecond optics” develops a wide variety of experimental methods providing information about the duration, the average intensity profile, the “instantaneous frequency” and even, in some cases, the complex amplitude distribution of optical pulses with pic0 or sub-picosecond resolution. These methods are based mainly on quantum detection of radiation combined with temporal or spectral processing.

-

-

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Excellent reviews cover this domain thoroughly (DE MARIA[ 19711, GREENHOW and SCHMIDT[1974], BRADLEY and NEW [1974], IPPENand SHANK [1977], BASOV[1977], BRADLEY[1977] and BRADLEY[1979] especially dealing with the progress of “streak cameras”). 15 4.2 of this text discusses recent progress in pulse duration measurements and Q 4.3 deals with deterministic analysis of the complex amplitude of short pulses from the point of view of coherent optics. 4.2. MEASUREMENTS OF PULSE INTENSITY PROFILES

There are many applications of picosecond light pulses (picosecond chemistry or photochemistry, laser fusion, optical communications, etc.) needing some knowledge of spectral and temporal distributions of the light energy. Simple measurements of spectral widths and temporal durations are often adequate for satisfactory characterization of interactions produced. Observation of spectral energy distributions involves the classical tools of instrumental spectroscopy; in the presence of single pulses or nonreproducible phenomena, only spectrographic analysis is possible; in the presence of periodic emission (cw modelocked lasers) spectrometric recording may also be performed with well known luminosity and resolution superiorities. Measurements of very short duration do not require sophisticated expensive apparatus. A number of methods is now available, based upon combinations of nonlinear optical interactions and two beam or multiple beam interferometry. These methods, called “intensity correlation”, only concern pulses of rather high intensity, at least a few tens or hundreds of kilowatts, depending on the magnitude of the optical nonlinear susceptibility involved in the measurement. The nonlinear phenomena giving rise to intensity correlations easily related to the pulse durations are mainly: -two photon processes, such as second harmonic generation, two photon fluorescence, and the two photon photoelectric effect, -three photon processes, such as saturable absorption, optical Kerr effect - that is self-induced birefringence of isotropic materials - and the three photon photoelectric effect. The highest resolution reported so far, about 10-14s, was reached by using three photon photoelectric detection (BURNHAM [1970]), but the pulse intensity had to be high, because the nonlinear interaction arises in the detection plane only.

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Fig. 4.1. Intensity correlation of a pulse (I) consisting of two rectangular subpulses (I1) and (I2) I,: temporal intensity distribution at fundamental frequency o ;IZw: spatial harmonic and HORVATH distribution at frequency 2w along the z axis (GYUZALIAN, SWOMONIAN [19791). a

Bulk interactions of waves in nonlinear optical materials require lower powers but lead to devices with lower temporal resolution because of various phase or group velocity mismatches, responsible for shape distortions of pulses to be observed. Second harmonic generation takes place among the most sensitive and resolving measurement techniques; its main drawback is its inability to pick up the whole intensity correlation curve of single pulses, because of working by sampling. This limitation was overcome in a simple way, too recently published (JANSKY, GORRADI and GYUZALIAN [ 19771, GYUZALIAN, SOGOMONIAN and HORVATH[ 19791) to appear in the previously mentioned reviews. Figure 4.1 illustrates the basic idea: the pulse under test (I), linearly polarized, propagates in a parallel beam of large size a ; after being split by the separator B.S. it will be recombined in an uniaxial LiI03 crystal of thickness e, with an angle p between the two interfering beams. The angle p and the direction of the optical axis z of the crystal are carefully adjusted, taking into account the crystal birefringence and dispersion, in order to satisfy the usual phase matching condition kezo = (k:), +(kz), between the wave vector k ; o of the harmonic extraordinary beam and the two wave vectors (k:), and (k:)* of the fundamental ordinary beams (1) and (2). The temporal profile of the intensity correlation function of the pulse (I) will be displayed along the geometrical coordinate z as a spatial distribution of illumination at the

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harmonic frequency 20. With crystal thickness e = 10 mm, beam size a = 15 mm and angle p = 19”40’,Nd/glass pulses shorter than 40 ps were measured with 0.1 ps resolution. Replacing the photographic detector by a multichannel analyzer, KOLMEDER, ZINTHand KAISER[ 19791 observed picosecond, ten nanojoules pulses with a temporal resolution of 0.1 ps. Such a sensitivity is orders of magnitude higher than the best actual performance of two photon fluorescence (TABOADA and VENABLE [ 19781). Among the recent work dealing with intensity correlations, let us also quote orthogonal cross correlation of pulses in saturable absorbers, which exhibit a much longer recovery time than the duration of the bleaching pulse (WIEDMANN and PENZKOFER [1978, 19791). Also let us note the multi-photon detection of FARKAS, HORVATHand KERTESZ[1972] and FARKAS and HORVATH [1974] when a comparison of electrical outputs from linear and nonlinear photocells leads to an estimation of pulse durations. Direct observation of picosecond temporal intensity distributions is only possible by using the sophisticated and expensive “streak cameras”, specially presented in the reviews of BRADLEYand NEW [1974] and BRADLEY [1977, 19791. These cameras give two-dimensional images of the space-time temporal intensity distribution i(x, t ) of a pulse versus time t and geometrical coordinate x, along the entrance slit of the camera. They are especially well suited to accurate imaging of intricate temporal structures, their high sensitivity being limited by the quantum yield of the photocathode. But their dynamic range is rather limited - of the order of two decades - and their temporal resolution has not yet really gone beyond the picosecond. 4.3. COHERENT OPTICAL ANALYSIS OF THE TEMPORAL STRUCTURE OF PICOSECOND PULSES

The optical characterization of a radiation is not complete as long as it does not specify both modulus and phase of each sample of its complex amplitude distribution (see D 1). None of the procedures mentioned above would alone lead to this entirely deterministic description, considering the information loss about the optical phase after energy detection, but combinations of energy detection with simultaneous temporal and spectral processing can provide information on the complex amplitude.

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4.3.1. Measurements of the number of temporal modes (samples) of a pulse This number, larger than 1, is equal ( 5 1) to the product At A v of the total duration A t by the bandwidth A u of the respectively temporal and spectral amplitude distributions of the pulse. Two measurements only, one in the time domain, the other in the frequency domain, are sufficient for a rough knowledge of the state of frequency modulation of a pulse. In the case where a regular frequency modulation causes a phase shifting of the order of 2 ~ radians m between the extreme frequencies of the spectrum, the product At A v is of the order of m. Although this information is of poor accuracy it is widely used as qualitative test of the phase relationship between the spectral components of a modelocked laser emission.

4.3.2. Measurements of phase (frequency) modulation by pulse compression TREACY[ 1968a, b; 1969a, b] demonstrated and measured linear chirping of Ndlglass laser pulses by observing their compression after anomalous dispersion at the output of grating pairs (see also § 3.3). In the temporal space {t, s}, introduced in § 2.3.2, this operation is a determination of centers of curvature of elementary cylindrical wavelets; since many frequency modulations co-exist (Fig. 4.2), many centers of curvature Ci of wavelets (Xi)are to be successively determined. This is possible using a

Fig. 4.2. Dispersion of a nonlinearly chirped pulse represented in the {t,s} space as diffraction of cylindrical wavelets (Xi) of various curvatures.

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streak camera, the slit of which is parallel to the axis z, in the arrangement previously described in Fig. 2.9 of 9 2.3.4, the centers of curvature Ci appearing as bright foci on the camera screen. This performs the decomposition of a pulse in its linear frequency modulations (COLOMBEAU [1980]), replacing the more usual frequency decomposition. This is a geometrical approach to the shape of “temporal wave surfaces”. 4.3.3. Measurements of ‘instantaneous frequencies” Again TREACY [1971] showed the nonlinearity of the frequency modulation of Nd/glass lasers in a famous experiment, comparing the arrival times of the successive frequencies by two photon fluorescence in a dye cell located in the back focal plane of a spectrograph. This is also a geometrical optics approach because it is based on the notion of “instantaneous frequencies” which behave as geometrical rays in the representation space {t, s}. The dispersion of light may be adequately described by ray propagation in this space only when the phase shifts, due to the frequency modulation, largely exceed 2 7 ~over short enough time intervals for intensity modulation to be neglected. Observations of “instantaneous frequencies” played an important part during the decade 1960-1970, at the time of studies about Q switched laser emission. Improvements in the picosecond camera and intensity correlation measurements allow a return to such methods with far greater sensitivity than in Treacy’s first experiments; picosecond scale “time resolved spectroscopy” is discussed in IPPEN and SHANK[ 19771. It could provide powerful tools for coherent analysis of temporal distributions of complex amplitude in optical pulses, on condition that temporal, spectral and spatial resolutions are high enough with respect to the number of modes of the pulse. These conditions will be more explicit in the following example: let a light beam of size a = 1 mm, divergence Q = 1 K 2rd, be carrying a temporal energy distribution spreading over duration At = 50 ps, bandwidth Au = 0.4 THz around the average frequency u, = 6 X Hz (wavelength A, = 0.5 pm). To each one of the K, = (acu/A,)’ = 400 spatial samples of the beam corresponds one particular pulse defined by K, = A t * Au = 20 temporal samples. One single such pulse should be spatially selected by transmission through an aperture of size ==A,/a= 50 pm (Fig. 4.3) and then spectrally analyzed by the spectroscope (Sp) with a resolving power lower than 3,= l/&, = 1500, I = c/Au = 0.75 mm denoting the correlation length of the radiation. The streak camera (S.C),the slit (S,) of

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Fig. 4.3. First approach of the temporal amplitude distribution by observation of its instantaneous frequency u ( t ) . (PI: pinhole; (Sp): spectroscope with horizontal slit (S, I ; (S.C.):streak camera with vertical slit (SJ.

which lies parallel to the frequency axis u of the spectroscope, displays the temporal evolution of the spectral distribution with a resolution higher than 6t = l/c = 2.5 ps. The energy distribution on the camera screen is an intensity modulated curve whose brightness increases with the instantaneous pulse power. The geometrical shape of the curve images the temporal variations of the frequency u( t) = uo + d4(t)/dt, u( t) characterizing the nonlinear phase modulation of the complex amplitude.

4.3.4. Temporal phase measurements by optical beating The interference of two vibrations of different frequencies u1 and u2 is a beating at frequency ( u 2 - ull = 6 v ; it may be observed by detectors of temporal resolution higher than 1/6u. GEX,SAUTERET, VALLAT, TOUIZBEZ and SCHELEV [ 19771 measured frequency modulations of Nd/glass mode locked laser pulses by streak camera recordings of the beatings between a pulse and a delayed image of itself. This is an experiment of differential interferometry analogous to the temporal Young’s experiment previously discussed (§ 2.3.4), which provides information about the rate du(t)/dt of the frequency modulation. It also may be considered as “differential interference constrast” (FRANCON [19521) in the time domain. AGOSTINELLJ, HARVEY, STONEand GABEI.[ 19791 extended the method to direct observation of phase variations by streak camera recordings of the beatings of the frequency modulated pulse with a well defined

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reference frequency. This reference is extracted from the pulse itself by filtering through a Fabry-PCrot interferometer. The close analogy of this method with Fourier holography was pointed out in § 3.1.3. Here a true representation of the complex amplitude distribution in the pulse with a camera limited resolution is obtained. This way could lead to the most straightforward approach to the complex field of a pulse. In fact, the actual limitations of the dynamic range and resolution of streak cameras unfortunately prevent using this full and elegant solution in the case of pulses shorter than a few picoseconds, or in the case of pulse structures exhibiting a At * A u product larger than a few tens of samples. In the subpicosecond domain or in presence of highly multimodal signals, coherent vibration analysis still remains possible by amplitude correlation or spectral analysis, as shown below.

4.3.5. Coherent pulse imaging by amplitude correlations or spectral analysis

Again suggested by holographic principles, the idea consists of following the unknown “object” temporal structure f(t) = If(t)l exp Cj(2nu0t + + ( t ) ) } with a delayed “reference” narrow pulse A 8(t - to), then performing the temporal autocorrelation function of these two signals. At the limit where the reference signal 6 ( t - to) would be a Dirac distribution, the autocorrelation function Y ( T )

I-,

+m

Y(T)=

g(t) *

dt = g ( t ) * g(t),

g*(t+T)

where g ( t ) = f ( t ) + A a(t-to),

would be expressed as:

+

Y ( T ) = A’S(T) (f(t)

* f(t)) + A

*

f*(to+

+A

7)

*

f(t0-

7).

If the delay to is larger than twice the duration At, the function Y ( T ) results from the juxtaposition of three distinct domains (Fig. 4.4). The central one, around T = 0, is the sum A* S ( T ) + f ( t ) * f (t ) and provides no direct information about the “object” structure; on the contrary the two others, centered on 7=*to, are symmetrical lateral images of the object distribution, phase conjugated from each other.

142

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, 54

Fig. 4.4. Real part Re(y(7)) of the autocorrelation of a At long pulse (I) followed by a much narrower pulse (R) with respect to (1,)); to = time delay.

If the “reference” distribution is not a Dirac distribution, but spreads over a duration (dt), its convolution with the lateral images causes some blurring, limiting to (dt) the temporal resolution at which the images may be considered identical to the object. The corresponding energy distribution lG(u)I2= F.T. [ y ( ~ )in ] the frequency space is a two beam interference pattern between the spectra IF(v)l exp &$(v)} of f ( t ) and A exp (-j27~t~u)of the reference pulse (Fig. 4.5): IG(v)I2= I IF(v>lexpW ( V ) I +A exp ( - J 2 7 ~ ~ ) 1 * = IF( v)\’

+ A*+ 2A IF( v)( cos ( 2 7 ~ t o ~4(v)). -

If the total energy of the reference pulse is larger than the “object”’ energy, this pattern is a Fourier hologram of the object (FROEHLY, LACOURT and VIENOT[1973], LACOURT, VIENOTand GOEDGEBUER [1976],

Fig. 4.5. Temporal power spectrum lC(v)\*of the object pulse accompanied by the delayed reference pulse (width dt, delay to).

11, 141

143

OPTICAL ANALYSIS OF PICOSECOND LIGHT PULSES

KUZNETSOVA [1977]). Its average fringe spacing is d u = l/to and the fringe contrast and phase are respectively proportional to (F(u)(and 4 ( u ) . A photographic recording of the spectrogram contains the information about the (coherent) amplitude and phase distributions in the spectrum of the pulse f(t). Amplitude autocorrelation of optical vibrations may be performed either directly, by two beam interferometry, or in two steps by spectroscopic detection and Fourier analysis of the spectrum. A two beam interference device frequently used for correlation analysis is the Michelson interferometer. After FELLGET [1958], JACQUINOT[19601 and CONNES [1961] it is well known that the fringe visibility function v(S) (versus the optical path difference S = C T ) images the real part of the complex correlation ~ ( 7 )The . autocorrelation function of a A t long pulse will be entirely scanned after 6 being continuously varied between -So = -c At and + S o = c . A t . This occurs (Fig. 4.6) in the image plane (T)of the mirrors (MI), (M,) through the lens (OJ, as “air wedge” fringes are imaged in this plane. The interferogram recording requires two conditions: - average illumination Eo of the pattern above the threshold of detectability , 9

Fig. 4.6. Amplitude autocorrelation by variable path difference two beam interferometry; E (6):intensity of the fringe pattern at path difference S in the image plane ( P ) of mirrors (MI), (M&,through the lens (OJ; (0,): collimating lens; f : focal length of the lenses.

144

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, § 4

-average spacing p of the fringes above the spatial limit of resolution of the recording material. These conditions are not met for low power pulses lasting over a few picoseconds, implying recording of thousands of fringes. A more versatile solution may exist (COLOMBEAU [1980]) by making use of the so called S.I.S.A.M. interferometer due to P. CONNES [1958, 1959, 19601, also described in BOUSQUET [1969]. The mirrors of a Michelson interferometer are replaced by two identical diffraction gratings (GI), (G,) working at opposite dispersions, the one in the order +k, the other in the order - k (Fig. 4.7). The point P of coordinate x in the image plane (a)of the gratings through the beam splitter (B.S.), is submitted to interference between two images of the pulse which spring from the source S, after reflections on the regions PI and P2 of the gratings. These images are temporally delayed by 7 = l/c, where 1 = 2x/tan 0; the autocorrelation function Y ( T ) of the pulse will be displayed in the plane ( T ) as an amplitude and phase modulated fringe pattern, the spacing of which may be adjusted independently of the delay by slight

Fig. 4.7. Application of S.I.S.A.M. interferometer to the recording of the amplitude autocorrelation of a pulse.

11,

141

OpllCAL ANALYSIS OF PICOSECOND LIGHT PULSES

145

Fig. 4.8. Single grating autocorrelation interferometer.

rotations of the gratings. The maximum delay is ~ ~ = 2 L /where c, L represents the “optical depth” of each grating. Amplitude autocorrelations of 600 ps long pulses were recorded in single laser shots by using gratings of depth L = 100 mm. Single grating devices, as shown in Fig. 4.8, give similar results, but they are of more difficult adjustment. The experimental set up in Fig. 4.9 gives images of the modulus and phase distributions in the coherent response of any pulse shaping system (F), which would work by temporal frequency filtering (§ 3.1). The curve of Fig. 4.10 is a photometric recording of one of the images (I1) or (I2), detected as the temporal response of the filter (F) exhibits approximately square law amplitude modulation, and no frequency modulation over about 300 ps.

Fig. 4.9. Correlation analysis of the temporal response of a shaping filter (F): the interferand a ence device consisting of a beam splitter (B.S.), two identical gratings (GI) and (G2), lens (0),which gives two images (I,) and (I2) of the filtered pulse (I) by usjng the initial pulse (I,) as a delayed reference pulse.

146

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

300ps .

~

~~~

III, 94

4

Fig. 4.10. Energy distribution in the fringe pattern representing a square envelope pulse of duration 300 ps.

Direct amplitude correlation has the advantage of rigorous linearity in the correspondence between the fringe visibility to be detected and the pulse amplitude under test. Moreover, they work in “quasi-real time”, as the time interval needed for interferogram formation is of the order of 2L/c, that is 1 ns if L = 15 cm. The main disadvantage lies in the fact that the interference visibility in the images (I1) and (I2) is nearly proportional to the inverse number of the image samples, and it frequently drops down to levels vanishing in the detection noise. The indirect two step Fourier analysis of spectrograms may yield better results. The power frequency spectrum IG(v)l’ of the sequence “object” and “reference” pulses will be first recorded - photographically for instance - then submitted to Fourier analysis - numerically or analogically - which reconstructs the autocorrelation function ~ ( 7 )As . recalled above, these are true steps of holographic recording and image restitution. One example of a hologram IG(v)I2 of a linearly chirped pulse is given in Fig. 4.11. A Desbois, Gires and Tournois dispersive delay line (§ 3.1) modulates the phase of the spectrum of a pulse (Io) according to the law @(v) = a ( v - v o ) + p ( v - vrJ2. The dispersive path {(Gl),(G2L (M1L (G2),(Gl)}average length is equal to 2(f, +Z2). By mirror reflection in the zero order of (GI), more than one half of the energy of (I,) is sent in a non-dispersive delay line ( G , ) , (Mo), (GI) of length 1,; along this path the frequency spectrum of (I,) undergoes a linear phase modulation a,(.) = ao(v- v,). After the two beams’ recombination on (GJ, one single parallel beam carries two successive pulses; (I,), only delayed, and (I), also frequency modulated. (I,) and (I) are of the same bandwidths exactly. A Fabry-Pkrot interferometer and a photographic camera record the spectrogram of the couple (Io), (I), care being taken to satisfy the resolution and free spectral range requirements needed for deterministic processing of each sample of IG(v)12(see § 1.3). These

OPTICAL ANALYSIS OF PICOSECOND LIGHT PULSES

147

Fig. 4.1 1 . Interferometric detection of spectral phase modulation at output of the Desbois, Gires and Tournois dispersive delay line: the Fabry-Wrot (F.P.) plate and the lens (0) display the coherent superposition of spectra of (I,,) and (I) pulses in plane ( n ) ;(H): photograph of the resulting spectrum IG(v)1’.

conditions are the following: -free spectral range at least equal to Av, -resolution S v at least equal to 1/T, where T is the whole duration of the pulse couple. In Fig. 4.12 the structure of the spectral interferogram is interpreted in terms of the interference between two spectra, whose phases G0(v) and @(v> have different curvatures. If the optical paths lo, I, and I, are adjusted in such a way that 1, = I, + I2 at a frequency vo existing in the pulse bandwidth Av, the interferogram looks like the fringe patterns of Figs. 4.11 and 4.12. On each side of a large central fringe, the interferences become more and more tightened, as for Newton’s fringes of a cylindrical lens. If 1, # 1, + l2 at any frequency inside the bandwidth Av, the spatial frequency of the fringe pattern linearly varies around an average frequency proportional to the path difference 2((1,+ 12) - lo).

148

S W I N G AND ANALYSIS OF PICOSECOND LIGHT PULSES

[II, 8 4

Fig. 4.12. Energy distribution in the spectral interference IG(v)1’ of two spectra of different phase distributions @&) and @(v).

Submitted to Fourier analysis by optical diffraction this interferogram behaves as a hologram, reconstructing two complex, conjugated, coherent images (I1)and (I2)of the response (I) of the dispersive line to the pulse (10). The holographic images of the “object” pulse (I) appear bright against the dark background of the optical noise. Their contrast is the highest possible. Unfortunately the intermediate photographic detection of 1 G(v)l’ introduces nonlinear distortions and noise in the holographic reconstruction. Figure 4.13 illustrates an application of this procedure to observation of the complex amplitude distribution in the temporal response of single mode and multimode short (less than 1m) optical fibers, [1980b], PIASECKI, COLOMBEAU, with a resolution of 0.03 ps (PIASECKI VAMPOUILLE, FROEHLY and ARNAUD [1980]). This high resolution and the recovery of the temporal phase evolution enabled measurement of single mode dispersion and group time differences between two distinct modes, which could apparently not be achieved by other methods at the present time (F’IASECKI and FROEHLY [1980], PIASECKI [1980a]). Obtaining images (I1) and (I2) of the temporal response from the spectrographic record assumes the v frequency scale to be quasi-linear versus the geometrical coordinate x on the detection plane. If not, the Fourier analysis of the spectrogram cannot be performed generally by diffraction. Nevertheless diffraction analysis of Fabry-PCrot spectrograms - exhibiting a nearly quadratic relationship between variables v and x - is possible by axial diffraction. This allows purely optical imaging of Nd/YAg pulse trains spreading up to about 7 ns with 30 ps temporal resolution (COLOMBEAU, VAMPOUILLE and FROEHLY [19801). These coherent optical techniques of spectroscopic or correlation

ACKNOWLEDGEMENTS

149

Fig. 4.13. Coherent imaging of the time impulse response of optical fibers (F)with temporal resolution- equal to l/Av -of the order of lo-’ ps. (a): spectrogram recording; (Sp): spectroscope; (H):photographic detector. (b): Fourier analysis of the spectrogram by diffraction; (I,), (Iz): coherent images of the temporal response of the fiber.

analysis are powerful tools capable of very high temporal resolution and sensitivity but need reference signals with spectra both wide and centered on the average frequency of “object” pulses under examination. In the examples presented above, where the object pulse will be deduced from an initial laser emission by temporal frequency filtering, the reference vibration is simply this initial emission, delayed and recombined by two beam interferometric devices. However, in general cases the reference pulse should be extracted from the object pulse itself. No systematic method presently exists, that allows pulse cutting and modulation in conditions of efficiency and reproducibility. This may be one of the ways for succeeding in an accurate experimental description of coherent subpicosecond interactions of laser pulses.

Acknowledgements The authors wish to express their gratitude to the D.R.E.T., Division Optique, for financial support of the work on optical processing of

150

SHAPING AND ANALYSIS OF PICOSECOND LIGHT PULSES

[I1

picosecond pulses performed by our Laboratory, and to Professor J. A. Arnaud for numerous enriching discussions.

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E. WOLF, PROGRESS IN OPTICS XX @ NORTH-HOLLAND 1983

I11

MULTI-PHOTON SCATTERING MOLECULAR SPECTROSCOPY BY

STAN1SLAW KIELICH Nonlinear Optics Diuision, Institute of Physics of A. Mickiewicz University, 60-780 Poznari. Poland

CONTENTS PAGE

8 1. HISTORICAL DEVELOPMENTS AND OUTLINE OF THE PRESENT REVIEW

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

82. NONLINEAR MOLECULAR ABILITIES . . . . . . . . . .

RAMAN

157

POLARIZ-

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

166

83. INCOHERENT AND NONRESONANT MULTIPHOTON SCATTERING BY FREE MOLECULES . . .

178

0 4. LINEWIDTH BROADENING

IN QUASI-ELASTIC MULTI-PHOTON SCA'ITERING BY CORRELATED MOLECULES . . . . . . . . . . . . , . . . . . . .

20 1

.

216

8 5 . COOPERATIVE THREE-PHOTON SCATI'ERING . .

$ 6 . RAMAN LINE BROADENING IN MULTI-PHOTON SCATTERING (CLASSICALTREATMENT) . . . . . . 226 § 7.

ANGULAR DISTRIBUTION AND POLARIZATION STATES OF MULTI-PHOTON SCATTERED LIGHT . ,

.. ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . APPENDIX A. IRREDUCIBLE CARTESIAN TENSORS . . . 8 8. CONCLUDING REMARKS, AND OUTLOOK . . .

233 246 249 250

APPENDIX B. ISOTROPIC AVERAGING OF CARTESIAN TENSORS . . . . . . . . . . . . . . . . . . . . . . .

25 1

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

254

REFERENCES..

0 1. Historical Developments and Outline of the Present Review Light scattering - one of the most important aspects of interaction between electromagnetic radiation and material systems - poses theoretical and experimental problems which, to this day, are unceasingly in the forefront of scientific investigation. The qualitative and quantitative evolution of numerous processes of light scattering is determined not only by the statistics and thermodynamical state of the scattering medium, but moreover - and, perhaps, chiefly - by the statistics of the incident light wave and the state of its polarization. This has become evident especially since the application of rapidly developing laser techniques in studies of light scattering.

1.1. THE DEFINITION OF SPONTANEOUS MULTI-PHOTON SCA'ITERING

In a quanta1 approach, some of the incident photons interact with the microsystems (electrons, atoms, molecules, particles, etc.) of the medium, or with collective excitations (phonons, excitons, magnons, polaritons, plasmons, etc.). As a result of this interaction the microsystem absorbs a photon of frequency ol(kl), and simultaneously emits a photon w2(k2), thus performing a transition from the initial quantum state li), by way of an intermediate state Ik)(kl, to the final quantum state (fl. The energy conservation law requires that

Ef-Ei=hwfi,

fi(wl-wz)

(1.1)

with an accuracy determined by the width of the line emitted. If, during the transition, the final state (fl and initial state li) are identical, Ef=Ei,one has, with regard to eq. (1.1) 0 2

=w,

(1.la)

meaning that, in this scattering process, the photon frequency remains unchanged although the direction of its motion generally undergoes a 157

158

[111, 8 1

MULTI-PHOTON SCATTERING

change, k, # k2. The elastic scattering process described above is referred to as Rayleigh scattering (Fig. 1.1). If the energy of the emitted photon E, differs from that of the incident photon Ei, the process is inelastic and is referred to as Raman scattering. In the quanta1 picture proposed by Smekal in 1923, if w2 = w 1 - lwfil we deal with a Stokes process, whereas if 02=01+(ofil we have an antiStokes process; i.e. by eq. (1.1): 0 2

(1.lb)

= 0 1 r Iofil.

As the microsystem is acted on by the complex analytic signal E(t) = E(w,) exp (-iwlt)

(1.2)

of the light wave vibrating at the circular frequency wlr and if the electric field is not excessively strong, the electric dipole moment for the transition (fI + li) is (in the case of linear response)

DL1)(t) = @(-w2, ol)E(w,) exp [-i(wl

+ wfi)f],

(1.3)

where u ~ ~ ) ( - 002, ,) is a tensor of the second rank determining the linear electric polarizability of the microsystem. One hundred and ten years have elapsed since Lord Rayleigh laid the foundations of the microscopic theory of light scattering and, at the same time, explained why we perceive the sky as blue on a cloudless day. Much later, the statistical-thermodynamical theory of Rayleigh scattering was formulated by Smoluchowski in 1908 and Einstein in 1910, based on the stochasticity of light scattering on thermal fluctuations of density and, in solutions, of the concentration. The next step was made by Brillouin in

111, $11

159

HISTORICAL DEVELOPMENTS

1922, who showed that time-variable statistical fluctuations of density cause a modulation of the spectrum of scattered light. In addition to the central Rayleigh-Smoluchowski line (due to isobaric fluctuations of density), two symmetrically disposed lines due to adiabatic fluctuations of density appear. This Bnllouin-Mandelshtam doublet was detected in liquids by Gross in 1930. These matters have been discussed in numerous monographs (see, for example, FABELINSKII [19681 and KIELICH[1980al). Just over half a century ago Raman and.Krishnan, and Landsberg and Mandelshtam, brought into being in 1928 what is now generally referred to as Raman spectroscopy. In these days of laser techniques, Raman scattering and the spectroscopic methods based on it have been generally accepted as a potent instrument of study, revealing to us the properties of atoms, molecules and macromolecules, as well as the microscopic structure of gases, liquids and solids. This completely autonomous discipline has been dealt with comprehensively in a number of books (see, for example, KONINGSTEIN [1972], LONG [1977], CARDONA [1975], and HAYE and LOUDON [1978]). Equations (1.1) and (1.3) determine a two-quantum process (one incident photon and one scattered photon). When a microsystem is in a radiation field with a high density of the photons, processes involving more than two photons can take place. Thus, as early as 1931, GoppertMayer considered theoretically three-photon processes involving the emission or absorption of two photons. Extending her theory BLATON [1931], and later NEUCEBAUER [1963] showed that a quanta1 system can produce elastic scattering at the doubled frequency 2w. GUTTINGER [1932] analyzed the possibility of inelastic three-photon scattering at the frequencies o3= w1 + o2F ofi(Fig. 1.2). The quantum-mechanical foundations of three- and four-photon Raman scattering have been formulated by KIELICH[ 1964bl for molecular systems of arbitrary symmetry. Accordingly, a molecular system in a strong electric field (1.2) generally exhibits a nonlinear response when its electric dipole moment of the nth order for a transition (fI t li) is

Dk'"(r)=K(w,+i) &)(-w,+i, on,. . . ,011 x[n]E(o,,)- * E(o,)exp[-i(w,+*

*

*+o,+ofi)t], (1.4)

where a r ) ( - w e + l ,w,, . . . ,w l ) is a tensor of the ( n + 1)th rank determining the nth order polarizability of the scatterer. In eq. (1.4), the symbol [n] stands for n-fold contraction of two tensors of rank n, whereas K(w,,+,) is a numerical expansion and frequency degeneracy coefficient.

160

MULTI-PHOTON SCATCERING

[III, 5 1

Fig. 1.2. Three-photon energy transitions: (a) hyper-Rayleigh, (b) Stokes hyper-Raman, and (c) anti-Stokes hyper-Raman.

Equation (1.4) shows that, if n photons with different frequencies . . . ,on are incident on the microsystem, then an (n + 1)th photon, having one of the frequencies wl,

wn+,=wn+.

*

.+w,Tlwfil,

(1.5)

is scattered in the quanta1 transition of transition frequency wfi. Equations (1.4) and (1.5) define (n + 1)-photon spontaneous Raman scattering, also referred to as non-degenerate hyper-Raman scattering of the (n - 1)th order, with n 2 2. In particular, if the photons have the same frequency w, we have for (n - 1)th order degenerate hyper-Raman scattering: %+1=

Tlwil.

(1.6)

The process, described by eq. (1.6), is also referred to as n-harmonic Raman scattering. For the case wfi = 0, eqs. (1.5) and (1.6) give the following frequencies: w,+~=wnf.*.+o~, %+l=

no,

(1.5a) (1.6a)

determining respectively spontaneous non-degenerate (n + 1)-photon Rayleigh scattering, and degenerate spontaneous Rayleigh (or (n - 1)order hyper-Rayleigh) scattering. Although various review articles on multi-photon spectroscopy have

111, 811

HISTORICAL DEVELOPMENTS

161

Fig. 1.3. Typical experimental geometry for the study of scattered light, of frequency o,, propagation vector k, and polarization vector e,. The incident laser light wave has the frequency o,propagation vector k and polarization vector e.

already appeared (see, for example, PETICOLAS [1967], VOGT[1974], FRENCH and LQNG[1976], and KIELICH [1977, 1980b]), we cannot refrain from giving a brief and in some respects actualized discussion of the matter. The laser light scattering experiment is represented in Fig. 1.3.

1.2. SPONTANEOUS HYPER-RAYLEIGH LIGHT SCATTERING STUDIES

The earliest experimental detection of Rayleigh nonlinear scattering at MAKERand SAVAGE [1965], who the frequency 2 0 is due to TERHUNE, used the light beam of a giant ruby laser and scattering liquids like CCI, and H 2 0 with non-centrosymmetric molecules, as suggested by the theory (KIELICH [ 1964a1). BERSOHN, PAO and FRISCH [1966], KIELICH [1964c, and KOZIEROWSKI [ 19741have analyzed the influence 1968a1, and KIELICH of radial and angular correlations of the scatterers in hyper-Rayleigh scattering by liquids. The preliminary observations of WEINBERG [19671 nonetheless led to a rather weak dependence on temperature of hyperRayleigh scattering in the case of CCl, and HzO, pointing to an insignificant role of angular correlations. This, in fact, could have been expected; the theory predicts that in liquids whose molecules lack a centre of inversion, incoherent three-photon scattering is the predominant effect, against the background of which a slight coherent (temperaturedependent) effect is weakly perceptible. Only liquids with molecules that have a centre of symmetry provide the

162

MULTI-PHOTON SCATI’ERING

[III, 9: 1

appropriate test of the role of correlations in hyper-Rayleigh scattering since for them incoherent three-photon Rayleigh scattering is forbidden in the electric-dipole approximation. In such liquids, only coherent three-photon scattering due to cooperative molecular effects in short-range regions (KIELICH [1968b]), or to molecular field inhomogeneities (SAMSON and PASMANTER [1974]) is possible. The demonstration of cooperative three-photon Rayleigh scattering by liquids such as cyclohexane, benzene and carbon disulphide has been provided by KIELICH,LALANNE and MARTIN[1971, 19731. The effect has also been analyzed theoretically by PASMANTER, SAMSON and BEN-REUVEN [1976]. In simple molecular liquids in the normal state, cooperative hyperRayleigh scattering is weak. Obviously, it can be expected to become stronger as the liquid approaches a critical point, when critical opalescence due to anomalous density fluctuations (according to Smoluchowski), or concentrational fluctuations (according to Einstein), sets in. With regard to nonlinear scattering, LAJZEROWICZ [19651 drew attention to this circumstance, suggesting that very considerable critical scattering can occur in liquids exhibiting phase transitions. The effect was first observed by FREUND [1967, 19681 for polycrystalline NH4CI at temperatures near the second-kind phase transition. Subsequent work by FREUND and KOPF [1970], and LUBAN,WISERand GREENFIELD [1970] has permitted the utilization of critical second-harmonic scattering as a source of information on order-disorder phase transitions. DOLINO,LAJZEROWICZ and VALLADE [1969, 19701 and DOLINO [1972] initiated studies of laser second-harmonic scattering on domain structures of ferro-electric crystals (triglycine sulphate). INOUE [19741, and WEINMANN and VOGT[1974] have carried out a detailed investigation of the second-harmonic of light, scattered in NaNO, crystals, whereas VOGTand NEUMAN [1978] have pursued the same work in single crystals of NaN03. Second-harmonic scattering in the electric dipole-quadrupole approximaand VOGT[1976] in the centrosymtion has been observed by ORTMANN metric crystal NaNO,, and by DENISOV, MAVRIN,PODOBEDOV, STERIN and VARSHAL [19801 in non-centrosymmetric TiO, crystal. KOSOLOBOV and SOKOLOVSKY [19771 have observed second-harmonic scattering on defects of LiIO, crystal structure. This type of light scattering by centrosymmetric crystals had been predicted by RABIN[ 19691. To MAKER[1970] are due the earliest line broadening observations for three-photon “quasi-elastic” scattering due to rotational-translational motions in molecular liquids. He simultaneously worked out the theory of

Ill, 9: I ]

HISTORICAL DEVELOPMENTS

163

its incoherent component, whereas the theory of the coherent component related with space-time angular molecular correlations has been proposed by BANCEWICZ and KIELICH [1976]. TANASand KIELICH [19751have considered second-harmonic scattering of light by a two-level system at two-photon resonance with the radiation field, when, in addition to the central line at 2w, two side lines appear with frequencies 2w f 6,where S is the level splitting, dependent on the beam intensity in the case of two-photon resonance. 1.3. SPONTANEOUS HYPER-RAMAN SCATTERING STUDIES

The work of PLACZEK [ 19341 provided the foundations of the polarizability theory of Raman scattering. His ideas have later been applied to the multi-photon case (KIELICH[1964b], AKHMANOV and KLYSHKO [19651, STRIZHEVSKY and KLIMENKO [1967], and LONGand STANTON [1970]). The selection rules for hyper-Raman scatterings differ from those of infrared absorption and usual Raman scattering. Those of vibrational transitions in three-photon Raman scattering have been established by CWIN,RAUCH and DECIUS [1965] for all point groups and types of vibrations. The theoretical formulation of hyper-Raman scattering is given in Cartesian representation, based on eq. (1.4) (see KIELICH[1964b], CYviN, RAUCH and DECIUS[1965], and ANDREWS and THIRUNAMACHANDRAN [1978]), as well as in spherical tensor representation (see, for example, OZGO [1975b]). Specifically, the methods of Racah algebra are well adapted to the description of hyper-Raman spectroscopy. They lead to general and at the same time clear results, comprising the angular dependences and different states of polarization of the incident and scattered photons (see CHIU[1970], KIELICH and OZGO[1973], OZGOand KIELICH [1974, 19761, JERPHAGNON, CHEMLA and BONNEVILLE [1978], and CHEMLA and BONNEVILLE [1978]). BANCEWICZ, OZGOand KIELICH [1973a, 19751 have calculated the rotational structure of hyper-Raman lines of gases consisting of spherical as well as symmetric top molecules. ALEXIEWICZ, BANCEWICZ, KIELICHand OZGO[1974] have worked out the theory of three-photon Raman line broadening caused by the rotational diffusional motion of molecules. The theory has been extended to multi-photon Raman scattering by KIELICH, KOZIEROWSKI and OZGO[1977]. Quite recently, ~ ~ A N A K O Vand OVSIANNIKOV [19801 have given a discussion of non-degenerate three-photon Raman scattering by atomic gases.

164

MULTI-PHOTON S C A m R I N G

[III,

P1

The resonantial amplification of hyper-Raman scattering discussed by KIELICH [1965a] and, on a numerical basis, by LQNGand STANTON [1970], WILSONand FRIEDMANN [1977] for has been analyzed anew by BEN-ZEEV, real coherent pulse shapes, causing the emergence of a novel resonance peak and a saturation effect such as Stark splitting, Stark shifts and optical nutation. AGARWAL [1979] has performed an analysis of saturation effects in hyper-Raman scattering by a four-level system at threephoton resonance (see also ARUTYUNIAN, PAPAZIAN, CHILINGARIAN, KARMENIAN and SARKISIAN [1974]). ALTMANN and STREY [19771 have analyzed the increase in intensity of hyper-Raman scattering due to the statistical properties of the field of radiation (cf. MANDEL and WOLF[19651). PERINOVA,PERINA, SZLACHETKA and KIELICH[1979], and SZLACHETKA, KIELICH, PERINAand PERINOVA [198011, have formulated a complete quantal theory of hyper-Raman scattering, in which they give an analysis of the dynamics of photon correlation and anti-correlation for arbitrary initial states. The effect of photon anti-bunching is purely quantal in nature (see WALLS[1979]). It has and MANDEL [1977] in resonance been detected by KIMBLE,DAGENAIS fluorescence, in accordance with the predictions of CARMICHAEL and WALLS [ 19761. Previously, SIMAAN [1978] considered the quantum statistical properties of Stokes hyper-Raman scattering using the master equation and Fock states (see LOUDON [1980]). MAKERand SAVAGE[1965] obIn their first experiment TERHUNE, served, in addition to hyper-Rayleigh lines, hyper-Raman spectra for water. VERDIECK, PETERSON, SAVAGE and MAKER[1970] observed hyperRaman spectra in some gases. Soon afterwards, SAVAGE and MAKER [1971], and FRENCH and LONG[1975] succeeded in perfecting the technique of recording hyper-Raman spectra, thus considerably shortening the time of exposition. FRENCH and LONG[1975] constructed a special spectrometer, operating on one or many channels, with which they recorded hyper-Raman spectra for the liquids HzO, CHCl, and CCl, and later (DINES,FRENCH, HALLand LONG[1976]), for a whole family of liquids comprising chloromethanes, bromomethanes and tetrachlorides. SCHMID and SCHROTTER [ 19771 have observed vibrational hyper-Raman spectra in solutions of C2CI, and CCl,. The past few years have witnessed an increase in the amount of work devoted to the hyper-Raman phenomenon in solids, the theory of which is due to STRIZHEVSKY and OBLJKHOVSKY [1970]. With regard to Scattering on phonons in ionic crystals the theory is due to JHA and Woo [1971]

111,

P I]

HISTORICAL DEVELOPMENTS

165

and, for polaritons in molecular crystals, to ZAVOROTNEV and OVANDER [1975]. BANCEWICZ, KIELICH and OZGO[1975] have given a discussion of three- and four-photon electric dipole and quadrupole Raman scattering and KIEIXH[1978] for crystals with symmetry Oh. OZGO,BANCEWICZ have analyzed the symmetry of the hyperpolarizability tensor for all symmetry classes and types of vibrations of crystals. BALAGUROV and VAKS[ 19781 have pointed out that intense critical hyper-Raman scattering can be observed in ferroelectrics. TERHUNE, MAKERand SAVAGE[1965], in their pioneering work, observed a hyper-Raman spectrum in molten quartz. SAVAGEand MAKER [197 11, using a many-channel spectrograph, recorded a hyper-Raman spectrum from NH4Cl single crystals (see also DINES, FRENCH, HALLand LONG[1976]). Of great importance is the report of Yu and ALFANO [1975] announcing the observation of three- and four-photon Raman and SAYAKHOV [ 19781 have observed spectra in diamonds. POLIVANOV hyper-Raman scattering on optical phonons in calcite crystals. VOGTand NEUMANN [1976, 19791 have performed systematic studies of such spectra for crystals of CsI, CsBr, RbI and SrTiO,. INOUEand SAMESHIMA [1979] and INOUE,ASAIand SAMESHIMA [1980, 19811, studying SrTiO, crystals, have proved hyper-Raman to be a simple and promising method for observing the phonon polariton mode, POLIVANOV and SAYAKHOV [1979a, b] have reported the first observation of hyper-Raman scattering on optical phonons in CdS. Quite recently, DENISOV, MAVRIN, PODOBEDOV, STEFUNand VARSHAL [19801 have published observations of both two-phonon and polariton hyper-Raman scattering in TiOa crystals subsequent to the results of DENISOV, MAVRIN, PODOBEDOV and STERIN [ 19781 for LiNb03 crystals. SCHREY,LYSSENKO,KLINGSHIRN and HONERLAGE [1979] in CdS and, more recently HONERLAGE, ROSSLER, PHACH,BIVA?,and GRUN[1980] in CuBr, have observed hyper-Raman scattering via virtually excited biexcitons. Hyper-Raman resonance scattering associated with excitonic molecules in CuCl was observed by NAGASAWA, MITA and UETA [19761, HENNEBERGER, HENNEBERGER and VOIGTr19771, and recently by GRUN[1980].

1.4.THE PURPOSE OF THIS PAPER

This review article is restricted to spontaneous multi-photon Rayleigh and Raman scattering. We dare not enter the vast, rapidly developing

166

MULTI-PHOTON SCATTERING

[III, 9 2

domain of stimulated Raman, Rayleigh or Brillouin scattering effects (see, for example, BLOEMBERGEN [1967, 19771, FABELINSKII [ 19681, and SCHUBERT and WILHELMI [1978]). Nor shall we deal with higher-order stimulated Raman processes (COMPAAN, WIENER-AVNEAR and CHANDRA [19783, Rayleigh and Raman resonant scattering by intense beams (PRAKASH, CHANDRA and VACHASPATI [19761, COHEN-TANNOUDJI and REYNAUD [1977], AGARWAL and JHA[1979], BALLAGH and COOPE[1980], FIUTAK and VANKRANENDONK [1980], and KNIGHT[1980]), Doppler-free [1977]), or the multiphotonic spectroscopy (GRYNBERG and CAGNAC parametric scattering of beams discussed in the monograph of KLYSHKO [ 19801. It is our intention to give a quantitative account of spontaneous multi-photon scattering in a uniform classical and semi-classical treatment, applying Cartesian representation for the description of integral scattering and irreducible spherical representation for spectral scattering. The formulae derived for the tensors of multi-photon scattering are adapted to the concrete particular cases encountered in actual experimental situations. In addition to considering the role of the nonlinear optical properties of free molecules apparent in incoherent scattering, we concentrate primarily on the stochastic aspects related to time-space correlations of the molecules and their translational-rotational motion within the scattering volume. We also expose the role played in dense media by electric molecular fields, the time and spatial fluctuations of which give rise to cooperative three-photon scattering. We adduce the essential experimental observations of the various scattering effects. Our discussion, moreover, comprises the angular distribution of integral intensities for arbitrary states of polarization (linear, circular and elliptical) of the incident and scattered photons, making no attempt to avoid the thorny problem of the choice of a univocal model of a natural light which, in application to nonlinear processes, is still controversial.

8 2. Nonlinear Molecular Raman Polarizabilities The quantum-mechanical theory of nonlinear Raman polarizabilities has been developed previously in the electric-dipole approximation (KIELICH [1964b]), and electric magnetic-multipole approximation (KIELICH [1965a1) for free molecules. Here, the problem will be extended [19651, and KIELICH to damping effects (BLOEMBERGEN [1965], BUTCHER

111. g21

MOLECULAR R A M A N POLARIZABILlTIFS

167

[1966b]). The problem is sometimes formulated in a very simple manner within the framework of classical Lorentz-Voigt electron theory, permitting the calculation of non-linear polarizabilities, taking into account the damping of electron vibrations as well as effects of nuclear vibrations (see, for example, KASPROWICZ-KIELICH and KIELICH[19751, and FLYTZANIS and BLOEMBERGEN [1976]).

2.1 THE MULTIPOLE INTERACTION HAMILTONIAN

We consider a microsystem composed of s point particles (nuclei and electrons), with electric charges ei, masses mi,and positional vectors ri referred to the centre Q of the micro-system (Fig. 2.1). Its electromagnetic field is observed at a point P, distant by R from Q; we denote the vector connecting P with a charge ei as Ri = R + ri. The Hamiltonian of the spinless microsystem, in the nonrelativistic case, is (HEITLER [1954]):

in which @(Ri,t) and A(& t ) are the scalar and vector potentials of the electromagnetic field at the time t and point Ri of the jth particle, having the generalized momentum operator pi. In eq. (2.1) we have neglected the potential-energy term of Coulomb interactions between the particles. We suppose that the microsystem is subjected to external space- and

Fig. 2.1. System of electric point charges of linear dimensions very small compared with the distance IRI >> lr,l at which the field of the system is studied.

168

[III, 8 2

MULTI-PHOTON SCATI'ERING

time-dependent electric and magnetic fields at the point (R, t) E(R, t ) = - ( l / ~ ) dA(R, t)ldt-V@(R, t ) ,

(2.2)

H ( R , t ) =V X A ( R ,t ) .

At the present stage, we do not specify a particular gauge. Quite generally, when the potentials @(Rj,t) and A(R,,t) are not constant within the region of the microsystem, one can expand these potentials at Ri = R+ri in a series in powers of ri (KIELICH[1966a]): m

@(R+ri,t ) =

1 (n!)-' r;[n]~" @ ( R ,t ) , n=O

A ( R + r ,t ) =

2 (n!)-'r;[n]Vn

A(R, t ) ,

(2.3)

n=O

where V is the spatial differential operator at the position R. After an appropriate canonical gauge transformation (see, for example, FIUTAK[1963] and POWER [1978]) we have, with respect to eqs. (2.1)(2.3):

H

H'(t)= H e + H ~ l ) ( t ) + H ~+) H ( tg) ) ( t )+

= H,,+

-

,

(2.4)

where Ho is the Hamiltonian of the non-perturbated microsystem, its first-order perturbation Hamiltonian being (KIELICH [1965al)

c [(2n m

H:') = -

- I)!!]-' l@)[n]

E'"'(R,t ) ,

(2.5)

[(2n - 1)!!]-' M',"'[n] H'"'(R, t)

(2.6)

n=l

m

H;'(t)

=n=l

with E'")(R,t ) = V"-'E(R,t) and H("'(R, t) = Vn-' H(R, t ) . We have introduced above the following 2"-pole electric moments of the microsystem S M?'= ejr; Vn)(ri) (2.7)

1

i=1

and 2"-pole magnetic moments of the microsystem S

M?)

+ I>C]-'

= n[(n

C ejrjP)(ri)x i ,

(2.8)

j=1

wherein the vector operator Y(n),of degree n (its properties resemble those of spherical harmonic functions), is given by Vn)(rj)= (-11~ (n!)-' r;+l V;(ry1).

(2.9)

111, 921

169

MOLECULAR RAMAN POLARIZAEZLITIES

The higher-order Hamiltonians of interaction between the microsystem and a strong classical electromagnetic field occurring in eq. (2.4) have been given in earlier papers (KIELICH11965, 1966a1). In commonly discussed problems it is sufficient to take only the first terms of the Hamiltonians (2.5) and (2.6):

H : " = - D " . E ( R , t ) - f Q e : V E ( R , f)-.

..,

H : ) = - D " - H ( R , ~ ) -...,

(2.5a) (2.6a)

where D,=M',') and Qe=lM',"' are operators of the electric dipole and is the magnetic dipole quadrupole moments respectively, and D , = moment operator. Obviously, in this approximation, one should take into consideration the first term of the Hamiltonian El:). It is given as follows (KIELICH 11965a]) :

-

HE' = -$H(R, t) x,,, * H ( R , t),

(2.10)

where S

xm=(4c2)-' C (e;/rnj)(riri - r;u)

(2.11)

j=1

is the operator of the magnetic polarizability of the microsystem and U the second-rank unit tensor. Similarly, one can calculate multipolar contributions to the electric and magnetic polarizabilities (KIELICH 11965a, 1966al and DE GROOT [1969]).

2.2. THE EQUATION OF MOTION FOR THE VECTOR OF STATE

The expectation value of a dynamic operator M ( t ) for a quantum transition (fl c li), under the influence of a perturbation, is Mfi(t) = (+dt)l M

I+i(t)),

(2.12)

where motion of the vector of state It,bi(f)) in Hilbert space is given by the Schrodinger equation

iNa/at) I+i(t)) + H'(t)I I+i(t)> (2.13) the Hamiltonian of the microsystem being given by (2.4). We express the wave function of the perturbated system +i(r, t ) in terms of the following expansion, involving known wave functions cli(r, 0)

170

[III, 8 2

MULTI-PHOTON SCATTERING

of stationary states of the non-perturbated system: (2.14) with ul = Ei/h being the circular vibration frequency corresponding to the eigen-energy El of the quantal state 1 of the non-perturbated system. The transition coefficients for a transition of the quantal system from the state li) to the state (21, under the influence of the perturbation H ’ ( t ) , are functions of time and can be expanded in a series as follows: m

cii(t)=clP’(t)+cll!’(t)+cif’(t)+. * * =

1 Cpyt).

(2.15)

n=O

The solution of the problem is dependent o n the initial conditions assumed, defining the state of the system while still not acted on by the perturbation. This reduces to finding the zeroth approximation of the expansion (2.15). Sometimes, one can assume the ideal situation when the levels of the states of the “isolated” microsystem are of zero width; then ciy’(t)= tili is independent of time. Regrettably, however, this situation is unrealistic, since even in the “isolated” case the levels almost always possess a non-zero width for various reasons (e.g. polarization of vacuum, heat reservoir, external fields, etc.). In fact, this finite width of the levels is the factor ensuring spontaneous emission - the action of the external electromagnetic field reducing to cause time-variations of the pre-existing level widths. We have thus in the zeroth approximation (LOUISELL [1973]) (2.16)

clP’(t) = tili exp (-T1t/2),

where r;’ is the lifetime of the quantum state 1 and r is real. With regard to the condition (2.16) we obtain, by (2.13)-(2.15), the following equation determining the transition coefficients for n = 1 , 2 , 3 , . . . (PLOCINICZAK [1980]): ih(a/at)cj:’(t)

= -ih(rI/2)ciy’(t)

+ 1 H i k ( t ) cK-’’(t) k

exp (-iwklt),

(2.17) where the matrix elements of the perturbation Hamiltonian H [ k ( t ) are defined as usual. The equations of motion (2.17) represent a modification of the approach of ORR and WARD[1971], as well as the respective equation of motion for the density matrix operator of a system in the presence of damping (see BLOEMBERCEN [1965], AGARWAL [1973], and APANASEVICH [1977]).

111, 821

MOLECULAR R A M A N POLAFUZABILITIES

171

In order to derive the transition coefficients for n = 1 , 2 , 3 , . . . one has to know the perturbation Hamiltonian in analytical form, or rather its explicit dependence on the time. From eq. (2.5), the first-order perturbation Hamiltonians are seen to be linear functions of the electromagnetic field which, in turn, can be expressed as a superposition of different monochromatic waves in Fourier representation

c

1 E(R, t ) = {E(R, wA)exp(-iw,t)+E(R, wA)*exp (iqt)}, (2.18) 2A

where summation extends over all discrete vibration frequencies, with E(R, WA)* = E(R, -Oh). Restricting our considerations to the Hamiltonian of first-order perturbation, with time-dependence of the type (2.18), we obtain, with regard to the solution of the inhomogeneous linear differential equation (2.17), the following first-order transition coefficients: 1 cj;)(t) = 2 {ci:)(uA)exp (-iwAt)

1 A

+cit'(-wA) exp (iwAt)}exp [i(wri+iTi/2)t],

(2.19)

where the transition coefficient amplitudes are of the form (2.19a) Above, rti = (I',- ri)/2 are the difference level widths of the system for transitions ( I 1 + li). Similarly, we obtain the transition coefficients for higher and higher approximations. Quite generally, in the nth order we have

and ko = 1. By eqs. (2.14) and (2.15), the transition operator (2.12) can be written

172

MCTLTI-PHOTON S C A W R I N G

[III, 8 2

in the form of the series (2.21) where the approximations of successive orders are (2.22) In particular, in the zeroth approximation, we hence have for the transition operator Mfi(t)(O)=Mfi exp[i(w,+iT$)t].

(2.23)

The above operator describes spontaneous emission and absorption. Here, however, = (&+ ri)/2is the summation level width. The transition matrix in the first-order approximation is, by eqs. (2.16), (2.19) and (2.22),

r;

We likewise express the transition matrices of higher orders, including the nth (2.22).

2.3. NONLINEAR POLARIZABILITIES IN THE ELECTRIC-DIPOLE APPROXNATION

We shall now consider the quantum transition matrices (2.22), restricting ourselves to the electric-dipole approximation in the perturbation Hamiltonian (2.5). In this procedure, we obtain by (2.24), for the component (+ (in a Cartesian reference system X,Y, 2) of the electric dipole transition moment, the following first-order approximations: 1 D:,(t)(')=S ~ { a ~E,(w,) ( ~exp~(-ioAt) ) A f

&(--~A)

E,(-oA)exp (ioht))exp [i(ofi+ir:)tI,

(2.25)

111, 821

MOLECULAR RAMAN POLARIZABILlTLES

173

where the second-rank tensors of the linear polarizability of the microsystem for the transition (fl t li) have the form (fl R lk)(kl 0: b> +(fl W lk)(kl E, b)}. Wkf fW, +irk,

(2.25a)

The preceding quantum-mechanical expressions proceed beyond the WEISSKOPF-WIGNER theory [1930] and, in the absence of damping, go over into the well known KRAMERS-HEISENBERG result [19251 (see EBERLY +ufi+irl. [1980]). By (2.23, the scattered frequencies are Proceeding along similar lines we obtain, by eqs. (2.20) and (2.22), the second-order transition matrix of the electric dipole moment

x exp [-i(o,

+ WA)t]+ 3 terms} exp [i(ofi+ir,+)t], (2.26)

where we have introduced the third-rank tensors b k , defining the second-order nonlinear polarizabilities for the transition (fl +li):

Here, S2 denotes the symmetrizing operation, which indicates that the expressions following it are to be summed over the 2! possible permutations of the pairs TW, and U O ~ . The third-order electric dipole transition moment is:

where the fourth-rank tensors c k p , defining the third-order nonlinear

174

1111, P 2

MULTI-PHOTON SCATTERING

The other 7 tensors of eq. (2.28) are hence derived by successively changing the signs at the frequencies w,, w, and wA. Finally, by (2.22) and (2.20), we have for the nth order electric dipole transition moment (cf. KIELICHr1966bl):

where the ( n + 1)th rank tensor of nonlinear nth order polarizability for the transition (fl + Ii) has the form n

C=o S n

aZm,...un(wA,, . . ., wAn>=fi-"

s

(4o:,, lk,) X

C

k,-..k,

ru

=I

(wk,f + wA1+ ' *

*

+

* *

(klD: lk+l) (kl0:" Ii>

+ i r k u f ) fl:

* *

= s + 1 (obi-

-*

*

- WA,,

- irki)

(2.31) Here, S,, is a symmetrizing operator, implying summation over all n! permutations of C T ~ O . . ~. ,~CT,W,,,. , As we see, the transition dipole moments (2.30) are in general complex quantities; to obtain their real parts, determining classical radiation of real dipole moments, one can have recourse to the postulate of KLEIN [1927] (see also PLACZEK [1934]).

111, s21

MOLECULAR RAMAN POLARIZABILITIES

175

In the non-resonance case, the nonlinear polarizabilities induced by vibrational transitions can be described by the method proposed by PLACZEK [1934] (see KIELICH[1964b]). Recently, PANDEYand SANTRY [ 19801 have evaluated vibrational contributions to the polarizabilities and hyper-polarizabilities of some simple molecules (CO, HCN and H20).

2.4. MULTIPOLE ELECTRIC AND MAGNETIC POLARIZABILITIES

Electric dipole transitions of even orders are forbidden for certain molecular symmetries, and one has to consider electric quadrupole or still higher multipole transitions. To deal with these situations, we insert the multipolar Hamiltonian (2.5) into the transition moment (2.24). This leads to the first-order electric multipole moment for the transition (fl li> +-

where the tensors of rank n + n, determine the linear multipole electricelectric polarizabilities (KIELICH [ 1965a, 19751):

(2.33) Strictly, (2.33) is the tensor of 2" -pole electric polarizability induced by 2"l-pole electric transitions. Obviously, for n = 1 and n , = 1 eqs. (2.32), (2.33) reduce to (2.25). In a similar way, one can write the tensors '"LAt;i' and (";At;) defining the tensors of 2"-pole electric polarizability induced by 2"l-pole magnetic transitions, and vice versa. Also, the magnetic multipole Hamiltonian (2.6) permits the calculation of the tensor (";Act determining the 2"-pole magnetic polarizabilities, induced in 2"l-pole magnetic transitions (see KIELICH [1965al). By having recourse to the expressions (2.20) and (2.22), we obtain, for

176

[III, 5 2

MULTI-PHOTON SCATTERING

the second-order 2" -pole moment, induced in multipole electric transitions (2.5) and magnetic transitions (2.6) (KIELICH[1965a, 1966b]),

X { [ ' " ~ B ~ ; i f n 2 ) ( OOh2) ~ , , [nl + t c n d ~ F ~ g n 2 ) ( ~ A@A,) ,,

+ nz] E(nl)(R,@A,) E(""(R,@AZ)

[n, -+ FZJ E("I'(R,@A,)

HYR, on2>

+ nz? H("')(R,@A,) E(")(R, WA,) +(nLB2A>)(U~,, O A ~[ f)l l + nz] H("I)(R, OA,)H'"'(R, @A,)] +'"LB$A~n2'(wA,, @A2) [a,

xexp [-i(wA,+q2)t]+. *}exp[i(ofi+ir:)t],

(2.34)

where, as an example, we write out the quantum-mechanical form of the (n + n1+ +rank pseudo-tensor: n t n )

'"LBkAfi

(@Al,

WA,>

= hp2S(n10Al,n2wA2)

(fl M F ) Il) ( I 1 MFI' ( k )(kl M$' li) I(wt~-@A,

+(fl

- O A 2 - ir(i)( @ki

- wAI- i r k i )

&$'I'Jk)(k(&[) 1l)(l1 Mg2) li)

(wkf+@Al

+irkf)(@ii-@A2-irLi) & )I: Ik)(kJ M,"' li)

1l)(l1 + (fl M(,"J

(Olf+ OA2+irff)(@kf+

Oh2

+ O A I +irkf)

]

(2.35)

determining the 2" -pole electric second-order nonlinear polarizability, induced by 2"t-pole electric and 2"z-pole magnetic transitions. Clearly, by the interchange of &!' and &$' in eq. (2.35) one arrives at the quantum-mechanical expression for the tensor (ndBz;i+n2)(~Al, oA2). Similar interchanges lead to the other tensors of the expansions (2.34). It is worth noting in particular that the second-order electric dipole transition moment (2.26) results from (2.34) for n = nl = n2 = 1. Likewise, eqs. (2.34) and (2.35) lead to the quadrupole and higher electric and electromagnetic transition moments. We have seen how expressions are derived for third-order and higherorder electric or magnetic multipole moments. However, in the calculations, one has to keep in mind the fact that, from the second-order approximation upwards, Hamiltonians of the second order and, in the relativistic case, Hamiltonians of higher orders, should be taken when calculating the vectors of state (2.27) (see KIELICH[1965a, 1966a1). In some cases, the contributions from the second-order Hamiltonian (2.10)

111, 521

MOLECULAR RAMAN POLARIZABLLlTlES

177

are highly essential, e.g. in nonlinear optical activity, as well as in the nonlinear Faraday effect of diamagnetic gases (KIELICH, MANAKOVand OVSIANNIKOV [ 19781, and MANAKOV, OVSIANNIKOV and KIELICH [19801). In particular, we get for the second-order electric dipole moment induced by a direct magnetic transition (2.10):

where we have introduced the third-rank pseudo-tensor (KIELICH [1965al)

determining the second-order nonlinear electric polarizability induced by a magnetic transition, due to the square of the magnetic field strength H 2 (quadratic transition). The third-order electric dipole moment, induced by a simultaneous electric dipole transition (2.5a) and magnetic transition (2. lo), has the form

x exp [-i(me where the fourth-rank tensor

+a A ) f ] +

+up

*

.} exp [i(wfi+iri)r], (2.38)

178

1111, § 3

MULTI-PHOTON SCA'ITERING

0 3. Incoherent and Nonresonant Multi-Photon Scattering by Free Molecules 3.1. THE ELECTRIC AND MAGNETIC FIELDS OF THE SCATTERED WAVE

We consider the molecular system defined in Fig. 2.1. Since we are concerned with the field in the wave zone, i.e. at distances from the radiating molecular system considerably exceeding the light wavelength ( R >>A), we can write (see HEITLER [ 1954]), at the space-time point (R, t ) :

E,(R, t)=(l/R3c2){RX[RXZ(t')},

(3.1)

Hs(R, t ) = - ( 1/R2c2){R X Z(t')},

(3.2)

where (3.3) is the Hertz vector at the retarded moment of time ti=t-(Rj/c)=t-c-'

lR-trjl,

(3.4)

since Rj = R + r j (Fig. 2.1). On expanding the vector (3.3) in a series in powers of rj, and taking into consideration (3.4), we obtain (KIELICH[1965a]): Z(t')= Z,(t - R/c) + Z & - R/c), where the first term of the Hertz vector,

(3.5)

m

Z,(t-R/c)=

1[(2n- I)!! (Rc)"-']-'

R"-'[n-l](a"-'/at"-')

Mc'(t-R/c)

n=l

(3.5a) describes electric multipole radiation and the second, m

z,(t - R/C)= -

1 [(2n- I)!! R~C"-'I-' Rn-'[n - 11 n=l

X { R X (a"-'/at"-')} 1M(!)(t - R / c )

(3.5b)

describe magnetic multipole radiation (see eqs. (2.7) and (2.8)). By the definition of the Poyntings vector of electromagnetic radiation:

S(R, t ) = ( 4 4 ~E)s ( R t ) X HJ.R, t ) and with regard to eqs. (3.1) and (3.21, we obtain for the mean value:

(S(R,t ) ) = (c/4?rR5)(R'S,

-%&)Is*.

(3.6)

111. B 31

179

SCA'ITERING BY FREE MOLECULES

Above, we have introduced the intensity tensor of light scattered by the molecular system:

I&= ( ~ 1 2( Z~u (~t -R/c)* ) z(t-R/~))n,E,

(3.7)

where the symbol ( denotes appropriate averaging over the orientations of the molecules and averaging over the states of the incident field of light E, whereas N is the number of molecules, reorienting freely in the scattering volume. The geometry of scattered light observation is shown in Fig. 1.3.

3.2. HARMONIC ELECTRIC-DIPOLE ELASTIC SCATTERING PROCESSES

We shall first consider the electric-dipole approximation, on the assumption of monochromatic incident light sufficiently intense to cause nonlinear polarization of the molecule. Taking into account purely harmonic terms, with frequencies o,20, 3 0 , . . . , we obtain m

Z,(t-R/c)

=

1 D,(no)cos[nw(t-R/c)],

(3.8)

n=l

where the amplitude of the nth harmonic of the electric dipole moment induced in the molecule is . unEul(~) * * * E-Jo).

D,(no)= (2"-'n!)-'

(3.9)

The ( n + l ) t h rank tensor defining the nth order nonlinear polarizability can be dealt with as completely symmetric in the nonresonant case. On the insertion of (3.8) into (3.7) we have:

where

IZ= 2- (n w / c ) 4 ~ ( ~ no) , * ( D, (nw))n,E

(3.11)

is the intensity tensor of the light scattered at the nth harmonic frequency; with regard to eq. (3.9), it has the following, explicit form:

IZ = Iq22"-'(n!)2]-' (no/c)4 * x (a2:. . . ," C, . . . TnEul(o) * *

EZw(o) ET1(o)* * . ETn(w))n,~. (3.12)

180

[Ill, § 3

MULTI-PHOTON SCAlTERING

The averaging of the Cartesian tensor products of (3.12) is readily performed for n = l , 2 , 3 (KIELICH [1961], and ANDREWSand THIRUNAMACHANDRAN [1977a]), but is in general a highly complex affair if n 3 4 . However, (3.12) is easy to calculate for arbitrary n if the incident beam is assumed to be linearly polarized, since in this case one obtains (KIELICH [1967a, 1968b], see also Appendix B):

I=

=(

n ~ / c( S> x~ w

+ e,e,G,,,)I"g("'

(3.13)

with: I = (lE1')/2 the intensity of incident linearly polarized light, and e the unit vector in the direction of the field E. Equation (3.13) involves two molecular parameters of elastic nharmonic scattering (KIELICH and KOZIEROWSKI [19721):

F,, = Na;. G,,

..

,. a%,. .

= N G E . ' . a, a;;,

. P, f o r P a , P , .

' . ' p.

gaP,,P,.

.. ' ' ,"&l

(3.14a) (3.14b)

where the tensor operators: f u p o l l S l . . . a,B, and g,,,,,, . . . are defined in Appendix B. We have also introduced in eq. (3.13) the degree of nth order coherence of the incident linearly polarized beam g(")= (lE(~)l"")E/(lE(~)l'~~,

(3.15)

and WOLF[19653) leading to (MANDEL g("' =

I

1

for coherent light,

n! for chaotic light.

(3.15a)

The molecular parameters (3.14) are valid for all symmetries of the tensor a=, . . . ,n. In the case of completely symmetric nonlinear polarizability tensors, the final results simplify considerably, and can be expressed in terms of the least number of irreducible invariants for the rank n + 1 under consideration. (i) Rayleigh scattering. In particular, for linear Rayleigh scattering (n = 1) we have:

F,

= (N/lO) latf'l',

G,

= (N/3)[)a~'1'+(1/10) la:)1]'

involving the parameters of isotropic and anisotropic scattering (a,(0)12 - a;za&/3, Jatf)12=(3a;,"a:p- a;;azp)/3.

(3.16) (3.17)

111, 8 31

181

SCATI’ERING BY FREE MOLECULES

(ii) Second-harmonic Rayleigh scattering. Similarly, on applying eq. (3.14) to second-harmonic scattering (n = 2) we have, for a completely symmetric tensor of second-order nonlinear polarizability a:;? = b$?

F2,

12 lbyi\2),

= (N/2520)(7

G2, = (N/1260)(28 lb$~l’+3

(3.18a) (3.18b)

with the two irreducible molecular parameters (KIELICH and OZGO[ 19731)

\b$zl’ = ( 3 / 5 ) b ~ $ b ~ ~ y ,

\

= (1/5)( 5 big:

b$?

- 3 b,$

(3.19a)

b::).

(3.19b)

(iii) Third-harmonic Rayleigh scattering. We shall still consider thirdharmonic elastic scattering (n = 3). Here, for a completely symmetric tensor of third-order nonlinear polarizability a:;+ = c:&, we obtain (KOZIEROWSKI [19701)

F3,

= (N/181444)(27 \c$ZI2+2O(c$:\’),

GSo= (N/60480)(84 Ic$212+391~$~1’+4 IC$:\’),

(3.20a) (3.20b)

where the following three irreducible molecular parameters, introduced by OZGO[1975a], intervene: (3.2 1a)

\ c $ Y= (1/5)cZ$~c~~titj, lC(32l’ = (2/7)(3C,&C$ki

-30

30

- c,,f3pCyy~G)~

1~$21~ = (1/35 ) ( 35 c~,$$c:;~~

- 30~,P3Y”yc$88

(3.2 1b)

+ 3c i:&&&J.(3.21C)

The irreducible molecular parameters (3.17), (3.19) and (3.21) are, in practice, immediately applicable to various symmetries of the molecule, since tables giving the nonzero and mutually independent components of the tensors a,,, bpsy and caPy8 for all point groups are available (see, for example, KIELICH [1972a, b, 1980al). The parameters react individually to the presence of various elements of symmetry in the molecule. If, for example, the latter, when in its ground state, possesses a centre of symmetry, the parameters (3.19) vanish, meaning that no secondharmonic scattering can take place in the electric-dipole approximation considered. Centrosymmetric molecules can give rise to second-harmonic scattering only if we go over to the electric-quadrupole approximation (see KIELICH, KOZIEROWSKI, OZGOand ZAWODNY [19741).

182

[111, 5 3

MULTI-PHOTON SCATTERING

3.3. MULTI-PHOTON VIBRATIONAL RAMAN SCATTERING (CLASSICAL APPROACH)

We shall now consider scattering processes due to the vibrations of molecular nuclei in the classical treatment of PLACZEK'S [19341 polarizability theory, which can be extended rather easily to second- and third[1964b], STFUZHEVSKY and KLIMENKO harmonic Raman scattering (KIELICH [1967], and LONGand STANTON[1970J). According to Placzek, the polarizability of a molecule is not constant, but varies with time, for example as a result of the vibrations of its nuclei. Let Qm(t), with m = 1,2, . . . , 3 N - 6 , denote the normal coordinates of the displacements of the nuclei with respect to their equilibrium position. Then, for harmonic vibrations with the frequency w, and phase shift (pm, we have:

where Qm(0)is the amplitude of the mth normal vibrational mode. In the expansion (3.9) the nonlinear polarizability tensors are now functions of the normal coordinates aZl... JQ), of an (in general) unknown analytical form. For small vibration amplitudes, however, one can write the following expansion:

.,(a) = a::,..

agl..

. JO)

+

a,:

. . . ,,.mQm

m

+ . . .,

(3.23)

where we have used the notation a,,nw . . .

n.. m =

( a a g l . . ,/aQm)o=o.

Hence, by eqs. (3.7H3.9) and (3.22), (3.23), we have for n-harmonic Raman scattering: p-?" rn = "22"-'(n!)2]-'

x (a:.

,

. am:m

[(nw T wm)/c]4 no

aml .. . T n : m

l a m l">n

x ( E z , ( w ). . * E z A o ) E T , ( o ) * * ETn(u))E. 9

(3.24)

Above, a g l . . is the first derivative of the nonlinear polarizability tensor with respect to the normal coordinate Q,. Concerning the calculation of the tensor (3.24), we proceed as follows: t in the representation we average over the orientations of the molecule L of spherical tensors, leaving the average over the fields in Cartesian basis. Transforming from laboratory Cartesian coordinates to the spherical basis

111, 831

183

SCA'ITERING BY FREE MOLECULES

we write; -

ruI)

aou,...a.:m-

C

JKL

(3.25)

)7

"'L( u,aK:m

~o,,,...

JKL

where the RLY... u. are transformation coefficients between the two representations. a(KJ)..(no) is the Kth component of the irreducible tensor of 5th order, with components transforming from the laboratory to the molecular system of coordinates in accordance with the relation:

a$!:=

f D J K MZg)frn (~)

(3.26)

M=-J

the D',,XL?) denoting elements of the unitary Wigner rotation matrix possessing the property:

(D',dn)* D;,w(n)), = (25 + 1)-1 6JJ,6mt6M&f,.

(3.27)

With regard to the expressions (3.24)-(3.27) we finally obtain 1Er'''m

= [2"-'(n!)']-'

[(no Tom)/c]4 N

\ Q mI2

1"g'"'

where we have introduced the following tensor:

(3.29) accounting for arbitrary polarization states of the incident and scattered light, and arbitrary geometries of observation. The coefficients RE:. . . cr" of the rotational transformation have been tabulated by MAKER[1970] for n = 2 and by OZGO[1975a] and STONE [1975] for n = 3. Like eq. (3.13), the tensor of n-harmonic vibrational Raman scattering (3.28) is applicable to first-, second-, third- and higher-harmonic light scattering processes. However, eq. (3.28) is more general than (3.13) in that, thanks to the tensor (3.29), it is valid for arbitrary polarization states of the fields. 3.4. ROTATIONAL, VIBRATIONAL AND ROTATIONAL-VIBRATIONAL MULTI-PHOTON SCATT'ERING PROCESSES (SEMI-CLASSICAL APPROACH)

When proceeding to the quanta1 (or rather semi-classical) treatment of multi-harmonic scattering, we wish to draw attention to the following

184

[III, P 3

MULTI-PHOTON SCATI'ERING

three aspects relating to the tensor (3.7). First, we have to deal with the matrix element 2: of the transition from the initial quantum state li) to the final state (fl. Second, in place of N we now have Ni, the number of freely orienting molecules in the initial state li), since only these molecules take part in scattering, accompanied by a change in quantum state from li) to (4. And third, averaging over the molecular orientations 0 has to be performed in a different way, since the rotational motion of the molecule is quantized so that it can only be in rotational states such that the projection of its angular momentum J onto the z-axis, shall be equal to M, with -J s M s J . Thus, instead of unweighted integration ( )n over all possible values of the Euler angles, we now have to carry out a summation over all the permitted quantum numbers Mi and Mf of the initial and final states of the molecule. We denote this average by the symbol ((. .))M = (25+ 1)-' CM,M, (. .) since, with regard to spatial degeneracy, each of the rotational levels is (25 + 1)-fold degenerate. With the above in mind, we have in place of (3.7) the following intensity tensor of light, scattered in the quantum transition (fl + li):

-

-

P,=

( ~ ~ 2 c(z:(t 4 ) - R / ~ )Z:(t *

-R

/~))~,~.

(3.30)

In the case of inelastic multi-harmonic electric-dipole scattering (3.8) we have an expansion of the type (3.10) where, now, the transition tensor for the nth scattered harmonic is:

~k(nw) = ~~[2"-'(n!)~]-I [(nw

(ab,. '.

T wif)/cI41"

* f i ' ' '

T,,(no)>M

(n) g U l ' ' ' U.71 ' ' ' 7,'

(3.31)

In (3.31), we have introduced the tensor of the nth order degree of coherence of the incident beam: (n)

gu, . . . u , , 7 1 . . . .,=(Ef,(w)

-

* *

Efn(o)E.,(o) *

*

- ET,,(w))E(lE(o)12)2"' (3.32)

In further discussions, for the sake of clarity, we shall apply certain simplifying assumptions which, by the way, have already been used by PLACZEK[1934] in his treatment of usual Raman scattering. (i) In our description of the wave function Gj of the molecule in the state li), we apply the adiabatic Born-Oppenheimer approximation, permitting the factorization

4i= 41Ar9 4 ) 4dQ) ki(0)4Jc+), with:

+E'(r,

(3.33)

q) the electron wave function, dependent on the coordinates r

111, § 31

185

SCAmEFXNG BY FREE MOLECULES

of the electrons and q of the nuclei; +I+(Q) the vibrational wave function, dependent on the normal coordinates Q of the vibrations and the vibrational quantum number V'; 1L,,(f2) the rotational wave function, dependent on the Euler angles 0 and the set of rotational quantum numbers R' = (f,?, AT);and t,br(u)the spin wave function, dependent on the spin coordinates u and spin quantum numbers I' of the nuclei. (ii) The vibration frequency of the incident light wave is far remote from regions of resonance, and is so high that the vibrational and rotational transition frequencies can be omitted in the denominators of the transition polarizabilities (2.27) and (2.29). (iii) In the act of scattering, the ground electron state g of the molecule remains unchanged; moreover, the state g is non-degenerate. On these assumptions, we are able to perform the summation over all the intermediate vibrational states V and rotational states R in the expressions (2.27) and (2.29) for the hyperpolarizability tensors; i.e., going over to the spherical basis, (3.25)-(3.26), we may write (nw) =

1 R::.

..

u ~

JKLM

x(R'1 D",f2)

IR') (Vf(ii$L(nq Q) IV'). (3.34)

Consider fust the matrix element of vibrational transitions. In the harmonic oscillator approximation (3.23) it can be written, by analogy to linear Raman scattering (see LONG[1977]), in the form

+Ii i ~ L , ( ~ ) ( VQ~, l IV,)+-

* .

m

(3.34a) In the latter approximation, these matrix elements have the following properties (LONG[1977]) for Rayleigh lines: for V'f V',

1

for V f = V i ,

(3.34b)

and for Raman lines: ( Vf( Q, IVL) =

I"

(Vi+1)"2(h/20,)112

for vf,= VL, for Vk = VL+ 1, (3.34~)

186

[III, 5 3

MULTI-PHOTON SCA"ER1NG

since the selection rules for the vibrational quantum number are: V',= Vm+1 for Stokes lines nw - w, and V', = VA- 1 for anti-Stokes lines no fw,, with rn = 0 , 1 , 2 . . . . We determine the vibrational and rotational wave functions of (3.34) by methods of quantum mechanics, applying the solutions of Schrodinger's equation for the Hamiltonian of the quanta1 system. The rotational quantum numbers R = ( J , T , M) correspond to the solutions for molecules of the freely rotating asymmetric top kind. Henceforth, we shall be considering symmetric top molecules, for which the quantum number 7 = K describes the projection of the angular momentum onto its symmetry axis. In this case, the solution of the Schrodinger equation gives the following rotational wave function:

Although these functions are of the same form for the spherical top and symmetric top, the respective eigenvalues of the Hamiltonian operator HR are different. For the spherical top we have EJ = h2AJ(J + l ) ,

(3.35a)

whereas for the symmetric top

EX

+ 1) + (A - B ) K 2 } ,

(3.35b)

= h2{BJ(J

with A, B the rotational constants (LONG[1977]). The selection rules for the rotational and rotational-vibrational transitions in Raman scattering have been discussed by PLACZEKand TELLER [1933], ALTMANN and STREY[1972], and KONINGSTEIN [1972], as well as by CHIU[19701, who moreover considered magnetic-dipole and electricquadrupole transitions. When calculating the intensities related to rotational transitions, one has to apply the formulae (EDMONDS [1957])

where the 33 Wigner coefficients fulfill the orthogonality condition

'

J3K3

Jl

J2

J3)

(2J3+1) ( K , K 2 K3

(J,

32

")-

M1 M2 K 3

8K2M2*

(3.36a)

Only those molecules act as scatterers for the transition VfRft V'R'

111, 931

SCATTERING BY FREE MOLECULES

187

which are in the initial state given by the Boltzmann distribution (3.37) Nv;Rt = Ng(ViRi)g, z;; exp (-Ev;Ri/kT), where g(VmRi) is the degree of degeneracy of the initial level, g, the nuclear statistical weight, and ZvR the vibrational-rotational partition function. A discussion of (3.37) for particular cases is to be found in handbooks, for example by KONINGSTEIN [1972] and LONG[ 19771. 3.4.1. Three-photon Raman scattering We begin with three-photon Raman scattering. We have, by eq. (3.31),

Ifi-,(20)= (Ni/8)[(20

T %)/CI

fi I (bevp(20)" b:Afi(2w))MgL:Aw-

4 2

(3.38)

This tensor is readily averaged in the Cartesian basis. In the general case of a non-symmetric tensor bLp(2w) one has to deal with five molecular parameters, of a rather complicated analytical form (see KIELICH [1964b1, ANDREWS and THIRUNAMACHANDRAN [1978], and STREY[1980]). We refrain from adducing them here, preferring to go over to a discussion of the case of completely symmetric bEVp(2w)in the treatment of spherical tensors proposed by BANCEWICZ, Ozco and KIELICH [1973a, 19751, and omitting the polarizational aspects and angular relationships discussed by OZGOand KIELICH[1974]. In order to simplify the discussion we assume the geometry shown in Fig. 3.1. On going over in eq. (3.38) to the spherical representation (3.25)

t"

Fig. 3.1. Geometry for the determination of the vertical and horizontal intensity components of scattered light observed in the YZ-plane, for vertically polarized I , and horizontally polarized I , incident light.

188

MULTI-PHOTON SCA7TERING

[III, 9 3

we obtain, for the vertical scattered (polarized) component and horizontal (depolarized) component (if the incident light wave was polarized vertically) (BANCEWICZ, OZGOand KIELICH[1973b3) rf;(20) = (NVrK3/280)[(2oF ~ , ) / c ] ~ l $ g ~ ’

(3.39a)

3‘ + 12(-Kf

)

3 F 2 s K’ ((Vf(b’F)(Q) \Vi)lz}.

(3.39b)

These expressions are for purely rotational hyper-Raman lines if Q = 0 and rotational-vibrational hyper-Raman lines if Q # 0. The properties of the 3j Wigner coefficients in eq. (3.39) impose the following selection rules on the rotational quantum numbers in hyper-Raman scattering: A J =.If -J’= 0, *l, *2, *3 and AK= 0, *l, *2, *3. For linear molecules (CO, NO), if K’ = K f = 0, the only permitted transitions are those with AJ=*l, *3 (BANCEWICZ, OZGOand KIELICH[1973a, 19751). The symmetry of the molecule and the symmetry of the vibration decide which of the nonlinear molecular parameters I( VfJ@(Q) (V‘)I2of eq. (3.39) are nonzero for J = 1, 3. Hence, moreover, we obtain the selection rules for the quantum number K, since the relation AK = M has to hold always. It is noteworthy that in three-photon scattering no isotropic intensity component, related to a spherical tensor of order zero, occurs. A Qbranch appears only if the selection rules A J = 0, AK = 0 are permitted, in the part described by spherical tensors of the ranks 1 and 3. At twophoton (linear) scattering, the intensity of the line for the transition JfKf t J’K’ depends (for any A J and well-defined AK) only on the one molecular parameter 1iiE))l”(see KONINGSTEIN [19721). Whereas at threephoton scattering we have two parameters, l@I2 and \bg)12, for the symmetric top; one 16$3’[2,for the symmetries D3, and C3,,;and one, \6$3)12,for the symmetries &, T, Td. The expressions for the molecular parameters \6g’12for all point group symmetries have been tabulated by

111, 8 31

189

SCA'ITERING BY FREE MOLECULES

BANCEWICZ, KIELICHand OZGO [1975] and ALEXIEWICZ, BANCEWICZ, KIELICHand OZGO[1974]. STANTON [1973] has given a complete discussion of the selection rdes governing rotational hyper-Raman transitions. (i) The rotational structure of the lines Let us now introduce, with regard to eq. (3.39a), the parameter of purely rotational structure for the vertical component of the lines (2w + ~ R w )

&(JfKf, J'K') = exp ( - E J ~ K ' / ~ T ) ( ~ ~ , K , / ~ ~ Z R ' )

(3.40) where ZRi denotes the rotational sum of states and gp,K' the nuclear statistical weighting factor, whereas EjiKi is given by (3.35b). We apply the rotational structure factor (3.40) to bi-atomic molecules (CJ, when K = 0

Fv(Jf,Ji) = (hB/35kT) exp [-hBJ'(J'+ l)/kT]

(3.40a) and molecules having the symmetry C3v

Fv(JfKf,J'K') =[&,K,/35(4Ii2+4I'+ I)] x [AB2h3/,rr(kT)3]1'2 exp (-E,,l/kT)(2Jf

+ 1)

(3.40b) The spectral density distributions S,(Aw,), calculated from eqs. (3.40a) and (3.40b) by BANCEWICZ, KIELICHand OZGO[1975], are plotted in Figs. 3.2 and 3.3. The purely rotational band distributions are strongly dependent on the value and sign of the hyperpolarizability tensor components

190

1111, § 3

MULTI-PHOTON SCAIITRING

1 I I

0

0

50

'do

1

Fig. 3.2. Rotational structure of the spectrum calculated theoretically (BANCEWICZ, KIELICH and OZGO[1975]) for the molecule CO applying hyperpolarizability component values of (a) O'HAREand HURST[1967], and (b) HUSHand WILLIAMS [1972].

b:zT for linear molecules. Herein we see an experimental method for checking the theoretically calculated quantum-mechanical values of the components b$&. It should also be stressed that the cross sections da(20), calculated for CO and NH3, are considerably in excess of those of CH,, for which MAKER[1966] performed observations of the rotational hyper-Rayleigh line structure (Fig. 3.4). The latter circumstance should be an encouragement to further experimental studies of the rotational structure of three-photon scattering processes in molecular gases. (ii) Vibrational hyper-RayIeigh and hyper-Raman lines With regard to eqs. (3.36a) and (3.37), we can carry out the summation in (3.39) over all the permitted rotational transitions J'K' t J'K', thus

111, § 31

191

SCATTERING BY FREE MOLECULES

I I

1 I

(6.69

I

Fig. 3.3. Rotational structure calculated by BANCEWICZ, KIELICHand OZGO[1975] for the molecule NH,, applying hyperpolarizability components after (a) HUSH and WILLIAMS [1972], and (b) ARRIGHINI, MAESTRO and MOCCIA[1968].

obtaining the integral intensities of the purely vibrational lines in threephoton scattering Ifivv(2~) = (N~/280)[(20+ ~vtvi)/cI 4Ivgv 2 (2) (7 I( Vfl 63’)(Q) 1 V’)12+ 2 I( Vfl dg“(Q) IVi>12}, (3.41a)

x S

(2)

I k ( 2 0 ) = (Nvi/2520)[(20 + ~ ~ v i )4Ivgv / 2c ] x (7 I( Vf( &)(a)1 V’)(*+ 12 l(Vfl 6 ~ ” ( 01)Vi>12). (3.41b) S

In the harmonic oscillator approximation (3.34a), and with regard to

192 192

[III, 00 33 [III,

MULTI-PHOTON SCATTERING SCATTERING MULTI-PHOTON

E,ul .05 -

METHANE HR 550 PSI C3.8~10~ Pa)

Fig. 3.4. The hyper-Raman spectrum of methane according to VERDIECK,PETERSON, SAVAGE and MAKER [1970]. Horizontal scale is the hyper-Raman shift AG = 2u- us. The predominant line, centered about 100 cm-', referring to the pure rotational hyper-Raman spectrum, was predicted and observed earlier by MAKER [ 19661. The other prominent line, centered about 3050cm-', corresponds to the hyper-Raman shift due to the C-H stretching frequency u3(F2).

(3.34b), the preceding expressions refer to hyper-Rayleigh lines at Q = 0 and to vibrational hyper-Raman lines for Q f 0 when, taking (3.34~)and the selection rules into account, we can write

c (7 \6i:,,lz

1 % 2 w )= (N/280)[(20 T w,)/cI4Z$g:2'fz

+ 2 1663i12},

S

( 3.42a)

Z$,(2w)

= (N/2520)[(20

o,)/~]~Z$g:2)f:

2 (7 16~1~,,,lz + 12 16:3,,lz}, s

(3.42b) where, applying (3.37) and (3.34c), we have introduced the statistical distributions for Stokes and anti-Stokes lines (see LONG[1977]):

fz= +((h/2wm)[1-exp (Thw,/kT)]-'.

(3.43)

KIELICHand OZGO[1973] have calculated the line intensities for the case of right-circularly polarized incident light. At forward scattering (Fig. 3 3 , they obtained for the right- and left-circularly polarized scattered components Z:1+1(2w)

= (N/2520)[(20

r u,)/~]~1:1gy;f;

1(28 16:1),,,12+3 16~~),,,lz), S

(3.44a)

111, 831

SCAWERING BY FREE MOLECULES

193

Fig. 3.5. Observation of the Reversal Ratio, on the angular momentum convention. Under the action of incident light, circularly polarized in the right sense, I + , , two circularly and the polarized components appear in the scattered light, the one right-circular I , other left-circular I - , + , .

Here, it is worth noting that the component (3.44b) is dependent only on the one molecular parameter \b:3),,,I2, providing the opportunity of an independent determination of its value and sign in experiment. The nonzero values of all these molecular parameters have been tabulated for all point groups and vibration symmetries by ALEXIEWCZ, BANCEWICZ, KIELICHand Oico [1974]. CYVIN, RAUCHand DECIUS [1965] have discussed the selection rules for the vibrational transition frequencies of molecules and lattice vibration frequencies of crystals. A new, complete classification of the hyper-Rarnan spectral lines has been given by ANDREWS and THIRUNAMACHANDRAN [1978]. Earlier, OZGO[ 1975a1, and more recently STREY [1980], have proposed a systematic discussion of rotational and vibrational selection rules, as well as a method for the determination of the five irreducible molecular parameters 16g)L\2(for the asymmetric tensor bapy), from appropriate measurements of the intensity and depolarization ratio of three-photon scattering for various states of polarization of the incident and scattered photons. Ozco and KIELICH [1974], as well as OZGO[1975b], have proposed a complete analysis of the polarization state and angular dependence of three-photon scattering, applying methods of Racah algebra. ILYINSKYand TARANUKHIN [1974, 19751 have studied the problem with regard to hyper-Raman scattering,

194

[III, § 3

MULTI-PHOTON SCATTERING

JE

ETHYLENE HR

P

.3-

4

200 PSI (i.~xtobmi 25,433 SHOTS

(0)

E

g .*3

s

gul .OL -

99 .03 E

ETHYLENE PHR CP IN.MRTK)AL PROP VP OUT, HORIZONTAL PROF 200 PSI (l.Lx106 pa) 12650 SHOTS

(b)

f3 .02 8 z

.

fi O

l

I

O -1000

0

J

,

1000

N

2000

3000

A 3cm-’ Fig. 3.6. The hyper-Raman spectra of ethylene observed by VERDIECK, PETERSON,SAVAGE and MAKER[1970] under different polarization conditions: (a) represents non-analyzed output scattering, (b) incident light circularly polarized and scattered light analyzed with horizontal polarization.

resonantially stimulated in IR (see also BLOK,KROCHIK and KRONOPULOS [ 19791). Figure 3.6 shows the hyper-Raman spectrum of ethylene observed by VERDIECK, PETERSON, SAVAGE and MAKER[1970], whereas Fig. 3.7 permits a comparison of the normal Raman spectra of (liquid) carbon tetrachloride and the hyper-Rayleigh and hyper-Raman spectra obtained by FRENCHand LONG[1975]. Our considerations concern electric-dipole scattering only, but can be extended to electric and magnetic multipole transitions on the basis of eq. (3.5). As shown recently by ANDREWS and THIRUNAMACHANDRAN [19791, the contributions from these transitions are particularly important

111, § 31

195

SCATTERING BY FREE MOLECULES

Fig., 3.7. Spectra of liquid carbon tetrachloride: (a) for normal Raman scattering, and (b) hyper-Rayleigh and Stokes hyper-Raman scattering observed by FRENCHand LONG[ 19751 with multi-channel devices. The spectra labelled (1)-(4)correspond to virgin data, obtained for various numbers of channels and laser shots.

-

in the case of chirial molecules, for which, beside the vibrational hyperRaman intensities calculated above, there appear additional cross-terms for electric-dipole c,magnetic-dipole as well as electric-dipole electricquadrupole (and vice versa) transitions. The contributions from electricquadrupole t,electric-quadrupole transitions calculated earlier by KIELICH, KOZEROWSKI, OZGOand ZAWODNY [1974] are insignificant in the visible range, but grow for UV and X-rays. They become significant in second-harmonic Rayleigh scattering, when the latter is permitted even

196

MULTI-PHOTON SCA'ITERING

[III, 8 3

for atoms and centrosymmetric molecules, but is forbidden in the electric dipole approximation. (iii) Vibrational-rotational hyper-Raman lines Returning once again to eq. (3.39), and taking into account the harmonic term of the expansion (3.34a), we get, for the intensity of vibrational-rotational three-photon Raman lines (BANCEWICZ [1976]),

x(2S+l)f+(

Ji)216:lL12 -Kf s K' (3.45a)

(3.45b) Similarly, we obtain for circular polarization (KIELICHand OZGO[19731)

(3.46a)

-Kf s K'

(3.46b)

If a vibration w, belongs to a representation rcm), of dimension k,>1, it is degenerate; then, several normal coordinates QE), j = 1 , 2 , . . . , k, correspond to the same frequency. In such cases the harmonic term in eq. (3.23) is given by the sum CFz, 6ikZ)Q$, where 6:"' is the sth component of the spherical tensor of order k, related to the jth normal coordinate of a mode of the type rn, belonging to the irreducible

111, 531

197

SCATTERING BY FREE MOLECULES

representation rcm) of the molecule. Accordingly, the molecular parameter 16ik,J2 related to the mode in question (neglecting coupling between rotational and vibrational motion) can be calculated from the formula (POULET and MATHIEU [1970]): (3.47) i=1

The nonzero molecular parameters (3.47) resulting from the vibrational selection rules have been tabulated by ALEXIEWICZ, BANCEWICZ, KIELICH and OZGO[1974] as quadratic functions of the Cartesian components of the tensor baPr, for all point group symmetries and for all types of vibrations of symmetric top and spherical top molecules. PASCAUD and POUSSIQUE [1978] have performed a detailed analysis of the vibrationalrotational hyper-Raman spectra of tetrahedral molecules. On defining the depolarization ratio Dv of a spectral line of scattered light as that of its horizontal component IHv,and vertical component Iw, we arrive with regard to eq. (3.49, at BANCEWICZ'S formula [1976]

for vibrational-rotational (as well as vibrational and rotational) lines. It will be remembered that, in linear scattering, the depolarization ratio of any line equals 3/4 (see PLACZEK [1934] and KONINGSTEIN [1972]). From eq. (3.48), we note that, for second harmonic scattering, the depolarization ratio is in general a function of J and K. However, for all rotational lines with AK=*2, &3 the depolarization ratio amounts to 2/3. Of special interest are vibrations for which only one of the molecular parameters (3.47) for k = 3 is nonzero, since here the depolarization ratio of any rotational line is 2/3. At the same time, for such a vibration, provided that it is completely symmetric, the depolarization ratio of the vibrational band or hyper-Rayleigh line is also equal to 2/3 (KIELICH [1964a1). Defining the reversal ratio as l!?1+1(20)/1~1+1(20) (see Fig. 3 . 9 , and with regard to (3.46), one obtains in the case of forward scattering (KIELICHand OZGO[1973]) 2

(3.49)

198

CIII, § 3

MULTI-PHOTON SCATTERJNG

For molecules with the point group symmetries Td, D3,, and C3,,only one parameter 1623),12 differs from zero and the reversal ratio (3.49) assumes the values 15. In general, one has (3.5Oa)

0 =GR‘(2w) c 15.

By comparison, in the case of usual Raman scattering (see LONG[1977]) O~R‘((o)~6.

(3.5Ob)

3.4.2. Four-photon scattering We now apply the tensor (3.31) to four-photon Raman scattering: 1 3 30 )= (Ni/144)[(30 + 0fi)I c ]“I3( c LVpA(30)* c;weq(3

))M gl;\psv

*

(3.51) The tensor has been averaged in Cartesian basis for arbitrary symmetries of cmPA for linearly polarized (KOZIEROWSKI[1970]), as well as [19741). We refrain, however, circularly polarized light (KOZIEROWSKI from adducing these highly complex results but restrict ourselves to writing out the vertical and horizontal component of (3.51) as obtained by OZGO[1975b], in spherical basis for linearly polarized light Ze”(3~)= (N~ji,i/45360)[(3~+ wfi)/C]41e’gc’

x(2Jf+ 1)[63(-zf

+ 36( -Kf ” + 8( -K‘ Jf

ii)21(VfI Zho’(Q) (Vi)12

y

” I( Vfl ZP’(Q) 1 Vi)I2

s K‘

” I(V‘l CY’(Q) I Vi)12},

s K’

(3.52a)

(3.52b) These components define the structure of the rotational lines as well as that of the vibrational-rotational lines at four-photon scattering. The

n1,831

199

SCATIERING BY FREE MOLECULES

matter is well adapted to a discussion similar to that of the components (3.39) of three-photon scattering. Equations (3.52) lead to the selection rules, discussed by IEVLEVA and KARAGODOVA [19671, OZGO[19681, and CHRISTIE and LOCKWOOD [19711 for vibrational transitions, and by OZGO[19751 for rotational transitions. ALEXIEWICZ, OZGOand KIELICH[I19751 have tabulated the molecular parameters \Eg):)2 as quadratic functions of the Cartesian tensor elements c, for all molecular symmetry point groups. Equations (3.52) show that the only permitted rotational transitions are those with A J = 0, *l, *2, *3, zt4 and AK = s. The permitted values of s are to be had from the condition of non-vanishing of the molecular parameters Ic6k,,12 for the vibrational transitions Vf, c Vk 1 under consideration. The branches with AJ=*3, *4 are dependent on the parameters lE:4L12 only. In the case of linear molecules only branches with even A J can occur. Especially easy to analyze are those types of scattering which are dependent on only one molecular parameter (e.g. for K, Y, Kh and Yh), or two (e.g. T, ThrTd, 0 and o h ) . On performing the summation in eqs. (3.52) over rotational transitions J f K ' t YKi, one obtains the integral intensities of the bands due to vibrational transitions V' c V' f1 only:

*

1&,(30)

= (Nv/45360)[(3w

+ ovv8)/cI4I:g:"{

63 I( V'l ELo'(Q) (Vi)I2

+ 1 [ 3 6 I(V'1 Ei2)(Q) IV')l"+8 I(V'( Er'(Q) \Vi)lz]},

(3.53a)

S

1:"(30)

+

= (Nvi/181440)[(3~ O W ~ ~ ) / C4Ivgv ] 3 (3)

x Z ( 2 7 )(VflEk2)(Q) (Vi)I2+2OI(Vf(E',4)(Q) IVi)lz}. (3.53b) S

OZGO[1975b] derived, as well, the vibration band intensities for circularly polarized light

200

[HI, 5 3

MULTI-PHOTON SCAlTERING

Thus, at circular polarization of the incident light wave, the parameter (EL0)/* defining isotropic four-photon scattering does not intervene. Equations (3.53) and (3.54) give for the depolarization and, respectively, reversal ratio of vibrational lines in four-photon scattering processes (OZGO[1975a])

(3.56) from which we have the following ranges of variability for the respective ratios: 0

O(3w

5/8,

0,)

(3.55a)

0 < R ( 3 w To,) s 2 8 .

(3.55b)

The same ranges of variability result as well for elastic scattering, both with regard to the depolarization ratio (KIELICHand KOZIEROWSKI [19701) and reversal ratio (KOZIEROWSKI [ 19741). The decomposition of all tensors c+, in irreducible representations of all point groups, has been given by OZGO and ZAWODNY [1970]. The

3c

2c 1c ,

16

18

, P . ,I

20

22

L' x

,

24 CM-'

,

4 I

26

,

I

28

;

30

Fig. 3.8. Observations of Yu and ALFANO [1975J, representing relative intensities of threeand four-photon scattering versus the frequency vL from diamond upon the passage of intense picosecond laser pulses (20 or more laser shots), with vp, the optical phonon frequency of the diamond lattice.

111, 841

20 1

LINEWIDTH BROADENING

properties of the tensors up to the fourth rank inclusively have been analyzed in full detail for the case of icosahedral molecules by BOYLEand OZGO[1973] and BOYLEand SCHAFFER [1974]. Atoms and molecules of icosahedral symmetry cannot scatter circularly polarized light elastically (rn = O ) , since in their case the parameters lC:2i12 and lZ:',I2 vanish. However, they cause four-photon scattering induced by linearly polarized light, since the parameter IZb")12, occurring in eq. (3.53), is nonzero. Hitherto, spontaneous four-photon scattering by molecular substances has not been observed. The only report by Yu and ALFANO [1975] concerns three- and four-photon elastic and inelastic scattering from diamond crystal upon the passage of intense picosecond laser pulses (Fig. 3.8).

J 4. Linewidth Broadening in Quasi-Elastic Multi-Photon Scattering by Correlated Molecules 4.1 THE ELECTRIC FIELD AND CORRELATION TENSOR OF SCATTERED LIGHT

We consider a macroscopic sample of volume V and electric permittivity E in an isotropic continuous medium of electric permittivity E,. The macroscopic electric field (Maxwellian field) E existing in the sample differs in general from the external field E", acting throughout the surrounding medium. The relation between the two fields is dependent on the structure and shape of the sample; in the particular case of an isotropic spherical sample it takes the form well known from electrostatics &

E"=-

+2&, E=RE. 3 ~ e

(4.1)

If the external field E is sufficiently strong the sample becomes electrically anisotropic and its permittivity is tensorial, sm. Instead of the vectorial relation (4.1) we now have the tensorial formula (KASPROWICZKIELICH[19751) : EZ = R-ET. (4.2) The tensor relating the field components E: and E, is, in general, for a dielectric ellipsoidal sample Rm = & i l [ & e L+ ( ~ c r v- & e L ) L v r I ,

(4.3)

202

[III, 9: 4

MULTI-PHOTON SCA?TERING

where L, is a field depolarization tensor, dependent on the shape of the dielectric sample, and defined so that its trace shall equal unity L, = Lxx+~,,+Lzz=l. In particular, for a spherical sample L, = 6,/3 and the tensor (4.3) becomes %, = (E, +2~,6,)/3~,.

(4.3a)

If, moreover, the sample is electrically isotropic, then (4.3a) becomes an isotropic tensor:

Rm

=R

L,

(4.3b)

where R is given by eq. (4.1). The above holds also for the electric fields E(t) of the incident light wave, and E,(t) of the scattered wave. However, the permittivities now become functions of the frequencies w and w,. We now assume that the scattering sample (volume V) contains N molecules, correlated in time and space. The electric field of the light scattered by the sample, and observed at a large distance R in the surrounding medium, is (4.4) p=l

where, for an isotropic spherical sample, we have by eq. (4.3b),

with ~ ( w , )the electric permittivity of the sample at the vibration frequency o,of the scattered light wave. The electric field strength vector of the light scattered by the pth t), molecule of the sample is, in the wave zone at the space-time point (R,

where Z(&)is the Hertz vector for the pth molecule at the retarded time tp = t - R,/c, and Rp= lR-r,l= R -s

- r,, + - .

*

(4.6a)

rp denoting the radius vector of molecule p, and s the unit vector in the direction of propagation (observation) of the scattered light, R = Rs. Similarly to the integral intensity tensor of scattered light we can

111, 841

203

LINEWIDTH BROADENING

introduce, on the basis of eqs. (4.4) and (4.6), the tensor of timecorrelation of the scattered light electric field (for processes stationary in time) N

1

Pm(R, t ) = $C4lR(ws)lz(

N

z,(b)*

+

(4.7)

Z T ( t q f).

p=l q=l

With regard to the theorem of Wiener and Khinchin, the Fourier transform of the time-correlation tensor (4.7) defines the spectral density m

S,(Ak, Aw) = ( 2 ~ ) - '

dtI&(R, t ) exp (io,t).

(4.8)

The range of applicability of this spectral approach to time-dependent processes has recently been the subject of an analysis by EBERLY and WODKIEWICZ [19771. We now proceed to define the Hertz vector in the electric dipole approximation, taking into account only time-dependent nonlinear components at harmonic frequencies. We thus write in complex analytic representation

Zg(6)= ( 2 " - ' n ! ) - ' R " ( o ) A. .~.&,, x E,,(o, k) -

*

0;)

- EwJo, k) exp [in(k - r;-w6)],

(4.9)

where the positional variables r; and orientational variables 0 ; determining the configuration of the molecule are taken at the retarded moment of time fp. The tensors A ~ , . . . u now m define effective nonlinear polarizabilities, dependent in general on the electric fields of neighboring molecules (KIELICH[1965b, c] and BEDEAUXand BLOEMBERGEN [1973]). By eqs. (4.7) and (4.9), the tensor of time-correlation of the electric field of n-harmonically scattered light is

ZZ(R,t ) = Q , I ~ (

1 1 A Z ~. .a.(rpr . 0,) a* N

N

0

p=l q=l

X

A:. . . T.(rb, 0;)exp [iAS, (rX- rb,]) K

x g,(n),... 4,71...T,

where the parameter 0,

Q,

=

1 2"-'(n!)'

is:

exp (-in ot),

(4.10)

204

[III, 6 4

MULTI-PHOTON SCATTERING

In (4.10), we have introduced the following tensor of the degree of nth order coherence of the incident electric light field (n)

gu, . . , ""7, . . .T,, = (EZl(o,k)

E:"C&, k) ET1(&9 k) * x (IHo,k)12)&". *

* *

ETnf@, k)>E (4.12)

When going over from eq. (4.7) to eq. (4.10) we assumed that, in a first approximation, statistical averaging over the configurations K of the molecules in the scattering medium (denoted by the symbol ( )K) can be carried out independently of the averaging over the states of the incident light field amplitudes (denoted by ( )E) (see LOUDON [1973]). The difference between the propagation vectors of the scattered wave and incident wave amounts to Ak, = k,, - nk,, and its module (Fig. 1.3) amounts to : Ak, =[(k,, -nk,)2+4nkmk, sin2 (0,/2)]"", (4.13) where 8, is the angle between the vectors k,

and k,.

4.2. LINEAR SCATTERING

Although a detailed discussion of the spectral theory of linear light scattering would lie beyond the scope of our present aims, we nonetheless adduce the equations which result from eq. (4.10), in order to provide a simple illustration of certain complex aspects of light scattering on correlated clusters of molecules. Accordingly, eq. (4.10) leads to

'p=l q = l

xexp [iAk * (r: - ri)]) g&) exp ( - i d ) ,

(4.14)

K

where 0, is given by eq. (4,11), with n = 1. A tensor of the second rank decomposes into three irreducible components: isotropic, antisymmetric and anisotropic (Appendix A) A- =

+A:;+

A:;.

Thus, on isotropic averaging (see Appendix B) eq. (4.14) can be reduced to the following form:

IZ(R, t ) = &Q,Z{lOA~(Ak, t) gb-l;"' + SAY(Ak, t) gbt;"+ A:(Ak, t ) gbt;") exp (-id),

(4.15)

111, 8 41

205

LINEWIDTH BROADENING

where we now have to deal with the following time-correlation functions ( h = 0, 1,2): Ar(Ak, t ) =

(

N

N

p=l q=l

0;). A:!(&

A @ :,

f i b ) exp [iAk * (r:-ri)])

K

(4.16) characterizing the statistical-molecular dynamics of isotropic (h = O), antisymmetric (h = l), and anisotropic ( h = 2) scatttering. The tensors of the degree of first order coherence are = (E%JE/(l~I2)E,

g%‘“

gbz;’) = (

~ 1 - E~ , E ~1) E / (~I E ( ~ ) E ,

gbz;” = (36,

(4.17)

(El2-k 3 E S f - 2 ~ ~ E T ) ~ / ( l E l z ) ~ .

If the linear polarizability tensor is symmetric AZB= A;-, antisymmetric scattering vanishes (A t ) = 0), whereas the time-correlation functions of isotropic and anisotropic light scattering become, with regard to eq. (4. l a ,

(c 1 A:=(r:,O:)* N

A:(Ak,

t)=j

N

A&(rb.fib)exp[iAk * (r:-rb)]

p=l q = l

>,

,

(4.18)

( c 1{3A:&:,

1 ” A W k , t ) =3

a:)*

0;)

A:,&,

p = l q=1

-A:Jr:,

OF)* A;&:,

. (4.19)

0:))exp [ihk * (ri-rb)]) K

4.2.1. Isotropic incoherent and coherent scattering To start with, we assume that the polarizabilities of the molecules are not dependent on the distances between the latter (i.e. we assume the approximation of isolated molecule polarizabilities), so that the correlation function of isotropic scattering (4.18) can be written in the form

Ag(Ak, t ) = 3N (a,(’ F(Ak, t ) , with a,

= a:J3

(4.20)

the mean polarizability of the isolated molecule, and

c c exp [iAk (r: -rb)] N

N

F(Ak, t ) = NP1(

p=l q=l

*

(4.21)

206

[In, 9; 4

MULTI-PHOTON SCA7TERING

the intermediate scattering correlation function, discussed in the theory of neutron scattering (see COPLEY and LOVESEY [19751). After VANHOVE[1954], we introduce the space-time binary correlation function: G(r, r’, t ) = Gs(rg, rb, t ) + GD(rB,rf, t ) . (4.22) where the self-correlation function Gs (rg, r;, t ) determines the probability of finding a (selected) molecule p in the point rk at the moment of time t, if it is known to have occupied the point r: at the moment of time t = 0. Similarly, the distinct correlation function GD(r:, rf, t ) expresses the probability of finding a molecule q in the point rb, if the fixed molecule p was in r: at t = 0. The evolution in time of the functions Gs(t) and G,(t) differs according to the time interval considered. Usually, we distinguish three different intervals, corresponding respectively to the short times of molecular collisions t, < s, the intermediate times of molecular < t, < lop6s, and the very long times of hydrodynamical relaxations relaxations th > s. It is important to find a reasonable and physically plausible analytical construction of Gs and, especially, GD In spite of the progress achieved, the problem of time-many-body correlation functions for the different time intervals has hitherto not been solved satisfactorily (see, for example, BERNE[1971], ROWLINSON and EVANS[1975], and EVANS [1977]). Here, of essential interest to us is the interval of times t,, for which one may apply the solution based on the model of d i h s i o n of translational and rotational molecular motion. By having recourse to the Van Hove function (4.22) we can split the intermediate scattering correlation function into two parts (see POWLES [1973]): a self-correlation part, describing incoherent scattering (p = q) Fi,,(Ak, t ) = (exp [iAk . (r: = V-’

- r;)])

exp [iAk (r:

- r;)]

Gs(rE, rf t ) dr: drb

(4.21a)

and a “distinct” part, describing coherent scattering on stochastically correlated molecules (p # q) Fcoh(Ak,t ) =

(

N

)

exp [iAk * (ri - rf)]

qfp

= (p1V)

exp [iAk * (I$-rf)]G&,

rk, t ) dr: dr:

with p being the average number density of molecules.

(4.21b)

111,

8 41

LINEWIDTH BROADENING

207

On the assumption of Einstein and Smoluchowski’s free translational diffusion model, we have (4.23) G&, i‘b, t) = (47TD~f)-~” eXp (-Ilb-l‘:lz/4&t). Thus, the correlation function (4.21a) finally takes the form (4.24) Fi,,(Ak, t) = exp (-(Akl* DTt), where DT is the coefficient of translational diffusion of Brownian particles. The calculation of the coherent scattering function (4.21b) is by no means simple for a lack of the analytical form of the correlation function GD(r:, rb, t). In some cases use can be made of VINEYARD’S convolution approximation [1958] GD(r:, ri, t) = g(rk) Gdr;, rt, t) dr:, (4.25)

I

where g(& is the (equilibrium) radial correlation function of two molecules p and q, the centres of which are distant by riq. The convolution approximation of Vineyard (4.25) has been criticized for a number of reasons La. because it does not lead to the MandelshtamBrillouin doublet, which appears in the hydrodynamical treatment (SINGWI and SJOLANDER [19641). Nonetheless, it is satisfactorily fulfilled within the interval of intermediate times t,, when the solution of the free diffusion equation can be applied to the description of the self-correlation function G,. Applying the correlation functions (4.23) and (4.25), we reduce the coherent scattering function (4.21b) to the following form (cf. NIJBOER and RAHMAN [1966]):

Fco,(Ak, t) = r ( A k ) Fin&Ak>t) involving the integral parameter

(4.26)

(4.27) introduced by ZERNIKE and PRINS[19271 in their theory of X-ray scattering by liquids. 4.2.2. Anisotropic incoherent and coherent scattering In 9 4.2.1 we have proved that, in the approximation of the polarizability of isolated molecules, the dynamics of isotropic light scattering is

208

[III, 8 4

MULTI-PHOTON SCA"ERING

restricted to translational motion of the molecules. We shall now show that, within the same approximation, the anisotropic scattering function (4.19) requires moreover the intervention of rotational molecular motion. However, here, eq. (4.22) can be replaced by generalized correlation functions, involving additionally the molecular orientations 0 :

a',

G(r, r'; 0, 2)

= Gs(r:,

rb;

a:, a;,t ) + GD(r;, rb; a:, a:, t ) . (4.28)

Regrettably, as yet, not much is known concerning the analytical form of eq. (4.28) and hardly anything concerning GD. Nonetheless, by having [1965], one can recourse to a procedure due to STEELEand PECORA expand eq. (4.28) in a series in spherical Wigner functions:

(4.28a) G&,

a:, a;,t ) = 1 C

rfi;

gi&,.

&%(rL, t )

J P r y M , Ja&%

X

r5~~~("n;)b~~.,,~(wnfi)*. (4.28b)

In eq. (4.28b), "0;and "0:determine the orientations of the molecules p and q in a system of coordinates pq, defined so that its positive z-axis coincides with rA = ri - r;. The analytical form of the functions f A , ( r A , t) and gi&p,KdM,(rh, J.6 t) can be specified for a given model of the molecular motions, the simplest model of this kind being that of translational-rotational diffusion. The correlation function of anisotropic scattering (4.19) has to be expressed in the same spherical representation as that used for the distribution function (4.28). With regard to the transformations (3.25) and (3.26), eq. (4.19) becomes

(4.29) where we have assumed, for the sake of simplicity, that the polarizability tensors c?g,in the system of reference of the molecule, are not dependent explicitly on the radial and angular variables of the other molecules. Obviously, the interference factor of eq. (4.29) has also to be written in the spherical representation, given by the Rayleigh expansion (ROSE

111, §

41

LINEWIDTH BROADENING

C19571) exp (ik r) = 4?r

1 iJjJ(kr) Y ~ L RY, ~J L R , ) *

209

(4.30)

JM

where jJ is a spherical Bessel function and the Yh are harmonic functions. The form of eqs. (4.28)-(4.30) is such as to convince us that the time-dependent problem of anisotropic light scattering is still, at this stage, enormously complex and that its effective solution requires the assumption of some model of stochastic molecular motion. (i) Incoherent scattering When dealing with incoherent scattering one is justified in applying the free diffusion model and assuming that the translation motions of the molecules are stochastically independent of their rotational motions. In this case, the expansion coefficients of the function (4.28a) can be expressed as follows: GS(rfPp,t ) exp (-t/7h),

fJ&(rfPP, t ) = 6-

(4.31)

where G&, t ) is defined by eq. (4.23) and 7L denotes the Mth component of the rotational relaxation time of the Jth order which, for the symmetric top, is given by

7h = {J(J + l)Dyl+ M * ( D-~Dy1)}-', ~

(4.31a)

and Dy3 being the principal values of the rotational diffusion tensor E D .

From eq. (4.29), by having recourse to the functions (4.28a) and (4.31), we derive the time-correlation function of incoherent anisotropic scattering (cf. KNASTand KIELICH [1979]) t2

A;(Ak, tIinc= NFi,,,(Ak,

t)

Irig')('exp (-t/&).

(4.32)

M=-2

(ii) Coherent scattering The expansion coefficients of (4.28b) can be expressed in the following [1965]): way (STEELEand PECORA g J d P&&(&, , t ) = exp ( - - t / ~ & )

'5

gk-k,q&(rw) G&, rb, t ) drz, (4.33)

where the equilibrium function of radial-angular correlations for two

210

MULTI-PHOTON SCATTERING

[III, § 4

molecules is, in general, defined as

In eqs. (4.28b) and (4.34), we moreover have to keep in mind the [ 19571): multiplication law for Wigner functions (EDMONDS

Thus, applying the function (4.28b) together with (4.23), (4.33) and (4.34) as well as (4.30) for J = O , we obtain the coherent part of the time-correlation function of anisotropic scattering (4.29) (KNASTand KIELICH [1979]):

(4.35) where we have introduced a generalized STEELEand PECORA[1965] radial-angular correlation parameter:

(4.36)

In the particular case of J = M = M’ = N = 0, it reduces to the parameter (4.27) for isotropic coherent light scattering. If Ak r<< 1 (short-range correlation), eq. (4.36) reduces to the STEELE parameter 119651:

which has been calculated numerically for concrete models of molecular interactions (see, for example KIELICH [1968c, 1972a1, ANANTH, GUBBINS and GRAY[1974], and HBYEand STELL[1977]).

4.3. THREE-PHOTON SCATI’ERING

Let us now apply the tensor (4.10) to three-photon scattering, assuming for simplicity the tensor AZ,, = BZ,,as completely symmetric. We finally

111, § 41

LINEWIDTH BROADENING

211

obtain (see Appendix B):

(4.37) where we have introduced the time-correlation functions

characterizing the molecular-stochastic aspects of second-harmonic scattering in dense fluids. The tensors of the degree of second-order coherence occurring in eq. (4.37) are of the form

We shall give a discussion of these tensors in § 7. When discussing in detail the correlation functions (4.38) and (4.39) we proceed as in § 4.2 for linear scattering. (i) Incoherent scattering For incoherent scattering of second-harmonic light we have, by eqs. (4.38) and (4.39) for J = 1 , 3 (with r, = rb- r:)

On the free translational-rotational diffusion model, for which the

212

MULTI-PHOTON SCATTERING

[III, 8 4

distribution (4.31) is valid, we finally obtain

B;“(Akz, t)inc = NFi,,(Ak2,

16&):’1”exp (-t/&).

t)

(4.42)

M

Taking, on the basis of eq. (4.8), the Fourier transforms of eqs. (4.37) and (4.42) we obtain the spectral expression first applied by MAKER [1970] to determine the relaxation times 7; and 7: from spectral linewidth measurements of “quasi-elastic” second-harmonic light scattering (see Fig. 4.1). ALEXIEWICZ El9751 has extended Maker’s theory to asymmetric top molecules, characterized in general by relaxation times &. The problem

L

Fig. 4.1. Spectral width of “elastic” second-harmonic light scattering observed by MAKER [1970], for N,N-dimethylformamide at room temperature: (a) Vertical and horizontal scattered intensity, the measured points being connected by a smooth line, (b) The molecular parameters @’) and b(3’ derived from the data (a), together with the best fit Lorentzian convolution.

111, § 41

LINEWIDTH BROADENING

213

simplifies considerably if only one of the molecular parameters of eq. (4.42) differs from zero corresponding to one relaxation time, e.g. T: for the molecular symmetries D3, D3hand CSh, or T ; for the symmetries T, Td and D,. (ii) Coherent scattering The coherent parts of eqs. (4.38) and (4.39) have been analyzed in detail by BANCEWICZ and KIELKH [19761. Here, we restrict ourselves to giving the results ( J = 1,3):

Above, the radial-angular correlation parameter has the form (4.36) on insertion of Ak, for Ak. In particular, it is analogical to that derived by [19661for hyper-Rayleigh scattering in liquids. BERSOHN, PAOand FRISCH Obviously, in order to calculate the parameter (4.36), one has to have available the molecular correlation function (4.34) in analytical form. In a satisfactory approximation, one may write (see KIELICH[1972al)

where U, = U(r,, 0,) is the potential energy of mutual radial-angular interaction of two molecules p and q, and has to be expressed by spherical harmonics (GRAY[1968] and MORAAL [1976]). Assuming for U(r,, 0,) in eq. (4.44) intrinsic dipole-intrinsic dipole interaction only, one obtains the first nonzero terms of the parameter (4.36) (BANCEWICZ [1976]):

involving the following radial averages (4.46) accessible to calculation for well defined molecular models (KIELICH [1972a] and STELLand WEIS[1977]). In the approximation considered, we finally obtain, with regard to eq.

214

MULTI-PHOTON SCAlTERING

[III, 8 4

(4.43), the results (BANCEWICZ and KIELICH[19761): (4.43a) (4.43b) which, for integral scattering, go over into those of KIELICH[1968a]. It is of interest to note that, on the model assumed, the coherent scattering parameter (4.43b) is negative; whereas, for incoherent scattering, the respective parameter resulting directly from (4.42) is always positive. Similar calculations of the parameters (4.36) can be carried out for other models of correlated molecules, leading in all cases to a stronger or weaker influence of temperature on the spectrum observed. Studies of

Fig. 4.2. Spectral width of “elastic” second-harmonic light scattering measured by MAKER [1970] for CCl, as a function of temperature from 2” to 67°C.

111, 0 41

21s

LINEWLDTH BROADENING

this kind have been performed by MAKER[1970] for CCl,, comparing his [19671, who results with the earlier integral observations by WEINBERG found a rather weak dependence on temperature (Fig. 4.2). Finally, it may be worth mentioning that ALEXIEWICZ [1976] succeeded in applying Mori’s formalism to the description of the spectrum of hyper-Rayleigh coherent scattering, i.e. as was done by -YES and KIVELSON [1971] for usual Rayleigh scattering in liquids (see also KEY= and LADANYI [1977]).

4.4. FOUR-PHOTON SCATTERING

Applying the tensor (4.10) to four-photon scattering ( n = 3), and as= C z A as completely symmetric, one obtains suming the tensor (see Appendix B)

AzvA

1 z ( R ,t ) = (Q3,/1260)r3{252C,””(Ak3,t) gff O’+9C2”(Ak3, t ) 82’)

+ C2”(Ak3,t ) g y ’ }

(4.47)

exp(-i 3ot).

The stochastic molecular mechanisms are determined by the following three irreducible time-correlation functions:

x C:!&(rfi,

a:)exp [iAk3

*

(rx - r:)])

K

,

(4.48)

216

MULTI-PHOTON SCATCERING

[111, 9: 5

the incident and scattered photons corresponding to the individual correlation functions (4.48)-(4.50) are of the form gEo) = (~3% l ~ ~ l ~ ) ~ / ( I E l ~ ) & ,

gff2’=((36, IE12-5EZET) IE2I2+6(E$ET!El2+EZEfE’ + EuETE*2)lE12)E/(IEIZ)&, gE4’ = (56,(7

]El4-3 )E2I2)IE(’+ 15(7EuEf - 2E$ET)(El4

(4.5 1)

-5(37EZET - 35EuEf) )E2I2 -

30(E,*EfE2+ E,ETP2) lE12)E/(lE12)&.

In the approximation of invariable molecular polarizabilities, the timecorrelation function of isotropic scattering (4.48) can be expressed in the form

C:”(Ak3, t) = 5 N

F(Ak3, t),

(4.52)

is the mean nonlinear polarizability of the isolated where c3, = c::,& molecule, and F(Ak3, t ) the “intermediate” scattering correlation function, which takes the form (4.21) if Ak is replaced by Ak,. The parameters (4.49) and (4.50) are well adapted to an analysis similar to that carried out for the parameters (4.38) and (4.39). However, with regard to volume, we refrain from pursuing the subject.

(I 5.

Cooperative Three-Photon Scattering

5.1. FLUCTUATIONAL VARIATIONS OF THE NONLINEAR MOLECULAR POLARIZABILITIES

In § 4, when considering multi-photon scattering processes in dense media, we wrote the Hertz vector in the form (4.9), where A ~ , . . . u , ( r p , O nstood p ) for the tensor of a certain effective nonlinear polarizability, differing in general from the polarizability a g , . . . un(O,) of the isolated molecule. The difference is due to the fact that, in statistically inhomogeneous media, the nearest neighborhood of a molecule presents regions of quasi-ordering engendered by various mechanisms of a microscopic or semi-macroscopic nature. We shall not, however, consider effects of short-range interactions, but shall concentrate essentially on the changes in polarizability of the molecule caused by fluctuations of the

111, B 51

COOPERATIVE! THREE-PHOTON SCATTERING

217

long-range electric multipole fields of the molecules surrounding it (see, and SPE for example, KIELICH[1965b, 1972al and VAN KRANENDONK [19771). In a stochastic medium, the Hertz vector can be written in the form

a$) = z(t'>+ SZ(tP),

(5.1)

where Z ( t') describes the electromagnetic properties of an individual microscopic scattering centre, whereas SZ($) moreover describes the variation in Hertz vector due to various processes, such as collisions and many-body interactions, or fluctuations of the molecular electric fields F(r, t ) in time and space. When determining the variations SZ($), characterizing the stochastic regions of correlated scatterers, we shall restrict our considerations to contributions from long-range electric fields F(r, t ) . At the centre of a molecule p, the field due to the polarized electric multipoles of the N- 1 surrounding molecules is (KIELICH[ 1965bl) N

F(")(rp, tp) =

-

1 1 (-1)"1[(2nl

- l)!!]-'(")T("~'(rps, w)[nl]M$)(rs, t s ) ,

s2p nl=l

( 5.2) where the tensor of rank (n+n,) W ) =v ;;'V;;-' (n)FnJ(rps,

[VpVs- (w/c)*U]r;2 exp [i(w/c)rps] (5.3)

describes the (2"-pole)-(2"1-pole) interactions between the molecules p and s, separated by a distance rps. The 2"-pole electric moment of (3.5a), induced in molecule p by the total electric field EF)(rL)+F("'(rp, tp) is, in h-order approximation (KIELICH [1965b]), j@)(rp,

$)(h)

5

1 " . . . C,,.,,h, (,)A(''+. . '+%) (rb, 0;) h!nI=l nh=l X[n,+* * .+nh][Epl'(r~)+F'"l'(r,, t,)]' * * (5.4)

=-

x [E',"")(r~)+PQ(r,, t,)]

c,,

with . . . ,h = [ ( 2 4 - 1)!!]-' * . * [ ( h h - I)!!]-'. The expressions (5.2) and (5.4) can be evaluated by the method of successive approximations, thus leading to the fluctuational contributions of (5.1) (for the linear multipole polarizability, see KIELICH[198Oc]). Since we shall be dealing only with three-photon scattering, given by the parameters (4.38) and (4.39), we are justified in writing the tensor of

218

"11,

MULTI-PHOTON SCAWERING

05

second-order nonlinear multipole polarizability (neglecting spatial dispersion, i.e. assuming the electric field of the incident light wave as homogeneous throughout the region of the molecule) in the form (KIELICH[1980dl) (n)B(l+l)(rr 0;) = (n) 2"

p,

( 1 + 1 ) (0;) + 6 (n)B(1+1) Zw (r;, b2w

a;),

(5.5)

where the tensor (")b92')of rank n + 2 refers to the (isolated molecule) single-body approximation (in particular, for n = 1, it becomes the tensor (')b(l+l) 2w = b:&). The variation caused in the tensor (5.5) by fluctuations of the electric multipole fields contains, in general, many-body contributions and, in the two-body interaction approximation, has the form

ccc N

6 (")B$!!')(r;, 0&

=

m

m

(-1)n2Cn,nz[(n)b~~ni)(0~)

s # p n l = l n2=l

+ (")bF2+')(09 3 [n1]("1)T("2)(rLs)[n2]("2)alf)(0L), (5.6)

where ("?P2)(rLS)= ( " ? Z ' 0 2 ) ( r p s , o)exp (ik rbs). It turns out that, in atomic fluids, essentially important are the threebody interaction contributions +

[(n)b$!w+f"l) (0;) + (n)bF:+l)(Olp)][nllcn?f'("2)(rbs) x [n2]'n2'a>'(0:)[n3] 'n?P4'(r~u)[n,]

' n 4 ' ~ 3 0 k )

N

+

1

(-l)n,+n.,(n)

b2d (n +n2) ( 0 ; ) [ n ,+ n2] ("?I"" )(rps)

U+P

x [n3] (n,)&)(n:) ( n2)T ( n4)(rp,)[n,](n4'al:)(~k)}. t (5.7)

The two- and three-body multipole contributions (5.6) and (5.7), derived above by the molecular-statistical method, are consistent with the results of the quantum-mechanical method developed by PASMANTER, SAMSON and BEN-F~UVEN [19761. In addition to the variations (5.6) and (5.7) due to multipole moments (5.4) of the first and second order, one has still to take into account many-body contributions from multipole moments of the third-order. Here, we shall restrict ourselves to the second-order approximation of

111, I51

COOPERATIVE THREE-PHOTON SCATIERING

nonlinear electric dipole polarizability (KIELICH[ 1968al) 6(l,p+l' t -(I) (I+l+l' (0;) * Fbb)+. * * , ZW (r;, 0,) -

219

(5.8)

where, by expression (5.2), the field of electrically polarizable multipoles (in the absence of external fields) is (KIELICH[1965c])

c 1 (-1)"l N

F(rL) =

-

Cnlcl'T'ni'(rbs>[nllMPi'(O~)

s f p n,=l

x[nJ 'ni'agJ(Ri)[n2] '".'T'n3'(rf.)[n,]~P3'(n:)+* .

(5.9)

Mpl'(0:) denoting the intrinsic 2"l-pole electric moment of molecule s. 5.2. THE TIME-CORRELATION FUNCTION FOR INTERACTING ATOMS AND CENTROSYMMETRIC MOLECULES

We have seen that, in the general case, when the nonlinear polarizabilities have the form (5.5) for n = 1, the time-correlation functions (4.38) and (4.39) split into three parts. The first is related with the intrinsic polarizability of the molecules b:",, and has been discussed in Q 4.3. The second part is related to the cross terms

b:;.';;,(fli)*

SB:&(rb, 0:) + 6B:&(ri, O",* b:&(0b)

and vanishes if b$;,(0;) = 0, as in fact is the case for molecules possessing a centre of symmetry. Obviously, in this case the first part also vanishes. As a consequence of this, for systems composed of centrosymmetric molecules, the time-correlation functions (4.38) and (4.39) take the form

(5.11) Since variations SB$jT exist only in the presence of well defined manybody molecular interactions, the time-correlation functions (5.10) and

220

MULTI-PHOTON SCATERING

[111, 9: 5

Fig. 5.1. Models of coherent three-photon scattering: (a) the electric field F(rPs,t ) of molecule s removes the inversion centre of molecule p, which now produces scattering at o3 (KIELICH[1968a, 1977]), (b) the dipole moment induced at the frequency o2 in molecule s gives rise to the electric field gradient VF(02, rps) in molecule p, which performs an electric SAMSONand dipole-quadrupole transition and produces a photon at o3 (PASMANTER, BEN-REUVEN [1976]).

(5.11) describe solely and exclusively three-photon coherent scattering, caused by cooperative effects in regions of quasi-ordering. We now proceed to discuss the two simple models shown in Fig. 5.1.

5.2.1. Many-body atomic multipole interaction SAMSON and PASMANTER [1974] have drawn attention to the fact that mixed interaction between a dipole induced in one atom, and the electric field gradient produced by the dipole induced in another atom (see Fig. 5. lb), causes three-photon coherent elastic light scattering. This effect is contained in our expansion (5.6) for n = 1, n, = 2 and n2 = 1:

where the fourth-rank tensor b;: :Se describes the second-order nonlinear electric-dipole polarizability induced by a mixed electric dipole-electric quadrupole transition. This tensor is of interest in that it is non-zero for atoms and centrosymmetric molecules. Its nonzero and mutually independent components have been tabulated by KIELICH,KOZIEROWSKI, OZGO and ZAWODNY [1974] for all point group symmetries. The tensor TsE,(rps) describes quadrupole-dipole interaction (2)T(1)(r PS ) * Let us consider the simplest case of atoms and molecules with the point

111, 8 51

22 1

COOPERATIVE THREE-PHDTON S C A m R I N G

group symmetries Y and K. The expression (5.6a) now reduces to N

aEzy(rb) = - 2 C qfarT,p,(rLs),

(5.12)

where a, =$a:= and qz" = &b$:a):ap. For the model considered in the approximation (5.12) the correlation function (5.10) vanishes, whereas the function (5.11) assumes the form N

B?'(Akz, f L h = 4 (

C

c11 N

N

N

qp2'"q~"a;"a:

p = l q = l s # p u#q

x Tap,(r;J* Tap&,) exp bAk2 (ri -.:)I).

(5.13)

The two-body contribution

1 c {1q3"aY12exp Ci& N

B?"Yk2, fLh = 4(

N

*

(r;-rL)I

TaP,Cr:J*TaP,(ris)

p = l sf-p

+ q,'"a;q?a;"

exp [iAk2 *
occurring in (5.13) vanishes for like atoms (a,= a, = a ; qp = qs = q) in the = -TapY(rip). absence of interference effects when Tupv(r;s) The three- and four-body contributions occurring in eq. (5.13) are in general nonzero even if the atoms are of the same species. However, evaluations are difficult, since the time many-body correlation functions are not available (see GROOME, GUBBINS and D m [1976] and KNAST, CHMIELOWSKI and KIELICH[19801). On applying eq. (5.7) to atoms we obtain the three-body contribution of interest to us:

c N

+

a; Ta,,(rbs) T&3(r;u)]

(5.7a)

U;+P

which now gives nonzero contributions to the two correlation functions (5.10) and (5.11). These many-body contributions to three-photon scattering by atomic systems have been analyzed and evaluated numerically by SAMSON and PASMANTER [1974]. Also, GELBART [1973] has considered the possibility of three-photon scattering by three-body clusters of atoms, taking into consideration electronic cloud distortion effects. Contributions

222

NLTI-PHOTON SCATTERING

[III, 9: 5

from long-range interactions of unlike atoms can be calculated as well (see GALATRY and GHARBI[1980]). Under normal non-resonance conditions, such predominantly collisional three-photon scattering effects are rather weak and their observation is beset with difficulties, as shown by the first and, hitherto, only attempts of MAKER[1972] in the liquids nitrogen, argon, oxygen, etc. Hitherto, only observations and studies of two-photon collisional Rayleigh and Raman scattering have been successful (see, for example, MCTAGUE and BIRNBAUM [1971], KNAAPand LALLEMAND [1975], FRENKEL and MCTACUE[1980], and TABISZ [1979]).

5.2.2. Molecules with centre of inversion destroyed by the field of electric multipoles KIELICH,LALANNE and MARTIN[19711 have proposed yet another mechanism leading to three-photon scattering by fluids composed of centrosymmetric molecules. It originates in the changes in polarizability determined by eq. (5.8), and resides in the fact that the time and spatially fluctuating electric field F of the intrinsic multipole moments lowers the symmetry of the molecule; if the latter possessed a centre of symmetry in its ground state, it loses its centre of symmetry under the influence of the field F, and is endowed with the ability to cause three-photon scattering (Fig. 5.la). Generally speaking, the molecular field F not only lowers the natural symmetry of the molecule by way of the nonlinear polarizability of the latter but, due to its very existence, causes the region of shortrange ordering to become anisotropic, with no local centre of symmetry. To provide a simple demonstration of the aforesaid, we neglect in a first ~ 2 in ; ~eq.~ (5.8) approximation the anisotropy of the tensor (')c~~~+')= so that, now, the correlation function (5.10) is nonzero

(5.14) whereas the correlation function (5.11) vanishes. One sees that, on this model, coherent three-photon scattering is in fact caused by the square of the time and spatially fluctuating electric multipole molecular fields, determined generally by eq. (5.9).

111, $51

COOPERATIVE THREE-PHOTON SCAITERINCr

223

(i) Quadrupolar molecules For centrosymmetric molecules with an intrinsic quadrupole moment Mi2' = 0 (neglecting hexadecapole moments) we obtain, by eqs. (5.9) and (5.14),

The function (5.15) leads to a result noteworthy for its simplicity when we consider integral scattering. Thus, for axially symmetric molecules having a quadrupole 8 = 4933 we ,obtain in the approximation of two-body LALANNE and MARTIN [19711) correlations (KIELICH, t?:"(O),h

=5 Ic2w120 ' iv'(r;:),

(5.16)

where the radial parameter ( I ; ) is given by (4.46). Dropping the assumption of isotropicity of the tensor c:&, we obtain by eqs. (5.8), (5.10) and (5.11) in the integral case (KIELICH[1968a, 1980al)

(ii) Multipole molecules The use of the time-correlation function (5.14) is especially justified in the case of molecules of a high degree of symmetry, e.g. tetrahedral, octahedral etc. molecules. Taking into account the multipole field (5.9), we now obtain for two-body correlation (KIELICH[1965c, 1968al)

For quadrupole molecules, eq. (5.19) immediately leads to the result (5.16). For tetrahedral molecules (CH,, CCl,), with an octupole moment

224

[111, § 5

MULTI-PHOTON SCATTERING

MF' = f1123,and a hexadecapole moment '@I regard to eq. (5.19),

= @1133, we

obtain, with

(5.19a) B:W(0)coh = 16 lczw1' N{fl:23(r&10) + (25/7)@:133(r;',")}. Obviously, one should keep in mind that tetrahedral molecules cause, in the first place, incoherent three-photon scattering (3.12). I n octahedral molecules (SF,), the first nonzero moment is a hexadecapole, and eq. (5.19a) leads to (5.19b) Since, at present, we have available the numerical values of the nonlinear polarizabilities czo, as well as the quadrupole, octupole and [ 19661 hexadecapole moments of various simpler molecules (STOGRYN and KIELICH[1972a, 1980a]), the formulae (S.16)-(5.18), (5.19a) and (5.19b) are directly applicable for numerical evaluation. The threephoton cooperative scattering evaluated in this way is, in some cases, by two orders of magnitude greater than the scattering effects caused by the many-body collisional effects discussed in P 5.2.1. The collisioaal contributions to three-photon scattering are accessible to evaluation for tetrahedral molecules, since the numerical values of their multipole polarizabilities are known (AMOS [19791). (iii) The depolarization ratio Going over in eq. (4.37) to integral scattering, we obtain, for the depolarization ratio of three-photon cooperative scattering (5.20) As mentioned in P 3, Bt"=O for incoherent scattering by tetrahedral molecules, so that (5.20) gives D$- = 3 (KIELICH[ 1964a1). For coherent scattering and the model described by eq. (5.19), the depolarization ratio (5.20) gives DCm= 4. We see that, depending on the type of scattering and the molecular model assumed, the depolarization ratio (5.20) takes a value ranging from 4 to 3 (KIELICH,LALANNE and MARTIN [1973]):

4 = 0.11 s DCms$= 0.66.

(5.20a)

Table 5.1 shows that the theoretical relation (5.20a) is satisfactorily confirmed by the existing experimental results for three-photon scattering, both for liquids composed of molecules without a centre of symmetry and for ones composed of centrosymmetric molecules.

III, 551

COOPERATIVE THREE-PHOTON SCA?TERING

225

TABLE5.1 Experimental values of the depolarization ratio D; for linear scattering and DZWfor second-harmonic scattering by molecular liquids Point group

D;

DZW

Authors

HZO CCI,

C,, T,

0.057 0.02

CHCI, C4HI"O C,H,4 CH,OH n-C,H,OH iso-C,H,OH

C,,

C6H12

D,,

0.11 0.038 0.058 0.025 0.025 0.02 0.025

CKI4 CZHZCh C6H6 CS, CH,CN

C,, D6, D,

0.1 16 0.345 0.5 1 0.45 0.65 0.16 0.10 0.17 0.45 0.21 0.12 0.20 0.24 0.17 0.21 0.10

TERHUNE,MAKER,SAVAGE[I9651 TERHUNE, MAKER, SAVAGE[1965] MAKER[19701 KIELICH,LALANNE, MARTIN[197 1 MAKER[1970] MAKER[1970] MAKER[1970] MAKER[1970] MAKER[1970] MAKER[I9701 KIELICH, LALANNE,MARTIN[197 9721 KIELICH,LALANNE,MARTIN[1971] KIELICH,LUANNE, MARTIN[1971] KIELICH,LALANNE, MARTIN[1973, 19721 KIELICH,LALANNE, MARTIN[1973, 19721 TERHUNE, MAKER,SAVAGE[1965]

Liquid

C, C,, C,

DZh

c,

0.25 0.27 0.48

LALANNE, MARTINand KIELICH [19751 have applied the earlier discovered and studied cooperative three-photon scattering (KIELICH, LALANNE and MARTIN [1971, 1972, 19731) to the numerical determination of the quadrupole moments of centrosymmetric molecules from eqs. (5.16)(5.18). Maybe in the near future eqs. (5.19a) and (5.19b) can be used to determine molecular octupoles and hexadecapoles, as hitherto done with success on the basis of collision-induced far infrared absorption (see, for example, GRAY[1971] and BIRNBAUM and COHEN[1975]) and dielectric measurements (KIELICH[1965b, c] and ISNARD,ROBERTand GALATRY [1980]). Quite recently, "ELLEand LAUBEREAU [1980] have observed a sharp increase of second-harmonic generation of picosecond laser pulses, suggesting this increase may be caused by a cooperative mechanism of clusters due to hydrogen bonding in water. In our approach, when dealing with this case, all successive contributions from intrinsic dipoles, quadrupoles and octupoles, the numerical values of which are available for H 2 0 (see STOGRYN [1966]), have to be taken into account in eq. (5.19), and the function B$u(0)coh(containing, like eq. (5.18) quadrupole contributions only) has to be calculated.

226

MULTI-PHOTON SCAITERING

[III, 8 6

W O ~ E J Kand O KIELICH [1975] have analyzed the influence exerted on three-photon scattering by statistical molecular reorientation induced by an intense laser beam. NITSOLOV[1977] has considered the influence of thermal fluctuations of density on three-photon scattering of the dipolequadrupole type. KIELICH,KOZIEROWSKI and LALANNE [19751 have developed a theory of three-photon scattering in solutions consisting of atoms and molecules with electric multipoles. The only hitherto known observations by LALANNE,KIELICH,KOZIEROWSKI and PLANNER[ 19761 concern CC14C6HI2solutions. Studies of solutions of this kind are highly relevant, since in CC14 we deal chiefly with incoherent three-photon scattering, whereas (as we have seen) C6HI2 gives rise to coherent three-photon scattering only. It is worth noting that SCHMID and SCHROTTER [1977] have reported observations of hyper-Raman spectra from CC14-C2C14 solutions. ANDREWS [1979al has analyzed hyper-Raman scattering by oriented molecules in liquid and molecular crystals. Four-photon scattering is less sensitive to cooperative effects, as was confirmed recently by KILDALand BRUECK[19801 when studying thirdharmonic generation in cryogenic liquids.

I 6. Raman Ljne Broadeningin Multi-Photon Scattering (Classical Treatment) The discussions in 994 and 5 make it clear that, in general, not only incoherent scattering on free molecules but moreover - and in some cases primarily - coherent scattering on statistically correlated molecules has to be taken into consideration. Mathematically, however, the description of coherent scattering processes, though formally feasible, is beset with difficulties when it comes to concrete numerical evaluations, chiefly for a lack of the analytical form of the many-body correlation functions. Luckily the situation is quite different with regard to the analysis of Raman multi-photon spectra since, in this case, coherent scattering may be neglected with sufficient accuracy. This is so because the normal vibrations of molecules (even in certain liquids) can be treated as statistically independent (BARTOLIand LITOVITZ[1972] and NAFE and PETICOLAS [19721). Within this approximation Raman line shape studies provide information regarding the translational and rotational motions of the individual molecules in contradistinction to Rayleigh scattering, where

,111, 8 61

227

RAMAN LINE BROADENING

molecular correlations play an outstanding role and may not be neglected, expecially in liquids (see, for example, 0 5 for a discussion of cooperative scattering). As in the case of usual Raman scattering, the normal vibrations are responsible for the vibrational hyper-Raman lines, whereas the translational and rotational motions of the molecules broaden the lines by Awn about the central values no+wm of the latter. In order to prove the aforesaid we have recourse to the classical treatment of § 3.3, which led us to the scattering tensor (3.28). In the spectral case of § 4 it takes the form IETmm(R,t ) =[Nf2/2"-1(n!)2] [(noT ~ , , , ) / CI"g'"' ]~ x((Qhl'>cm(t) KZF"4R, t ) ,

(6.1)

where we have introduced the normalized vibrational autocorrelation function

cw ,

= (Q:*QL)/(~QAI~)

(6.2)

and the tensor of translational-rotational and polarizational autocorrelations

KET"m(R, t ) =

(25+ l)-' ( f h , ( r t ,t ) exp (-iAk, JKLMM'

- rt))

with @EL given by eq. (3.29). On calculating the Fourier transform of (6.3) we obtain, by (4.8), the spectral density tensor of multi-photon Raman scattering with the frequency change Awn = o,- (nw 5 w,) SE'"m(Ak,,

Am,)

=

1

(25+ l)-' fLM,(Akn,Awn)

JKLMM'

x 6 UM):Lm ( n w ) *

6&?Lm(nm)

(6.4)

where we have introduced the following spectral function:

xexp[i(Aw,t-Ak,

r,)].

(6.5)

[1965], for Raman processes, the spectral function (6.5) After GORDON can be calculated separately for molecular motion at short and long times.

228

[III,

MULTI-PHOTON SCATTERING

(i 6

On taking STEELE and PECORA’S solution [1965] for the short-time approximation we can write (6.5) in the form

f&’(Aknr

= aMM* G g A k n ,

Awn),

(6.6)

where, for a molecule of mass m and inertia moment I,

G;IxAk,, Am,,) = {27rmI/kT[ mJ(J + 1) + I JAk,>1]:’

xexp{-~Ao,)2[mI/2kT[ml(J+1 ) + I IAk,(’]B. In particular, for A h + 0 and J = 0, we have lim G 2 A k , , , Am,,) = 2 d ( A m , ) .

(6.6a) (6.6b)

Ak,-O

Thus, in the present case, the total scattered spectrum consists of a sum of Gaussian shape functions centered about Am,, with width determined by the mass and moments of inertia of the scatterer. The properties of the function (6.5) at long times may be simulated by a stochastic (Markov) process. As mentioned in P 4, the sole stochastic model for which the complete analytical form of fLMf(rt,t) is known is that of free diffusion of translational-rotational motions of Brownian particles, giving eqs. (4.23) and (4.31). In spite of numerous objections the model is still in common use due to its heuristic value and simplicity, permitting the expression of (6.5) in the form

fLMr(Akn,Am,,)

=

aMM’L;IxAk,,

(6.7)

Am,,).

Above, we have introduced the generalized Lorentz function for the translational-rotational shape of the spectrum

The half-width of the preceding Lorentzian lines amounts to = 2(1/&+ IAknI2 DT). Under normal experimental conditions I/&>> lAk,I2 DT= 71.’ (e.g. for molecules TT- lo-’ s), whereas &--lo-’’ s and the effect of translational motion of the molecules in broadening the only: spectral lines is determined by their rotation relaxation times

TL

(6.7b) where, now, ( A U , , ) ~ ,=~ I/&. In fact, the spectral functions (6.5)-(6.7) are applicable both to multiphoton Raman scattering processes and to the incoherent Rayleigh scattering processes discussed in $ 4 . In this sense, the applicability of the

111,s 61

RAMAN LINE BROADENING

229

diffusion model (6.7) can be justified by arguments put forward by STARUNOV [1965], who considers the broadening of the central part of the spectrum of the depolarized component of Rayleigh scattering, adjacent on the central line and extending to 3-15 cm-’ on both sides of the latter, to be due to rotational Brownian motions. All the other processes taking place in the liquid (in short-time approximation) affect the shape and fine structure of the wings. Maybe, in some cases, it would be profitable to replace the model of continuous diffusion (4.31) by some other model of molecular motion, e.g. that of “diffusion by jumps” (see VALIYEVand IVANOV[1973]), or the J - or M-diffusion model (see, for example, STEELE [1976] and MCCLUNG [1977]). However, distribution functions for jumpwise diffusion models applicable to molecular spectroscopy are available, strictly speaking, for the spherical top only; in the long run, they too lead to a Lorentzian spectral distribution, albeit with other relaxation times. Hence, on the free diffusion model and with regard to eq. (6.7), the spectral density tensor (6.4) for the nth order hyper-Raman is, finally, SgFwm(Ak,,Ao,) =

1 (2J+ l)-’L2Ak,, Am,,) \iif,)k(no)l’@.”, JKLM

(6.8) The preceding spectral theory of hyper-Raman scattering differs essentially from the theory of incoherent “quasi-elastic” multi-photon scattering processes of § 4 in two ways. First, the molecular parameters lii~&(no)l’ of eq. (6.8) are defined via the nonzero components of the spherical nonlinear Raman polarizability tensor, related to the normal vibration Q, of frequency om. Their selection rules are dependent on the of the vibration considered (cf. symmetry of the molecule and type eq. (3.47)). Second, to determine the Fourier transform of (6.1), one has to be in a position to separate the intrinsic shape of the natural vibrational line, determined by the correlation function (6.2), from the observed nth order Raman spectrum. The vibrational autocorrelation functions (6.2) are accessible to determination from IR absorption and usual Raman scattering measurements by measuring the spectral distribution, whence the part related with vibrations of the molecule can be separated (NAFIEand PETICOLAS [ 19721): t m dt C,,,(t) exp (iAw,,,t). (6.9) c m ( A o m ) = L2T This Fourier transform is a function of the intrinsic vibrational line

230

[III, § 6

MULTI-PHOTON SCATTERING

shape. Its independent determination from three- and five-photon scattering is not possible because, as we know, processes with even harmonics produce no isotropic scattering; the latter, however, does take place in processes involving odd harmonics. Hence, having available the total correlation tensor (6.1) or Fourier transform of the hyper-Raman scattering spectrum observed and, independently, the vibrational functions (6.2) or (6.9), one is able to determine the translational-rotational tensor (6.3) and its Fourier transform (6.4) or (6.8). KIELICH, KOZIEROWSKI and OZGO[19771 have proposed a more general treatment of the problem taking into account, among other things, the fact that one has to consider the Fourier transform of the total tensor (6.1) which, generally, is not a product of the spectral density tensor (6.4) or (6.8) and Fourier transform of the vibrational functions (6.9). In other words, we have in general a convolution of the intrinsic vibration line shape with the translational-vibrational spectrum (see, for example, BARTOLI and Lrrovm [1972]). Equations (6.1)-(6.8) are applicable to two-, three-, four- and morephoton Raman scattering. 6.1. THREE-PHOTON RAMAN SCATTERING

For the sake of simplicity, we shall discuss the spectral density tensor (6.8) only. In the three-photon case it becomes

@z}.

+ 3L3Ak2,Am2) 16g):m12

(6.10)

Here, we have assumed the nonlinear polarizability tensor 6g)k in the simpler, completely symmetric form bK):m. We note that the hyper-Raman line shape (6.10) is in general a superposition of several Lorentz lines (6.7a), among which one can distinguish spectra for J = 1 and J = 3. If the anisotropy of the rotational diffusion tensor in (4.31a) is considerable for the molecule under consideration, several lines with different M-values appear within the same value of J. The nonzero molecular parameters \6g):mlz for all groups of molecular symmetry and all types of vibrations, active in hyper-Raman, BANCEWICZ, are to be found in tabulated form in papers by AEXIEWICZ, KIELICH and Ozco [1974] and BANCEWICZ [1976].

111, 561

R A M A N LINE BROADENING

23 1

For linearly polarized incident light, the spectral density tensor (6.10) becomes

(6.11a)

+ 12L3Akz,Aw,) \6g):m\z},

(6.11b)

whereas for light circularly polarized in the right sense

+ 3L;fi(Akz.A o 2 ) 16~):mlz}, S?yTy-(Ak,, Awz)

=71

c

LL(Ak2, A w ~ 16g):m12. )

(6.12a) (6.12b)

M

When studying the rotational motion of the molecules of a liquid, special importance should be attached to those normal vibrations to which only one molecular parameter \6!&):,,lzcorresponds, since in this case the line-width is dependent on one relaxation time & only, and the latter can be determined from eq. (6.7). For example, such is the case of the molecules of C2H6(point group symmetry D6h) and their vibrations of the types Blu, BZuand E2,,, for which eqs. (6.11) and (6.12) lead to the relation sz$-om=3 2 W T O m = 6S20ro'"=2ss2"'"m=4 2SH" +t+1 s 1+t 3sLL16k?:m12, (6.13) where one has to put M = 3 for m=B,, and m=B,,, and M = 2 for m =E2,,.In this way we derive from eq. (6.7) the rotational relaxational times T: or T:. In the case of the group C6hrone can also determine T: for vibrations of the type B, and T ; for E,,. Similarly, TL and TL are accessible to determination for appropriately selected types of vibrations and molecular symmetries. The hyper-Raman lines of molecules without a centre of inversion are much weaker than the hyper-Rayleigh line (cf. Fig. 3.4), and are thus more difficult to observe. However, the outlooks become quite promising in the case of centro-symmetric molecules for which elastic three-photon scattering is forbidden in the electric-dipole approximation. Here, one can observe, solely or chiefly, hyper-Raman lines (see Fig. 3.6); and

232

[HI, 8 6

MULTI-PHOTON SCATTERING

certain of them are forbidden in IR spectroscopy or in the spectroscopy of usual Raman scattering, which provides information on the relaxation times only. Also, the complete absence or faintness of coherent scattering is a factor in favor of hyper-Raman, as compared to hyperRayleigh, spectroscopy.

TL

6.2. FOUR-PHOTON RAMAN SCA'ITEFUNG

On the assumption of a completely symmetric tensor G!$L,,(3w) = EZtrn, eq. (6.8) leads to the following spectral density tensor of four-photon Raman scattering:

SZFw-(Ak3, A03) = Lz(Ak3, Am,)

lZh0),l2

@z

@z + 5 L 3 A k 3 , Aw3) lZg):m12 @z}. 1

+- C {9L&(Ak3, Au3) lE$$):rn12 45 M K

(6.14)

With the values, tabulated by OZGO[1975a1, of the transformation coefficients REpAoccurring in the tensor (3.29), we obtain, by (6.14), for incident linearly polarized light

{

S % Y A k 3 , Aw3) = fLg(Ak3, Au3)

(6.15a)

+ 2 0 L 3 A k 3 , Au3) JcI~):m12},

(6.15b)

and for light circularly polarized in the right sense

S:";?-(Ak,,

1 A 4 =C { 5 4 L 3 A k 3 , Ao3) lZ~):rn12 1260

+ 5 L 3 A k 3 , Am3) lZg):mlz},

(6.16a) (6.16b)

111, $71

233

ANGULAR DISTRIBUTION, POLARIZATION STATES

Equation (6.15a) conveys to us that third-harmonic Raman, like linear Raman, contains isotropically scattered light, related to the parameter I~bo),,,\~.This enables us to separate the contribution to the spectral line shape due to vibrational motion of the molecules (6.9). The component (6.16b) is also of interest, since it involves only one term, permitting the determination of the new rotational relaxation time hitherto inaccessible by other methods. By comparison, in usual Raman the intensity of isotropic scattering can be determined from the relation (BARTOLIand Lrrovrrz [1972])

TR

I;--,=

IW w-_-4

3IH O-wm V

.

(6.17)

Similarly, for third-harmonic Raman we get, by eqs. (6.1), (6.15) and (6.161, the relation (6.18) permitting the determination of the isotropic third-harmonic scattering component, and hence the vibration function (6.2). In particular, for = 0 and eqs. (6.15) and (6.16) lead to the spherical top molecules relation = I -3lw+r1o , = 28I:";Tm 7pu&Fwrn (6.19) with regard to which the relation (6.18) reduces to (see KIELICH, KOZIEROWSKI and OZGO[ 19771): p'-" = I;"V'", - -1 a 3wr o,,, (6.20) 1s 5 HV * The determination of the isotropic component of third-harmonic scattering, in this case, requires the measurement of two components only, as in that of the usual Raman scattering effect given by the relation (6.17).

8.7. Angular Distriiiition and Polarization States of Multi-Photon Scattered Light

7.1. THE SCATTERING TENSORS IN TERMS OF STOKES PARAMETERS

The state of polarization of a plane quasi-monochromatic light wave, propagating along the Z-axis with electric vector

E ( t )= E,(t) + E , ( t ) ,

(7.1)

234

MULTI-PHOTON SCATTERING

[Ill,

(i 7

is usually expressed in terms of the Stokes parameters (BORNand WOLF [ 19681): S o = E2Ex+ E t G , S2 = E$E,

+ ETE,,

S1==EZEx- ETE,,

S3= i(EtE, -EZE,).

(7.2)

These, in fact, are integral Stokes parameters. By analogy, one can introduce spectral Stokes parameters (see PERINA[ 19721). Of the four parameters, only three are mutually independent, since the identity

s; = s: + s; + sg is fulfilled. With regard to (7.2), we obtain (see BORNand WOLF[1968]):

ETE, = $ ( S , + S , ) ,

ETE, = $ ( & - S , ) ,

EZE, =$(S2+iS,),

E;E,=$(S2-iS3).

(7.3)

We apply this relation to express the multi-photon scattered intensities in terms of Stokes parameters. The versors of the incident and scattered wave fields are now, respectively (Fig. 7.1), e=xsin++ei'ycos+, e,

x sin cp + eiS*(ycos 8 - z sin 0) cos cp,

(7.4) (7.5)

Fig. 7.1. Systems For the calculation of the angular distribution and polarization states of scattered light.

111, $71

ANGULAR DISTRIBUTION. POLARIZATION STATES

235

where x, y and z are unit vectors in the direction of the axes X,Y and Z of the coordinate system attached to the vector E = Ee of the incident light wave. Here, obviously, we have e e* = 1 and e, * ez = 1. Equations (7.4) and (7.5) account for all possible states of polarization of incident and scattered light. For example, at 6 = O the incident wave is linearly polarized at an arbitrary angle to the plane of observation Y Z , and we have

-

e = x sin + + y cos +.

(7.4a)

If the y-component of the field is shifted in phase by 6 = *.rr/2, we have for the elliptical polarization e, = x

sin $*iy cos JI.

(7.4b)

With regard to the angular momentum convention a phase shift + ~ / 2 refers to right elliptical polarization of the wave and -m/Z to left elliptical polarization. If, in addition, = .rr/4, then eq. (7.4b) gives, for a circularly polarized wave,

+

e, = T"'(x kiy).

(7.4c)

It is our aim to determine the angular distribution and polarization state of the scattered wave. They are given by the intensity tensor components, measured by the analyzer:

CJt)

= e,,

I-0) ex7,

(7.6)

where the scattering tensor in Cartesian representation is given by eq. (4.10). Thus, the problem reduces essentially to an analysis of the coherence degree tensor (4.12) or, rather, of its components (7.7) where the irreducible components for n = 1, 2 and 3 are given respectively by eqs. (4.17), (4.40) and (4.51). Thus, by (4.37) and (7.6), we obtain the intensity of integral secondharmonic scattering in the form

Iz;(O) = (Q,,Z2/315){7B~"(0) gi?:'+

3B:"(0) .}):g::

(7.6a)

With regard to the aforesaid, we express the polarization tensors of

236

[III, 8 7

MULTI-PHOTON SCATTERING

linear scattering (4.17) in terms of Stokes parameters: gk:;”’

s,)sin’

= (2(so))-’ ((So+

cp +(So- S , ) cos2 e cos2 cp

+ ( S , cos 6, + S3sin 6,) cos 8 sin 2q), gI,~;1)=(2(So))-1(2So-(S,+S,)sinZ cp-(sO-sl) cos2e cos2 cp - (S2 cos 6, - S3sin 6,)

(7.8)

cos 8 sin 2 q ) ,

g&2)=(2(S0))-1(6So+(So+S1)sin2c p + ( S , - S , ) cos2 8 cos2 cp

+ ( S , cos 6, - 5S3sin 6,) cos 8 sin 2q), and, similarly, those of symmetric three-photon scattering (4.40) : gLt:’=(So)-2

( S : , - S : + 2 ( 2 S 2 , + 2 S 0 S , - ~ 2 , ) sin’cp

+ 2(2s; - 2sos, - s:) COS, e COS, cp + 2S0 (2S2cos 6,+ S3sin 6,) cos 8 sin 2q), gL:;3’=(So)-2

( ~ S ~ + S : + ( S ~ + S , S , + sin’cp ~S~)

(7.9)

+ (s;- sosl+ 2s;) COS, e COS, cp + So (S, cos 6, - 7S3sin 6,) cos 8 sin 2cp). We now proceed to consider the polarization parameters (7.8) and (7.9) for various states of polarization of the incident e and scattered e, photons.

7.2. NATURAL INCIDENT LIGHT

Hitherto, no unequivocal model of natural light is available (see, for and CHANDRA [19711). We shall accordingly consider example, PRAKASH some plausible models (WOKEJKO, KOZIEROWSKI and KIELICH[1978]): (a) On the traditional model, natural light is considered to be a superposition of two waves, linearly polarized, or polarized circularly in opposite senses, with amplitudes equal and constant but with uncorrelated phases. This is the equivalent of a single wave with fluctuating direction of polarization and constant amplitude (see, for example, BORN and WOLF[1968]). In this case, by (7.4a), one has for the linear Stokes parameters: (7. lo) (So) = (IEI”), (S,) = (S,) = (S,) = 0, and for the nonzero nonlinear Stokes parameters:

(s;)= 2(S3 = 2(S3 = ( s o y ,

(7.11a)

111, 171

237

ANGULAR DISTRIBUTION. POLARIZATION STATES

so that the parameters (7.8) reduce to the form well known from the literature : g&O’ = $(sin2cp + cos2 cp cos‘ e), (7.8a) ga:;”=$(6+sin2 cp+cos2cp cos’ 0). On the same model the nonlinear scattering parameters (7.9) take the form ga:;” = 1+ 4(sin2 cp + cos2 cp cos’ e), (7.9a) g‘2*3’ e,u = 4 +sin2 cp + cos’ cp cos’ 8. With regard to eq. (7.6a), this leads to the results of CYVIN, RAUCHand DECIUS [1965] (see also KIELICH and KOZIEROWSKI [1972]). (b) Natural light is treated as the superposition of two waves with mutually orthogonal polarizations, equal amplitudes, and independently fluctuating phases (STRIZHEVSKY and KLIMENKO [1967]), so that the linear Stokes parameters have the form (7.10), whereas the nonzero nonlinear ones take the form

(s;)= 2 ( S 3 = 2 ( S 3 = ( S ” ) 2 .

(7.11b)

In this case, the linear parameters (7.8) have the form (7.8a), whereas the nonlinear ones (7.9) assume the following form:

+ cos’ cp cos’ e)}, = ${9+ 4(sin2 cp + cos2 cp cos2 e)},

gat:) = ${1+ 6(sin2 cp

g;;”

(7.9b)

and eq. (7.6a) leads to the formulae of STRIZHEVSKY and KLIMENKO [1967] for the depolarization ratios and their angular dependence. (c) Natural light is a superposition of two waves, orthogonally (linearly or circularly) polarized, with independently fluctuating phases and mutually independent Gaussian amplitudes. One now has, in addition to (7.10), the following nonzero nonlinear Stokes parameters: (7.11~)

(Si) = 3 - 3 3= 3 ( S 3 = 3(S3 = $(So)’.

Equation (7.8a) still holds, whereas the nonlinear polarizational parameters have the form: g:’

)::g:

+ 5(sin2 cp +cos2cp cos’ e), = &13 + 5(sin2 cp + cos’ cp cos2 O)], =1

and, for the depolarization ratios, THIRUNAMACHANDRAN [1977bl relation.

one

has

(7.9c) the

ANDREWS-

238

m r , 51

MULTI-PHOTON SCATITRING

(d) Natural light is a multi-mode light (ALTMAN and STREY [1977] and STREY [1980]). On the assumption that all N incoherent modes have the same intensity, and that N is infinite, this model leads to the same results as the model (c). This is so because, as shown by PRAKASH and CHANDRA [ 19711, if two orthogonally polarized components of unpolarized light are statistically independent, the radiation is necessarily chaotic. We note that, in linear scattering, all four models of natural light lead to identical results; however, nonlinear scattering is strongly modelsensitive and can be considered as a test of the correctness of the natural light model assumed.

7.3. LINEARLY POLARIZED INCIDENT LIGHT

(i) Incident light, polarized in the vertical plane (Ex# 0, E, = 0) is, with regard to (7.2), characterized by the following Stokes parameters:

s"=s,=(E,12,

s2=s3=0,

(7.2a)

causing the polarizational parameters of linear scattering (7.8) to take the form: (7.12) and those of nonlinear scattering (7.9) to become ggy = gv( 2 )(1+8sin2cp), gLt$)=ge)(4+2sin2cp).

(7.13)

These parameters take the following values for the vertical scattered component (cp = 90"): g&o' = (1.2)- 4 (7.12a) 1, gvv gc2,1) vv = 9gv (2) (7.13a) gc$) = 6ge), 3

9

and, for the horizontal scattered component (cp g&y

=0

( 2 1) -

(2)

gHb - gV

I

= OO),

g g )= 3

(7.12b)

g g / = 4g$'.

(7.13b)

In this case, by (7.6a), the depolanzation ratio is given by (5.20). (ii) For incident light, polarized in the horizontal plane (Ex= 0, Ey# 0), the Stokes parameters (7.1) take the values So = - S ,

= IE,

12,

S2

= S3 = 0.

(7.2b)

In this case, the polarizational parameters of linear (7.8) and nonlinear

239

ANGULAR DISTRIBUTION. POLARIZATION STATES

111, 971

(7.9) scattering assume the form: ghtg'

= COS2 Q COS2

6,

g:;&) = g g ( 1+ 8 C O S ~Q C O S ~e),

gz:$) = 3 + COS2 Q COS2 6, g:;$)

=g

3 4 + 2 C O S ~Q C O S ~6 ) . (7.15)

One obtains, for the vertical scattered component

and for the horizontal scattered component (cp g g = cos2 8,

(7.14)

(Q

= 90")

= 0")

ggz' = 3 + cos2 6,

(7.14b)

g'2," HH = gH (2' (I+8cos2e), g~~'=g(H2)(4+2cosz e). (7.15b) Defining the depolarization ratio as Dh" - 12" /p HH,we have by eq. (7.6a) (KIELICH[1968al) :

Oh"(0) =

7B:"(O)+ 12B,2"(0) 7B:"(0)

+ 12B:"(0) + 2[28B:"(0) + 3B;"(0)]

cos2 8

(7.16)

yielding, at perpendicular observation, Dh(90") = 1.

7.4. CIRCULARLY POLARIZED INCIDENT LIGHT

In the case of circularly polarized incident light, eq. ( 7 . 4 ~ leads ) to only two Stokes parameters,

so= \ E + (+~I E - I ~ ,

s3= p-I2 I E + ( ~ . -

(7.17)

When calculating the reversal ratio for incident light circularly polarized in the right sense only (e = +l), one has to put 6, = F7r12, cp = ~ 1 4 and , e, = *1 in eqs. (7.8) for forward scattering. This leads to the following results, well known from the literature, referring to isotropic, antisymmetric, and anisotropic scattering (see PLACZEK [1934]): g?&/g$!&

= tg48/2,

(7.18a)

(7.18b) (7.18~)

240

MULTI-PHOTON SCATITRING

[III, 8 7

Let us return to the nonlinear polarization parameters (7.9). By (7.17), for right-circularly polarized incident light, they take the form gk:?:

= 2gyl(sin2 cp +cos2 8

g::?,)

= g!,?j[5

cos2 cp -sin 6, cos 8 sin 2q),

+ 3(sinz cp + cos'

8 (30s' cp)

+ 7 sin 6, cos 8 sin 291.

(7.19)

(i) Depolarization ratio At 6, = 0, the expressions (7.19) take the following form for the vertical (cp = 90") and horizontal (cp = 0") component, respectively: (7.19a)

from which (7.6a) now leads to the depolarization ratio of circularly polarized incident light (KiELrcH and KOZIEROWSKI [ 19741): D:T(8) =

15B~"(O)+[l4B~"(O)+99B:"(O)] cos' 8 14B:"(0) + 24B:"(O)

(7.20)

In the particular case of tetrahedrally symmetric molecules in the absence of cooperative scattering B:"(O) = 0, and the depolarization ratio (7.19) becomes D:y(e) = ( 5 + 3 C O S ~8)/8. (7.20a) At perpendicular observation of the scattered light (8 = 90") we obtain the value D:";90") = 5/8, to be considered as the upper limit. For cooperative three-photon scattering one has generally B:"(O) >> B:"(O), so that eq. (7.19) leads to the simple result D",";(8) -- cos' 8,

(7.20b)

signifying that, in the cooperative case, vertically scattered (8 = 90") light is completely polarized, whereas light scattered into the propagation direction of the incident wave (8 = 0") is unpolarized. With regard to (7.19a) and (7.19b), the range of variability of the scattered light depolarization ratio for circularly polarized incident light is

111, 171

ANGULAR DISTRIBUTION. POLARIZATION STATES

24 1

(ii) Reversal ratio On putting cp = ~ / and 4 8, = T T / ~in (7.19), one obtains

(7.19b) from which, by (7.6a), one gets, for the angular dependence of the [19741): reversal ratio (KIELICHand KOZIEROWSKI R:y( 8 ) =

8 ) BZ"(0) 56B5"(0)sin4 8/2+3(13+ 1 4 ~ 0 8+3cos2 s 56~:"(0) C O S ~8/2+ 3(13 - 14 cos 8 + 3 C O S ~8) B:-(o) ' (7.21)

In particular, at 8 = O", eq. (7.21) leads to

R"(oo)

45B$'"'O) + 3BS"(O)

= 28B:"(0)

(7.2 1a)

corresponding to the previously discussed formula (3.49). For B;"(O)=O, formula (7.21) leads to ~ 2 , " ; (= 8 )tg40/2.

(7.21b)

In the opposite case of B:"(O) = 0, one obtains R;Y(d) =

13+ 14cos 8 + 3 cos2 8 13- 14 cos 8 + 3 cos2 8

(7.2 lc)

Thus, the reversal ratio (7.21b) is the same as for the case of isotropic linear scattering (7.18a). In particular R;y(O") = 0, meaning that cooperatively scattered light is polarized circularly. Equations (7.21b) and (7.21~) lead to the relation (3.50a).

7.5. FOUR-PHOTON LIGHT SCATTERING

The rapid progress achieved recently in the domain of various fourphoton spectroscopies stimulates us to supplement this article with an analysis of the angular and polarizational properties of light, scattered in the four-photon processes described by the tensors (4.51). With regard to

242

[III, § 7

MULTI-PHOTON SCAlTERlNG

eqs. (7.3) and (7.7), the latter assume the form g&”

+ S,Sl - $1 + S,(S$- S:)]

= (4(~,)~)-~([2S, (Sg

+ [2S0 (sg- SOSl

-

sin2 cp

Sf)- s1 (s;- Sf)]cos2 8 cos2 cp

+ 2(Sg - S:) (S, cos 6, + S3sin 6,) cos 8 sin 2q), g:y’

= (4(s0)3)-’(12S0

cs; - s:)

+[2S0 (13Sg+ 13S0S1-7s;)-

+ [2s,

(13s; - 13S0S1- 7s:)

5S1(Sg- S;)] sin2 cp

+ 5 s1(s;- s:)] COS’ e C O S ~cp

+2[(13Sg+5S:) S2cosS S + ( S g + 5 S f ) S3sin S,] cos 8sin 2q),

(7.22)

g~~p’=(4(S,)3))-’ (20S0(4S2,+3S;)+3[2S0(4Sg+4SoSl+21S$)

+ 3[2S0 (4s;- 4S0S,+ 21s;) 8 sin’ cp + 6[(4S;+ Sf)S2cos 6,

- S,(S; - S?)]sin’ cp

+ S,(S; - S:)] cos’

S3sin 6,] cos e sin 2cp).

- 9(7Sg- 2s:)

(i) Vertical polarization For vertically polarized incident light one has (7.2a), from which the nonlinear parameters (7.22) become

so that, in this case, the expression (4.47) gives a formula for the depolarization ratio DCw analogical to eq. (3.55) (cf. KIELICHand KOZIEROWSKI [19701). (ii) Horizontal polarization With regard to eq. (7.8b), we bring the nonlinear polarizational parameters (7.22) to the form = gH (3) cosz 8 cos2 q,

&I ge=H

= gH (3) ( 3 + 13 cos2 8 cosz Q),

o):g;

= 4 gH (3) ( 5 + 3 c 0 s 2 ~ c 0 s 2 ( p ) .

(3,’)

(7.22b)

Thus, the depolarization ratio of four-photon scattered light at horizontal polarization of the incident light wave is (KIELICHand KOZIEROWSKI

243

ANGULAR DISTRIBUTION, POLARIZATION STATES

In particular, at perpendicular observation D$"(90")

= 1.

(iii) Circular polarization Applying the nonlinear parameters (7.22) to circular polarization of the incident light wave, we obtain g::,?! = 3 gy; (sin' cp + cos' 8 cos' cp -sin 6, cos 8 sin 2cp),

(3 0)-

- 0,

,;g ,

'g+, [14+ 15 (sin' cp + cos' 8 cos' cp)

(3.4)= 5 (3)

g,,+l

+ 27 sin 6, cos 0 sin 2p]. (7.22~)

Insertion of these parameters into eq. (4.47) leads to the following depolarization ratios for the vertical and horizontal component (cf. KOZIEROWSKI [19741):

D:y( 0 ) =

COS' 8 70 c:yo) + 3 [ 18 c;yo)+ 25 cp(o)] , 54 C;w(0)+ 145 Cz'(0)

(7.24)

from which, at perpendicular observation D:";90°)

=

70 Ci"(0) 54 Clw(0)+ 145 C:'"(O) '

(7.24a)

whereas at collinear observation ( 0 = 0") D:y(O") = 1, proving that the scattered light is natural light. Similarly, we calculate with eqs. (4.47) and (7.22~)the reversal ratio of four-photon scattering (cf. KOZIEROWSKI [ 19741) R:";(8) =

216 C2°(0)sin48/2+ 5 (43+ 54 cos 0 + 15 cos' 8 ) C:-(O) 216 Cqo(o)C O S ~e/2 + 5 (43- 54 cos e + 15 COS' e) c:yo) (7.25)

This leads, for observation at 8 = 0", to a result corresponding to the formula of Oigo (3.56), whereas, for observation at 0 = 90", it leads to R:";(9Oo) = 1 for arbitrary media. In cases of incoherent scattering by atoms in their ground state

244

[III, § 7

MULTI-PHOTON S C A m R I N G

C:w(0) # 0 and C;"(O)= C:-(O) = 0, showing that, here, four-photon scat-

tering of circularly polarized incident light cannot occur. In the case of cooperative scattering due to many-body interaction the situation is quite different (see the discussion in 0 5 ) .

7.6. RECIPROCITY RELATIONS

With regard to the definition (7.6), the tensor of n-harmonic scattering (3.13) may be written as follows: (7.26) where, by (7.4a) and ( 7 . 3 , one has for linearly polarized light (8, = 0, see Fig. 7.1): e,

- e = s i n cp sin ++cos 8 cos cp cos

+.

Hence, the four components take the form (see Fig. 3.1)

I& = ( w / c ) ~(F,,,,,+ G,) I: g$', I G / I f i gg) = I z / I ; g$" = ( n ~ / cF,,,,,, )~ IE(e) = ( ~ o I c (F, ) ~ +COS' e G,,,,,)I ; gg).

(7.26a) (7.26b) (7.26~)

Thus, at perpendicular observation, the Rayleigh-Krishnan reciprocity relation holds for all scattered harmonics:

I K / I k g g ) = IrH(9Oo)/If;gg) = (nolc)" F,,,,,.

(7.27)

Equation (7.26), moreover, leads to the following relation between the depolarization ratios of arbitrary scattered harmonics (KIELICH and KOZIEROWSKI [1972]): D r ( 8 )= 0;"(0;"+ (1- D Y ) COS' 8}-',

(7.28)

where

DT = F,,,,,/(F,,,,,+ G,,,,,),

(7.28a)

D P (8 ) = F,,,,,/(F,,,,, +COS' 8 G,,,,,).

(7.28b)

Similarly, for unpolarized incident light, one has the relation (KIELICH and KOZIEROWSKI [ 19721)

D r ( 8 )= DF(90")+ [1- DT(9O0)]COS* 8.

(7.29)

Obviously, in the case of linear scattering, the relations (7.28) and

111, 171

ANGULAR DISTRIBUTION, POLARIZATION STATES

245

(7.29) are fulfilled unrestrictedly ; in addition, the following relation also holds: D",90°) = 207/(1+ DV). (7.30) [1977b, 19781 have shown that, at ANDREWS and THIRUNAMACHANDRAN nonlinear scattering, no immediate relation of the type (7.30) exists between D;l;"and D y , since the circularly-polarized light intensity has to be taken into account. Let us consider the matter more closely for the case of second-harmonic scattering. We obtain from eqs. (7.6a) and (7.9c), for unpolarized incident light: 16%= & [14 B:"(O) + 9Bg"(O)]I&,

I$%(6)

(14 B:"(O) + 39 BP(0)

=

+ 5 [ i 4 B:"(o) + 3 B:"(o)]

(7.31) C O S ~e}I;.

On the other hand, from eqs. (7.6a), (7.13a), (7.13b), (7.15a), (7.15b) and (7.19a) we obtain

I;;

=

& [7 B:"(O) + 2 Bf"(O)]1:

g:",

r$yI$ g:") = I&/I& gg) = I+/I; g p

& [7B:"(0) + 12 B3"(0)], I k X 8 ) = & (7 B:"(O) + 12 Bg"(0)+ 2 [28 Bf"(0) + 3 Bf"(O)]cos2 8 ) I; gg', =

(7.32)

=&{i5 B ~ ( o ) + [B:"(o) I~

+ 9 B;"(o)] C O S ~e} I; g p . Since g:") = gg) = gz' = g'2', Iv = IH = Ic = I", we have, with regard to (7.31) and (7.32),

(7.33)

where gc2)= 2 for chaotic light and g'2' = 1 for coherent light. By (7.32) and (7.33) we obtain the following relations for perpendicular observation: 1&%(90") 1+ D&"(90") (7.34) De(90") = = 20;" 1% 1+3D$" '

246

MULTI-PHOTON SCA'ITERING

[III, 5 8

This proves that, in second-harmonic scattering, the relation between De(9Oo) and Dzo involves, additionally, DgU(9O0)as determined by eq. (7.20) for 8 = 90". Obviously, on the light model (c), and with regard to (7.16) and (5.20), the relation (7.28) as well as (7.29) are fulfilled, with (cf. ANDREWS and THIRUNAMACHANDRAN [I1977bl): D&"(90")=

14 B:"(O) + 39 Bg"'(0) 84 B:"(O) + 54 B:"(O) '

(7.35)

A similar analysis can be performed for the reciprocity relations of the third-harmonic scattering effects, discussed in P 7.5. For a complete polarizational analysis, it is very important to know the symmetry properties and selection rules for nonlinear responses of matter to circularly or elliptically polarized light, as determined by TANGand RABIN [19711, and OZGOand KIELICH [19763, on the basis of group theory and irreducible spherical tensors.

8 8. Concluding Remarks, and Outlook As we have seen, the investigation of spontaneous multi-photon incoherent scattering processes provides direct information concerning the nonlinear polarizabilities of atoms and molecules. This data can be compared with that derived from studies of optically induced birefringence (KIELICH [1958, 1972a, b] and HELLWARTH [1977]), DC secondharmonic generation of laser beams (LEVINE [1977] and KIELICH[1979]), third-harmonic generation (WARDand NEW [ 1969]), and n-harmonic generation by free molecules (ANDREWS [ 1980]), as well as theoretical calculations (LEUL~TITE-DEVIN and LOCQUENEUX [19751, HAMEKA [ 19771 and SUNDBERG [19771). Particularly valuable are studies of three-photon scattering effects, which are highly sensitive to the ground state symmetry of the molecules. Coherent multi-photon scattering by stochastic inhomogeneous media are a source of information concerning the many-body correlation function. Of special importance are cooperative scattering effects, caused by fluctuations of the molecuIar fields, as a source of data for the electric multipoles and polarizabilities of molecules with various point group symmetries for which the components are known in spherical representation (GRAYand Lo [1976]). This type of cooperative scattering constitutes

111, §81

CONCLUDING REMARKS. AND OUTLOOK

241

a fine example of a self-organizing process (HAKEN[1978]). In the description of cooperative scattering, it appears that the consequences of the Ewald-Oseen “extinction theorem” have to be taken into account explicitly. This by now classical problem (see BORNand WOLF[19681) has and WOLF[1972] as well been considered in a new light by PATTANAYAK as DE GOEDEand MAZUR[1972] and, more recently, by many others (SEIN [1975], PATTANAYAK [1975] and VAN KRANENDONKand SIPE [19771). Statistical-fluctuation processes exert a strong influence on spontaneous multi-photon scattering. In fact, molecular field fluctuations are sometimes its sole origin, for example, in the case of cooperative three-photon scattering effects. Under extremal conditions, nonlinear light scattering is a particularly potent method of investigation, as when occurring in optical inhomogeneities near a critical point, in phase transitions, and in the domain structure of a ferroelectric, as well as in defects of crystal structure. The truly unlimited possibilities provided by laser techniques will permit a fuller investigation of multi-photon elastic, as well as inelastic, scattering processes as sources of data concerning translational and rotational stochastic molecular motion. In addition to the first-order correlation tensor of scattering electric fields discussed in this article, one can analyze second-order correlation tensors of multi-photon scattering (KIELICH, KOZIEROWSKI and TANAS[19751) which provide finer information on the stochastic motions of molecules. Three- and more-photon incoherent spontaneous Raman scattering effects provide new information on the structure of rotational, vibrational and rotational-vibrational spectra, since the latter obey selection rules other than those of usual Raman and absorption in the infrared. Much is to be expected from the coherent hyper-Raman scattering processes recently analyzed by BONNEVILLE and CHEMLA [1978], BJARNASON, HUDSON and ANDERSEN [ 19791, and BJARNASON, ANDERSEN and HUDSON [1980]. To keep within the space allotted to the present review, we refrain from discussing the theory of multi-photon scattering by molecular crystals and, in general, solids. The subject has to be dealt with by the methods of crystal lattice dynamics discussed in the comprehensive monograph of BIRMAN [1974], and applied to infrared absorption and Raman scattering. We have refrained from an analysis of non-degenerate multi-photon scattering processes in which the scattering frequencies are given by sum

248

MULTI-PHOTON SCA?TERING

[III, 8 8

frequencies (1.5) or difference frequencies (see KIELICH [1964b, 1965a, 1966b], PERINOVA, FERINA, SZLACHETKA and KIELICH[19791, ANDREWS [1979b], and MANAKOV and OVSIANNIKOV [1980]). Nor have we considered the hyper-parametric scatterings discussed in the monograph of KLYSHKO [ 19801 (see also SCHUBERT and WILHELMI [ 19801). We hope to have given a detailed analysis of spontaneous hyperRaman scattering processes. However, the work on its stimulated electronic counterpart in metal vapors, initiated by BADAWAN, IRADJAN and MOVSESJAN [1968] and YATSIV, ROKNIand BARAK[1968] and developed in the past few years (VREHEN and HIKSPOORS [1977], COTTER,HANNA, TUTTLEBEE and YURATICH [1977], REIF and WALTHER [1978], HARTIC El9781 and BERGER [1978]), should not be left unmentioned. This novel type of scattering has already been applied for obtaining infrared and far infrared radiation (KIM and COLEMAN [1980]), as well as in studies of ultra-short laser pulse propagation in nonlinear media (HERMAN and THOMPSON [1981]). DNEPROVSKY, KARMENIAN and NURMINSKY [ 19721 and PENZKOFER, LAUBEREAU and KAISER[19731 observed stimulated hyperRaman scattering in water. Perhaps, too, studies of higher-order Brillouin scattering can achieve a similar status, as suggested by BAROCCHI [1971]. Since multi-photon scattering processes are of a stochastic nature (GABRIEL [19731 and SPOHN[19801) their complete quantitative description has to include the statistics of matter as well as the statistics of the radiation field in conjunction with a model of its state of polarization. When treated on a quantum-theoretical basis, the statistical and polarizational properties of the electromagnetic field should be described in terms of the nth order correlation tensors introduced by GLAUBER [1963]. They represent a generalization to quantal fields of the correlation tensors of WOLF[1954] and MANDEL and WOLF[1965] for classical fields. In the quantal case, the polarization density matrix for n photons can be extracted from the nth order correlation tensors and then put in a relationship with the Stokes parameters (ATKINS and WILSON[19721 and TANAS [19791). Although this review does not deal with the results of quantum theories of stimulated Raman scattering, we nonetheless have to mention the fundamental papers of SHEN[1967], WALLS[1973], MCNEILand WALLS [1974], SIMAAN [1975] and, more recently, GUITA and MOHANTY [1980]. Here, the difference between spontaneous and stimulated coherent Raman scattering should be kept in mind (see DESIDERIO and HUDSON [ 19791). SIMAAN [1978] and, independently, SZLACHETKA and KIELICH

111, §81

ACKNOWLEDGEMENTS

249

[1978] have drawn attention to the possibility of photon antibunching occurring in hyper-Raman scattering. Effects of correlation and anticorrelation of incident and scattered photons in the presence of phonon PERINA, SZLACHETKA fluctuations have been analyzed closely by PERINOVA, and KIELICH[19791 and SZLACHETKA, KIELICH,PERINAand P E ~ N O V A [1979, 19801, for various initial statistical properties of laser and Stokes or anti-Stokes modes, e.g. coherent, chaotic or in vacuum state. The dynamics of photon antibunching, in processes of multi-photon scattering as well as in processes of nonlinear light propagation, are largely dependent on the photon polarization state (not only on the field statistics) (ATKINSand WILSON[1972], TANAS and KIELICH[1979], and RITZE[1980]). Under certain conditions, antibunching of laser photons in spontaneous hyper-Raman scattering processes occurs in a similar way to their antibunching in harmonic generation processes (WALLSand TINDLE [ 197 11, KOZIEROWSKI and TANAS[19771, MOSTOWSKI and R Z ~ ~ E W S K I [ 19781, KIELICH,KOZIEROWSKI and TANAS [ 19781, DRUMMOND, MCNEIL, KIELICH,PERINAand PERINOVA [19801, and WALLS[19791, SZLACHETKA, and PERINA [1980]). Although the spontaneously scattered multi-photon intensities are, under normal conditions, weaker than those scattered at stimulation, the labor spent on their observation is nonetheless highly rewarding, due to the importance of the information gained. This is so because the essence of spontaneous light scattering resides in the very foundations of stochastic physics as well as quantum mechanics and electrodynamics (see, for example, KLAUDER and SUDARSHAN [1968], RISKEN[1970], MEHTA[1970], [1980]). Accordingly, the SENITZKY [1978], MANDELEl9761 and SPOHN spontaneous effects discussed provide a test of the correctness of those foundations, e.g. of the purely quanta1 structure of light apparent in the phenomenon of photon anticorrelation (antibunching) (WALLS[19791 and LOUDON [1980]).

Acknowledgements I wish to thank Dr M. Kozierowski, K. Knast M.Sc., M. Kaimierczak M.Sc., K. Pk6ciniczak M.Sc., and Dr L. Wokejko for their valuable remarks and discussions. I express my gratitude to Dr T. Bancewicz for reading the typescript and for his remarks. I am sincerely indebted to K. Flatau MSc. for the English translation

250

MULTI-PHOTON SCATTERING

[III, App, A

of my review and for discussions which have led to a clearer presentation of certain points. I wish to thank all Authors who had the kindness to make their preprints and reprints available to me, thus helping me to write this review.

Appendix A. Irreducible Cartesian Tensors A Cartesian tensor T(")of the nth rank has 3" components Ti,,..h. Since a tensor has to be independent of the coordinate system XYZ,its components have to obey the following transformation law when we go over from one coordinate system to another (rotated) system:

T 11. ...c. = R .

. . * RimamTal ...an-

&loll

(A.1)

The transformation (rotation) coefficients R , are functions of the mutual orientation of the two systems of reference. An important operation on tensors consists in expressing the components of an nth rank Cartesian tensor in weight-5 irreducible representation in terms of their 25+ 1 independent components (COOPE,SNIDER and MCCOURT [19651, COOPEand SNIDER [1970], and JERPHAGNON, CHEMLA and BONNEVILLE [1978]). Thus, we have the representation of a second-rank tensor in the form of the sum of three irreducible components: Tii= 'I$ + T!;)) +'T',;',

(A.2)

7'::)= f T,, Sii

(A.2a)

where

is an isotropic tensor of the second rank (trace of the tensor) obtained by unweighted averaging of (A.2) over all directions in space. The antisymmetric part of the tensor is

TI!' = 4(T.. 11 - T..) 11

(A.2b)

and its anisotropic part (or deviator) is

Ti;) = 4( Ti + Tii) - 4 T k k Sii.

(A.2c)

A tensor of the third rank has 33 = 27 independent components and can be represented as the sum of one pseudo-scalar ( J = O ) , three vectors

111, App. B]

ISOTROPIC AVERAGING OF CARTESIAN TENSORS

25 1

(J=l ) , two pseudo-deviators ( J = 2 ) , and one septor ( J = 3 ) (see, for example, JEFWHAGNON, CHEMLA and BONNEVILLE [19781). In particular, a completely symmetric tensor of the third rank has ten independent components and is the sum of one vector (3) and one septor (7): Tijk

= ?-$j

f

qi3k),

(A.3)

where

T$d = $(6 i j T k l l f 6 j k T i l l + 6 k i T,II), T(3) ilk = T.. ilk - p?) ilk*

(A.3a) (A.3b)

A tensor Tiikl,which in general has 34= 81 independent components, has only fifteen in the completely symmetric case, and is the sum of one scalar ( J = 0 , 2 J + 1= l), one deviator (J= 2 , 2 J + 1= 5 ) , and one nonor (J= 4,25+ 1= 9) (JERPHAGNON, CHEMLA and BONNEVILLE [1978]):

where

with the notation uijkl

= 6 i j 6kI

+ 6 i k 6j, + 6 i l 6 j k .

(A.5)

More complete information concerning irreducible Cartesian tensors and the transition leading from Cartesian to spherical tensors is to be found in the original papers (see, for example, COOPE[1970], STONE [1975], O ~ Gand O KIELICH[1974, 19761, Ozco [1975b], and JERPHAGNON, CHEMLA and BONNEVILLE [1978I).

Appendix B. Isotropic Averaging of Cartesian Tensors If the systems of coordinates in which the tensors Til...hand Tat...%are expressed are both orthogonal, the rotation transformation coefficients Ria are directional cosines, i.e. cosines of the angles between the axes i and a of the two systems of coordinates. Thus, isotropic averaging of

252

[III, APPB

MULTI-PHOTON SCAlTERING

Cartesian tensors reduces to the averaging of products of the directional cosines, and we have, with regard to (A.11, (Ti l...i,,)o=(Rilul *

* *

Ri,,u,,)n T,,...u,,~

(B.1)

where the symbol ( ) n stands for unweighted averaging over all orientations: 1 (B.2) (RiluI* * * Rinu,,),=, j R i l u l . . Rinund0. *

If n s 4, the averaging procedure is trivial, but becomes complicated starting from n = S (KIELICH[1968d] and HEALY [1975]) and has been carried out generally for n = 6 (KIELICH [1961] and MCCLAIN [1972]) and n = 7 (ANDFEWSand THIRUNAMACHANDRAN [1977a]). The results are of a high degree of complexity, and shall not be given here. The procedure of eq. (B.l) is also applicable when it comes to the isotropic averaging of a product of tensors. For example, the following result is obtained for the isotropic average of the product of two secondrank tensors: (AijAkl)o =+A:

&j

6kl +,$A: (& 8ji - a i l

8jk)

+&jAs(3Sik8j1+36i, ajk-28ij

&I),

03.3)

where we have the following irreducible components: A; = A!$ A$ =+A,, A,,, A21 A(’) =1 u p 4Mu, - APJ

(A, - A@)> (A,+ A,) - 4&u A,@.

AZ=A(2) A(Z’=’ or, a, ‘%(A,,+

03.4)

The isotropic average of the product of two symmetric third-rank tensors is (Bijk Blmn)R = &B:

u$!lmn+d@z

(+$ilmn,

03.5)

where we have introduced the isotropic tensors:

fl$ilmn

- aij u k l r n n +

+ aj&u i l m n r = 5[6l (aim a k n + ajn a k m ) + aim ( a k l ajn + a k n + ain (aim & I + a k n i 11- 2u:j’Annijklmn

6 k ujlmn

(B.6)

The irreducible components have the form B:

E

Bgiy B(&

BZ=Bf& BL&=$(S B,,

= SB

PPBB~yV B,,,-3 B,,, Buv).

5

03.7)

253

ISOTROPIC AVERAGING OF CARTESIAN TENSORS

111, APPBl

Similarly, we find the isotropic average of a product of two symmetric tensors of the fourth rank:

with @iiklmn

= 6ij

U k I m n + 6 i k gjImn+

(+jkmn+

gjkln+

sin fljklm?

and irreducible components

c;= c$+5c%*= &C,,,, C,,&€i, (B.lO) ct = C & 8 C & s = %3 co,yy Capss- cso,p C,,ss), 4' c7 = C&S(4) c'upys = &35 c,,, c o p , - 30 cogyy G p s s + 3 c o o , , cyyss>. Isotropic averaging (B.2) becomes highly complicated for n a 8. HOWever, the general solution can be derived for the following isotropic average (KIELICHand KOZIEROWSKI [19721): (%

Re em, . *

eonep,.

*

ep,)a

= 8,

P~P,,P,...~,B,+e,e,qOl~olp1...o,p,,

(B.ll) where we have introduced the unit tensor operators: Poaa,a,...un,n=r2(2n + 3

) [(2n ~ +3) s,,

-

G,~~...~,,,

(+~po,O1...a,Bnl, (B.12)

+3)!!1-' ~o,slg,...a,f3,=[2(2~ e

[3~apalp,...s,pn

-(2n

3,

b g

ga~@~...a,,@J

being a real unit vector. We have made use of (B.ll) when going over from the scattering

254

[III

MULTI-PHOTON SCATTERING

tensor (3.12) to (3.13). Thus, performing on (3.12) the tensorial transformation ( A . l ) and then putting E z , = E*e,,, we have

*

aK...TnEwl * * * Ez" ETl* -

*

a

ETn)n,E

(B.13) a:&...@,,(&,k&,, * * * ea,,e,, . . . e p , h (IEI2"h

and with regard to ( B . l l ) we arrive at (3.13), on having introduced the tensors

into the molecular parameters (3.14). From ( B . l l ) we easily find the useful expression

(eal . . . eane,,

*

e,,>n

= [(2n

+ I)!!]-' u,,,

@,

(B.15)

with the isotropic tensor -

(+a,B*...a,Pn -

L@,

u o r 2 f 3 2 . . .or"@.

f . . . ~a,,,

-

+ sor ,B2 (+a,@,...a,@.

+.

~ a 2 B 2 . . . L I " @ . ~ a,

(B.16)

+

and (2n + l)!!= 1 3 * 5 * . . (2n 1).

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E. WOLF, PROGRESS IN OPTICS XX 0 NORTH-HOLLAND 1983

IV

COLOUR HOLOGRAPHY BY

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

CONTENTS PAGE

6 1. INTRODUCTION .

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

265

6 2. EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY . . . . . . . . . . . . . . . . . . . . . . .

268

. MULTICOLOUR RAINBOW HOLOGRAMS . . . . . .

283

§3

$ 4. VOLUME REFLECTION HOLOGRAMS: NEW TECH-

NIQUES

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

6 5 . PSEUDOCOLOUR IMAGES .

295

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

300

6 6. ACHROMATIC IMAGES . . . . . . . . . . . . . . .

303

. APPLICATIONS OF COLOUR HOLOGRAPHY. . . . .

307

§7

6 8 . CONCLUSIONS. . . . . . . . . . . . . . . . . . . .

321

ACKNOWLEDGEMENTS .

321

REFERENCES

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

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

321

0 1. Introduction One of the goals of workers in holography has been the production of holograms that can reconstruct a bright three-dimensional image in natural colours. While considerable progress was made initially towards this objective, early techniques suffered from several drawbacks. This led to a slackening of interest in this field until a few years ago, when new approaches were explored. This article reviews the problems of early workers and describes recent advances which have made multicolour holography practical.

1.1. BASIC PRINCIPLES

LEITHand UPATNIEKS [1964] were the first to point out that a multicolour image can be produced by a hologram recorded with three wavelengths. The object is illuminated with three beams of coherent light corresponding to suitable primary colours, while three reference beams, one of each colour, are incident on the photographic plate and interfere with light of the same colour reflected from the object. The resulting hologram can be considered as made up of three incoherently superimposed holograms. To view the image, this hologram is replaced in the same position that it occupied during the exposure and illuminated with the three reference beams used to make it. Each beam is then diffracted by the hologram recorded with it to give, at the position originally occupied by the object, a reconstructed image in the corresponding colour. The superposition of these three images results in a multicolour image. However, each hologram diffracts not only light of the wavelength used to record it, but the other two wavelengths as well. As a result, a total of nine primary images and nine conjugate images are produced. Three of these images, as mentioned earlier, give rise to a full-colour reconstructed image at the position originally occupied by the object. The remaining 265

266

COLOUR HOLOGRAPHY

",

81

images resulting from light of one wavelength diffracted by a component hologram recorded with another wavelength (cross-talk images) are formed in angularly displaced positions and, in general, overlap with and degrade the multicoloured image. One of the major problems of colour holography has been the elimination of these cross-talk images. 1.2. THE CROSS-TALK PROBLEM

The cross-talk problem has been analyzed by MANDEL[196S] and by MAROM[1967] and this discussion is based on their treatment. Consider the hologram recording system shown in Fig. 1.1 and assume that all rays are confined to the xz plane, and that the object is illuminated from the left with plane waves at three wavelengths A,, AZ, A3, corresponding to the three primary colours. Let the complex amplitude at a point P in the hologram plane due to a wavefront of wavelength A, that has been scattered by the object be O,(x), while the complex amplitude at the same point due to the reference beam of the same wavelength, is

R,(x) = R,(x) exp [-ik,x sin 81,

(1.1) where k, = 27r/A,, x is the distance of P from the centre of the hologram, and 8 is the mean angle between the object and reference beams. The irradiance at P due to these two wavefronts is

+ Rm(X)('

Irn(X) = lOm(X)

= I o m f x ) I z + [ R m ( ~ ) 1 2 + O ~Rm(x)+Om(x) (~) R:(x).

(1-2)

-

Multicolour Laser

beam

4@

Hologram

Fig. 1.1. Optical setup for recording a hologram of a multicoloured object (MAROM[1967]).

IV, 9: 11

267

INTRODUCTlON

Since light waves of different colours are not mutually coherent, I ( x ) , the resultant irradiance at P due to all three wavelengths, is obtained by summing the irradiances in the individual interference patterns. This implies that, in general, the three holograms can be recorded either simultaneously or sequentially. If we assume linear recording, the amplitude transmittance of the hologram can be written as

where the Pm are parameters determined by the exposure times and the characteristics of the recording medium for the wavelengths used. When the hologram is illuminated once again with the three reference wavefronts, the complex amplitude of the transmitted wave is V ( x )= R ( x )T(x),

(1.4)

where R ( x ) Rn(x). Accordingly, from eqs. (1.4), (1.3) and (1.2), n=l m=l

n=l

1

1

n=l

1

n=l m=l

n=l m=l

The terms of the first two summations represent the directly transmitted beams and a halo surrounding them, while those of the second double summation represent the conjugate reconstructed images which are diffracted to one side of the direct beam. Only the terms of the third double summation, nine in all, which are diffracted to the other side of the direct beam, are of interest. From eq. (l.l),these can be written out in the form n=l m=l 3

=

3

2 pmOm(x)Rn(x)exp [-iknx

sin 61

n=l m=l

x Rm(x)exp [ i k x sin 61.

(1.6)

268

COLOUR HOLOGRAPHY

[IV, cj 2

In eq. (1.6), the three terms obtained by setting n = m correspond to waves of the three primary colours travelling at an angle 8 to the direct beam, which reconstruct three superimposed images of the object. Provided the &, are properly chosen, these will give rise to a virtual image with the same colours as the object. The six remaining terms in the double summation, obtained when n m, are the unwanted cross-talk images arising out of the diffraction of light of wavelength A, by a hologram formed with light of wavelength A,,.

+

I 2. EarIy Techniques for Colonr Holography Several methods have been used to eliminate or minimize the effect of these cross-talk images.

2.1. THIN HOLOGRAMS

We shall examine, in the first instance, methods which can be used with thin holograms. These can be defined as holograms recorded in a medium whose thickness is small enough compared to the fringe spacing for volume effects to be neglected.

2.1.1. Frequency multiplexing The simplest method of separating the cross-talk images from the true images is to encode them on different spatial carriers. To do this LEITH and UPATNIEKS [1964] proposed to introduce the reference beam for each primary colour at a different angle. To reconstruct the image, the hologram is illuminated once again by beams of the same wavelengths at the same orientations to the hologram. The cross-talk images are then diffracted out of the plane containing the directly transmitted beam and the multicoloured image and present less of a problem. Later, MANDEL[1965] showed that such an arrangement was not necessary if the image was viewed over a limited field. From eq. (1.6) it can be seen that, even for collinear reference beams, the wavefronts corresponding to the cross-talk images do not propagate at the same

IV, § 21

269

EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

Multicolour image

f

/ 011; 4 2 2 ; a 3 3

Crosstalk

images

Hologram

Multicolour laser beam

Screen Fig. 2.1. Optical setup for reconstructing a multicolour image free from cross-talk by frequency multiplexing (MAROM[1967]).

angle to the directly transmitted beam as the three wavefronts constituting the true image, but at angles a,,, given by the relation a,,, = arc sin [(&/A,)

sin 81.

(2.1)

Hence, by a proper choice of the value of 8, it is possible to make the angular separations of the cross-talk images and the true images large enough that they do not overlap. In addition, it was shown by MAROM[1967] that when the angle 8 is made sufficiently large, some of the cross-talk images disappear. This occurs when eq. (2.1) has no real solutions, that is to say, when sin 8 > 1. (2.2) Typically, if 8 = 60°, and if A l , A2 and A, are 633, 514 and 442 nm, the three true images are formed at an angle of 60" to the directly transmitted beams, while three cross-talk images corresponding to the directions ( ~ ~ ~ and , a 3 disappear. ~ The remaining three cross-talk images are 3 and (~13=37.2',as shown in Fig. formed at angles al2=44.7O, ( ~ 2 =48.1" 2.1, and can be eliminated by means of a stop, leaving a useful angular field of about *6". (&,/A,)

2.1.2. Spatial multiplexing Another method of eliminating cross-talk, used by COLLIER and BNN[1967], is spatial multiplexing. In this, the hologram is made up

INGTON

270

COLOUR HOLOGRAPHY

[IV, § 2

of many small elements, each of which records fringes formed with only one wavelength. During reconstruction each portion of the hologram is illuminated only with the wavelength with which it was recorded. The simplest way of doing this is to use a mask consisting of a number of red, green and blue colour filter strips placed over the hologram while it is being recorded. This is replaced again, in register, during reconstruction of the multicolour image. If the spectral bandwidth of the filters is narrow enough, such a mask has the advantage that a white light source can be used to reconstruct the multicolour image. A better alternative makes use of the fact that a hologram is formed only on those areas of the recording medium on which an object wave and a reference wave of the same wavelength are incident at the same time. In this case, the object is illuminated with all three wavelengths, while the filter mask is placed in the reference beam and a lens is used, as shown in Fig. 2.2, to image it onto the hologram plate. The reference wavefront is thus divided into discrete areas, each illuminated by a single wavelength, and the holograms formed are spatially separated. To reconstruct the image, the processed hologram is replaced in register with the Mosaic of colour filters

Hologram Lens to image mosaic on hologram

Multiwavelength laser light (Reference beam)

1

Multiwavelength laser Light

Fig. 2.2. Optical system used to record a spatially multiplexed hologram of a multicoloured &ject (COLLIER, BURCKHARDT and LIN[1971]).

IV, B 21

27 1

EARLY TECHNlQUES FOR COLOUR HOMGRAPHY

image of the filter mask and illuminated with the same multicolour reference wavefront. Spatial multiplexing has the advantage that the angle between the object and reference beams can be made fairly small (-1S"), permitting the use of recording materials with relatively low resolution. It has the disadvantage that the contrast and hence the resolution of the image suffers.

2.1.3. Coded reference beams

A third method which has been shown to reduce the effects of crosstalk, even though it cannot eliminate it, makes use of coded reference beams (COLLIER and PENNINGTON [1967]). In this technique, the amplitude and phase of the reference wave are made to vary across the hologram plate in a manner that is significantly different for each of the colours used to make the hologram. One way to do this is shown in Fig. 2.3. Laser light of three wavelengths is combined into a single beam and used Diffusing screen

Multiwave length laser light

/

Hologram plate

Fig. 2.3. Setup for recording a hologram with a coded reference beam (COLLIER,BURCKHARDT and LIN[1971]).

272

COLOUR HOLOGRAPHY

rw, § 2

to illuminate the object as well as a ground-glass diffuser which provides

the reference wavefront. To reconstruct the multicolour image, the processed hologram is relocated in exactly the same position in which it was recorded and illuminated once again with the same reference beam used to record it. If the complex amplitude of each reference wave can be represented by a random function of large bandwidth, its autocorrelation function is sharply peaked. Accordingly when any one of the component holograms is illuminated with the same reference wave used to make it, it reconstructs an image identical to the original object, except that it is superimposed on a weak background of noise which is nearly uniform and extends beyond the image in any direction for a distance equal to the corresponding dimension of the coded source. The cross-talk images, as always, are displaced to either side of the true multicolour image. In addition, since the cross-correlation function of any two of the coded reference waves is a broad-band random function, the cross-talk images are spread out in the image plane as relatively uniform areas of noise. The degree to which the cross-talk images interfere with the multicolour image depends on the dimensions of the diffuser used to code the reference wavefront. If the solid angle subtended by the coding plate at the hologram is small, the cross-talk images are localized and therefore most annoying; if it is fairly large, the cross-talk images are spread out as more or less uniformly distributed noise. A drawback of this technique is that, as in any experiment involving ghost imaging (COLLIER and PENNINCTON [1966]), it is essential to preserve the geometry of the system and replace the hologram after processing in its original position with an accuracy better than half the fringe spacing in the diffraction pattern produced by the coded reference source at the hologram. Only then is an image of the object reconstructed when the coded reference wave illuminates the hologram.

2.1.4. Division of the aperture field Another method achieves coding of the coloured object wavefronts through a multiplexing technique based on division of the aperture field (LESSARD, SOMand BOIVIN [1973], LESSARD, LANGLOIS and BOIVIN [1975]). An optical setup that can be used with three-dimensional objects is shown in Fig. 2.4.

IV, 5 21

EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

273

Fig. 2.4. Experimental arrangement for eliminating cross-talk by division of the aperature field (LESSARD,LANGLOIS and BOIVIN [1975]).

In this, a beam consisting of laser light of three wavelengths is used to illuminate the object, while a collimated reference beam obtained from the same sources is incident on the hologram plate. The object wavefronts are relayed towards the hologram plate, through a coding mask, by a lens. The object is located at a distance 2f from the lens (where f is its focal length), as shown in Fig. 2Sa, so that an image of the object with unit magnification is formed in the vicinity of the hologram plate, which is located at a distance (2f-Az) from the lens. The coding mask is located in the first focal plane of the lens, so that it is imaged at infinity. As shown in Fig. 2.4, the aperture of this mask is divided into nine horizontal slits arranged in three groups of three. Each slit of a group is covered with a narrow-band interference filter so that it transmits only one of the wavelengths used - red, green or blue. After processing, the hologram is replaced in the same position in which it was recorded and illuminated once again with the same collimated reference beam, as shown in Fig. 2.5b. Under these conditions, an orthoscopic real image, which exhibits normal perspective, is reconstructed in the same position which the image projected by the lens occupied in the recording setup. If the lens is now moved to the other side of the hologram at a distance (2f+Az) from it, a number of orthoscopic images are formed at unit magnification on the other side. At the same

274

[IV, li 2

C O M U R HOLOGRAPHY

Film

iI

f

z

= 2f -

A d

-2f

(a)

Real image /

\

--\ J

-\ I

jJ I

LII

(b) Fig. 2.5. Schematic of the optical system showing relative positions of the object, mask, lens and film (a) while recording the hologram and (b) while reconstructing the image (LESSARD, LANGLOIS and BOWIN[1975]).

time, an image of the coding mask is formed in the back focal plane of the lens. Hence, to eliminate the cross-talk images, all that need be done is to insert the coding mask in the back focal plane of the lens so that it coincides with its image. The advantage of this technique is that it produces orthoscopic real images of three-dimensional objects, free from cross-talk, even when the angle between the object and reference beams is small. However, as in the previous case, the relatively complicated setup required for reconstruction limits its application.

2.1.5. Separation of spectra in image holograms Another technique involving separation of the spectra applicable to an image hologram has been described by TATUOKA [1971]. In this, a lens forms a real image of the object close to the hologram plate. The object is

N ,8 21

EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

275

illuminated by laser beams of three different colours, and three reference beams derived from the same lasers are incident on the hologram plate. When the processed hologram is replaced in the same position in which it was recorded, and illuminated with the same three reference beams, eighteen images of the pupil defined by the aperture of the lens are formed, corresponding to the nine virtual images and the nine real images that are reconstructed. In every case, the light flux diverging from a reconstructed image is confined within the corresponding reconstructed pupil. A field lens is now placed in front of the hologram so that these reconstructed pupils are focused on a plane which is at a suitable distance from the hologram in the viewing space. A stop placed in this plane will pass only the three images of the pupil which correspond to the multicolour image. If the diameter of the projected image of the pupil is made larger than the interocular distance, a three-dimensional colour image can be seen.

2.2. VOLUME HOLOGRAMS

While the effects of cross-talk can be minimized or eliminated in thin holograms by a number of techniques, none of them is entirely satisfactory. Apart from more or less close tolerances on the geometry of the setup for reconstructing the image, a penalty is always involved; this is either a restricted image field, a reduction in resolution or a decrease in the signal-to-noise ratio. The first methods to eliminate cross-talk which did not involve such penalties were based on the use of volume holograms. In volume holograms any incident wave is diffracted successively from a large number of regularly spaced fringe planes, and the net diffracted amplitude for a given illuminating wavelength A. is a maximum only when the angles of incidence and diffraction satisfy the Bragg condition

2n A sin Bo = Ao,

(2.3)

where n is the average refractive index of the recording medium, A is the spacing of the hologram fringes and O0 is the angle of incidence (and diffraction) in the recording medium. For small values of A (large values of 6,) and a thick recording material (thickness >> A), a relatively small change in the incident wavelength.extinguishes the reconstructed image.

276

COLOUR HOLOGRAPHY

",

82

In a hologram recorded with several wavelengths in a thick medium, a three-dimensional grating is created containing a set of such fringe planes for each wavelength. When this hologram is illuminated once again with the original multiwavelength reference beam, each wavelength is diffracted with maximum efficiency by the set of fringe planes created originally by it, resulting in a multicoloured reconstructed image. However, the cross-talk images, which are formed by light of one wavelength diffracted from the fringe planes produced by another wavelength, are severely attenuated, since they do not satisfy the Bragg condition. As a result, a volume hologram can give a multicolour image free from cross-tal k.

2.2.1. Volume transmission holograms This principle was first applied by PENNINGTON and LIN [1965] to produce a two-colour transmission hologram. Subsequently, it was extended by FRIESEM and FEDOROWICZ [1966,1967] to multicolour imaging of diffusely reflecting objects, using a three-colour beam for both recording and reconstruction. The optical setup used by them is shown schematically in Fig. 2.6. In it, light of two wavelengths (488 nm and 514 nm) derived from an argon laser was mixed at a beamsplitter with light of a third wavelength (633 nm) from a He-Ne laser to produce two beams each containing light of three wavelengths. One beam was used as a reference beam, while the other was used to illuminate the object. The

Fig. 2.6. Setup used by F-EM and FEDOROWICZ [1966] for recording a multicolour hologram of a diffusely reflecting object in a thick recording medium.

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EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

277

three interference patterns due to the scattered waves from the object and the reference waves were recorded in a thick photographic emulsion. When this hologram was replaced in its original position and illuminated once again with a similar multicolour beam, a multicolour reconstruction of the three-dimensional object was obtained. However, while there were no spurious images arising from cross-talk between the 633 nm wavelength of the He-Ne laser and the two shorter wavelengths, it was found that the spurious images resulting from cross-talk between the 514 nm and 488 nm wavelengths of the argon laser were not completely eliminated. This was because of the closeness of these two wavelengths and the insufficient thickness of the emulsion of the photographic plate used to record the hologram. The diffraction of light at thick gratings has been analyzed by KOGELNIK [1969],who has discussed their wavelength selectivity. He has shown that the diffraction efficiency of a volume phase transmission hologram, is given by the equation q = sin2 ( v 2+ 52)1’2/(1+ t 2 / u 2 ) ,

(2.4)

in which the parameters v and 5 are defined by the relations

eo,

= ~n ( T ~ / A cos ~)

(2.5)

and

5 = -(AA/A,) (27rnd/Ao)tan Oo sin Oo,

(2.6)

where AA is the deviation of the diffracted wavelength from Ao, the wavelength for peak diffraction efficiency at the Bragg angle Oo, d is the thickness of the recording medium, n is its average refractive index and An is the amplitude of modulation of the refractive index. The parameter v defined by eq. (2.5)is equal to half the phase shift of a diffracted ray due to the change in refractive index An. The diffraction efficiency at the Bragg angle attains a maximum of 100 per cent when Y reaches a value of r/2.Over this range of values of v, the diffraction efficiency of the hologram drops to zero when 151= 3. This corresponds to the condition

IAA/Ao( = (A cot Oo)/d,

(2.7)

where A is the spacing of the hologram fringes. The nominal thickness of the emulsion used (Kodak 649F)was 12 p,m; substituting appropriate values for A and Oo shows that while the wavelength difference between the red and green lines is larger than that

278

COLOUR HOLOGRAPHY

[IV, 4 2

required to satisfy eq. (2.7), the separation between the green and blue lines meets it only marginally. On the other hand, with photochromic glass, which is an extremely thick recording medium (d = 1.6 mm), very high wavelength selectivity was obtained, so that cross-talk could be eliminated completely even in transmission holograms made with an angle of only 10" between the object and reference beams (FRIESEM and WALKER[1970]). However, this material requires an exposure lo4 times that for photographic plates.

2.2.2. Volume reflection holograms KOGELNIK'S theory [1969] shows that the wavelength response of a volume hologram narrows with increasing values of the Bragg angle OO. The narrowest wavelength response, and hence maximum attenuation of the cross-talk images, is possible when the angle between the reference and object beams is made close to 180", so that they are incident on the recording medium from opposite sides. The fringe planes then lie approximately parallel to the surface of the recording medium, and the image is formed by reflection of the incident light. The major advantage of the high wavelength selectivity of a reflection hologram is that the image can be reconstructed with a white light source. The diffraction efficiency of a volume phase reflection hologram can be conveniently expressed, as before, in terms of a parameter & !. which is defined by the relation [R

= (AA/Ao)

(2~ndlA") sin 80.

(2.8)

In this case, the difiaction efficiency drops to zero when IcR[=3.5, corresponding to the condition

IAA/AO( = A/d.

(2.9)

With a typical photographic emulsion having a nominal thickness d = 12 pm, and taking h0 = 514 nm, the diffraction efficiency drops to zero when AA -7.3 nm; this is sufficiently small to permit reconstruction of a reasonably sharp image. This technique was first used by DENISYUK [1962] and later by STROKE and LABEYRIE [1966] to produce reflection holograms that reconstructed a monochromatic image when illuminated with white light. Its extension to multicolour imaging followed directly.

IV, 5 21

EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

279

For this, a volume reflection hologram is recorded with several wavelengths, so that one set of fringe planes is produced for each wavelength. When such a hologram is illuminated with white light, each set of fringe planes, because of its high wavelength selectivity, diffracts only a narrow band of wavelengths centred on the original laser wavelength used to record it, giving a multicolour reconstructed image free from cross-talk (LIN, PENNINGTON, STROKE and LABEYRIE [1966], and FEDOROWICZ [1966], STROKE and ZECH[1966]). UPATNIEKS, MARKS 2.3. PROBLEMS WITH EARLY TECHNIQUES

While the techniques of colour holography described in P2.1 and in were all developed several years ago, their practical application lagged; colour holography remained a laboratory curiosity for many years. It is appropriate therefore to take a look at some of the problems which held up further progress. With techniques using transmission holograms the obvious drawback was the need for multiple laser wavelengths (or, at least, equivalent monochromatic light sources) in reconstruction. This difficulty was avoided by the use of volume reflection holograms; however, these had other problems.

P 2.2

2.3.1. Diffraction efficiency The most serious of these was the low diffraction efficiency of conventionally processed reflection holograms recorded on commercial photographic emulsions. This was aggravated by the fact that when more than one hologram is recorded in the same emulsion layer, the available dynamic range is shared between the recordings. As a result, the diffraction efficiency of each recording is reduced by a factor approximately equal to the square of the number of holograms (CHOMAT [1970], COLLIER, BURCKHARDT and LIN[197 11).

2.3.2. Emulsion shrinkage Another problem was the reduction in the thickness of a photographic emulsion layer which occurs with conventional processing, due to removal

280

COLOUR HOLOGRAPHY

[IV, 8 2

of part of the silver halide. Since the fringes in a volume reflection hologram lie nearly parallel to the surface of the emulsion, the main effect of this shrinkage, which can amount to between 15 and 20 per cent, is a decrease in the fringe spacing and a consequent shift in the colour of the reconstructed image towards shorter wavelengths. Thus, a hologram recorded with red light (A = 633 nm) typically reconstructs a green image (A == 530 nm). One method to reduce this shrinkage was to omit the fixing process (LIN,PENNINGTON, STROKE and LABEYRIE [1966]). However, an unfixed photographic emulsion, apart from being unstable, has a high noise level due to scattering by the undeveloped silver halide grains. Lm and Lo BIANCO[1967] found this scattering to be particularly noticeable at the blue end of the spectrum, where an unfixed photographic emulsion exhibited a noise level almost an order of magnitude higher than one that had been fixed. Accordingly, they adopted the technique of swelling the emulsion after fixing to restore it to its original thickness. This was done by soaking it in an aqueous solution of triethanolamine, (CH20HCHJ3N, before drying. Since the emulsion returns to its original state when the triethanolamine is washed out in water, the correct concentration could be determined by a series of trials. A detailed study of this technique has been made by NISHIDA 1119701. He showed that the diffraction efficiency of the hologram increased significantly when emulsion shrinkage was corrected, a conclusion supported by later experiments (DZYUBENKO, PYATIKOP and SHEVCHENKO [1975]). He also showed that the diffraction efficiency of a hologram treated with triethanolamine was affected by changes in ambient humidity. To minimize the effect of these changes, it was necessary to isolate the treated emulsion from the atmosphere by a cover glass sealed at the edges.

2.3.3. Colour rendering A third problem was the optimum choice of light sources for recording the hologram and its effects on colour rendering. Obviously, to be suitable for colour holography, a laser must furnish a reasonable output at a suitable wavelength with an adequate coherence length. The output required is determined primarily by the sensitivity of the recording

rv, 8 21

EARLY TECHNIQUES FOR COLOUR HOLOGRAPHY

281

material and the dimensions of the object and the hologram, while the coherence length must be greater than the depth of the object. The most commonly used lasers for colour holography have been the He-Ne laser (633 nm) and the argon laser which has two strong output lines at 514 nm and 488 nm. Typical power levels available are up to 50 mW with the He-Ne laser, and 1W with the argon laser. The argon laser normally has a much shorter coherence length than the He-Ne laser, but single-mode operation can be obtained relatively easily with an intra-cavity etalon. The range of colours that can be reconstructed with these three wavelengths can be determined by plotting them on the C.I.E. chromaticity diagram (THEOPTICAL SOCIETY OF AMERICA [1953]) shown in Fig. 2.7. In this, points representing monochromatic light of different wavelengths constitute the horseshoe-shaped curve known as the spectrum locus; all other colours lie within this boundary. New colours obtained by mixing any two colours, such as laser light with wavelengths of 633 nm and 514 nm, lie on the straight line AB joining these primaries. When laser light with a wavelength of 488 nm is added to these two components, any colour within the triangle ABC can be obtained. A wider range of hues can be obtained if other laser lines are used, permitting a better choice of primaries. Some of these are listed in Table 2.1. The He-Cd laser line at a wavelength of 442nm is a very attractive blue primary, but it involves the use of one more laser and the power available at this wavelength is limited. A better alternative is the blue line at a wavelength of 477 nm produced by the argon laser. While the power available at this wavelength is less than that at 488 nm, it is adequate for most purposes and results in a significant improvement in colour rendering in the blue and purple regions of the chromaticity diagram, as shown by the broken lines in Fig. 2.7. The krypton laser has attracted some attention since it has outputs at wavelengths of 647 nm, 521 nm and 476 nm, so that, in principle, it could furnish all three primaries. However, the power available at the latter two wavelengths is comparatively low, and close control of the operating pressure of the plasma tube is necessary to ensure stable operation at these wavelengths. On the other hand, commercially available krypton lasers have a much higher output in the red than He-Ne lasers, and single-mode output can be obtained with an etalon. This makes them a

282

COLOUR HOLOGRAPHY

Fig. 2.7. C.I.E. chromaticity diagram. The triangle ABC shows the range of hues that can be produced by a hologram illuminated with primary wavelengths of 633 nm, 514 nm and 488nm, while the broken lines show the extended range possible if the blue primary is replaced by one at 477nm. The chain lines enclose the range of hues which can be reproduced by a typical colour-televisiondisplay.

good choice, in combination with an argon laser, for recording large holograms. While the deepest blues cannot be reproduced with these primaries, the range of hues available is greater than that produced by colour television systems and most colour films. This is also very nearly the range of surface colours encountered in typical scenes (HUNT [19771). Problems with colour distortion can arise with subjects having sharply

N ,§ 31

283

MULTICOLDUR RAINBOW HOLOGRAMS

TABLE 2.1 Laser wavelengths for colour holography Wavelength (nm)

Laser

Typical power (mW"

Colour

442 458 476 477 488 514 521 633 647

He-Cd Argon Krypton Argon Argon Argon Krypton He-Ne Krypton

25 200 50 400 1000 1400 70 50 500

Violet Blue-violet Blue Blue Green-blue Green Green Red Red

*With commercial lasers

peaked spectral transmittance or spectral reflectance curves, due to the need to illuminate them with three discrete monochromatic sources rather than with white light. However, an analysis by KOMARand OVECHKIS [1976] suggests that such a situation is exceptional, since most colours encountered in nature exhibit smooth spectral reflectance curves. An experimental study by NOGUCHI [1973] also confirms that reasonably good colour reproduction is possible with multicolour holograms. In this, volume reflection holograms of four colour transparencies were recorded using four primary wavelengths, three from a krypton laser (477 nm, 521 nm and 568 nm), and one from a He-Ne laser (633 nm). It was found that the red and yellow test patches were reproduced almost perfectly. A slight loss of saturation was observed with the green and blue test patches; this was partly due to the primary wavelengths chosen, though a contributing cause was the greater influence of background noise due to the low transmittance of these patches.

I 3. Multicolour Rainbow Holograms As outlined in $ 2 , even though holographic techniques capable of producing multicolour images of three-dimensional objects were developed at a relatively early stage, their wider application was held up by several problems.

284

[Iv,3: 3

COLOUR HOLOGRAPHY

3.1. THE RAINBOW HOLOGRAM

The first step towards the solution of some of these problems was the development by BENTON[1969,1977] of a new type of transmission hologram capable of reconstructing a bright, sharp, monochromatic image when illuminated with white light. In this technique, part of the information content of the hologram is sacrificed to gain other advantages. What is given up is parallax in the vertical plane; this is relatively unimportant, since depth perception depends essentially upon horizontal parallax. On the other hand, a white light source can be used for reconstruction, and there is a considerable gain in the brightness of the reconstructed image. In BENTON’S technique [1969, 19771, a hologram is recorded of a real image of the object formed by a conventional hologram. The aperture of (a) Reconstruction with monochromatic Light

/Hex

n

Monochromatic Light source

I

I

Image of slit

(b)

Image of object

Reconstruction

with w h i t e light

Image

Fig. 3.1. Reconstruction of the image by a rainbow hologram, (a) with monochromatic light, (b) with white light.

IV, B 31

MULTICOLOUR RAINBOW HOLOGRAMS

285

this hologram is limited by a horizontal slit to eliminate vertical parallax. If the final hologram is illuminated with a coherent source it forms, in addition to an image of the object, a real image of this slit. All the light diffracted by the hologram is concentrated in this slit pupil, as shown in Fig. 3.1. If the hologram is viewed from this point, a very bright image is seen. With a white light source, the slit image is dispersed in the vertical plane to form a continuous spectrum. An observer whose eyes are located anywhere within this spectrum sees an image of the object formed by a narrow range of wavelengths. This image exhibits normal horizontal parallax when the observer moves his head from side to side; vertical motion only produces changes in the colour of the image without vertical parallax. For obvious reasons, these holograms have been called rainbow holograms. Because all the diffracted light from the hologram is concentrated into a fairly small solid angle, the images formed are very bright. Since the wavelength selectivity does not depend on volume diffraction effects, surface-relief holograms, which can be reproduced cheaply, can be used.

3.2. MULTICOLOUR IMAGES WITH RAINBOW HOLOGRAMS

The extension of this technique to three-colour recording (HARIHARAN, [1977], TAMURA [1977], SUZUKI, SAITO and MATSTEELand HEGEDUS SUOKA [1978]) made it possible for the first time to produce holograms that reconstructed bright multicolour images when illuminated with a white light source. For this, three primary holograms (colour separations) are made from the object with red, green and blue laser light. In the second stage these primary holograms are used with the same laser sources to make a single hologram consisting of three superposed recordings. When this multiplexed hologram is illuminated with a white light source, it reconstructs three superposed images of the object. In addition, three spectra are formed in the viewing space. These are, as before, dispersed real images of the limiting slit. However, these spectra are displaced vertically with respect to each other, as shown in Fig. 3.2 so that, in effect, each component hologram reconstructs an image of the limiting slit in its original position, in the colour with which it was made. Accordingly, an observer viewing the hologram from the point where the spectra overlap

286

[IV, 8 3

COLOUR HOLOGRAPHY

rce

displaced spectra

Mu It icolour image

Fig. 3.2. Reconstruction of a multicolour image by superimposed rainbow holograms.

sees three superposed images of the object reconstructed in the colours with which the primary holograms were made. Figure 3.3 shows an optical system which permits both steps of the process to be carried out with a minimum of adjustments. To record the three primary holograms, as shown in Fig. 3.3a, a collimated reference beam is used, and a mirror on a kinematically located, removable mount serves to select the appropriate laser source for each hologram. A red-sensitive plate (Holotest 10E75) is used with the He-Ne laser, while orthochromatic plates (Holotest 10E56) are used to record the green and blue colour separations. For the second stage of the process, the optical system is modified, as shown in Fig. 3.3b. Each of the primary holograms is used in turn with the appropriate laser source to form a real image that serves as the object for one of the component recordings that make up the final hologram. Since a collimated beam is used as reference in the first stage, it is only necessary to turn the primary hologram through 180" about an axis normal to the plane of the figure and replace it in the plate holder; an undistorted real image is then projected into the space in front of the primary hologram. Vertical parallax is eliminated by a limiting slit, a few millimetres wide, placed over the collimating lens with its long dimension normal to the plane of the figure (this corresponds to the horizontal plane in the final viewing geometry). A convergent reference beam is used to record the final hologram; the latter, after it has been processed, is reversed for viewing. When it is illuminated with a divergent beam from a point source, an orthoscopic image of the object is formed, and three dispersed real images of the limiting slit are projected into the viewing space. Multicolour rainbow holograms give bright images with an ordinary tungsten lamp. The images exhibit high colour saturation and are free

IV, 9: 31

MULTICOLOUR RAINBOW HOLOGRAMS

287

Removable mirror

-hologram

(a) Removable mirror

(b)

reference beam

Fig. 3.3. Optical system (a) used to record the three primary holograms of the object, and [1977J). (b) modified to make the final hologram (HARIHARAN, STEELand HEGEDUS

288

COLOUR HOLOGRAPHY

",

03

from cross-talk. Problems with emulsion shrinkage are eliminated, since volume effects are not involved, and shrinkage primarily affects the thickness of the emulsion. However, as with any rainbow hologram, the colours of the image change with the viewing angle in the vertical plane. This change can be utilized effectively in some types of displays. Where necessary, it can be kept within acceptable limits by optimization of the length of the spectra projected into the viewing space and the use of baffles to define the available range of viewing angles in the vertical plane.

3.3. ONE-STEP MULTICOLOUR RAINBOW HOLOGRAMS

A rainbow hologram can also be produced in a single step from real images of the object and the slit produced by an optical system (BENTON [1977]). This results in a considerable simplification of the process. An optical setup for this (CHEN,TAIand Yu [1978]) is shown in Fig. 3.4. An orthoscopic image of the object is formed by a lens, just in front of the hologram. A narrow slit is placed between the object and the focal plane of the lens, so that it is imaged into the viewing space at a suitable distance from the hologram. A diverging reference beam can be used in

tf

'1 A

[ I

Real Hologram

Fig. 3.4. Optical setup for the production of multicolour rainbow holograms in a single step (CHEN,TAIand Yu [1978]).

IV,

P 31

289

MULTICOMUR RAINBOW HOLOGRAMS

this case, corresponding to that used for reconstruction. The final hologram is made directly from three exposures to the three primary wavelengths used. The disadvantage of this technique is that the field of view is limited by the aperture of the lens. To achieve a reasonable field of view, an imaging lens with a large aperture and a relatively small focal length is required; this makes a setup for the production of large holograms prohibitively expensive. While methods have been suggested to minimize this problem (BENTON, MINGACE and WALTER[1979]), a better alternative is to use a large concave spherical mirror. Apart from being free from chromatism, this gives aberration-free imagery on-axis at unit magnification. Since the longitudinal magnification is the square of the transverse magnification, unit magnification must be used if a three-dimensional image is to be free from distortion in depth. The choice of the mirror parameters is a compromise between object size and viewing angle. The greater the radius of the mirror, the larger the image that can be formed without serious off -axis aberrations or variations of magnification with depth, while the larger the numerical aperture, the greater the angle over which the reconstruction can be viewed. A typical setup (HARIHARAN, HEGEDUS and STEEL[1979]) is shown in Fig. 3.5. The mirror used had a diameter of 600mm and a radius of Beam splitter reflects red transmits green and blue

-

laser

/

plate

I

-

Image of slit (about 0.5 1 m away from hologram)

--/--

Fig. 3.5. Layout of the optical system used to produce multicolour rainbow holograms in a HEGEDUS and STEEL[1979]). single step with a concave mirror (HARIHARAN,

290

COLOUR HOLOGRAPHY

IN, 9: 3

curvature of 550 mm, giving an angle of view of approximately 50". The object, turned sideways, was placed on one side of the axis of the mirror so that its image was formed on the other side, at the same distance from the mirror. A vertical slit was placed between the object and the mirror; the distance of this slit from the mirror was adjusted so that a magnified image of it was formed in the viewing space at a convenient distance (= 1 m) from the hologram. To make the final hologram, three successive exposures were made, using as the three primary colours, light of wavelengths 633 nm from a He-Ne laser and 514 nm and 488 nm from an argon laser.

3.4. IMAGE BLUR

Although the image reconstructed by a well-made rainbow hologram appears sharp to the naked eye, there is always some image blur. The extent of this blur depends on the recording geometry and the size of the source used to reconstruct the image and has been analyzed by WYANT [1977] and by TAMURA [1977]. CHEN[1978] has presented a more general analysis for object and reference beams of arbitrary curvature. The discussion in this section is based on the analysis carried out by WYANT [1977] and can be used to select the significant parameters of the setup, so as to keep the blur within acceptable limits.

3.4.1. Wavelength spread The primary cause of image blur is the finite wavelength spread in the image. To calculate this, consider a rainbow hologram made with the setup shown in Fig. 3.6. If the angles made with the axis by the rays from the primary hologram to the final rainbow hologram are small compared to the reference-beam angle 8, it can be shown that the wavelength spread observed when the rainbow hologram is illuminated with white light is 6A = (h/sin 8) [(b/D)+ (a/D)],

(3.1)

where A is the mean wavelength of the reconstructed image, b is the width of the slit, a is the diameter of the pupil of the eye, and D is the distance between the primary hologram and the final rainbow hologram. The two terms, (b/D) and (a/D) in eq. (3.1) correspond to the angular

IV, P 31

MULTICOLOUR RAINBOW HOLOGRAMS

29 1

/" I

Fig. 3.6. Schematic of the optical arrangement used to produce a rainbow hologram (WYANT [1977]).

subtense of the slit and the eye pupil measured from the hologram during recording and reconstruction, respectively. Typically, if A = 500 nm, 8 = 45", D = 300 mm and a = b = 3 mm, then 6A = 14 nm. Using the value of the wavelength spread given by eq. (3.1) it is possible to calculate the image blur 6y,, due to it; this is given by the relation 6y8,

= z0 (SA/A)

= zo ( a

sin 8

+ b)lD,

(3.2)

where zo is the distance of the image from the hologram. If the image is formed at a distance of 5 cm from a hologram with the parameters listed earlier, the image blur is approximately 1mm. This corresponds to an angular blur of about 3mrad at the eye pupil, which is acceptable. GROVER and TREMBLAY [19801 have described an alternative one-step recording system, which does not use a real masking slit. Instead, the object is displaced at a constant speed during the exposure. The resulting hologram diffracts light in such a manner that, at a plane in the viewing space, it gives rise to an irradiance distribution described by a sinc' function; the image blur can be related to the width of this distribution in the same way as for a slit. 3.4.2. Source size When a light source of finite size is used for reconstruction, the image exhibits an angular blur equal to the angular spread R, of the source, as viewed from the hologram. The resultant image blur is 6ys = zol2,.

(3.3)

292

COLOUR HOLOGRAPHY

[Iv,9: 3

If this is not to exceed the blur due to wavelength spread,

0,< ( a + b)fD.

(3.4)

3.4.3. Diffraction The final source of image blur to be considered is diffraction; this is noticeable only when the width of the slit is very small. Assuming that the slit is imaged onto the eye pupil and its width b S D , the image blur due to diffraction is approximately 6 y b = 2 h ( z o +D)/b.

(3.5)

For the hologram parameters listed earlier, the image blur due to wavelength spread is greater than that due to diffraction, as long as the slit width b 3 1 mm. Further improvements in the technique of rainbow holograms permit an increased field of view (TAMURA[1978b]) and reduced image blur when the image is formed at a large distance from the hologram (LEITH and CHEN[1978], ZHUANG, RUTERBUSCH, ZHANGand Yu [1981]). CHEN [19791 has also described a one-step technique that combines the advantages of both these schemes. The extension of such methods to multicolour imaging is quite logical.

3.5. RECORDING MATERIALS

In principle, the three superimposed holograms used to reconstruct a multicolour image can be recorded on a single plate. An ideal recording material for this purpose should have balanced sensitivity and low scattering at all three primary wavelengths, as well as reasonable speed. The only photographic material that is about equally sensitive to all three colours, Kodak 649F, falls far short of the other two requirements. 3.5.1. "he sandwich technique

HARMARAN, STEELand HEGEDUS [ 19771 therefore developed a sandwich technique that permits the use of two types of photographic plates to record the final hologram. As shown in Fig. 3.7a on the left, the red component hologram was recorded on a red-sensitive plate (Holotest

n/,$ 3 1

MULTICOLOUR RAINBOW HOLOGRAMS

293

Green/ BLue

Red

111

I l l

7-

10E56 plate

1 0 E 7 5 plate Glass

(no anti-halo)

(a) Component

White

Processed 10E 75 plate

holograms

light

Processed 10E56 plate

(b) Final sandwich hologram Fig. 3.7. Schematic of the sandwich technique used to make multicolour rainbow holograms (HARIHARAN, STEELand HEGEDUS [1977]).

lOE75), which was loaded into the plate holder with the emulsion facing forward and with a clear glass plate of the same thickness in front of it. After this, the blue and green component holograms were recorded on an - . ,__~. . _ ._ . . - . orthochromatic plate (Holotest 10E56) without any antihalo coating. This was loaded into the plate holder as shown in Fig. 3.7a on the right, with the emulsion side facing backward and with a clear glass plate of the same thickness behind it. The individual exposures were adjusted by trial to bring the diffraction efficiencies of the three component holograms into balance. Finally, as shown in Fig. 3.7b, the two processed plates were cemented together with their emulsions in contact to give the final multicolour hologram. Registration of the two plates need be done only to an accuracy determined by the residual image blur and is automatic if the plate holder is used as an assembly jig. This sandwich technique was worked out initially to surmount the problems of finding a photographic plate with suitable characteristics. In addition, it made it much easier to match the diffraction efficiencies of the three individual holograms and permitted a better yield of finished holograms. However, further trials with bleached holograms showed that ~

294

COLOUR HOLOGRAPHY

[IV, 3: 3

distinctly brighter images were obtained when the individual holograms were recorded on separate plates, even when the same emulsion (Kodak 649F) was used for all the exposures (HARIHARAN, HECEDUSand STEEL [1979]). The reason for this is discussed below.

3.5.2. Gain in image luminance with the sandwich technique HARIHARAN [1978] has shown that with a recording setup such as that shown in Fig. 3.6, the average luminance Lv of the image reconstructed at a wavelength A by a rainbow hologram is given by the expression

where A, is the area of the hologram that diffracts light, AI is the area of the image, 9 = l/(D+ zo) is the range of viewing angles in the horizontal plane, A is the average spacing of the fringes in the hologram, q is the diffraction efficiency of the hologram, EA is the spectral irradiance of the beam illuminating the hologram and KA is the spectral luminous efficacy of the radiation. If the image is close to the hologram plane, zo becomes negligible and A H = A I ,since only the area of the hologram corresponding to the image diffracts light. The image luminance is then reduced to (3.7)

To maximize its luminance, the image should be located at the maximum distance from the hologram plane consistent with the permissible image blur, and the interbeam angle should be made as small as possible, consistent with the required field of view in the vertical plane. Under these conditions, the object beam is effectively diffuse and the emulsion thickness can be neglected. Because the irradiance of the object beam has a Rayleigh probability distribution, the diffraction efficiency of the resulting thin phase hologram is, as shown by UPATNIEKS and LEONARD [1970], an average taken over the range of values of its complex transmittance. Accordingly, while the theoretical maximum diffraction efficiency from a grating produced by the interference of two plane wavefronts is 34 per cent, the theoretical maximum diffraction efficiency in this case is only 22 per cent. Because of residual absorption and the need to minimize nonlinear effects, the diffraction efficiencies obtained in practice are only about one half of the

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295

theoretical maximum, around 12 per cent. If two holograms are recorded in the same emulsion, the available dynamic range is shared between the recordings, and the maximum diffraction efficiency of each recording is cut by a factor of almost 4, to about 3 per cent. Multiplexing three holograms on the same plate results in a cut in the maximum diffraction efficiency by a factor close to 9, to around 1.3 per cent. Under these conditions, the major part of the light incident on the hologram remains in the directly transmitted beam which, in the case of a singly-exposed thin phase hologram, contains (depending on residual absorption) between 50 and 60 per cent of the incident light. If, therefore, another thin phase hologram recorded on a separate plate with another colour and exposed for maximum diffraction efficiency is superimposed on the first, an overall diffraction efficiency between 6 and 7.2 per cent can be obtained for each hologram. This corresponds to an improvement in diffraction efficiency by a factor of 2 or more over that possible if the two holograms are recorded in the same emulsion. In the case of a three-colour hologram, an improvement in image luminance of the same order can be obtained if the three recordings are divided between two plates. However, the best results are obtained if the three recordings are made on separate plates and superimposed; in this case, an overall diffraction efficiency of about 4 per cent, corresponding to an improvement by a factor of 3, is possible. In this case, to compensate for the thickness of the glass plates and to ensure that the reconstructed images coincide in depth, the plateholder must be moved forward or backward through a distance (tln), where t is the thickness of the plates and n is the refractive index of the glass, between exposures.

8 4. Volume Reflection Holograms: New Techniques Volume reflection holograms have two advantages over rainbow holograms: vertical parallax is retained, and the colours of the image change only slightly with the position of the observer’s eyes in the vertical plane. As discussed in § 2.3 the most serious problem with early multicolour reflection holograms was the relatively low luminance of the reconstructed image. This section describes some improvements in technology that yield a substantial gain in image luminance and promise to make the production of multicolour volume reflection holograms a practical proposition.

2 96

COI.OUR HOLOGRAPHY

[IV, 5 4

4.1. ALTERNATTVE RECORDING MATERIALS

One reason for the low diffraction efficiency of early multicolour volume reflection holograms was the lack of suitable recording materials. Most of the early work on such holograms was carried out with Kodak 649F plates, which have the advantage that they have approximately equal sensitivity at all three primary wavelengths. However, the drawbacks of this plate for recording volume reflection holograms were pointed out at a very early stage by LINand Lo BIANCO[1967]. Many of these arise because its grain size is not small enough, and its silver halide content is higher than optimum. Because of this, the scattering in the emulsion is high, and the diffraction efficiency is low, especially for holograms recorded with green and blue light. LIN and Lo BIANCO[1967] were able to obtain significantly better results with a similar emulsion with a lower silver halide content, while MuZ~Kand ROZEK[1974] used an Agfa-Gevaert 8E56 emulsion, specially sensitized to extend its sensitivity into the red. One possible solution is the use of other recording materials. KURTZNER and HAINES[1971] used a photopolymer with an argon laser to produce volume transmission holograms that reconstructed multicolour images when illuminated with white light. Photopolymers can give a diffraction efficiency in excess of 45 per cent, and it is possible to sensitize them to red light (see, for example, the review by HARIHARAN [1980b]). However, they have drawbacks such as low sensitivity and relatively short shelf life. An alternative is dichromated gelatin, which is also well suited to the production of phase holograms. It has very low absorption and scattering, and CHANGand LEONARD [1979] have shown that with suitable processing, it can produce quite high modulation of the refractive index, with a modulation transfer function that is flat, out to spatial frequencies of the order of 5000 mm-'. Dichromated gelatin was not used earlier, because normally it can be exposed only with blue light. However, it has been shown that it can be sensitized to longer wavelengths with dyes (GRAUBE [1973], KUBOTA,OSE, SASAKIand HONDA [1976]), and KUBOTAand OSE [1979] have used dichromated gelatin sensitized with methylene blue to record multicolour reflection holograms. Since these holograms have peak diffraction efficiencies as high as 40 per cent, they show great promise. 4.2. BLEACHED REFLECTION HOLOGRAMS

Because of the relatively low sensitivity of other materials, photographic materials still remain an attractive recording medium for volume

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297

reflection holograms; in addition, improvements in technique have now made it possible to produce much brighter images with them. Thus, it has been shown that low-noise bleaches can be used to produce reflection holograms with improved diffraction efficiency (HARIHARAN [1972], PHILLIPS,WARD,CULLEN and PORTER [1980]). In addition, the use of relatively thin emulsion layers (=6 p m thick), which diffract a wider spectral bandwidth, can give as high an image luminance as a thicker layer, with the advantage of lower scattering (HARIHARAN [1972,1979]). A further improvement in diffraction efficiency should be possible when photographic emulsions with a significantly smaller grain size become available. This is because scattering due to the grains in the virgin emulsion (which is proportional to the sixth power of their diameter) during exposure and results in reduced modulation at high spatial frequencies (BUSCHMANN METZ[197 11). With commercial photographic materials having an average grain size of 50 nm, JOLY and VANHOREBEEK [1980] have shown that there is a drop of 65 per cent in the modulation transfer function at the spatial frequencies involved in making reflection holograms, but experimental emulsions with grain sizes as small as 1 0 n m have been made (DENISYUK [1978]), for which this drop should be negligible. Triethanolamine cannot be used to correct the shrinkage in thickness normally encountered with such bleached holograms, since they then darken rapidly when exposed to light because of the formation of printout silver. This is due to the well-known hypersensitizing action of triethanolamine. A solution (-10 per cent) of D(-) sorbitol [CH20H(CHOH)&H20H] can be used without adverse effects (HARTHAFUN [1980a]); alternatively, a tanning developer can be used to minimize, or even eliminate, this shrinkage (JOLY and VANHOREBEEK [1980]).

4.3. SANDWICH TECHNIQUE

Since such bleached reflection holograms are completely transparent at wavelengths outside the relatively narrow band which is diffracted, HARIHARAN [1980a] has shown that it is quite feasible, in this case as well, to record the three component holograms for different primary wavelengths on two separate plates and superimpose them to make up the final multicolour hologram. This permits the use of different types of plates to record the component holograms, one with optimum characteristics for the red, and the other with optimum characteristics for the green and the blue. In addition, an improvement in image luminance by a

298

rw, Q 4

COLOUR HOLOGRAPHY

factor of 2 or more is obtained if the three component holograms are divided between two plates in this manner, instead of being recorded in the same emulsion layer. Typically, the red component hologram is recorded with a He-Ne laser (A = 633 nm) and Holotest 8E75 plates, while an argon ion laser is used with Holotest 8E56 plates for the green (A = 5 15 nm) and blue (A = 488 nm) exposures. As shown in Fig. 4.1, the 8E75 plate is exposed with the emulsion side towards the reference beam, while the 8E56 plate is exposed with the emulsion side facing the mirror. To compensate for the thickness of the plates, the plate holder is mounted on a micrometer slide and moved normal to its plane through a distance equal to t [ 1- (lh)],where t is the thickness of the plates and n is the refractive index of the glass, between the two sets of exposures. ( a ) Red image E m uIsi on

8E75 plate

&

( b 1 Green / blue image Emu'siOn\ Object beam

8E56 plate

( c ) Final hologram

! @ I

I

8E75 plate

plate

Observer

Image of aperture

Fig. 4.1. Schematic showing how a multicolour reflection hologram is built up from [1980aD. exposures on two plates (HARIHARAN

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299

VOLUME REFIECTION HOLOGRAMS

After drying, the plates are assembled with the emulsion layers in contact and the reconstructed images are viewed, with the 8E56 plate in front and the 8E75 plate behind. This helps to equalize the diffraction efficiencies of the holograms.

4.4. CONCENTRATION OF THE DIFFRACTED LIGHT

A further improvement in image luminance can be obtained if the diffracted light from the hologram is concentrated into a smaller solid Ar

k

+

I He -Ne

-

\

r \

I Reflects red ; transmits blue and green

I

ll

Fig. 4.2. Optical system used for recording multicolour reflection holograms with increased image luminance (HARIHARAN [1980a]).

300

COLOUR HOLOGRAPHY

",

Ei 5

angle. For this, a hologram is recorded, not of the original object, but of a real image of the object projected either by another hologram or by an optical system whose effective aperature is limited by a suitably shaped stop as shown in Fig. 4.2. This is the same principle which has been exploited in the rainbow hologram. The gain in image luminance is proportional to the reciprocal of the available solid angle of viewing (HARIHARAN [1978]). Normally, a gain in image luminance by a factor of 3 or 4 can be obtained, without any loss in convenience, if the range of viewing angles in the vertical plane is restricted to about 15".

0 5. Pseudocolour Images All the techniques described so far for producing holograms that reconstruct an image in more than one colour have involved the use of light of two or more wavelengths in the recording setup. However, the holograms themselves are not coloured ; the colour information is recorded only in the form of specific carrier fringe frequencies. This suggests the possibility of using a single laser wavelength to generate the different carrier frequencies by some other means. Such pseudocolour techniques are a cheaper alternative which can, within certain limits, produce holograms that reconstruct a multicolour image.

5.1. COLOUR CODING

The simplest method of generating the different carrier frequencies with a single wavelength (say, from a He-Ne laser) is to change the angle of the reference beam between exposures. A different object is used for each exposure; alternatively, the area illuminated, or the reflectance of particular areas, is changed; this corresponds to the required colour coding of the scene. If the angles of the reference beams in the recording setup are properly chosen, the finished hologram can be illuminated by a single beam of light of three wavelengths corresponding to three primary colours, to produce a multicolour image (FRIESEM and FEDOROWICZ [1967]). While cross-talk images can be eliminated by using a thick photographic emulsion, a problem which remains and makes it difficult to synthesize an image of a single multicoloured object is the lack of exact

IV, 5 51

PSEUDOCOLOUR IMAGES

30 1

spatial superposition of the different coloured reconstructions. This is due to the difference in the wavelengths of the light used to make the hologram and the sources used to reconstruct this image. This problem will be analyzed in more detail later; however, the relative displacements of the images of different colours can be minimized by the use of an image hologram (GADDISand WELTER [1978]). The colour dispersion in the image formed by a hologram illuminated with white light is proportional to the distance between the plate and the image. Hence, if the image is formed in the plane of the hologram, the dispersion is reduced to zero, and the images formed by the beams of different colours will coincide.

5.2. RAINBOW HOLOGRAMS

A rainbow hologram can be made to produce pseudocolour images with a white light source. For this, light of a single wavelength is used to record three superimposed rainbow holograms, but the angle of the reference beam is changed between exposures (TAMURA [1968al). An alternative with two-dimensional transparencies is to use different positions of the limiting slit for the exposures (VLASOV,RYABOVAand SEMENOV [1977], YAN-SONG, Yu-TANGand BI-ZHEN[1978]). In this case also, the images reconstructed at a different colour from that used to record the hologram are displaced with respect to the image of the same colour. The magnitude of this displacement is zero in the horizontal plane, while, with the setup shown in Fig. 3.6, the vertical displacement is Ay

= zo (2AAlA) tan3 8.

(5.1)

In eq. (5.1) zo is the distance of the image from the hologram, A is the wavelength with which the hologram is recorded and AA is the difference between A and the wavelength at which the image is reconstructed. The image also suffers a longitudinal displacement Az = zo [2Ah/A]. A comparison of eqs. (5.1) and (5.2) shows that the longitudinal displacement is more severe than the vertical displacement. Both can be reduced to acceptable limits if zo, the distance of the image from the hologram, is made sufficiently small.

302

COLOUR HOLOGRAPHY

[IV, 9 5

Besides multicolour imaging, such pseudocolour techniques have been adapted for encoding fringe patterns in hologram interferometry as well as for encoding spatial frequency information (see Yu, TAI and CHEN [ 19801). 5.3. VOLUME REFLECTION HOLOGRAMS

Pseudocolour techniques can also be applied to volume reflection holograms. With volume reflection holograms, changing the angle between the reference and object beams has little effect on the colour of the reconstructed image. However, its colour is affected by changes in the thickness of the recording medium, and these changes can be controlled and used to produce pseudocolour images (HARIHARAN [198Oc]). To record in the same emulsion layer two holograms that reconstruct images of different colours, the first exposure is made, say, with red light, with the emulsion in its normal condition. The emulsion is then soaked in a 3% solution of triethanolamine and dried in darkness. The second exposure is made on the swollen emulsion with the same laser. Normal processing eliminates the swelling produced by the triethanolamine and produces the usual shrinkage. Accordingly, the first exposure yields a green reconstructed image, while the second produces an image at an even shorter wavelength, that is to say, a blue image. If the normal loss of thickness of the emulsion is corrected, red and green images are obtained. Much brighter images can be obtained with volume phase reflection holograms. In addition, as mentioned in § 4.3, such holograms are effectively transparent at wavelengths outside the relatively narrow band which is diffracted. Hence, it is possible to obtain improved diffraction efficiency by recording the component holograms on two separate plates, which are processed to obtain reconstructed images in the desired primary colours and then superimposed to produce a multicolour image. For three-colour images, it is most convenient to use a combination of these two techniques. The red component hologram is recorded on a plate exposed with the emulsion side towards the reference beam. This plate is processed to eliminate emulsion shrinkage. The green and blue holograms are recorded on another plate exposed with the emulsion side towards the object beam. The green component hologram is exposed with the emulsion in its normal condition, while the blue component hologram is exposed after swelling the emulsion with triethanolamine. This plate is processed without any correction for emulsion shrinkage.

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ACHROMATIC IMAGES

303

After drying, the plates are cemented together with the emulsion layers in contact. The images are viewed with the hologram reconstructing the green and blue images in front and the hologram reconstructing the red image behind. In this case also, the images reconstructed at wavelengths differing from that used to record the holograms undergo shifts which depend on p , the ratio of these wavelengths. These shifts have been analyzed by HARIHARAN [1976, 1980a], who has shown that to avoid lateral misregistration the lateral magnification MI,, must be independent of F ; this is possible if the hologram is reconstructed with a parallel beam. Similarly, to eliminate longitudinal misregistration, the longitudinal magnification, M,,,, = (l/p)M&,must be independent of p . However, to eliminate longitudinal distortion it must be equal to MI,,. While these conditions normally cannot be satisfied at the same time, the images formed at different wavelengths coincide in the hologram plane, for which MI,,= 1 and Mlong = (l/p).Since the eye is fairly tolerant of longitudinal misregistration, acceptable results can be obtained over a limited depth centred on this plane.

0 6. Achromatic Images The production of holograms that reconstruct an almost white or achromatic image when illuminated with white light is, in a sense, complementary to the pseudocolour techniques described in 9 5 . Such achromatic images have the advantage that they can be very bright, because they use the entire output of a white light source. Very nearly achromatic imaging of an object of limited depth is possible with an image hologram. When such a hologram is illuminated with white light, sharp images of all the points in the hologram plane are formed at all wavelengths. However, for other image points, there is a residual colour blur which increases with their distance from the hologram.

6.1. DISPERSION COMPENSATION

Early attempts to make holograms that could produce achromatic images of objects with significant depth were based on dispersion compensation.

304

[IV,5 6

COLOUR HOLOGRAPHY

One method described by PAQUES [19661 uses the well-known 'thin-lens achromatization technique of separating two lenses by a distance equal to half the sum of their focal lengths. Consider a hologram which produces a real image at a distance Fl when illuminated with a collimated beam produced by a point source of white light placed at the focus of a holographic lens (a hologram of a point source recorded with a plane reference wave) with a focal length F2. If the holographic lens is located at a distance D = (Fl+ F2)/2from the hologram, an achromatic real image is reconstructed. An alternative method described by DE BITTETO[1966] employs a plane diffraction grating, with a line spacing equal to the average fringe spacing in the hologram, to provide equal but opposite angular dispersion. As shown in Fig. 6.1, the hologram is illuminated with white light from a point source and a transmission grating is placed in the diffracted beam forming the virtual image. The wave diffracted by this grating in the opposite sense then produces an image in which dispersion is almost completely compensated. The doubly-diffracted wave reconstructing the image now propagates in the same direction as the wave illuminating the hologram. As a result, directly transmitted light spills into the field of view unless the latter is severely restricted, or a light shield consisting of a set of parallel baffles, rather like a venetian blind, is used between the hologram and the grating (BURCKHARDT [1966]). While the lateral chromatic aberration of the image can be cancelled out by this method, the longitudinal chromatic aberration and variation of magnification with wavelength are unaffected. One method to minimize these aberrations is to use an image hologram in conjunction with a Hologram

-1 Real, dispersed

0-Order primary

/-

( White light source

Virtual di s persod

+l,-1

Virtual, Undisper sed

Diffraction grating

Fig. 6.1. Production of an achromatic image by dispersion compensation using a diffraction grating (DEBITI'ETO[1966]).

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ACHROMATIC IMAGES

305

grating (BRYNGDAHL and LOHMA” [1970]). Another method is to record the hologram with a convergent reference beam and illuminate it with a similar converging beam of white light (BURCKHARDT [19663. An observer viewing the image from the centre of convergence receives rays passing through corresponding points on all the different coloured images, so that an achromatic image is seen.

6.2. RAINBOW HOLOGRAMS

The use of a compensating grating necessarily implies a lower overall diffraction efficiency. Much brighter reconstructed images can be obtained with techniques based on the use of rainbow holograms. The simplest way of obtaining a partially achromatized image with these is to use, instead of a point source of white light, a vertical line source to illuminate the hologram (BENTON [1969, 19771). This results in a series of overlapping spectra projected into the viewing space, so that a near-white reconstructed image is obtained over a considerable viewing area. However, with an object of appreciable depth there is some colour blur. This is because the images reconstructed by different wavelengths have different magnifications and are formed at different distances from the hologram. The red image is the smallest and is formed closest to the hologram, while the blue image is the largest and is formed at the greatest distance. This is also true of the slit images constituting the spectrum projected into the viewing space, which lie along an inclined line. In order to produce a truly achromatic image, it is necessary to have red, green and blue images which coincide precisely. For this, the corresponding slit images must also coincide; this means that the overlapping spectra must lie along a single line. This is possible if the final hologram is made with a series of reference beams with suitably chosen convergences and angles of incidence. Alternatively, the primary hologram can be illuminated with multiple beams. BENTON[1978] has used a multiply exposed holographic lens, which produces the effect of a series of point sources of light located at suitable angles and distances, in combination with a narrow strip of the first hologram to make a second hologram which is again illuminated with monochromatic light to reconstruct an image for recording a third hologram. When this hologram is illuminated with white light it reconstructs a

306

COLOUR HOLOGRAPHY

[IV, 9: 6

One -dimensional

Fig. 6.2. Setup using a one-dimensional diffuser to produce a hologram that reconstructs an achromatic image (LEITH,CHENand ROTH[1978]).

set of overlapping spectra in the viewing space which coincide at appropriate wavelengths, so that an achromatic image is obtained over a wide range of viewing angles. Another technique described by LEITH,CHENand ROTH[1978] uses a one-dimensional diffuser to generate multiple reference beams. The setup for this is shown in Fig. 6.2. The primary hologram is masked by a horizontal slit and forms a real image at some distance from the hologram plate. However, the introduction of a cylindrical lens causes this image to be focused in the vertical plane at the hologram itself. The reference beam passes through a phase plate which diffuses the beam in the vertical direction and transmits without scatter in the horizontal direction. When the final hologram is illuminated with white light, it reconstructs an image essentially free from colour. The operation of this system can be understood best by considering separately what happens in the vertical and horizontal directions. In the vertical direction, as shown in Fig. 6.3a, because the aperture of the cylindrical lens is restricted by the slit, the entire depth of the object is in focus at the hologram. Accordingly, in this plane we have an image hologram which reconstructs a sharp image when illuminated with white light. In the horizontal direction, as shown in Fig. 6.3b, the system corresponds to an in-line hologram, and the colour blur is small, because

IV, 9: 71

307

APPLICATIONS OF COLOUR HOLOGRAPHY

Hologram plate

Slit

4

lens

Primary hologram

Referencebeam

/ (b)

Primary hologram

Hologram plate

image

Fig. 6.3. Achromatic hologram recording setup in (a) the vertical plane and (b) the horizontal plane (LEITH,CHENand ROTH[1978]).

the angles of diffraction involved are small. Since light of any colour incident on the hologram is scattered through a range of angles in the vertical plane determined by the extent of the diffuser, the colour effects normally obtained with a rainbow hologram are washed out. The image shows some astigmatism but is still acceptably sharp.

0 7. Applications of Colour Holography 7.1. STORAGE OF COLOUR IMAGES

A significant application of multicolour holography, which grew out of the studies outlined in P2, was its use for storage of two-dimensional

308

COLOUR HOLOGRAPHY

",

rj 7

colour information. Typically, archival storage of colour movie films has been a major problem, because the organic dyes used are not stable and fade with time. The use of spatial carrier frequencies to encode colour images is, of course, not new; it goes back as far as 1899, when WOOD[1899] used three superposed modulated diffraction gratings in an additive colour projection system. Most of the early work o n such techniques has been covered in a review by BIEDERMANN [1970]. One advantage of the holographic approach is the possibility of using higher carrier frequencies, giving increased resolution. Another is that the hologram permits recording phase information as well; this makes reconstruction possible without an imaging lens, giving a simpler system for readout. Surface-relief phase holograms have the further advantage that copies can be produced quite cheaply by replication on thermoplastic film. Such replicated holograms have excellent archival properties.

7.1.1. Systems using image holograms Systems using image holograms have been described by BURTON and CLAY[1972] and by GALEand KNOP[1976]. The recording arrangement used by them is shown in Fig. 7.1. Because a suitable panchromatic photoresist is not available, three colour-separation transparencies are used, and a colour-encoded image hologram is recorded by means of three successive exposures using a He-Cd laser. The three discrete carrier frequencies required are obtained by using a different angle between the

Green

Fig. 7.1. Recording system for colour-encoded image holograms (GALEand KNOP [1976]).

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APPLICATIONS OF COLOUR HOLOGRAPHY

309

object beam and the reference beam for each exposure. These angles are chosen so that when the hologram is illuminated by an off-axis white light source, a single slit aperture mounted on the optical axis in front of the projection lens selects the three diffracted beams producing the required primary colours. A detailed analysis of the colour reproduction possible with this system has been made by GALEand KNOP[1976], who have shown that the chromaticity of each of the three primaries is determined to a good approximation by its mean wavelength and its spectral width. By a careful choice of these parameters, a colour range similar to that of most colour films and colour television was obtained.

7.1.2. Systems using spatial filtration An alternative technique described by IH [1975] uses Fourier holograms and a low-pass filter (a simple aperture) placed in the Fourier plane during reconstruction. If the recording setup is illuminated with a polychromatic laser beam containing three wavelengths corresponding to three primary colours, three superposed Fourier holograms are formed on the recording medium. When this composite hologram is illuminated once again with the same polychromatic reference beam, the three diffracted wavefronts corresponding to the multicolour reconstructed image are focused on the aperture in the Fourier plane. The other diffracted wavefronts corresponding to the unwanted cross-talk images are focused into different locations and are blocked. An optical system which eliminates the need for an imaging lens at the reconstruction stage is shown in Fig. 7.2. This system is suitable for displaying large colour transparencies that can be viewed without magnification. It also has the advantage that the spatial frequency content of the image can be limited in the recording process itself, avoiding any possible overlap of the spectra of the reconstructed images. If a converging reference beam is used to record the hologram, a diverging reference beam can be used for reconstruction, permitting a very simple setup. Such holograms have the fundamental advantage that, since no colour filters or masks are used in the process, the primary colours used to reconstruct the image correspond to the laser wavelengths themselves. Hence, the only factors that could affect the reconstructed image are

310

[Iv,9: 7

COLOUR HOLOGRAPHY

Hologram

Reference beam

(a)

Recording

rconstructed image Spatial filter

(b 1

Y

Reconstruct ion

Fig. 7.2. Optical setup used to record and reconstruct multicolour images from a plane hologram without an imaging lens (IH [1975]).

defective colour registration, changes in image contrast, and image distortion. A study by IH [1978] has shown that colour registration was not affected by film bending, film shrinkage and changes in the angle of incidence and divergence of the beam used to reconstruct the image. Image contrast was also preserved if a recording material with a low negative gamma was used. Image distortions could be minimized by the use of image holograms.

7.1.3. Systems using rainbow holograms The use of rainbow holograms for storage of multicolour images has been described by Yu, TAIand CHEN[1978]. This has the advantage that a white light source can be used for reconstruction. They use a one-step

IV, § 71

APPLICATIONS OF COLOUR HOLOGRAPHY

311

Fig. 7.3. Optical system for archival storage of colour images using rainbow holograms (Yu, TAIand CHEN[19783.

recording technique with an imaging lens. As pointed out in 03.3, a disadvantage of this technique, when recording holograms of threedimensional objects, is the limited field of view. However, this is relatively unimportant when recording holograms of two-dimensional colour images. As shown in Fig. 7.3, their setup uses a He-Ne laser emitting light at a wavelength of 633 nm and two argon lasers emitting light at wavelengths of 514 nm and 477 nm, respectively. Individual neutral filters are used to adjust the outputs at the three wavelengths to suitable levels, so that the three component holograms can be recorded simultaneously with a single exposure on the same film. The image is reconstructed with an optical system symmetric to that used in recording as shown in Fig. 7.4. When the hologram is illuminated with white light, only the appropriate range of wavelengths diffracted by each of the three component holograms is transmitted by the slit to reconstruct a rnulticolour image at the output plane. This can be recorded on a fresh colour film for use in a conventional projector. The need to limit the aperture of the imaging lens by a slit, in the system shown in Fig. 7.3, limits the resolution of the image in the direction perpendicular to the slit. Improved resolution can be obtained, as shown by Yu, RUTERBUSCH and ZHUANG [1980], if the difiser and slit

312

COLOUR HOLOGRAPHY

'

I Slit

",

87

film

I

1

white light

Fig. 7.4. Optical setup for reproduction of colour images from rainbow holograms (Yu, TAI and CHEN[1978]).

are removed, and a cylindrical lens is placed behind the object transparency. There is still a marginal loss of resolution due to the slit used in the reconstruction setup, but this is less severe.

7.2. COLOUR HOLOGRAPHIC STEREOGRAMS

To make a hologram of a three-dimensional object it is normally necessary to illuminate it with coherent light. However, this is not essential for holographic visual displays; a composite hologram that reconstructs an acceptable image of an object illuminated with white light can be synthesized in two steps (MCCIUCKERD and GEORGE [1968], DE B ~ T [1969]). O In the first step, a series of photographs of the subject is taken from equally spaced positions along a horizontal line. In the second step, as shown in Fig. 7.5, contiguous, narrow, vertical strip holograms are recorded of each of these photographs on a high-resolution photographic plate. When the final holographic stereogram is illuminated with a point source of monochromatic light, the viewer sees a three-dimensional image. This image lacks vertical parallax but it exhibits horizontal parallax over the range of angles covered by the original photographs. The obvious advantage of this technique over recording a hologram directly is that a laser is required only for the second step; white light can be used to illuminate the subject, so that holograms can be made of quite large scenes and even of living subjects.

IV, § 71

313

APPLICATIONS OF COLOUR HOLOGRAPHY

Photographic plate

WI

2D transparencies Translucent screen

W

- /

I

LiN

Fig. 7.5. Optical system for recording a holographic stereogram from a series of twodimensional transparencies (DEBITTETO[1969n.

MCCRICKERD and GEORGE [1968] pointed out the possibility of producing multicolour images with this technique. This can be done by using a colour transparency film in the first stage of the process and multiwavelength laser light, as described in 0 2.2.1, to record holograms of these transparencies. To view the image, the final holographic stereogram must be illuminated with a similar multiwavelength source. 7.2.1. White-light holographic stereograms An interesting development of these techniques has been the production by Cross of cylindrical holographic stereograms which can be illuminated with white light and can reconstruct an almost monochromatic image (see BENTON [1975]). This is achieved by an adaptation of the rainbow hologram. In the first step, the subject is placed on a slowly rotating turntable and a movie camera is used to make a record of a 120" or 360" rotation. Typically, three movie frames are exposed for each degree of rotation, so that the final movie sequence may contain up to 1080 frames. The optical system used to produce a holographic stereogram from this movie sequence is shown schematically in Fig. 7.6. In this system, each frame of the movie film is imaged in the vertical plane onto the hologram film. However, in the horizontal plane, the cylindrical lens brings all the rays leaving the projector to a line focus on the film. By means of the reference beam incident from below, a contiguous sequence of vertical

314

COLOUR HOLOGRAPHY

[IV, ci 7

Fig. 7.6. Setup used to produce a Cross hologram (Hun: and FUSEK [1980]).

strip holograms is then recorded of successive movie frames, covering the full range of views of the original subject. When the processed film is formed into a cylinder and illuminated with a monochromatic light source, the reconstructed images can be seen over only a narrow range of angles in the vertical plane, corresponding to the vertical spread of the object beam; with white light, a monochromatic image is seen which changes colour, as with any rainbow hologram, when the observer moves his head up or down. Due to the very large number of frames recorded, a modest amount of subject movement can be accommodated without destroying the stereoscopic image. When the cylinder is rotated, or the observer walks past, a convincing impression of a moving, three-dimensional figure is created. Techniques which could be used to produce multicolour images with [19801. Cross holograms have been discussed by HUFFand FUSEK The simplest method is to make use of the colour dispersion of the rainbow hologram. For this, all that would be necessary, as described in P 3.3, would be to record three superposed holograms of each frame of a colour film, using three suitable laser wavelengths. A similar technique has been used, as described in § 7.1.3 for the archival storage of colour images, but it has the disadvantage for displays that the colours of the image change with the viewing position in the vertical plane, limiting the useful range of viewing angles. HUFFand FUSEK[19801 have therefore examined techniques that can produce multicolour images with a wide vertical viewing range. In order to do this, it is necessary to produce individual images whose colours do

IV,

I 71

315

APPLICATIONS OF COLOUR HOLOGRAPHY

not change over a range of angles. One technique which could be used for this is that developed by LEITH,CHENand ROTH[1978], and described in fi 6.2, which uses a one-dimensional dispersive element in the reference beam path. When a hologram made by this technique is illuminated with monochromatic light, it produces an image which can be viewed over a wide range of angles in the vertical plane. An alternative technique is to disperse the subject beam itself in the vertical plane when recording the hologram. Since a dispersing element placed in contact with the hologram film would also disperse the reference beam, HUFFand FUSEK [1980] have proposed imaging the dispersing element onto the hologram film. Figure 7.7 shows a vertical section of the optical system used. In this, the movie film transparency is imaged in the vertical plane onto the vertical dispersing element by the projection lens, and this real image is then relayed to the hologram film by the cylindrical lens L3. is merely a cylindrical field lens. In the horizontal plane, as in the setup of Fig. 7.6, another cylindrical lens brings all the rays leaving the projector to a line focus on the hologram film. This system permits the reference beam to illuminate the hologram film without having to traverse the dispersing element. Three-colour holograms have been successfully recorded on Kodak 649F film using such an optical system with the 633 nm line of the He-Ne laser, and the 514 nm and 477 nm lines of the argon laser. The reference beams at wavelengths of 477nm and 633nm were incident on the hologram at angles of 40" and 65" to the normal, while the reference beam at a wavelength of 514nm was incident at an angle of -45". A drawback is that reconstruction of the image requires three colour-filtered incandescent sources placed at appropriate angles. Vertical p

M

I ?

L3

Irn

Reference beam

Fig. 7.7. Modified optical arrangement to produce achromatic images using a relayed image of a unidirectional diffuser (HUFFand FUSEK[1980]).

316

COLOUR HOLOGRAPHY

",

I7

7.2.2. Achromatic holographic stereograms Very recently, BENTON [1981] has described a simple technique for producing a synthetic hologram that reconstructs a black and white image with appreciable depth. In this, a series of perspective views of the subject are recorded by translating a movie camera sideways in front of it. Successive frames are then projected on to a diffusing screen and holograms are recorded of these views on adjacent narrow strips of a photographic plate H1 which is tilted so that it makes an angle 4 with the axis of the system. If a fresh photographic plate Hz is placed in the plane formerly occupied by the diffusing screen and H1 is illuminated with the conjugate of the reference beam used to record it, each of the strip holograms recorded on HI projects the corresponding view of the subject on to H2.A hologram of this synthetic real image is then recorded on H2 using a collimated reference beam making an angle 8 with the axis. To view the image, Hz is illuminated with a collimated beam of white light which is effectively the conjugate of the reference beam used to make it. Different wavelengths then form a series of overlapping images of H1 in the viewing space. If the angles 4 and 8 satisfy the condition

4 =tan-' sin 8,

(7.1)

all these images of HI lie in the same plane, so that an observer whose eyes are positioned in this plane sees a three-dimensional image of the subject which is almost free of colour.

7.3. COMPUTER-GENERATED COLOUR HOLOGRAMS

A multicolour image can be produced from three computer-generated holograms, corresponding to three primary colours, provided two basic problems can be solved. The first is, as outlined in Q 1.2, the formation of cross-talk images; the second is that since the three desired images are reconstructed by light of different wavelengths, their magnifications differ. The simplest method to ensure that the three reconstructed images are of the correct size is to scale the holograms suitably during synthesis, though it is also possible, with a suitable optical system, to scale them during reconstruction. Techniques to overcome this problem have been reviewed by FLENUP and GOODMAN [1974]. Cross-talk images can be

IV,8 71

APPLICATIONS OF COLOUR HOLOGRAPHY

317

eliminated by the use of spatial-frequency multiplexing techniques or by the use of colour filters to ensure that each hologram is illuminated only by light of the appropriate colour (DALLAS, ICHIOKA and LDHMANN [19721).

7.3.1. Techniques using multilayer colour film Since computer-generated holograms can be recorded on materials of relatively low resolution, interesting possibilities are opened up by the use of multilayer colour film. The simplest method is to record three binary detour-phase holograms with light of appropriate colours. Each hologram only transmits light of the desired colour, so that false images are eliminated. However, such holograms have very low diffraction efficiency and require a large number of display elements. An alternative technique is the use of on-axis holograms. One such is the referenceless on-axis complex hologram (CHU,FIENUP and GOODMAN [1973]). In this, different layers of the film are exposed selectively by light of different colours. When illuminated with light of a given colour, one layer of the film will absorb, while the other layers which are effectively transparent can cause phase shifts due to variations in film thickness and refractive index. Thus, both the amplitude and phase can be controlled by a single element. Since all the light is diffracted into a single image, the diffraction efficiency is very high. An alternative technique, the parity sequence hologram (CHU and GOODMAN [1972]), eliminates the need for an absorbing layer to control the amplitude at each point. In this, auxiliary elements are added to the image elements, resulting in a level spectrum, these auxiliary elements being chosen so that the reconstructed images formed by them are at a distance from the desired image. As a result, the recording medium need only control the phase of the transmitted light. Both these types of on-axis holograms require the individual holograms to be illuminated through appropriate colour filters to avoid cross-talk images. The need for such filters can be avoided by a phase-null method (FIENUP and GOODMAN [19741). For a given hologram illuminated with white light, one colour, say blue, produces the desired image, while the two others produce false images. One of these, say green, can be completely absorbed by one layer of the colour film. To eliminate the red image, each cell on the hologram is

318

COLOUR HOLOGRAPHY

[IV, i 7

divided into two halves, and the right-hand half is exposed to red and blue light to give the correct complex transmittance for blue light. The left-hand half is given a much smaller exposure to blue light, so that, while it is almost opaque to blue light, it transmits the same amount of red light with a phase difference of T.As a result, the on-axis red image is eliminated, and only the blue image is reconstructed by the right-hand half. An advantage of this technique is that the three holograms can be spatially multiplexed. However, its implementation requires very accurate exposure control during synthesis of the hologram. 7.3.2. Techniques using holographic stereograms A completely different approach which can be used to produce computer-generated holograms of three-dimensional objects and which can, in principle, be extended to multicolour images, has been followed by KING, NOLL and BERRY[1970]. This technique is related to the techniques used in making holographic stereograms described in 0 7.2 and has the advantage that it requires much less computer time than conventional techniques. A computer is used to produce a series of perspective projections of the object as seen from a number of angles in the horizontal plane. These are then optically encoded as a series of vertical strip holograms on a single plate. The real image formed by this composite hologram, when it is illuminated by the conjugate reference beam, is then used to produce an image hologram. Since this real image is actually two-dimensional, it is located entirely in the plane of the final hologram, which can therefore be illuminated with white light to reconstruct a bright, almost achromatic image. 7.4. HOLOGRAPHIC CINEMATOGRAPHY

Another application of holography which has attracted considerable attention is holographic cinematography. The major problem faced is that the hologram has to be small for economy, while the image has to be sufficiently large to permit comfortable viewing. Some of the early schemes proposed for this purpose have been described by LEITH,BRUMM and HSIAO[1972]. Later techniques using projection holography have been reviewed by OKOSHI [1977]. In some, the hologram area is reduced first by discarding

IV, 9: 71

APPLICATIONS OF COLOUR HOLOGRAPHY

319

vertical parallax information and then by horizontal sampling. Typical of these is a three-dimensional multicolour movie reported by TSUNODA and TAKEDA [19751 using projected images from a horizontally sampled hologram. Such a hologram has the advantage that it can be built up from colour photographs of the object taken from different directions. The real image reconstructed by this hologram is projected onto a lenticular screen which is direction selective in the horizontal direction and diffusing in the vertical direction. However, this system is limited to showing rotation of the object. A system which does not require laser light for reconstruction has been described by YANOand MATSUMOTO [1973]. In this, an image hologram illuminated by a beam of white light projects a real image through an ordinary lens onto a transmission-type, horizontally direction-selective screen, the projected image being observed from the rear. When the image hologram is recorded, the three images corresponding to the three primary colours are focused on the film from three different directions in the vertical plane. A multicolour image is obtained during reconstruction by dividing the lens aperture into three parts, corresponding to these directions, which are provided with appropriate primary colour filters. Probably, holographic movie processes have been studied most extensively in the USSR. Proposals for multicolour, three-dimensional, holographic movies have been discussed in two articles by KOMAR[197Sa, b] as well as in a recent review (KOMAR[1977]). Figure 7.8 is a schematic diagram of the recording setup. In this, the object is illuminated by three pulsed lasers producing red, green and blue light. Scattered light from the object is picked up by a lens of large aperture (-200mm) which forms a reduced image on the holographic film. A multicolour reference beam derived from the same three lasers is also incident on the holographic film, producing an image hologram of the object. The film is moved intermittently between laser pulses to record the holographic movie. In the projection setup shown in Fig. 7.9, a movie frame containing a hologram is illuminated by a quasi-coherent source emitting light at three wavelengths corresponding to the wavelengths used to record the hologram. A small multicolour image of the original object is formed near the film, cross-talk images being eliminated by the angular selectivity of the relatively thick emulsion used. This image is magnified and projected onto a holographic screen, a key feature of the system, which forms a series of images of the pupil of the projection lens in the viewing space.

320

COLOUR HOLOGRAPHY

",

87

Plane of focus I

Multicolour

Fig. 7.8. Schematic of the recording system used for holographic movies (KOMAR[1975a]).

Each of these secondary pupils constitutes a viewing zone, about 200 mm wide, at which an observer can see a three-dimensional image. One of the problems in this system is the production of the screen, which is actually a multiply exposed hologram of a pair of point sources. At present its size is limited to about 0.8 m X 0.6 m, and four spectators can view a monochromatic image simultaneously. For a multicolour Screen

\I

I

\

\

//

J

K \

reconstruction \ beam Viewing zone

Fig. 7.9. Optical arrangement used to project holographic movies (KOMAR [1975a]).

IVI

REFERENCE5

32 1

movie, it would be necessary to use a thick emulsion and make the exposures with light of the three primary wavelengths. Another problem is the pulsed lasers required to record holograms of large scenes (SUKHMAN, KOMAR,OVECHKINA and SOBOLEV [ 19771). Twocolour pulsed laser holograms have been recorded by ALPIN, FLEURET and GACCIOLI [1971] using a ruby laser (A = 694 nm) and a frequencydoubled neodymium glass laser (A = 530 nm). However, problems of heat dissipation due to the high pulse-repetition rate have to be overcome, and a suitable, pulsed laser with an output in the blue region of the spectrum has to be developed. A possibility is the third harmonic from a yttriumaluminium garnet (YAG) laser. An alternative would be the use of pulsed dye lasers. It is also possible, especially where only a limited range of perspectives is required, to photograph the scene in white light, using colour film in contact with a lenticular raster. This film can then be used with a matched lenticular raster and a suitable lens to project an image which is recorded holographically, using coherent light from three lasers. There is little doubt that these difficulties can be overcome, and that the production of a multicolour, three-dimensional, holographic movie is not very far off.

J 8. Conclusions While colour holography had a promising start, it made little progress for many years because of several problems. Recent research has revealed solutions for these problems as well as a number of promising applications. This has resulted in a resurgence of interest in this field. We can look forward with confidence to new and interesting developments which will exploit its full potential.

Acknowledgements I wish to thank Dr W. H. Steel for his encouragement, advice and helpful criticism. References AFJLIN, C., J. FLEURET and N. GAGGIOLI,1971, C. R. Acad. Sci. B (France) 273, 173. BENTON,S. A., 1969, J. Opt. SOC.Am. 59, 1545. BENTON,S. A., 1975, Opt. Engg. 14, 402.

322

COLOUR HOLOGRAPHY

"

BENTON, S. A., 1977, in: Applications of Holography and Optical Data Processing, eds. E. Marom, A. A. Friesem and E. Wiener-Avnear (Pergamon Press, Oxford) p. 401. BENTON, S. A., 1978, J. Opt. SOC.Am. 68, 1441. BENTON,S. A., 1981, Achromatic Holographic Stereograms, in: Abstracts, Congress and Twelfth Assembly of the International Commission for Optics, Graz, 1981 (University of Graz, Graz) p. 129. Jr. and W. R. WALTER,1979, One-Step White-Light BENTON,S. A., H. S. MINGACE Transmission Holography, in: Proc. SPIE Vol. 212, Optics and Photonics Applied to Three-Dimensional Imagery, eds. M. Grosmann and P. Meyrueis (SPIE, Bellingham) p. 2. BIEDERMANN, K., 1970, Opt. Acta 17, 631. 0. and A. LOHMANN, 1970, J. Opt. Soc. Am. 60, 281. BRYNGDAHL, BURCKHARDT, C. B., 1966, Bell Syst. Tech. J. 45, 1841. BURTON,G. T. and B. R. CLAY, 1972, RCA Eng. 18, 99. BUSCHMANN,H. T. and H. J. METZ,1971, Opt. Commun. 2, 373. 1979, Appl. Opt. 18,2407. CHANG, B. J. and C. D. LEONARD, CHEN,H., 1978, Appl. Opt. 17, 3290. CE~EN, H., 1979, Appl. Opt. 18, 3728. CHEN,H., A. M. TAIand F. T. S. Yu, 1978, Appl. Opt. 17, 1490. M., 1970, Opt. Commun. 2, 109. CHOMAT, and J. W. GOODMAN, 1973, Appl. Opt. 12, 1386. CHU, D. C., J. R. FIENUP 1972, Appl. Opt. 11, 1716. CHU,D. C. and J. W. GOODMAN, R. J., C. B. BURCKHARDTand L. H. LIN, 1971, Optical Holography (Academic COLLIER, Press, New York). COLLIER, R. J. and K. S. PENNINGTON, 1966, Appl. Phys. Lett. 8, 44. R. J. and K. S. PENNINGTON, 1967, Appl. Opt. 6, 1091. COLLIER, DALLAS, W. J., Y. ICHIOKA and A. LBHMANN, 1972, J. Opt. SOC.Am. 62, 739. DE BITETO, D. J., 1966, Appl. Phys. Lett. 9, 417. DE BITETO, D. J., 1969, Appl. Opt. 8, 1740. Yu. N., 1962, Sov. Phys. Doklady 7, 543. DENISWK, Yu. N., 1978, Sov. Phys. Tech. Phys. 23, 954. DENISWK, M. I., A. P. PYATIKOP and V. V. SHEVCHENKO, 1975, Sov. Phys. Tech. Phys. DZYUBENKO, 20, 965. FIENUP,J. R. and J. W. GOODMAN, 1974, Nouv. Rev. Opt. 5, 269. 1966, Appl. Opt. 5, 1085. FRIESEM, A. A. and R. J. FEDOROWICZ, -EM, A. A. and R. J. FEDOROWICZ, 1967, Appl. Opt. 6, 529. FRIESEM,A. A. and J. L. WALKER,1970, Appl. Opt. 9, 201. GADDIS,M. W. and D. D. WELTER,1978, Synthesized Color Holography, in: Digest of Technical Papers, OSNIEEE Cod. on Lasers and Electro-optical Systems, San Diego, 1978 (IEEE, New York) p. 34. GALE,M. T. and K. KNOP,1976, Appl. Opt. 15, 2189. GRAUBE, A,, 1973, Opt. Commun. 8, 251. 1980, Appl. Opt. 19, 3044. GROVER,C. P. and R. TREMBLAY, HAIUHARAN, P., 1972, Opt. Commun. 6, 377. HAIUHARAN, P., 1976, Opt. Commun. 17, 52. P., 1978, Opt. Acta 25, 527. HARIHARAN, I&WHAMN, P., 1979, Opt. Acta 26, 1443. HAIUHARAN, P., 1980a, J. Optics (Paris) 11,53. HARIHARAN, P., 1980b, Opt. Engg. 19, 636. HARMARAN, P., 1 9 8 0 ~ Opt. Commun. 35, 42. and W. H. STEEL,1979, Opt. Acta 26,289. HARIHARAN, P., Z. S. HECEDUS

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1977, Opt. Lett. 1, 8. HARIHARAN, P., W. H. STEEL-and Z. S. HEGEDUS, HUFF, L. and R. L. FUSEK,1980, Opt. Engg. 19,691. HUNT,R. W.G., 1977, Rep. Progr. Phys. 40, 1071. IH, C. S., 1975, Appl. Opt. 14, 438. IH,C. S., 1978, Appl. Opt. 17, 1059. JOLY,L. and R. VANHOREBEEK, 1980, Phot. Sci. and Engg. 24, 108. KING, M. C., A. M. KNOLLand D. H. BERRY,1970, Appl. Opt. 9, 471. KOGELNIK, H., 1969, Bell Syst. Tech. J. 48, 2909. V. G., 1975a, Tekh. Kino i Telev. (USSR) No. 4, 31. KOMAR, KOMAR,V. G., 1975b. Tekh. Kino i Telev. (USSR) No. 5, 34. KOMAR, V. G., 1977, Progress on the Holographic Movie Process in the USSR, in: Proc. SPIE Vol. 120, Three-DimensionalImaging, ed. S. A. Benton (SPIE, Bellingham) p. 127. KOMAR,V. G. and Yu. N. OVECHKIS, 1976, Tekh. Kino i Telev. (USSR) No. 9, 18. KUBOTA.T. and T. OSE, 1979. Opt. Lett. 4. 289. KUBOTA,T.,T. OSE,M. SASAKI and K. HONDA,1976, Appl. Opt. 15, 556. KURTZNER,E. T. and K. A. HAINES. 1971, Appl. Opt. 10, 2194. and S. S. H. HSIAO,1972, Appl. Opt. 11, 2016. LEITH,E. N., D. B. BRUMM LEITH,E. N. and H. CHEN,1978, Opt. Lett. 2, 82. LEITH,E. N., H. CHEN and J. ROTH,1978, Appl. Opt. 17, 3187. 1964, J. Opt. SOC.Am. 54, 1295. LEITH,E. N. and J. UPATNIEKS, LESSARD,R. A., P. LANGLOIS and A. BOIVIN,1975, Appl. Opt. 14, 565. LESSARD. R. A,, S. C. SOMand A. BOIVIN,1973, Appl. Opt. 12, 2009. 1967, Appl. Opt. 6, 1255. LIN,L.H., and C. V. Lo BIANCO, G. W. STROKE and A. E. LABEYRIE, 1966, Bell Syst. Tech. J. LIN,L. H., K. S. PENNINGTON, 45, 659. MANDEL, L., 1965, J. Opt. SOC.Am. 55, 1697. MAROM, E., 1967, J. Opt. SOC.Am, 57, 101. 1968, Appl. Phys. Lett. 12, 10. MCCRICKERD, J. T. and N. GEORGE, MuZ~K,J. and J. R~TZEK,1974, lemna Mech. and Opt. (Czech.) 19,284. NISHIDA, N., 1970, Appl. Opt. 9, 238. NOGUCHI, M., 1973, Appl. Opt. 12, 496. OKOSHI, T., 1977, Projection-TypeHolography, in: Progress in Optics Vol. XV, ed. E. Wolf (North-Holland, Amsterdam) p. 141. PAQUES, H., 1966, IEEE Proceedings 54, 1195. K. S. and L. H. LIN,1965, Appl. Phys. Lett. 7, 56. PENNINGTON, N. J., A. A. WARD,R. CULLENand D. PORTER,1980, Phot. Sci. and Engg. 24, PHILLIPS, 120. G. W. and A. E. LABEYRIE, 1966, Phys. Lett. 20, 368. STROKE, STROKE,G. W. and R. G. ZECH,1966, Appl. Phys. Lett. 9, 215. SUKHMAN, Y.P., V. G. KOMAR, T. G. OVECHKLNA and G. A. SOBOLEV, 1977, Tekh. Kin0 i Telev. No. 11, 31. SUZUKI, M., T. SAITO and T. MATSUOKA,1978, Kogaku (Japan) 7, 29. TAMURA, P. N., 1977, Multicolor Image from Superposition of Rainbow Holograms, in: Proc. SPIE Vol. 126, Clever Optics, eds. N. Balasubramanian and J. C. Wyant (SPIE, Bellingham) p. 59. TAMURA, P. N., 1978a, Appl. Opt. 17, 2532. TAMURA, P. N., 1978b, Appl. Opt. 17, 3343. TATUOKA, S., 1971, Japan J. Appl. Phys. 10, 1742. THEOWICAL SOCIETYOF AMERICA,Committee on Colorimetry, 1953, The Science of Color (Crowell, New York) p. 244.

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TSUNODA, Y. and Y. TAKEDA, 1975, IEEE Trans. Electron Devices ED-22, 784. J. and C. D. LEONARD, 1970, IBM J. Res. Dev. 14, 527. UPATNIEKS, UPATNIEKS, J., J. MARKSand R. J. FEDOROWCZ, 1966, Appl. Phys. Lett. 8, 286. VLASOV,N. G., R. V. RYABOVA and S. P. SEMENOV, 1977, Zh. Nauch. Prikl. Fotogr. Kinematogr. 22, 384. WOOD,R. W., 1899, Lond. Edinb. Dubl. Phil. Mag. 47, 368. WYANT,J. C., 1977, Opt. Lett. 1, 130. YANO,A. and T. MATSUMOTO, 1973, Proc. 34th Fall Meeting of Japan SOC.Appl. Phys., p. 94. YAN-SONG, C., W. Yu-TANGand D. BIZHEN,1978, Acta Phys. Sin. (China) 27, 723. Yu, F. T. S., P. H. RLITERFJUSCH and S. L. ZHUANG, 1980, Opt. Lett. 5, 443. Yu, F. T. S., A. M. TAI and H. CHEN,1978, Opt. Commun. 27, 307. Yu, F. T. S., A. M. TAIand H. CHEN,1980, Opt. Engg. 19,666. S. L., P. H. RLWERBUSCH, Y. W. ZHANGand F. T. S. Yu, 1981, Appl. Opt. 20, ZHUANG, 872.

E. WOLF, PROGRESS IN OlTICS XX @ NORTH-HOLLAND 1983

V

GENERATION OF TUNABLE COHERENT VACUUM-ULTRAVIOLET RADIATION? BY

WIESLAW JAMROZ" and B. P. STOICHEFF Department of Physics, Uniuersity of Toronto, Toronto, Ontario, M5S 1 A7, Canada

I'Research supported by the Natural Sciences and Engineering Research Council of Canada, and the University of Toronto. * Visiting scientist from: Institute of Physics, Technical University of Lodz, Wolczanska 219, 93-005 Lodz, Poland.

CONTENTS PAGE

$ 1. INTRODUCTION

$ 2. THEORY

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

327

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

328

0 3. EXPERIMENTAL RESULTS . . . . . . . . . . . 349 §4

. CONCLUSION . . . . . . . . . . . . . . . .

REFERENCES

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

377 377

8 1. Introduction The availability of tunable dye lasers in the visible and near-infrared wavelength regions has had a profound effect on spectroscopy, and holds promise for many new applications in atomic and molecular studies. At the present time, the important vacuum-ultraviolet (VUV) region, below 2 0 0 ~ 1 lacks , tunable lasers. In fact only a few lasers operate in this region, and these emit at discrete wavelengths or are tunable over limited wavelength ranges. While substantial effort has been expended in attempts to develop V W (200 to 1 O O n m ) and XUV (100 to -2Onm) lasers, without matching success, there has been notable progress in the use of nonlinear, frequency-mixing techniques for generation of tunable, coherent radiation below 200 nm. The observation of second harmonic generation (SHG) by FRANKEN, HILL, PETERSand WEINREICH [1961] was a crucial step leading to the eventual production of coherent radiation in the V W region. This observation was quickly followed with the classic theoretical paper on second and third order nonlinear susceptibilities by ARMSTRONG, BLOEMBERGEN, DUCUING and PERSHAN [19621. Third harmonic generation (THG) at 231 nm was demonstrated by MAKER,TERHUNE and SAVAGE [1964] in crystals, glasses and liquids. The major problem of generating even shorter wavelengths (due to the limited transparency of many nonlinear solids to the region above -200 nm) was resolved when NEWand WARD [1967] succeeded in producing THG in a number of gases. HARRISand MrLEs [1971] then demonstrated that high conversion efficiency of THG and of sum-frequency mixing could be obtained by using phase-matched metal vapors as nonlinear media, and that efficiency could be improved further by resonance enhancement (MILESand HARRIS [19733. Tunability is achieved by the use of tunable pump lasers, but invariably with reduced efficiency since resonance enhancement cannot be maintained. To date, coherent THG has been generated with peak powers up to -1 M W , and with limited tunability to wavelengths as short as -57 nm, and seventh harmonic radiation at 38nm. A new method, providing 327

328

rv, 5 2

TUNABLE COHERENT VUV

tunability over broad regions, and using four-wave sum-mixing (4-WSM), 2w, + o2+w3, with the advantage of resonance enhancement, was introduced by HODGSON, SOROKIN and WYNNE[1974]. They used two dye lasers, one tuned to a two-photon allowed transition of a nonlinear metal vapor, and the other tunable over a broad frequency range w 2 , such that 2wl + w2 corresponded to a transition from the ground state to a broad auto-ionizing state of the metal vapor. In this way they succeeded in generating tunable coherent radiation over broad regions of the VUV. Such radiation is now produced from 200 to 106 nm by this technique, with line-width <0.1 cm-' and with peak powers >10 W. The purpose of the present chapter is to review the general principles of frequency mixing, and to describe the experimental progress towards attainment of tunable coherent radiation in the VUV and XUV regions. The process of frequency mixing and third harmonic generation is best understood by means of the nonlinear susceptibility of atomic systems when irradiated by intense electromagnetic waves. The relevant theory is presented in the next section together with discussions on resonance enhancement, conversion efficiency and limiting processes. A brief extension of the theory to higher order nonlinear effects is also included. In the following section on experimental results, descriptions of 4-wave summixing and third harmonic generation experiments are given. In particular, experimental techniques which have achieved extensive tunability are reviewed. These are essentially of two classes: those making use of rare gases for nonresonant sum and difference mixing, and those using 2electron atoms for resonant sum mixing via auto-ionized states. A brief discussion of the accomplishments of spontaneous anti-Stokes scattering is included, and mention is made of experiments on higher order mixing which have produced coherent radiation to 38 nm.

8 2. Theory 2.1. NONLINEAR SUSCEPTIBILITIES

A nonlinear optical medium is characterized by an induced polarization on the electric field B of incident radiation. This polarization may be described by a power series expansion

F, which is dependent

...

~ = ~ ( l ) B + ~ ( 2 ) : B ~ + ~ ( 3 ) ~ (2.1) ~ ~ ~ + .

v,

21

329

THEORY

The quantities ?i'" are macroscopic susceptibility tensors, %(') being the usual linear susceptibility tensor, and %('), %(3) the nonlinear susceptibility tensors of second- and third-order, respectively. With the incident radiation assumed to have a finite number of frequency components, mi, the electric field of the wave E(T,t ) can be expanded as a Fourier sum

The electric field will induce in the medium a time-dependent polarization p(F,t ) given by (2.3)

-

The induced polarization component at frequency os= w1 + w2 + * is related to the applied electric field components by the nonlinear susceptibility tensor %(os; wl, 02, . . .): P r s = K ( ~ s w; I ,

02,.

. .)X+JOS;

w~,oZ,

. . .)EY'E>.

* *

.

(2.4)

The coefficient K arises from the Fourier expansions of the polarization and also from the composition of the set of field frequencies. The tensor %(') only gives rise to polarization components at the incident frequencies. The lowest order term producing nonlinear effects is However, this tensor has nonzero components only in noncentrosymmetric systems (for example, piezo-electric crystals); isotropic media, such as cubic crystals, liquids and gases, by virtue of their centrosymmetry do not display quadratic nonlinearities. The next coefficient %"' has 3 independent and nonzero elements for isotropic media. In this case, eq. (2.4) may be written in the following form:

x(').

Pi3' = KX::',,E, (I? * I?).

(2.5)

It is this term which gives rise to four-wave sum-mixing (4-WSM) processes. Numerical values of K for some third-order nonlinear polarization effects are presented in Table 1. Calculations of the nonlinear susceptibilities for atomic systems have been carried out by several authors. ARMSTRONG, BLOEMBERGEN, DUCUING and PERSHAN [1962] were the first to calculate nonlinear susceptibilities using quantum theory. In their work, the atomic system was assumed to have a finite number of energy levels of zero width. O m and WARD [1971] later presented a theory of nonlinear polarization taking into account the nonzero linewidth of atomic energy levels. The interacting

330

TUNABLE COHERENT VUV

TABLE 1 Numerical values of K(o,; ol, 0 Process Nonlinear d-c polarization Electro-optical Kerr effect Optical Kerr effect Third-harmonic generation D-C electric field induced second-harmonic generation 4-WSM

~

~

~

3

)

"3

K

0

0

1

0

0

0

3

0

0

0

-w

3

30

0

0

w

i

20

0

0

"

2w,+w2

o1

o1

w2

0 s

0 1

0 2

0

0

0

$

3

electromagnetic field was treated as having a Dirac delta-function frequency spectrum. Their result for the microscopic susceptibility describing the third order nonlinear processes, us = 3 0 (THG) and w s = 201+ 0 2 (4-WSM) may be expressed as follows:

v. 6 21

33 1

THEORY

Here, e is the electronic charge, h is Planck’s constant/2~,Z1,2 represents the average of all terms generated by permuting frequencies, and is the pth Cartesian component of the dipole matrix element between two (and fl* the states ( b ) and la). In the denominators, f l a b = 0 , b -i(rab/2), complex conjugate) with hwab= Eabthe energy difference between states la) and Ib), and r , b the linewidth of the transition. In general, the microscopic nonlinear susceptibility 2(3)in eq. (2.6) is related to the macroscopic %3) of eq. (2.4) by:

rs)

%ws;

w1702,.

*

.I =“R(os)R(w,)R(wz)

*

*12(ws;w19w2,

*

f

.I

where N is the number density of microscopic systems and R ( w i )is the ratio of the local to the applied field at frequency wi (ARMSTRONG, DUCUING and PERSHAN r1962-J. For gases at pressures less BLOEMBERGEN, than 10 atm all R(wi) are unity, so that % = Nf (WARDand NEW[19695). One may use eq. (2.6) for numerical calculations of the third-order susceptibilities of atoms. Absolute values of the matrix elements rab may be found from tabulated oscillator strengths (e.g. WIESE, SMITHand GLENNON [1966], WIESE, S m and Mms [1969]) by using a formula given by MILESand HARRIS[1973]:

(2.7) where f n l ; n 9 1 + l is the oscillator strength for a transition between 2 states In, I ) and In’, I + 1). If the energy E in eq. (2.7) is in rydbergs, then r is given in Bohr radii. WANG[1970] has found an empirical relation between the linear x“’ and third-order x‘~’susceptibilities of gases at low pressures. He has shown that at relatively low frequencies, eq. (2.6) reduces to

in the approximation where a single oscillator with absorption frequency and oscillator strength feff replaces the series of oscillators with different absorption frequencies w,b. g is a dimensionless quantity involving the properties of the ground state as well as the excited states of the system. For gases g = 1.2. wo

2.2. RESONANT ENHANCEMENT AND TUNABILITY IN GASES

In general, the nonlinear susceptibilities X(3)for gases are much smaller than the corresponding values for solids and liquids. (Some examples of

332

TUNABLE COHERENT W V

TABLE 2 Values of 3rd-order nonlinear susceptibilities X'"(0 -+ 0) Material

X'3'(in cm3/erg) 2 x 102'*

Xe (gas)

Mg (vapor) Cs (liquid) NaCl (monocrystal) Ge (monocrystal)

2X 4 x lo-"

* Calculated for the

number density

N = 1OI6 atoms/cm3.

values of nonlinear susceptibilities X(3' for different materials are presented in Table 2.) However, the use of nonlinear processes in gases provides a number of important advantages in comparison with solids. First, many gases are transparent in the vacuum ultraviolet region, while the short wavelength limit for crystals is about 200 run. Secondly, the isotropy of the gases does not show the Poynting vector walk-off observed for birefringent phase matching in crystals. Thirdly, the maximum applicable power densities are higher in gases, and in addition, if breakdown does occur the material is not destroyed. Finally, in gases it is possible to realize significant enhancement of the value of x'~'by using transition resonances. From eq. (2.6) it is seen that such enhancements may be obtained when the incident radiation frequency is tuned to one-, two-, or three-photon allowed transitions. Resonances occur whenever the applied frequencies w1 and w2 are such that the real part of the resonance denominator vanishes, for example when Re 10og-~ll =O Re (Ong-201)= O Re (0,,,,-20,-w21

=O.

We can see these resonances in Fig. 1, which depicts the behaviour of the nonlinear susceptibility for third-harmonic generation ~ ' ~ ' ( 3of0 )lithium vapor calculated on the basis of eqs. (2.6) and (2.7). For the 2-photon resonance, a new feature arises which is absent in lower-order processes; the resonance enhancement of x ' ~ can ) occur without strong linear absorption of the incident beam. For the resonance 4-WSM process, the third-order susceptibility ~ ' ~ ' ( + 20 02 , )evaluated

v, 5 21

THEORY

333

5 INCIDENT WAVELENGTH ( p1 Fig. 1 . Nonlinear susceptibility x"'(3w) in esulatom for lithium vapor versus incident wavelength (MILESand HAMS [19731.

from eq. (2.6) may be separated into a resonant part containing the resonant transition 18) + In), and a nonresonant part including contributions of all the other transitions. The resonant parts are orders of magnitude larger than the nonresonant ones. PUELL,SCHEINGRABER and VIDAL[1980] have shown that for the resonant case eq. (2.6) simplifies to

(2.9)

The calculated values of ~ ' ~ ' ( 3 w for) 2-photon resonances in several metallic vapors are given in Table 3. A comparison of N ~ ' ~ ' ( 3for 0 ) Mg given here and in Table 2 shows an increase of lo4 for the resonant over the nonresonant susceptibility. If the levels Im) correspond to energies that exceed the ionization limit, there can be an additional enhancement of the generated signal. (All of these cases are illustrated on Fig. 2.) This enhancement may arise from interaction of levels Im)with the continuum

334

TUNABLE COHERENT VUV

TABLE3 Calculated values of the resonant susceptibility ~ ‘ ~ ’ ( 3 w ) Material Cs Mg

Sr

Zn

2-photon resonance state

Ix(~’(~o)~,,,

(in

esulatom)

6~9d(~D,,,) 3.0 (LEUNG, WARDand ORR[19743) 3s3d(’D2) 0.0897 (JUNGINGER, PLJELL, SCHEINGRABER and VIDAL [19801) 5s5d(’D2) 0.51 (SCHEINGRABER, PUELLand VIDAL[19781 4s5s(’S0) 0.31 (JAMROZ, LAROCWEand STOICHEFF [t982])

state of a gas or vapor, and also with auto-ionizing levels which can exist in this energy range. SOROKIN, WYNNE,ARMSTRONG and HODGSON [1976] have discussed the contribution of the auto-ionizing states and of the continuum with which it interacts. They have shown that in such situations it is possible to present the nonlinear susceptibilities x ‘ ~ ’in the following way

(2.10) where rgw, and rT., are the matrix elements of the dipole moment operator from the ground state Ig) to the perturbed continuum state ZY’, and from the state ZY’ to the intermediate states In), respectively. C is a normalization constant, and the integration takes into account the contribution of the auto-ionizing states as well as of the continuum. (a 1

g:; v:: (b)

(C)

w.z////////////h

lg’

Fig. 2. 4-level systems indicating the relevant energy levels for four-wave sum-mixing processes: (a) nonresonance, (b) 2-photon resonance, (c) incident energy (20, +o,)h, includes 2-photon resonance and reaches the continuum or auto-ionizing level.

v, § 21

335

THEORY

When (20, + w2) corresponds to a broad auto-ionizing level, the integral in eq. (2.10) may have a large contribution to the value of x ' ~ ' Another . important result of such interactions with broad auto-ionizing levels is that tunable radiation may be generated by keeping 2w1 fixed, and by varying w2 so that 20, + w2 scans through the auto-ionizing states or the continuum. ARMSTRONG and WYNNE[1974] have shown that the integral in eq. (2.10) may be written in terms of the so-called Fano parameters qg and qn, making use of the following relationships:

rgy,= rgy(qgsin A -cos A)

(2.1la)

ryPn= ryn(qn sin A -cos A).

(2.11b)

Here rgy and ryn are the matrix elements from the ground state 18) to the unperturbed continuum state F and from the state P !' to the intermediate states In), respectively. Also A=-arctg-;

1 &

and

E=-

- OAR rAR/2

is the frequency offset from the Fano resonance WAR normalized to its halfwidth rm/2. FANO [1961] has shown that the parameter qg may be calculated by using the ratio of the actual absorption cross section, a,,,, in the vicinity of an auto-ionizing line to that part of the continuum, a,, which interacts with the discrete state (Irgy12 = aa):

(2.12) Here aunis a part of the continuum which does not interact with the discrete state. By knowing the absorption cross section for a vapor and using eq. (2.12) one is able to find the parameter qg. For example, Fig. 3 presents the absorption cross section for Sr-vapor and Table 4 the results of calculations of the parameters qg (GARTON, GRASDALEN, PARKINSON and REEVES [1968]). On the basis of the discussion of ARMSTRONG and BEERS[1975] we may replace eq. (2.10) by the following:

336

[V, 8 2

TUNABLE COHERENT VUV

:li A E

-g

I

2050

I

I

2000

2025

40

I

1975

I

1950 p50

II

1925

I

1900

1

1875

MA)

Fig. 3. Auto-ionization resonances seen in absorption in SrI (CARTON,CRASDALEN, PARKINSON and REEVES [19683.

where X = 2(w - w R ) / r R is the normalized frequency offset of the 4-WSM signal from its resonance wR. The real and imaginary parts of ( ~ ( ~ ' are 1 given by

(2.14) When the 4-WSM signal is far from the auto-ionizing resonance (i.e. TABLE4 Parameters characterizing auto-ionizing resonances in SrI (CARTON,GRASDALEN,PARKINSON and REEVES[1968]). Resonance No. hAR[A] (Fig. 3)

1 2 3 4 5 6

2024 2018 1970 1891 1878 1867.9

u, (arbitrary units)

qe

too narrow for analysis 0.18 0.29 0.11 0.15

-5.2 0.10 4.60 3.56

v, B 21

THEORY

337

EXF€RIMENT-

1.0-

0.806-

(b)

X Fig. 4. Comparision of experimental data of 4-WSM signal and theory for Sr vapor, with 2-photon resonance at (a) the 5pz IDz state, and (b) the 5s5d ID, state (ARMSTRONG and

BEERS[19751).

when X is large), Re I x ( ~ ) ( = 0 and Im I x ( ~ ) ) = a constant. This means that the nonlinear susceptibility of the smooth continuum is constant. On the other hand, when the 4-WSM signal is on resonance, the signal is enhanced by a factor -(q,qn)2. When qg is known for the transition between the ground state and a particular auto-ionizing level, then eq. (2.13) and an experimentally observed 4-WSM spectrum allow one to determine the parameter q,, for the transition between an intermediate discrete state In) and the given auto-ionizing level. Another way of determining the parameter qn has WYNNE,ARMSTRONG and HODGSON [1976]. been suggested by SOROKIN, The bound-continuum transition matrix elements rgY, and r, may be calculated by using known quantum defects and the numerical tabulations of PEACH [1967]. Figs. 4a and b present theoretical calculations of 4-WSM signals obtained by ARMSTRONG and BEERS[1975] for Sr-vapor using eq. (2.13).

2.3. CONVERSION EFFICIENCY

The power conversion efficiency in 4-WSM generation is the most important factor from the practical point of view. The relationship between the nonlinear susceptibilities x ( ~ and ) the power conversion coefficient q has been analyzed by several authors (BJORKLUND [1975],

338

[V,§ 2

TUNABLE COHERENT VUV

MILESand HARRIS [1973], BINTJES, SHEand ECKARDT [1978], WARDand NEW [1969]). BJORKLUND [1975] in his calculation has assumed that the incident electric fields I?( F) are lowest-order Gaussian modes which propagate concentrically along the z-axis with identical waist location and identical confocal beam parameters. With such assumptions, the fundamental electric field may be given as

E,(F)=E, exp(ikz)(l+iP)exp

[

-

k$f+:L;].

(2.15)

Here b is the confocal parameter defined by b = 27ra2n/h0 (with a, the beam-waist radius; n, the index of refraction; Ao, the vacuum wavelength), k, is the wave vector of the incident beam, and @ = 2(z - f ) / b is a normalized coordinate along the z-axis (f is the position of the focus along the z-axis). For incident electric fields given by eq. (2.15) and with limiting processes such as pump depletion, absorption, saturation, breakdown and thermal defocusing neglected, the power conversion factor q for the process w1 + o2+ o34o4 is found to be (BJORKLUND [1975]):

Here M"' are the total powers (in watts) of each of the fundamental (i = 1,2,3) and generated (i = 4) beams, ko and k4 are wavevectors of the generated radiation in vacuum and in the nonlinear medium respectively, and k' = k , + k2+ k3 (= k" for 4-WSM processes). If all wave vectors are expressed in units of cm-', N in atoms per cubic centimeter, x(3' in esu per atom, then the numerical constant A in eq. (2.16) for 4-WSM is equal to 1.58 x The function Fl depends only on the dimensionless parameters b Ak, f/L, b f L (where L is the length of an experimental cell). It characterizes the nonlinear medium and the experimental geometry, and is a figure of merit in the optimization of conversion efficiency. Fl is defined as the integral:

where the value -2flb is the location of the vacuum-medium interface. As one may see from eq. (2.16), the function Fl accounts for the effects of focusing and dispersion, which play important roles in this process. The integral of eq. (2.17) can be evaluated at two limiting regimes. One is the

v, P 21

339

THEORY

(a1

(b)

~,

““I,

,

,

;/lq,J

I

,

FI (Mk,IO,20, I 0)X 100

0 02 0 01

,/-

5,

0

-100

*

-60

-20 0 20

60

100

-12

-8

-4

0

4

bAk +

bAk-

Fig. 5. F , versus b Ak for (a) b/L = 10 and f/L =0.5, 20 and (b) blLs0.1 and f/L =0.5 (BJORKLUND [1975D.

plane-wave approximation for b>>L.In this case, the function Fl is proportional to s i x 2(AkL/2), and maximum conversion efficiency is attained with Ak = k4-k’=O (Fig. 5). The changes in Fl are connected with a slip in phase between the driving polarization and the generated radiation leading to destructive interference between radiation generated in different portions of the active region. The second limit occurs when the incident beam is focused in the medium: for b << L we have the “tight focusing” regime, and when b = L the “confocal focusing” regime. BJORKLUND [1975] has calculated the function PI for such cases and his result is shown in Fig. 5b. For “tight focusing”, the nonlinear process may be observed in negatively dispersive media only, since for A k r O the function F1=Oand according to eq. (2.16) the conversion factor q = O also. In the case of “tight focusing”, when the entire generating region is contained within the cell, the function Fl is given by:

[rrb Ak exp ($bAk)I2, 0,

for A k < O for Ak 2 0 .

The optimum value of Ak (which corresponds to the maximum of Fl) may be achieved when b Ak = -2. However, for 4-wave differencemixing processes, different optimum values of Ak have been found (BJORKLUND [19753>,namely b Ak = 2{,:

for for

01+02-03=04 01-02-~3=04.

As one may see from eq. (2.16), in order to optimize the process of

340

W,S2

“UNABLE COHERENT VUV

4-WSM, one should maximize the quantity N2Fl (assuming that parameare constant) by varying the parameters N, b Ak. blL ters M2)and M3) and f/L. For the condition where the confocal parameter b is much longer than the gas cell (the plane-wave approximation), Ak must be made equal to zero. This may be achieved by selecting an appropriate buffer gas and varying its density to satisfy the phase-matching relation:

N1 _

~SX$”(+J

c

-Ci

oixY’Coi)

Nz - wix:”(oi)-osx:l)(os).

(2.18)

i

HARRISand MILES [1971] proposed this technique for third-harmonic generation in a mixture of rubidium and xenon. As an example, the refractive index of a phase-matched Rb and Xe mixture is shown in Fig. 6. One can see from Fig. 6 that for rubidium the dispersion Ak = k, - 3kl (for A l = 1.06 pm as an input wavelength and A, =0.35 pm as THG signal) is negative. Since the refractive index of Xe gas demonstrates normal dispersion over this region (dotted line in Fig. 6), it is possible to make Ak = 0 by adding a predetermined pressure of Xe, according to eq. (2.18). In the same way it is possible to achieve phase-matching in the focusing regime. When the parameters b Ak, b/L and f/L are independent of N, the optimization procedure simplifies to increasing N to the highest possible value, while independently maximizing Fl.A different optimization procedure is required when N is a function of b Ak. BJORKLUND [1975] has considered the case when Ak is proportional to N, while b is constrained to be constant and N is a free parameter. He defined a new,

-! I

106,~1

IPART Rb

’

412 PARTS X e l

Ii

-

0.35~

Fig. 6. Refractive indices of rubidium and xenon in the wavelength region 1.06 to 0.3 Frn (MILES and HARRIS [19733).

v, § 21

THEORY

341

Fig. 7 . G, versus b Ak for b/L 50.1 and f/L =0.5 (BJORIUUND [19751).

dimensionless function G,: GI = ( b Ak)2F1

represented by the graph of Fig. 7. In such a case the quantity to be maximized is G J b 2 (=Ak2F1).

2.4. SATURATION EFFECTS AND OTHER LIMITING PROCESSES

Various effects and processes may seriously limit the intensity of the generated signal or even completely cancel the production of harmonic or sum-mixing radiation. Here, we present some results of the influence of Doppler broadening, absorption, and saturation effects on the efficiency of nonlinear conversion. It is easily seen from eq. (2.9) that the linewidth r,,, (s2,,=0,,,i(I’,,,J2)) of the level In) plays a very important role in the 2-photon resonance enhancement of xC3’.Thus one should remember that the experimental resonance curve is modified by Doppler broadening arising from thermal motion of the atoms. The effect is essentially to broaden the resonance curve symmetrically from r,, (natural width) to Doppler width r,. It has been found, that at resonance

Here, rDis given in cm-’, k is the Boltzmann constant, T the temperature (“K), m, is the atomic mass, and c is the speed of light. If the energy of the generated signals exceeds the ionization limit of the nonlinear medium (see Fig. 2c), then significant (one-photon) absorption may occur which is characterized by the absorption cross section &)(as).

342

TUNABLE COHERENT VUV

rv.92

The influence of this absorption on the intensity I, of the generated third-harmonic signal is easily seen from the following expression (SCHEINGRABER, PUELLand VIDAL[1978]):

x[1 +exp{-NLa‘”(3w)}-2

exp {-NLa‘”(3w)/2} cos (AkL)]

(2.19)

where I, is the intensity of the incident beam and the ni are refractive indices. The above result assumes a linearly polarized incident wave and a homogeneous nonlinear medium, that is N(z) = A, a constant. For a“’(3w) = 0, eq. (2.19) yields the relation I3 -sinc2 (AkL/2). This result is the same as that obtained from the solution of eq. (2.17) for the plane-wave approximation (see Q 2.3). Saturation effects have been investigated by several authors (GEORGES, LAMBROPOULOS and MARJWRCER [19771, JUNGINGER, PUELL,SCHEINGRABER and VIDAL [1980], LEUNG, WARDand ORR[1974], MILESand HARRIS[1973], PUELL,SCHEINCRABER and VIDAL[1980], PUELLand VIDAL [1978], SCHEINGRABER, PUELLand VIDAL[1978], VIDAL[1980]). In their analysis of saturation of ~ ‘ ~ ’ ( 3 0PUELL, ), SCHEINGRABER and VIDAL [1980] have considered a number of effects at frequency 3 w and at the incident frequency w. These effects are described by the equations:

N Pc3’(w)= - [3x?’(w)E(3w>E*(~)E*(0) + xL3’(w)E(o) lE(w)I2 4 + xL3’(w, 3w)E(w) lE(34l21

(2.20a)

N P3’(3w) = - [ x ~ ’ ( ~ o ) E ( o ) E ( o ) E+ ( ~xL3’(3w)E(3w) ) IE(3w)I2 4

+ xg’(30,o)E(3w) IE(w)l”].

(2.20b)

All of the effects considered in eqs. (2.20) are responsible for saturation. ~ ;w ) is responsible for third-harmonic The term xy’(3w) = ~ ‘ ~ ’ ( 3w,0 w, generation; the term xy’(w)= ~ ‘ ~ ’ ;( 30, w -w, -a) describes the inverse process by which some of the third-harmonic radiation is transferred back to the fundamental wave. The real parts of xf’ represent intensity dependent changes of the refractive index, namely, xf’(3o)=

v, 8 21

343

THEORY

x(~’(w w,; -w, w ) is the optical Kerr effect at frequency o,and

xL3’(o)=

~ ( ~ ’ ( 3w, 3 0 ;- 3 o , 3 o ) is the optical Kerr effect at frequency 3 0 . The imaginary parts of xL3’(o) and xk3’(30) are responsible for 2-photon , the fundamental frequency is absorption; the cross section, ~ “ ’ ( w ) at given by

and a“’(3o) at the harmonic frequency, by a corresponding equation. 3 q;3 w ) describes the change of refracThe term xk3’(o,30) ~ ( ~ ’w,(- 0 tive index at the fundamental frequency due to the harmonic intensity; the term xb3’(3w,o)= ~ ‘ ~ ’ ( 30, 3 ~ 0-0, ; o) describes the change of the refractive index at the harmonic frequency due to the fundamental intensity. The imaginary parts of xk3’(w,3 0 ) = xL3’(30,o) give rise to Raman-type gain or losses. A detailed analysis of the optimum conditions for resonant and nonresonant third-harmonic generation including the above effects has led to the following result (e.g. VIDAL[1980]):

fq

I

XF’(34

= (n3,nJX;3’(0)+(n*,n3)x&3’(3w) -2 X & 3 b , 3 0 )

I

(2.21)

where the ratio f,, represents a parameter which characterizes the influence of saturation effects on the efficiency of the nonlinear conversion process. It has been shown that complete conversion is only possible for (f,,)-’ = 0. In this extreme limit, the maximum possible conversion efficiency is given by (2.22) For the nonresonant situation, eqs. (2.21) and (2.22) are dominated by the real parts of xL3’, whereas for the resonant case -the imaginary parts are the most important. This means that, for 2-photon resonant generation, saturation occurs due to 2-photon absorption, which leads to a depletion of the fundamental wave and to changes of level populations. For the nonresonant case, the generated signal is limited by the Kerr effect which gives rise to intensity-dependent changes of the refractive index destroying the phase matching conditions. The nonlinear parameters which appear in eq. (2.21) may be calculated by using eq. (2.6) or simply approximated by taking into account only the electronic part of

344

[V, s 2

TUNABLE COHERENT VUV

x‘~’(as done by JAMROZ[1980a, b])

to obtain (2.23)

where E~ is the permittivity of free space, me is the electron mass and p is the anharmonic force constant. [Q. (2.23) is written in MKS units. To convert it to esu units, one should multiply the RHS by the factor 0.7 x lo8.] It is seen from eq. (2.22) that the saturation limiting processes are due to the structure of energy levels of an atom. Thus, for resonant THG in vapors and gases, the conversion efficiency q of eq. (2.22) is determined by the ratio

where “m” is the highest energy level in eq. (2.9). Values of this ratio were calculated for several elements and are given in Table 5 . It is interesting to note that 2-photon saturation processes are stronger for the higher resonant levels. Also, this parameter for the lowest allowed 2-photon resonant transitions is nearly the same for different atoms, leading to the conclusion that all of these gases exhibit similar saturation limitations for resonant 4-WSM. Recently SCHEINGRABER and VIDAL[1981] have investigated theoretically the influence of a-c Stark shifts on THG in the vicinity of a two-photon resonance. They have assumed input intensities ranging from TABLE5 Theoretical values of the parameter (Cm.n,,-w)/(Emn,,-3~) Element

2-photon resonant level

Ern 0 m g - w

1, om,-3w

He

ls2s

1.6

Li

2s3s

Mg

3s4s 3s3d 4s5s 4s4d

1.7 1.7

Zn Sr

5s6s

Hg

6s7s

5s5d

1.3

1.7

1 .0 1.6

0.9 1.7

v, 5 21

THEORY

345

FUNDAMENTALWAVE (ern-' )

Fig. 8. Maximum achievable third harmonic intensity versus fequency of the fundamental wave in the region of the 5s2-5s5d two-photon resonance in strontium (SCHEMGRABER and VlDAL [198ln.

lo6 to 10" W/cm2. They found that the resonant enhancement in the small signal limit is reversed into a minimum of the conversion efficiency at high input intensities. Their result for Sr vapor is presented in Fig. 8. One can see that for the lowest input intensity the maximum achievable third-harmonic intensity reproduces the shape of the two-photon resonant transition. With increasing input intensity the two-photon transition begins to saturate, and for @=108W/cm2 the conversion efficiency slightly off -resonance becomes larger than that on resonance. This is an example of a 'self-induced resonance effect', i.e. an effect for which the dynamic Stark shift caused by an intense field can lead to enhancement or, as in this case, can shift the level away from resonance. At the actual Stark shifted resonance frequency (Fig. 8) a deep minimum appears. This minimum is caused by the population density of the two-photon resonant state, since only the population difference of the two-levels contributes to the THG. An influence of the Kerr effect is also visible in Fig. 8, namely that the wings of the high input intensity curves are slightly asymmetric with respect to the observed Stark-shifted resonance-frequency. This asymmetry is due to the real part of the xp' in eqs. (2.20) to (2.22). The

346

TUNABLE COHERENT VUV

rv, pi 2

contribution of the Kerr effect to the mismatch Ak, in eq. (2.17) is different for the two sides of the resonance transition.

2.5. HIGHER ORDER NONLINEAR EFFECTS

Higher order nonlinear processes are of interest for the generation of coherent light in the extreme ultraviolet region. Generation by thirdorder effects becomes increasingly difficult because of the scarcity of intense coherent sources at the required pumping wavelengths. The development of frequency conversion techniques using higher-order nonlinearities offers an alternative to this approach. In the general case, for an arbitrary combination of frequencies one may obtain the following expression for the microscopic nonlinear susceptibility of order p ( ~ ~ I T E VPAVLOV , and STAMENOV [19793>

where the symbols have the same meanings as those in eq. (2.6). The sums are performed over all t, states of the discrete energy spectrum and over all a, frequencies of the electromagnetic field. Some values of the 5th, 7th and 9th order susceptibilities obtained by using eq. (2.24) and by summing over the 3s-l0s, 3p-lop and 3d-10d levels for sodium vapor at h l = 106 nm are given in Table 6. It can be shown by comparison of eqs. (2.6) and (2.21) that additional resonances involving three- and four-photons may be used to enhance higher-order generation. In general one expects that the higher-order TABLE6 Nonlinear susceptibilities for sodium vapor, in esu/atom ( m v , PAVLOV and STAMENOV [1979])

v, 4 21

347

THEORY

harmonics which involve higher-orders of perturbation theory will be weaker than those of lower order. But, if one considers the ratio: rl = MP+Z)/MP=) x(P+Z)/X(P)(E/2)2

(2.25)

where (E/2)2 is related to the intensity at the incident field, then it is possible that the higher-order polarization can be comparable to, or even larger than lower-order polarizations (REINTJES, SHE and E~KARDT [19781). HARRIS [19731 has discussed the nonresonant higher-order nonlinear processes at the sum frequency w, = w1 + w 2 + * + w, and has found that the conversion factor q may be written as:

-

(2.26) where J(wn) is the incident energy density at the highest applied frequency on.The quantity hwS/2o"'(w) which appears in eq. (2.26) is called the saturation energy density, i.e. the density which would approximately saturate the transition, As we may see from eq. (2.26), the conversion efficiency is independent of the order of the nonlinear effect, and also of the oscillator strengths and positions of the intermediate levels. In Table 7 some results of the calculated conversion factors for higher-order harmonics are presented. A second interesting feature of the higher-order nonlinear effects should be pointed out; namely, one expects the saturation threshold of the 3-photon resonance to be higher than that of 2-photon resonances, and the highest for 4-photon resonances. REINTJES, SHEand ECKARDT [19781 have analyzed the optimum conditions for higher-order processes, and have calculated for 3rd, 5th and 7th harmonic generation a function

TABLE 7 Calculated conversion factors for some higher-order nonlinear processes (HARRIS [19733) Process

Material

3X5320A-1773A 5 x 5320 A -+ 1064 Sx1182A-+236A 7x1182A-169A 15 x 2660 + 177 A

Xe Xe Li+ Li+ Li+

q(%)

0.084 0.05 1 0.002 0.004 4x

348

TUNABLE COHERENT VUV

[V, 5 2

- bAk Fig. 9. Variation of the quantity ( b AkF(* as a function of b Ak for third-harmonic (a), fifth-harmonic (b), and seventh-harmonic (c), generation (REINTJES,SHE and ECKARDT [1978n.

F which is equivalent to F, in eq. (2.16) used for third-order processes. Their results are shown in Fig. 9. The function ( b AkF)’ plotted in Fig. 9 was calculated for “tightfocusing”, with the beam focused not in the centre, but at the exit window of the gas cell. This function plays the same role as the function GI (5 2.3), which is valid for focusing in the centre of the cell. A significant difference between the functions F and GI is evident from a comparison of Figs. 9 and 7; namely, that GI = 0 for Ak > 0, while F# 0 when Ak is positive (or negative). Thus there is no possibility of producing gain in third order processes when focusing in the centre of the sample cell and when A k r O . However, nonzero conversion is possible for media with positive dispersion when focusing at the exit window. This is particularly advantageous for XUV generation (
v, I 31

EXPERIMENTAL RESULTS

349

nonlinear interactions in two separate steps, i.e. first, tripling, w + w + w = 30, and second mixing, w + w + 3 0 = 5 0 . In the plane-wave approximation, -90 percent conversion efficiency of the pumping energy into the fifth harmonic is shown to be possible. However, saturation effects and other limiting processes have not been considered.

Q 3. Experimental Results 3.1. GENERAL TECHNIQUES OF FREQUENCY CONVERSION

Frequency conversion into the VUV and XUV regions has been achieved by a variety of laser systems. Powerful pulsed lasers such as ruby, Nd:YAG, Nd in glass, flashlamp pumped dye (FPD), rare gas excimer (RGE) and rare gas halide exciplex (RGHE) lasers provide the primary coherent radiation. Visible radiation (>400nm) is usually doubled once or twice in nonlinear crystals to produce coherent radiation in the UV to about 200nm. Subsequently, the coherent UV radiation is converted to coherent V W and X W by THG or frequency mixing in rare gases or metal vapors. Specific atomic (and molecular) systems are selected because of their large third order nonlinear susceptibility, negative dispersion for phase matching, suitability of energy levels for resonance enhancement, and low absorption at the desired VUV or XUV wavelength. Limited tunability has been obtained by tuning over the bandwidth of the primary laser itself, and over broader regions by the use of parametric generators and dye lasers in mixing experiments. Extensive tunability has been achieved by using frequency mixing to reach broad auto-ionized states of atomic metal vapors. Here, we describe briefly, two early, yet general methods of generating VUV and X W radiation which have essentially set the pattern for f u t u e work. In the sections which follow, we review specific systems and experiments based on third order nonlinear processes with phase matching and resonance enhancement, and which have achieved some success in producing VUV and XUV radiation with high efficiency, narrow linewidth, and broad wavelength tunability. The pioneering work of Harris and his co-workers at Stanford University led to the first announcement of efficient THG and tunable frequency mixing in the VUV and X W . Their experiments are shown in Figs.

350

R,§3

TUNABLE COHERENT VUV

APERTURE, WEN

/

I

'

SATURABLE 7ABSORBER

COPPER WICK

I

aLmm

PARAMETRIC WATER COOLER

/CsI PHOTOMULTIPLIER

WPL'FIER 75cm F L 0 LENS 1

MODE LGCKED QUARTZ WINDOW

LiF WINDOW

m

GAS MANIFOLD

(a 1

(b)

Fig. 10. Experimental methods for generating harmonic and sum- and difference mixing VUV radiation by third order nonlinear processes (a) in a phase-matched metal vapor, at fixed wavelengths (KUNG,YOUNG, BJORKLUND and HARRIS [19723), and (b) in xenon gas, and broadly tunable (KUNG [19741).

10a, b. The primary laser source was a mode-locked Nd :YAG laser and amplifier producing 1.06 km radiation in several pulses each of 50 psec duration, and peak power of -2OMW. This radiation was doubled to 532 nm in a KDP crystal, and then used directly, or doubled again, or mixed with 1.06 p.m radiation in a second KDP crystal, to produce radiation at 532, 266, or 355 nm with -10% efficiency. In this way, high-power radiation at several fixed wavelengths was available for nonlinear mixing t o shorter wavelengths in metal vapors and rare gases. Initial experiments in a Cd :Ar phase-matched mixture produced summing of 1.064 and 2 x 354.7 to yield 152 nm, and tripling of 532 nm and 354.7 nm to yield 177.3 and 118.2 nm radiation, respectively (KUNG, YOUNG, BJORKLUND and HARRIS [1972]). In a phase-matched mixture of Xe:Ar, THG of 354.7 to 118.2 nm was achieved (KUNG,YOUNGand HARRIS [1973]), and in Ar, T H G of 266 to 88.7nm (HARRIS,YOUNG, KUNG,BLOOM and BJORKLUND [1974]). Tunability was added to this coherent source by using the 266 nm radiation to pump an ADP parametric generator (Fig. lob). Variation of the crystal temperature from 50 to 105°C provided a tuning range of 420 to 720 nm, with >lo0 p J output in a bandwidth of 0.5 to 2 nm. The pump (266 nm) and resulting signal and idler radiation was then focused into Xe gas (at pressures up to 1 atm). THG and sum and difference mixing produced V W radiation tunable over portions of the region from 118 to 147 nm and continuously from 163 to 194 nm (Fig. ll), at peak

V, 8 31

351

EXPERIMENTAL RESULTS

ANGSTROMS 2wp

+

w5

2wp

+

w,

LYMAN a

1 0 6 4 loloa ~ 1122!?0648 wp

lxiImJ

+ 2ws

1330% wp

1173%

+ 2wi

2wp - w j

nOBSERVED 2wp

-

PREDICTED

wg

3%

Fig. 11. Tuning ranges of VUV radiation generated by third harmonic and sum- and difference-frequencymixing, using a parametric source (o,, wp, mi are the signal, pump, and idler frequencies, respectively). (KUNG[1974].)

-

-

powers of 1W corresponding to lo7 photons in a 20 psec pulse (KUNG [19741). Recent variants of these experiments are presented in 9 3.2. At about the same time as the above experiments on tunable VUV radiation, HODGSON, SOROKIN and WYNNE[1974] of the IBM Thomas J. Watson Research Center reported a new 4-WSM technique providing a more extensive range of continuous tunability in the V W , albeit at considerably lower peak power. Their experimental arrangement is shown in Fig. 12. A Nzlaser emitting -10 nsec pulses of -1 M W peak power at 337 nm was used to pump simultaneously, two dye lasers of the H ~ S C H [1972] design. These in turn, emitted up to 100 kW radiation of linewidth 0.1 to 1cm-' and tunable from 470 to 700 nm depending on the dye or dye mixture used. The two laser beams were orthogonally polarized, then spatially overlapped in a Glan-Thompson prism, and finally circularly polarized in opposed senses by a quarter-wave plate. These collinear beams of frequency v1 and v2 were focused into a heated cell of Sr (at -800 to 900°C). The radiation of frequency vI was tuned so that 2vl was in resonance with various excited states of even parity, and vz was swept to enable 2v, vz to reach into the ionization region. As already noted in

+

352

rv, 5 3

TUNABLE COHERENT W V

-

i,/

DYE LASER

c

1

N2

-LASER

SOLAR BLIND PHOTOMULTIPLIER

Fig. 12. Method of combining beams from two dye lasers to produce tunable coherent VUV radiation by 4-WSM in a metal vapor (HODGSON, SOROKIN and WYNNE [1974J).

P 2.2, additional resonance enhancement in x ( ~ arises ) when 2v, + v2 corresponds to term values of auto-ionizing states (Fig. 2) embedded in the continuum, eq. (2.10). Also, th.e simple technique of using left and right circular polarization for v l and v2 prohibits frequency tripling in isotropic media (since angular momentum is not conserved), and makes the 4-WSM process, v = 2ul + v2, the exclusive nonlinear effect to be observed. It is easily seen from eq. (2.5) that no radiation at the third harmonic, 3vl or 3v2, will be generated for pure circularly polarized incident radiation at v 1 and v2. In this way, Hodgson, Sorokin and Wynne used Sr vapor to generate the first tunable VUV radiation. Spectral ranges of -1200 cm-' around each of the wavelengths 181.5 and 193 nm (Fig. 13) were covered, with the shortest wavelength reached at -155 nm. The extension of this method to other 2-electron atomic systems having broad auto-ionizing levels is described in P 3.3. In the process of developing the tunable sources reviewed in 05 3.2, 1

1

1

,

1

I

I

r

/

2y1+y SIGNAL FROM 55 TORR STRONTIUM VAPOR DYE LASERS

1959 51046

1911 A u , ( % l 52329 v,,(crnll

Fig. 13. Intensity variation of tunable VUV radiation generated by 4-WSM in Sr vapor, in and WYNNE[1974]). the range 191 to 196 nm (HODGSON,SOROKIN

v, s 31

EXPERIMENTAL RESULTS

353

3.3, many investigations at fixed wavelengths (or with limited tunability) have been carried out. Some have stressed techniques and others, a better understanding of the concepts reviewed above. A list of these relevant experiments is given in Table 8. Brief mention of some of these contributions is warranted prior to more detailed descriptions of the tunable sources in Tables 9 and 10. The quantitative measurements and analyses by Vidal and colleagues (SCHEINCRABER, PUELLand VIDAL[1978], JUNGINGER, PUELL,SCHEINGRABER and VIDAL[1980], VIDAL[19803 of resonant THG in Sr and in Mg have laid the foundations for a detailed understanding of the enhancement due to resonant and phase matching processes, and to the loss mechanisms inherent in THG. The work of Freeman and Bjorklund and associates on enhancement of THG in electric (FREEMAN and BJORKLUND [1978]) and magnetic fields (ECONOMOU, FREEMAN and BJORKLUND [1978]), and in phase matching by frequency adjustments of the three input beams (BJORKLUND, BJORKHOLM, FREEMAN and LIAO[ 19773, led to BJORKLUND, ECONOMOU, LIAO generation of cw V W radiation (FREEMAN, and BJORKHOLM [1978]). The production of coherent VUV radiation in the spectral region around Lyman-a (121.6nm) was reported by three et al. [1977], groups, two using tripling experiments in Kr (BATISHCHE MAHON, MCILRATH and KOOPMAN[19783, and the third using resonant 4-WSM in Mg vapor (McKEE, STOICHEFF and WALLACE [1978J). Further developments in this interesting region were quickly reported by LANGER, PUELLand ROHR [1980], MAHONand YIU [1980] and WALLENSTEIN [1980]. Preliminary experiments with Hg at fixed wavelengths by BOKOR, PANOCK and WHITE[1981] and TOMKINS and MAHON [1981] laid FREEMAN, the groundwork for extensive tunability in the VUV and XUV regions (TOMKINS and MAHON[1982], FREEMAN, JOPSONand BOKOR[1982]). Finally, the broad range of experiments on nonresonant tripling and mixing in Ar, He and Ne by REINTJES, SHEand ECKARDT [1978] may be mentioned, since these helped in the inexorable push to shorter wavelengths and to the observation of seventh harmonic generation at 38 nm (REINTJES, SHE,ECKARDT, KARANGELEN, ANDREWS and ELTON [1977]).

3.2. TUNABLE GENERATION IN RARE GASES

The early work of HARRIS,YOUNG,KUNG,BLOOMand BJORKLUND [1974] on nonlinear mixing in Xe and Ar was quickly followed by work

wl W P

TABLE 8 VUV and XUV generation at fixed wavelengths

A (nm)

Nonlinear medium

Process

200 200

Ca Na:Xe

3 X 600 [R]” 3x602 [R]

195 194 194

TI Yb

3x5853 [R] 3 X 582 [R] 3 X 582 [R]

192 185.5

Sr:Xe Eu

3x576 [R] 2 x 576.7 +520 [R]

184.9, 143.5 183.3, 125.1 177.3, 152 171.2

Hg Hg Cd:Ar Sr

2x280.3-A [R] 2x312.9-A [R] 3x532; 2X355tA 652+433+501 [R]

169.7 155 143.7 130.7, 125.9 127.8

Sr Sr Mg:Kr Hg CaII

477+572+488 [R] 3 x 465 [R] 3x431 [R] 2 x 248.7 - A [R] 3 X 383.3 [R]

Ba

Primary laser

Reference

FPDb F~RGUSONand ARTHURS [1976] FPD; Nd : G l ~ s - D y c ~TAYLOR[1976], DXABOVICH,METCHKOV,MITEV, PAVLOVand STAMENOV[1977] FPD WANGand DAVIS[I9751 N,-Dye HEINRICH and BEHMENBURG[1980] N,-Dye SOROKIN, -STRONG, DREYFUS, HODGSON, LANKARD, MANGANARO and WYNNE [1975] FPD SCHEINGRABER, PUELLand V ~ A [1978] L SOROKIN, ARMSTRONG, DREWS, HODGSON, LANKARD, ~ G A N,-Dye NARO and WYNNE [1975] Nd :YAG-Dye BOKOR,FREEMAN, PANOCKand WHFE [1981] Nd :YAG-Dye T o w s and WON [1981] Nd:YAG KUNG,YOUNG,BJORKLLJND and HARRIS[1972] Nd :Y AG-Dye BJORKLW, BJORKHOLM, FREEMAN and LIAO [1977], ECONOMOU, FREEMAN and BJOFXLUND[1978] cw Ar++cw Dye FREEMAN, BJOFXLUND,&ONOMOU,LIAOand BJORKHOLM [1978] N,-Dye HODGSON, SOROKIN and W m [1974] N,-Dye JUNGINGER, PUELL,SCHEINGRABER and VDAL 119801 KrF BOKOR,FREEMAN, PANOCKand W m [1981] N,-Dye SOROKIN, ARMSTRONG, DREYFUS, HODGSON, LANKARD, MANGANARO and WYNNE[1975]

i

$ 2 g

8 E 3 z rn

2< -

-.< ;m

W

125.9,101.7 121.6 121.6 120.3 118.2

Xe Kr:Ar Kr:Ar Hg Cd :Ar, Xe:Ar

[RI 3x3643 3X364.8 2 x 313+ 521 [R] 3x355

Nd :YAG-Dye Ruby-Dye FPD Nd :YAG, FPD Nd :YAG

r19821 Ynr, B o r n and MCILRATH “C m BATISHCHE et al. [1977], LANGER, PUELLand ROHR[1980] Ll MAHONand YIU [1980] Hsu, KUNG,ZYCH,YOUNGand H w s [1976] KUNG,YOUNG,BJORKLUNDand HARRIS[1972], KUNG,YOUNG and HARRK [1973]

118.2 118.2, 106.4 112.4 97.2 89.6 88.7

Xe:Ar He Kr H

Ar,He, Ne

3x355 2X266.1 + A 3x337 2 x 243+ 486 3 x268.8 3X266.1

Nd :YAG Nd:YAG N2 FPD Nd :Glass Nd :YAG

ZYCHand YOUNGt-19781 REINTJES,SHE and E ~ K A R D T[1978] CO~TER[1979aI TROSHIN, CHEBOTAEV and CHERNENKO [1978] SLABKO, POPOVand LUKINYKH [19771 HARRIS,YOUNG,KUNG,BUW~M and BJORKLW[1974]; REINT m x,m m, SHEand ECKARDT [I9781

He, Ne

4X266.1TA

Nd :YAG

REWJES, SHEand

He, Ne,

5~266.1

Nd :YAG

REINTJES,

]

76’ 70.9 62.6, 59.1 53.2

Hg

Ar,Kr 38

a

He

7x266.1

Y

ECKARDT [1978]

ECKARDT, SHE, KARANGELEN,ELTON and ANDEWS [1976], REINTS, SHE and ECKARDT [1978]

; z

4

F m

c!

Nd:YAG

R E I N T ~SHE, , ECKARDT, KARANGELEO, ANDREWS and ELTON5 [1977] 2

[R] Resonant process. FPD Flash lamp pumped dye laser.

W ‘A

TABLE9 Tunable generation in rare gases Nodiiear medium

Processes

Xe 206160" 195-163 Xe 147-118 Xe 147-140 Xe:Kr Kr 130-110" 123.6-120.3 Kr Kr:Ar 123.5-120 Kr 121.6' Kr 121.6' Xe 106' Ar 102.7d Xe 83d 79' HZ 64d Ar, Hz, KI

2Aw-AL 2x266*AS, A: 2x266*Ah,,A: 3Ah, 21, + A, 3A, 3Ahe 3 X 364.8 3 x 364.8 3x318 3 x 308 3 x 248 2 x 193.6 + A,,, 3 X 193.6

Nd :YAG-Dye Nd :YAG and POb Nd:YAG and POb Nd :YAG-Dye Nd :YAG-Dye KrF-Dye Nd :YAG-Dye FPD Nd :YAG-Dye KrF-Dye XeCl cw Dye, Kr-Dye ArF, Dye cw Dye, ArF-Dye

3 x 171

Xe,

A (nm)

57d

AI

Primary laser

Auv from a range of laser dyes. PO =parametric oscillator with signal and idler wavelengths A,, hi. 'Tunable over a small region. dTunable over the RGE and RGHE laser or amplifier bandwidth.

a

Reference

HILBIGand WALLENSTEIN [1982] KUNG[1974] KUNG[1974] HLBIGand WALLENSTEIN 119811 HILBIGand WALLENSTEIN [1982] COTIER [1979b] HILBIGand WALLENSTEIN [198l] MAHON,M C I ~ T and H KOOPMAN [1978] WALLENSTEIN [1980] ZAPKA, COTTERand BRACKMAN [198l] REINTJES119791 EGGER, HAWKINS, BOKOR,PUMMER, ROTHCHILD and RHODES[1980] EWER, SRINIVASAN, BOYER,PUMMER and RHODES[1982] EGGER,ROTHSCHILD, MULLER,PUMMER, SRINNASAN, ZAVELOVICH and RHODES[1981] HUTCHINSON, LINGand BRADLEYr19761

V, I31

357

EXPERIMENTAL RESULTS

on Kr and Xe (Table 8). These investigations focused attention on the possibilities of efficient VUV generation with these gases, and culminated in the achievement of broad tunability over the region 100 to 200 nm. Nonresonant tripling in Kr was used by several groups to generate tunable radiation in the region of Lyman-a at 1 2 1 . 6 ~ 1(MAHON, MCILRATH and KOOPMAN[1978], COTTER[1979b] and WALLENSTEIN [19803. Various dye-laser systems were used having different beam quality, linewidth, and peak output power (1 to -10 M W ) . Each experiment used tight focusing, with Kr pressure adjusted to optimize Ak, the wave vector mismatch, over the region 120 to 124nm where Kr is negatively dispersive (§ 2.3). Conversion efficiencies of -lop6 were obtained, with a maximum output of -60 W (or -10" photons/pulse) in a 10 ns pulse (MAHON, MCILRATH and KOOPMAN [1978]), and a linewidth of -0.05 nm. More monochromatic output (-lop4 nm) was reported by COTTER [1979b], and by WALLENSTEW [1980]. In all three experiments, the peak power output was limited by optical breakdown. Experiments with nonresonant frequency tripling in Kr (as well as in Xe) were continued by HILBIG and WALLENSTEW [1981] who studied phase matching of Kr with positively dispersive Ar and Xe (and of Xe, with Kr). The conversion efficiency was increased substantially, but again limited by dielectric breakdown and by absorption of the V W in the gas mixture. Extensive wavelength tunability with rare gases was achieved by HILBIG and WALLENSTEIN [1982]. They used sum-frequency mixing (ovuv = 2oUvfoD) in Kr and Xe to generate VUV radiation of -20 W power, tunable over most of the range 110 to 130nm where these gases are negatively dispersive (Fig. 14). These results are summarized in Fig. 15. They also generated VUV radiation of -50 W power at longer wavelengths in Xe by difference-frequency mixing. The process ow= 2ow-wD resulted in radiation from 185 to 207 nm, and wwv = 2ww-oL at shorter wavelengths, from 160 to 190 nm. (The frequencies oL,oDand ow refer to the output of a Nd :YAG laser, a dye laser and harmonic of the dye laser respectively, as described below.) For these processes, generation may take place in a medium with Ak > O (§ 2.3) so that efficient tripling in Xe was possible at A > 146.9 nm (the first resonance transition). Since the positive Ak decreases continuously with increasing VUV wavelength, the pressure was increased from 100 Torr at 160 nm to >300 Torr at 186 nm (this being close to the dielectric breakdown limit at the laser focus in Xe). The tuning range and V W output power is shown in Fig. 16. The conversion efficiency in these

-

358

TUNABLE COHERENT VUV

rv, 8 3

Fig. 14. Wave vector mismatch per atom, A k = CN = 27r(n, - n,)/A, for frequency tripling in Kr and Xe versus wavelength of generated third harmonic, A, (MAHON,MCILRATH, MYERSCOUGH and KOOPMAN[1979]).Here N is the number of atoms per cm3 and n, and n3 the refractive indices at the incident (A,) and tripled (3A,-A,) wavelengths. The shaded areas are regions of negative dispersion used in generating VUV radiation shown in Fig. 15. The clear regions of positive dispersion correspond to the tuning curves of Fig. 16.

V, 8 31

359

EXPERIMENTAL RESULTS

1150

I100

-

I200

I250

[A]

-A,

I300

Fig. 15. Output power and tuning range of VUV generated by sum-frequency mixing in Kr and Xe (HUSK and WALLENSTEIN [1982D.

nonresonant mixing experiments was reported to be -lop6. In similar experiments with Xe, YIU, BONINand MCILRATH [1982] observed enhancement by a factor of -20, below 130nm, when 2wuv was scanned through a two-photon resonant state, and HAGERand WALLACE [19821 obtained an efficiency of 0.4% at 162nm (corresponding to an output power of 360 W) by using resonantly enhanced difference frequency mixing. Thus VUV radiation -10 times as intense as that of Figs. 15 and 16 appears to be within reach. For these experiments, HILBIGand WALLENSTEIN [1982] used the second harmonic radiation of a Nd:YAG laser (AL) to excite a dye-laser

1600

I700

IBOO

-

PI

1900

2000

2 100

-A"", Fig. 16. Output power and tuning range of V W generated by difference-frequency mixing in Xe (HILBIG and WALLENSTEIN [19821).

360

TUNABLE COHERENT VUV

[V, 8 3

oscillator-amplifier system (WALLENSTEIN and ZACHARIAS [19801). This system operated at AD = 550 to 650 nm, with output powers of 3 to 5 M W in pulses of -6ns duration, and with bandwidth of -0.02cm-'. This tunable visible radiation was then doubled in KDP to produce tunable UV radiation (A,) at powers of -1 M W . Both visible and ultraviolet radiation was focused in the rare gas, and the resulting VUV radiation analyzed with a monochromator and detected by a solar-blind photomultiplier and NO ionization cell. This multi-laser system, with further flexibility in order to fill in the few gaps in tuning range, will be a useful source for spectroscopy in the region 100 to 200nm (see for example ZACHARIAS, ROTTKE and W E L G E [1980]). With careful attention to detail, Rhodes and his colleagues have developed KrF and ArF laser systems of extremely high brightness for use in THG and 4-WSM below 100 nm. A schematic outline of a typical ROTHSCHILD, multi-laser R G H E system is shown in Fig, 17 (EGGER, MULLER,PUMMER, SRINIVASAN, ZAVELOVICH and RHODES[198lD. The tunable output of a single-frequency, cw dye laser at -580nm was

-

193.6nm -6MW @.u< 260 MHZ E-30mJ 5x l5p rod

%=

i

ArF AMPLIFIER (EMG 200)

Fig. 17. A schematic diagram of a tunable, high spectral brightness ArF laser system for use MULLER, PUMMER, SRINIVASAN, in generating VUV radiation (EGGER, ROTHSCHILD, ZAVELOVICH and RHODES [1981J).

v, § 31

361

EXPERIMENTAL RESULTS

pulse-amplified in a three-stage XeF-pumped amplifier. The -10 ns, 20 mJ pulses were focused into Sr vapor to generate third harmonic radiation at -193 nm, in 5 ns pulses of 200 mW peak power. These pulses were then amplified in two ArF laser amplifiers, to produce -5 ns pulses of 6 M W peak power and -0.01 cm-' bandwidth, tunable within the ArF gain profile. Another system, with a final KrF laser amplifier, produced similar output at 248 nm (HAWKINS, EGGER, BOKOR and RHODES [19803. The KrF radiation was frequency tripled in flowing Xe (EGGER, HAWKINS, BOKOR, PUMMER, ROTHSCHILD and RHODES [19803. Tight-focusing was used, with the input beam focused at the exit end of a windowless cell, through a 350 p,m diameter pinhole. The third harmonic radiation at -83 nm was dispersed by a monochromator and detected with a windowless photomultiplier. In similar experiments, ArF radiation at 193.6 nm was focused and tripled to 64 nm in Kr, H2 and Ar (EGGER, ROTHSCHILD, PUMMER,SRINIVASAN, ZAVELOVICH and RHODES [19813. In later MULLER, BOYER,PUMMER and RHODES[1982] used work, EGGER,SRINIVASAN, 4-WSM ( w ~ ~ ~ = ~ w with~ radiation ~ + w of ~ ,an ArF laser and of a tunable dye laser) in flowing H2 to generate tunable VUV in the region of 79 nm. These V W sources have been used in spectroscopic studies of H2 (ROTHSCHILD, EGGER, HAWKINS, BOKOR, PUMMER and RHODES [1981]) and Ar (EGGER, SRINIVASAN, BOYER,PUMMER and RHODES[1982]).

-

3.3. TUNABLE GENERATION IN METAL VAPORS

The experiments of HODGSON, SOROKIN and WYNNE[1974] with Sr vapor clearly demonstrated that resonantly enhanced 4-WSM could be used with modest powers of -1OkW at the fundamental frequencies, to generate single-frequency, coherent V W radiation which was tunable over broad wavelength regions. They showed that auto-ionizing states played an important role in achieving wide tunability and high efficiency of the third order nonlinear process, and suggested that other systems with auto-ionizing states, namely 2-electron atoms, may be promising media for VUV generation. The vapors of various 2-electron atoms have been investigated for this purpose, including Eu and Yb (SOROKIN, ARMSTRONG, DREYFUS, HODGSON, LANKARD, MANGANARO and WYNNE [19753>, Ca (SOROKIN, ARMSTRONG, DREYFLJS, HODGSON, LANKARD, MANGANARO and WYNNE [1975], ZDASIUK [1975]) and Be (MAHON,MCILRATH,

TABLE10 Tunable generation in metal vapors

Medium (and ionization limit in nm) Sr(217.8) Ca(202.8) Mg(162.1)

Be(133.0) Zn( 132.0) Hg( 11 8.0)

primary

A (nm) 200-190 195.7-177.8 176.7-176.3 174-145 160-140 129-121 123-121 144-106 125.1-1 17.4 115.0-93.0

laser Nd :Glass-Dye N,-Dye N,-Dye N,-Dye N,-Dye KrF-Dye Nd :YAG-Dye XeCI/KrF-Dye Nd :YAG-Dye Nd :YAG-Dye

Reference

2&

R o n and LEE[1977]

HODGSON,SOROKIN and WYNNE[I9741 ZDASIUK [I9751 S T O I ~BANIC, , &m, JAMROZ,LAROCQUEand -ON WALLACE and ZDASIUK119761 MCKEE, STOICHEFF and WALLACE [1978] MAHoN, MCILRATH,T o m s and KELLJZHER [I9791 JAMROZ, LAROCQUE and STOICHEFF [1982] T o m s and MAHON[19821, WON and T o m s [I9821 FREEMAN, JOPSON and BOKOR [I9821

G n

[I9821

0

g

E

4

2<

V, 8 31

EXPERIMENTAL RESULTS

363

TOMKINS and KELLEHER [1979]). The best results have been obtained with Sr, Mg, Zn and Hg (Table 10).

3.3.1. Strontium An energy-level diagram for Sr atoms is given in Fig. 18. It illustrates the typical generation of resonantly-enhanced radiation at 2vl + v2 using a 2-photon allowed intermediate state (5s5d 'D2) with tunability provided by scanning v2 over a broad auto-ionizing state, 4d6p 'P: (the peak 3 in Fig. 3). The radiation generated by this process is shown in Fig. 13. By the use of several dyes and the resonant states 5s5d1D2 (with 2 x 575.7 nm radiation) and 5p2 'Dz (with 2 X 540.9 nm radiation) HODGSON, So~om and WYNNE[19741 produced VUV radiation continuously tunable from 178 to 182 nm and from 183 to 196 nm. In these experiments, the importance of resonance enhancement was

Fig. 18. Partial energy-level diagram of Sr atoms showing resonantly enhanced tunable 2v, + v, generation (Fig. 13) using a 5s5d ID, intermediate state and a 4d6p 'P: autoionizing state.

364

rv, 8 3

TUNABLE COHERENT VUV

dramatically shown by a signal increase of -lo4 when 2 v , was tuned through these and other resonance states. For the tuning range of Fig. 13, laser powers of -20 kW focused to -100 p,m diameter in -5 Torr of Sr vapor, produced conversion efficiencies of With pulse duration of -10 nsec, this corresponds to 10' photons per pulse of VUV radiation; and with linewidths <1 cm-', this represents a source of extremely high brightness. Tunability to longer wavelengths (-200 nm) was obtained by Row and LEE[1977], and shown to be possible of extension to -150 nm SOROKIN and WYNNE[19743) by tuning through the many (HODGSON, auto-ionizing states of Sr (GARTON, GRASDALEN, PARKINSON and REEVES [ 19681). Spectroscopic studies with this tunable source have been reviewed by WYNNE[1979], and the 4-WSM process itself has been used by ARMSTRONG and WYNNE [1974] to study the spectral profiles of auto-ionizing lines of Sr.

3.3.2. Magnesium One of the most promising candidates for 4-WSM is atomic Mg, judging from its broad absorption bands resulting from the ionization continuum and auto-ionizing levels shown in the spectrum of Fig. 19. Mg

1026A MgII

3pns'qo

"i4 3p nd'P:

9 t iP? "7 4 7 7

Fig. 19. Mg absorption spectrum in the region 145 to 100nm, somewhat beyond the [19693). ionization limit at 162.1 nm (MEHIM-BALLOFFETand ESTEVA

v, 831

EXPERIMENTAL RESULTS

365

vapor was indeed found to be a very efficient nonlinear medium by [1976] who demonstrated a conversion efficiency WALLACEand ZDASIUK of -0.2% for THG, an increase of -lo3 over that in Sr. They used SOROKIN and WYNNE[1974] essentially the same technique as HODGSON, with 2-photon resonance enhancement and tuning v2 through the ionization continuum (by the use of seven different dyes) to obtain coherent VUV radiation tunable from 162 to 140nm. They used tight-focusing, and examined the power dependence of THG at 144 nm as a function of the incident dye laser power all the way to the onset of saturation at -30 kW input. An output of 10l2 photons per pulse was obtained in a linewidth of -0.2cm-' (STOICHEFF and WALLACE [1976]). The high conversion efficiency in Mg vapor was confirmed by JUNGINGER, PUELL, SCHEINCRABER and VIDAL [1980] in a detailed study of THG in a phase-matched Mg-Kr mixture, using a parallel incident beam. These authors attributed the marked efficiency to low one and two photon absorption in Mg. TOMKINS and Lu [1979] also obtained photons per pulse at 145 nm, and observed significant increase (by lo3) when 15 Torr of Xe was used for phase matching. The tuning range for Mg was extended to shorter wavelengths, 129 to 121 nm,by MCKEE,STOICHEFF and WALLACE [1978] who used a KrF laser for pumping UV dye lasers at -340 nm in order to tune 2v, + u2 through the broad auto-ionizing 3p4s 'P: level centered at 127nm. Tuning to longer wavelengths, as far as 174 nm, well beyond the ionization limit at 162 nm, was also shown to be possible (BANIC,LIPSON, EFTHIMIOPOULOS and STOICHEFF [19811, STOICHEFF, BANIC,HERMAN,JAMROZ, LAROCQUE and LIPSON[1982]). An indication of the generated VUV signal intensity in the region of the ionization limit of Mg is given in Fig. 20. There is no dramatic change in intensity as 2u, + u2 is swept below the ionization limit, suggesting that the overlap of pressure- and Stark-broadened levels (with high principal quantum numbers) are as effective as auto-ionizing levels in the enhancement of 4-WSM. Large enhancement at n p resonances is clearly evident. The coherent VUV radiation generated in Mg has been used by PROVOROV, STOICHEFF and WALLACE [1977] to measure lifetimes of individual rovibronic levels in CO in a region of strong interactions (-155 nm), and by BANIC,LJPSON,EFTHIMIOPOULOS and STOICHEFF [1981] to measure radiative lifetimes of rovibronic levels of the B', F and N, 2A states of NO by fluorescence excitation over the wavelength range 174 to 145 nm. With dye laser linewidths of <0.01 cm-' (STOICHEFF, BANK,

W

m

3-

Mg 4WSM ( 3 ~ ~3 -~ 4 s )

-

? 0 - 2

>

! I v)

z w I-

z I

1

0

I

1630

I

IONIZATION

1625

X(

A,

I

1620

LIMIT

Fig. 20. Intensity of W V generated by 4-WSM in the vicinity of the ionization limit of Mg at 162.1 nm (STOICHEFF, BANK,HERMAN, JAMROZ, LA ROCQUEand LIPSON[1982&

V, 6 31

367

EXPERIMENTAL RESULTS

HERMAN,JAMROZ, LAROCWEand LIPSON[ 19821, the corresponding VUV linewidths of -0.03cm-' will provide a source having a resolving power >loh,much higher than obtained by any other source in the VUV.

3.3.3. Zinc Several characteristics of Zn make it a prime candidate for generating even shorter wavelength radiation by 4-WSM. For example, its ionization limit at 132 nm is relatively high in comparison with that of other 2-electron atoms; it has a relatively large third-order susceptibility esu/atom, Table 3); and its absorption spectrum (Fig. 21) indicates a small absorption cross-section just above the ionization limit and very strong cross-section for auto-ionizing levels at 105 nm. JAMROZ,

-

150r

(b)

4P37 I

WAVENUMBER (ern-')

Fig. 21. Zn absorption cross-section data, (a) from 75 000 to 88000cm-', and (b) from 82 000 to 103 000 cm-' (MARRand AUSTIN[1969J).

368

TUNABLE COHERENT VUV

W,§3

LAROCQUE and STOICHEFF [1982] have used Zn vapor to generate tunable V W radiation from 144 to 106 nm, the LiF transmission cut-off. They used XeCl and KrF lasers for pumping dye lasers (at -10 kW in 12 ns pulses) in essentially the same arrangement used for Sr and Mg. Initial studies with THG produced radiation at 119.5 and 106.7 nm when the dye laser was tuned, respectively, to the 2-photon allowed transitions 5s 'So t 4s 'So at 358.5 nm and 4d ID2 t 4s 'So at 320.2 nm. These experiments were followed by 4-WSM, with attainment of tunable radiation from 144 to 106 nm at a conversion efficiency of lov5 to producing -lo6 to lo' photons per pulse for input powers of -10 kW. The W V intensity in the vicinity of the ionization limit is shown in Fig. 22. A comparison with Fig. 20 indicates that the results for Zn and Mg (and perhaps all metallic vapors) are similar. Firstly, the intensity is essentially constant above the ionization limit, in agreement with theory [eq. (2.14)] that xC3)remains constant in the ionization continuum. Secondly, generation of VUV radiation continues below the ionization limit, with strong 3-photon resonant enhancement at np 'PI levels, which provides conversion efficiencies >lop3. Several of the np resonance lines are shown in Figs. 20 and 22. Their shapes are determined by two processes. One is three photon resonance enhancement of x ' ~ 'with a singlet np 'PI level [Re 100,-2w,-w2( = O in eq. (2.6)]; and the other is the variation of phase-matching conditions with wavelength [described by

IOP'P

Zn

\

1

73956

Fig. 22. Intensity of VUV generated by 4-WSM in Zn, showing the region near the ionization limit, and the shapes of some np 'P, resonances (JAMROZ, LA ROCQUEand STOICHEFF [19821).

V, 8 31

369

EXPERIMENTAL RESULTS

(b)

WO

WO

Fig. 23. Examples of line shapes of np'P, resonances in the 4-WSM spectrum of Zn, showing the correlation with the wave vector phase matching Ak, when the optimum value of Ak (AkoDJ is closer to (a) the short-wavelength wing, and (b) the long-wavelength wing, of the resonance.

the function Fl in eq. (2.17)]. The reason for the observed asymmetry of resonance lines is graphically explained in Fig. 23 by the correlation between values of Ak and VUV intensity. Finally, for these experiments with Zn and Mg (also with Hg, TOMKINS and MAHON [1982]) and 4-WSM using tight-focusing, VUV generation occurs on both sides of the np resonances and over broad regions between resonances. This means that Ak is negative over the whole region except for small sections where the signal falls to zero. This result is in marked contrast to the results with Kr and Xe (Fig. 15) where tunability is limited by large regions of positive Ak (HILBIG and WALLENSTEIN [1982]).

3.3.4. Mercury Mercury vapor is one of the most efficient media for resonantly enhanced 4-WSM, and has been used to generate tunable VUV and

370

[V, $ 3

TUNABLE COHERENT VUV

XUV radiation (Table 10). The first reports on the use of Hg were concerned with THG at 89.6 nm (SLABKO, POPOV and LUKINYKH [1977]), with four-wave parametric oscillation (4-WPO) to produce intense radiation at 184.9, 193.3, 143.5, 130.7 and 125.9 nm (BOKOR,FREEMAN,PANOCK and WHITE [1981], TOMKINS and MAHON[1981]), and with sum- and difference-mixing, 2v,*v2 (Fig. 24) to generate -5 kW peak power at 125.1 nm and somewhat less at 208.6 nm (TOMKINS and MAHON[1981]). These authors quote a conversion efficiency of 0.3% and linewidth -O.O4cm-' at 125.1 nm, and estimate the generation of -10'' photons sec-' k'which they compare to the 5 X lo1' photons sec-' k'sr-' delivered by the National Bureau of Standards U S A . synchrotron facility (SURF). In an extension of this work, TOWNSand MAHON[1982] and MAHON and TOMKINS [1982] used resonantly-enhanced 4-WSM (v3 = 2vl + v2)to generate continuously tunable radiation in the wavelength region 121.8 to 119.7 nm, all below the ionization limit at 118.0 nm. They used second and third harmonic radiation from a Nd:YAG laser to pump dye lasers at v 1 and v2. The two collinear beams were focused in Hg vapor 2v, 6s7s 'So at 2 Torr to generate radiation at v3 via the transitions 6s2 ' S o

-

WI

=31285

8 (IrnJ)

?w, +w2

8 (6mJ)

! =62570

I

20857%

I

10496i

I

12514%

Fig. 24. Hg term levels involved in four-wave parametric generation of 184.9 nm resonance radiation, and sum and difference generation at 125.1 and 208.6nm. and the resultant spectrum (TOMIUNS and UAHON [1981]).

v, 8 31

371

EXPERIMENTAL RESULTS

;'.6snp 'P, y 1 \ 6 s 2 'So. The v3 radiation was generated in the region of the 12p through 16p 'P, resonances with substantial increase in power in the vicinity of the np 'P, transition frequencies (as observed for Mg and Zn, Figs. 20 and 22). FREEMAN,JOPSONand BOKOR[1982] used T H G and 4-WSM in Hg vapor (kept in a windowless chamber purged with He) to generate radiation from 115 to 93 nm (above the ionization limit at 118 nm). They carefully investigated the dependence of power output at 93 nm on low and high incident laser intensities and on low and high Hg densities, and and VIDAL[1981]. In c o n k e d the predictions (Fig. 8) of SCHEINGRABER particular, at high laser intensities and high Hg densities it was found that peak output power occurred for tunings slightly off the exact 2-photon resonance. The use of KrF pumped dye lasers should permit continuous tuning to wavelengths as short as -83 nm.

3.3.5. Beryllium,calcium

In principle, one would expect Be vapor to be an efficient source of tunable radiation in the region of Ly-a, because of a broad auto-ionizing level between 112 and 130 nm in the absorption spectrum (MEHLMANBALLOWET and ESTEVAN [19691). However, because of the high reactivity of Be with most metals, instabilities in heat-pipe operation have limited its usefulness. MAHON,MCILRATH, TOMKINS and KELLEHER [1979] managed to generate radiation between 121 and 123 nm by 4-WSM, but with limited efficiency (-3 x lo-') because of these difficulties. The use of calcium vapor has also met with little success. A strong THG signal at 200nm has been obtained by FERGUSON and ARTHURS [1976], and ZDASIUK [1975] has generated VUV radiation at 176.5 nm tunable over --5Ocm-', using the same experimental technique as with Mg (WALLACE and ZDASIUK [1976]). However, Zdasiuk was unable to generate any signal in the region of a broad and intense auto-ionizing feature centred at 188.7 nm in the absorption spectrum (NEWSOMand SHORE[1968]) even when incident powers as high as 50 kW were used. A possible explanation is suggested by eq. (2.14); namely, since Ix'~'~* 4294: at resonance, both parameters qg and qn must be large for good conversion efficiency. While it is known that qg is large from the absorption cross-section, nothing is known of the magnitude of qn: a low value would account for the null result.

-

372

TUNABLE COHERENT VUV

[V. P 3

3.4. TUNABLE GENERATION IN MOLECULAR GASES

As an alternative to metal vapors, INNES, STOICHEFF and WALLACE [1976] and WALLACE and INNES[1980] explored the use of molecular systems as non-linear media for VUV generation. They found that nitric oxide, NO, was admirably suited for these experiments. Considerable information was available on its electronic states and oscillator strengths, and a strong 2-photon transition was available for resonant enhancement of x'~'.Also, the experiments were simplified, since a simple gas cell replaced the heat pipe used with the metal vapors, and high gas pressures could be readily obtained. As shown in an energy level diagram of some of the N O states (Fig. 25), one dye laser was tuned so that 2vl was in resonance with the A "2'+X2n transition. When a second dye laser was tuned to the C21Z manifold, a rich rotational structure was observed in the VUV output (Fig. 26). VUV radiation was generated in the y bands (of breadth -600cm-') at 151, 143, 136 and 130nm. At an NO gas pressure of -90 Torr, it was estimated that a photon yield of -10' photons/pulse was obtained for incident laser powers of 20 kW. Significant pressure broadening occurred at 10 atm, and the rotational structure of the 2-photon transition used for resonance enhancement was

Fig. 25. Energy level diagram of NO showing the resonances used in thud harmonic and WALLACE[19763). generation (STOI~HEFF

4 4 070 t

67105

66090

66075

66060

66045

66030

66015

66000

Fig. 26. Recordings of coherent tunable radiation generated near 151 nm in NO at pressures of 50 Tom (bottom) and 10 atm (top) (INNES, STOICHEFF and WALLACE [1976]). The wave number scale with larger numerals refers to THG and that with smaller numerals to the two-photon transitions of the (0.0) band of the AZPf+ Xz173,z band system which provide resonance enhancement.

65985

374

TUNABLE COHERENT VUV

W,S3

essentially eliminated. This provided continuously tunable VUV radiation by simple THG using a single laser (Fig. 26). In similar experiments with CO at a few Torr, LUKASIK, WALLACE, GREENand VALLI~E [1982] used 2-photon resonantly enhanced 4-WSM to generate VUV radiation in the region of 115 nm and tunable over the rotational lines of the B-X(0, 0) band, and GLOWNIA and SANDER [1982] demonstrated resonantly enhanced THG (via the z)' = 2 vibronic levels of the A'Il state), in the 147.5 nm region. 3.5. VUV AND XUV GENERATION BY HIGHER ORDER PROCESSES

Higher order frequency conversion (0 2.5) has been used to generate radiation at fixed frequencies to wavelengths as short as 38 nm. Reintjes and co-workers at the Naval Research Laboratory in Washington, D.C. have used the fundamental, second and fourth harmonics of a modelocked Nd :YAG laser to generate X W radiation in rare gases (Table 8) through fifth and seventh harmonic conversion and by 6-wave mixing (u3=4vlfu2) (REINTIES, ECKARDT,SHE,KARANGELEN, ELTON and ANDREWS [1976], REINTJES, SHE, ECKARDT, KARANGELEN,ANDREWS and ELTON [1977], REINTJES, SHE and ECKARDT [1978], SHE and REINTIES [19771). Their powerful laser system and experimental arrangement are shown in Fig. 27. Fifth harmonic of 266.1 nm radiation produced radiation at 53.2 nm (Fig. 28) in He, Ne, Ar and Kr. The highest conversion efficiency was -lo-' in He (at -50Torr ) with strong tight-focusing, and yielded peak pulse powers of -1 kW. Seventh harmonic at 38 nm was observed only in He (Fig. 28), with peak power of -100 W. As discussed by REINTJES [1980] and REINTJES, SHE and ECKARDT [1978], the relative conversion efficiency to harmonics of different order POCKELS CELL PULSE SELECM(

w

2B6nm

m M w

MOM- LOCKED Nd YAG OXILLATOR

90.

e 5cm

T

MOKICHROMATOR

m

WLARIZATION ROTATOR

Fig. 27. Schematic diagrams of the laser system and experimental arrangement used to generate and detect higher harmonics in rare gases (REINTJES,SHE and EKARDT [1978].

V,

B

31

375

EXPERIMENTAL RESULTS

I

I

yi=53.2nm I

L

I

54nm 5 0 n m I

:

:

:

46nm :

;

:

4 2 n m 38nm ;

:

'

,

WAVELENGTH

Fig. 28. Partial energy level diagram and spectrum of He showing the generation of fifth and seventh harmonic radiation at 53.2 and 38.0 nm, respectively (REINTJES,SHE,ECKARDT, KARANGELEN, ANDREWS and ELTON[19771).

is dependent on phase matching parameters (Fig. 9), incident laser intensities, and on the magnitudes of nonlinear susceptibilities. The last of these is very important since the presence of resonant enhancements in the higher order susceptibilities can lead to larger nonlinear polarizations in the higher order than in the lower order interactions. Experimental investigations of this problem in He showed that the fifth harmonic signal was larger than the third harmonic by a factor of -20 at incident power of 12MW and a factor of -2 at incident power of 300MW. Conversion to the fifth harmonic was favored over conversion to the third by a 5-photon resonant enhancement, until saturation by competing processes such as the quadratic Kerr effect limited its growth. Conversion to the seventh harmonic was less than that to the fifth harmonic by two orders of magnitude probably because of a decrease in the nonlinear susceptibility.

376

TUNABLE COHERENT VUV

[V, § 3

These results demonstrate that conversion to higher order harmonics can be as effective as conversion to a lower order for some media. Six-wave mixing ( u 3 = 4 v l f v 2 ) was used with Ne and He to generate radiation at 76.0, 71.0, 62.6 and 59.1 nm with conversion efficiencies of lop7. These results indicate that continuously tunable radiation can be expected in this region when the fixed laser frequency v2 is replaced by powerful and tunable dye lasers.

-

3.6. GENERATION OF TUNABLE XW RADIATION BY ANTI-STOKES RAMAN SCAlTERING

A different type of laser-induced light source which is incoherent, yet tunable and of extremely high brightness merits discussion here. It is based on spontaneous anti-Stokes scattering from atoms stored in metastable states (HARRIS [1977J). Such a source using a He glow discharge with atoms stored in the 2s ' S metastable state (at Y = 166272 cm-') has been developed by ZYCH,LUKASIK, YOUNGand HARRIS[1978]. In their experiments, pulsed laser radiation at 1.06 bm (vp- 9395 cm-') was focused in the He discharge, and spontaneously-emitted X W radiation at Y f vp or 56.9 and 63.7 nm was observed at right angles to the incident laser beam (Fig. 29). This radiation exhibited several unique properties: linear polarization, narrow linewidth (1.3cm-' at 56.9nm and 1cm-' at 63.7nm, He DISCHARGE LASE-

vuv LASER

Fig. 29. Energy level diagram (and experimental arrangement) for laser-induced XUV light source based on spontaneous anti-Stokesscattering in He (HARRIS [19773).

VI

REFERENCES

377

compared to the 5.6 cm-' of the 58.4 nm resonance line of the discharge) and high peak spectral brightness (with the 56.9nm radiation being at least 100 times brighter than that of the resonance line emitted from the He discharge). Finally, such spontaneous anti-Stokes radiation should be tunable over a range of -60 000 cm-' in the vicinity of 58 nm in He by pumping atoms out of the metastable state with tunable laser radiation.

S

4. Conclusion

It is evident that past theoretical and experimental exploration of third-order nonlinear processes has led to basic understanding and useful knowledge of nonlinear processes in atomic and molecular systems. As a result, it is now possible to generate tunable coherent radiation over the wavelength region 200 to 106 nm, and in limited regions at even shorter wavelengths. Generation of V W radiation by third harmonic and sumand difference-mixing has been demonstrated in the rare gases and in many metal vapors. This has been done with and without phase matching or resonance enhancement with corresponding conversion efficiencies of 10-3 to 10-5. Much remains to be done in the X W region before generally useful laboratory sources can be readily constructed, and the soft X-ray region has yet to be penetrated. Most promising are the higher-order nonlinear processes such as 6- and 8-wave sum and difference mixing. Preliminary results have shown some success and have indicated that effort is required in choosing systems with suitable resonant enhancement and phase matching. Further developments will undoubtedly improve the efficiencies, intensities, range of tunability and monochromaticity of these light sources. However, it is important to stress that laser sources of high brightness are now available for application to a wide variety of scientific uses in the VUV and XUV regions.

References ARMSTRONG, J. A., N. BLOEMBERGEN, J. DUCUING and P. S. ~ERSHAN, 1962, Phys. Rev. 127, 1918. ARMSTRONG, J. A. and J. J. WYNNE, 1974, Phys. Rev. Lett. 33, 1185. Jr., L. and B. L. BEERS,1975, Phys. Rev. Lett. 34, 1290. ARMSTRONG

378

TUNABLE COHERENT VUV

[V

BANK,J. R., R. H. LIPSON,T. EFTHIMIOPOULOS and B. P. STOICHEFF, 1981, Opt. Lett. 6,461. S. A. et a]., 1977, Sov. Tech. Phys. Lett. 3, 473. BATISHCHE, BJORKLUND, G. C., 1975, IEEE J. Quantum Electron, QE-11, 287. R. R. FREEMAN and P. F. LIAO,1977, Appl. Phys. BJORKLUND, G. C., J. E. BJORKHOLM, Lett. 31, 330. and J. C. WHITE, 1981, Opt. Lett. 6, 182. BOKOR,J., R. R. FREEMAN,R. L. PANOCK COTTER,D., 1979a, Opt. Lett. 4, 134. COTTER, D., 1979b, Opt. Commun. 31, 397. DRABOVICH, K. N., D. I. METCHKOV, V. M. MITEV,L. I. PAVLOVand K. V. STAMENOV, 1977, Opt. Commun. 20, 350. ECONOMOU, N. P., R. R. FREEMAN and G. C. BJORKLUND, 1978, Opt. Lett. 3, 209. J. BOKOR,H. PUMMER,M. ROTHSCHILD and C. K. RHODES, EWER, H., R. T. HAWKINS, 1980, Opt. Lett. 5, 282. D. MULLER,H. F'UMMER, T. SRINIVASAN, J. ZAVELOVICH and EWER, H., M. ROTHSCHILD, C. K. RHODES,1981, The Study of Atomic and Molecular Processes with Rare-Gas Halogen Lasers, in: Laser Spectroscopy V, Roc. 5th Intl. C o d . Laser Spectroscopy, Jasper, Canada, 1981, eds. A. R. W. McKellar, T. Oka and B. P. Stoicheff (SpringerVerlag, Berlin) p. 446. K. BOYER,H. PUMMERand C. K. RHODES,1982, Generation of EWER,H., T. SRINTVASAN, Tunable, Coherent 79 nm Radiation by Frequency Mixing, in: Proc. Laser Techniques for Extreme Ultraviolet Spectroscopy, Boulder, Colorado, 1982, eds. R. R. Freeman and T. J. Mcnrath (American Institute of Physics, New York). FANO,U., 1961, Phys. Rev. 124, 1866. FERGUSON, A. I. and E. G. ARTHURS, 1976, Phys. Lett. 58.4, 298. FRANKEN, P. A., A. E. HILL,C. W. PETERSand G. WEINREICH, 1961, Phys. Rev. Lett. 7, 118. FREEMAN,R. R. and G. C. BJORKLUND, 1978, Phys. Rev. Lett. 40, 118. FREEMAN,R. R., G. C. BJORKLUND, N. P. ECONOMOU, P. F. LIAOand J. E. BJORKHOLM, 1978, Appl. Phys. Lett. 33, 739. FREEMAN, R. R., R. M. JOFSONand J. BOKOR,1982, Generation of Coherent and Incoherent Radiation Below l O O O A in Hg, in: Proc. Laser Techniques for Extreme Ultraviolet Spectroscopy, Boulder, Colorado, 1982, eds. R. R. Freeman and T. J. Mclkath (American Institute of Physics, New York). W. H. PARKINSON and E. M. REEVES,1968, J. Phys. GARTON, W. R. S., G. L. GRASDALEN, B (Proc. Phys. SOC.)1, 114. and J. H. MARBURGER, 1977, Phys. Rev. A15,300. GEORGES,A. T., P. LAMBROPOULOS GLOWNIA,J. H. and R. K. SANDER,1982, Appl. Phys. Lett. 40, 648. HAGER,J. and S. C. WALLACE, 1982, Chem. Phys. Lett. 90, 472. H ~ S C HT., W., 1972, Appl. Opt. 11, 895. HARRIS,S. E., 1973, Phys. Rev. Lett. 31, 341. HARRIS,S.E., 1977, Appl. Phys. Lett. 31,498. HARRIS,S. E. and R. B. MUES, 1971, Appl. Phys. Lett. 19, 385. HARRIS,S. E., J. F. YOUNG,A. H. KUNG,D. M.BLOOMand G. C. BJORKLUND, 1974, Generation of Ultraviolet and Vacuum Ultraviolet Radiation, in: Laser Spectroscopy I, Proc. Intl. Conf. Laser Spectroscopy, Vail, Colorado, 1973, eds. R. G. Brewer and A. Mooradian (Plenum Press, New York) p. 59. 1980, Appl. Phys. Lett. 36, 391. HAWKINS, R. T., H. EGGER,J. BOKORand C. K. RHODES, HEINRICH,J. and W. BEHMENBURG, 1980, Appl. Phys. 23, 333. HILBIG,R. and R. WALLENSTFJN, 1981, IEEE J. Quantum Electron. QE-17,1566. HILBIG,R. and R. WALLENSTEIN,1982, Appl. Opt. 21, 913. HODGSON, R. T., P. P. SOROKIN and J. J. WY"E, 1974, Phys. Rev. Lett. 32, 343.

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380

TUNABLE COHERENT W V

Vl

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a

AUTHOR INDEX A ACKERMAN, M. 30,59 ACTON, L. W. 41,42,60 AGARWAL, G. S. 164, 166, 170,254 J. 110,118, 121, 140, 150, 153 AGOSTINELLI, AKHMANOV, S. A. 163,254 W. 163, 189, 193, 197, 199, ALEXIEWICZ, 212, 215, 230, 254 ALFANO, R. R. 122, 150, 165,200,201,261 ALTMANN, K. 164, 186, 238,254 AMOS,R. D. 224,254 ANANTH, M. S. 210,254 ANDERSEN, H. C. 247, 255 D. L. 163, 180, 187, 193, 194, ANDREWS, 226, 237,245,246,248,252,254 ANDruzws, R. A. 353, 355, 374,375, 379 APANASEVICH, P. A. 170, 255 APLIN,C. 321 ARECCHI,F. T. 121, 150 J. A. 327, 329, 331, 334, 335, ARMSTRONG, 337, 354,361, 364, 377, 380 ARMSTRONG.JK., L. 335, 337, 377 J. A. 68, 148, 150, 152 ARNAUD, ARRIGHINI, G. P. 191, 255 ARSAC,J. 80, 150 ARTHURS, E. G. 354, 371, 378 ARTZNER, G. 41, 60 ARUTYUNIAN, V. M. 164, 255 ASAI,N. 265, 257 ASHKIN,A. 122, 152 H. 22, 30, 34, 60 ATKINS, ATKINS,P. W. 248, 249, 255 AUSTIN,J. M. 367, 379 AUSTON,D. H. 121, 124, 150 AZEMA,A. 122, 150

B BADAWAN, N. N. 248, 255 BALAGUROV, B. YA. 165, 255 BALLAGH, R. J. 166,255 BANCEWICZ, T. 163,165, 187,188, 189, 190,

191, 193, 196, 197, 213, 214, 230, 254, 255,259 BANIC,J. R. 362, 365, 366, 378, 380 BARAK,S. 248, 261 BARANNE, A. 6,59 F. 248, 255 BAROCCHI, BARTOLI, F. J. 226, 230, 233, 255 BASOV,N. G. 135,150 S. A. 353, 355, 378 BATISHCHE, BEDEAUX, D. 203, 255 BEERS,B. L. 335,337, 377 BEHMENBURG, W. 354, 378 BEN-REUVEN, A. 162, 218, 220, 259 S. A. 284,288,289, 305, 313,316, BENTON, 321,322 BENTZ,W. 34, 35, 36, 60 P. 48,61 BENVENUTI, T. 164,255 BEN-ZEEV, BERGER,H. 248, 255 BERGES,J . C. 24, 60 BERNE,B. J. 206,255 BERRY,D. H. 318,323 R. 161,213, 255 BERSOHN, BIAUME,F. 30, 59 BIEDERMANN, K. 308, 322 BIRMAN,J. L. 247, 255 G. 222, 225, 255, 259 BIRNBAUM, BIVAS,A. 165, 257 BI-ZHEN,D. 301, 324 BJARNASON, J. 0. 247, 255 J. E. 106, 133, 150, 353, 354, BJORKHOLM, 378 BJORKLUND, G. C. 337, 338, 339, 340, 341, 350,353, 354, 355, 378, 379 J. E. 41, 59, 60 BLAMONT, BLANC-LAPIERRE, A. 73, 150 BLATON, J. 159,255 BLISS,E. S. 132, 150 BLOEMBERGEN, N. 121, 128, 150, 166, 167, 170, 203, 255, 256, 327,329, 331, 377 BLOK,V. R. 194, 255

38 1

382

AUTHOR INDEX

BLOOM,D. M. 350, 353, 355, 378 A. 4, 61 BOGGESS, BOHLIN,R. C. 5 , 19,43,61 BOIVIN, A. 272, 273, 274, 323 J. 353, 354, 356, 361, 362, 370, 371, BOKOR, 378, 379 BOKSENBERG, A. 51, 60 BONIN,K. 355, 359, 380 BONNET, G. too, isn BONNET, R. M. 41, 42, 59, 60 R. 163, 247, 250, 251, 255, BONNEVILLE, 257 BORGMAN, J. 4,60 BORN,M. 67, 70, 73, 79, 127, 150, 234,236, 247, 255 J. 122, 150 BOTINEAU, BOUSQUET, P. 144, 150 BOYER,K. 356, 361, 278 BOYLE,L. L. 201, 255 BRACEWELL, R. 68,150 U. 356,380 BRACKMAN, BRADFORD, J . N. 124, 132, 151 BRADLEY, D. J. 133, 135, 137, 150, 356,379 BREWER,R. G. 128, 150 M. J. 124, 151 BRIENZA, BROWN, R. Hanbury, 77, 151 BROWN, W. A. 41,42,60 S. R. J. 226, 258 BRUECK, BRUMM, D. B. 318, 323 BRUNER,41,42, 60 0. 305,322 BRYNGDAHL, BURCKHARDT, C. B. 270, 271, 279, 304, 305,322 BURNHAM, D. C. 135, 150 BURTON,G. T. 308, 322 BUSCH,G. E. 122, 150 H. T. 297, 322 BUSCHMANN, P. N. 166, 255 BUTCHER, C

CAGNAC, B. 166,257 M. 159,255 CARDONA, H. J. 164, 255 CARMICHAEL, CARNOCHAN, D. J. 4, 7, 60 G. R. 5, 19,21,25,34,43,60 CARRUTHERS, CAUGHEY, S. J. 133, 150 CAULFIELD,H. J. 115, 151 CHANDHA, N. 166, 236, 238, 260 S. 166, 255 CHANDRA, CHANG,B. J. 296, 322

CHEBOTAEV, V. P. 355, 380 CHEMLA,D. S. 163, 247,250, 251, 255, 257 CHEN,H. 288, 290,292, 302, 306, 307, 310, 311, 312, 315, 322, 323,324 A. A. 355, 380 CHERNENKO, CHILINGARIAN, Yu. S. 164, 255 CHINLON LIN122, 135, 153 CHILI,Y.N. 163, 186, 255 W. 221, 258 CHMIELOWSKI, M. 279, 322 CHOMAT, CHRISTIE, J. H. 199, 255 CHU,D. C. 317, 322 CLAY,B. R. 308, 322 CODE,A. D. 5 , 34,60 COHEN,E. R. 225, 255 COHEN-SABBAN, J. 30,60 COHEN-TANNOUDJI, C. 166,255 COLEMAN, C. I. 51, 60 COLEMAN, P. D. 248, 258 COLLES,M. J. 122, 151 COLLIER,R. J. 269, 270, 271, 272, 279, 322 COLOMBEAU, B. 93, 95, 110, 112, 114, 115, 120, 128, 139, 144, 148, 151, 152, 153 A. 166, 255 COMPAAN, CONNES, J. 97, 111, 143, 151 CONNES, P. 144, 151 COOPE,J. A. R. 250, 251,255,256 COOPER,J. 166, 255 COPLEY,J. R. D. 206, 256 CORER, D. 248, 256, 355, 356, 357, 378, 380 COURT& G. 4, 5 , 6, 7, 11, 16, 19, 21, 22, 24, 30, 34, 35, 36, 41, 48, 57, 60 CRUVELLIER, P. 4, 5 , 34, 60, 61 R. 297, 323 CULLEN, CYVIN, S. J. 163, 193, 237,256

D DAGENAIS, M. 164, 258 DALLAS,W. J. 317, 322 R. 41, 61 DANIELSON, DAVIS,L. I. 354, 380 DAVIS,R. J. 3, 60 DE BITETO, D. J. 304, 312, 313, 322 DECAUDIN, M. 41, 42,60 DECIUS,J. C. 163, 193, 237, 256 DE GOEDE,J. 247,256 DE GROOT,S . R. 169, 256 J. M. 4, 5 , 22, 23.24, 34, 35, DEHARVENG, 36, 60, 61

AUTHOR INDEX

DE MARIA,A. J. 135, 151 V. N. 162, 165, 256 DENISOV, DENISYUK, Yu. N. 278, 297, 322 J. 109, 110, 118, 120, 121, 133, 151 DESBOIS, DESHAYES, J. P. 24, 60 R. A. 248, 256 DESIDERIO, DETAILLE, M. 6, 10, 21, 31,48, 59, 60, 61 DINES,T. J. 164, 165, 256 DNEPROVSKY, V. S. 248, 256 DOHNALIK, T. 93, 95, 115, 151 DOLINO,G. 162,256 DONAS,J . 34, 35, 36, 60 DONOHUE, P. J. 112, 151 DRABOVICH, K. N. 354,378 K. H. 133, 151 DREXHAGE, R. W. 354, 361, 380 DREYFUS, P. D. 249,256 DRUMMOND, M. 18, 19, 61 DUBAN, DUCUING, J. 327, 329, 331, 377 D m , .I.221, 256 DUGUAY, M. A. 106, 121, 124, 132, 133, 151 R. J. 4, 60 DUINEN, DWORETSKY, M. M. 4, 7,60 M. I. 280, 322 DZYUBENKO,

E EBERLY, J. H. 173, 203, 256 ECKARDT, R. C. 124, 132, 151, 152, 338, 347,348,353,355,374,375,379 ECONOMOU, N. P. 353, 354, 378 EDMONDS, A. R. 186, 210, 256 EFIXIMIOPOULOS, T. 365, 378 EGGER,H. 356, 360, 361, 378,379 K. B. 133, 151 EISENTHAL, ELTON,R. C . 353, 355, 374, 375, 379 L. 14, 43, 61 EPSTEIN, ESTEVA, J. M. 364, 371, 379 M. W. 206, 256, 260 EVANS, F FABELINSKII, I. L. 159, 166, 256 FANCONI, B. 256 FANO,U. 335, 378 FARKAS, G. 137, 151 R. J. 276, 279, 300, 322, 324 FEDOROWICZ, P. 143, 151 FELLEGETT, FERGUSON, A. I. 354, 371, 378 FIENUP, J. R. 316, 317, 322 FISHER,P. A. 133, 151

383

J. 166, 168, 256 FIUTAK, FLEURET, J. 321 FLYTZANIS, Chr. 167, 256 FORTET,R. 73, 150 P. A. 327, 378 FRANKEN, FRANCON, M. 140, 151 R. R. 353, 354, 362, 370, 371, FREEMAN, 378 FRENCH, M. J. 161, 164, 165,194, 195, 256 FRENKEL, D. 222, 256 I. 162,256 FREUND, FRIEDMA",H. 164, 255 A. A. 276, 278, 300, 322 FRIESEM, FRISCH,H. L. 161, 213,255 FROEHLY, C. 100, 112, 114, 128, 142, 148, 151, 152, 153 FUSEK,R. L. 314, 315, 323

G GABEL, C. W. 121, 150 GABEL,G. 110, 118, 140, 150 GABOR, D. 72, 151 GABRIEL, G. J. 248, 256 GADDIS,M. W. 301, 322 N. 321 GAGGIOLI, L. 222, 225, 256, 257 GALATRY, GALE,M. T. 308, 309, 322 GARSIDE, B. K. 123, 151 W. R. S. 334, 336,364, 378 GARTON, W. M. 221, 256 GELBART, GEORGE, N. 312, 313, 323 A. T. 342, 378 GEORGES, GEX,J. P. 140, 151 GHARBI, T. 222, 256 J. A. 106, 151 GIORDMAINE, GIRES,F. 108, 109, 110, 118, 120, 121, 122, 133, 150, 151 GLAUBER, R. J. 248, 256 GLENN,W. H. 124, 151 GLENNON, B. M. 331, 380 J. H. 374, 378 GLOWNIA, GOEDGEBUER, J. P. 115, 142, 152 M. 256 GOEPPERT-MAYER, GOLAY, M. 34, 35, 36, 60 GOODMAN, J. W. 80, 100,151,316,317,322 R. G. 227,256 GORDON, GORRADI, G. 136, 152 G O ~ B R O ZP. E 41, , 60 GRASDALEN, 335, 336, 364, 378 GRAUBE, A. 296, 322

384

AUTHOR INDEX

GRAY,C. G. 210, 213, 225, 246, 254, 256 GREEN,W. R. 374, 379 A. J. 162, 259 GREENFIELD, GREENHOW, R. C. 135, 151 D. 106, 121, 133, 151, 153 GRISCHKOWSKY, L. 221, 256 GROOME, GROVER, C. P. 291,322 GRUN,J. B. 165, 257 GRYNEIERG, G. 166,257 K. E. 210, 221, 254, 256 GUBBINS, GUINN,K. R. 112, 152 GUFTA,P. S. 248, 257 GURZADYAN, G. A. $ 6 1 T. K. 122, 125, 133, 151, 152 GUSTAFSON, G ~ N G E P. R 159, , 257 GYUZALIAN, R. N. 136, 151, 152

H HAGER,J. 359, 378 K. A. 296, 323 HAINES, HAKEN, H. 247,257 HALL,R. J. 164, 165, 256 HALLOCK, H. B. 3,61 HAMEKA, H. F. 246,257 HANBURY BROWN, R., see BROWN, R. Hanbury D. C. 248,256 HANNA, HANSCH, T. W. 351, 378 HANSEN, J. W. 106, 121, 124, 132, 133, 151 P. 285, 287, 289, 292, 293, HARIHARAN, 294, 296, 297, 298, 299, 300, 302, 303, 322, 323 S. E. 327, 331, 333, 338, 340, 342, HARRIS, 347,350,353,354,355,376,378,379,380 W. 248, 257 HARTIG, HARVEY, G. 110, 118, 140, 150 HAUS,H. H. 123, 151 R. T. 356, 361, 378, 379 HAWKINS, HAYES, W. 159,257 HEALEY, W. P. 252,257 HECKATHORN, H. M. 19,60 Z. S . 285, 287, 289, 292, 293, HEGEDUS, 294, 322, 323 J. 354, 378 HEINRICH, W. 173, 258 HEISENBERG, W. 167, 178,257 HEITLER, HELLWARTH, R. W. 246,257 HENIZE, K. G. 5, 20, 43, 61 F. 165, 257 HENNEBERGER, K. 165,257 HENNEBERGER,

J. A. 248, 257 HERMAN, HERMAN, P. 362, 365, 366, 367, 380 HERSE,M. 41, 61 HIKSPOORS, H. M. J. 248, 261 HILBIG,R. 356, 357, 359, 369, 378 HILL,A. E. 327, 378 T. 115, 151 HIRSCHFELD, HODGSON, R. T. 328, 334, 337, 351, 352, 354, 361, 362, 363, 364,365, 378, 380 K. 296, 323 HONDA, HONERLAGE, B. 165,257,260 Z. G. 136, 137, 151 HORVATH, HBYE,J. S. 210, 257 HSIAO,S . S . H. 318, 323 Hsu, K. S. 355, 379 HUDSON, B. S . 247,248, 255, 256 HUFF,L. 314, 315, 323 HUGHES, J. L. 112, 151 D. 33, 61 HUGUENIN, HUNT,R. W. G. 282, 323 HURST,R. P. 190,259 HUSH,N. S . 190, 191, 257 HUTCHINSON, M. H. R. 356, 379

I ICHIOKA, Y. 317, 322 L. D. 199, 257 IEVLEVA, IH, c. s. 309, 310, 323 Yu. A. 193, 257 ILYINSKY, INNES, K. K. 372, 373, 379, 380 INOUE, K. 162, 165, 257 IPPEN, E. P. 122, 125, 133, 135, 139, 151, 152 IRADJAN, V. A. 248, 255 P. 225, 257 ISNARD, E. H. 229,261 IVANOV, J JACQUINOT, P. 143, 152 JAIN, R. K. 122, 152 P. 34, 60 JAKOBSEN, JAMROZ, W. 334, 344, 362, 365, 366, 367, 368, 379, 380 JANSKY, J. 136, 152 E. B. 5 , 19, 43, 61 JENKINS, JERPHAGNON, J. 163,250,251,257 JHA,S . S . 164, 166,254,257 JOHNSON, B. C. 112, 152 JOLY,L. 297, 323 JONES,R. P. 122, 150

AUTHOR INDEX

JOPSON,R. M. 353, 362, 371, 378 A. 41, 60 JOUCHOUX, H. 334, 342, 353, 354, 365, 379 JUNGINGER,

K

385

198, 200, 220, 226, 230, 236, 237, 240, 241,242,243,244,247,249,253,258,261 H. A. 173,258 KRAMERS, KROCHIK, G. M. 194, 255 KRONOPULOS, Yu. G. 194, 255 T. 296, 323 KUBOTA, KUNG,A. H. 350, 351, 353, 354, 355, 356, 378,379 J. E. 4, 61 KUPPERIAN, KURTZNER, E. T. 296, 323 T. I. 143, 152 KUZNETSOVA, KWOK,H. S . 122, 152

KAISER,W. 122, 137, 152, 248, 259 KARAGODOVA, T. Ya. 199, 251 KARANGELEN, N. E. 353,355,374,375,379 KARMENIAN, A. V. 164,255 KARMENIAN, K. V. 248, 256 B. 167, 201, 257 KASPROWICZ-KIELICH, KELLEHER, D. W. 362, 363, 371, 379 P. L. 133, 151 KELLEY, KERTESZ, I. 137, 151 L KEYES,T. 215,257 LABEYRIE, A. E. 278, 279, 280, 323 KIELICH,S. 159, 161, 162, 163, 164, 165, LACOURT, A. 100, 115, 142, 151, 152 166, 167, 168, 169, 174, 175, 176, 177, LADANYI, B. M.215,257 178, 180, 181, 182, 187, 188, 189, 190, LAGET,M. 21,22,23, 30, 34,35,36,60, 61 191, 192, 193, 195, 196, 197, 199, 200, LAIZEROWICZ, J. 162,256, 258 201, 203, 209, 210, 213, 214, 217, 218, LALANNE, J. R. 162, 222, 223, 224, 225, 219, 220, 221, 222, 223, 224, 225, 226, 226, 258 230, 233, 236, 237, 239, 240, 241, 242, LALLEMAND, P. 128, 150, 222, 258 244, 246, 247, 248, 249, 251, 252, 253, LAMBROPOULOS, P. 342, 378 254,255, 257,258, 259, 260, 261 H. 353, 355, 379 LANGER, KILDAL,H. 226, 258 P. 272, 273,274, 323 LANGLOIS, KIM,D. J. 248, 258 LANKARD, J. R. 354, 361, 380 KIMBLE,H. J. 164, 258 P. E. 334, 362, 365, 366, 367, LAROCQUE, KING,M. C. 318, 323 368,379, 380 D. 215,257 KIVELSON, LAU,A. 122, 153 KLAUDER, J. R. 249, 258 LAUBEREAU, A. 122, 125, 133, 152, 225, KLEIN,0. 174, 258 248, 259, 261 KLIMENKO, V. M. 163, 182, 237, 261 LAURENCE, R. J. 53, 61 C. 165, 260 KLINGSHIRN, LECKRONE, D. S. 47,61 KLYSHKO, D. N. 163, 166, 248, 254, 258 LEE, C. H. 124, 132, 151,362, 364, 380 H. F. P. 222,258 KNAAP, LEHMBERG, R. H. 125, 132, 133, 152 KNAST,K. 209, 210, 221, 258 LEIBACHER, J. W. 41, 60 KNIGHT,P. L. 166, 258 LEITH, E. N. 265, 268, 292, 306, 307, 315, KNOLL,A%.M. 318, 323 318, 323 KNOP,K. 308, 309, 322 P. 41, 60 LEMAIRE, KOCKARTS, G. 30,59 LEMARRE, G. 6, 14, 15, 43, 44, 46, 49, H. 277, 278, 323 KOGELNIK, 61 KOLMEDER, C. 137, 152 LENZ,K. 122, 153 KOMAR,V. G. 283, 319, 320, 321, 323 LEONARD, C. D. 294, 296, 322, 324 J. A. 159, 186, 187, 188, 197, LESSARD, KONINGSTEIN, R. A. 272, 273, 274,323 258 E. 246, 259 LEULIETTE-DEVIN, KOOPMAN, D. W. 353,356, 357, 358, 379 LEUNG,K. M. 334, 342, 379 J. 4, 60 KOORNNEEF, LEVINE,B. F. 246, 25? KOPF,L. 162, 256 LIAO,P. F. 353, 354, 378 S. N. 162, 258 KOSOLOBOV, LIM,T. K. 123, 151 KOZIEROWSKI, M. 161, 163, 180, 181, 195, LIN,C. 122, 152

386

AUTHOR INDEX

LIN, L. H. 270, 276, 279, 280, 296, 322, 323 LING,C. C. 356, 379 LIPSON,R. H. 362, 365, 366, 367, 378, 380 LITOVITZ,T. A. 226, 230, 233, 255 Lo, B. W. N. 246, 256 Lo BIANCO,C. V. 280, 296,323 D. J. 199, 255 LOCKWOOD, R. 246, 259 LOCQUENEUX, LOHMANN,A. 305, 317, 322 LONG,D. A. 159, 161, 163, 164, 165, 182, 185, 186, 187, 192, 195, 198, 256, 259 LOREE,T. R. 123, 124, 153 LOUDON,R. 159, 164, 204, 249,257, 259 LOUISELL,W. H. 170, 259 LOVESEY,S. W. 206, 256 W. H. 112, 152 LOWDERMILK, Lu, K. T. 365, 380 LUBAN,M. 162, 259 LUKASIK, J. 374, 376, 379, 380 LUKINYKH, V. F. 355, 370, 380 LYSSENKO, V. G. 165, 260

M MACCHEITO,F. 53, 61 MAESTRO,M. 191, 255 MAGNAN, A. 33,61 MAHON,R. 353, 354, 355, 356, 357, 358, 361, 362, 369, 370, 371, 379, 380 MAKER,P. D. 161, 162, 164, 165, 183, 190, 192, 194, 212, 214, 215, 222, 225, 259, 260, 261, 327, 379 MANAKOV, N. L. 163,' 177, 248, 258, 259 MANDEL,L. 77, 152, 164, 180, 248, 249, 258, 259, 266, 268, 323 MANGANARO, L. H. 354, 361, 380 J. H. 88, 123, 126, 152, 342, MARBURGER, 378 MARKS,J. 279, 324 MAROM,E. 266, 269, 323 MARR,G. V. 367, 379 MARTIN,F. B. 162, 222, 223, 224, 225, 258 MARTIN,0. 114, 152 MARTIN,W. E. 112, 152 MARTY,J. 126, 128, 153 MARWITT,5, 61 MASSEY,G. A. 133, 152 MATHIEU,J. P. 197, 260 T. 314, 324 MATSUMOTO, T. 285, 323 MATSUOKA, MAUCHERAT~OUBERT, M. 4, 5, 61

MAURON, N. 57, 58, 61 MAVRIN,B. N. 162, 165, 256 MAZUR,P. 247, 256 MCCLAIN,W. M. 252, 259 MCCLUNG,R. E. D. 229, 259 MCCOURT,F. R. 250, 256 J. T. 312, 313, 323 MCCRICKERD, MCGRATH,J. F. 5, 61 MCILRATH,T. J . 353, 355, 356, 357, 358, 359, 361, 362, 371, 379, 380 MCKEE,T. J. 353, 362, 365, 379 MCLELLAND, G. 122, 152 MCMAHON,J . M. 125, 133, 152 MCNEIL,K. J . 248, 249, 256, 259 MCTAGUE,J. P. 222, 256,259 MEHLMAN-BALLOFFET, G. 364, 371, 379 MEHTA,C. L. 249, 259 MEINEL,A. B. 25, 26, 48, 61 D. I. 354, 378 METCHKOV, METZ,H. J. 297, 322 MILAM,D. 112, 152 MILES,B. M. 331, 380 MILES,R. B. 327, 331, 333, 338, 340, 342, 378, 379 MILLIARD,B. 34, 35, 36, 60 T. E. 4, 61 MILLIGAN, MINGACEJr., H. S. 289, 322 MITA,T. 165,259 MITEV,V. M. 346, 354, 378, 379 MOCCIA,R. 191, 255 B. K. 248, 257 MOHANTY, MONNET,G. 19, 22, 23, 24, 34, 60, 61 MONTRY,G. R. 124, 132, 153 H. 213, 259 MORAAL, MORTON,D. C. 5 , 19, 43,61 MOSTOWSKI, J. 249, 259 MOUROU,G. 121, 150 J . 24, 60 MOUTONNET, MOVSESJAN, M. E. 248, 255 MULLER,D. 356, 360, 361, 378 MULLER,R. 123, 152 MUZfK, J. 296, 323 MYERSCOUGH, V. P. 358,379

N NAFIE,L. A. 226, 229, 259 NAGASAWA, N. 165, 259 NEEF, E. 123, 152 NEUCEBAUER, Th. 159, 259 G. 162, 165, 261 NEUMANN,

AUTHOR INDEX

NEW,G. H. C. 123,133, 135, 137, 150, 152, 246, 261, 327, 331, 338, 379, 380 NEWSOM, G. H. 371,379 B. R. A. 207,259 NIJBOER, NISHIDA,N. 280, 323 NITSOLOV, S. L. 226, 259 M. 283, 323 NOGUCHI, NOVARO, M. 114, 152 I. I. 248, 256 NURMINSKY, 0 OBUKHOVSKY, V. V. 164,261 F. G. 5,20,43, 61 ~'CALLAGHAN, O'HARE,J. M. 190, 259 OKOSHI,T. 318, 323 OPAL,C. B. 19, 60 ORR,B. J . 170, 259, 329, 334, 342, 379 ORTMANN, L. 162, 259 OSE, T. 296, 323 OSTERBROCK, D. 11, 61 L. N. 165,261 OVANDER, T. G. 321, 323 OVECHKINA, OVECHKIS, Yu. N. 283, 323 V. D. 163, 177,248,258,259 OVSIANNIKOV, OZGO,2. 163, 165, 181, 183, 187, 188, 189, 190, 191, 192, 193, 195, 196, 197, 198, 199, 200, 201, 220, 230, 232, 233, 246, 251, 254, 255, 258, 259

387

PERSHAN, P. S. 327, 329, 331,377 PETERS,C. W. 327, 378 S. H. 164, 192, 194, 261 PETERSON, W. L. 161, 226, 229, 256, 259, PETICOLAS, 260 PFEIFER,M. 122, 153 PHACH,V. D. 165, 257 PHILLIPS, N. J. 297, 323 J . 106, 148, 152 PIASECKI, PICINBONO, B. 152 PLACZEK,G. 163, 174, 175, 182, 184, 186, 197, 239, 260 PLANNER, A . 226, 258 PZ6CINICZAK, K. 170,260 V. B. 162, 165,256 PODOBEDOV, POLIVANOV, Yu. N. 165, 260 POPOV,A. K. 355, 370, 380 PORTER,D. 297, 323 POULET,H. 197, 260 POLJSSIGUE, G. 197, 259 POWER,A. E. 168, 260 J. G. 206, 260 POWLES, H. 166, 236, 238, 260 PRAKASH, PRINS,J. A. 207, 261 A. C. 365, 379 PROVOROV, PUELL,H. B. 333, 334, 342, 353, 354, 355, 365, 379, 380 PUMMER, H. 356, 360, 361, 378, 379 A. P. 280, 322 PYATIKOP,

P PANDEY,P. K. K. 175, 259 R. L. 353, 354, 370, 378 PANOCK, PAO, Y. H. 161, 213, 255 T. A. 164, 255 PAPAZIAN, PAQUES, H. 304, 323 W. H. 335, 336, 364, 378 PARKINSON, E. 197, 259 PASCAUD, PASMANTER, R. A. 162, 218, 220, 221, 259, 260 D. N. 247,259 PAITANAYAK, PAUL,F. W. 25, 61 L. I. 346, 354, 378, 379 PAVLOV, PEACH,G. 337, 379 PEARSON, D. B. 106, 133, 1.50 PECORA,R. 208, 209, 210, 228,260 PENNINGTON, K. S. 269, 271, 272, 276, 279, 280, 322, 323 PENZKOFER, A. 122,137, 152,153,248,259 PERINA,J . 164, 234,248,249, 259, 260,261 V. 164, 248, 249, 260, 261 PE~INOVA,

R RABIN, H. 121, 152, 162, 246, 260, 261 RAHMAN, A. 207, 259 RAUCH,J. E. 163, 193, 237, 256 RAYLEICH, Lord 105, 152 REEVES,E. M. 335, 336, 364, 378 REIF,J. 248, 260 REINTJES,J. 132, 152, 338, 347, 348, 353, 355, 356, 374, 375, 379, 380 RENTZEPIS, P. M. 122, 150 REYNAUD, S. 166, 255 RHODES,C. K. 356, 360, 361, 378, 379 M. C. 124, 132, 152, 348, 380 RICHARDSON, RISKEN,H. 249, 260 RITZE,H. H. 249, 260 G. P. 24, 60 RIVIERE, ROBERT, D. 225, 257 RODDIER, F. 101, 152 R ~ H RH., 353, 355, 379 ROKINI, M. 248, 261

388

AUTHOR INDEX

ROSE,M. E. 208, 260 U. 165, 257 ROSSLER, ROm, J. 306, 307, 315, 323 M. 356, 360, 361, 378, 379 ROTHSCHILD, ROTTKE,H. 360, 380 ROUSSIN, A. 4, 34,60 J. S. 206, 260 ROWLINSON, ROYT,T. R. 362, 364, 380 RUSTZI,0. M. 3, 60 RUTERBUSCH, P. H. 292, 311, 324 RUZEK,J. 296, 323 RYABOVA, R. V. 301,324 K. 249, 259 RZAZEWSKI, S

SAME,M. 21, 27, 31, 53, 60, 61 SAISSY, A. 122, 150 SAITO,T. 285, 323 T. 165, 257 SAMESHIMA, SAMSON, R. 162, 218, 220, 221, 259, 260 A. 9,61 SANDAGE, SANDER, R. K. 374, 378 SANTRY, D. P. 175, 259 S. M. 164, 255 SARKISIAN, M. 296, 323 SASAKI, C. 114, 140, 151, 152 SAUTERET, SAVAGE, C. M. 161, 164, 165, 192, 194,225, 260, 261,327, 379 SAYAKHOV, R. Sh. 165, 260 SCANDONE, F. 48.61 M. G. 121, 153 SCEATS, SCHAFFER, C. E. 201, 255 H. 333, 334, 342, 344, 345, SCHEINGRABER, 353, 354, 365, 371, 379, 380 M. 140, 151 SCHELEV, SCHMID, W. J. 164, 226, 260 A. J. 135, 151 SCHMIDT, SCHREY, H. 165,260 D. J. 43, 61 SCHROEDER, S C H R O ~ EH. R , W. 164, 226, 260 M. 166, 248, 260 SCHUBERT, E. 0. 121, 150 SCHULTZ-DUBOIS, SCHWARZSCHILD, M. 30,41,61 SEIN,J. H. 247, 260 SEMENOV, S. P. 301, 324 SENITZKY, I. R. 249. 260 SHACK,R. V. 25, 26, 61 C. V. 122, 125, 133, 135, 139, 151, SHANK, 152 SHANMUGANATHAN, K. 133, 152

C. E. 72, 152 SHANNON, SHAPIRO, S. L. 122, 124, 132, 150, 151 S. 11, 61 SHARPLESS, SHE, C. Y. 338, 347, 348, 353, 355, 374, 375, 379, 380 SHELTON, J. W. 124, 152 SHEN,Y. R. 88, 123, 124, 126, 152,248, 260 SHEVCHENKO, V. V. 280,322 SHORE,B. W. 371, 379 L. D. 111, 124, 132, 153 SIEBERT, H. D. 164, 248, 260 SIMAAN, W. W. 132, 150 SIMONS, K. S. 207, 260 SINGWI, SIPE,J. E. 217, 247, 261 SIVAN, J. P. 6, 22, 30, 34, 35, 36, 60, 61 A. 207, 260 SJOLANDER, A . 41, 60 SKURMANICH, V. V. 355, 370, 380 SLABKO, SMITH,M. W. 331, 380 SMITH, S. D. 122, 152 SNIDER, R. F. 250, 256 G. A. 321,323 SOBOLEV, S. B. 136, 151 SOGOMONIAN, SOKOLOVSKY, R. I. 162, 258 SOM, s. c. 272. 323 A. 97, 153 SOMMERFELD, SOROKIN, P. P. 328,334, 337, 351, 352, 354, 361, 362, 363, 364, 365, 378, 380 SPECK,D. R. 132, 150 L. 5, 19, 61 SPITZER, SPOHN,H. 248, 249, 260 T. 356, 360, 361, 378 SRINIVASAN, STAMENOV, K. V. 346, 354, 378, 379 L. 163, 164, 182, 189, 259, 260 STANTON, V. S. 229, 260 STARUNOV, M. 121, 153 STAVOLA, STEEL,W. H. 285, 287, 289, 292, 293, 294, 322, 323 STEELE,W. A. 208, 209, 210, 228, 229, 260 STELL.G. 210, 213, 257, 260 STERIN,Kh. E. 162, 165, 256 A. P. 224,225,260 STOGRYN, STOGRYN, D. E. 224, 225,260 STOICHEFF, B. P. 334, 353, 362, 365, 366, 368, 372, 373, 378, 379,380 STOLEN, R. H. 122, 125, 152, 153 STONE,A. J. 183, 251, 260 STONE,T. 110, 118, 140, 150 G. 164, 186, 187, 193, 238, 254, 260 STREY, STRIZHEVSKY, V. L. 163, 164, 182, 237, 261

AUTHOR INDEX

STROKE,G. W. 278, 279, 280, 323 SUDARSHAN, E. C. G. 249, 258 SUKHMAN, Y. P. 321, 323 K. R. 246, 261 SUNDBERG, SWDAM,B. R. 132, 153 SUZUKI,M. 285, 323 SVELTO,0. 123, 126, 153 P. 164, 248, 249, 260, 261 SZLACHETKA,

T TABISZ,G. C. 222, 261 J. 137, 153 TABOADA, TAI,A. M. 288,302, 310, 311,312,322,324 Y.319, 324 TAKEDA, P. N. 285, 290, 292, 301, 323 TAMURA, TANAS,R. 163, 247, 248, 249, 258, 261 TANG,C. L. 121, 152, 246, 261 TARANUKHIN, V. D. 193, 257 S. 274, 323 TATUOKA, TAYLOR,J. R. 354,380 TELLE,H. R. 225, 261 TELLER,E. 186,260 TERHUNE,R. W. 161, 164, 165, 225, 261, 327, 379 THAG, C. D. 122, 153 THE OPTICAL SOCIETY OF AMERICA 281, 323 THIRUNAMACHANDRAN, T. 163, 180, 187, 193, 194, 237, 245, 246, 252,254 THOMAS,C. E. 111, 153 B. V. 248, 257,261 THOMPSON, THORNE, J. M. 123, 124, 153 TINDLE,C. T. 249, 261 TODD,J. J. 4, 7, 60 F. S. 353, 354, 362, 363, 365, 369, TOMKINS, 370, 371, 379, 380 I. V. 348, 380 TOMOV, H. 140, 151 TOURBEZ, P. 108, 109, 110, 118, 120, 121, TOURNOIS, 133, 151 E. B. 94, 106, 107, 121, 124, 132, TREACY, 133, 138, 139, 153 R. 291, 322 TREMBLAY, TROSHIN,€3. I. 355, 380 Y. 319, 324 TSUNODA, TURNER,E. H. 106, 133, 150 TLTITLEBEE, W. H. W. 248, 256 TWISS,R. Q. 77, 151 U UETA,M. 165, 259 UPATNIEKS, J. 265,268,279, 294, 323, 324

389

V VACHASPATI, 166, 260 VAKS,V. G. 165, 255 VALERIO,Y. 4, 34, 60 K. A. 229, 261 VALIYEV, M. 162, 256 VALLADE, VALLAT,P. 140, 151 VALLBE,F. 374, 379 VAMPOUILLE, M. 112, 114, 126, 128, 134, 148, 151, 152, 153 R. 297, 323 VANHOREBEEK, VANHOVE,L. 206, 261 VAN KRANENDONK, J. 166, 217, 247, 256, 261 VANNESTE, C. 122, 150 VARSHAL, B. G. 162, 165, 256 D. D. 137, 153 VENABLE, J. F. 164, 192, 194, 261 VERDIECK, VIAL,J. C. 41, 60 VIDAL,C. R. 333, 334, 342, 343, 344, 345, 353,354, 365, 371, 379, 380 VIDAL-MADJAR, A. 41, 60 VIENOT,J. Ch. 100, 142, 151, 152 G. H. 207, 261 VINEYARD, VITON,M. 6, 22, 30, 34, 35, 36, 60, 61 VLASOV,N. G . 301, 324 VOGT,H. 161, 162, 165, 259,261 VOIGT,J. 165, 257 VON DER LINDE,D. 122, 124, 132, 152 VREHEN,Q. H. F. 248, 261 A. 22, 23, 30,34, 60 VUILLEMIN,

W WACKERLING, L. R. 5, 61 WALKER,J. L. 278, 322 WALLACE,S. C. 353, 359, 362, 365, 371, 372, 373, 374, 378, 379, 380 R. 353, 356, 357, 359, 360, WALLENSTEIN, 369,378, 380 WALLS,D. F. 164, 248, 249, 255, 256, 259, 261 WALTER,W. R. 289, 322 H. 248,260 WALTHER, WANG,C. C. 331, 354, 380 WARD,A. A. 297, 323 WARD,J. F. 170, 246, 259, 261, 327, 329, 331, 334, 338, 342, 379, 380 J. H. 122, 153 WEIGMANN, D. L. 161, 215, 261 WEINBERG, WEINMANN, D. 162, 261

390

AUTHOR INDEX

WEINREICH, G. 327, 378 WEIS,J. J. 213, 260 V. 173, 261 WEISSKOPF, WELGE,K. H. 360, 380 WELTER,D. D. 301, 322 WERNCKE,W. 122, 153 J. A. 48, 61 WESTPHAL, WHITE,J. C. 353, 354, 370, 378 WIEDMANN, J. 137, 153 E. 166, 255 WIENER-AVNEAR, WIESE,W. L. 331, 380 WIGMORE,J. K. 106, 121, 133, 153 WIGNER,E. 173, 261 WILHELMI,B. 166, 248,260 WILLIAMS, M. L. 190, 191, 257 WILLIS,A. J. 4, 7, 60 WILSON,A. D. 164, 248, 249, 255 WILSON,R. 4, 7, 60 WISER,N. 162, 259 W6DKIEWICZ, K. 203, 256 WWEJKO,L. 226, 236, 261 WOLF,E. 67, 70, 73, 79, 127, 150, 164, 180, 234, 236, 247, 248, 255, 259, 261 Woo, J. W. F. 164, 257 WOOD, R. W. 308, 324 WRAY,J. 5, 20, 43,61 WYANT,J. C. 290, 291, 324 WYNNE,C. 23, 61 WYNNE,J. J. 328, 334, 335, 337, 351, 352, 354,361,362,363,364,365, 377,378,380

Y YABLONOVITCH, E. 122, 152, 153 YANO,A. 319, 324 YAN-SONG,C. 301, 324 YATSIV,S. 248, 261 YIU, Y. M. 353, 355, 359, 379, 380 YOUNG,J. F. 350, 353, 354, 355, 376, 378, 379, 380 Yu, F. T. S. 288, 292, 302, 310, 311, 312, 322, 324 Yu, W. 165, 200, 201, 261 YURATICH, M. A. 248, 256 Yu-TANG,W. 301, 324

2 ZACHARIAS, H. 360, 380 ZAPKA, w. 356, 380 ZAVELOVICH, J. 356, 360, 361, 378 ZAVOROTNEV, Yu. D. 165, 261 ZAWODNY, R. 181, 195, 200, 220, 258, 259 ZDASIUK, G. 361, 362, 365, 371, 380 ZECH,R. G. 279, 323 ZERNIKE,F. 207, 261 ZHANG,Y. W. 292, 324 ZHUANG,S. L. 292, 311, 324 ZINTH, W. 137, 152 ZYCH,L. J. 355, 376, 379, 380

SUBJECT INDEX A

E

analytic signal, 67 anti-Stokes line, 186 --- process, 158 scattering, spontaneous, 328, 376 autocorrelation, 72, 141-144, 272 auto-ionizing level, 334, 335, 371 - _resonance, 336

electric-dipole approximation, 172, 181 Euler angles, 184 Ewald-Oseen extinction theorem, 247

F

B Bessel function, spherical, 209 birefringence, 135, 136 blackbody distribution, 30 Boltzmann constant, 341 - distribution, 187 Born-Oppenheimer approximation, 184 Bragg angle, 277 - condition, 275 Brillouin scattering, 248 C

camera, Faint Object, 47, 54, 55, 57 -, Schmidt, 9, 10 -, Very Wide Field, 7-9, 11-19, 22, 44 -, Wide Field and Planetary, 47, 48, 49 -, Wynne, 25 caustic fringes, 129, 132 chaotic light, 180 coherence length, 70, 79, 80 partial, 76 coherent light, 180 - optics, 65, 71 correlation function, 206, 207, 219, 246, 272

Fabry-Ptrot interferometer, 108, 111, 141, 146 Fourier analysis, 65, 115 - synthesis, 116 -transformation, 68, 86, 203 four-wave-sum-mixing, 328, 332, 337, 338, 340, 364, 367, 369 frequency conversion, 349 - dispersion, 85 -, instantaneous, 99, 139 - modulation, 89 - spectrum, 85 Fresnel-Kirchhoff diffraction, 83 - length, temporal, 88, 89 -zone plates, 115 fusion, nuclear, 65

G

-.

gauge transformation, 168 Gaussian beam, 70, 89, 123, 134 - distribution, 50 - mode, 338 Gegenschein, 11 geocorona, 17 Glan-Thompson prism, 351 Goddard space flight center, 4, 6

D

H

depolarization ratio, 238, 239, 243 Dirac distribution, 75, 82, 141 -pulse, 98, 100, 101, 112 Doppler broadening, 341

harmonic generation, second, see second harmonic generation - -, third, see third harmonic generation Hertz vector, 178, 203, 216, 217 391

392

SUBJECT INDEX

Hilbert space, 169 hologram, 118, 146, 265-279, 299, 304, 306, 308 -, bleached reflection, 296, 297 -, computer generated, 316, 317 -, Fourier, 142, 309 -, rainbow, 283-286, 288, 290-292, 300, 301, 305, 310-313 -, volume reflection, 295, 302 holographic cinematography, 318 - fringes, 117 - grating, 18, 19, 114 holography, colour, 268, 279, 280, 307, 316, 321 -, Fourier, 141 Huygens principle, 65, 84, 88

Morgan’s photometry, 3 Mount Palomer Observatory, 58 multiphoton scattering, 182, 183, 246 --, incoherent, 178 --, spontaneous, 157 multiplexing, 272 -, frequency, 268 -, spatial, 269

N nonlinear medium, 248 -optics, 121 - susceptibility, 327, 328, 331-334, 337, 346, 349

I

0

intensity correlations, 137 - fluctuations, 77 interferometer, Michelson, see Michelson interferometer) -, S.I.S.A.M., 144 irradiance, 267

Observatoire de Marseille, 6 Optical bistability, 65, 133 - coherence theory, 73 - communication, 65 -fibers, 148 - Kerr effect, 135 - nutation, 164 -switching, 122 orthoscoscopic image, 273, 274 oscillator strength, 331

K Kerr effect, 330, 343, 345, 346, 375 KDP crystal, 350

L

P

Laboratoire d’Astronomie Spatiale, 6, 7, 24, 29 laser, mode locked, 65, 70, 123, 135, 350 Lorentz function, 228 --Voigt electron theory, 167 Lyman CY, 16, 17, 353, 357, 371

partial coherence, 72 Petzval curvature, 10 phase conjugation, 65 - matching, 340, 349, 353, 368, 374 -transition, 162 photodetector, 75 photodiode, 72, 73 photoelectric effect, 135 photographic emulsion, 73 photon antibunching, 164, 249 - counting detector, 51 photopolymer, 296 picosecond pulse, 63, 66, 102, 106, 134,201, 225 piezo-electric crystal, 329 polarizability, nonlinear, 166, 172-174, 177, 216, 220 power spectrum, 129 Poynting vector, 178, 332

M Magellanic clouds, 5, 21, 29, 30 Mandelshtam-Brillouin doublet, 207 Markov process, 228 Michelson interferometer, 75, 95, 143, 144 Milky Way, 3, 5, 6, 9, 11, 16 molecule, multipole, 223 -, quadrupolar, 223 -, tetrahedral, 223 monochromatic discrimination, coefficient of, 16

SUBJECT INDEX

Q quasi-monochromatic pulse, 78 radiation, 79

R Racah algebra, 193 Raman line broadening, 226 - scattering, 158, 159, 183-186, 228, 232, 233 --, hyper-, 160, 163-165, 193, 226, 229-231 --, multiphoton, 163, 165, 187 --, spontaneous, 160, 163, 247-249 --, stimulated, 65, 122, 166 -spectroscopy, 159 Rayleigh-Krishnan reciprocity relation, 244 -probability distribution, 294 -scattering, 158, 160, 162, 180, 181, 226, 228, 229 --,hyper-, 161, 164, 215, 231 --, spontaneous, 160, 161, 165 --, stimulated, 166 -Smoluchowski line, 159 resonance fluorescence, 164 reversal ratio, 241, 243 Rochon polarizer, 50 S

saturable obsorber, 122, 137 Schrbdinger’s equation, 186 second harmonic generation, 135, 136, 225, 246, 327, 330, 359 --scattering, 162, 163, 181, 214 self focusing, 125 - phase modulation, 122 Seyfert galaxy, 41 Skylab, 20-22, 28, 31 Space Sciences Laboratory, 24 spectral distribution, 67, 189 - finesse, 79 spectroscopy, Doppler-free, 166 -, infrared, 232 -, multiphoton, 160 -, Raman, see Raman spectroscopy

393

-, time resolved, 139 spherical harmonic function, 168 Stark shift, 164, 344, 345 - splitting, 164 Steele parameter, 210 Stokes line, 186 - parameters, 234, 238 - process, 158 susceptibility tensor, 329

T telescope, Maksutov, 5 -, Meinel and Shack, 26, 27 -, Schmidt, 5 , 6, 11, 19, 39, 43, 44, 45,

59 -, - Cassegrain, 31, 32 -, Wynne, 27 thermal sources, 72 third harmonic generation, 327, 328, 330, 332, 340, 342, 343 tunable generation in beryllium, 371 --_ calcium, 371 --_ magnesium, 364 --- mercury, 369 - _ - strontium, 363 zinc, 367 two-photon absorption, 343 resonance, 344

-__ W

Weisskopf-Wigner theory, 173 Wiener-Khinchine theorem, 203 Wigner coefficients, 33, 186 - function, 208, 210 - rotation matrix, 183

Y Young’s formula, 95

- fringes, 96 - slit experiment, 83, 98 2

Zodiacal light, 3, 6, 11, 12

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CUMULATIVE INDEX - VOLUMES I-Xx ABEGS, F., Methods for Determining Optical Parameters of Thin Films 11, 249 ABELLA,I. D., Echoes at Optical Frequencies VII, 139 ABITBOL,C. I., see J. J . Clair XVI, 71 AGARWAL,G. S., Master Equation Methods in Quantum Optics XI, 1 ACRANOVICH, V. M., V. L. GINZBURC, Crystal Optics with Spatial Dispersion IX, 235 ALLEN. L., D. G. C. JONES,Mode Locking in Gas Lasers IX, 179 E. 0.. Synthesis of Optical Birefringent Networks AMMANN, IX,123 ARMSTRONG, J. A,, A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI,211 ARNAUD, J. A,, Hamiltonian Theory of Beam Mode Propagation XI,247 BALTES,H. P., On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium Environment XII, 1 BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images I, 67 BASHKIN, S., Beam-Foil Spectroscopy XII, 287 BECKMANN, P., Scattering of Light by Rough Surfaces VI, 53 BERRY,M. V., C. UPSTILL,Catastrophe Optics: Morphologies of Caustics and their Diffraction Patterns XVIII, 259 BEVERLY111, R. E., Light Emission from High-Current Surface-Spark Discharges XVI, 357 BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements IX, 1 BOUSQUET. P., see P. Rouard IV, 145 BRUNNER, W., H. PAUL,Theory of Optical Parametric Amplification and Oscillation xv, 1 O., Applications of Shearing Interferometry BRYNGDAHL, IV, 37 O., Evanescent Waves in Optical Imaging BRYNGDAHL, XI,167 BURCH,J. M., The Metrological Applications of Diffraction Gratings 11, 73 H. J., Principles of Optical Data-Processing BUTTERWECK, XIX,211 CAGNAC.B., see E. Giacobino XVII, 85 CASASENT,D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 CLAIR,J. J., C. 1. ABITBOL,Recent Advances in Phase Profiles Generation XVI, 71 CLARRICOATS, P. J. B., Optical Fibre Waveguides-A Review XIV, 321 COHEN-TANNOUDJI, C,, A. KASTLER,Optical Pumping v. 1

395

396

CUMULATIVE INDEX

XV, 187 COLE.T. W., Quasi-Optical Techniques of Radio Astronomy XX, 63 COLOMBEAU, B., see C. Froehly M. DETAILLE,M. SAISSE,Some New Optical COURT& G., P. CRUVELLIER, Designs for Ultra-Violet Bidimensional Detection of Astronomical Objects xx, 1 CREW, A. V., Production of Electron Probes Using a Field Emission Source XI, 223 CRUVELLIER, P., see C. G. Courtks xx, 1 Light Beating Spectroscopy VIII, 133 CUMMINS, H. Z., H. L. SWINNEY, XIV, 1 DAINTY,J. C., The Statistics of Speckle Patterns DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 XII, 101 DECKERJr., J. A,, see M. Hanvit VII, 67 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters IX, 31 DEMARIA,A. J., Picosecond Laser Pulses DETAILLE,M., see G. Court& 1 xx, X, 165 DEXTER,D. L., see D. Y. Smith K. H., Interaction of Light with Monomolecular Dye Layers DREXHAGE, XII, 163 DUGUAY, M. A., The Ultrafast Optical Kerr Shutter XIV, 161 VII, 359 EBERLY,J. H., Interaction of Very Intense Light with Free Electrons ENNOS,A. E., Speckle Interferometry XVI, 233 FIORENTINI, A., Dynamic Characteristics of Visual Process I, 253 FOCKE, J., Higher Order Aberration Theory IV, 1 FRANCON,M., S . MALLICK,Measurement of the Second Order Degree of Coherence VI, 71 FRIEDEN, B. R., Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 311 FROEHLY,C., B. COLOMBEAU, M. VMOUILLE, Shaping and Analysis of Picosecond Light Pulses XX, 6 3 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 I, 109 GABOR,D., Light and Information GAMO,H., Matrix Treatment of Partial Coherence 111, 187 XIII, 169 GHATAK,A. K., see M. S. Sodha GHATAK, A,, K. THYAGARAJAN, Graded Index Optical Waveguides: A Review XVIII, 1 GIACOBINO, E., B. CAGNAC, Doppler-Free Multiphoton Spectroscopy XVII, 85 GINZBURG, V. L., see V. M. Agranovich IX, 235 R. G., Diffusion Through Non-Uniform Media GIOVANELLI, 11, 109 GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the DiffracIX, 281 tion Theory of Elastic Waves GOODMAN, J. W., Synthetic-Aperture Optics VIII, 1 GRAHAM,R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 HARIHARAN, P., Colour Holography XX, 263 HARWIT,M., J. A. DECKERJr., Modulation Techniques in Spectrometry XII, 101

CUMULATIVE INDEX

397

HELSTROM, C. W., Quantum Detection Theory X, 289 HERRIOT, D. R., Some Applications of Lasers to Interferometry VI, 171 HUANG,T. S., Bandwidth Compression of Optical Images x, 1 JACOBSON,R., Light Reflection from Films of Continuously Varying Refractive Index V, 247 Apodisation JACOUINOT, P., B. ROIZEN-DOSSIER, 111, 29 J A M R O ~ .W., B. P. STOICHEFF, Generation of Tunable Coherent VacuumUltraviolet Radiation XX, 325 JONES,D. G. C., see L. Allen IX, 179 KASTLER, A., see C. Cohen-Tannoudji v. 1 KIELICH,S., Multi-Photon Scattering Molecular Spectroscopy XX, 155 K., Surface Deterioration of Optical Glasses KINOSITA, IV, 85 KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, 1 KOTIZER,F., The Elements of Radiative Transfer 111, 1 KOTTLER, F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory IV, 281 KOTI-LER, F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory VI, 331 KUBOTA,H., Interference Color I, 21 1 LABEYRIE,A.. High-Resolution Techniques in Optical Astronomy XIV, 47 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XI, 123 LEE, W.-H., Computer-Generated Holograms: Techniques and Applications XVI, 119 Recent Advances in Holography LEITH,E. N., J. UPATNIEKS, VI, 1 LETOKHOV,V. S., Laser Selective Photophysics and Photochemistry XVI, 1 LEVI,L., Vision in Communication VIII, 343 LIPSON,H., C. A. TAYLOR,X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 MALLICK,S., see M. Francon VI, 71 MANDEL,L., Fluctuations of Light Beams 11, 181 MANDEL,L., The Case for and against Semiclassical Radiation Theory XIII, 27 MARCHAND, E. W., Gradient Index Lenses XI, 305 MEESSEN,A., see P. Rouard xv, 77 ~IEHTA, C. L., Theroy of Photoelectron Counting VIII, 373 Quasi-Classical Theory of Laser MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Radiation VII, 231 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction XVII, 279 MILLS, D. L., K. R. SUBBASWAMY, Surface and Size Effects on the Light Scattering Spectra of Solids XIX, 43 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design I, 31 MOLLOW,B. R., Theory of Intensity Dependent Resonance Light Scattering and Resonance Fluorescence XIX, 1 MURATA,K., Instruments for the Measuring of Optical Transfer Functions V, 199 MUSSET,A., A. THELEN, Multilayer Antireflection Coatings VIII, 201

398

CUMLTLATIVE INDEX

XV, 139 OKOSHI,T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image PAUL,H., see W. Brunner xv, 1 PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 PEGIS,R. J., see E. Delano VII, 67 PERINA,J., Photocount Statistics of Radiation Propagating through Random and Nonlinear Media XVIII, 129 V, 83 PERSHAN, P. S., Non-Linear Optics IX, 281 PETYKIEWICZ, J., see K. Gniadek V, 351 PICHT,J., The Wave of a Moving Classical Electron XVI, 289 PSALTIS,D., see D. Casasent RISEBERG,L. A., M. J. WEBER, Relaxation Phenomena in Rare-Earth XIV, 89 Luminescence VIII, 239 RISKEN,H., Statistical Properties of Laser Light XIX, 281 RODDIER,F., The Effects of Atmospheric Turbulence in Optical Astronomy 111, 29 ROIZEN-DOSSIER, B., see P. Jacquinot ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical XIII, 69 Aberration Measurements of the Human Eye IV, 145 ROUARD,P., P. BOUSQUET, Optical Constants of Thin Films xv, 71 ROUARD,P., A. MEESSEN,Optical Properties of Thin Metal Films IV, 199 RUBINOWICZ, A., The Miyarnoto-Wolf Diffraction Wave XIV, 195 RUDOLPH, D., see G. Schmahl SA~SSE, M., see G. Court6.s XX, 1 VI, 259 SAKAI,H., see G. A. Vanasse XIV, 195 SCHMAHL, G., D. RUDOLPH,Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence between Coherence XVII, 163 Properties of Light and Nonlinear Optical Processes XIII, 93 Interferometric Testing of Smooth Surfaces SCHULZ,G., J. SCHWIDER, XIII, 93 SCHWIDER, J., see G. Schulz X, 89 SCULLY,M. O., K. G. WHRNEY,Tools of Theoretical Quantum Optics I. R., Semiclassical Radiation Theory within a Quantum-Mechanical SENITZKY, XVI, 413 Framework XV, 245 SIPE,J. E., see J. Van Kranendonk X, 229 SITTIG,E. K., Elastooptic Light Modulation and Deflection XII, 53 SLUSHER, R. E., Self-Induced Transparency VI, 21 1 SMITH,A. W., see J. A. Armstrong SMITH, D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in X, 165 Insulators x, 45 SMITH,R. W., The Use of Image Tubes as Shutters SODHA,M. S., A. K. GHATAK, V. K. TRIPATHI, Self Focusing of Laser Beams in XIII, 169 Plasmas and Semiconductors V, 145 STEEL, W. H., Two-Beam Interferometry XX, 325 STOICHEFF,B. P., see W. Jamroz

CUMULATIVE INDEX

STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere STROKE,G. W., Ruling, Testing and Use of Optical Gratings for HighResolution Spectroscopy SUBBASWAMY, K. R., see D. L. MILLS

SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser Beams S ~ YH. H., , see H. Z. Cummins TANGO, W. J., R. Q. TWISS,Michelson Stellar Interferometry Strong Fluctuations in Light Propagation TATARSKII, V. I., V. U . ZAVOROTNYI, in a Randomly Inhomogeneous Medium TAYLOR, C. A., see H. Lipson TER-MIKAELIAN, M. L., see A. L. Mikaelian THELEN,A,, see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light THYAGARAIAN, K., see A. Ghatak TRIPATHI,V. K., see M. S. Sodha TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and by Spatial Frequency Filtering TWISS, R. Q., see W. J. Tango UPATNIEKS, J., see E . N. Leith UPSTILL,C., see M. V. Berry USHIODA,S., Light Scattering Spectroscopy of Surface Electromagnetic Waves in Solids VMOUILLE, M., see C. Froehly Fourier Spectroscopy VANASSE,G. A., H. SAKAI, VAN HEEL, A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J . E. SIPE, Foundations of the Macroscopic Electromagnetic Theory of Dielectric Media VERNIER,P.. Photoemission WEBER,M. J., see L. A. Riseberg WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings WELFORD,W. T., Aplanatism and Isoplanatism WILHELMI,B., see M. Schubert WITNEY,K. G., see M. 0. Scully

399

Ik, 7 3 11, 1 XIX, 43

XII, 1 VIII, 133 XVII, 239 XVIII, 207 V, 287 VII, 231 VIII, 201 VII, 169 XVIII, 1 XIII, 169 11, 131 XVII,239 VI, 1 XVIII, 259

XIX, 139 XX, 6 3 VI, 259 I, 289 XV, 245 XIV, 245 XIV, 89 IV, 241 XIII, 267 XVII, 163 X, 89

WOLTER,H., On Basic Analogies and Principal Differences between Optical I, 155 and Electronic Information X, 137 WYNNE,C. G.. Field Correctors for Astronomical Telescopes VI, 105 YAMAJI,K., Design of Zoom Lenses YAMAMOTO,T., Coherence Theory of Source-Size Compensation in Interference Microscopy VIII, 295 YOSHINAGA, H., Recent Developments in Far Infrared Spectroscopic Techniques XI, 77 XVIII, 207 ZAVOROTNYI, V. U., see V. I. Tatarskii

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