Progress in Optics: v. 16

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

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PROGRESS IN OPTICS VOLUME XVI

EDITORIAL ADVISORY BOARD L. ALLEN,

Brighton, England

M. FRANCON,

Paris, France

E. INGELSTAM,

Stockholm, Sweden

K. KINOSITA,

Tokyo, Japan

A. KUJAWSKI,

Warsaw, Poland

A. LOHMANN,

Erlangen, Germany

M. MOVSESSIAN,

Armenia, U.S.S.R.

G. SCHULZ,

Berlin, D.D.R.

W. H. STEEL,

Chippendale, N.S.W., Australia

W. T. WELFORD,

London, England

PROGRESS I N OPTICS VOLUME XVI

EDITED BY

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

Contributors V . S . LETOKHOV, J . J . CLAIR, C . I. A B ITBO L W-H. LEE, A . E . ENN O S

D . CASASENT, D . PSALTIS R . E . BEVERLY 111, I . R. SENITZKY

1978

NORTH-HOLLAND PUBLISHING COMPANYAMSTERDAM. NEW YORK . OXFORD

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C O N T E N T S OF V O L U M E I ( 1 9 6 1 ) I. I1.

THE MODERN DEVELOPMENT OF I-IAMILToNIm O m a . R . J . PEGIS . . . WAVE Omcs AND GEOMETRICALO m a IN Omcm DESIGN. K . ~~IYAMOTO

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

111. THE INTENSII 'x D I S T R I B ~ O AND N TOTALILLUMINATION OF ABERRATIONFREE DIFFRACTION IMAGES.R . BARAKAT. . . . . . . . . . . . . . IV. LIGHTANDINFORMATION. D. GABOR . . . . . . . . . . . . . . . V. ONBASICANALOGIES AND PRINCIPALDIFFERENCES BETWEEN O m c m AND ELECTRONIC "FORMATION. H . WOLTER. . . . . . . . . . . . . . . VI . INTERFERENCE COLOR. H . KUBOTA . . . . . . . . . . . . . . . . CHmm~rncs OF VISUAL ~OcESsEs.A . F~ORENTINI . . . VII . DYNAMIC VIII. MODERN ALIGNMENTDEVICES. A. C. S. VAN HEEL . . . . . . . . . .

1-29 31-66 67-108 109-153 155-210 211-251 253-288 289-329

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

RULING.TESTING AND USEOF O ~ C AGRATINGS L FORHIGH-RESOLUTION SPEC~ROSCOPY. G. W. STROKE . . . . . . . . . . . . . . . . . . 1-72 I1. THE h4ETROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS. J. M . BURCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-108 111. DIFFUSIONTROUGH NON-UNIFORM MEDIA.R . G. G I o v m u r . . . . . 109-129 IV. CORRECIION OF Omcm IMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING. J . TSUJIUCHI . . . . . . . . . . . 131-180 V. FLUCTUATIONS OF LIGHT BEAMS. L. . . . . . . . . . . . . 181-248 VI . METHODSFOR DETERMINING O ~ C A PARAMETERS L OF THIN FILMS.F. &EL& ............................ 249-288

C O N T E N T S OF V O L U M E I11 (1964) I. I1.

THE ELEMENTSOF RADIATIVETRANSFER. F. KOTIZER . AFWDISATION. P. JACQUINOT AND B . ROIZEN-DOSSIER. 111. lMATRM TREATMENT OF P m n a COHERENCE .H . GAMO

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

1-28 29-186 187-332

C O N T E N T S OF V O L U M E I V (1965) I. I1. 111.

HIGHERORDERABERRATION THEORY. J . FOCKE

. . . . . . . . . .

. . . .

APPLICATIONS OF SHEARING INTERFEROMETRY. 0. BRYNGDAHL

SURFACE DETERIORATION OF O m c m GLASSES. K . JSINOSITA . . . . . Iv. OPnCAL CONSTANTS OF THIN FILMS.P. ROUARD AND P . BOUSQUET. . .

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

V. THE MNAMOTO-WOLF DIFFRACTION WAVE.A . Rusr~ow~cz . . . . . VI . ERRA AT ION THEORY OF GRATINGS AND GRATING MOUNTINGS. w. T . WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . . 241-280 VII . D ~ C T I OATN A BLACKSCREEN.PART I: KXRCHHOFFSTHEORY.F. KOTIZER . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281-314

C O N T E N T S OF V O L U M E V (1966) I. I1. 111.

O ~ C APUMPTING. L C. COHEN-TANNOUDJI AND A . KASTLER . . . . . . NON-LINEAR Omcs. P. S. PERSHAN . . . . . . . . . . . . . . . . TWO-BEAM INTERFEROMETRY W. H . STEEL . . . . . . . . . . . . .

.

1-81 83-144 145-197

IV. INSTRUMENTS FOR THE MEASURING OF O ~ C ATRANSFER L FUNCTIONS, K. 199-245 MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . LIGHTREFLECTION FROM F m s OF CONTINUOUSLY VARYMC R E m m INDEX, R. JACOBSSON. . . . . . . . . . . . . . . . . . . . . . 247-286 VI. X-RAY CRYSTAL-STRUCTURE DETERMINATION AS A BRANCH OF PHYSICAL Omcs, H. LIPSONAND C. A. TAYLOR . . . . . . . . . . . . . . . 287-350 VII. THEWAVEOF A MOVING CLASSICAL ELECTRON, J. PIcm . . . . . . . 35 1-370

V.

C O N T E N T S OF V O L U M E V I ( 1 9 6 7 ) RECENTADVANCE~INHOLOGRAPHY, E. N. L E ~ A NJ.DUPATNIEKS. . 1-52 SCATIBRING OF LIGHT BY ROUGH SURFACES, P. BECKMA” . . . . . . 53-69 MEASUREMENT OF THE SECONDORDERDEGREEOF COHERENCE, M. AND S. MALLICK . . . . . . . . . . . . . . . . . . . . FRANCON 7 1-104 IV. DESIGN OF ZOOM LENSES, K. Y W I . . . . . . . . . . . . . . . . 105-170 V. SOMEAPPLICATIONS OF LASERS TO INTERFEROMETRY, D. R. HERRlorr . 171-209 VI. EXPERIMENTALS m m OF I m s m FLUCTUATIONS IN LASERS,J. A. ARMSTRONG AND A. w. . . . . . . . . . . . . . . . . . . 21 1-257 259-330 VII. FOURIER SPECTROSCOPY, G. A. VANASSE AND H. S m . . . . . . . . VIII. DIFFRACTION AT A BLACK SCREEN,PART11: ELECTROMAGNETIC THEORY, . . . . . . . . . . . . . . . . . . . . . . . . . . . 331-377 F.KOT~LER I. 11. 111.

C O N T E N T S OF V O L U M E V I I ( 1 9 6 9 ) MULTIPLE-BEAM INTERFERENCE AND NATURAL MODESIN OPEN RESONATORS, G. KOPPELMAN . . . . . . . . . . . . . . . . . . . . . 1-66 11. METHODSOF SYNTHESIS FOR DIELECI’RIC MULTILAYER FILTERS, E. DELANO m R . J. PEGIS . . . . . . . . . . . . . . . . . . . . . . . . . 67-137 111. ECHOES AT O m c a FREQUENCIES, I. D. ABELLA . . . . . . . . . . 139-168 IV. IMAGE FORMATION WITH PARTIALLY COHERENT LIGHT,B. J. THOMPSON . 169-230 V. QUASI-CLASSICAL THEORY OF LASER RADIATION, A. L. MVIIKAELIAN AND M. L. TER-MIKAELIAN. . . . . . . . . . . . . . . . . . . . . . . 231-297 VI. THEPHOTOGRAPHIC IMAGE, S. O o m . . . . . . . . . . . . . . . 299-358 VII. INTERACTION OF VERY INTENSELIGHTWITH FREE ELECTRONS,J. H. EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359-415 I.

C O N T E N T S OF V O L U M E V I I I ( 1 9 7 0 ) 1-50 SYNTHETIC-A~ERATURE OPTICS, J. W. GOODMAN . . . . . . . . . . 51-131 11. W O ~ C APERFORMANCE L OF THE HUMAN EYE, G. A. FRY . . . . . 133-200 S P ~ O S C O PH. YZ. , CtTMMINs AND H. L. SWINNEY . . 111. LIGHTBEATING 201-237 ANTIREFLECTION COATINGS, A. MUSSET AND A. THELEN . IV. MULTILAYER STATISTICAL PROPERTIES OF LASER LIGHT,H. RISKEN . . . . . . . . . 239-294 V. THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE VI. COHERENCE MICROSCOPY, T. YAMAMOTO . . . . . . . . . . . . . . . . . . . 295-341 L. LEw . . . . . . . . . . . . . . . . 343-372 VII. VISIONIN COMMUNICATION, OF PHOTOELECTRON C o u m c , C. L. MEHTA . . . . . . . . 373-440 VIII. THEORY I.

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

GASLASERS AND THEIR APPLICATION TO PRECISE LENGTH m s m m , A. L.BLOOM . . . . . . . . . . . . . . . . . . . . . . . . . .

1-30

RCOSECOND LASERPUISES,A. J. DEMARIA . . . . . . . . . . . . OPTICALPROPAGATION THROUGH THE TURBULENT ATMOSPHERE,J. W. STROHBEHN . . . . . . . . . . . . . . . . . . . . . . . . . . . m. SYNTHESISOF OPTICALBIREFRINGENT NETWORKS, E. 0. AMMA” . . . V. MODELOCKING IN GASLASERS, L. ALLENAND D. G. C. JONES . . . . . VI. CRYSTAL OPnCS WITH SPATIAL DISPERSION, V. M. AGRANOVICH AND V. L. GINZFIURG. . . . . . . . . . . . . . . . . . . . . . . . . . . THEORY OF VII. APPLICATIONS OF OPTICAL METHODS IN THE DIFFRACTION ELASTICWAVES,K. GNIADEK AND J. PETYKEWCZ . ........ . VIII. EVALUATION. DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL SIGNALS, BASEDON USEOF THE PROLATE FUNCTIONS, B. R. FRIEDEN . . 11. III.

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

3 11-407

C O N T E N T S OF V O L U M E X ( 1 9 7 2 ) I. BANDWIDTH COMPRESSION OF OFTICALIMAGES,T. S. HUANG . . . . . 11. THEUSEOF IMAGE TUBESAS SHUTTERS,R. W. SMITH . . . . . . . . . 111. TOOLSOF THEORETICALQ U A N T UOFTICS, M M. 0. SCULLYAND K. G. WHITNEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . Iv. FIELDCORRECTORSFOR ASTRONOMICAL TELESCOPES, C. G. WYNNE . . V. OPTICALABSORYIIONSTRENGTH OF DEFECTSIN INSULATORS, D. Y.s m ANDD.L. DEXTER . . . . . . . . . . . . . . . . . . . . . . , . VI . ELASTOOPTIC LIGHTMODULATION AND DEFLECTION, E. K. S ~ .G . . VII. Q U A N T DETECTION UM THEORY, C. W. HELSTROM . . . . . . . . . .

1-44 45-87 89-135 137-164 165-228 229-288 289-369

C O N T E N T S OF V O L U M E XI ( 1 9 7 3 ) 1-76 MASTEREQUATION METHODS IN Q U A N TOPTICS, U M G. S. AGARWAL . . RECENTDEVELOPMENTSIN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . . . 77-122 111. INTERACTION OF LIGHT AND ACOUSTIC SURFACE WAVES,E. G. LEAN . . 123-166 Iv. EVANESCENTWAVESIN OPTICALIMAGING, 0. BRYNGDAHL . . . . . . 167-221 V. PRODUCTION OF ELECTRON PROBES USINGA FIELD EMISSION SOURCE, A. 223-246 v.cREwE . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . HA~~ILTONIAN 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 OF V O L U M E XI1 (1974) I. 11. 111. IV. V. VI .

SELF-FOCUSING, SELF-TRAPPING, AND SELFPHASE MODULATION OF LASER BEAMS,0. SVELTO . . . . . . . . . . . . . . . . . . . . . . , SELF-INDUCED TRANSPARENCY, R. E. SLUSHER . . . . . . . . . . . MODULATION TECHNIQUES IN SPECTROMETRY, M. HARWIT,J. A. DECKER JR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS,K. H. DREXHAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . THE PHASETRANSITION CONCEFTAND COHERENCEIN ATOMIC EMISSION, .......................... R . G w BEAMFOE S P E ~ O S C O PS.YBASHKIN , . . . . . . . . . . . . . . .

1-51 53-100 101-162 163-232 233-286 287-344

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

ON THE VALIDlTY OF KIRCHHOFWS LAWOF HEAT RADIATION FOR A BODY IN A NONEQLJEIBFXJM ENVIRONMENT,H. P. BALTES . . . . . . . . .

1-25

11. 111.

IV. V. VI.

THE CASE FOR AND AGAINSTSEMICLASSICAL RADIATION THEORY,L. MANDEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-68 SPHERICAL ABERRATION MEASUREMENTS OF OBJECTIVEAND SUBJECITVE THE HUMAN EYE,W. M. ROSENBLUM, J. L. CHRISTENSEN. . . . . . 69-91 INTEFWEROMETRIC TESTINGOF SMOOTHSURFACES,G. SCHULZ,J. SCHWIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-167 OF LASERBEAMS IN PLASMAS AND SEMICONDUCTORS, M. S. SELFFOCUSING SODHA, A. K. GHATAK, V. K. TRPATHI . . . . . . . . . . . . . . 169-265 APLANATISM AND ISOPLANATISM, W. T. WELFORD . . . . . . . . . . 261-292

C O N T E N T S OF V O L U M E X I V ( 1 9 7 7 ) THESTATISTICS OF SPECKLE PATTERNS,J. C. DAINTY . . . . . . . . . 1-46 HIGH-RESOLUTION TECHNIQUES IN OFTICAL ASTRONOMY, A. LABEYRIE . 47-87 RELAXATION ~ O M E N IN A RARE-EARTH LUMINESCENE,L. A. RISEBERG, M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . 89-159 IV. THEULTRAFAST O ~ C AKERR L SHUTER,M. A. DUGUAY. . . . . . . 61-193 HOLOGRAPHIC DIFF~~ACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . V. 95-244 VI. PHOTOEMISSION, P. J. VERNIER . . . . . . . . . . . . . . . . . . 245-325 VII. O ~ C A-RE P. J. B. CLARIUCOATS. . . 327-402 L WAVEGUIDES-AREVIEW,

I.

11. 111.

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

THEORY OF OFTICAL PARAMETRICAMPLIFICATION AND OSCILLATION, W. H. PAUL . . . . . . . . . . . . . . . . . . . . . . . 1-75 BRUNNER, O ~ C APROPERTIES L OFTHINMETALFILMS,P. ROUARD, A. MEESSEN . 77-137 PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI. . . . . . . . . . . . . 139-185 QUASI-OFTICAL TECHNIQUES OF RADIO ASTRONOMY, T. w. COLE . . . 187-244 OF THE MACROSCOPIC ELECTROMAGNETICTHEORYOF FOUNDATIONS DIELECTRIC MEDIA,J. VANK R A ~ W W OJ. N E. K , SWE . . . . . . . . . 245-350

PREFACE The review articles which are being presented in this volume of Progress in Optics demonstrate once again that optics has become one of the most dynamic fields of contemporary science and engineering. The article cover a wide range of topics: laser selective physical and chemical processes, phase profile generation, computer-generated holograms, speckle interferometry, optical pattern recognition, light emission from spark discharges and semi-classical radiation theory. A measure of the rapid advances is indicated by the fact that some of the subjects discussed in this volume were quite unknown a decade ago. The great amount of research activities in the areas surveyed in this volume is evident from the simple observation that of the approximately 130 references cited in the first article about 50% are to publications which appeared within the last four years; and that the corresponding percentages pertaining to references given in several of the other articles are of the same order of magnitude. It is my hope that this volume will be as favorably received as its fifteen predecessors.

Department of Physics and Astronomy University of Rochester Rochester, N.Y. 14627 October 1978

EMUWOLF

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CONTENTS I . LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY by V . S. LETOKHOV (Moscow) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. GENERAL CONCEFTIONS . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Selectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic processes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General requirements . . . . . . . . . . . . . . . . . . . . . . . 2.4 Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Laser radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 3. ELEMENTARY SELECTIVE PHOTOPROCESSES . . . . . . . . . . . . . . . . 4 . CLASSIFICATION OF LASER PHOTOPHYSICAL AND PHOTOCHEMICAL METHODS . . 4.1 Type of photoexcitation . . . . . . . . . . . . . . . . . . . . . . 4.2 Type of photochemical process . . . . . . . . . . . . . . . . . . . 5 . PHOTOPHYSICAL METHODS OF ISOTOPE SEPARATION . . . . . . . . . . . . 5.1 Selective multi-step photoionization . . . . . . . . . . . . . . . . . 5.2 Selective two-step (ir+uv) photodissociation . . . . . . . . . . . . . 5.3 One-step selective photopredissociation . . . . . . . . . . . . . . . 5.4 Multiple photon dissociation of polyatomics . . . . . . . . . . . . . . 6. PHOTOCHEMICAL METHODSOF ISOTOPE SEPARATION. . . . . . . . . . . . 6.1 Electronic photochemistry . . . . . . . . . . . . . . . . . . . . . 6.2 Vibrational photochemistry . . . . . . . . . . . . . . . . . . . . . 7 . PURIFICATION OF hhTEIUALs AT ATOMIC-MOLECULAR LEVEL . . . . . . . . 7.1 Selective atomic photoionization . . . . . . . . . . . . . . . . . . 7.2 Selective molecular dissociation . . . . . . . . . . . . . . . . . . . 8. SELEC~VE LASERBIOCHEMISTRY. . . . . . . . . . . . . . . . . . . . 8.1 General requirements . . . . . . . . . . . . . . . . . . . . . . . 8.2 Some possibilities . . . . . . . . . . . . . . . . . . . . . . . . . 9. SELECrrvE DEECTIONOF NUCLEI,ATOMS AND MOLECULES . . . . . . . . . 9.1 Detection of single nuclei and atoms . . . . . . . . . . . . . . . . . 9.2 Detection of complex molecules . . . . . . . . . . . . . . . . . . . 10. SPATIAL LOCALIZATION OF MOLECULAR BONDS. . . . . . . . . . . . . . . REFERENCES

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

3 4

4 5 6 6 8 10 16 16 18 21 21 26 29 31 41 42 44 46 46 49 50 50 52 55 55 58 62 65

I1. RECENT ADVANCES IN PHASE PROFILES GENERATION b y J . J . C~AIRand C. I. h m o (Paris) ~ 1.INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. PRINCIPLESAND TECHNIQUES USEDIN PHASE ~OFILES GENEFLUION. . . . . . 2.1 Classical methods based on the transformation of a reference profile . . .

73 74 74

xii

CONTENTS

2.1.1 Matter removal methods . . . . . . . . . . . . . . . . . . . 2.1.2 Matter supplying methods . . . . . . . . . . . . . . . . . . 2.1.3 Thermal deformation methods . . . . . . . . . . . . . . . . . 2.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Off-axis phase profiles photofabrication by coherent light . . . . . . . 2.3 Computer generated phase profiles . . . . . . . . . . . . . . . . . 2.3.1 Synthetic holograms or complex-spatial filter . . . . . . . . . . 2.3.2 Mathematical models and coding schemes (review of computersynthesis processes) . . . . . . . . . . . . . . . . . . . . . 2.4 On-axis phase profiles . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Kinoform and related computer generated holograms . . . . . . . 2.4.2 Photolithographic and evaporating methods . . . . . . . . . . . 2.4.3 Optical synthesis processes . . . . . . . . . . . . . . . . . . 3. OFTICAL MATERIAL FOR PHASE INFORMATION STORAGE. . . . . . . . . . . 3.1 Non-erasable materials . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Silver halide emulsions and bleaching . . . . . . . . . . . . . . 3.1.2 Bichromated gelatin . . . . . . . . . . . . . . . . . . . . . 3.1.3 Photoresists . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Photosensitive polymers plexiglass (PMMA) . . . . . . . . . . . 3.2 Erasable materials . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Thermoplastics . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Photochromes . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Electro-optical materials . . . . . . . . . . . . . . . . . . . 3.2.4 Liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Acoustical media . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Miscellaneous media . . . . . . . . . . . . . . . . . . . . . 4. PHASEPROFILESAPPLICATIONS--CURRENT.ANDFUTUREPROSPECTS . . . . . 4.1 Video-disc imaging system . . . . . . . . . . . . . . . . . . . . . 4.2 New gratings for coupling and spectroscopy . . . . . . . . . . . . . . 4.3 Optical processing of mathematical operations . . . . . . . . . . . . 4.4 Phase profile for infrared technology . . . . . . . . . . . . . . . . . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . COMPLEMENTARY REFERENCES . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

74 76 77 77 78 79 80 82 86 87 89 90 96 96 97 97 98 98 100 100 101 101 102 103 105 106 107 110 111 111 113 114 114 116

I11. COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS by W.H . LEE (Palo Alto, California) 1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . TECHNIQUES FOR MAKING COMPUTER-GENERATED HOLOGRAMS. . . . . . . 2.1 Detour phase holograms . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Other d i n g techniques . . . . . . . . . . . . . . . . . . . 2.2 Modified off-axis reference beam holograms . . . . . . . . . . . . . 2.2.1 Burch’s method . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Huang and Prasada’s method . . . . . . . . . . . . . . . . . 2.2.3 Lee’s delayed sampling method . . . . . . . . . . . . . . . . 2.3Kinoforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Fourier transform-type kinofoms . . . . . . . . . . . . . . .

121 126 126 133 136 136 138 139 143 143

CONTENTS

2.3.2 Phase Fresnel lens . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Extensions of kinoform technique . . . . . . . . . . . . . . . 2.4 Computer-generated interferograms . . . . . . . . . . . . . . . . . 2.4.1 Generation of binary holograms . . . . . . . . . . . . . . . . 2.4.2 Considerations in making binary holograms . . . . . . . . . . . 2.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Generalization of computer-generated interferograms . . . . . . 3. QUANTIZATIONS IN COMPUTER-GENERATED HOLOGRAMS. . . . . . . . . . 3.1 Discrete phase kinoforms . . . . . . . . . . . . . . . . . . . . . . 3.2 Quantization noise . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Phase error from quantization . . . . . . . . . . . . . . . . . . . . 4 . APPLICATIONS OF COMPUTER-GENERATED HOLOGRAMS . . . . . . . . . . . 4.1 3-D imagedisplaywith computer-generated holograms . . . . . . . . . 4.2 Optical data processing . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Edge enhancement . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Image deblurring . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Matched filters optical processor . . . . . . . . . . . . . . . . 4.3 Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Shaping a reference wave in optical testing . . . . . . . . . . . 4.3.2 Shearing interferometry in polar coordinates . . . . . . . . . . 4.4 Optical data storage and random phase coding . . . . . . . . . . . . 4.5 Laser beam scanning . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Computer-generated holograms for laser beam scanning . . . . . 4.5.2 Interferometric grating scanner with aberration correction . . . . . 5 . SUMMARYAND COMMENTS . . . . . . . . . . . . . . . . . . . . . . . REFERENCB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

144 148 152 152 154 158 166 168 169 171 172 173 173 178 179 182 186 195 195 197 205 211 212 220 227 229

IV . SPECKLE INTERFEROMETRY by A . E . ENNOS(Teddington, Middlesex)

1. INTRODUC~ON. . . . . . . . . . . . . . . . . . . . . . . . . . . . .................... 2 . LASERSPECKLE CHARA~RISTICS 2.1 Objective speckle pattern . . . . . . . . . . . . . . . . . . . . . . 2.2 Subjective speckle pattern . . . . . . . . . . . . . . . . . . . . . 3. INTERFERENCE OF LASERSPECKLE . . . . . . . . . . . . . . . . . . . . 4 . DIRECTOBSERVATION SPECKLE INTERFEROMETRY ............. BASEDUPON SPECKLE CORRELATION . . . . . . . . . . . 5 . INTERFEROMETRY 6 . SPECIAL PLTRPosE CORRELATTON INTERFEROMETERS . . . . . . . . . . . . 6.1 Out-of-plane displacement interferometers . . . . . . . . . . . . . . 6.2 In-plane displacement interferometers . . . . . . . . . . . . . . . . 6.3 Speckle shearing interferometers . . . . . . . . . . . . . . . . . . 6.4 Speckleinterferometryfor three-dimensionalcontouring . . . . . . . . 7 . ELECTRONIC SPECKLEPATIERNINTERFEROMETRY . . . . . . . . . . . . . 7.1 On-line operation for vibration analysis . . . . . . . . . . . . . . . 7.2 On-line displacement measurement . . . . . . . . . . . . . . . . . 8. INTERFEENCE E F F EWITH ~ RECORDED SPECKLEPATTERNS . . . . . . . . 9 . SPECKLEPHOTOGRAPHY ........................ 10. FURTHERTECHNIQUES OF SPECKLE PHOTOGRAPHY . . . . . . . . . . . . . 10.1 Surface tilt measurement . . . . . . . . . . . . . . . . . . . . . 10.2 Analysis of object motion by time-averaged speckle photography . . . .

235 236 237 238 240 243 246 249 249 250 254 257 259 262 264 266 268 272 272 27.5

C‘O~IlwTS

XIV

11. APPLICATIONS OF SPECKLE PHOTOGRAPHY IN METROLOGY. . . . . . . . 12. “WHITELIGHT”SPECKLEPHOTOGRAPHY . . . . . . . . . . . . . . . 13. MEASUREMENT OF SURFACE ROUGHNESS BY SPECKLE INTERFEROMETRY. . 14. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

. .

278 281 283 285 286

V . DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATIERN RECOGNITION by D . CASASENT and D . PSALTIS(Pittsburgh, Pennsylvania)

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Optical pattern recognition . . . . . . . . . . . . . . . . . . . . . 1.2 Non-linear and space variant systems . . . . . . . . . . . . . . . 1.3 Scope of chapter . . . . . . . . . . . . . . . . . . . . . . . . . 2 . SPACE-VARIANT O ~ C APROCESSING L . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Space-variant correlation . . . . . . . . . . . . . . . . . . . . . . 2.3 Coordinate transformation selection . . . . . . . . . . . . . . . . 2.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Space bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 3. SCALE-INVARIANT SPACE-VARIANT OFTICAL PROCESSOR . . . . . . . . . 3.1 General analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Scale-invariantcorrelation . . . . . . . . . . . . . . . . . . . . . 3.4 Space bandwidth and accuracy requirements . . . . . . . . . . . . 3.5 Implementation (using non-linear scanning) . . . . . . . . . . . . . 3.6 Implementation (using computer-generated holograms) . . . . . . . . 3.7 Implementation (by digital computer) . . . . . . . . . . . . . . . 4 . APPLICATTONS OF SCALE INVARIANT SYSTEMS . . . . . . . . . . . . . . 4.1 Priorapproaches . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mellin transform correlator . . . . . . . . . . . . . . . . . . . . . 4.3 Scale invariant pattern recognition . . . . . . . . . . . . . . . . . 4.4 Exponentiated coordinate distortions . . . . . . . . . . . . . . . . 4.5 Doppler signal processing . . . . . . . . . . . . . . . . . . . . . 4.6 Correlation of non-vertical imagery . . . . . . . . . . . . . . . . 4.7 Other space-variant systems . . . . . . . . . . . . . . . . . . . . 5. ROTATIONAL INVARIANT SPACE-VARIANT SYSTEMS. . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Polar transformation . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Space bandwidth product requirements . . . . . . . . . . . . . . . 5.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Rotation invariant pattern recognition . . . . . . . . . . . . . . . 5.6 Rotation and scale invariant pattern recognition . . . . . . . . . . . 6 . MULTWLE INVARIANT SPACE-VARIANT PRoc~sso~s. . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Approaches to multiple invariant space-variant processing . . . . . . 6.3 Phase extraction from a complex wave front . . . . . . . . . . . . 6.4 Multiple invariant optical processor . . . . . . . . . . . . . . . . I . SUMMARY AND CONCLUSION . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

. .

. .

. . . . . . .

. .

. . . .

. . .

291 292 294 295 296 296 297 299 300 301 302 302 303 304 306 307 311 313 314 316 316 317 319 323 326 331 332 332 333 335 336 337 339 345 345 346 348 352 354 355 355

xv

CONTENIS

VI . LIGHT EMISSION FROM HIGH-CURRENT SURFACE-SPARK DISCHARGES by R . E . BEVERLY 111 (Columbus, Ohio)

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 History ............................. 1.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. CHANNELDEVELOPMENT AND GASDYNAMICS . . . . . . . . . . . . . . . 2.1 Breakdown mechanisms . . . . . . . . . . . . . . . . . . . . . . 2.2 Channel expansion and substrate vaporization . . . . . . . . . . . . . 2.3 Particle density distribution and channel conductivity . . . . . . . . . . 3. RADIATIVEPROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Dependence of optical properties on circuit parameters . . . . . . . . . 3.2 Qualitative description of radiative characteristics . . . . . . . . . . . 3.3 Luminance, spectral radiance and temperature measurements . . . . . . 3.4 Role of evolved substrate species . . . . . . . . . . . . . . . . . . 4 . APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Spectroscopic sources . . . . . . . . . . . . . . . . . . . . . . . 4.2 Laser uv preionization . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Spark uv preionization of pulsed CO, lasers . . . . . . . . . . . 4.2.2 Spark uv preionization of pulsed exciplex lasers . . . . . . . . . 4.3 Light sources for laser pumping and photoinitiation . . . . . . . . . . 4.3.1 Pumping of solid-state lasers . . . . . . . . . . . . . . . . . 4.3.2 Iodine photodissociation laser . . . . . . . . . . . . . . . . . 4.3.3 Photoinitiated chemical lasers . . . . . . . . . . . . . . . . . 4.4 Population inversions in surface discharges . . . . . . . . . . . . . . 4.5 Nonoptical applications . . . . . . . . . . . . . . . . . . . . . . 5. CONCLUDINGREMARKS . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

359 359 359 361 362 362 366 368 370 370 374 381 382 389 389 392 392 398 398 398 399 404 405 406 406 407 408

VII . SEMICLASSICAL RADIATION THEORY WITHIN A QUANTUM-MECHANICAL FRAMEWORK by I. R . S E ~ K (Haifa) T

1. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . 2. DESCFWIION OF SEMICLASSICAL m o m . . . . . . . . . . 2.1 Semiclassical theory I . . . . . . . . . . . . . . . . . 2.2 Semiclassical theory I1 . . . . . . . . . . . . . . . . . 2.3 Semiclassical theory 111 . . . . . . . . . . . . . . . . . 2.4 Semiclassical theory IV . . . . . . . . . . . . . . . . . 2.5 Other theories .................... 3. SIhlPLE EXAMPLE OF FIELD-ATOMS INTERACTION . . . . . . . . 4 . DISCUSSION OF THE SEMICLASSICAL m o m . . . . . . . . . 4.1 Semiclassical theory I . . . . . . . . . . . . . . . . . 4.1.1 Photoelectric detection . . . . . . . . . . . . . . 4.2 Semiclassical theory I1 . . . . . . . . . . . . . . . . . 4.3 Semiclassicaltheory I11 . . . . . . . . . . . . . . . . . 5 . CLASSICAL LIMITOF QUANTIUM-MECHANICAL RADIATION THEORY 5.1 Boson-second-quantization formalism . . . . . . . . . .

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

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

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

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

. . . . . . . .

. . . .

415 416 416 416 417 418 418 419 426 426 428 430 434 434 435

xvi

CONTENTS

5.2 Classical limit (semiclassical theory N)and relationship to semiclassical theories I1 and I11 . . . . . . . . . . . . . . . . . . . . . . . . . 438 445 5.3 Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATrvE INDEX .VOLUMES I-XV1 . . . . . . . . . . . . . .

.

449 458 461

E. WOLF, PROGRESS IN O€TICS XVI @ NORTH-HOLLAND 1978

I

LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY BY

V. S. LETOKHOV Institute of Spectroscopy, Academy of Sciences USSR, Moscow, 142092, Podol’slki rayon, Akademgorodok, USSR

CONTENTS PAGE

1. INTRODUCTION . .

. . . .

. . . . . . . . .

3

5 2. GENERAL CONCEPTIONS.. . . . . . . . . . . .

4

$

,

5 3. ELEMENTARY SELECTIVE PHOTOPROCESSES.. . . 10 $ 4. CLASSIFICATION OF LASER PHOTOPHYSICAL AND

. . . . . . . . . .

16

5 5. PHOTOPHYSICAL METHODS OF ISOTOPE SEPARATION.. . . . . . . . . . . . . . . . . . . . .

21

PHOTOCHEMICAL METHODS.

8 6. PHOTOCHEMICAL METHODS OF ISOTOPE SEPARATION.. . . . . . . . . . . . . . . . . . . . . 41 OF MATERIALS AT ATOMICMOLECULAR LEVEL. . . . . . . . . . . . . . .

46

. . . . . . .

50

$ 7. PURIFICATION

8 8. SELECTIVE LASER BIOCHEMISTRY.

5 9. SELECTIVE DETECTION OF NUCLEI, ATOMS AND MOLECULES. . . . . . . . . . . . . . . . . . 5 5

0 10. SPATIAL LOCALIZATION OF MOLECULAR BONDS.

62

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

65

Q 1. Introduction The creation of coherent laser light source, that have tunable radiation, opened the prospect of selective excitation of practically any quantum state of atoms and molecules with an excitation energy in the range 0.1-10 eV. At present it is possible to obtain coherent radiation in the range of wavelengths from 2000 A to 2 x lo6 A with sufficient intensity to excite the significant part of atoms and molecules into a chosen quantum state. It is this qualitative progress in the art of quantum electronics beginning in 1969-70, that provided the means for systematic studies of the effects of selective laser radiation on matter. Presently the investigations in the field of selective laser photophysics and photochemistry are being conducted in a great number of laboratories in several countries. This field lying on the border of quantum electronics, optics and spectroscopy, chemistry and technology of materials, attracts many investigators. It is being actively discussed at international conferences [1-3]*. Laser isotope separation is undoubtedly one of the most important current problems of laser selective photophysics and photochemistry. An attractive field of study for scientists and engineers was provided by the uniquely important role that materials whose isotope composition differs from that naturally occuring play in nuclear technology and energetics since all existing methods of isotope separation had considerable disadvantages and new laser separation methods, seemed possible. These studies apply the latest data on molecular and atomic structures and their interaction with the coherent laser light and the latest achievement in the development of tunable lasers. Researchers sought to develop new isotope separation methods that would be cheaper, more productive, more flexible, and less power-consuming, than the existing one. Considerable progress has been achieved in this field and pilot installations for laser isotope separation are being created on a practical scale in some countries. The present state of this field of selective laser photophysics

* Numbered

references refer to conference proceedings, see p. 6 5

3

4

LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY

[I, 5 2

and photochemistry is considered in several special reviews (LETOKHOV and MOORE[1976, 19771; ALRIDGE, BIRELY, CANTRELL and CARTWRICHT [1976]; AMBARTSUMIANand LETOKHOV [ 19771). However, to my mind, the meaning of ideas, methods and processes that are worked out for this purpose, far exceeds the bounds of the isotope separation problem. One can expect in the near future the elaboration of new approaches applying the selective effect of laser radiation in many other fields of science and technology: chemical technology, biochemistry, molecular biology and others. I think that now a new field of coherent light application is opened up, when optical radiation is used not for the passive analysis but for the active effect on a substance to change its composition and structure. Both approaches of light usage are already well known in nature: passive -vision, active photosynthesis. However until now in the technology we have used only the passive approach. In this review* I’ll try to consider the main ideas and trends of this new active approach of the light usage.

0 2. General Conceptions

2.1. SELECTIVITY

The term “selectivity” in laser photophysics and photochemistry has two meanings. First, it may deal with selective photophysical or photochemical conversion of a particular molecule in a mixture with others. Such selectivity may be called “intermolecular”. We always mean it, for instance, when speaking about isotope wparation. Secondly, it may deal, in principle, with selective photoexcitation of any one molecular bond. If we ensure the breaking of a specific bond or chemical reaction of such a molecule before excitation transfer to many molecular bonds, we may hope that the photochemical reaction will be selectively controlled. Such selectivity can be conveniently called “intramolecular”. It is sensitive not only to energy but also to a type of molecular excitation. This potentiality of selective laser photophysics and photochemistry has not been studied experimentally yet. Therefore, when using the term “selectivity” in the present article we almost always mean “intermolecular selectivity”. *Many of these ideas and trends are considered briefly in papers (LETOKHOV [1976a,

1977al).

I, I21

GENERAL CONCEPTIONS

0 Chemical reaction

l

5

Dissociahon or ionization of excited melccules (atoms)

Resonant transfer of excitation

Relaxation

Thermal excitation

Fig. 1. General conception of the selective laser excitation of A particles and classification of processes causing the loss and conservation of selectivity.

2.2. BASIC PROCESSES

The basic processes of selective action on matter by laser radiation, which is often called “selective laser photophysics and photochemistry”, are as follows: I. Selective separation of substances at atomic-molecular level; 11. Selective chemical reactions of atoms or molecules of desired sorts (photochemical separation) or in a desired direction (photochemical synthesis); 111. Selective detection of atoms, molecules or molecular bonds. The generalized concept of laser separation is illustrated by Fig. 1. When two types of atoms or molecules of different isotopic composition (A and B) have even one spectral line that does not overlap with others, it is possible to excite selectively with laser light an atom or molecule (“particle” A) of a required isotopic composition. The excitation of the A particles changes their chemical and physical properties and, hence, may be used to separate substances by any method based on differences in the characteristics of excited and unexcited particles. Selective laser excitation of A particles is followed by chemical reaction of A* to give products designated AR, or by absorption of a second photon, to ionize or dissociate A. Nearly resonant transfer of energy to B, yielding B*, and loss of energy to the thermal reservoir with subsequent thermal population of

6


[I, § 2

B, are schematically in Fig. 1. The excitation energy is meant here, of course, to be much higher than the thermal energy of the particles kT,,, which is responsible for nonselective thermal population of the lower energy states of all particles in the mixture. This is the general concept of laser separations at the atomic and molecular levels.

2.3. GENERAL REQUIREMENTS

Successful application of a specific concept of photoseparation requires that the following four conditions be met: I. There must be at least one absorption line, wg), of the A particles being separated that does not overlap significantly with any absorption lines of the other particles in the mixture. 11. Monochromatic radiation at the chosen frequency of selective absorption, wg), must be available with the characteristics of power, duration, divergence, and monochromaticity necessary for the separation method in use. 111. A primary photophysical or photochemical process must be found that transforms the excited particles into species easily separated from the mixture. IV. The selectivity obtained for the A particle must be maintained against all competing photochemical and photophysical processes throughout the entire separation.

2.4. OBJECTS

Difference in the absorption spectra of atoms or molecules which allows selective excitation of atoms, molecules or molecular bonds of a desired kind is fundamental for selective action of laser radiation on matter. Since the spectral linewidths of atomic and molecular absorption in gases are rather small, the spectral criterion of particle difference is very sensitive to the smallest details of atomic and molecular structure. Spectral differences in atoms may be caused by the following effects: 1. A difference in atomic numbers, or in the number of protons 2, being responsible for a cardinal difference in atomic absorption spectra. This difference is fundamental for atomic absorption spectral analysis. 2. A difference in the number of neutrons N, the number of protons 2

I , § 21

GENERAL CONCEFTIONS

1

being the same, resulting in a small isotope effect which shows up clearly in atomic spectra. 3. A difference in nuclear excitation level with its consequent change in the nuclear spin causing an isomeric effect in the hyperfine structure of the atomic spectrum. Here the other parameters of the nucleus ( Z , N ) and the electron shell are unchanged. The value of the isomeric effect is usually even larger than that of the isotope effect. Spectral differences in molecules may be caused by the following effects: 1. A difference in chemical composition, that is, in the number and the type of atoms in a molecule, resulting in a substantial change of the electronic and vibrational molecular spectra. This difference is basic for molecular absorption spectral analysis. Very small substituent changes which give negligible chemical effects may cause useful spectral shifts. 2. A difference in the isotopic composition of molecules, their other parameters being the same, bringing about an isotope effect in vibrational and, in some cases, in electronic spectra of molecules. 3. A difference in the three-dimensional structure of molecules, their chemical composition being the same, resulting in a difference of vibrational and electronic spectra. Of course, the spectra of many structual isomers are completely different, cyclopropane versus propylene. For cis and trans geometrical isomers the differences are still quite apparent. Differences in the secondary and tertiary structure of biomolecules give some spectral shifts. Right- and left-handed stereoisomers exhibit spectral differences in polarized light. 4. A difference in the nuclear spin orientation of atoms in a molecule, that is, the difference in the total nuclear spin of a molecule, resulting in a change of the hyperfine structure in the molecular spectrum. This difference appears most vividly in the structure of the spectra of molecules containing identical atoms in symmetrically equivalent positions. The ortho and para forms of homonuclear diatomics are a familiar example. CH,, NH, and H,CO are similar. Thus, we may consider the possibility of atomic and molecular separations by lasers of: 1. Chemical elements 2. Isotopes 3 . Nuclear isomers 4. Molecular isomers and 5. Ortho- and para-molecules.

8


[I, 8 2

So, using laser radiation it is possible to realize processes I to I11 (separation, chemical reaction, detection) in substances at atomicmolecular level selectively with respect to various chemical elements, molecular bonds, isotopes, nuclear isomers, molecular stereoisomers, ortho- and para-molecules, etc. The variety of processes of selective laser photophysics and photochemistry is drawn very conditionally in Fig. 2, where the types of selective processes are given down and their objects across. One must keep in mind though that this classification is rather conventional. For instance, selectivity in photochemical reactions may mean selective participation of chosen atoms or chosen molecules in a reaction or selective participation of some molecular bond for specific chemical synthesis. Under molecular bonds we mean e.g. in some cases the sections of macromolecules. Presently isotopically selective photophysical and photochemical processes are intensely developing. The reports at the recent international conferences [2, 31 on lasers and their applications testify clearly about the continuous rapid progress in this field. There is hardly any one who doubts the large perspectives for science and technology in this trend. The present paper gives a review of not only present but also future less probable applications of selective laser photophysics and photochemistry. We’ll also consider the perspectives of development of some nonisotopic selective photoprocesses indicated o n Fig. 2 also. Realization of selective laser photoprocesses opens quite new possibilities to some relative subjects, such as: nuclear physics, spectroscopy, molecular biology, chemical technology, biochemistry. Some of the processes considered below, such as laser identification of nucleotide sequence in desoxiribonucleic acids (DNA) and laser selective biochemical processes, are no less important than isotope separation by lasers.

2.5. LASER RADIATION

Tunable laser radiation is now attainable at almost any wavelength from the vacuum ultraviolet through the infrared. In principle, the fulfillment of requirement I1 is thus a standard problem in quantum electronics. In practice it is often difficult to produce sufficient laser

0 V I C

+aJ n W- o

GENERAL CONCEFTIONS

/

1

I

t

I I I

t

I I

t

aJ

9

10


[I, § 3

energy for laboratory separation research. Of course, years will be needed until the production of laser radiation at a given frequency with desired parameters becomes a routine problem solved with standard devices without involving Quantum Electronics I’h. D’s. Lasers sufficiently powerful and efficient for commercial production purposes are now available only for rather limited wavelength ranges and applications. In practice the relative merit of different methods of isotope separation depends very much on the type of laser required. Let us enumerate the essential properties of laser radiation which make lasers very valuable and efficient tools in the field of selective laser photophysics and photochemistry. They are: 1. Radiation frequency tunablity, which permits the production of radiation at any frequency in the infrared, visible, ultraviolet and vacuum ultraviolet spectral regions. 2. High intensity, sufficient to saturate the absorption of quantum transitions, i.e. to excite a large portion of the atoms or molecules. 3. Controlled radiation duration, which can be made shorter than the lifetime of excited atomic or molecular states. 4. Spatial coherence of radiation, which makes it possible to form directed beams of radiation and to irradiate in longpath cells. 5. Monochromaticity and temporal coherence, which allow an extremely high selectivity of excitation with a very small difference in absorption frequencies of species being separated. The combination of all these valuable properties in an efficient source of optical radiation makes development of optical methods for materials technology at the atomic and molecular level most promising.

0 3. Elementary Selective Photoprocesses In view of the problem of isotope separation a good deal of elementary processes has been recently suggested, discovered and successfully demonstrated for selective action of laser radiation on matter in various states of aggregation: (1) atomic and molecular gases; (2) condensed medium; (3) heterogeneous medium (gas-condensed medium boundary). The most of the experiments however has been conducted in a gas phase. When exciting the quantum state of an atom or a molecule, their properties may change as follows (Fig. 3):

1,s 31

11

ELEMENTARY SELECTIVE PHOTOPROCESSES

11) Chemical reaction

activity

0 1 )lonisainon enerqv

l m l Dissociation

( v J Spatial structure rearrange rnent (isomerization)

( v i ) Photodeflection of tralectory

energy

A’+ M-AM

(iv) Spontaneous breaking 01 bond loredissociation)

(recoil eflectl

Fig. 3. The atomic and molecular properties varying during laser excitation: i) enhancement of reactivity; ii) reduction in ionization energy; iii) reduction in dissociation energy; iv) predissociation; v) isomerization; vi) change in mechanical trajectory.

1. The chemical reactivity of atoms or molecules is often increased by excitation. 2. The ionization energy of an excited atom or molecule is smaller than that of unexcited ones. 3. The dissociation energy of an excited molecule is smaller than that of an unexcited one. 4. Predissociation occurs when an excited molecule passes spontaneously into a dissociative state. 5. The excitation of a molecule may result in isomerization. The isomer, because of its different internal structure, has different chemical properties. 6. The recoil of an atom or molecule occurs when it absorbs a photon with its momentum hale. This gives a very small but observable particle photodeflection. 7. Excited atoms and molecules may have a higher polarizability, a different symmetry of wave function, etc. This may cause changes in cross sections for scattering by other particles, in its motion in external fields, etc. Many possible photochemical and photophysical methods of laser separation follow from this classification.

12


[I, 8 3

There are many methods for laser separation of atoms and molecules of different type. A general classification of possible methods is given above. At the same time they can be classified from another standpoint, according to the type of particles used, and thus we may subdivide all the methods into atomic and molecular ones. Figure 4a shows schematically possible general approaches of laser isotope separation based on selective excitation of atoms, i.e. different atomic methods: (1) photochemical reactions; (2) ionization by a second photon and/or by an external electric field or by collision; and (3) velocity change of selectively excited atoms. Figure 4b gives possible approaches of laser isotope separation based on selective excitation of molecules. Of course, the excitation of molecules offers new possibilities: selective photodissociation and photoisomerization. The field of selective photochemistry has a long history. Changes in the reactivity of atoms and molecules due to photon absorption are well known and have long been used in photochemistry, including isotopically selective photochemistry. Isotopically selective excitation of atoms and molecules followed by photochemical reaction was conceived as a method of isotope separation soon after isotopes and the isotope effect in atomic

+

C

-

'AC

Photochemical Reoc t ion

Selective E xci to tion

(a) A(piA

+

'LIB

+nw,=

'AB.

/ -

C

-*

'AC

Photochemicol Reoct ion

'A. 'A+ Photodissociolion or Photoionizotion

+

'A%

Selective Exci la t ion

+

D e f l e c t i o n of Trojectory

(TI}

A

"ine

Photoisornerizotion

Deflection of Trajectory

Fig. 4. Different selective laser photophysics and photochemistry approaches for atoms (a) and molecules (b).

1, § 31

ELEMENTARY SELECTIVE PHOTOPROCESSES

13

and molecular spectra were discovered (MERTONand HARTLEY [1920]). The first attempt to bring about photochemical isotope separation was PONDER, BOWENand MERTON made as far back as 1922 (HARTLEY, [1922]). In this work 37Cl, molecules were exposed to the light of a common source which passed through an absorbing filter containing mainly 35Cl, molecules. The first successful experiment was conducted ten years later by KUHNand MARTIN[1932, 19331 who exposed phosgene (CO’’Cl,) molecules to a strong spectral line of an aluminium spark at h = 2816.2 A. At about the same time, MROZOWSKI [1932] proposed a method of selective excitation of mercury isotopes by the 2537 A mercury line after passage through a resonantly absorbing mercury filter. He suggested that this method should be used in isotopically-selectivephotochemical reactions of excited mercury atoms with oxygen. The method [1935, 19361. was first realized and investigated in 1935 by ZUBER With the advent of laser sources of intense monochromatic radiation it became possible to selectively excite many atoms and molecules without depending upon accidental coincidences between strong lines of spontaneous radiation and absorption lines of atoms and molecules. With lasers, photochemical isotope separation entered a new experimental domain. The first attempt to realize photochemical isotope separation with a laser was made in 1966 by TIFFANY, Moos and SCHAWLOW [1967]. High-power IR lasers permitted excitation of molecular vibrational levels and opened the possibility of vibrational photochemistry. The first attempt at laser isotope separation by vibrational photochemistry was and SPENCER [1970]. Both of the carried out by MAYER,KWOK,GROSS experiments relied on the photochemical approach-increased chemical reactivity of excited atoms and molecules. Both failed because the overall chemistry did not preserve the initial excitation selectivity (Requirement IV). Since 1969 we at the Institute of Spectroscopy USSR Academy of Science have been developing mainly the photophysical approach in the selective action of laser light on matter particularly for isotope separation. The photophysical approach differs considerably from the well known photochemical one. The differences are shown in Fig. 5. Let us consider atom (‘)Awith the ground state, excited states and the continuum, corresponding to the atomic ionization (Fig. 5a). An atom of another isotopic composition has somewhat shifted energetic levels, which makes it possible to excite atoms of the selected isotope by monochromatic radiation. Photochemical separation is based on the enhancement of

14


[I, 8 3

II

TRANSFER

RELAXATION

I/hu4

a.

b Fig. 5. Universal processes of selective two-step action by laser radiation: a) selective two-step photoionization of atoms and b) selective two-step photodissociation of molecules and their comparison with photochemical processes.

the reaction rate of excited atoms ci)A*with scavenger R. In 1969 we suggested (LETOKHOV [1977b]) a new approach based on the ability of laser light to transfer a considerable number of atoms into any given excited state. It was suggested that photoionization of selectively excited atoms be obtained by additional laser light before the atoms return to the ground state or transfer their excitation while colliding with atoms of other isotopic composition. It can always be achieved, principally, because photoionization probability is proportional to the intensity of additional laser radiation. Thus, the process of selective photoionization, as against photochemical reaction, does not require collisions with another particle and hence, is fully controlled by laser light.

I , § 31

ELEMENTARY SELECXIVE PHOTOPROCESSES

15

A similar situation is possible for molecules as well. Let us consider a molecule with the ground state and the excited electronic state (stable and unstable) (Fig. 5b). At the photochemical approach a selectively excited molecule (')AB* must, in principle, involve into chemical reaction with another particle (scavenger R), during the collision, with the rate exceeding the reaction rate for unexcited atoms. The process of photochemical reaction is completed by the excitation relaxation and excitation scrambling during the collisions of molecules of different isotopic composition. In 1969 we suggested (LETOKHOV [1977c]) a way to realize photodissociation of selectively excited molecules by additional laser light with the rate, exceeding the one of all parasite competitive processes, and then to bind chemically the radicals formed as the result of photodissociation. Such an approach allows us to control fully the process of selective action on molecules by laser light. The first successful experiment on selective two-step atomic ionization (e.g. of rubidium atoms) was carried out at the beginning of 1971 in the Institute of Spectroscopy USSR Sc. Ac. (AMBARTZUMIAN, KALININand LETOKHOV [19711; LETOKHOV and AMBARTZUMIAN [19711; AMBARTZUMIAN and LETOKHOV [1972]). Later on, similar experiments were performed in the summer of 1971 with uranium atoms in Avco Everett Research Laboratory. "he results of these experiments were published in 1975 ITZKAN, PIKE,LEVYand LEVIN[1975, 19761) after the publication (JANES, of experimental results obtained in Livermore Laboratory, also with and SNAVELY [1974]). The uranium atoms (TUCCIO,D ~ R I NPETERSON , first experiment on isotopically-selective two-step photodissociation of molecules (e.g. of ammonia molecules and nitrogen isotopes) was carried out also in the Institute of Spectroscopy, USSR Sc. Ac. (AMBARTZUMIAN, LETOKHOV, MAKAROV and PURETSKII [1972, 19731). At the same time Prof. C . B. Moore at the University of California, Berkeley, carried out the first experiment (YEUNG and MOORE[1972]) on isotope separation by photodissociation (e.g. formaldehyde molecules and hydrogen isotopes). Selective processes in condensed media are as yet imperfectly understood and seem to be less promising because of line broadening at normal temperatures and high-rate relaxation of vibrational excitation. This dictates certain stringent experimental conditions to selective action: (1) excitation of the electron states the relaxation rate of which is lower than that of vibrational ones; (2) the use of low temperatures at which the spectral line broadening and V - T relaxation rate drop drastically; (3) the use of ultrashort pulses. For the time being we have only a good example

16


[I,

I4

of isotopically-selective photoprocess (KING and HOCHSTRASSER [19754, but the case of condensed medium seems to be of prime importance in realizing selective photobiochemical processes. In case of heterogeneous medium it is possible to act selectively both on the particles in the gas phase and on the atoms or molecules on the condensed medium surface. In the first case such processes as photochemical reactions, photoadsorption (GOCHELASHVILI, KARLOV, ORLOV,PETROV and PROKHOROV [1975]) and condensation of excited atoms or molecules can be used (BASOV,BELENOV, ISAKOV, LEoNov, MARKIN,ORAEVSKII and ROMANENKO [1975]). As the atoms or molecules on the condensed medium surface are excited, their selective breaking off, photodesorption and evaporation are possible. Selective breaking off of particular atoms or molecules from the surface (DJIDJOEV,KHOKHLOV,KISELEV,LYGIN, NAMIOT,Osrpov, PANCHENKO and PROVOTOROV [1976]) is a very important though the most hard-to-realize and underdeveloped process. Selective breaking off of electrons or protons from certain parts of macromolecules may prove to be rather important for a direct identification of the spatial-chemical structure of biological molecules (LETOKHOV [1975a1).

0 4. Classification of Laser Photophysical and PhotochemicalMethods

4.1. TYPE OF PHOTOEXCITATION

In Fig. 6 various types of selective molecular photoexcitation are shown in a very simple form. The classical (prelaser) photochemical method is based on one-step excitation of an electronic state of an atom or a molecule. This type of molecular excitation has a serious disadvantage for selective photochemistry. Most molecules, especially polyatomic, have comparatively wide structureless bands of electronic absorption at normal temperature. So this scheme can not be used for, say, isotopically selective excitation of molecules. Only a limited number of simple (in the main two- or three-atom) molecules have narrow lines of electronic absorption suitable for isotopically selective excitation. On the other hand, the excitation of electronic states is beneficial because of a high quantum yield of the photochemical reaction. One-step excitation of a molecular vibrational state (photochemistry in the ground electronic state) features rather high excitation selectivity both

I, 0 4

CLASSIFICATION OF LASER PHOTOPHYSICAL AND PHOTOCHEMICAL METHODS

17

\

a

b

C

Fig. 6. Types of selective molecular photoexcitation: a) single-step excitation of electronic or vibrational states; b) two-step excitation of electronic state through intermediate vibrational or electronic states; c) multiple photon excitation by IR radiation.

for simple and complex molecules. The main disadvantage of this method is the fast relaxation of vibrational excitation to heat and hence a low quantum yield of the subsequent photochemical process. Besides, the method can be used only for photochemical reactions with low activation energy . Two-step excitation of a molecular electronic state through an intermediate vibrational state by joint action of IR and UV radiation (Fig. 6b) combines the advantages of one-step IR and W excitation processes and [ 1977~1).In two-step photoexcitremoves their disadvantages (LETOKHOV ation by a two-frequency (IR+UV) laser field it is possible to separate the functions of selective excitation, when a molecule derives rather low energy (IR photon), and absorption of much larger energy (UV photon) by the selectively-excited molecule. This type of two-step photoexcitation has high enough selectivity together with a high quantum yield of photochemistry. The merits of two-step (IR+ UV) excitation show themselves most vividly in the case of condensed media where the discrepancy between the high selectivity and the high quantum yield at normal temperatures becomes unsolvable. Two-step excitation of molecules through an intermediate electronic state (Fig. 6b) is not so universal as I R + U V excitation. Its only advantage over one-step excitation of electronic states is the strong possibility of exciting states with specific properties and high-lying states without V W radiation. For polyatomic molecules it is possible to excite selectively high vibrational and even excited electronic states under the action of sufficiently

18


[I, § 4

powerful IR radiation alone (AMBARTZUMIAN, LETOKHOV, RYABOV and CHEKALIN [19741; AMBARTZUMIAN, GOROKHOV, LETOKHOV and MAKAROV [ 197Sal). Due to multiple absorption of IR photons of the same frequency a molecule derives an energy comparable to the typical energy of electronic excitation (Fig. 6c). Therefore, we can realize at the same time the excitation selectivity which suffices to separate isotopes and a rather high quantum yield of the subsequent photochemical process. In this case it is also possible to separate the functions of selective excitation and subsequent absorption of high energy by a selectively excited molecule in a two-frequency IR field (AMBARTZUMIAN, GOROKHOV, LETOKHOV, PURETZKII and FURZIKOV [ 19761 which provides enhancement MAKAROV, in the selectivity of the process. A limitation of the method of multiple photon I R radiation absorption is that it can be applied only to polyatomic molecules having a high density of excited vibrational levels in the ground electronic state. The last two methods of selective photochemistry illustrated in Figs. 6b and 6c have been realized only by means of laser radiation since in this case high population of intermediate quantum levels is required. Basically, this cannot be realized with any efficiency using conventional incoherent light sources because of the low temperature of their radiation. Conventional light sources are applicable to one-step processes only, with higher efficiency €or electronic states than for vibrational ones. 4.2. TYPE OF PHOTOCHEMICAL PROCESS

An excited molecule can participate in subsequent photochemical processes. Several different mechanisms of this participation may be classified in the most general and simplified form thus: (1) photochemical reaction of an (electronically o r vibrationally) excited molecule with a proper acceptor; (2) photodissociation (or photopredissociation) of an excited molecule; (3) photoisomerization, i.e. rearrangement of the spatial structure of an excited molecule. All these types of photochemical conversions are well known in photochemistry. But as far as the attainment of high selectivity in the photochemical process and its universality are concerned, these mechanisms are far from being equivalent. Each of them has its pros and cons for selective photochemistry. The first process (chemical reaction of excited molecules) is potentially rather universal. Really, any molecule has an excited electronic state with increased reactivity which may be selectively excited by laser radiation

I, 0 41

CLASSIFICATION OF LASER PHOTOPHYSICAL AND PHOTOCHEMICAL METHODS

19

with the use of a proper scheme (Fig. 6). This process requires, however. a suitable acceptor which would react at a higher rate with excited molecules than with unexcited ones. The rate at which excited molecules react with an acceptor must be much higher than the rate of excitation transfer during their collisions with unwanted molecules as well as the rate of excitation relaxation. Another fundamental requirement consists in conserving the photochemical reaction selectivity in the inevitable subsequent secondary photochemical reactions. These are rather rigid requirements which, as a rule, can be complied with non-trivially. It suffices to say that quite recently high isotopic selectivity of photochemical reactions has been demonstrated (an electronically excited molecule, ICI, reacted with the C,H,Br, acceptor molecule, DATTA, ANDERSON and ZARE[1976]). All of these requirements are very difficult for vibrationally excited molecules when the difference in the reaction rates of excited and unexcited molecules is relatively small. Molecular photodissociation is as universal a process as the photochemical reaction of excited molecules. Photodissociation can be realized either by exciting an unstable electronic state (repulsive term) of the molecule o r through strong vibrational excitation within the ground electronic state. Until very recently, only photodissociation through an excited electronic state was known, and only the creation of high-power pulsed IR lasers allowed the second possibility to be realized. These two methods of selective molecular photodissociation differ greatly from one another. Every molecule has excited unstable electronic states suitable for molecular photodissociation. Since a molecule on a repulsive electronic surface dissociates very quickly (about to sec), relaxation and excitation transfer in such a short time are of no importance, of course. Because the band of electronic absorption at the transition to a dissociative state is wide, excitation selectivity may be ensured only by means of a two- (or multi-) step process (Fig. 6b). So the requirements of small losses of excitation because of relaxation and transfer of excitation apply to the intermediate excited state. These requirements can always be met by the choice of the intensity and duration of the second-step UV radiation. Since radicals are always dissociation products it is necessary to use scavengers which react with the radicals without affecting the initial molecules. The choice of such scavengers which do not result in significant losses of selectivity through secondary photochemical processes is a simpler task than for chemical reactions of excited molecules.

20


[I, 5 4

Sometimes electronically excited molecules dissociate due not to the repulsive nature of the electronic state, but due to the intersection of the stable and unstable states. The dissociation in this case, called photosec, for different predissociation, occurs more slowly, say, in lop6to molecules in excited electronic-vibrational-rotational states. By virtue of the fact that the absorption lines of the transition to a predissociated state are narrow, we can make the high selectivity of excitation useful for one-step photoexcitation (YEUNGand MOORE[1972]; LETOKHOV [1972]). This gives a new advantage to the photodissociation method, though at the expense of universality because of the limited number of molecules which exhibit the photopredissociation phenomenon. Photodissociation in the ground electronic state (Fig. 6c) even has advantages over that through excited electronic states. First, it requires less energy; secondly, it produces, in principle, lower energy radicals. But it was impossible to carry out this process with its advantage for lack of a proper method of molecular photoexcitation. The question is seemingly only one of the excitation of molecular transitions. However, this would require the use of multifrequency IR laser radiation for photoexcitation up to the dissociation limit since the series of vibrational-rotation transitions are not equidistant. This is possible in principle but the present level of tunable laser engineering does not enable us to do it for the time being. The situation became much easier after the effect of collisionless dissociation of polyatomic molecules in intense IR laser fields had been revealed (ISENOR,MERCHANT, HALLSWORTH and RICHARDSON [19731; AMBARTZUMIAN, CHEKALIN, DOUIKOV, LETOKHOV and RYABOV [1974]). A rich structure of the vibration-rotation transitions in polyatomic molecules allows absorption of a very large number of I R photons if the laser field is sufficiently intense. And what is more, such a process of photodissociation in a single-frequency, intense IR field is so selective that it enables isotope separation (AMBARTZUMIAN, LETOKHOV, RYABOVand CHEKALIN [19741; ~ A R T Z U M I A N , GOROKHOV, LETOKHOV and MAKAROV [197.51). Thus, photodissociation in the ground electronic state in an intense IR field proves to be a simpler process but again at the expense of some universality. (It cannot be applied to simple, two- or three-atom molecules.) Its other pros and cons are common to any photodissociation met hod. Molecular photoisomerization, like photodissociation, is a unimolecular photoprocess which requires no collisions with other particles. This is an advantage of both methods over the chemical reaction of excited

I,§ 51

PHOTOPHYSICAL METHODS OF ISOTOPE SEPARATION

21

molecules which necessitates collisions with all the ensuing limitations (loss of excitation and selectivity). But photoisomerization, unlike the first two methods, does not need any acceptor, and so there is almost no selectivity loss in secondary photochemical processes for this method. The only limitation of this method is the relatively small number of the molecules which exhibit this effect. Since the final state of the molecular phototransition is stable, selective photoisomerization in the case of a narrow absorption line may be done by one-step photoexcitation (BRAUMAN, O'LEARYand SCHAWLOW [1974]). Otherwise two-step photoexcitation can be applied. We will consider in detail the main selective photophysical and photochemical processes for a particular important problem - isotope separation. It goes without saying that their application fields are wider than isotope separation. Some of the possible applications are considered in the consequent sections of the present article.

0 5. Photophysical Methods of Isotope Separation Among the basic photophysical methods of practical interest under active development in many research laboratories are the following: (1) selective multi-step photoionization of atoms; (2) selective two-step photodissociation of molecules by IR and UV radiation; ( 3 ) photopredissociation of molecules; (4) multiple photon molecular dissociation in an intense IR field. Each of these methods has been demonstrated by a laboratory experiment; each of them has its advantages and disadvantages, and is either at the stage of providing evidence for possible application in pilot installation (methods 2" and 3"), or is at the stage of direct realization in pilot installation (methods 1" and 4").

5.1. SELECTIVE MULTI-STEP PHOTOIONIZATION

Selective photoionization of atoms is the most universal photophysical method for selective separation of substances, particularly isotopes, at the atomic level. A common feature of all schemes for selective ionization is the sequence of the two processes: (1) isotopically selective excitation; (2) ionization of the excited atoms. Figure 7 illustrates some schemes of selective atomic ionization of special interest for laser isotope separation.

22

[I, 0 5


; f=

IF?

dc electit

fieid

A

(a1 (bl (c 1 (dI (el Fig. 7. Schemes of selective step-wise photoionization of atoms by laser radiation: a) two-step photoionization; b) three-step photoionization; c) two-step photoionization through an autoionization state; d) two-step selective excitation of Rydberg state and its photoionization by IR laser radiation; e) two-step selective excitation of Rydberg state and its ionization by electric field.

The two-step photoionization scheme is the simplest (LETOKHOV [ 1977bl; [ 19711). The three-step scheme AMBARTZUMIAN, KALININ and LETOKHOV may be of use, say, for atoms with a high ionization potential. The photoionization cross-section can be increased by tuning the frequency of last step radiation to that of the transition to an autoionization (spontaneous (LEVY and JANES [1973]) or electric-field-induced (IVANOV and LETOKHOV [ 19751)) state (Fig. 7c). Finally, high-flying Rydberg states can and LEVIN[1973]) or pulsed be ionized by IR radiation (NEBENZAHL electric field (IVANOV and LETOKHOV [1975]), (Fig. 7 d, e). All these schemes have already been tested for isotopically-selective ionization. A discussion of them, their advantages and disadvantages, is not the subject of this article. Those interested in it can find information concerning the above mentioned problems in the special review [1977a, b]) or in corresponding, more (LETOKHOV, MISHINand PURETZKII compact paragraphs of reviews (LETOKHOV and MOORE[1976, 19771; ALDRIDGE, BIRELY,CANTRELL and CARTWRIGHT [1976]). Let us give as examples just general requirements, which should be satisfied by a scheme of the selective ionization for isotope separation in practical scale: (1) All atoms in an unexcited beam should be in the ground state, and there should be no ions. If atoms of a selected isotope are distributed over several levels or sublevels, multifrequency radiation is required to excite atoms from every sublevel in order to completely remove the selected isotope from the mixture. Any thermal ions existing in the atomic vapor must be removed before laser excitation.

1, § 51


23

(2) The laser radiation should perform selective photoionization for each atom of a selected isotope. This requirement determines the power of the exciting and ionizing radiation and depends on the cross-sections for the excitation and ionization processes. ( 3 ) The laser radiation intensity should practically be used in full to excite and ionize atoms of a selected isotope. Certain requirements arise in this connection concerning the geometry of the atomic beam (flow) and laser beams and the density of the atoms. (4) There should be no transfer of excitation or charge between the isotopes being separated. This condition limits the density allowed. Most attention has been paid to uranium isotope separation. Results have been published from research programs at Avco Everett Research Laboratory and the Lawrence Livermore Laboratory. In 1974, the results of the first Livermore experiments were presented at the VIII Intern. PETERSON and SNAVELY Conf. on Quantum Electronics (TUCCIO,DUBRIN, [1974]). In these experiments, CW dye lasers were used for excitation of 235 U and UV-radiation of a mercury lamp was used for ionization. The selectivity separation factor was about 100. In 1975 Livermore Laboratory presented (TUCCIO,FOLEY,DUBRIN and KRIKORIAN [1975]) the results of its experiments on two-step ionization of uranium atoms by xenon and krypton ion lasers. The ion yield rate of 23sU+in their experiments was 2x g/hr, that is lo7 times higher than the rate obtained in the early experiments. An industrial research and development program is being carried out jointly by the Avco Everett Research Laboratory and the Exxon Nuclear Co. Some results of this work, obtained, according to the authors, in 1971 in the early stage of investigation of the method were reported (JANES, ITZKAN, PIKE,LEVYand LEVIN[1975, 19761). In an experiment in which the exciting pulsed dye laser was scanned over a range broad enough to cover both the 235Uand 238Utransitions, and a pulsed N2 laser was used to photoionize excited U, the selectivity separation factor K(235/238)was about 70. The process, involving three steps (Fig. 7b) is similar to the two-step process but permits the use of orange dyes such as rhodamine which are more efficient for use in the dye lasers. This process has been realised recently in the frame of the same joint project (JANES,FORSEN and LEVY[1977]). The data shown in Fig. 8, based on individual particle counts, illustrate 70% enrichment (the separation factor about 100) for a three-step selective photoionization process at low density of uranium

24


10000

a

I

[I, § 5

I

'""1

Fig. 8. Data obtained from low density experiment illustrating 70% enrichment for a three-step photoionization process. The abscissa represents the wavelength. The two peaks are located at the proper wavelengths for mass 238 and 235, respectively. The ratio of the heights of the two peaks corresponds to the natural isotope ratio of the feed material (0.7%),(from JANES, FORSENand LEVY[1977]).

vapor. The abscissa is the wavelength of the exciting laser. The ordinate is the ion yield for 235Uand 238U,respectively. Besides the uranium isotope separation, which demands the rather difficult conditions pointed out above, laboratory experiments are being carried out on selective ionization of other isotopes, potentially useful in much smaller quantities such as K, Ca, Rb, rare-earth and, of course, transuranium elements. An effective ionization of excited atoms at a moderate average intensity of the ionizing radiation presents a serious problem in these cases. The ionization schemes given in Figs. 7a-c cannot be practically applied here, so greater attention is paid to the ionization scheme of highly excited atoms shown in Figs. 7 d , e (see LETOKHOV, MISHINand PURETZKII [1977]).

I , § 51

25


From the viewpoint of the comparative ease of ionization, the highly excited states of atoms are of great interest fof isotope separation. The first successful experiment on increasing the ionization cross section by means of the electric-field-induced autoionization of the Rydberg state BEKOV, LETOKHOV and MISHIN[1975] was carried out in AMBARTZUMIAN, using Na vapors. A comprehensive study of the autoionization of highly and excited Na atoms was carried out in DUCAS,LITTMAN,FREEMAN KLEPPNER [1975] and LITTMAN, ZMMERMAN and KLEPPNER [1976]. A quantum state with the principal quantum number y1 will fall within the continuum if electric field strength (in at. un.):

gCr 2 ( 16n4)-'

(1 at. un. = 5 x 10" V/cm).

(1) The results of critical ionization field measurements for different principal quantum numbers, obtained in DUCAS, LI~TMAN, FREEMAN and KLEPPNER [1975] and BEKOV, LETOKHOV and MISHIN [ 1977al for sodium, in PAISNER, WORDEN, JOHNSON,MAYand SOLARZ [1976] for uranium and in CARLSON, BEKOV,LETOKHOV and MISHIN[1977b] for rubidium are compared to the theoretical dependence (1)shown in Fig. 9. All these experiments demonstrate the efficiency of using the electric field to ionize highly excited states.

qN 0.2

0

10

15

20

30

40

n"

Fig. 9. Log-log plot of critical ionization electric field versus effective principal quantum N, and KLEPPNER number n" for n = 2 4 , . . . ,50 for Na (data from L ~ ~ T M AZIMMERMAN [1976]), U (data from PAISNER, CARLSON, WORDEN.JOHNSON, MAY and SOLARZ[1976]) and Rb (data from BEKOV, LETOKHOV and MISHIN[1977b]).

26


[I, 8 5

5.2. SELECTIVE TWO-STEP (IR + W) PHOTODISSOCIATION

The process of selective two-step molecular photodissociation is possible if the excitation of a molecule shifts the band of continuous photoabsorption and results in the photodissociation of a molecule. Then, selecting the laser frequency o2in the region o f the shift where the ratio of the absorption coefficients of the excited and unexcited molecules has its maximum, we can accomplish photodissociation of molecules excited selectively by laser radiation of frequency o1 via an excited, unstable electronic state. The intermediate state may be a stable., excited vibrational or electronic state. Each scheme has its advantages and shortcomings. When a vibrational state is excited, the shift of the electronic absorption band is sometimes fairly small, and the low vibrational levels are populated appreciably by nonselective thermal excitation. Therefore, it is sometimes difficult to achieve preferential photodissociation of the laser-radiationexcited molecules. On the other hand, the isotopic shift is manifested clearly in the vibrational spectrum. The photoabsorption band may exhibit a very large shift for an electronically excited intermediate state but the electronic spectra of many molecules do not have lines with a sharp structure in which the isotopic shift would appear clearly. Moreover, when the isotopic structure does appear, the method of two-step photodissociation competes with the methods of one-step selective photopredissociation and selective electronic photochemistry. Therefore, in

\

\

a

b

C

d

Fig. 10. Schemes of selective multi-step photodissociation of molecules by laser radiation through an excited electronic state (a, b) and ground electronic state (c, d): a) two-step IR+ UV photodissociation; b) multi-selective excitation of high vibrational levels and their photodissociation by W radiation; c) multiple photon selective excitation and dissociation by single frequency intense IR field: d) multiple photon selective excitation of vibrational levels by resonant TR field and their multiphoton dissociation by nonresonant intense IR field.


27

practice, the most interesting method is the photodissociation via an intermediate vibrational state by IR and UV radiation (Fig. 1Oa). The process of two-step photodissociation of molecules is more complex than the process of two-step photoionization of atoms because of the following effects which influence the selectivity and rate of the process (LETOKHOV and MARTZUMIAN [ 19711; AMARTZUMIAN and LETOKHOV [ 19721): (1) the thermal nonselective excitation of vibrational levels; (2) the broadening of the edge of the electronic photoabsorption band of the molecule; ( 3 ) the “bottleneck” effect due to the rotational structure of the vibrational levels. The first two effects decrease the dissociation selectivity, and the third effects limits the absorption rate of IR radiation by the and MAKAROV [19721) and, consequently, the rate of molecule (LETOKHOV the two-step photodissociation process. These effects were discussed in detail in the reviews LETOKHOV and MOORE[1976, 19771 and in the original papers. Figure 11 illustrates the typical form of continuous absorption bands for transitions from the ground state and the vibrationally excited state to a repulsive excited electronic state. The photodissociation band spectrum

Fig. 11. The difference between the dissociative continua for the ground state and excited vibrational states of a diatomic molecule.

28


[I, 8 5

of a diatomic molecule is given by the formula

where 'Pv(x) is the nuclear wave function of a vibrational level of ground electronic state, Tu(x) is the nuclear wave function of a level in the continuum of an excited electronic state. The excited state absorption band may be shifted to the red by as much as the vibrational excitation energy, h o , ; however, the bandwidth is usually greater than the shift. For selective photodissociation it is necessary that there should be a frequency Estimates show that this is possible only on the range in which It,, >> IoOY. side of the absorption band where the photodissociation cross section decreases sharply. So, the photodissociation selectivity may be increased by tuning to the edge of the band, this greatly decreases vpdand the photodissociation rate. Therefore it is advisable to apply other methods for increasing the band shift by the selective excitation of high-lying vibrational levels (Fig. lob). This may be performed by one of several methods: (1) direct excitation of high levels with laser radiation at the overtone transition frequency as was done, for example, in AMBARTZUMIAN, APATIN and LETOKHOV [1972] on HC1 molecules; (2) subsequent step-wise excitation of high levels with multifrequency IR radiation; (3) resonance excitation of high levels of polyatomic molecules due to multiple absorption of IR photons from a pulsed IR field; this was demonstrated for the molecule BC1, (AMB~RTZUMIAN,LETOKHOV, RYABOV and CHEKALIN [1974]), SF, (AMBARTZUMIAN, GOROKHOV, LETOKHOV and MAKAROV[ 1975al). OsO, (AMBARTZUMIAN, GOROKHOV, LETOKHOV and MARKAROV [1975b]) and many others using a CO, laser. The first experiments on the separation of isotopes by the two-step selective photodissocation method were described in AMBARTZUMIAN, LETOKHOV, MAKAROV and PURETSKII [1972, 1973, 19741. These experiments were carried out on I4NH3 and 15NH3 molecules because, first, they could be excited selectively by CO, laser radiation and, second, IR and UV absorption spectra and photochemical decomposition were thoroughly investigated. The enrichment coefficient, K(15N/14N) for the final product N2 varied from 2.5 to 6 . These results were confirmed by Japanese investigators (NOGUCHI and IZAWA[19741). Experiments in which the boron isotopes loB and "B were separated by the two-step selective photodissociation of BCI, molecules were carried out (ROCKWOOD and RABIDEAU [1974]) using similar apparatus (a CO, laser and a

I, 0 51


29

conventional UV source emitting in the 2000A region). Only a 10% enrichment of a mixture with the light boron isotope, comparable to the typical value of the kinetic isotopic effect, was achieved in these experiments. The universal character of two-step photodissociation of a molecule by joint action of I R and UV radiation is limited to molecules having an unstable excited electronic state with the energy 2000 A). From this point of view the multiphoton dissociation of polyatomic molecules by an intense IR field due to vibrational transitions within the ground electronic state is more universal. However, as the two-step IR-UV dissociation is applicable in principle to a number of molecules with heavy isotopic atoms which are of practical interest this method is being actively developed in many laboratories. TO my mind the method of a selective two-step excitation of the electronic state of the molecule via the vibrational state is of great importance in selective laser photobiology and photobiochemistry. We’ll return to the discussion of this method further in 9 8.

5.3. ONE-STEP SELECTIVE PHOTOPREDISSOCIATION

Isotope separation by photopredissociation requires a molecular excited state that exhibits a resolvable isotope shift, decays primarily by dissociation, and whose dissociation products are simply removed from the starting material. This method is not as general as the two-step photoprocesses. Moreover, for most molecules, sufficient spectroscopic and photochemical data are not available to determine whether these requirements are satisfied. Photopredissociation has been studied most extensively in formaldehyde, primarily be C. B. Moore and co-workers. Near the origin of the first excited singlet state, H,CO dissociates with high quantum yield to H2 and CO, i.e. absorption of a single photon leads to chemically stable dissociation products. Separation of hydrogen from deuterium has been demonstrated using 1 : l mixtures of H,CO and D,CO in YEUNGand APATIN,LETOKHOV and MISHIN MOORE[1972, 19731, AMBARTZUMIAN, [19751 and BAZHIN,SKUBNEVSKAYA, SOROKIN and MOLIN[1974]. Enrichments, limited by the excitation selectivity of the laser source, were as APATIN,LETOKHOV and MISHIN[1975]). An high as 9 : 1 (AMBARTZUMIAN,

30


[I, 8 5

experiment on hydrogen isotope separation in the natural mixture of H,CO and HDCO has been conducted (MARLING [1975]), in which the enrichment coefficient K(DIH) is about 14 under irradiation by CW He-Cd laser line at 325.03 nm. The 80-fold enrichment of CO in isotope 12C has been obtained (CLARK, HAAS,HOUSTON and MOORE[1975]) by means of photopredissociation of a mixture H, "CO :H, l3C0 = 1 : 10. Spectroscopic and photochemical research is under way (BARANOVSKI, CABELLO, CLARK, HAAS,HOUSTON, KUNG,MOORE,REILLY, WEISSHAAR and ZUGHUL [1976]) which should lead to a full understanding of the photoprocesses in formaldehyde and to the development of practical systems for the separation of 13C, " 0 , "0. A number of experiments have been done with other molecules LEONE and MOORE[1974] excited Br, to the predissociated 3110+ustate, and avoided the problems of many of the potential scrambling processes by observing the IR chemiluminescence from HBr formed in vibrationally excited states in the reaction of the Br fragment with HI. The enrichment coefficient K(81Br/79Br)was about 5. By selective predissociation of ortho-I, molecules with 5 14.5 nm argon ion laser light BAZHUTIN, LETOKHOV,MAKAROV and SEMCHISHEN [19731 and BALIKIN, LETOKHOV, MISHINand SEMCHISHEN [ 19761were able to convert ortho-I, to para-I, with enrichment factor about 2-4. As an example, the typical time dependence of the concentration of ortho-I, and para-I, is shown in Fig. 12. In the

Time (min)

Fig. 12. Kinetics of selective photopredissociation o f ortho-I, in a natural mixture with para-I,. The mixture is photolyzed with 514.5 nm argon laser radiation. Concentrations of ortho-I, and para-I, are proportional to fluorescence. intensity due to 514.5 and 501.7 nm excitation, respectively (from BALIKIN, LETOKHOV, MISHINand SEMCHISHEN [1976]).

1, § 51


31

experiment the total concentration of I2 is falling during the irradiation as iodine atoms formed by predissociation are adsorbed by the walls of the gas cell. This process can also be used to separate iodine stable and radioactive isotopes. KINGand HOCHSTRASSER [1975] and KARLand INNES [ 19751 have independently demonstrated high enrichments of carbon and nitrogen isotopes in low-temperature condensed-phase (KINGand HOCHSTRASSER [1975]) and gas-phase (KARLand INNES[1975]) irradiation of s-tetrazene having naturally occurring isotope composition. Evidence is given for the dissociation reaction s-tetrazene -+ N2 + 2HCN. Photopredissociation promises to become a practical method of isotope separation. Work on H2C0 and probably s-tetrazene may lead to economically viable methods of enriching I3C, I4C, "0 and '"0.While the method is not as generally applicable as two-step excitation methods, in some situations it may be more practical. In spite of a considerable simplicity of the photopredissociational approach, it has not yet entered the stage of pilot plants for any single isotope. It may be connected with the energetics of narrowband tunable lasers of UV range. The creation of excimer lasers with high efficiency can greatly contribute to rapid practical implementation of this method.

5.4. MULTIPLE PHOTON DISSOCIATION OF POLYATOMICS

All the laser isotope separation methods discussed above are based on the excitation of electronic states of atoms and molecules by visible or ultraviolet laser radiation. The isotope separation method discussed below is quite different because it uses only intense IR laser radiation for direct excitation of very high vibrational levels in ground electronic states (Fig. 10 c-d). The method is based on the isotopically-selective dissociaLETOKHOV, RYABOV tion of polyatomic molecules (BCl, (AMBARTZUMIAN, and CHEKALIN [19743, SF, (AMBARTZUMIAN, GOROKHOV, LETOKHOV and [1975a]), OsO, (AMBARTZUMIAN, GOROKHOV, LETOKHOV and MAKAROV MAKAROV [1975b]), etc.) by intense C0,-laser pulses. The effect was discovered in 1974 in our laboratory. Though for some people our discovery was perhaps unexpected, for us it was, in fact, a logical and related result of our work on the isotopically-selective dissociation of molecules by laser radiation. The discovery of this effect was preceded by [19731; several studies (ISENOR, MERCHANT, HALLSWORTH and RICHARDSON AMBARTZUMIAN, CHEKALIN, DOUIKOV,LETOKHOV and RYABOV[1974];

32


[I, 0 5

ISENORand RICHARDSON [197 11; LETOKHOV, RYABOVand TUMANOV [1972a, bl; LYMAN and JENSEN[1972]) of the interaction of powerful IR radiation pulses with molecular gases. The comprehensive discussion of these early works has been presented in a review (AMBARTZUMIAN and LETOKHOV [19771). The essence of the effect consists in the following. When the COz TEA laser radiation frequency is tuned to the molecular vibrational band, the isotope shift of which is comparable to, or larger than the width of the Q-branch of the vibrational band (the intensity being about lo7 to lo9 W/cm2), irreversible dissociation of the irradiated isotopic molecules occurs. This is reflected by changes in the isotopical composition (enrichment) of the undissociated molecules and of those formed by dissociation. An enrichment of over 3000 of the residual SF, gas with the isotope 34Swas LETOKHOV obtained in the first experiments (AMBARTZUMIAN, GOROKHOV, and MAKAROV [1975a]). The chemical composition of some pure gases (BCI,, OsO,, etc.) remains constant even under rather intense radiation at which visible molecular luminiscence can be observed and, thus, their dissociation continues steadily. Investigations have shown that this is caused by the reverse reaction, i.e. recombination of dissociation products forming the initial molecule. When an acceptor, reacting with the dissociation products before their recombination, is introduced, irreversible isotopically selective dissociation of the initial molecules occurs as high-power IR radiation acts in the mixture (BC1, + 0, ( ~ A R T Z U M I A N , LETOKHOV, RYABOV and CHEKALIN [19741; AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROV, RYABOV and CHEKALIN [ 1976]), OsO, +C,H, (AMBARTZUMIAN, GOROKHOV, LETOKHOV and MAKAROV [1975b]), etc.). In this more general sense, dissociation under high-power IR laser radiation is typical of all polyatomic molecules rather than of some of them. At present investigators have obtained much information on isotope separation by the dissociation caused by the multiple IR photon absorption method in a large number of polyatomic molecules (see the review ~ A R T Z U M I A Nand LETOKHOV [1977]). The dissociation process of SF, was studied most carefully (AMBARTZUMIAN, GOROKHOV, LETOKHOV and MAKAROV [1975~1;AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROV and PURETZKII [1976]). These studies enabled us to understand the nature and basic characteristics of both selective dissociation and isotope separation by this process. Comprehensive description of multiple IR photon laser photochemistry has been presented in AMBARTZUMIAN and


I, 0 51

33

LETOKHOV [1977]. Here we are discussing briefly some of the most important features of this approach only. The multiphoton dissociation is a nonlinear process with a sharp intensity threshold. Figure 13 illustrates the experimental dependence of SF, molecule dissociation yield in relative units per one pulse on the CO, laser radiation intensity, the frequency of which is tuned either to the u3 fundamental vibrational band or to the u,+ u6 weak compound vibrational band. A dotted horizontal line shows a measurement sensitivity. The dissociation threshold region of about 23 f.2 MW/cm2 is shaded. We shall mark, that in some experiments with pure SF, other authors did not observe a sharp threshold. It is explained by “dirty” experiment conditions, when the homogeneity of laser beam intensity is not controlled. The dissociation threshold exists for many other polyatomic molecules as well. For example, the collisionless appearance of C, radicals during C,H, molecule dissociation, which are detected by means of a dye laser, has also a distinct threshold (CHEKALIN, DOLJIKOV, LETOKHOV, LOKHMAN and SHIBANOV [ 19771).

8 . 6 -

4 -

D i S S O C I A T ION R A T E

2 . io-3

-

8 6 -

4 -

2 io4--

8. 6 .

2 .

(o-~

LASER 1

INTENSITY I

I

Fig. 13. T he dependence of the dissociation rate Wdi,, in SF, on the laser intensity when a laser pulse excites the v 3 fundamental band or the v 2 + v f , compound band of SF,. Unfocused beam measurements, no scavenger added. Th e laser pulse length was the same in these cases-90 nsec FWHM, the SF, pressure -0.2 torr (from AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROVand PUFETZKII [1976]).

34


[I, § 5

Of most interest are the dependence of the dissociation selectivity S on the IR laser frequency at which dissociation occurs and its correlation with the low-power IR absorption bandshape of SF, (the dissociation selectivity S = W,/ W, is the ratio of dissociation yields of two isotopically different molecules “a” and “b”). Such a frequency dependence of the enrichment coefficient of SOF, formed as a result of dissociation SF, is given in Fig. 14a (AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROV and PURETZKII [ 19761). This figure shows the dissociation yield W(”SF,) dependence of 32SF, on laser frequency also. It is obvious that when the dissociation yields W ifor both isotopical molecules become equal, no enrichment is expected. The dashed line in this figure is for the assumed frequency dependence of the “SF, dissociation yield. The comparison with Fig. 14a shows that near the intersection point (at 931 cm-’) of the dissociation yield curves the enrichment coefficient is unity, which means that there is no enrichment here. A shift up or down from this frequency gives enrichment of one or other isotope in the reaction product (SOF,). The highest dissociation selectivity measured at 0.05 torr is S = 14. 10-

I

I

I

1

I

1

,

930

940

I

-

I*

864-

2-

--

.‘,

0.01 -

z l 0.02 - 0.03-

\

U

?

5004-

3

W 1

910

.

1

920

I

o

950

Fig. 14. Frequency characteristics for multiphoton selective dissociation of SF, molecules by IR laser radiation. The measurements were made in a focused beam, the average radiation power density was 31 MW/cmZ, and the SF, pressure in the cell was 0.2 torr. a) Dependence o f the enrichment coefficients K(32/34) (.) and K(34/32) (A) on the laser frequency. The enrichment coefficients were measured in the dissociation products with the fragmentation ion SOF;. b) Dependence of 32SF6dissociation rate on laser radiation frequency (dashed curve). The linear absorption spectrum of ”SF, and 34SF6 molecules (solid curves) (from A M B A R I Z U M IGOKOKHOV, AN, LETOKHOV, MAKAROV and PURETZKII [1976]).

I, o 51


35

Actually S is much higher, and the value obtained depends on the natural 34S/’80 ratio, since the mass analysis of SOF, has been carried out with low resolution. Having learned the main characteristics of multiphoton absorption phenomenon and polyatomic molecule dissociation, we tried to create a simple model of the process that could at least qualitatively explain these characteristics. The absorption of comparatively large number of quanta in the field of moderate intensity (less than 1h4W/cm2), which was observed by us for many molecules (SF,, OsO, and others) proved, that the anharmonicity, at least, at low transitions is not an essential obstacle for molecule excitation. Therefore we assumed, that some kind of mechanism of “soft” anharmonisity compensation exists at low transitions; this compensation does not require a “rough force”, that is, a very strong field. As a means of finding such a mechanism we have researched a rotational anharmonicity compensation at three consequent P-Q-R transitions (triplerotational-vibrational resonance) (AMBARTZUMIAN, GOROKHOV,LETOKHOV,MAKAROV and PURETZKII [19761). As a result of a stepwise excitation at the consequency of three vibrational transitions uo = 0 4 1 -+ 2 --+ u’ = 3 the molecule reaches the region where the vibrational level density is rather high. In this region of so called vibrational quasi-continuum the frequency of the strong infrared field almost always coincides with one of the numerous, but weak vibrational-rotational transitions. Hence a sufficiently intense infrared field can excite the molecule up to the energy levels enough for its dissociation. Canadian research workers have paid attention to the possible role of the high density of polyatomic molecule vibrational levels in the multiphoton dissociation mechanism in one of the first works on this and RICHARDSON [19731) and problem (ISENOR,MERCHANT,HALLSWORTH more recently BLOEMBERGEN [19751 has given estimates of stimulated transition probability in vibrational quasicontinuum. The molecule excitation at the transitions in the vibrational quasicontinuum leads to the excitation of many vibrational modes (AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROV and PURETZKII [ 19761). The distribution of vibrational energy among many modes provides its dissociation at the energy, which is slightly above the dissociation energy of the weakest bond (GRANT,SCHULTZ,SUDBO,COGGIOLA,SHEN and LEE [1977]). Thus, this simple model provides some ideas about the existence,

36

LASER SELECXIVE PHOTOPHYSICS AND PHOTOCHEMISTRY

[I, 8 5

firstly, of resonance excitation at low vibrational transitions in the field of moderate intensity and, secondly, about the existence of nonresonance excitation at transitions in vibrational quasicontinuum in sufficiently intense field. To check this model we studied the molecule excitation and dissociation on the two-frequency infrared field (AMBARTZUMIAN, GOROKHOV, LETOKHOV, MAKAROV, PURETZKII and FURZIKOV [19761; AMBARTZUMIAN, FURRIKOV, GOROKHOV, LETOKHOV, MAKAROV and PURETZKII [1976]). In these experiments o1radiation frequency of the first laser with moderate intensity ( lo4- lo6W/cm2) scanned in the v, absorption band of the SF6 molecule. The o2 radiation frequency of the second laser with much larger power (lo7- lo8 W/cm2) scanned at some range far away from the absorption band, i.e. in the region of the supposed absorption at vibrational quasi-continuum transitions. Our experiments have shown that this simple model is correct but only in the first approximation. Figure 15a illustrates the dependence of the SF, molecule dissociation yield during o1 radiation frequency tuning of the first laser in the presence of a strong field at o2 frequency far from the resonance. Frequencies and intensities of both laser beams are chosen in such a way, that the dissociation is possible only at joint action of the two laser pulses. In fact, it turned out that the dissociation takes place when o1resonance radiation intensity is at the level of 104--106W/cm2, that is much lower than the threshold dissociation intensity in the single frequency field. This safely proved the existence of the “soft” compensation mechanism of anharmonicity at low vibrational transitions. However, the resonance width at frequency scanning of the resonance field has turned out to be resonance model. much larger than it should be from the triple P-Q-R Having supposed, that this broadening is connected with the “hot” bands contribution, we have carried out the experiments at lower temperature and got some harrowing of the resonance curve. Nevertheless, the resonance width exceeds the width of u = 1+ v = 2 transition Q-branch required for the triple P-Q-R resonance. Figure 15b illustrates the SF6 molecule dissociation yield dependence on o2 frequency of the strong nonresonant infrared field at o1 fixed frequency of the “weak” resonance field. In fact, in accordance with the vibrational quasicontinuum model, dissociation takes place even for strong field frequency detuning far from the absorption band. This is a proof of the absorption existence at the transitions in vibrational quasicontinuum. However, the dissociation yield increases when the strong field frequency approaches the absorption band, pointing to the


v,

0.020( b l

5a

. D ?2

'b,

,

1

I

,

I

,

37

(ern-') I

I

I

,

,

1

I

1

(b) \

0.015

t

2 0.010 '"

P

Fig. 15. The dependence of the dissociation yield W of SF, by two IR pulses on the frequencies w1 and oz:a) the dependence on the frequency w 1 which is in resonance with the vj IR absorption band; the off-resonant frequency w 2 = 1084cmp*. Intensities of both IR pulses: I, = 4 h4W/cm2, Z2 = 60 MW/cm2 (averaged over the irradiation volume; focused beam). The curves 1 and 2 correspond to two different temperatures of SF,, 300 and 190°K respectively. The linear absorption of SF, and the dissociation rate dependence on frequency in a single frequency case at T = 300°K (curve 3) is also shown (AMBARTZUMIAN, FURZIKOV, GOROKHOV, MAKAROV and PUFETZKII [197h]); b) the dependence on the frequency o2of the nonresonant field; o,was fixed at 942.4 cm-'. The linear absorption of SF, is also shown (AMBARTZUWJAN, FURZIKOV,GOROKHOV, LETOKHOV, MAKAROV and PURETZKII [1976]).

probability of wide resonance existence, shifted to the red side relative to v3 SF, absorption band (AMBARTZUMIAN, FURZIKOV, GOROKHOV, LETOKHOV,MAKAROV and PUFETZKII[19761). Thus, the experiments in the two-frequency infrared field confirm the simple model of resonant excitation at low transitions and quasiresonant excitation at transitions in vibrational quasicontinuum. However, there are further essential points to be considered in this simplest model, at least, in two aspects: 1) resonance absorption width at low transitions is

38

LASER S E L E n I V E PHOTOPHYSICS A N D PHOTOCHEMISTRY

[I, 8 5

noticeably larger than it should be from the three-step P-Q-R resonance model; 2) absorption in vibrational quasicontinuum is uniform and has a wide but quite distinct resonance, shifted into the “red” side. It’s already clear now how to correct our simple model in order to explain CANTRELLand LARSEN[19761; these peculiarities (BLOEMBERGEN, LETOKHOV [1977d]). Thus, by relying upon the numerous experiments already carried out as well as upon theoretical considerations it becomes convenient to consider the process of multiphoton exitation and dissociation of polyatomic molecules as a sequence of three processes: 1) a resonance (isotopically selective) multistep and multiphoton absorption in the sequence of several low transitions; 2) a wide resonant absorption in vibrational quasicontinuum; 3) a dissociation of super excited molecule and a consequent multiphoton absorption by polyatomic dissociation products. However, such a division of the process at three sequent stages is to a considerable extent very conventional. Though we have no quantitative picture of the polyatomic molecule dissociation, the information we have obtained about the main properties of this phenomenon is quite adequate to formulate the principles of optimal isotope separation by this method. Selective dissociation of polyatomic molecules by the infrared radiation at single frequency has already been successfully employed in the enrichment of a number of isotopes. As a method of isotope separation, the dissociation of molecules by a single IR frequency suffers several disadvantages ( ~ A R T Z U M I A Nand LETOKHOV [1977]). The requirement of a strong IR field for the dissociation contradicts the requirement of high selectivity of the excitation process. Though the dissociation rate is more sensitive to the pulse energy rather than the intensity, the V-V transfer rate determines the maximal pulse length, and therefore the minimal intensity of the field. The selectivity of the excitation falls off with the rise of the field intensity due to power broadening, and this makes the method not applicable for the separation of heavy isotopes, i.e. when the isotope shift is small compared to the width of the absorption band. The next disadvantage is that it is difficult from the technological point of view to create a high field intensity in a large volume. This makes the process of the separation difficult to scale. The above considered dissociation by two IR pulses of different frequencies (Fig. 10d) avoids both of the problems mentioned above. The two-frequency dissociation method with separation of functions of the selective excitation and selectively excited molecule dissociation allows


39

essentially for the decrease of power broadening of the resonance on the excitation selectivity. This, firstly, allows the use of the method for the separation of heavy elements isotopes with the isotopical shift of only Avlsotope 10-3vvi,, and, secondly, to decrease the requirement to the energetics of the finely tunable laser. As it has been shown above (Fig. 15b) the multiphoton absorption spectrum, at the transitions in vibrational “quasicontinuum” through which the dissociation is performed represents a broad maximum shifted to the red from the absorption band of the mode. The rise of the dissociation cross section means the decrease of the dissociation threshold. Therefore the proper choice of the frequency w2 at which the dissociation is performed allows for the use of beams of much lower intensity compared with the threshold value in the single frequency case. For the first time this was realized in the experiments on separation of osmium isotopes by the radiation of two IR pulses (AMBARTZUMIAN, FURZIKOV, GOROKOV, LETOKHOV, MAKAROV and PURETZKII [19771). In these experiments the intensities of the both fields at w1 and at w2 were significantly lower compared with the threshold value, and the dissociation of OsO, took place in unfocused laser beams of moderate intensities. The osmium isotope enrichments in the natural abundance in the experiments with single-frequency and two-frequency dissociations are entirely different. In the case of single-frequency dissociation the enrichment is completely absent, but makes a noticeable magnitude (-60%) for the two-frequency method. These experiments confirm the conclusion that the optimal scheme for isotope selective dissociation of molecules is the dissociation by two IR pulses. The first pulse at wIof very moderate intensity ( 104-10’ W . cmp2) should be tuned to the frequency of maximum isotope selective excitation. The second one at w2 with the intensity 106-107 W . cmP2 (or energy density 0.1-1 J/cm2) should be tuned to the frequency of maximum dissociation cross section of excited molecules, i.e. the minimum of the threshold intensity. The multiple photon excitation of vibrational levels opens a new possibility for isotopically selective excitation without isotopic shift in linear IR absorption (CHEKALIN, DOLJIKOV, KOLOMIISKY, LETOKHOV, LOKHMAN and RYABOV [ 19761).Figure 16a shows linear absorption spectra of molecules CH3’4N0, and CH,”NO, over the range 900-110 cm-’ (v, and v I 3 band) taken with resolution about 0.5 cm-’. There is no isotopic shift for vI3 band within the experimental errors. In the intense IR field (above lo7 W/cm’) both molecules’ absorption spectra change in form

-

40


[I, § 5

LINEAR IR b~SORPTlON

j-

i'

900

:; '2

\.

950

1000

1050

1100

cm-l

- . . i . , . ...rw . .. .._ , _ . , I , . . , _ 900

950

1000

1050

1100 cm-'

Fig. 16. Isotropical effects for the v I 3 band of CH,NO, (solid curve: CH315N0,, dashed curve: CH,14N0,): a) linear absorption spectrum, pressure: 20 torr; b) multiple photon absorption spectrum at power density lo9 W/cm2, pressure: 2 torr (CHEKALIN, DOUIKOV, KOLOMIISKY, LETOKHOV, LOKHMAN and RYABOV[1976]).

differently which is equivalent to isotopic shift appearance about 5 cm-' (Fig. 16b). This effect has been used for nitrogen isotopes separation in the isotopical mixture of molecules of nitromethane. Let us underline (or stress) that, far not all laser methods, which were applied in laboratories for isotope separation in indicator or even weight quantities, are perspective for industrial isotope separation. The method, potentially fit for the industrial realization, should be characterized at least by two features: 1) The possibility of generating the necessary laser radiation with the level of average power in the range from 1kW up to 1 MW (depending on needed productivity); 2) Laser technique that is simple and efficient in construction and exploitation. These two requirements limit significantly the number of methods, which can be applied at the industrial level with lasers known to date. Powerful IR radiation of molecular lasers is easy and cheap to obtain, and multi-photon dissociation by I R field is quite a simple method; hence, this method is the one most ready to be developed in pilot plants. The first experiments (BAGRATASHVILI, BARANOV, VELIKHOV, GZAKOV, KOLOMIISKY, LETOKHOV, NIZ'EV,PISMENNY, RYABOV and STARODUBTZEV [1978]) o n isotope separation by the radiation of a pulse CO, laser with a

I, 0 61

PHOTOCHEMICAL METHODS OF ISOTOPE SEPARATION

41

high average power (up to 1 kW) and a high rate of pulse repetition (up to 180 hertz) have shown that comparatively high isotopic selectivity of dissociation is preserved in these conditions. Figure 17 shows the experimental dependence of isotopical selectivity of the SF6 molecule dissociation (on the mass-analysis of SOF, dissociation product) on the rate repetition of CO, laser pulses for equal total number of pulses. These results show the possibility of development of a rather productive installation for isotope separation by means of a CO, laser. The simplicity and technological qualities of the laser isotope separation method must permit in rather short terms the beginning of its introduction into industry. Proceeding from the parameters of the processes already achieved under laboratory conditions, at the average power about one KWCO, laser (pulses with the energy of 10 J and a repetition rate of 100 hertz) at ten percent radiation usage, it will be possible to obtain about 0.3 g/hour of 34 S isotope enriched at 80%. The expected cost of an enriched stable I3C, "N, " 0 , " 0 , 34S, 36Sisotopes on the experimental installations must be dozens of times lower than that of the existing cost. § 6. Photochemical Methods of Isotope Separation

To date, the possibilities of photochemical isotope separation, have been successfully demonstrated using the excitation of atomic and molecular electron states and of molecular vibrational levels. However, in

T

2ol -+i-

KSOFZ

T

T

T

1

15

0

50

1

I

100

150 f [Hzl

Fig. 17. Enrichment coefficient K(32S/34S)in product (SOF,) of dissociation of the molecule SF, as a function of rate repetition of pulses f of TEA CO, laser. for various pressures of SF, and the same total number of laser pulses ( N = 1000)(1:0.1 torr; 2:0.24 torr; 3:0.5 torr) (from BAGRATASHVILI, BARANOV,VELIKHOV, KAZAKOV,KOLOMIISKY, LETOKHOV, NIZ'EV,PISMENNY, RYABOV and STARODUBTZEV [ 19771).

42


[I, 8 6

spite of the optimism of early research, the actual advances made appear to be much less impressive than for photophysical methods discussed above .

6.1. ELECTRONIC PHOTOCHEMISTRY

The chemical reactions of electronically excited atoms and molecules have been an active subject of research for many years-especially before creation of the laser. The photochemical isotope separation of Hg excited by the 253.7 nm resonance line has been successfully demonstrated with a variety of reagents (ZUBER[1935, 19361; PERTEL and GUNNING [1959]; BILLINGS, HITCHCOCK and ZELIKOFF [1953]). For example, PERTELand GUNNING [1959] were able to enrich 202Hgfrom 30% natural abundance to 85% in mixture of Hg, H 2 0 and butadiene. The kinetics of the photochemical reactions of Hg are sufficiently complex that even the extensive work of Gunning and his collaborators (see review of GUNNING and STRAUSZ[1963]) does not give a complete mechanism. Isotopic enrichment in diatomics has been carried out by Harteck and coworkers (Lrmr, DONDES and HARTECK [1966]; DUNN, HARTECK and DONDES [1973]), who used atomic resonance lamps for excitation: an 'NO molecule with a Br lamp and 'CO molecuk with an iodine lamp. The enrichment factor was about 4-6. Obviously, laser sources for photochemical isotope separation have many advantages over incoherent sources. The broad tunability and high ultimate resolution of lasers gives a comparatively free choice of absorption lines in the visible UV and probably in the VUV ranges, while permitting the highest possible selectivity. Several successful schemes of laser photochemical enrichment have been reported mostly by R. Zare and coworkers. The most interesting results have been obtained in experiments on photochemical separation of 35Cl and 37Cl through selective excitation of 137Cl molecules by CW dye laser radiation (LIu, DATTAand ZARE[1975]). The laser radiation only excited the I 37Cl molecules to states below the predissociation limit. The excited molecules were subjected to two reactions. In one case, the I 37Cl molecules reacted with trans-CIHC=CHCI, forming cisCIHC=CHCl and causing 10% enrichment with 37Cl. In the other case, they reacted with 1.2-didromoethylene, forming t r a n s - C I H e C H C 1 and causing 50% enrichment with 37Cl. Much higher enrichment in this

1.561

PHOTOCHEMICAI. METHODS OF ISOTOPE SEPARATION

J3.

scheme at pressure of mixture 7.5 torr was reported by ZAKE [1976]. Recently a more successful experiment with ICI molecule has been carried out in STUKEand SCHAFER [1977] where high enrichment coefficients of the chlorine isotope have been obtained. The isotopically selective photoaddition of IC1 to acethylene giving cis- 1,2-iodochloroethylene results in an enrichment factor of 37Clin C,H21C1 of about 48 o r 94% of 37Cl. This result is most promising in isotopically selective electronic photochemistry. Another successful experiment for a diatomic halogen has been reported in BALIKIN, LETOKHOV, MISHINand SEMCHISHEN [1976] and LETOKHOV and SEMCHISHEN [1975]. The ortho-I, molecules in the mixtures with 2-hexene were excited with the 514.5 nm line of the CW Ar laser. The excited ortho-I, molecules react with 2-hexene and para-I, remains inreacted. The selective photochemical reaction of ortho-I, molecules studied in BALIKIN, LETOKHOV, MISHINand SEMCHISHEN [19761 and LETOKHOV and SEMCHISHEN [1975] is a reproduction, at a new level, of the experiment in prelaser selective photochemistry (BADGERand URMSTON [1930]), and it can be applied directly to iodine isotope separation. Enrichment with C1 has also been achieved by selective excitation of C1,CS in mixtures with diethoxyethylene (LAMOTTE, DEWEY, KELLERand RIVER [19751). Mass-spectroscopic analysis of the remaining C1,CS after irradiation with either argon or dye laser light showed the concentration of 35Cl altered from its natural abundance of 75% to 64% or 80% depending on the isotopic species initially excited. The great prospects of electronic photochemistry for laser isotope separation have still not been as successfully demonstrated as might be expected for such an old and classical approach. There are some possible explanations. First, as noted in § 4 of this review, the problem of selectivity loss in the secondary photochemical processes is more difficult for photochemical methods than for photophysical ones. For example, even in such an ideal case as the reaction of selectively excited metastable mercury atoms, the enrichment coefficient due to secondary processes did not exceed 14, in spite of numerous experiments (GUNNING and STRAUSZ [ 19631). Second, the method of electronic photochemistry, judging by the publications, shows no prospect for uranium isotope separation, which is the focus of interest for most investigators. Third, the energetics of visible and UV tunable lasers is far from being economically effective. and the powerful lasers of this range are still hardly accessible.

44

LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMSTRY

[I, 8 6

6.2. VIBRATIONAL PHOTOCHEMISTRY

The rate of a chemical reaction may be substantially enhanced by vibrational excitation of the reactant molecules. GIBERT [19631 suggested this as a method for isotope separation. However, the first successful experiment by this approach was not done until more than ten years later KAUFMANN and WOLFRUM [1975]). (ARNOLDI, A wide variety of vibrational excitation processes may be used (Fig. 18). The IR active fundamental vibrations of a molecule may be excited by absorption of a single photon (Fig. 18a). Excitation of combination and overtone bands gives two or more quanta of vibrational excitation on absorption of a single higher-energy photon (Fig. 18b). Higher vibrational levels may also be reached by step-wise excitation through one or more intermediate levels by multifrequency radiation (Fig. 18c). Raman excitation may also be used (Fig. 18d). It is the only method for excitation of vibrations with a zero-transition dipole (e.g., homonuclear diatomics). The selective excitation of high vibrational levels by multiple IR photon absorption at comparatively moderate intensities (lo6- lo7W/cm2) give us one more - and probably the only effective - method for direct excitation of vibrational levels with energies of several eV (Fig. 18e). All these excitation schemes were tested in the experiments on laser isotope separation. Positive results were obtained by the latter three excitation schemes (ee). However, the actual experiments are small in number, so we cannot state the fundamental disadvantages of any of them. The

e Fig. 18. Schemes of selective excitation of vibrational levels of molecules by laser radiation; a) single photon absorption on the fundamental band; b) single photon absorption on the second overtone band; c) two step excitation by two-frequencv IR field; d) Raman excitation of an IR unactive absorption vibration by two-frequency visible laser field; e) multiple photon excitation of highly-excited levels by single frequency intense IR field.

I.

P 61

PHOTOCHEMICAL METHODS OF ISOTOPE SEPARATION

45

success or failure of an experiment is evidently connected more with the experimental conditions and the right scavenger than with the particular excitation scheme used. But, of course, the schemes that excite high vibrational levels are more favorable, because they provide potentially a freer choice of chemical reactions and a lesser influence of heat mechanism on the separation selectivity. ARNOLDI, KAUFMANN and WOLFRUM [1975], have reported the enrichment of 3sCl by enhancement of the reaction of Br atoms with HC1. They used an HCl pulsed chemical laser to excite HC1 sequentially from u = 0 to u = 1 to u = 2 (Fig. 18c). Selective excitation of HCl led to an acceleration of the reaction Br + H 3sCl ( u = 2) + HBr + 35Cl by a factor of eleven orders of magnitude over the rate of reaction of Br with HC1 ( u = 0). Isotopical enrichment was demonstrated by time-resolved massspectroscopy of BrCl formed in the secondary process C1+ Br, -+BrCl + Br. For equal pressures of Br and HCl a separation factor of 2 was observed. Raman excitation of N, in air at 77°K was reported by BASOV, ISAKOV, MARKIN, ORAEVSKII, ROMANENKO and BELENOV,GAVRILINA, FERAPONTOV [19741 and BASOV,BELENOV, ISAKOV, MARKIN, ORAEVSKII, ROMANENKO and FERAPONTOV [1975] to produce NO enriched 100 times in ''NO. This might result from N,+O, -+ 2N0. Basov et al. interpreted their results in terms of a mechanism (DUBOST, ABOUAF-MARGUIN and MARKIN, ORAEVSKII and ROMANENKO [19731) LEGAY[19721; BELENOV, that requires considerable vibrational energy transfer among N, molecules in a Treanor pumping process (TREANOR, RICH and REHM [1968]). The same separation method and a similar interpretation was used by Basov et al. in the experiments on isotope separation in an electric discharge, where the enrichment of NO by "N isotope reached 10. But the results of this work contradict those of MANUCCIA and CLARK [ 19761, their interpretation of the results in BASOV, BELENOV, GRAVILINA, ISAKOV, MARKIN, ORAEVSKII, ROMANENKO and FERAPONTOV [1974] and BASOV, BELENOV, ISAKOV, MARKIN, ORAEVSKII, ROMANENKO and FERAPONTOV [1975] cannot be considered final. Pumping of BCl, molecules in mixture of BCl,+H,S and BCl,+D,S by a focused, pulsed CO, laser, which probably excites high vibrational levels, caused separation of boron isotopes (FREUND and RITIER [1975]). Starting with natural boron isotopic composition, irradiation of the mixture with either P(16) or the R(20) line of the lO-"A CO, laser gave enrichment of the residual BC1, K(10/11) of the order 1.7 and

46


[I, 8 7

K(10/11)=0.7, respectively. It is clear now that this experiment is an intermediate one between single-photon I R photochemistry in a weak field and multiphoton I R photochemistry in an intense field. The attractive feature of vibrational photochemistry for isotope separation is the prospect of using low-energy IR photons from an efficient molecular laser to get a good yield of the product. However, from the point of view of the efficient usage of IR radiation, the method of I R molecular photochemistry will be able to compete with the method of multiphoton excitation and dissociation of polyatomic molecules in an intense I R field. The multiple-photon approach has a number of disadvantages in comparison with vibrational photochemistry (reaction area restricted by the intense field area and application to polyatomic molecules only), but is much less restrictive in many other aspects. Future experiments will undoubtedly indicate the applications preferable for each approach.

0 7. Purification of Materials at AtomieMolecular Level The methods of selective photophysics and photochemistry being developed to separate isotopes make it possible to elaborate a new approach to materials technology at atomic-molecular level, when by laser radiation one can directly manipulate atoms and molecules of a particular sort, that is, collect macroscopic amounts of a substance “by one atom, by one molecule”. The most important process of universal laser atomicmolecular technology of materials is, no doubt, the production of highly pure substances in atomic state, alloys and molecular compounds. The processes of selective atomic photoionization and selective molecular photodissociation may be used to produce highly pure substances or to purify a substance. Of course, the fields of applications for both approaches differ greatly.

7.1. SELECTIVE ATOMIC PHOTOIONIZATION

This approach to materials technology is the most universal and flexible. An optimal scheme for selective atomic photoionization under the action of two (or more, in principle) laser beams with properly tuned frequencies and chosen intensities enables every atom to be ionized

1 . 4 71

PURlFlCAl ION OF MATERIALS A I ATOMIC-MOLLCULAK LEVk1.S

31

within to lO-'sec. When 20% of the energy of the radiation with an average power of lo3W is used to photoionize atoms with an ionization potential E , = 10 eV, it is possible to ionize about one mole of substance per hour. Thus a comparatively moderate scale setup may ensure production of several tons of a pure substance per year. The method of selective atomic ionization in combination with tunable lasers, with average output of 100 W to 1000 W, can therefore be considered as a sufficiently efficient method for fine substance separation at atomic level. Laser purification of substances by selective ionization (LETOKHOV [1975b]) must have a number of material advantages over the existing methods of purification on the basis of differences in any chemical or physical properties of a substance and its impurities: 1. High selectivity or high degree of purification in single-step process. The degree of purification of a desired element from any admixtures may be higher than lo3. This value depends on the process of exchange of charge during collisions between the ion of the given element and the neutral atom of the impurity. Basically, by decreasing the atomic density in the beam we may achieve a separation selectivity much higher than lo3, with the efficiency reduced respectively. In particular, if we take a mass-production material with its purity of 10-70/~,it is possible to purify it up to lo-"% by the method of selective atomic ionization. 2. Universality. Selective ionization may be realized through proper selection of laser beam frequencies on any element, independent of their physical and chemical properties (melting and boiling temperatures, reactivity, etc.). When a substance is to be purified of one or more specified elements, it is possible to ionize selectively its impurities only and to remove them from the atomic beam of the substance. Under this regime a maximum efficiency of the method is achieved with a minimum of coherent light energy. 3 . Flexibility which makes it possible to use ion beams directly to produce pure films o r to implant ions into a homogeneous substance (ion implantation). Ion beams can be directed onto the substrate surface to produce a pure film of a specified element, as shown in Fig. 19. We think it is possible simultaneously to carry out independent selective ionization of two or three elements in different atomic beams and deposit these elements on the same surface. Thus, it will probably be possible to produce films of complex atomic compounds with their stoichiometric composition controlled by the intensity of photoioii beams. The whole process of selective atomic ionization, extraction of ions from the atomic

48


C o o l e d T r a p of N e u t r a l A . B Atoms

B + Ions

1

L a y e r of B Atoms

Vacuum Chamber

Collector Electrode Atoms l A . 0 1 L o s e r B e a m for Excjtatlon o f B Atoms

I o m z a t t o n of B Atoms

Diaphragm

I

'1--Atom,c Source ( A , 8)

Fig. 19. Possible scheme of atomic purification by selective photoionization of atoms.

beam and their deposition on the substrate can be brought about in a high vacuum. It is not necessary for the process that the substance to be purified should make contact with any reagents or material apart from the substrate, for which we may always use a material without undesirable impurities. We should concentrate our attention on the use of selectively formed photoions of boron, arsenic, phosphorus and other elements in setups for MISHINand PURETZKII ionic implantation in semiconductors (LETOKHOV, [1977b1). Electrodeless laser production of certain ions eliminates, first of all, the necessity of using an electromagnetic mass-separator and, secondly, makes it possible to insulate the high-temperature source of atoms from the ionizer. The latter is of no small importance since this enables the atoms to be photoionized near the high-voltage electrode and hence the construction of electrostatic ion accelerators with an energy of about MeV and over, can be substantially simplified. The basis €or successful development of the photoionization method is elaboration of optimal schemes for multistep selective ionization of various elements and also creation of rather efficient tunable lasers in the W and visible range with a high average power and long lifetime. The multistep resonant excitation of the states near the ionization limit and subsequent autoionization of highly excited atoms by a pulsed electric

I . ii 71

PURIFICATION OF MATERIALS AT ATOMIC-MOLECULAR LEVELS

39

field suggested by IVANOV and LETOKHOV [1975] is a universal and optimal ionization scheme providing a high excitation cross-section and high ionization yield. Recent experiments (AMBARTZUMIAN, BEKOV, LETOKHOV and MISHIN[1975]; DUCAS, LITTMAN, FREEMAN and KLEPPNER [1975]; LITMAN, ZIMMERMAN and KLEPPNER [1976]; BEKOV, LETOKHOV and MISHIN [ 1977a, b]; PAISNER, CARLSON, WORDEN, JOHNSON, MAYand SOLARZ [ 19761) have proved the feasibility of this approach (see section 5.1). As to lasers, the main difficulty when operating with high average power UV lasers will be overcome probably by the use of excimer lasers. 7.2. SELECTIVE MOLECULAR DISSOCIATION

The process may be used to purify a substance in a gas phase of molecular impurities, the removal of which by standard techniques is not efficient. Purification by the dissociation method is based on differences in the physical - chemical properties of the basic substance and the dissociation products. This enables us to use the standard techniques of purification at the end of the process, after the mixture is irradiated. The possibility of substance purification in a gas phase through dissociation of admixed molecules by intense IR radiation has been recently demonstrated experimentally (AMBARTZUMIAN, GOROKHOV, GRIGOR'EV, LETOKHOV, MAKAROV, MALININ,PURETZKII,FILIPPOVand FURZIKOV [1977]) where arsenic trichloride (AsCl,) was purified of 1.2dichloroethane (C,H,Cl,) and carbon tetrachloride (CCl,). The minimum content of these impurities given by the standard techniques of purification is of the order 10-2-10-3%. The absorption bands of the admixed molecules C,H,Cl, and CC1, fall within the oscillation region of a CO, laser where there are no absorption bands of the basic substance molecules AsCl, (Fig. 20). Therefore, the effect of dissociation of " C q Laser Bands DissociationProducts

AsC13 CCI, %%C'2

I00 300

500 700 900 v

gc1,.

CZCI,

q&.

gH3CI.

HCI

1 1 0 0 1300

bn-9

Fig. 20. Purification of AsCI, by selective multiple photon dissociation of impurities (CCI,,

C,H,ClJ.

50


[I, 0 8

polyatomic molecules in an intense field of a CO, laser can be used for selective dissociation. The final dissociation products were identified from I R absorption spectra (for C2H4C12)and mass spectrum (for CCl,). In experiments (AMBARTZUMIAN, GOROKHOV,GRIGOR’EV,LETOKHOV, MAKAROV, MALININ, PURETZKII, FILTPPOV and FURZIKOV [19771) selective dissociation of C2H4C12and CCl, admixed in AsCl, was clearly observed, the pressure of AsCl, being about 10 torr. The initial content of the admixed molecules was comparatively high, which was conditioned not by the limitations of the method but only by detection selectivity. In one case of 1.2-dichloroethane the final products differ greatly from AsC1, in their physical properties, and this enables them to be separated easily and AsCl, to be purified. The method of selective molecule dissociation in a two-frequency intense I R field, which has been discussed in section 5.4 looks especially promising. Controlling intensities and frequencies w1 and w2 of two infrared laser beams, we may provide simultaneously a high selectivity and a low intensity threshold of a dissociation effect. It means, that in principle it is possible to remove certain unwanted molecular admixtures from a multicomponent molecular mixture. The perspectivity of this new technology is undoubtful. The method of selective molecular dissociation seems to be applied not only to technology of pure materials but also to removal of toxic and canceregeneous substances from gas mixtures, that is, to selective atmospheric photochemistry. If dissociation of such impurities converts them to inactive forms, the method becomes rather simple and independent.

S 8. Selective Laser Biochemistry 8.1. GENERAL REQUIREMENTS

Selective action of laser radiation on complex molecules in a condensed medium is a promising possibility for molecular biology, which however has not been carefully studied and is therefore uncertain. In a condensed medium at normal temperature the discrepancy between the requirements for excitation selectively and conservation of selectivity becomes more aggravated. The electron transitions of biomolecules are grouped in the UV region, and a good excitation selectivity of particular molecules in

I,§ 81

SELECTIVE LASER BIOCHEMISTRY

51

a mixture can hardly be expected. On the other hand, electron excitation sec) that is sufficient for a is conserved, as a rule, within a time (= photochemical reaction with an appreciable quantum yield. Vibrational excitation of molecules is more selective but it relaxes to heat in a short and time (at 300°K the relaxation time T;lbe lo-’’ sec) (LAUBEREAU KAISER[1977]). Since the energy of one vibrational quantum is just a few times greater than the thermal energy kT, the contribution to the biochemical reaction rate by short-lived vibrational selective excitation cannot be as large as that by permanent thermal nonselective excitation. Successful experiments on isotopically-selective photoprocesses in gas media show that there are at least two ways of eliminating this discrepancy: 1) a combination of selective vibrational excitation with subsequent electron excitation from vibration-excited states, that is, two-step IR-UV excitation, 2) multiple photon vibrational excitation in an intense IR field. In both cases the process, unlike the case of gas medium, should of course be realized under the action of picosecond laser pulses so that a molecule can absorb a considerable energy of several eV prior to thermal relaxation of vibrational excitation. The process of two-step IR-UV excitation, which offers additional advantages compared to vibrational or electron excitations, is believed to be especially universal. In each particular case, of course, we must first prove the potential feasibility of the two-step I R - W selective photoprocess for a chosen molecule in the solution or for the molecular bond in a macromolecule. The scanty spectral information available on excited states, in particular for biomolecules, makes the answer to this question far from trivial. Below we shall consider several potentialities for different molecules and bonds. But besides this general question we have to deal with other “tricky” problems, i.e.: 1) heating of the medium during selective photoexcitation, 2) absorption of IR radiation by solvent molecules. These problems are particularly essential for experiments “in vivo”. [1975c] show that it is possible to The estimations in LETOKHOV eliminate heating when the molar concentration of molecules in a solution is below M and the absorption of IR radiation by solvent molecules is rather weak. To reduce the IR absorption by molecules in water, a typical solvent, in experiments “in vivo” one may try to excite the overtones and compound vibrations in the near IR region, even though the selective absorption cross-section is decreased. In the case of a homogeneous molecular solvent one can probably reduce the absorption when operating in the regime of “self-induced transparency” (MCCALL

52

LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMSTRY

[I, 9 8

and HAHN[1967, 19691) for the solvent molecules. It is also possible to excite selectively the vibrational levels by the stimulated Raman process in the field of two-frequency visible laser radiation, which is quite transparent for the solvent, as has been done in experiments (LAUBEREAU, VON DER LINDE and KAISER[1972]). ‘fie estimations show that the required intensities of IR and UV pulses from lo-’’ to lo-’’ seconds in duration range between lo8 and 109W/cm2. No doubt, cooling of molecules decreases the V-T relaxation rate and so makes all these difficulties less problematic. The first successful experiment on a two-step IR-UV excitation of the electronic state of a complex molecule (coumarin 6 in CCI4) via the vibrational state has been carried out by LAUBEREAU and KAISER[1977] and LAUBEREAU, SEILMEIER and KAISER[1975]. This experiment has been done not for selective photochemistry, but in order to elucidate a vibrational relaxation of a molecule (the same method has been used to investigate the vibrational relaxation of the NH, molecule (AMBARTZUMIAN,LETOKHOV, MAKAROV, PLATOVA, PURETZKII and TUMANOV [19731; AMBARTZUMIAN, LETOKHOV,MAKAROVand PURETZKII[19751). A picosecond tunable parametric oscillator excited the vibrations of the CH, group at about 2970 cm-’; while the second harmonic (18950 cm-’) of a picosecond Nd-glass laser transferred vibrationally excited molecules further into a singlet electronically excited state, whose population was detected on a consequent fluorescence.

8.2. SOME POSSIBILITIES

Let us now consider some specific possibilities in laser selective biochemistry. They seem quite obvious “on paper” but may prove to be most difficult in experiments. Selective excitation of DNA bases. Polynucleotide chain molecules DNA and RNA are important objects of investigation as far as both the possibility of selective laser mutations and their simple structure are concerned. Both molecules, despite their large dimensions, contain five repetitive nucleotides: guanine ( G ) , cytosine (C), thymine (T), adenine (A) and uracil (U). All these five bases are purine and pyrimidine rings which have similar W spectrum with two maxima. Figure 21 presents the values of energies for UV band maxima, from CLARKand TINOCO [1965]. We think that it is possible to bring about selective electron

1,s 81

53

SELECTIVE LASER BIOCHEMISTRY

uv . ABSORPTION

GUANINE

4.20

4.90

CYTOZINE

4.30

6.10

THYMINE

4.70

5.90

ADENINE

4.90

6.00

URACIL

5.10

6.00

Fig. 21. Energy excitation of the two first intense absorption UV bands ot nucleotide bases.

excitation of the long-wave bands of the bases of G and C which corresponds to excitation of the rr-electron system of the base pair “G-C” (LADIK[1972]). At the same time we cannot excite Anucleotides T-nucleotides without exciting the others. The difference in the molecular structure of these nucleotides gives us hope of finding for each of them a specific vibrational band showing itself in the UV absorption spectrum. We wish to say that the I R spectra of DNA are yet imperfectly understood (SUSI [1969]), so we cannot specify now the frequencies for all the nucleotides. In our opinion, the excitation of DNA nucleotide vibrations by picosecond I R tunable pulses and simultaneous probing of changes in the UV spectrum may be a good method to study and resolve the vibrational spectrum of DNA as well as a necessary intermediate stage in studying the possibilities of selective action on DNA bases. Selective excitation and breaking of hydrogen bonds in DNA. The double helix of DNA is formed by hydrogen bonds between the bases (guanine-cytosine and adenine-thymine). The breaking of the hydrogen bonds must result in splitting of the double helix into two single helixes and subsequent replication of DNA. Selective excitation of hydrogen bonds and their selective breaking seem to be of interest for laser control over the process of DNA replication. I think that this is essential not only as a potentiality of laser stimulation of a biological process rate but also as a basically new possibility for an external controlled “start” of the DNA replication process, which has not yet been studied in detail. Two pairs of bases in DNA have slightly different hydrogen bonds. The pair A-T is linked by two hydrogen bonds N-Ha . .O, and an energy of about 7.0 kcal/mol is required to break this pair of bases. The pair G-C is N, and linked by two hydrogen bonds N-H . C and one bond N-H about 9.0 kcal/mol must be consumed to break them (LADIK[1972]).

--

54

LASER SELECTIVE PHOTOPHYSICS 24ND PHOTOCHEMISTRY

[I, § 8

An IR absorption band of about 1720cm-’ corresponds to the bond G-C in native DNA and a band of about 1700 cm-’ to the A-T bond. In DNA denaturation when the hydrogen bonds are broken, both bands [1975c] it was therefore proposed disappear (SUSI[1969]). In LETOKHOV to act by picosecond powerful IR pulses on these frequencies (5.814X lo5 and 5.888 x lo5A) to stimulate hydrogen bond breaking. There are, of course, many more possibilities for experiments “in vivo”, especially with regard to the choice of more convenient wavelengths not absorbed by the solvent. The potential function of a hydrogen bond has two characteristic minima corresponding to two potential spatial and energy positions of a proton (Fig. 22). Though not experimentally revealed, the energy levels of a proton in the hydrogen bonds N-H . 0 and N-H . N have been calculated. The transition of a proton (or rather of two protons in a pair of the next hydrogen bonds) to a higher energy tautomeric state is believed to result in mutations (LOWDIN[1964, 19681 mechanism). It is evident that using a laser at the wavelength of 1.8 x lo58, we may try to transfer a proton to an excited state and thereby stimulate its tunneling to a higher energy minimum. This possibility has been discussed in LADIK [1977] and with the progress in picosecond tunable IR lasers must become a subject of experimental studies. Proton excitation must also show itself in the spectrum of electron tJV absorption and thus can be used in two-step selective processes by the scheme: “selective IR excitation of a proton +UV excitation of an electron”.

- -

t

I

1

I

r(N-H...O)

(8)

Fig. 22. Potential function of hydrogen bond in DNA from LADIK[1972] and possible scheme of IR field-induced tunneling of a proton.

1, 91

SELECTIVE DETECTION OF NUCLEI, ATOMS AND MOLECULES

55

0 9. Selective Detection of Nuclei, Atoms and Molecules The methods of selective laser photophysics solve the problem of physical extraction of a particular atom or molecule from mixtures the chemical properties of which are very similar to those of other atoms and molecules. The primary and relatively simpler part of this problem is selective detection of single atoms and molecules. Selective two- (or multi-) step photoionization of atoms and molecules is best suited to this purpose. Let us consider some of these potentialities.

9.1. DETECTION OF SINGLE NUCLEI AND ATOMS

At present, excited (metastable) nuclei are being detected in the process of their radioactive decay. But the specific features of an excited nucleus affect not only nuclear transitions but also the hyperfine structure of optical transitions of the electron shell around the nucleus. As the isomeric structure usually considerably exceeds the Doppler broadening of spectral lines, we think, it is quite possible to selectively ionize not only nuclei of a particular isotopic composition but also excited nuclei with a specific nuclear spin and quadrupole moment. This possibility has been discussed as far as separation of isomeric nuclei and preparation of the active medium of the future y-laser (LETOKHOV [1973]) are concerned. Here I would like to point toward the possibility of selective detection of excited nuclei as a new approach to studying and searching for metastable nuclear levels. It is possible to define the quantum nuclear state from the electron shell of the nucleus without its de-excitation in the process of detection. After each selective extraction of an electron and its detection we may exchange the charge of the ion and thus repeat the whole process many times. A possibility of the existence of relatively stable superheavy and so called superdense nuclei is being now under active discussion in nuclear and POLIKANOV [1977]). In the first case, the task is physics (KARNAUKHOV to detect spectral lines, which do not belong to any of the known elements. The spectrum of optical atomic transitions corresponds only to the one nuclear structure. So the development of the detection methods for single atoms will provide a new method of nuclear detection of nuclei for nuclear physics. It seems especially interesting to apply laser methods

56

LASER SELECTIVE PHOTOPHYSICS .4ND PHOTOCHEMISTRY

[I, § 9

for the detection of atoms with superdense nucleus, at anomalously large (-100 cm-.’) isomeric shift in electronic spectra. In all these cases simultaneously a high selectivity, and an ultimate sensitivity of detection (single nuclei and atoms), as well as the possibility of extraction of detected nucleus or atoms are required. Obviously the only reliable method for their purpose is the method of selective multistep photoionization of the atom. We have paid attention to this possibility in LETOKHOV [1976a, b; 1977el. By the method of selective stepwise ionization of atoms it is possible to obtain extremely high selectivity determined only by the excitation selectivity of atoms and molecules. The resolution lo6 is standard at laser excitation of atoms and molecules and when special methods are used to eliminate the Doppler broadening, the resolution of the order 108-109 (see SHIMODA [1976] and LETOKHOV and CHEBOTAYEV [1977]) is quite realizable. When a selectively excited transition is saturated by laser radiation, an atom stays in the excited state half of the time. Additional laser radiation with quantum energy hw,, satisfying the condition

Ei - h o , < ha2 < Ei,

(3)

where Ei is the potential of ionization of atom, and h q , the energy of an excited level, can ionize an excited atom with the quantum yield of the order of 1. In this case the energy density of the second laser pulse 8, must exceed the saturation energy density 8!Jt of the transition into continuum:

with ai,the cross section of the photoionization from the excited state and the duration of the second laser pulse being less than the relaxation time T of the excited state. A t the typical value of the photoionization = 2 eV, the energy density of cross section mi== 10p’7-10-18cm2 for the photoionizing laser pulse must lie in the range g2= 0.03 - 0.3 J/cm2. To detect each individual atom it is very important to fulfill the condition (4) above, which provides the ultimate photoionization yield. The dependence of the ionization yield on the energy of ionizing pulse was experimentally investigated by AMBARTZUMIAN, APATIN, LETOKHOV, MAKAROV, MISHIN, PURETZKII and FURZIKOV [1976] for R b atoms. Figure 23 shows the experimental dependence of the total ion yield on the energy density of the ionizing pulse. The crossing point of the linear part

1, I 91


51

10'6

E,

(photonlcrn')

Fig. 2 3 . Photoion signal as a function of the energy density E , of laser pulse, which ionize Rb atoms excited into 6 'P states, for two different values of extracting electric field strengths: I : 2.4 kV/cm; 2 : 1.O kV/cm (from AMBARTZUMIAN, A P A T I N , LETOKHOV, and FURZIKOV [1976]). M A K A R O V ,M I S H I N ,PURETZKII

of the curve of the transformation in the plateau corresponds to the energy density %'!it,at which 63% of excited atoms are ionized. The first successful experiments on detecting single atoms (Cs) by the method of a two-step photoionization have been carried out in HURST, NAYFEH and YOUNG[1977], where an ionized atom caused a signal in the chamber of a proportional counter. The signal was quite intense to detect a single act of selective photoionization. Such a method can be used for selective single atom detection when the conditions of saturation of exciting and ionizing the stimulated transitions are carried out, and hence, the ionization quantum yield is approaching the ultimate. The cross section of the phototransition from a discrete state into continuum a,,is small as compared to the cross section of the excitation at resonance transitions wI2. So, to ionize each excited atom the energy %$2t is required which exceeds uI2/uietimes the energy density, = hol/2uI2 needed for the excitation of an atom. The selective photoionization of an atom with the maximum quantum yield and low expense of laser energy requires the use of the transitions into quasicontinuum with high photoionization cross section, for example, the transitions into sharp autoionization states. Another approach is much more universal. It is based on the selective resonant stepwise excitation of the high-lying (Rydberg) state and the successive ionization of a high-excited state by a pulse of a d.c. electric field (IVANOVand LETOKHOV [1975]). The critical strength of the electric field (in at. un.) is determined by an expression (2)

58


[I, I 9

c

Fig. 24. Ion signal as a function of the strength of the electric field ionizing Na atoms from the 17 2S state (from BEKOV,LETOKHOV and MISHIN [1977a]).

(see Fig. 9). Figure 24 shows the experimental dependence of ion yield on the electric field strength for Na atoms, excited into the state 17 ’S. Full ionization of the excited atom is obtained at the strength of d.c. electric and MISHIN pulse of several kV/cm. In the experiment (BEKOV,LETOKHOV [1977a]) by the method of a selective step excitation of the Rydberg state and its subsequent ionization by the d.c. electric pulsed field rather high quantum yield (about 1.0) of sodium atom ionization and detection have been achieved. The photoionization method can provide the detection of individual atoms, but it is “a destroying” detection method, as against the fluorescent one. It is favourably applied for the detection of atoms in metastable states, for which the fluorescence is very weak, and their photoionization requires low intensity. Besides, the photoionizing method of the selective detection of atoms has a very important advantage, as compared to all other ones, which lies in the possibility of extracting the detected atom from the mixture by external electric and magnetic fields.

9.2. DETECTION OF COMPLEX MOLECULES

As is known, selective detection of trace amounts of polyatomic complex molecules is a very difficult problem as yet not solved by physical methods. Mass-spectral analysis is now a common method for detection and identification of complex molecules but its sensitivity is inadequate and there is practically no selectivity of detection for complex molecules

1,s 91


59

that differ only in spatial structure. Therefore, the development of new methods to solve this problem is today very urgent. The idea of the selective photoionization of molecules by laser radiation is illustrated in Fig. 25 for the simplest case of two-step photoionization. One laser with the tunable frequency ol,excites selectively the vibrational (or for some molecules electronic) state. Due to such an excitation the edge of the photoionization band of a molecule, lying usually in the vacuum ultraviolet, is shifted by a small value. The second lases with frequency w2 in VUV range causes the photoionization of molecules, its frequency being selected in the point of the maximal slope of the photoionization band edge. The preliminary selective excitation of molecules by a tunable laser to a comparatively, small value E,,, = hw, = 0.1-0.5 eV causes a decrease by a small value A u of the ionization cross section of a molecule . The systematic investigation of selective photoionization of molecules became possible after the development of a simple VUV H, laser (KNYAzEv,LETOKHOV and Movsmv [1975]). As the first step one-step photoionization of molecules of dimethylaniline and methylaniline by H, laser radiation in the range 1600 A, was realized, and also that of the NO molecule in the range 1200A (ANDREEV, ANTONOV, KNYAZEV, LETOKHOV and MOVSHEV [19751). The first experiments on selective two-step photoionization of molecules have been carried out recently. ANDREEV, ANKNYAZEV and LETOKHOV [1977] report the experiments with a TONOV, H,CO molecule, the photoionization of which was carried out by joint e-

(R,Pz)++

R: + R~

a.

+

e-

- PHOTOIONI7ATlON -DISSOCIATIVE

IONIZ!ATION

b.

Fig. 25. The simplified scheme of two-step selective photoionization of a molecule through the intermediate vibrational (or electronic) state (a) and the selection of the frequency of V W radiation o2in the range of the maximum variation of photoionization cross-section for excited molecules (b).

60


[I, I 9

action of the radiation pulse of an N, laser with the wavelength A , = 3371 A, exciting state 'A, and a radiation pulse of an H, laser at A2 = 1600 A, causing the photoionization of excited molecules (Fig. 26a). The ionization potential of H,CO molecules is Ei=10.87 eV, and the summary energy of two laser photons ha, +ha,= 3.7 eV+7.7 eV= 11.4 eV, which is quite sufficient for the photoionization of this molecule. A temporal delay of H, laser pulse (pulse duration shorter than 1 nsec) was varied relatively to N, laser pulse (pulse duration about 2 nsec) which allowed the dependence in ion yield on the delay time to be measured (Fig. 26b). The experimental curve is close to the exponential dependence with decay constant about 15 nsec which is equal to the molecule H,CO lifetime in the excited 'A, state. Of course, the H,CO photoionization signal is absent under reverse order of N2 and H2 lasers pulses. KNYAZEV, LETOKHOV and MOVSHEV In the next experiment (ANTONOV, [1977]) we have investigated the two-step photoionization of molecule (NO,) under excitation of the intermediate electronic state by tunable dye laser radiation (transition 'A2 -+ 'B1 in the region 4470-4970 A). The photoionization of the excited molecules was realized by H, VUV laser pulse in the region 1600A. Since NO, ionization potential is equal to 9.78 eV, two laser photon energy is 2.7 eV+7.7 eV = 10.4 eV and hence the NOz molecule could be photoionized only by means of two-step process. By measuring the dependence of photoion current value on the dye laser frequency, the spectrum of the molecule NO, absorption was TWO-STEP

10.87eV

Hz- Laser i6OOA

1 4.0

-10

t

pwolo,o)l

0

10

CURRENT

20

nsec

a.

H&O

PHOTOIONIZATION

30

40

pulse delay

b.

Fig. 26. The scheme of transitions at two-step photoionization of molecules by H,CO pulses of N2- and H,-lasers (a) and the dependence of photoion yield of two-step photoionization on the delay AT of a H, laser pulse relative to a N, laser pulse (b).

I, 8 91


61

measured at the electron transition which coincided well enough with the well-known absorption spectrum. The intermediate vibrational states should be used for realization of highly selective photoionization of molecules by laser radiation. In this case it is hoped that high selectivity and high sensitivity may be combined in the single method of detection of complex molecules. The possibility of combining the molecule selective photoionization method giving the information on molecule optical spectrum, with massspectrum analysis giving information on the mass of photoionization products of the detected molecule is particularly promising. Following this method it is possible to create, in principal, a laser selective detector and for single molecules or so called laser mass-spectrometer (LETOKHOV AMBARTZUMIAN [19713, AMBARTZUMIAN and LETOKHOV [19721, LETOKHOV [1976b, 1977d, el), the simplified diagram of which is shown in Fig. 27. The selective ionization of molecules in molecular beam is carried out by, at least, two (IR + VUV) laser radiations instead of ordinary unselective photoionization of molecules by electron beam (or by continuous VUVradiation). The formed ions are directed into the mass-spectrometer. Thus, the tuning of IR laser radiation permits both mass-spectrum and optical spectrum of the molecule to be measured by means of the photoion signal measurement. The first experiments on the combination of VUV-laser with massMOVSHEV, LETOKHOV, KNYAZEV spectrometer were described by POTAPOV, and EVLASHOVA [1976]. During these experiments an H2 laser at 1600 A Electromagnetic

Photo-ion Current I = f(M/e.w,) Molecules and

V U V Laser Rodiotion

Photo-Ions

Mass Spectrum

w 2 = = b

Tunable

Opticol Absorption Spectrum

Loser Radiation /

Fig. 27. A possible simplified scheme of a two-dimensional laser optical mass-spectrometer.

62


[I, § 10

performed a one-step photoionization of some molecules in the molecular stream of a mass-spectrometer under pressure 10-5-10-3 torr. The H2 laser operated at pulse repetition rate of 50 to 300 Hz. Pulse duration was no more than 1 nsec, and its energy was equal to approximately 20 pJ of laser output and 5 pJ in photoionizing volume. Laser radiation average power for 1600 A achieved several mW, i.e. corresponded to the photon flow 1015 ph/sec. At pulse repetition rate 300Hz the photoion current value is equal to 7 x lo-’’ A which exceeds 100 times the value obtained under usage of continuous spark gap of hollow cathode as a source of VUV radiation. It proves that two-step photoionization of molecules in mass-spectrometer beam may be really performed. The infra-red mass-spectrometer may become a universal highselectivity and high-sensitivity detector of complex molecules, which is useful when solving a great number of scientific and applied problems. Indeed as the detection of the excited molecules by photoionization method is highly sensitive, then, as was shown above by means of the simple evaluations, it may be hoped to measure IR spectrum of extremely small amounts of substance - amounts which are much smaller than it is required for the best existing classical and laser IR spectrometers. It should be underlined that there is the possibility of simultaneous realization of the extremely high spectral resolution which is defined, in principal, only by the residual Doppler broadening because of the finite angle divergence of molecular beam. Probably, this is the very way to approach the solving of the principal problem - the advent of a physical method on detection and identification of trace amounts of molecular impurities which can be competitive only with human and animal senses of smell.

0 10. Spatial Localization of Molecular Bonds Selective action of laser radiation on molecular bonds opens up a principal possibility for spatial localization of particular molecular bonds, that is, for macromolecular “mapping”. The idea of this approach can be understood from the so-called photoelectronic (photoionic) laser micro[1975a]) which is schematically shown in Fig. 28. Unlike scope (LETOKHOV the common field-electron or field-ion MCJLLER[1969] projectors, an electron or an ion breaks away selectively here from the molecule under the selective action of laser radiation, rather than because of nonselective field-induced ionization by a strong electric field. The only function left to

I, D 101

SPATIAL LOCALIZATION OF MOLECULAR BONDS

r

+

DC

63

-

VoI 1 oge

~Mocr0rn01c:ule

' - L a s e r U l t r o s h o r t Pulses

,-104H n=iocm

Fig. 28. Possible laser- ion microscope for spatial localization of molecular bonds (from LETOKHOV [1975a1).

the electric d.c. field is to transport electrons or ions along radial paths to the projector screen. Selective photoionization of certain molecular bonds in a macromolecule, which is adsorbed o n the needle top of the projector, can be done by the multi-step scheme, under the action of several picosecond laser pulses at special frequencies. In the case of a selective breaking away of an electron it is possible to attain a resolution of about 25 A, which is limited by such fundamental causes as the presence of a tangential velocity component of an emitted electron and the principle of uncertainty. In some cases after the electron breaks away the resultant positive molecular ion becomes unstable and spontaneously gives off the proton. If we change the polarity in the projector, it is possible to transport electrons instead of protons to the screen. The spatial localization of the ionization point for protons must be (m,/m,)lR = 40 times higher than for photoelectrons. Such a laser photoionic microscope may have a resolution quite sufficient to resolve atomic details in molecular structures. Further increase in the resolution of laser photoemission microscopy is also possible due to selective photoionization of a molecule to heavier molecular ions. The idea of this new approach to atomic-resolution microscopy is based on the combination of two important properties which usually belong to quite different methods. Corpuscular (electronic, for instance) microscopy ensures a high spatial resolution if the particle has a high energy. But in

64


[I, 8 10

this case the “contrast” of image is lost automatically because a variation in the particle energy due to its interaction with an object being observed is difficult to distinguish against the background of the higher initial energy. Vice versa, as the particle energy drops, the image “contrast” may increase considerably and spectral selectivity becomes possible in observing the image details. But in this case the spatial resolution decreases. The method of selective photoionization makes it possible to combine the high selectivity (or contrast) of an “optical channel” with the high spatial resolution of a “corpuscular (ionic) channel”. Determination of the nucleotide sequence in a DNA molecule, which carries hereditary information on every individual organism, is one of the most important applications of the laser ion microscope. This problem consists in selective break-away of a proton (or of a heavier molecular ion), either from the base pair A-T or G-C. Of course, we may try to realize the method based on two-step selective I R - W excitation of an electron state through intermediate vibrational state of a particular nucleotide and subsequent ionization of the electron-excited state. For good distinction between the pair of bonds A-T and G-C it is essential that the selectivity of this process is adequate. Particular attention should be drawn to the principal possibility of preliminary chemical “marking” of either base pair, to make selective excitation and ionization easier. For KOHLHAGE and ZILLIG [1961]) that one instance, it is known (WERWOERD, of the bases of DNA (custosine) selectively reacts with hydroxylamine. The reaction is followed by changes in the UV spectrum of cytosine absorption, the value of which is much larger than that of the typical difference between the spectra of the bases (see Fig. 21). In this case selective two-step photoionization through electron-excited state of the product of cytosine-hydroxylamine reaction may be sufficient. When the magnification coefficient of the laser ionic projector M = 3 x lo5 and the screen dimension R = 10 cm, there will be imaged a section of linear DNA chain with about lo3 nucleotides (adjacent nucleotides are spaced 3.3 A apart), i.e. a length of about 3300 A. To record the sequence of nucleotides in long DNA chains, we shall need, of course, successive projections and “sewing” the images of subsequent sections. For successful realization of the laser ionic microscope for observing biological molecules one will have to investigate a number of rather difficult problems: 1) selective excitation of macromolecules absorbed on the surface, 2) search for schemes of selective dissociative ionization resulting in breakaway of protons of heavier molecular ions, 3 ) spatial

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scanning of microscope needle point along the chain of a macromolecule, etc. The solution of the problem of direct observation of nucleotide sequence in genes, including those of man, would open up such large possibilities in heredity study and control that this field of selective laser photophysics seems to me to be worthy of our closest attention.

References Recent conference proceedings [1] Laser Spectroscopy, Proc. Second Intern. Conf. (France, Megeve, June 23-27, 1975), eds. S. Haroche, J. C. Pebay-Peyroula, T. W. Hansch and S. E. Harris (Springer-Verlag, Berlin-Heidelberg-New York, 1975). [2] Tunable Lasers and Applications, Proc. Intern. Conf. (Loen, Norway, 6-11 June, 1976), eds. A. Mooradian, T. Jaeger and P. Stokseth (Berlin-Heidelberg-New York, Springer, 1976). [3] Multiphoton Processes, Proc. Intern. Conf. (USA, Rochester, 6-9 June 1977), eds. J. Eberly and P. Lambropulos (Wiley-Interscience, 1978).

111, J. P., J. H. BIRELY,C. D. CANTRELL I11 and D. C. CARTWRIGHT, 1976, in: ALDRIDGE Laser Photochemistry, Tunable Lasers, and Other Topics, Physics of Quantum Electronics Vol. 4, eds. S. F. Jabobs, M. Sargent 111, M. 0. Scully and C. T. Walker (Addison-Wesley Publ. Co., Reading) p. S7. 1972, Pis’ma Zh. Eksp. Teor. AMBARTZUMIAN, R. V., V. M. APATINand V. S. LETOKHOV, Fiz. (Russian) 15, 336 [Sov. Phys.-JETP Lett. 15, 2371. R. V., V. M. A~ATIN,V. S. LETOKHOV, A. A. MAKAROV, V. I. MISHIN,A. AMBARTZUMIAN, A. PURETZKIIand N. P. FURUKOV,1976, Zh. Eksp. i Teor. Fiz. (Russian) 70, 1660. and V. I. MISHIN,1975, KvanAMBARTZUMIAN, R. V., V. M. APATIN,V. S. LETOKHOV tovaya Eiektronika (Russian) 2, 337 [Sov. J. Quant. Electron. 5, 1911. AMBARTZUMIAN, R. V., G. I. BEKOV,V. S. LETOKHOVand V. I. MISHIN,1975, Pis’ma Zh. Eksp. Tea. Fiz. (Russian) 21, 598 [Sov. Phys.-JEFT Lett. 21, 2791. V. S. DOLJIKOV,V. S. LETOKHOV and E. A. AMBARTZUMIAN, R. V., N. V. CHEKALIN, RYABOV,1974, Chem. Phys. Lett. 25, 515. G. N. AME~ARTZUMIAN, R. V., N. V. CHEKALIN,Yu. A. GOROKHOV, V. S. LETOKHOV, MAKAROV and E. A. RYABOV, in [l], p. 121. V. S. LETOKHOV, G. N. AMBARTZUMIAN, R. V., N. P. FURZIKOV, Yu. A. GOROKHOV, MAKAROV and A. A. P~RETZKII, 1976, Optics Comm. 18, 517. R. V., N. P. FURZIKOV,Yu. A. GOROKHOV, V. S. LETOKHOV, G. N. AMBARTZUMIAN, MAKAROV and A. A. -I, 1977, Optics Letters 1,22. S. L. GRIGOR’EV, V. S. LETOKHOV, G. N. AMEIARTZUMIAN, R. V., Yu, A. GOROKHOV, MAKAROV, Yu. A. MALININ, A. A. PURETZKII, E. P. FILIPPOV and N. P. FURZIKOV, 1977, Kvantovaya Electronika (Russian) 4, 171. V. S. LETOKHOV and G. N. MAKAROV, 1975a, AMBARTZUMIAN, R. V., Yu. A. GOROKHOV, Pis’ma Zh. Eksp. Teor. Fiz. (Russian) 21, 375 [Sov. Phys.-JETP Lett. 21, 1711. AMBARTZUMIAN, R. V., Yu. A. GOROKHOV, V. S. LETOKHOV and G. N. MAKAROV, 1975b, Pis’ma Zh. Eksp. Teor. Fiz. 22, 96.

66


[I

AMBARTZUMIAN,R. V., Yu. A. GOROKHOV, V. S. LETOKHOV and G. N. MAKAROV, 1975c, Zh. Eksp. Teor. Fiz. (Russian) 69, 1956 [Sov. Phys.-JETP 42, 9931. ~ A R T Z U M I A N ,R. V., YU. A. GOROKHOV, V. S. LETOKHOV, G. N. MAKAROV and A. A. PURETZKII, 1976, Zh. Eksp. Teor. Fiz. (Russian) 71, 440. . ~ E I A R T Z U M IR. A NV., , Yu. A. GOROKHOV, V. S . LETOKHOV,G. N. MAKAROV, A. A. PURETZKII and N. P. FURZIKOV,1976, Pis’ma Zh. Eksp. Teor. Fiz. (Russian) 23, 217 [Sov. Phys.-JEPT Lett. 23, 1941. AMBARTZUMIAN, R. V., Yu. A. GOROKHOV, V. S. LETOKHOV, G. N. MAKAROV, E. A. RYABOVand N. V. CHEKALIN,1976, Kvantovaya Elektronika (Russian) 2, 2197. and V. S. LETOKHOV, 1971, Pis’ma Zh. Eksp. Teor. ~ARTZUMIA R.Nv ,. , V. P. KALININ Fiz. (Russian) 13, 305 [Sov. Phys.-JEFT Lett. 13, 2171. 1972, Appl. Opt. 11,354. AMBARTZUMIAN, R. V. and V. S. LETOKHOV, 1977, in: Chemical and Biochemical ApplicaAMBARTZUMIAN, R. V. and V. S. LETOKHOV, tions of Lasers, Vol. 3. ed. C. B. Moore (New York, Academic Press) p. 167. AMBARTZUMIAN, R. V., V. S. LETOKHOV,G. N. MAKAROV,A. G. PLATOVA,A. A. PURETSKIIand 0. A. TUMANOV,1973, Zh. Eksp. i Teor. Fiz. (Russian) 64, 770 [Sov. Php.-JETP 37, 3921. AMBARTZUMIAN, R. V., V. S. LETOKHOV, G. N. MAKAROVand A. A. PURETSKII,1972. Pis’ma Zh. Eksp. Teor. Fiz. (Russian) 15, 709 [Sov. Phys.-JETP Lett. 15, 5011. AMBARTZUMIAN, R. V., V. S. LETOKHOV,G. N. MAKAROV and A. A. PURETSKII, 1973. Pis’ma Zh. Eksp. Teor. Fiz. (Russian) 17, 91 [Sov. Phys.-JEW Lett. 17, 631. AMBARTZUMIAN, R. V., V. S. LETOKHOV, G. N. MAKAROV and A. A. PURETZKII, 1974, i n : Laser Spectroscopy, eds. R. G. Brewer and A. Mooradian (New York, Plenum Press) p. 611. AMBARTZUMIAN, R. V., V. S. LETOKHOV, G. N. MAKAROV and A. A. PURETZKII, 1975, Zh. Eksp. i Teor. Fiz. (Russian) 68, 1736. E. A. RYABOV and N. V. CHEKALIN, 1974, Pis’ma AMBARTZUMIAN, R. V., V. S. LETOKHOV, Zh. Eksp. Teor. Fiz. (Russian) 20, 597 [Sov. Phys.-JETP Lett. 20, 2731. and V. S. LETOKHOV, 1977, Chem. Phys. ANDREEV,S. V., V. S. ANTONOV,I. N. KNYAZEV Lett. 45, 166. ANDREEV,S. V., V. S. ANTONOV,I. N. KNYAZEV, V. S. LETOKHOV and V. G. MOVSHEV, 1975, Phys. Lett. 54A, 91. V. S. LETOKHOV and V. G. MOVSHEV,1977, Zh. Eksp. i ANTONOV, V. S., I. N. KNYAZEV, Teor. Fiz. (Russian) 73, 1325. ARNOLDI,D., K. KAUFMANN and J. WOLFRUM,1975, Phys. Rev. Lett. 34, 1597. BADGER,R. M. and J. W. URMSTON,1930, Proc. Nat. Acad. Sci. USA. 16, 808. BAGRATASHVILI, V. N., V. YU. BARANOV,E. P. VELIKHOV,s. A. KAZAKOV,YU. R. KOLOMIISKY, V. S. LETOKHOV, V. G. NIZ’EV,V. D. PISMENNY, E. A. RYABOVand A. I. STARODUBTZEV, 1977, Appl. Phys. 14,217. V. I. MISHINand V. A. SEMCHISHEN, 1976, Chem. Phys. BALIKIN,V. I., V. S. LETOKHOV, 17, 111. A. H. KUNG,C. BARANOVSKI, A. P., A. CABELLO, J. H. CLARK,Y. HAAS,P. L. HOUSTON, B. MOORE,J. REILLY,J. C. WEISSHAAR and M. B. ZUGHUL,1976, in [2], p. 404. V. A. ISAKOV,E. P. MARKIN,A. N. BASOV,N. G., E. M. BELENOV,L. K. GAVRILINA, Ow+EvsKn, V. I. ROMANENKO and N. B. FERAPONTOV, 1974, Pis’ma Zh. Eksp. Teor. Fiz. 20, 607 [Sov. Phys.-JEFT Lett. 20, 2771. BASOV,N. G., E. M. BELENOV,V. A. ISAKOV,Yu. S. LEONOV,E. P. MARKIN,A. N. 1975, Pis’ma Zh. Eksp. Teor. Fiz. 22, 221 [Sov. ORAEVSKII and V. I. ROMANENKO, Phys.-JETP Lett. 22, 1021. BASOV,N. G., E. M. BELENOV,V. A. ISAKOV,E. P. MARKIN,A. N. ORAEVSKII,V. I.

I1

REFERENCES

67

ROMANENKO and N. B. FERAPONTOV, 1975, Kvantovaya Elektronika (Russian) 2, 938 [Sov. J. Quant. Electr. 5, 5101. N. I. SOROKIN and Yu. N. MOLIN,1974, Pis’ma Zh. BAZHIN,N. M., G. I. SKUBNEVSKAYA, Eksp. Teor. Fiz. (Russian) 20, 41 [Sov. Phys.-JETP Lett. 20, 181. S. A,, V. S. LETOKHOV, A. A. MAKAROV and V. A. SEMCHISHEN, 1973, Pis’ma BAZHW~IN, Zh. Eksp. Teor. Fiz. 18, 515 [Sov. Phys.-JETP Lett. 18, 3031. BEKOV,G. I., V. S. LETOKHOV and V. I. MISHIN,1977a, Zh. Eksp. Teor. Fiz. (Russian) 73, 157. and V. 1. MISHIN,1977b, Optics Comm. 23, 85. BEKOV,G. I., V. S. LETOKHOV and V. I. ROMANENKO, 1973, Pis’ma BELENOV, E. M., E. P. MARKIN, A. N. ORAEVSKII Zh. Eksp. Teor. Fiz. (Russian) 18, 196 [Sov. Phys.-JETP Lett. 18, 1161. BILLINGS, B. H., W. J. HITCHCOCKand M. ZELIKOFF,1953, J. Chem. Phys. 21, 1762. BLOEMBERGEN, N., 1975, Optics Comrn. 15,416. BLOEMBERGEN, N., C. D. CANTRELLand D. M. LARSEN,1976, in [2], p. 162. BRAWN, J. I., T. J. O’LEARYand A. L. SCHAWLOW, 1974, Optics Comrn. 12, 223. Yu. R. KOLOMIISKY, V. S. LETOKHOV, V. N. LOKHMAN CHEKALIN, N. V., V. S. DOUIKOV, and E. A. RYABOV,1976, Phys. Lett. 59A, 243. CHEKALIN, N. V., V. S. DOLJIKOV, V. S. LETOKHOV, V. N. LOKHMAN and A. N. SHIBANOV, 1977, Appl. Phys. 12, 191. CLARK, J. H., Y. HAAS,P. L. HOUSTON and C. B. MOORE,1975, Chem. Phys. Lett, 35,82. CLARK,L. B. and I. TINOCO,1965, J. Arner. Chem. SOC.87, 11. and R. N. ZARE, 1976, J. Chern. Phys. 63, 5503. DATTA,S.,R. W. ANDERSON A. V. KISELEV,V. J. LYGIN,N. A. NAMIOT,A. J. DJIDJOEV,M. S., R. V. KHOKHLOV, and B. I. PROVOTOROV, 1976, in [2], p. 100. Osrpov, V. J. PANCHENKO and F. LEGAY,1972, Phys. Rev. Lett. 29, 145. DUBOST,H., L. ABOUAF-MARGUIN R. R. FREEMAN and D. KLEPPNER, 1975, Phys. Rev. Lett. DUCAS,T.W., M. C. LITTMAN, 35, 366. and S. DONDES,1973, J. Phys. Chem. 77, 878. DUNN,O., P. HARTECK FREIJND,S. M. and J. J. RITI-ER, 1975, Chem. Phys. Lett. 32, 255. GIBERT,R., 1963, J. Chem. Phys.-Chem. Biol. 60,205. K. S., N. V. KARLOV, A. N. ORLOV,R. P. PETROV,Yu. N. PETROV and A. GOCHELASHVILI, M. PROKHOROV, 1975, Pis’ma Zh. Eksp. Teor. Fiz. 21, 640 [Sov. Phys.-JETP Lett. 21, 3021. Y. R. SHENand Y. T. LEE, GRANT,E. R., P. A. SCHULTZ, Aa. S. SUDBO,M. J. COGGIOLA, 1977, in [3], p. 359. GUNNING, H. E. and 0. P. STRAUSZ, 1963, Adv. Photochem. 1, 209. HARTLEY, H., A. 0. PONDER, E. J. BOWEN and T. R. MERTON, 1922, Philos. Mag. 43,430. HURST,G. S., M. H. NAYFEHand J. P. YOUNG,1977, Appl. Phys. Lett. 30, 229. and M. C. RICHARDSON, 1973, Can. J. ISENOR,N. R., V. MERCHANT,R. S. HALLSWORTH Phys. 51, 1281. ISENOR,N. R. and M. C. RICHARDSON, 1971, Appl. Phys. Lett. 18, 224. IVANOV, L. N. and V. S. LETOKHOV,1975, Kvantovaya Elektronika (Russian) 2, 585 [Sov. J. Quant. Electron. 4, 13911. JANES, G. S., H. K. FORSENand R. H. LEVY,1977, Research Report N442, Avco Everett Res. Lab. C. T. PIKE,R. H. LEVYand L. LEVIN,1975, IEEE J. Quant. Electr. JANES,G. S., I. ITZKAN, QE-11. 101 D. C. T. PIKE,R. H. LEVYand L. LEVIN,1976, IEEE J. Quant. Electr. JANES,G. S., I. ITZKAN, 12, 111. KARL,R. R. and K. K. INNES,1975, Chem. Phys. Lett. 36,275.

68


[I

KARNAUKHOV, V. A. and S. M. POLIKANOV, 1977, Pis’ma Zh. Eksp. i Teor. Fiz. (Russian) 25, 328. KING,D. S. and R. M. HOCHSTRASSER, 1975, J. Amer. Chem. SOC.97,4760. W A Z E V , I. N., V. S. LETOKHOV and V. G. MOVSHEV, 1975, IEEE J. Quant. Electr. 11, 805.

K W , W. and H. MARTIN,1932, Natunviss. 20,772. KUHN,W. and H. MARTIN,1933, Z. Phys. Chem. Abt. B21,93. LADIK, Janos, 1972, Quantenbiochemie fur Chemiker und Biologen (Akademiai Kiado, Budapest). L A M m , M., H. J. DEWEY,R. A. KELLER and J. J. -R, 1975, Chem. Phys. Lett. 30, 165. LAUEIEREAU, A. and W. KAISER,1977, in: Chemical and Biochemical Applications of Lasers Vol. 2, ed. C. B. Moore (Academic Press, N.Y.) p. 87. LAUBEREAU, A., A. SEILMEIER and W. KAISER,1975, Chem. Phys. Lett. 36, 232. LAUEIEREAU, A., D. VON DER LINDEand W. KAISER,1972, Phys. Rev. Lett. 28, 1162. LEONE,S. R. and C. B. MOORE,1974, Phys. Rev. Lett. 33, 269. V. S., 1972, Chem. Phys. Lett. 15, 221. LETOKHOV, V. S., 1973, Optics Comm. 7, 59. LETOKHOV, LEMKHOV,V. S., 1975a, Kvantovaya Elektronika (Russian) 2,930; Phys. Lett. 51A, 231. LETOKHOV, V. S., 1975b, Spectroscopy Lett. 8, 697. LETOKHOV, V. S., 1975c, J. of Photochemistry 4, 185. LEMKHOV,V. S., 1976a, in ref. [2], p. 122. LETOKHOV, V. S., 1976b, Uspekhi Fiz. Nauk (Russian) 116, 199. LETOKHOV, V. S., 1977a, Physics Today 30, N5, 23-32. LETOKHOV, V. S., 1977b, Soviet Patent N 65743 (Appl. in 30.03 1970). LETOKHOV, V. S., 1977c, Soviet Patent N 65744 (Appl. in 30.03. 1970). L m o m o v , V. S., 1977d, in [3] p. 331. LETOKHOV, V. S., 1977e, in: Frontiers in Laser Spectroscopy, Proc. Les-Houches Summer School on Theoretical Physics, July 1975, France Vol. 2. (North-Holland Publ. Co.) p. 771. 1971, IEEE J. Quant. Electr. QE-7, 305. LETOKHOV, V. S. and R. V. AMBARTZUMIAN, V. S. and V. P. CHEBOTAYEV, 1977, Nonlinear Laser Spectroscopy (SpringerLETOKHOV, Verlag, Berlin-Heidelberg-New York). LETOKHOV, V. S. and A. A. MAKAROV, 1972, Zh. Eksp. Teor. Fiz. (Russian) 63,2064 [Sov. Phys.-JETP 36, 10911. LETOKHOV, V. S., V. I. M s m and A. A. PURETZKII,1977a, Progress on Quantum Electronics Vol. 6, eds. J. H. Sanders and S. Stenholm. L m o m o v , V. S., V. I. MISHINand A. A. PURETZKII,1977b, in: Chemistry of Plasma Vol. 4 (Russian), (Atomizdat, Moscow) p. 3. LETOKHOV, V. S. and C. B. MOORE,1976, Kvant. Elektron (Russian) 3,248; 3,485 [Sov. J. Quant. Electron. 6, 129; 6, 2591. L ~ O K H O V. V , S. and C. B. MOORE,1977, in: Chemical and Biochemical Applications of Lasers Vol. 3, ed. C. B. Moore (New York, Academic Press) p. 1 . LETOKHOV, V. S., E. A. RYABOV and 0. A. TUMANOV, 1972a, Optics Comm. 5, 168. 1972b, Zh. Eksp. Teor. Fiz. LETOKHOV,V. S., E. A. RYABOVand 0. A. TUMANOV (Russian) 63, 2025 [Sov. Phys.-JETP 36, 10691. 1975, Dokl. Akad. Nauk SSSR 222, 1071 [Sov. LETOKHOV, V. S. and V. A. SEMCHISHEN, Phys.-Dokl. 20, 4231; Spectroscopy Lett. 8, 263. LEVY,R. H. and G. S. JANES, 1973, USA Patent N3. 722. 519. L m , M. C., M. L. ZIMMERMAN and D. KLEPPNER, 1976, Phys. Rev. Lett. 37, 486.

I1

REFERENCES

69

LIU, D., S. DATTA and R. N. ZARE, 1975, J. Am. Chem. SOC.97, 2557. L r m , G., S. DONDESand P. HARTECK,1966, J. Chem. Phys. 44, 4052. LOWDIN,P. O., 1964, in: Electronics Aspects of Biochemistry, ed. B. Pullman (Academic Press, New York) p. 167. LOWIN, P. O., 1968, Adv. Quant. Chem. 2, 213. LYMAN,J. L. and R. J. JENSEN,1972, Chem. Phys. Lett. 13, 431. MANUCCIA, Y. J. and M. D. CLARK, 1976, Appl. Phys. Lett. 28, 372. MARLING,J. B., 1975, Chem. Phys. Lett. 34, 84. MAYER,S. W., M. A. KWOK,R. W. F. GROSSand D. L. SPENCER,1970, Appl. Phys. Lett. 17, 516. M c C w , S. L. and E. L. HAHN, 1967, Phys. Rev. Lett. 18, 908. MCCALL,S. L. and E. L. m,1969, Phys. Rev. 183, 457. MERTON,T. R. and H. HARTLEY,1920, Nature (London) 105, 2630. S., 1932, Z. Phys. 78, 826; 72, 844. MROZOWSKI, E. W. and TIENTZONTSONG,1969, Field Ion Microscopy (Amer. Elsevier Publ. M-ER, Co., Inc., New York). NEBENZAHL,I. and M. LEVIN,1973, FRG Patent, N2. 312. 194. NOGUCHI,N. and Y. IZAWA.1974, Progress Report X, Osaka University, p. 63. C. A. MAY and R. PAISNER,J. A., L. R. CARLSON,E. F. WORDEN,S. A. JOHNSON, W. SOLARZ,1976, Preprint UCRL-78034, Lawrence Livermore Laboratory. PERTEL,R. and H. E. GUNNING,1959, Can. J. Chem. 37, 35. POTAPOV, V. K., V. G. MOVSHEV,V. S. LETOKHOV,I. N. KNYAZEVand T. U. EVLASHOVA, 1976, Kvantovaya Elektronika (Russian) 3, 2610. ROCKWOOD, S. and S. W. RABIDEAU, 1974, IEEE J. Quant. Electr. QE-10, 789. SHIMODA,K., editor, High Resolution Laser Spectroscopy, Topics in Appl. Phys. Vol. 13 (Springer-Verlag. Berlin-Heidelberg-New York, 1976). STUKE, M. and F. P. SCHAFER,1977, Chem. Phys. Lett. 48, 271. Susr, H., in: Structure and Stability of Biological Micromolecules, eds. S. N. Timasheff and G. D. Fasman (Marcel Dekker Inc., New York, 1969). 1967, Science 157,40. TIFFANY,W. B., H. W. Moos and A. L. SCHAWLOW, TREANOR,C. E., J. W. RICH and G. G. REHM,1968, J. Chem. Phys. 48, 1798. and B. B. SNAVELY,1974, IEEE. Quant. TUCCIO,S. A., J. W. DUBRIN,0. G. PETERSON Electr. QE10, 790. TuCCIO, S. A., R. J. FOLEY,J. W. DUBRIN and 0. KRIKORIAN, 1975, IEEE J. Quant. Electr. QE-11, 101 D. WERWOERD, D. W., H. KOHJ-HAGEand W. ZILLIG,1961, Nature 192, 1038. YEUNG,E. S. and C. B. MOORE, 1972, Appl. Phys. Lett. 21, 109. YEUNG,E. S. and C. B. MOORE, 1973, J. Chem. Phys. 58, 3988. ZARE, R. N., 1976, Invited Report at IX Intern. Conf. on Quantum Electronics, Amsterdam, Netherlands. ZUBER,K., 1935, Nature (London) 136, 796; Helv. Phys. Acta 8, 487. ZUBER,K., 1936, Helv. Phys. Acta 9, 285.


E. WOLF, PROGRESS IN OPTICS XVI @ NORTH-HOLLAND 1978

RECENT ADVANCES IN PHASE PROFILES GENERATION BY

J. J. CLAlR and C. I. ABITBOL Institut d’optique, Universite P. et M. Curie, Tour 13, 3 ?me Etage, 4 Place Jussieu, 75230 Paris Cedex 05, France

CONTENTS PAGE

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

73

$ 2 . PRINCIPLES AND TECHNIQUES USED IN PHASE PROFILES GENERATION . . . . . . . . . . . .

74

$ 3 . OPTICAL MATERIAL FOR PHASE INFORMATION STORAGE. . . . . . . . . . . . . . . . . . .

96

0 1 . INTRODUCTION . . .

5 4. PHASE PROFILES APPLICATIONS - CURRENT AND FUTURE PROSPECTS . . , . . . . . . . . . . . 106

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

113

COMPLEMENTARY REFERENCES . . . . . . . . . .

114

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

116

CONCLUSION . .

REFERENCES . . . . .

§

1. Introduction

The interest in optical components whose phase o r complex transmittance is nonuniform, began many years ago. It seems Descartes and Kepler were the first to consider the advantages of aspherical surfaces. A report of Descartes describes this subject in the eighth discourse of “La Dioptrique” published in 1638. He entitled it: “Des figures qui doivent avoir les corps transparents pour detourner les rayons en toutes les faGons qui servent B la vue”. Descartes calculated the shape of a surface giving a rigourous stigmatism. Since this approach many methods of calculation and fabrication have been developed and new research fields have been introduced regarding the properties of particular surfaces. Whether it concerns stellar observation, testing or correction of any optical surface, o r optical or digital information processing, the solution consists primarily of creating a particular complex wavefront from a phase profile or a hologram. There are several methods of realisation of optical components capable of modifying the phase of an incident wavefront. The purpose of this review article is to present the current and most significant methods of fabrication of phase profiles used in the more recent fields of application of optics. However, to enable the reader to have an historical and global view of the subject, it appeared interesting to us to outline the different steps which led to the current research and applications. Any wavefront propagating in any medium is depicted as an amplitude and phase distribution. If we wish to act on the wave, it is necessary to modify these two parameters. Holography offers a solution for coding the phase by introduction of a reference wavefront, but new coding processes enables one to act directly on the phase by creating a transparent relief or a local transformation of the electro-optical properties of a material. We shall distinguish two main parts in the following report. The first is based on the coding of the phase modulation by introducing a reference wave (characterizing the “off -axis” diffraction phase profiles), while the second part will concern the creation of the phase relief directly on the plate (giving the “on-axis’’ image).

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The third section will report the new materials used to obtain a phase or complex distribution, and describe the respective recording phenomenon involved. The last section will give an outline of the main current applications of phase retardation used for information storage and transmission both in visible and infrared range, signal processing, and for performing particular mathematical operations.

0 2. Principles and Techniques Used in Phase Profiles Generation Until recently, the production of phase profiles, aspherics or any other optical element of revolution, was based principally on different processes involving more or less the wear or the transformation of a reference profile, generally in glass. With the introduction of the laser, new capabilities of computers, and recent photosensitive materials, many methods of fabrication of synthetic filters o r quantified phase profiles have been developed. These latter have considerably simplified the previous fabrication processes whose main disadvantage was the cost to quality-dimension ratio for a specified profile. We report in this section the various existing methods and shall establish a comparative study of their respective capabilities. 2.1. CLASSICAL METHODS BASED ON THE TRANSFORMATION OF A REFERENCE PROFILE

Several methods have been proposed for the construction of aspherics, e.g. the Schmidt corrector plate. They can be classified into three groups depending on the way they are performed; viz. (i) by matter removal, (ii) by matter supply, (iii) by plastic deformation of a spherical surface of specified radius.

2.1.1. Matter removal methods These can be subdivided into two groups according to the manner in which the polishing is performed i.e.; either by hand, with local retouching or by mechanical shaping.

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PRINCIPLES AND TECHNIQUES

75

Local removal by hand This is an empirical method mainly applicable to aspherical mirrors and astronomical objectives. Although it is a very costly and slow method, a high precision may be achieved. Mechanical shaping devices Several methods have been proposed and many machines are still fabricated, e.g. a machine to realize 2nd order surfaces by generation; the machines of Descartes, and of Schubert-Werke in 1888, Gullstrand’s method of 1919, Dourneau’s apparatus of 1945 for generating parabolas, etc. J . P. MARIOGE[1965] of the Institute of Optics of Paris produced very accurate aspherical surfaces without Petzval’s limitation. More recently, a judicious choice of the trajectory and the ratio of the diameter of the tool to the diameter of the sample, permitted the production of a machine-tool with corrected wear. The local wear is almost proportional to the number of passages of the tool at each point. When the sample to be deformed has a large diameter, a very accurate crank mechanism is used for guiding the tool; such a machine has been constructed by M. D. Loomis at Tucson (Arizona). The trajectories are various and the statistical path calculations lead to many improvements of the method. Regarding the precision, some authors have evaluated the fabricating errors and established mathematical models giving the best trajectory for a specified profile, with respect to the material and the cinematic parameters of the apparatus and the tool (SHANNON and WAGNER [1974]). We mention also a method using the electrostatic conductivity of a substance spread onto the final profile to detect the shaping error by a measurement of its capacitance. Finally real-time and two-wavelength holography are shown to permit more accurate interferometrical testing. Also related to the mechanical shaping devices, we mention the machines of Ross of 1939, Zeiss, and Philips, where the surface is eroded by application of a mould having the desired shape. Some techniques use also the shaping of a profile previously deformed by mechanical constraints. After the removal is achieved and the mechanical forces withdrawn, the desired profile is obtained. This latter process has been used by B. Schmidt for the manufacture of his plates. Schmidt fabricated his first plates as follows (Fig. 2.1). Placing a plane parallel plate on the top of a cylindrical cavity he produced a vacuum underneath. He then polished the upper surface of the deformed plate with a spherical tool. At the atmospheric pressure the plate had the

76

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RECENT ADVANCES IN PHASE PROFEES GENERATION

“1, § 2

I

--__ _---

F Fig. 2.1. Principle of the plastic deformation used by B. Schmidt with utilization of one vacuum cavity.

Fig. 2.2. LEMAITRE’S method [1972], using two cavities instead of one.

desired profile. Recently LEMAITFE [ 19721 at the Marseille observatory, used the same process but introduced a second cavity (Fig. 2.2).

2.1.2. Matter supplying methods It is also possible to transform spherical surfaces by supplying transparent material for transmitting optical components, and opaque material for reflecting components, such as a mirror. This material can be deposited either: (i) by moulding, with respect to the shrinking due to the cooling or to the material phase transformations (SCHLJLZ [1947a]); or and GAVIOLA [ 19361 were the (ii) by evaporation under vacuum. STRONG first to employ this technique. For correcting the defects of a mirror or to transform its surface into an aspheric, he deposited thin films of aluminium onto particular areas of the profile. A rotating mask placed in front of the sample assured the localization of the successive deposits. This method was limited to a maximum thickness of 3 to 6 microns, because for thicker layers, aluminium crystallizes resulting in a fall of the reflection coefficient. SCHULZ [1947b] solved the problem of thick layers by using the “Fresnel zones” principle of modulation (Fig. 2.3). However, the precision thus obtained is limited by the fact that a certain space must be left between the sample and the mask for the rotation of one of

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77

Fig. 2.3. Depositing of thin film by evaporation for a continuous or Fresnel zones profile (SCHULZ [1947b]).

two elements. Other evaporating devices have been proposed by [1955]. DOBROWOLSKY and WEINSTEIN

2.1.3. Thermal deformation methods Here one uses an optical surface, generally spherical, whose curvature has been previously determined and which is deformed by heating. One can, for example, heat a spherical lens previously set on a heating ring. By a judicious choice of the radii of curvature and thickness of the lens, temperature and the manner of heating, it is possible to obtain different [1974]). This method has been used by Zeiss aspherical shapes (EMMONY [1952] for his 1.96 m astronomical mirror. The Cie in 1909 and COUDERC sample can also be retouched while heating, but strict precautions must be taken for the dissipation of heat.

2.1.4. Conclusion This non-exhaustive review seems to indicate that general mechanical methods of shaping are characterized by the good optical quality of the phase profiles obtained; however, this quality is achieved only by a relatively long time of fabrication and a very high cost. These two parameters contribute to make the mechanical methods less competitive

7x


[II, Q 2

than the methods of holography and synthetic optics which we summarize below. We should differentiate here between wavefront reconstruction devices formed with experimental wave interference phenomena and those formed with the aid of a computer. In general any physical wave phenomena, e.g., sound waves, microwaves, light waves and perhaps even X-rays, can be used to form holograms. We shall lump the formation of these physical holograms under the term “holography”, and use the terms “digital holography” and “computer holography” to denote the formation of holograms constructed by computer techniques ; these holograms are frequently reduced to holograms suitable for optical reconstruction.

2.2. OFF-AXIS PHASE PROFILES PHOTOFABRICATION BY COHERENT LIGHT

The introduction of lasers some twenty years ago, has shown coherent light sources can be a suitable phase information carrier for optical processing. The first holograms obtained in coherent light were applied to spatial filtering systems (ZERNIKE [19351, MARI~CHAL and CROCE[1953], TSUJUICHI [1963], VANDER LUGT[1968]). One of the simplest approaches is the binary filter whose principle is still broadly used in data processing. A binary filter has an amplitude transmittance value equal to 1 or 0. Binary filters are particularly easy to construct because they are simply apertures and stops. Nevertheless, they can perform many important operations. Binary filters are useful for detecting extended periodic signals in the presence of random noise. The required filter passes the diffracted light at those positions in the frequency domain where the spectral orders of the signal occur. Because the noise spectrum is scattered more or less uniformly in the frequency domain, the signal to noise ratio is increased [1960]). Binary filters can be used to increase the considerably (CUTRONA contrast of a photograph by partially or completely eliminating the zero frequency term in the spectrum (O’NEILL[1956]). Many techniques to produce these binary filters have been used, since it is easy to record an interference fringes pattern, in which the phase is coded by different spacing of the fringes. One of the many possible interferometric filter-generating systems is shown in Fig. 2.4. If the total light amplitude distribution G(p,q ) in plane P2 is recorded,

11, § 21


point source of monochromatic hqht

Fig. 2.4. Holographic realization of phase filter. On plane Pz we record the Fourier transform of S(x, y).

the effective transmittance of filter is proportional to

IG(p, q>?= a2+lS(p,q)12+aS(p,s>* exp(-jpb)+aS*(p, 41. exp(jqb), where a is the amplitude of the reference beam at the plane Pz, b is the distance between the centre of the signal and the point reference beam, and S ( p , q ) is the Fourier transform of the signal s(x, y). The last term of this equation is proportional to the desired function, and although this term is mingled with the other terms, in all cases the information is diffracted “off-axis’’ due to the linear phase factor exp (jpb). The value of b must be large enough to avoid overlap of the output images. This effect reduces the diffraction efficiency of holographically generated phase profiles. However, the bleaching technique can improve this efficiency by keeping the photographic emulsion on the phase information only, but the related processing parameters are in this case critical. Complex valued spatial filters can also be constructed by using hard-clipped functions. One reason for doing so is that we can use a digital computer to generate the Fourier spectrum of a given signal, digitize the spectrum, modulate a spatial carrier frequency and automatically plot the required filter so that it can be recorded photographically.

2.3. COMPUTER GENERATED PHASE PROFILES

In ordinary interferometric holography, the hologram is made by recording the interference pattern of an object complex wave and a phase coherent reference wave. Upon illumination of the hologram, an image of the original object can be reconstructed. In synthesizing holograms by a

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computer, no physical object exists. Instead one starts with a mathematical description of the object, usually in the form of an array of phase coherent point sources. The complex wave-front from such an array on a fixed plane some distance away can be computed. If a transparency whose complex transmittance is equal to the computed wavefront on that plane can be synthesized then, upon illumination of such a transparency (or hologram) by a plane wave, light will be modulated by the hologram so that it emerges exactly as though it came from the object. With this technique, therefore, a visible image of an object which does not exist physically can be constructed, in three dimensions if desired. The problem, of course, is the realization of the transparency whose transmittance is proportional to a computed complex wavefront. Black and white film can be used to control the amplitude o r modulus of a light wave, and a bleached film can be used to control the phase of light, due to changes in refractive index and surface relief effects. A transparency of a given complex transmittance can be systhesized by making a “sandwich” of two control transparencies, one controlling the amplitude and the other controlling the phase. In practice, this technique is seldom used because of the difficulty of obtaining proper registration between the two transparencies. To overcome this problem, many ingenious approaches were devised to represent a complex function by using a recording medium which controls only amplitude or only phase. Some of these are described in the third section. These holograms of non-physical objects are termed “synthetic holograms”. Since a digital computer is often used in their synthesis, they are sometimes also called “computer-generated holograms”. We mention briefly here the principal solutions in vogue.

2.3.1. Synthetic holograms or complex-spatial filter

The general problem of synthesizing a complex optical wavefront with a specified amplitude and phase distribution, encompasses both the problems of computer-generation of holograms and synthesis of complex spatial filters. While these problems are often treated separately in the literature, their differences result mainly from their intended use (holograms for imaging and spatial filters for optical processing). Applicable to both problems is the general method of synthesis comprising three basic processes, viz.:

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81

(i) - Computer mathematical processing. Here the specified complex function is encoded in a sampled version of a non-negative real function (herein called the hologram function) in order to be simulated by a drawing device, e.g. plotter, scanner, etc. . . . (ii) - Photographic processing. In this case a transparency (the synthetic hologram) is produced with a transmittance proportional to the simulated function. (iii) - Optical processing. In optical processing the synthetic hologram is illuminated and the desired wavefront filtered out from the total real wavefront emanating from the hologram. All methods of synthesis can be classified according to the mathematical scheme by which information of the complex function is encoded in the hologram function. Any coding scheme is characterized by two quantities, the number of samples required to represent the complex function, and the product of drawing device resolution (in sampleslunit length) and length (per dimension) of the simulation. The former quantity indicates the required computer capability/cost while the latter, a quantity invariant under photoreduction, is a measure of the drawing device’s capability. These quantities are derived for each scheme-coding in terms of the space-bandwidth product (a measure of the information content) of the complex function. TABLE 2.1 Comparative characteristics of synthetic image-type hologram and synthetic Fourier transform-type hologram. Type I Synthetic image-hologram

Type I1 Synthetic Fourier transform-hologram

- Reconstructed image un-modified

- Dimension

of hologram inversely proportional to the plotter resolution.

8M

by translation.

- Averaging effect on the reconstructed

c

image deffects (Fourier transform property). - Reconstructing set-up requires one lens only.

If I

I

I

- Reconstructed

S

5 2B

image varying by translation. - Reconstructing set-up requires two lenses

- Size of hologram is a function of the resolution of the plotter. -Requires the computation of the F.T of the desired function before coding.

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S C E N T ADVANCES IN PHASE PROFILES GENERATION

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For any coding scheme, the information related to the complex function may be encoded in two alternative types of synthetic holograms, viz., (i) The complex function itself is encoded in the hologram function (synthetic image hologram), or (ii) The Fourier transform of the coniplex function is encoded in the hologram function (synthetic Fourier transform hologram). Table 2.1 summarizes the different characteristics of these two types of holograms. In this paper we do not attempt to present a detailed account of the numerous coding schemes available for use. The reader interested in these problems will find a general analysis in the review articles by HUANC[I9711 and RANSOM[1972]. But we will briefly summarize the most significant coding schemes which have been used in recent years to produce good quality synthetic holograms.

2.3.2. Mathematical models and coding schemes (review of computer-synthesis processes) In this subsection we will briefly outline the history of “off-axis’’ computer generated holograms, and, in particular, discuss the techniques used by KOZMAand KELLY[1965], WATE.RS [1960], BROWNand LOHMANN [1966] and MEYERand HICKLINC [1967]. Each of the methods for the computer-synthesis of holograms starts with a digital representation of the object. Once the computation is completed there is the common problem of plotting. Some techniques have used a “photographic” plotter. Since the plotters do not have adequate spatial resolution to draw usable holograms, the plots are photographically reduced. The following examples are outlined. (I) Kozma and Kelly’s procedure: Here, the computer techniques are used to construct “matched” filters which enhanced the frequency spectrum of a known object while filtering o u t background noise. The authors increased the signal-to-noise ratio by as much as 10.5 db. The actual filters used were binary in nature. The Fourier spectrum of the desired wavefront was computed and the resultant continuous function was binarized. (11) Extension of the binary mask technique: In 1966, Lohmann et al. introduced a method called “detour-phase’’ the principle of which is illustrated in Fig. 2.5. The binary synthetic holograms made by using their techniques comprise an array of rectangular apertures, the areas of which

11, § 21

X3


Fig. 2.5. Principle of “detour-phase”. In the direction of difiaction a little variation h in the period of the grating, introduces a phase shift q.

are proportional to the amplitudes of the wavefront as calculated at the corresponding sampling locations in the hologram plane. The positions of the apertures are shifted from the centre of the sampling interval by an amount proportional to the phase of the wavefront at the corresponding sampling locations in the hologram plane (see Fig. 2.6). The function of the aperture in a Lohmann-type binary hologram is similar to that of the fringes in a conventional hologram formed in the interference of a wavefront, exp ( j 4 ( x , y)), by a tilted plane wave, exp (j2rx/Ax), where the positions of the fringes satisfy the following equation :

(x/Ax)- ( d x , y ) / 2 r )= n, where

IZ

(2.1)

is an integer. If q ( x , y ) is constant, the hologram thus formed is a

I

n I

I

’ m

I

m Fig. 2.6. Lohrnann’s coding technique. The size of the spot ( P m f l )is proportional to the amplitude, and its position ( W,,,,,) in the cell ( m n ) ,is proportional to the phase, according to the “detour-phase” effect.

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periodic grating with grating period Ax. If the value of exp (jq(x,y ) ) within an area A x * A y centred at ( n Ax, m A y ) is approximated by exp (j+(n Ax, m Ay)), then eq. (2.1) shows that the positions of the fringes are given by:

x = n Ax +{[Ax q ( n Ax, rn Ay)]/2~}.

(2.2)

This is the equation used by Lohmann to determine the positions of the apertures in his binary synthetic holograms. In eq. (2.2), II Ax gives the location of the centre of each of the sampling cells along the x direction, and the second term gives the deviation of the aperture from the centre of the sampling cells. As can be seen, the deviation from the centre of the sampling cell is proportional to the phase of the wavefront. Because of the approximation, a binary hologram cannot produce exactly the desired wavefront. The error due to the approximation can be reduced by using a smaller and smaller period x. Various types of binary synthetic hologram have been proposed, e.g., HASKELL [1973], BURCKHARDT [1970], etc. (111) Lee’s hologram (1970): Another interesting coding scheme for binary holograms was proposed by W. E. LEE[1970]. He represented the complex function by four components (Fig. 2.7), i.e. the projections of the function on (i) - the real positive axis; (ii) -the real negative axis; (iii) -the imaginary positive axis; (iv) - the imaginary negative axis. The complex transmittance value is obtained by varying the transmittance of each of the four adjacent cells. The number of levels possible with this model varies from 1 to 256. (IV) Zone plate synthetic phase profiles: A different technique was used by WATERS[1960], to achieve the first published reconstruction of a computer generated three-dimensional image. The author observed the similarity between holograms formed by illuminating a point aperture and Fresnel zone plates, whose imaging properties are well-known. The diffraction pattern from a point aperture has annular rings, or zones, of positive or negative phase. Fresnel zone plates are plates in which the zones of positive phase are transparent, while those of negative phase are opaque. When a Fresnel zone plate is illuminated by coherent light from a point source, it does image that light. One way to analyze the holographic process is to consider that the object comprises discrete point

11, § 21

+ PRINCIPLES AND TECHNIQUES

+i

-1

t1

Fig. 2.7. Lee’s coding scheme: the complex function to be synthesized is decomposed into its four real components.

apertures, so that the hologram may be thought of as the superposition of many Fresnel zone plates. Waters used the Fresnel zone plate approach to calculate his holograms. Fresnel zone plates are by nature binary; depending upon the sign of the phase, they are either transparent or opaque. The actual intensity of the diffracted wave is not recorded, and it is not necessary to do so, providing one is using a point source. After calculating the appropriate superposition of Fresnel plates, Waters examined the phase of the synthesized diffraction pattern. At a satisfactory sampling rate, he plotted, using a mechanical plotting table, those points having positive phase. This plot was photoreduced and illuminated with coherent light to form reconstructed images. Waters succeeded in forming the three-dimensional image of the edges of a tetrahedron. This computer-generated hologram had the properties of a true optical hologram, made with a diffuser. The three-dimensional properties of the virtual image could be sensed by an observer. (V) The C.R.T. plotting technique: MEYERand HICKLING [1967] constructed holograms by displaying the computer output of their calculations on the face of an on-line oscilloscope. They formed holograms having about 64 000 resolution elements, while simulating the “Fourier holography” geometry, i.e., using a point source as a reference. The objects used were simple geometric figures, e.g. lines and points. The holograms constructed by Meyer and Hickling exhibited 12 grey levels, obtained by “hitting” each CRT raster point as many as twelve times, depending on the scaled values of the computed hologram. The CRT screen was photographed and the resultant transparency was illuminated with laser light. Meyer and Hickling’s holograms were limited in frequency content to 22 lines/mm, although this is a fundamental limitation of their technique.

86


[II, 0 2

Unlike BROWN and LOHMANN’S [19661 binary masks, the holograms constructed by Meyer and Hickling were constructed from points of variable intensity. Each resolution element in the hologram was a single point, and thus was simpler to construct than the rectangular apertures of a binary mask. Since the CRT is limited to so few resolution points (1024 x 1024 in the extreme), this dot-by-dot method is imperative if the CRT technique is used. ICHIOKA, IZUMI and SUZUKI [1969] were able to increase the number of grey levels to 32. In summary, the holographic photofabricating technique and the computer-synthesis technique reported above, are amongst the best possible solutions for phase profile generation. However they are all “off-axis” techniques and the light diffracted to the desired image is typically about 1%of that incident on the hologram. This limitation has been particularly studied in the following alternative solutions to produce high diffraction efficiency phase profiles giving an “on-axis’’ wavefront.

2.4. ON-AXIS PHASE PROFILES

If high diffraction orders are to be avoided, a technique which gives only a transmitted or diffracted wavefront must be found. For this purpose this section will be concerned with “on-axis’’ profiles. It is first necessary to produce a phase plate, but here difficulties arise because of thickness limitation and the difficulty of fabrication, e.g., polishing, evaporating, etc. Early workers produced phase profiles in two steps, with the techniques described previously, using computer holography. This required the synthesis of two hologram transparencies, one producing the real component of the object and the other the imaginary component (RANSOM [19721). The two components were combined interferometrically but in general, this was difficult to achieve experimentally. Other experiments and were performed with polarized light and vectograph film (HOLLADAY GALATIN [1966], MARATHAY [1969]). In practice, the ideal case would be to produce the phase profiles on photosensitive emulsions, but the emulsions are generally very thin usually a few microns-so it seems difficult to record very high phase variations on them. Such high variations may occur, e.g., when it is required to synthesize phase profiles to correct large aberrations. As it will be shown later, the refractive index variation of commercially

11, § 21

87


available emulsions is about lop2, which implies very accurate precautions while processing the emulsion or the crystal to supply a suitable media.

2.4.1. Kinoform and related computer generated holograms A solution to the problem of obtaining the required thickness of phase relief was suggested by MIYAMOTO [1961]. He proposed the mathematical solution of modulating the phase from 0 to 27r (Fig. 2.8). As in the computer holography method the wave required is previously introduced in the computer, digitalized, and its diffraction pattern calculated in a specific plane. Similarly the phase is encoded and modulated from 0 to 23-r to avoid too high values of this phase. The pattern is recorded by a plotter or is displayed on a C.R.O. screen, and photographed. This photograph may be produced on a photographic emulsion, which is then optically reduced and bleached, or directly recorded on a phase relief material to increase its efficiency (this will be described in the next section). These pure phase profiles are called kinoforms and have been produced by LESEM, HIRSH, JORDAN and PATAU[1970] for I.B.M. Figure 2.9 depicts this modulation procedure. The kinoform looks like a Fresnel lens because of the modulation of the phase from 0 to 2rr, but the profile is more adaptable. When the phase is matched correctly, for a specified wavelength, the efficiency is 100%. One of the simplest kinoforms is a

0

8X

0

2%

Fig. 2.8. These two representations of the phase variation are equivalent; the phase being , kinoform looks like a Fresnel lens. modulated at 2 ~ r the

88


[II, § 2

Fig. 2.9. Example of kinoform filtering for enhancement of data obscured by noise. (a) unfiltered image; (b) filtered image obtained by using a least squares kinoform fan-filter (LESEM,HIRSH, JORDAN and PATAU[1970]).

blazed grating, which is equivalent to a prism if the profile is modulated to 27r. Thus if the incident wave is plane, the emerging wave through this phase profile will also be a plane wave propagated in the direction a, to the incident wave, and the efficiency is theoretically 100%. Another interesting property of the kinoform is that it allows a reconstruction in incoherent light, in contrast with holographic filters (Fig. 2.10). If the phase matching is not perfect, the phase kinoform profile will behave like a profile illuminated with a wavelength different from that used for its conception. This chromatism or phase mismatching effect will give false images. This aspect has been thoroughly investigated by Myamoto in the reference cited. H e proposed the use of this peculiar effect to produce dispersing profiles like common gratings. Another more important source of noise is the quantization of the information, which especially occurs in the technology of the computer holography. The discrete number of levels gives a quantization in the phase level recorder. There are many possibilities of having a phase quantized into N equal levels, e.g., if the phase varies between 0 and 2rr, the elementary phase level will be 2rr/N. The influence on the reconstructed wave and its relative noise introduced

11, § 21


Fig. 2.10. A kinoform filtering system using incoherent light (LESEM, PATAU[1970]).

89

HIRSCH, JORDAN

and

by the quantization has been studied by many authors, some of whom have even proposed different models of quantization to minimize the SCVESTRI[19701, DALLAS noise at the reconstruction (GOODMAN, [1971], CLAIR[1972] and LOHMANN [1973]). The phase quantization introduces high order images whose amplitudes are proportional to Sinc (rn + l / w , rn being the order of the false image and N the number of levels, e.g., for N = 2, 40% of the incident light is in the good image, while for N = 10 this factor reaches 97%. Figure 2.11 gives an example of a computer generated hologram. 2.4.2. Photolithographic and evaporating methods

If the use of the computer is to be avoided, one may use the step-bystep approximation of the phase profile desired, and by applying the

Fig. 2.11. Two-level phase quantized binary hologram (DALLAS [1971]).

90


[II, 8 2

photolithographic technique broadly utilized for the production of microintegrated circuits. For example, PUECH[1973] and S~rrz,D’AURIA, HUIGNARD and ROY [1972], by using this principle, created lenses and other optical components. Figure 2.12(a, b) depicts the different processes involved in producing a four-level spherical lens. These lenses may be used to form matrix lenses for holographic storage. Some evaporating techniques can also be utilized, in particular those used by FIRESTER [1973], DUPOISOT and MORIZET[1974] to obtain zone plates and quantified profiles used for integrated optics and correction of aberrations. These techniques use sequentially different masks whose number and shape are related to the desired profile, but this additional preparation increases the cost of the method. Moreover the setting of these masks must be performed accurately, and their number is necessarily limited. This affects, especially at the discontinuities, the precision of the final profile obtained. We now present some alternative solutions based on optical synthesis by light ray or pattern tracing methods.

2.4.3. Optical synthesis processes If more phase or grey levels are required in the sampling, one may use directly and optically the photosensitive material itself ; this has been achieved by the authors ( A B ~ Oand L CL.AIR [1975]). The first step is to produce the simplest kinoform possible, that of a point source. If a plane wave is incident on a lens kinoform, it will emerge as a converging Emosure

Binary mask

. Second step

Third stcp

(a)

(b)

Fig. 2.12. (a) Quantization of the phase with 4 levels. (b) Realization of the successive HUIGNARD and ROY [1972]). contacts with binary masks (SPITZ, D’AURIA,

11, 0 21

91

PRINCIPLES A N D ECHNIQUES

spherical wave having the same properties as that emerging from a classical glass lens. The repartition of the different zones for the corresponding kinoform has been given in Fig. 2.9.

Use of scanning by optical pattern techniques A method which we have developed in the laboratoire FranGon. utilized the annular fringes obtained with a Fabry-Perot interferometer in which the repartition of the annulii follows the same law as that for the Fresnel zones (CLAIR[1972]). The principle is to utilize successive recordings of the F.P. pattern corresponding to different values of the path difference within the etalon. After a scanning corresponding to one order of interference we complete, step by step, all the possible positions of the interference system. If the sweeping is well chosen, a great number of zones whose profile is the profile desired for a kinoform lens, may be recorded directly onto a photographic plate (in case of amplitude variation) or on phase relief material (in case of phase). In our original experiments of 1972, the optical path difference within the etalon was modified by variation of the air pressure, the geometrical thickness remaining constant. Figure 2.13 depicts the apparatus, while Fig. 2.14 gives the intensity pattern of a kinoform spherical lens obtained by the set-up described above. KORONKEVITCH, LENKOVA, MIKHAL'TOOVA, R E ~ N I FATEEV K, and TSUKERMAN [19761, developed a similar apparatus but with a variation of reflection coefficient of the plates. Another interesting procedure is the graphical process used by CAMUS, and CLARK[1967]. One of the faces of a black box 3 cm thick, GIRARD containing a fluorescent tube is covered by a glass plate on which are drawn small areas of circles, opaque or transparent following the Fresnel

P

,

pan

H , photo plate I

M

, manometer

V C , valve I

and pump

gatc-o;r odmksion

T R , tape Lcodcr C , control

Fig. 2.13. Scanning set-up for recording Fabry-Perot fringes to synthesize the desired amplitude distribution (CLAIR[1972], CAMATINI [1972]).

92


[II, P 2

Fig. 2.14. Intensity pattern of a kinoform spherical lens according to Fig. 2.13.

law. When this box rotates in front of a photographic plate, the recorded pattern is the classical annulii of a Fresnel lens. It is also easy to photograph the successive diffraction patterns of a circular aperture moving perpendicularly to the photosensitive plane. STROKE [19671 related each diffraction pattern to one resolution element of the object to be created. CHUTJIAN and COLLIER [1967] applied this idea to visualize a 3 dimensional object by the use of multiple exposures and for different planes, under computer control. If the use of interference patterns is to be avoided, and a more general profile produced, each element of the desired pattern may be considered separately and the scanning performed by use of optical deflectors.

Use of optical scanning deflectors The “set-up” which we utilize allows us to simulate optically the kinoform principle of modulating the Fresnel zones from 0 to 2 ~ A. revolving prism makes the image of a pinhole source on the photographic plate move in the form of a circle; different positions of the prism correspond to different diameters of the circular patch produced on the plate (ARITBOL and CLAIR [1975]). Figure 2.15 depicts the mounting. The time of exposure needed at each annular zone is determined by the degree of blackness, in case of amplitude variation, or thickness in case of phase relief. One can then obtain elementary annular zones which collectively correspond to the final profile. We use photoresists as the photosensitive surface to avoid the need for bleaching.

II,6 21

93

PRlNCIPLES AND TECHNIQUES

t

@$%==+.

trensletion

c----

rotation

point source monochromatic

light rotating prism recording plane

Fig. 2.15. Revolving prism set-up for synthesizing two-dimensional phase profile

Experimental arrangement The kinematic slide on which the revolving prism is mounted may be displaced laterally by a micrometer screw arrangement driven by a stepping motor and the displacement of the prism controlled to e.g., a 1/10 mm for one pulse passing through the motor. The photosensitive surface is exposed for each position of the prism. Preliminary experiments allowed us to determine the dynamic curves of the photographic emulsion or photoresist. Generalized wave-front profiles may be produced as follows: since only two parameters, viz., the position of the prism and the time of exposure for each position of the prism, determine the nature of the elementary annular zones on the photosensitive surface, a combination of any type of elementary annular zone may be synthesized by controlling these two parameters. Hence, any profile of revolution may be produced, e.g., phase or amplitude profile, apodisers, testing and correcting plate, etc. The two parameters mentioned above may be calculated by a computer program which takes account of the profile under consideration, the modulation of phase from 0 to 2 7 ~in different zones (if necessary) and the number of steps constituting each zone. We applied this optical arrangement to produce a Schmidt plate for the correction of the spherical aberration of a F12.7 concave mirror. Figure 2.16 recalls the principle of modulation of the desired phase profile. Figure 2.17 depicts the step-by-step annular rings approximation of each zone by the scanning mentioned above, while Fig. 2.18 gives the Schmidt piate in terms of amplitude variation obtained on a photographic plate. It is particularly interesting to note that the step-by-step annular approximation does not affect the phase variation emerging from the profile, see Figs. 2.19a and 2.19b. One-dimensional gratings have been produced by

94


[II, § 2

Fig. 2.16. Kinoform representation for the Schmidt plate.

PI Fig. 2.17. Optical quantization of the phase variation

Fig. 2.18. Photograph of the Schmidt plate in terms of amplitude instead of phase distribution. Diameter of the plate: 46 mm, diameter of the scanning light spot: 70 microns.

11, 9 21

PRINCIPLES AND TtCHNIOlJES

95

Fig. 2.19. Equal thickness fringes obtained by a Michelson interferometer for the corresponding phase filter obtained by contact o n photoresist from the plate shown in Fig. 2.18.

this technique, but we are also concerned with the possibility of producing non-revolution phase profiles. These will be critically investigated in the last section. Achromatism is an inherent disadvantage of phase profiles calculated for a specific wavelength. However it seems possible with different methods to produce phase profiles whose function varies in a similar fashion to optical achromatic doublets. These phase profiles are then used as tests for aspherical surfaces during their manufacture. To the best of our knowledge, one of the latest developments of phase and complex profiles, is the so-called R.O.A.C.H. (Referenceless OnAxis Complex Hologram) which has been developed by GOODMAN and FIENUP [1973]. The ROACH uses multiple emulsion films, such as Kodachrome, in which different layers can be exposed independently by light of different colors. Upon illuminating with light of a given color, one layer will absorb while the other two layers will be predominantly transparent but will cause phase shifts due to variations in film thickness and refractive index (Fig. 2.20). Since the complex transmittance, both amplitude and phase, can be controlled at each point, only one resolvable element is needed to encode one complex Fourier element. For example, if we wish to reconstruct in red light, the amplitude pattern is exposed in this light (since only the red sensitive layer will absorb red after processing) and the phase pattern is exposed in blue-green light to generate the phase relief. By this very simple method the authors could store and reconstruct color complex variation on commercialized slide films (Kodachrome 2 and 25).

96


/ ‘Phase

magc

cmhl

plow

Amplitude contrd

[II, § 3

Fig. 2.20. A schematic diagram showing the reconstruction of a ROACH. The amplitude and phase are independently controlled by different emulsion layers (GOODMAN and FIENUP [1973]).

In summary to this section we have given a selection of the methods and techniques which have been commonly used over the last few years for the production of phase or complex profiles. This general compendium contains the classical mechanical methods of shaping a surface, the coherent or conventional photo-fabrication, and also some computer optical synthesis techniques used for the same purpose. In the following section we will be concerned with the possibility of producing phase or complex variations by the use of the physical and chemical properties of the matter. Recent advances in solid state physics have produced new materials (including photo-, thermo- and electro-sensitive media), on which phase and complex variations may be accurately recorded. A general analysis of materials suitable for phase information storage and retrieval follows. These materials are classified into two main categories; viz., erasable and non-erasable media.

0 3.

Optical Material for Phase Information Storage

3.1. NON-ERASABLE MATERIALS

In many applications a non-erasable storage in which the information is retained permanently, is acceptable, and in some cases may have definite advantages, especially where security against loss of the stored information is important. The terms “non-erasable”, “permanent”, or “semipermanent”, “fixed” or “read-only” are strictly relative, but all are used to qualify systems in which the information cannot be changed physically.

11, §

31

PHASE INFORMATION STORAGE

97

Another advantage of non-erasable materials is that, in any storage, it must be possible to write, but since this writing is often carried out by some mechanical operation, its time may be many orders of magnitude longer than the reading time. Since the information is permanent, there is no “write” phase in the access cycle which may therefore be considerably shorter than the cycle in a conventional destructive read-out storage. We will now briefly survey a selection of materials most frequently used because of their physical properties and commercial availability. We shall describe the bleached silver halide emulsions, bichromated gelatin, photoresists, photopolymers, and photosensitive plexiglass (PMMA). 3.1.1. Silver halide emulsions and bleaching This article is concerned only with phase profiles. If phase relief variations are to be recorded on silver halide emulsions with maximum diffraction efficiency, then bleaching techniques must be used. Silver halide photographic emulsions were used to form the first holograms, and they remain the most commonly used recording materials today. They have high sensitivity and are readily available, but require multi-step processing including bleaching and costly processing equipment. In this general study of the phase relief images on photographic emulsions, SMITH[1968] has shown that, for a given process and emulsion, the differential height of the surface relief between the minimum and maximum density point is strongly dependent on the spatial frequency, e.g., for 25 lines/mm and A =590mm, this relief is about 2 pm, i.e. 7% of the total height for a 16 pm thick emulsion. Although photographic relief images have been used for microdensitometry of spectral lines, the principal application concerns holography. [19741, Detailed bleaching procedures have been outlined by GRAUBE NISHIDA [1974] and others. At times it is also desirable to remove residual dyes from the emulsion, e.g., a useful procedure for removing the green residual from Kodak S.0.173 (also type 120) has been presented by COBLITZ and CARNEY [1974]. 3.1.2. Bichromated gelatin Once properly sensitized (in blue or U.V. light) the gelatin responds to exposure by swelling and changing its solubility. The unexposed gelatin is

98


[II, D 3

dissolved in warm water. This results in a surface relief and forms a phase profile for which the diffraction efficiency can reach 100°/~with quite a low noise level. However, the film preparation procedures are rather involved and must be carefully and individually performed. These materials are also somewhat limited by their restricted spectral response (230500 mm) and their relative sensitivity to high humidity. 3.1.3. Photoresists Photoresists are photosensitive materials on which the phase relief image can be chemically obtained either, (a) by the areas unexposed to the light (positive photoresist) or, (b) by the exposed areas (negative photoresist). Photoresists have been optically developed for application in the photoetching industry. A new field of application has been introduced by the masking technology applied to the manufacture of microintegrated circuit boards. In the field of optoelectronics, photoresists seem to be a valuable solution to the problems of phase holography and for the mass duplication of holograms, e.g., by a method similar to the moulding technique used in audio-recording industry. The ability of photoresists of satisfying a specific application depends on several parameters, e.g., sensitivity, adherence to the substrate, thickness and uniformity of the coating, etc. Photoresists, contrary to silver halide emulsions, are not commercially available coated on a substrate ready for use. They are, however, available in liquid form, so that the experimentalists are free to perform the coating and choose the processing. The physical and chemical properties, and optical behaviour of these materials have been thoroughly investigated by numerous authors, and in the laboratoire FranGon, we have particularly studied these materials for the production of phase profiles, e.g., gratings and phase plates: TORRES [1975], A B ~ Oand L CLAIR[1975], FRE~~LICH and CLAIR[1976]. We have given, in the tabulation (Table 3.1) some of the most commonly used photoresists with their respective applications.

3.1.4. Photosensitive polymers plexiglass (PMMA) Photopolymers typically consist of incompletely polymerized polymer which is activated during exposure through the intermediary of a photosensitizer. This causes a polymerization chain reaction which takes several minutes to complete.

TABLE 3.1 Optical characteristics and applications of four commercialized photoresists.

z


z a

99

100


[II, § 3

The illuminated areas have a higher density and therefore a higher refractive index than the basic material; thus, photopolymers form efficient phase relief profiles. However the primary present limitation of this material is the high exposure energy requirement. Photopolymers require no processing and therefore form holograms which can be viewed immediately following exposure, thus offering obvious advantages for realtime holography. A fixing procedure is required to bleach the dyes and complete polymerization of the remaining monomer.

3.2. ERASABLE MATERIALS

Erasable real-time phase profiles are of particular interest because of the possibility of modifying the stored information during the recording stage. The requirements for such materials are: (i) - High resolution (> 1000 l/mm) to record a fine structure. (ii) - Good sensitivity ((1 mJ/cm2). (iii) - Recording and erasure without degradation of the material. (iv) - Short writing-erasure cycle time (- 1ms). (v) - Long lifetime of the stored information. (vi) - Non-destructive read-out with the wavelength used for recording. (vii) - Good dynamic range. (viii) - High optical quality. (ix) - Room temperature operation. This list is not exhaustive. Some commonly used materials are described below.

3.2.1. Thermoplastics Currently used devices are in the form of a thermoplastic photoconducting layer coated onto a glass substrate having a transparent electrode. The entire thermoplastic surface is uniformly electrically charged. With a non-uniform exposure (e.g. from an interference pattern), the photoconductor discharges in a manner following the intensity distribution. After recharging, a short heating pulse increases the temperature almost to the melting point of the thermoplastic (-60°C) which deforms under the effect of the local electric field, and produces a phase pattern when cooled (Fig. 3.1). Erasure is achieved by re-heating the material to

11, P 31


Step 2

----_ Step 4 d ev el o p emcn t heat treatment

101

step 3 second charging

Step 5 erasure heating

*F====l Fig. 3.1. Recording-erasure cycle of the thermoplastic profile.

a similar temperature as in the previous step. The most serious problems are the degradation of stored information after about 100 writing erasures cycles, and limited spatial frequency imposed by the film thickness (-1 km). The sensitivity is about 100 J cm-'. Thermoplastics have been [1840], CZERNY and MOLLET [1938], particu€arly studied by HERSCHELL BILLINGS [1951], TAQW [1963] and LOULERGUE [1971]. 3.2.2. Photochromes Photochromism is a reversible color change in a material which occurs with exposure to light. It is a property exhibited by many solid, liquid and crystalline organic and inorganic compounds. Holograms recorded in these materials have no grain, require no development, and can be re-used for many cycles after either thermal or optical erasure (FRIESEM and WALKER [19701). However, the materials are relatively insensitive, and after exposure must be refrigerated to avoid information loss through thermal relaxation. They remain sensitive after exposure and will therefore fade in the read-out beam. They also fatigue to insensitivity. However, the thin film form of the material minimizes these disadvantages and shows some promise as a real-time data storage medium.

3.2.3. Electro -optical materials Many materials are anisotropic when submitted to electrical fields. In some crystals, such as lithium niobate o r strontium barium niobate,

102

RECENT ADVANCES IN PHASE PROFEES GENERATION

[II, § 3

exposure to light frees electrons which migrate and become trapped in regions of low exposure, creating a space charge pattern whose field modulates the crystal’s refractive index and forms a phase profile. This effect is reversible. Electro-optical crystals thus exhibit many of the same advantages as photochromic materials. However, holograms formed with electro-optical crystals are more efficient than those formed in photochromics; they can be fixed and do not fatigue. Sensitivities are about 10 to 100mJ/cm2 (SERAPHIN [19761). New recording devices using electro-optical effect have been proposed recently; we mention particularly the PROM material set-up in Fig. 3.2 and the phototitus apparatus for converting incoherent to coherent imaging, to permit an optical filtering, Fig. 3 . 3 . Open or short electrical circuit configurations for the electro-optical crystal give similar effects and are depicted in Fig. 3.4.

3.2.4. Liquid crystals Three molecular arrangements, or mesophases, of liquid crystals may be considered. Of these mesophases, only the nematic and cholesteric have practical application. In the cholesteric state, the molecule forms a screw-like structure, while in other phases the material scatters light, producing a variation of transmittance. Figures 3.5, 3.6 and 3.7 illustrate these properties. Phase variations may be produced by external electromagnetic fields. Generally available materials have not as yet been found suitable for use in coherent light. Although the resolution is low, and the

Fig. 3.2. Set-up using PROM for correlation and convolution of image 1 and image 2 (BALKANSKY [19731).

11, 5 31


Constant current

103

K D2P04

(0-1 2 5 ~ ) conductive layers Fig. 3.3. Principle of operation of TITUS (BALKANSKY [1973]). The video-signal is applied between one side of the crystal and a grid placed close to the other side. The scanning is effected independently by a constant current electron beam. The input light beam is polarized and after reflection on the back side of the target, the modulated beam is analyzed by means of a crossed polarizer.

delay-time long (-10 ms), some of these materials have been used successfully to store binary information in data arrays. Figure 3.8 shows the structure of such a liquid crystal (CAMATINI [1975] and KMETZ [ 19761).

3.2.5. Acoustical media The propagation of acoustical waves in some materials gives rise to a refractive index variation and surface modulation. Acoustical holography

Fig. 3.4. Spatial index change of refractive index after hologram grating recording and total index change after recording-coherent selective erasure cycle; (a) open circuit configuration, (b) short circuit configuration.

104

[II,

RECENT ADVANCE!? I N PHASE PROF’LES GENERATION

OFF

Planar Cholestrric Clear -

ON

ST ORE

Dynamic Scattering Scattering

Disordered Cholestcrk Scattering

53

ERASE

Planar Choltstcric Clear

-

Fig. 3.5. Storage effect in the liquid crystal. In the initial state the material adopts an ordered cholesteric state and appears clear. When a low-frequency field is applied the nematic component of the mixture shows dynamic scattering and the material is disordered. This disorder is retained after the field is removed so that a scattering state is stored. The scattering is “erased” by applying a high-frequency field which re-establishes the ordered 119751). state (CAMATINI

is well developed and phase generation may be performed with the deformation of a surface by acoustical operations. This has been reported, and STROKE. [19761. Table 3.2 tabulates the particularly by NESTERIKHIN new devices used for real-time operations, while Fig. 3.9 depicts a general [1975]). mounting for acoustical operations (CAMATINI

6On ~raosmisston

P

’s

Fig. 3.6. Binary transmission by variation of the polarization with phase structure modification (CAMATINI [1972]).

11, 0 31


OFF Hamcotropic ncmakic

105

ON nrmolic birtfringtnt

Fi 3.7. The variable birefringence effect in luid crystal. When a DC or AC field is ap,,.ied, the molecules rotate towards a direction perpendicular to the field as shown, and [ 19751). the film becomes birefringent (CAMATINI

3.2.6. Miscellaneous media These include magneto-optical materials and biological substances sensitive to the light. The development of opto-electronics and integrated optics is based on the technology of such new, transparent, erasable materials. A summary of erasable and non-erasable phase storage materials which are most commonly used to perform phase variation for optical applications, is given in Tables 3.3 and 3.4.

cristol

Fig. 3.8. A liquid crystal cell (BALKANSKY [1973]).

106


HI,

04

TABLE 3.2 Page composer devices for real-time operations (after HUIGNARD, MICHERONand SPITZ [ 19761). Physical effect Polarization change by induced birefringence electrooptic effect

Material

I

Addressing techniques

PLZT ceramic Bi,Ti,O,, 0 GdMoO, 0 SBN

Electrode matrix Threshold and memory

Liquid crystal

Electrode matrix Threshold and no memory (multiplexing)

Scattering induced by poled and non-poled regions of ferroelectric

PLZT ceramic

Electrode matrix Threshold and memory

Scattering induced by electrical agitation

Liquid crystal

Electrode matrix, no threshold, no memory

Thermally induced shift in absorption band edge

CdS

Electrode matrix for heating and heat sink substrate for cooling Multiplexing

Phase disturbance by piezoelectric excitation of mirrors

Piezoelectric mirrors

Individual addressing of drivers

Water Fused quartz PbMoO,

Transverse interaction of light and acoustic waves Individual addressing

Reflection changes from thin deformable membranes

Metallic film Thin polymer film

Electronic memory o n the back of the substrate

Optically induced birefringence changes

Bi,,SiO,,

Laser beam scanning C.R.T.

Optical access

KOP +photoconductor

Laser beam scanning C.R.T.

Acoustooptic interaction

0 0

0 4. Phase Proliles Applications - Current and Future Prospects In recent years, phase informations storage and retrieval has found a wide field of application, and the introduction of new materials has permitted research workers to develop new methods in optical processing.

11, (i 41

PHASE PROFILES MPLICATIONS-CUFUENT

AND FUTURE PROSPECTS

107

MONITOR

rdurce

transducer

lenses

Fig. 3.9. Acoustical operation in holography. The surface modulation is read by a laser and then filtered.

As an illustration we have grouped in the following section some of the most significant applications, based on the phase profile utilization, which have been investigated or practically utilized. This summary concerns principally the video-disc imaging system, current application of new coupling gratings, phase profiles for the solution of particular mathematical problems, and new recording devices used in the infrared.

4.1. VIDEO-DISC IMAGING SYSTEM

Although holographic T.V. is not yet fully developed, the phase profile gives a good solution to the problem of storage and retrieval of information with respect to sound and color, The T.V. signal is modulated in frequency and coded, in the form of a micropit, the size of which gives the replica of the binary coding. These pits are recorded on a disc and are read with coherent light. The video-disc is produced on plastic from a replica. The matrix is a glass substrate covered with a resist, which is then “graved” with a laser beam (Figs. 4.1 and 4.2). The size of the pits is about 2 Fm and the actual recording-time may be of about 20 to 40 minutes. Commercial systems

TABLE 3.3 and Phase optical materials non-erasable and erasable. a comparative study of their characteristics (HIJIGNARD, MICHERON SPITZ[19761).

Sensitivity (mJ/cm2)

Material

I

High resolution photographic plates 649F

1

Bichromated gelatine Photoresist Photosensitive plexiglass PMMA

0.1

Efficiency (%)

Amplitude: 6 Phase: 15-30 60-80 20-40 50-70

10 100-1000 2000

Resolution (lines/mm) >4000

>4000 >3000 >4000

Erasable holographic storage materials Material - Photochromic

-Thermoplastics - Magnetooptic MnBi -Bismuth titanate + photoconductor - Amorphous semiconductor glasses

I

~ _ _

Sensitivity (mJ/cm*)

Efficiency (YO)

Resolution (lines/mm)

100

3

>2000

0.1 10-100 1

10-34 Faraday: 0.01 Kerr: 0.05 1-0.1

100-1000

1-5

Erasure

1000

Optical heating Heating Magnetic field Electric field

1000

Heating

1000

> 1000

> 1000

v 00 0

Phase optical materials. a comparative study of their current applications

110


Frequency modulated ngnol Binary coding

Engraved signal on the

disk

[II, § 4

.

n 0 0 8

0

Y

micrrpisls

Fig. 4.1. Coding of the phase modulation in micropits for the video-disc (HUIGNARD, MICHERON and S P ~ [1976]). Z

have been developed by C.S.F., Philips and R.C.A. (COMPAAN-KRAMER [1973], BROADBEN [1974], BROUSSAUD [1974], HRBECK [1974] and HUG NARD, MICHERON and SPITZ[1976]). Other applications of phase profiles include the possibility of producing colored image in real-time, with utilization of liquid crystals. One can also use a common resist profile in which the zero order diffracted pattern transforms information of thickness into intensity o r color (GALEand KNOPP[19761). 4.2. NEW GRATINGS FOR COUPLING AND SPECTROSCOPY

One of the more modern applications, phase profiles, is based on phase matching. For a given wavelength, where the field distribution in the structure is known, one must couple the external field in that structure. The best solution seems to be a coupling grating which, by diffraction, gives a wave which may be guided in the thin film. These gratings are of high resolution, if sinusoidal, o r low resolution, if blazed. Gratings are used in spectroscopy, and it is possible to obtain a incident signal Micropit HF signal

I

Tronsporcnt disc

Fig. 4.2. Read-out of the video-disc, using diffraction of the micropits by coherent light.

11, 5 4)

PHASE PROFILES APPLICATIONS-ClJRREhT A N D FUTURE PROSPECTS

111

surface mapping for the correction of aberrations. Concave and holographic gratings are now used as optical components at an industrial level, particularly by Jobin-Yvon Cie., France. Gratings and periodic structures are used for multiplexing and data storing; such applications are described in detail by SCHMAHLand RUDOLPH[1976] in Progress in Optics Vol. XIV, and applications for [ 19761 in waveguides and coupling devices particularly by CLARRICOATS the same volume. In Table 4.1 we have summarized the advantages of the holographically produced gratings (FLAMAND et al. [1975]). 4.3. OPTICAL PROCESSING OF MATHEMATICAL OPERATIONS

O’NEILL[1956] demonstrated the usefulness of spatial filtering for data processing. The same objectives were pursued extensively by the optics group at the University of Michigan, particularly by CUTRONA[1960], LEITHand UPATNIEKS [1964], VANDERLUGT[1968] and by KOZMAand and WERLICH [1971]. KELLY[1965] and LOHMANN In our group at the Institute of Optics (Paris-Orsay), we have followed a different approach, as reported in 8 2, for making object or spatial filters, such as corrector plates and kinolenses. Another interesting application is the performance of mathematical operations using coherent optical filtering systems and phase profiles. This activity has developed from our work on phase profile generation. It is thus possible to produce an inverse filter capable of indicating the presence of signal u(x - xo, y - yo) in the input, by a light point at (x,,,yo) in the output plane. This filter can also be used for restoring an image which has been blurred, by a process which can be described as a convolution of the object with the blur function. Image restoration with an inverse filter was achieved by MARECHAL and [1967] for deblurring turbulCROCE[1953], by MUELLERand REYNOLDS and PARIS[1968]. ence and also by LOHMANN Some other mathematical operations, such as partial differentiation or Laplacian and high order derivatives, are currently performed (S. H. LEE [19741). 4.4. PHASE PROFILE FOR INFRARED TECHNOLOGY

In the search for new, low weight and low cost materials capable of making good phase profiles for the infrared, it has been found that some

TABLE 3.1 Holographic grating solutions to improve the signal to noise ratio (FLAMAND, HAYAT,LACROIX and CRILLO[1975]).

+ Decreases

t

Increases efficiency

Increases groove

Increases size

Changes design to a concavc aberration corrected HRDG

Decreases N I

Limited to a few percent

Not realistically possible

No increase in noise Moderate increase

Always available Moderate increase in cost

r 7LA Possible

Possible

Improves performances Reduces cost Reduces size Increases reliability

Noise level at least one order of magnitude lower with HRDG’s at n o extra cost

Very possible Brings only advantages

Brings only advantages

P

111

113

CONCLUSION

negative photoresists exhibit very good transmitting properties when L CLAIR illuminated over the 2-15 km range of radiation ( A B ~ Oand [1977]). It is particularly interesting to investigate the production of such phase profiles with the technical process developed in 0 2.3. Figure 4.3 depicts the infrared transmittance spectrum of a sample of negative photoresist Kodak 747, 11 microns in depth, coated onto a potassium bromide substrate and polymerized in white light. The four examples, briefly outlined in this section are illustrations of some of the applications of phase profiles and phase materials in optics. We are conscious of the fact that there are many other possibilities actually commercialized. The reader interested in this subject will find several references which we have grouped in the complementary references concerning the infrared materials.

Conclusion In summary we have presented a non-exhaustive review of the recent advances in phase profile generation. The techniques we have reported concern the production of optical components used in current applications, e.g., holographic gratings in spectroscopy, waveguide structures and coupling devices for integrated optics, generalized phase profiles used in optical processing, etc. In these fields of investigation we have been concerned with the utilization of new optical materials capable of producing the desired phase variation. Liquid crystals, electro-optical systems, and other new

ioa

t

0/0 Transmittance

4

6

8

10

12

14 Wave 1c nq t h

( rr 1 c ro n s 1

Fig. 4.3. Infrared transmittance spectrum of thin film of 747 Kodak micro-resist, recorded after polymerization.

114


[I1

transparent, erasable and non-erasable materials used in optics and opto-electronics have been reviewed. Integrated optics, for example, is a science which is deeply connected with solid-state physics, chemistry and electronics. Because of this fact, our subject may be considered as a quite vast and new one. Although new techniques and new materials are used to produce very classical optical devices, the final aim is to carry, to store, and to retrieve the phase information transported by a light beam. After a brief outline of some of the latest applications of phase information processing, we have given a list of complementary references on phase profiles generation, for 1975 and 1976, in order to enable the reader to obtain supplementary information on the subject. These references have been obtained by the computer of the C.N.R.S. (France).

Acknowledgements The authors wish to thank Professor M. FranGon in whose laboratory they have worked for many years under his guidance in this field. They are also indebted to R. Shaw for his help in the translation of the paper, and J. Rosiu for having performed the drawings.

Complementary References (1) Devices for filter generation BRYNGDHALL, 0. Optic. Corn. Netherl. 1975. 15. no. 2. p. 237-240 “Optical Scanner - Light Deflection Using Computer-Generated Difiactive Elements”. ICHIOKA, Y. Rev. Sci. Instrum. U.S.A. 1974. 45. no. 2. p. 261-263 “Circular Plotter and Kinoform Lens”. FIRESTER,A. H. HOFFMAN, D. M. JAMES,E. A. HEL.MER, M. E. Optic. Corn. Netherl. 1973. 8. no. 2. p. 160-162 “Fabrication of Planar Optical Phase Elements”. KURTZ,C. N. HOADLEY, H. 0. DEPALMA, J. J. Josa. U S A . 1973. 63. no. 9. p. 108Cb1092 “Design and Synthesis of Newdom-Phase Diffusers”. FEDOTOWSKY, A. LEHOVEC, K. Optik. Dtsch. 1975. 42. no. 4. p. 303-314 “Filter Design for Maximazing Sensor Response’’.

111

COMPLEMENTARY REFERENCES

115

HASKELL,R. E. Opt. Eng. U.S.A. 1975. 14. no. 3. p. 195-199 “Synthetic Holograms and Kinoforms”. B E N N E J. ~ ,J. Appl. Opt. U.S.A. 1976. 15. no. 2. p. 542-545 “Achromatic Combinations of Hologram Optical Elements”. SIROHI, R. S. BLUME,H. Opt. Acta G.B. 1975. 22. no. 11. p. 943-946 “On the Diffraction Efficiency of Synthetic Binary Holograms”. GUPTA,S. D. Indian J. Pure Appl. Phys. India. 1975. 13. no. 2. p. 116-120 “Frequency Response of Complex Amplitude Annuli”.

(2) Phase storing materials - infrared materials ALPHONSE, G . A. R.C.A. Rev. U.S.A. 1975. 36. no. 2. p. 213-229 (RCA Lab. Princeton). “Time Dependent Characteristics of Photo-Induced Space Charge Field and Phase Holograms in Lithium Niobate and Other Photo-Refractive Media”. BADER,T. R. Appl. Optics U.S.A. 1975. 14. no. 12. p. 2818-2819 “Hologram Gratings: Amplitude and Phase Components”. FRAZIER, G. F. WILKERSON, T. D. LINDSAY, L. M. Appl. Optics U S A . 1976. 15. no. 6. p. 1350-1351 “Infrared Photography at 5 p n and 10 wm”. BARNET, M. E. GAY,P. F. Opt. Corn. Netherl. 1975. 14. no. 1. p. 46-50 “Matched Filtering of Continuous Tone Transparencies Using Phase Media-Thermoplastics”. STEVENSON, W. H. MATEKUNAS, F. SWEENY,D. W. Appl. Optics U.S.A. 1976. 15. no. 6. p. 1541-1549 “Infrared Holography. An Analysis of the Thermal Behavior of Thin Films”. MATTHIJSSE, P. J. Opt. SOC.Amer. 1975. 65. no. 11. p. 1337 “Sufficient Conditions for a Thin Filter Description of Thick Phase Filters”. MCLELLAN, J. H. IEEE Trans. Circuit Theory U.S.A. 1973. 20. no. 6. p. 697-701 “A Unified Approach to the Design of Optimum FIR Linear-Phase Digital Filters”. SIEVERS, A. J. MON, K. K. Appl. Optics U.S.A. 1975. 14. no. 5. p. 1054-1055 “Plexiglass: A Convenient Transmission Filter for the FIR Spectral Region”. PERRYMILES Opt. Engineering 1976. 15. no. 5. p. 451 “High Transparency Infrared Materials-A Technology Up Date”. SCUDIERI, F. BERTOLOTTI, M. FERRARI, A. APOSTO,D. Opt. Com. Netherl. 1975. 15. no. 1. p. 54-56 “Nematic Liquid Crystal PI-Filter for Difference Holography”. SU, s. F. GAYLORD, T. F. J. Appl. Phys. U.S.A. 1975. 46. no. 12. p. 305-308 “Unified Approach to the Formation of Phase Holograms in Ferro-electric Crystals”.

116


[I1

YOUNG,L. WONG,W. K. Y. THEWALT, M. L. W. CORNISH, W. D. M. Appl. Phys. Letters. U.S.A. 1974. no. 6. p. 264-265 “Theory of Formation of Phase Holograms in Lithium Niobate”.

References ABITBOL, C. I. and J. J. CLAIR,1975, Opt. Acta 2, 145. A B m O L , .C. I. and J. J. CLAIR,1977, Appl. of Holog. and Opt. Data Process, eds. Marom and Friesem (Pergamon Press, Oxford and New York). BALKANSKY, M., 1973, Photonics, ed. Lallemand (P. Gauthier Villars, Paris). B. H., 1951, Research techniques leading to the development of a wavelength BILLINGS, infrared viewer evaporograph (Baird Associates Inc., Cambridge, Mass.). D. K., 1974, 115th SMPTE Tech. Conf. and Equip. Exhibit. (Univ. Los BROADBEN, Angeles). G., 1974, SID. Inter. Symp. and exhibition (San Diego). BROUSSAUD, BROWN,B. R. and A. W. LOHMANN, 1966, Appl. Optics 5,967. BURCKHARDT, C. B., 1970, Appl. Optics9, 1949. CAMATINI, E., 1972, Optical and Acoustical Holography, ed. Camatini (Plenum Press, New York). CAMATINI, E., 1975, Progress in Electro-optics, ed. Camatini (Plenum Press, New York). CAMPAAN-KRAMER, P., 1973, Philips Tech. Rev. 33, 178. CAWS, J., F. GIRARDand R. CLARK,1967, Appl. Optics 8, 1433. CHUTJIAN, A. and R. J. COLLIER,1967, J. Opt. SOC.Amer. 57, 1405. CLAIR,J. J., 1972, Appl. Optics 11, 480-481. CLAIR,J. J., 1972, Opt. Communications 6, 135. CLARRICOATS, P. J. B., 1976, Progress in Optics, Vol. X I V , ed. E. Wolf (North-Holland, Amsterdam) p. 327-402. COBLITZ,D. B. and J. A. CARNEY,1974, Appl. Optics 13, 1994. CUTRONA, L. J., 1960, I.R.E. Trans. Inf. Theory 6, 386. COUDERC, A., 1952, C.R. Seance Acad. Sciences, Paris 235, 491. P. and Z. MOLLET, 1938, Physik 85, 108. CZERNY, DALLAS, W. J., 1971, Appl. Optics 10, 673. 1955, Nature 175, 646. DOBROWOLSKY, J. A. and W. WEINSTEIN, DUPOISOT, H. and J. MORIZET,1974, Opt. Communications 10, 316. EMMONY, D. C., 1974, Optics and Laser Technology, June 74, p. 104. FLAMAND, J., G. S. HAYAT,M. LACROIX and A. CRILLO,1975, Opt. Engineering 14,420. FIRESTER,A. H., 1973, Appl. Optics 2, 198. FREJLICH,J. and J. J. CLAIR, 1977, J. Opt. Soc. Amer. 67, 92-96. FRIESEM,A. A. and J. L. WALKER,1970, Appl. Optics 9, 20. GALE,M. T. and K. KNOPP,1976, Appl. Optics 15, 2189. GOODMAN, J. W. and A. M. SILVESTRI,1970, I.B.M. Journ. of Research and Develop. 14, 478. GOODMAN, J. W. and J. R. FIENUP, 1973, J. Opt. Soc. Amer. 63, 1325. GRADE, A,, 1974, Appl. Optics 13, 2942. HASKELL,R. E., 1973, J. Opt. Soc. Amer. 63, 504. HERSCHELL, J. F. W., 1840, Phil. Trans. Roy. Soc. London 131, 52. HOLLADAY, T.M. and J. D. GALATIN,1966, J . Opt. SOC.Amer. 56, 869. HRBECK, G. W., 1974, see BROADBEN [1974].

111

REFERENCES

1 I7

HUANG,T., 1971, Proc. IEEE. 59, 1335. HUIGNARD,J. P., F. MICHERON and E. SPITZ, 1976, Optical Properties of Solids. New Developments, ed. N. Y. Seraphin (North-Holland, Amsterdam). ICHIOKA, Y., M. IZUMIand T. SUZUKI,1969, Appl. Optics 8, 2461-2471. KORONKEVKCH, V. P., G. A. LENKOVA, I. A. MIKHAL.TOOVA, V. J. REMESNIK, V. A. FATEEV,and V. G. TSUKERMAN, 1976, Opt. Inf. Processing, eds. Y. E. Nesterkhin and G . W. Stroke (Plenum Press, New York). KOZMA,A, R. and D. L. KELLY, 1965, Appl. Optics 4, 395. KMETZ, A. R., 1976, Non-emissive Electro-optic display, ed. F. K. Von Wullisen (Plenum Press, New York). LEE, S. H., 1974, Optical Engineering 13, 196. LEE, W. E., 1970, Appl. Optics 9, 639. LEITH.E. N. and J. UPATNIEKS, 1964, J. Opt. Soc. Amer. 54, 1295. LEMAITRE, G.,1972, Appl. Optics 11, 1630. LESEM,L. B., P. M. HIRSH, J. A. JORDAN Jr. and J. C. PATAU.1970, 1.B.M. Journ. of Research and Develop. 14, 485. LOHMANN, A. W. and D. P. PARIS,1968, Appl. Optics 7, 651. LOHMANN, A. W. and H. W. WERLICH,1971, Appl. Optics 10, 670. LOHMANN, A. W., 1973, Optical Holography (Plenum Press, New York). LOULERGUE,J. C., 1971, Thesis, Paris. Fac. Sciences Orsay. MARATHAY, A. S., 1969, J. Opt. Soc. Amer. 59, 748. MARECHAL, A. and P. CROCE,1953, C.R. Seance Acad. Sciences, Paris 237, 607. MARIOGE, J. P., 1965, Nlle. Rev. d’Optique 44, 57. MEYER,A. J. and R. HICKLING, 1967, J. Opt. SOC.Amer. 57, 1388. MIYAMOTO, K., 1961, J. Opt. SOC.Amer. 51, 17. MUELI-ER, P.F. and G. 0. REYNOLDS, 1967, J. Opt. Soc. Amer. 56, 1438. NESTERIKHIN,Yu. E., and G. W. STROKE,1976, Opt. Inf. Processing (Plenum Press, New York). NISHIDA,N., 1974, Appl. Optics 13, 2769. O’NEIIL,E. L., 1956, I.R.E. Trans. Inf. Theory 2, 56. PUECH,C., 1973, Opt. Communications 7, 135. RANSOM,P. L., 1972, Appl. Optics 11,2554. SCHMAHL, G. and D. RUDOLPH. Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland, Amsterdam) p. 195-244. SCHULZ,L. G., 1947a, J. Opt. SOC.Amer. 37, 509. SCHULZ, L. G., 1947b, J . Opt. SOC.Amer. 37, 349. SERAPHIN, €3. o., editor, 1976, Optical Properties of Solids. New Developments (NorthHolland, Amsterdam). SHANNON, R. P. and R. E. WAGNER,1974, Appl. Optics 13, 1683. SMITH,H. M., 1968, J. Opt. Soc. Amer. 58, 533. Sprrz, E.,L. d’AuRIA, J. P. HUIGNARD, and A. M. ROY, 1972, Opt. Communications 5, 232. STROKE,G. W., 1967, Scien. Research 41, 256. STRONG,J. and E. GAVIOLA,1936, J. Opt. Soc. Amer. 26, 153. TAQUET,M., 1963, Sciences et Industries Photographiques, Vol. 34, No. 9, p. 10. TORRES,L. H., 1975, Opt. Acta 22, 963. TSUJUICHI, J., 1963, Progress in Optics, Vol. 11, ed. E. Wolf (North-Holland, Amsterdam) p. 133. VANDERLUGT,A., 1968, Optica Acta 15, 1. WATERS,J. P., 1960, Appl. Phys. Letters 9, 405. ZERNIKE,F., 1935, Z. Tech. Phys. 16, 454.



111

COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS BY

WAI-HON LEE Xerox Palo Alto Research Center 3333 Coyote Hill Road, California 94304, U.S.A.

CONTENTS PAGE

9 1. INTRODUCTION . . . . . . . . . . . . . . . . 121 § 2. TECHNIQUES FOR

ATED HOLOGRAMS

8 3. QUANTIZATION HOLOGRAMS

MAKING COMPUTER-GENER-

. . . . . . . . . . . . . . 126

IN

COMPUTER-GENERATED

. . . . . . . . . . . . . . . . . 168

§ 4. APPLICATIONS OF COMPUTER-GENERATED HOLO-

GRAMS. . . . . . . . . . . . . . . . . . . . 173

9 5. SUMMARY AND COMMENTS . . . . . . . . . . . 227 REFERENCES

. . . . . . . . . . . . . . . . . . . 229

0 1. Introduction Computer-generated holograms, synthetic holograms and computer holograms are terms used to refer to a class of holograms which are produced as graphical output from a digital computer. Given a mathematical description of a wavefront or an object represented by an array of points, the computer can calculate the amplitude transmittance of the hologram and display the result on a CRT or plot it on paper. Just as for conventional holograms, computer-generated holograms can be classified as image holograms, Fourier transform holograms or Fresnel holograms, depending on the relationship between the object and the complex wavefront recorded in the hologram. Using photoreduced copy of the graphical output from a computer as holograms is only one of many things that distinguishes computergenerated holograms from conventional ones. When the digital computer is used to calculate the transmittance function of a hologram, the object wavefront is just a mathematical description inside the computer. In practice, it may not even be realizable with optical components. Consequently, with the digital computer we can create optical elements that cannot be fabricated by conventional methods. Another distinction between a computer-generated hologram and a conventional hologram is in the way that the complex wavefront is recorded. In off-axis reference beam holograms as developed by LEITH and UPATNIEKS [1962], the amplitude transmittance of a hologram recorded under ideal conditions is proportional to t(x, y) = 1ReJZmax +A(x, y)eJ'p(x,Y)12

In eq. (1.11, ReJzmrxrepresents the tilted reference wave and A(x, p)eJs(x*y) the object wave. t ( x , y) is the resulting intensity variation of the interference pattern between the two waves. In computer-generated holograms, the transmittance of the hologram and the object wave is not 121

122

COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES AND APPLICATIONS

[III. 5 1

restricted to the relationship specified by eq. (1.1). In fact, most of the work in computer-generated holograms has dealt with the problem of coding the complex object wavefront for convenient production on computer graphic devices. Coding as used here means the conversion of a complex valued function into a real, nonnegative function in such a way that the complex valued function can be retrieved intact by optical means at a later stage. Over the last decade, because of the interest in coherent optics and the availability of digital computers as scientific tools, this special field of using computer and graphic devices to make holograms has attracted the attentions of many researchers, and has resulted in many publications in the scientific journals. The references at the end of this chapter contain a listing of the publications that are related to computer-generated holograms. Studies on computer-generated holograms can be roughly divided into three main categories: (1) Coding techniques, (2) Applications, and (3) Techniques for improving the quality of computer-generated holograms. In the third category are problems such as that of finding the best random phase code or algorithm for reducing the dynamic range in the Fourier transform of an image (AKAHORI [1973], ALLEBACH and LIU [1975], DALLAS[1973a], GABELand LIU [1970], GABEL[1975], GALLAGHER and LIU [1973], GALLAGHER [1974]), and the problem of quantization noise in the hologram (ANDERSON and HUANG[19691, GOODMAN and SILVESTRI [1970], DALLAS[1971], DALLASand LOHMANN [1972], NAIDU[19751). Quantization noise occurs in computer generated holograms because the computer graphic devices have limited gray levels and a limited number of addressable locations in their outputs. Depending on the techniques used to make the computer-generated holograms, quantization will limit accuracy in reconstructing the phases, the amplitudes or both of the desired wavefronts. The coding of complex wavefronts to make computer-generated holoand LOHMANN [19661 with their grams was first demonstrated by BROWN detour phase hologram. An interesting aspect of their technique is that the computer-generated hologram is made without explicit use of a reference wave or a bias. Also, their holograms have only two levels of amplitude transmittance (0 or 1). This makes the holograms very easy to make, and they can be copied many times without degradation. Because computer-guided plotters are available in most computing centers, Brown and Lohmann’s method has been widely used for making binary

111, § 11

INTRODUCITON

I23

computer-generated holograms. Their approach to the coding problem is evident in the many techniques that developed afterwards. A different coding method for making computer-generated holograms was later described by BURCH [1967]. Since the term A*(x, y) in eq. (1.1) does not contribute to the reconstruction of the object wavefront, Burch suggested that the computer calculate only the values of the last term in t ( x , y ) at regular sampling intervals. A constant bias would then be added to all the samples to make them nonnegative. Because the data samples in the computer-generated holograms are all equally spaced, Burch's holograms can be recorded from CRT displays. HUANCand PRASADA [1966] suggested a similar modification of eq. (1 1) for making computergenerated holograms. Another type of computer-generated holograms is the kinoform (LESEM, HIRSCH and JORDAN [1967, 1968, 1969, 1970]), which uses the relief images recorded on film to record the phase variations of the complex wavefronts calculated by the computer. The relief heights of the kinoforms are proportional to the residues of' the phase variations after taken modulo 2 m As a result, phase variations in kinoforms are restricted to a range of [0,27~].Because kinoforms are made without using a carrier frequency, wavefronts reconstructed from kinoforms are centered on the optical axis. When properly made, kinoforms can have a diffraction efficiency of 100%. However, the kinoform, as originally conceived, cannot record the amplitude variation of the wavefront. LEE [1970a] recognized that if two real functions were sampled at periodic intervals and if there were a delay in the sampling of one of the functions, a constant phase difference in the Fourier transforms of the two functions could be created. Therefore, by combining four real nonnegative functions that have been sampled with quadrature phase delays, one can produce the Fourier transform of the desired complex valued function in the Fourier transform plane. Complex valued functions can also be decomposed into three positive components with phase differences of 120" (BURCKHARDT [1970]). Lee's method, like Burch's, is designed for displaying computer-generated holograms on a CRT. KIRKand JONES [1971] described a method €or incorporating amplitude information into kinoforms. Their method requires the use of a carrier frequency which results in a reduction in the efficiency of the kinoform. and GOODMAN [1973] used Kodachrome slide transparency CHU,FIENUP film to make kinoforms that can have both phase and amplitude variations. They called their computer-generated holograms ROACH for

124

COMPUTER-GENERATED HOLOGRAMS: TECHNIQUES A N D APPLICATIONS

[I& $ 1

“Referenceless On- Axis Complex Holograms”. Another method of modifying the kinoform to record the amplitude information was also described by CHUand FIENW[1974]. The most recent technique for making computer-generated holograms, developed by LEE [1974], is for making image holograms of wavefronts having only phase variations. This type of hologram is similar to an interferogram and is made by finding the location of the fringes corresponding to the interferograms and plotting them on paper. WYANTand BENNETT [19721 used a similar approach to generate aspheric wavefronts for testing optical surfaces. However, they used a ray tracing program for locating the fringes for holograms. These various techniques for making computer-generated holograms are summarized in Table 1. Except for the kinoform-type holograms which are already phase-relief recordings, amplitude-type holograms can be converted into phase recordings by bleaching the developed holograms. Applications of computer-generated holograms can be divided into five areas: (1) 3-D image display, (2) optical data processing, ( 3 ) interferometry, (4) optical memories, and (5) laser beam scanning. Some of the early work in computer-generated holograms was on the problem of computing the wavefronts for three dimensional objects (WATERS [1966, 19681). This problem is far more complicated than the coding problem. Most of the computer graphic devices are rather limited in resolution and cannot be used to display holograms containing high degrees of complexity. For displaying three dimensional objects generated by computer, the method developed by KING, NOLL and BERRY [1970] is an attractive alternative. Recently, YATAGAI [1974, 19761 used the same method to make mosaic computer-generated holograms for 3-D image display. In the early development of computer-generated holograms, their application to matched filtering, code translation and spatial frequency LOHMANN and PARIS filtering was investigated quite extensively (BROWN, [1966], LOHMANN, PARISand WERLICH[1967], LOHMANN and PARIS [19671). More recently, computer-generated holograms have been used as matched filters for processing synthetic aperture radar data (LEE and

TABLE1 Different types of computer-generated holograms

Type of hologram

On-axis reconstruction

Amplitude transmittance

Other features

References

Detour phase holograms

No

Binary

Amplitude and phase of wavefront are coded separately

BROWNand IDHMANN [1966, 19691

Modified off -axis reference beam holograms

No

Gray levels

Require reference beam and bias

BURCH[1967], HUANCand PRASADA [1966], LEE[1970]

Kinoforms

Yes

Constant

Wavefronts are recorded as surface relief on film

LESEM,HIRSCH and JORDAN [19671, KIRKand JONES[19711, CHU,FIENUP and GOODMAN [ 19731, CHUand ~ ~ E N U[:97Sj F

Computergenerated interferogram

No

Binary

Hologram has many fringes similar to in terferograms

LEE [ 19741

126


[III, 5 2

GREER[19741). Deblurring experiments using computer-generated holograms were reported by CAMPBELL, WECKSUNG and MANSFIELD [19741. The potential of computer-generated hologram in interferometry was [1969]. Experiments demonstrating the use of pointed out by PASTOR computer-generated holograms in interferometers were reported by MACGOVERN and WYANT [1971] and WYANTand BENNETT[1972]. Inclusion of computer-generated holograms in interferometers has become an established technique for testing optical surfaces (BIRCHand GREEN[19721, ICHIOKA and LOHMANN [1972], FERCHER and KRIESE [1972], TAKAHASHI, [1976]). KONNOand KAWAI[1974], SIROHI,BLUMEand ROSENBRUCH Recently, BRYNGDAHL [ 19731 used computer-generated holograms to provide reference wavefronts for displaying interferograms with radial and circular fringes. Computer-generated holograms have also been used in a shearing interferometer (BRYNGDAHL and LEE[1974]). One dimensional computer-generated holograms have been used in optical data storage (KOZMA,LEE and PETERS[1971], KOZMA[1973]). Fourier transforms of segments of the digital data are calculated in real time and the holograms are recorded on film using a scanning laser beam recorder. This solves the problem of having to use page composers and elaborate optics to form the Fourier transform holograms. Computer-generated holograms can be viewed as variable-frequency gratings. By controlling the frequency variations of the hologram, the hologram can be used to deflect the laser beam in any desired pattern (BRYNGDAHL and LEE [1975, 19761, BRYNGDAHL [1975]). Raster scans or two dimensional scan patterns could be produced by moving a computergenerated hologram across the laser beam. We will review the different techniques for making computer-generated holograms and the five areas of applications of computer-generated holograms. The effects of quantization on reconstructed wavefronts from computer-generated holograms will also be discussed. § 2. Techniques for Making Computer-Generated Holograms 2.1. DETOUR PHASE HOLOGRAMS

In 1966, BROWNand LOHMANN described a detour phase method for making binary computer-generated holograms for complex spatial filtering. Their holograms are distinguished by three unique properties: (1)the transmission of the hologram is binary, (2) the hologram can record both

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MAKING COMPUTER-GENERATED HOLOGRAMS

127

the amplitude and phase of any complex valued function and (3) the hologram is not recorded with the explicit use of a carrier frequency o r a bias as in the off -axis reference beam holograms. To make a binary hologram of this type, the wavefront represented by the complex valued function, A(x, y)e’s(x,y),is first sampled at equally spaced intervals in accordance with the sampling theorem. That is, the sampling distance must be smaller than 1/U where U is the spatial bandwidth of the wavefront in the direction of sampling. In plotting the hologram the paper is divided into equally spaced cells. Rectangular apertures are drawn inside each cell. Each aperture is determined by three parameters: its height, h,,,, its width, w,,, and its center with respect to the center of the cell, c,,. Figure 1 shows one of the sampling cells in the detour phase hologram. The index nm indicates the relative location of the cell in the hologram plane. The parameters of the aperture are selected as follows:

and

:

y=md ._ _ _ - _ _

I

w hnrn

-.;___..-._ __ __

:“I I 1

Cnl?

x =nd,

-

128


[III, 5 2

with many small apertures is plotted and recorded on photographic film, it creates an amplitude transmittance on the film given by

The function p ( x ) is given by

P(X> =

[XIS+

1 0

for otherwise.

(2.3)

That this binary hologram works can be verified by putting it in the optical system in Fig. 2. The hologram is illuminated by a collimated laser beam. The Fourier transform of the desired wavefront occurs in an off-axis region on the Fourier transform (back focal) plane of lens L,. The aperture mask shown in Fig. 2 passes only one diffracted wave from the hologram through the optical system. Lens L2 performs the inverse Fourier transformation to produce the wavefront A (x,y)eJ'+'p(x,y , at the back focal plane of lens L2. The lens system in Fig. 2 is a telecentric system. Without the aperture mask, the optical system images the hologram one to one at the back focal plane of lens L,. The aperture mask in the optical system converts it to a bandpass system. The wavefront at the back focal plane of the lens is the bandpassed output of the diffracted waves from the hologram. This property of the binary hologram can be demonstrated analytically by examining the Fourier transform, Tl(u,v), of the function t,(x, y):

Tl(u, v) = =

I

m

m

-m

-m

t,(x,

y)e-12T(ux+uy) d x dY

c c G,,(u, exp {-j2.rr(ndxu + mdyv)>, V)

n

(2.4)

m

Fig. 2. Optical system for reconstructing wavefronts from computer-generated holograms.

I n , § 21

129


where Gnm(u, u

sin T W U sin m k m v

)

=

TU

p

m

exp (-~~Tc,,,u).

If the coefficients, G,,,(u, v ) , are independent of u and u, the function T,(u, v) is a Fourier series expansion of a periodic function. To continue, we expand the terms in G,,,(u, u ) that are dependent on the index, n, m, about u, = k/d, and v, = 0, the center location of the aperture in Fig. 2. As will be seen, the spatial frequency, uc, is equivalent to the carrier frequency in the off-axis reference beam holograms. The function Gnm(u,v), in a region where l u - u , l ~ l / d , and lu1-f,-(x,

y>)+jcf,+(x, Y>-fi-(x, Y ) } . (2.25)

The functions on the right-hand side of eq. (2.25) are, in order, the positive and negative portions of the real and imaginary parts of the complex function. The phase information of these four functions can be recorded by sampling each function with delays, E , given by 0, dJ4, d,/2 and 3dJ4. The smallest value of E is d,/4, this is 1/4 of the sampling

111, § 21


141

interval. Therefore, the complex function must be sampled at a rate equal to four times the spatial bandwidth of the function along the x coordinate. This is in contrast to the double bandwidth required by Burch’s method. The transmittance of the hologram resulting from this method is equal

to

Because each of the component functions is real and nonnegative, the function t(x, y) in eq. (2.26) is also real and nonnegative. WECKSUNG and MANSFIELD [19741 observed that Recently, CAMPBELL, Lee’s method could be interpreted as a modified off-axis reference beam hologram with amplitude transmittance function given by t(x, y ) = A ( x , Y){COS [2raX-cP(X7

Y)I+ICOS[ ~ T ~ ~ x - vy)II>, (x,

(2.27)

where a = Ud,. The bias term in such a hologram is the function A(x, y) (cos[ ~ T C Y X q ( x , y)]l. The equivalence of the transmittance function in eq. (2.26) and eq. (2.27) can be demonstrated by sampling the function t(x, y) in eq. (2.27) at locations x = nd, + kdJ4 and y = md,, where k = 0, 1 , 2, 3 and showing that its value at these locations are the same as the values given by eq. (2.26). For example, the sampled value of fi+(x,y ) at x = nd, + dJ4 and y = md, is equal to fi+(ndx+ dJ4, mdy) = O.SA(nd, + dx/4,m&){sin cp(nd, + dJ4, m4)+ lsin q(nd, + dJ4, rnd,,)l>. (2.28)

On the other hand, the sampled value of t(x, y) in eq. (2.27) is equal to t(&

+ &/4, m 4 ) = O.SA(n4+dJ4,m4) x{cos[2~/4-cp(nd,+dJ4,m4)I+(cos [ 2 ~ / 4 - c p ( n d+dx/4, , md,)lI} = fi+(ndx+

m4).

(2.29)

I42

C'OMPLTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS

[III, $ 2

Fig. 9. (a) Original continuous-tone picture. (b) Enlargement of a small section of the hologram made by Lee's method. (c)The image reconstructed from the hologram in (b).

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MAKING COMPUTER-GENERATED H O L O G M S

143

The equivalence of the two functions at other sampling locations can be shown in a similar way. With this interpretation o f Lee’s method it can be shown that by choosing CY to be 2/3dx, the transmittance of the hologram will be equal to that suggested by BURCKHARDT [1970] for reducing the sampling rate in the Lee hologram. In this case, the complex function is decomposed into only three components rather than four components as used by Lee. An example of Lee’s hologram is shown in Fig. 9. The object for this Fourier transform hologram is the continuous tone picture with 128 X 128 pixels as shown in Fig. 9(a). The Fourier transform of the picture has 128x 5 12 samples. Four times more samples are used along one direction than the other as required by this coding technique. The value of the transmittance of the hologram is quantized into 4 levels and is recorded by the different heights of the narrow bars in the hologram. This creates a hologram having 512 x 5 12 samples. Figure 9(b) is a magnified portion of the hologram. The reconstructed image is shown in Fig. 9(c). The speckles in the image are due to the random phase factor applied to the original picture for reducing the dynamic range in the Fourier transform of the picture. 2.3. KINOFORMS

2.3.1. Fourier transform-type kinoforms The kinoform is a different form of computer generated hologram. It is not recorded as amplitude transmittance on film as are the previous types of computer-generated holograms but as relief patterns on film. Most of the computer-generated holograms discussed so far rely on a diffraction effect to reconstruct a complex wave field. The kinoform, however, like a Fresnel lens, changes the phase of the illuminating wave by its thickness variation. The kinoforms made by LESEM,HIWCHand JORDAN [1967], were Fourier transform type holograms. To make such a kinoform, the discrete Fourier transform of an object is first calculated by the computer. The phase angles of the complex samples of the Fourier transform are determined and the amplitude of the transform is set equal to 1. When the kinoform is used to display the image of an object, a pseudorandom phase array is used with the original object to reduce the effect of losing the amplitude information in the kinoform. The phase angles of the

Fourier transform are obtained by taking the arctangent of the ratio bJb, where bi and b, are the imaginary and real part of the complex sample of the transform. The angles determined this way have values between - 7 ~ to 7~ radians. The variations of the phase of the transform are then displayed on a CRT and recorded on film as an intensity variation. An example of the kinoform obtained up to this point is shown in Fig. lO(a). The exposed film after development then goes through a bleaching process to create a relief pattern on the film emulsion. When the kinoform is inserted in the optical system shown in Fig. 2, an image of the digital picture will be reconstructed at the back focal plane of lens L1. Because the amplitude variation is not recorded, the reconstructed image tends to be noisy. Moreover, if the relief height in the kinoform is not matched to the phase variation of the Fourier transform of the object, there will be, as can be observed in Fig. 10(b), a focused spot in the center of the reconstructed image.

2.3.2. Phase Fresnel lens

A similar procedure for creating a phase image has been described by MIYAMOTO [1961] for making the phase Fresnel lens. For this application, the wavefront to be recorded has only phase variation. Hence, the kinoform technique can be used to record this type of wavefront without error. Recently, BRYNGDAHL [1973] used the kinoform technique to record helicoid wavefronts for use in an interferometer. The phase variation in a helicoid wavefront is linear in the azimuthal direction as shown in Fig. ll(a). The interferogram in Fig. l l ( b ) and (c) illustrates the phase variation of the wavefront reconstructed from the kinoform. The interferograms are the interference pattern between the reconstructed wave from the kinoform and two different phase objects. The spokes in the interferograms are the constant-phase contours of the phase variation recorded in the kinoform. How well does the kinoform work is dependent on the precise control of the relief heights on the developed film emulsion. If the relief heights in the kinoform produce phase variations exceeding 277 rad., multiple waves will be generated by the kinoform. The bright spot in the center of the reconstructed image in Fig. 10(b) is the result of the phase mismatch in the kinoform. This phase mismatch problem of the kinoform can be studied by modelling the phase recording process in the kinoform as a nonlinear

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145

Fig. 10. (a) Phase variation for making a kinoform. (b) Image reconstructed from the kinoform with the phase variation in (a). (Courtesy of Lesem, Hirsch and Jordan.)

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COhQ'UTER-GENERAF2D HOLOGRAMS: TECHNIQUES A N D APPLICATIONS

[III, 5 2

Fig. 11. The phase variation in (a) is made into a kinofonn. The interferograms (b) and ( c ) are obtained by using the wavefront from the kinofom as a reference wave.

operation on the phase function. The nonlinear limiter characterizing the kinoform recording process is shown in Fig. 12(a) with Z being the input to the limiter and 2' as the output. 2' is linear in 2 for a short interval and its value is limited to the range 0 to 1+ p. Figure 12(b) shows a phase function q ( x ) and the resulting phase variation after it passes through the nonlinear limiter. The constant p is the parameter determining the amount of phase mismatch in the kinoform. If the phase variation of the original wavefront is Z, the wavefront reconstructed from the kinoform is equal to eJz'. Since 2' is a periodic

111, 8 21


147

\ (b)

Fig. 12. A nonlinear limiter with the input-output relationship as shown in (a) can convert a continuous-phase function into a discrete valued function such as the one shown in (b).

function in 2, we can expand eJZ’in a Fourier series:

where

(2.31)

Equation (2.30) is valid for any real function 2. Therefore, we can substitute &, y) for 2 in eq. (2.30) and get (2.32) Note that if p = 0, the only nonzero term in eJZ’is c l . Hence ejZ’

-e

j d x , Y)

(2.33)

as desired. For /3 an integer, the nonzero term in eq. (2.30) is shifted to n = 1+ p. This produces a wavefront in the reconstruction with IZ times more phase variation. However, if p is not equal to an integer, there will be more than one nonzero term in elz’. The n == 0 term in eq. (2.30) will cause a bright spot such as the one shown in Fig. 10(b) to occur in the

148


[III, 6 2

reconstruction. One way to solve the phase mismatch problem of the kinoform is to add a linear phase term to the phase variation before it is recorded in the kinoform. This is similar to the use of a carrier frequency in the off-axis reference beam hologram. With this linear phase term the higher order diffracted waves generated by the mismatch of the relief heights in the kinoform will propagate in different directions and can be separated in the reconstruction process. If there is perfect phase matching in the kinoform, the addition of the linear phase term will not affect the diffraction efficiency of the kinoform. It will change only the direction of propagation of the reconstructed wave.

2.3.3. Extensions of kinoform technique Since kinoforms cannot record the amplitude variation of the wavefronts, the natural extension to the kinoform technique is to add an extra function to the phase variation so that in the reconstruction this extra term in the phase can modify the amplitude of the incident wavefront. To introduce amplitude variation to the reconstructed wavefront, light has to be taken away from the wavefront. This results in a kinoform with less light efficiency. A phase function that can modify the phase and the amplitude of an incident wave is (KIRKand JONES [1971])

qo,(x,y) = a(x, y ) cos 2TffX+ d x ,Y),

(2.34)

where a ( x , y) is a function of the amplitude of the wavefront. With a plane wave incident on the kinoform with q l ( x , y ) as its phase variation, the waves emerging from the kinoform are terms in the expansion of eiv,(x.

Y).

eIm,(*.Y)=e"P'l.L)

C b , ~ , [ a ( xy)] , cos 2 ~ n a x ,

(2.35)

n

where J,(x) is the Bessel function and b,, is equal to 1 for n = 0 and 2 for nfO. Consequently, if a(x, y) is such that

the diffracted wave from the n = 0 term in eq. (2.36)will have the proper

111, 0 21


149

amplitude and phase information. The wavefront can also be reconstructed from the higher order diffracted waves of this kinoform provided that

J J a ( x , y ) l = A(x, Y ) .

(2.3'7)

Another way to introduce amplitude control to the kinoform is to use an additional layer of emulsion. Some photographic films, such as Kodachrome 11, have more than one layer of emulsion. The different layers of the emulsion can be exposed independently by light of different wavelengths. After the film is processed and is illuminated with monochromatic light, one layer can be made to modify the amplitude of the incoming wave; while other layers which are transparent can cause phase and GOODMAN [1973] used this shifts of the incoming wave. CHU,FIENUP technique to produce kinoforms that can record both the amplitude and phase of the wavefronts. To make this type of multiemulsion kinoform which is called ROACH (Referenceless -On-Axis Complex Hologram) by its inventors, the film is first exposed to the brightness variation corresponding to the amplitude of the wavefront through a red filter. The phase variation will be exposed to the film through a blue-green filter to generate a relief pattern. If this transparency after development is illuminated by the red light from a helium-neon laser, the layer exposed to red light will modulate the amplitude of the incident wave and the layers exposed to blue-green light will modulate the phase. C m and FIENW[1974] also proposed two other methods that use parity sequence to make kinoforms that can have amplitude control with a single layer of emulsion. In one of their methods, the phase recorded in the kinoform is equal to (2.38) The phase angle O(x, y) will be determined by the amplitude of the wavefront. This particular kinoform will produce a wavefront given by - ejv(x.

Y )

{cos O(x, y ) + j sin O(x, y)}.

(2.39)

Hence, if cos O(x, y ) = A ( x , y), the first term in eq. (2.39) will be the desired complex wavefront. To determine O ( x , y) we assume that the maximum value of A(x, y) is normalized to 1. Although eq. (2.39) contains the correct wavefront, it also contains a term that has an

150


[III, 8 2

improper amplitude variation. To get rid of the second term, Chu and Fienup used a spatial multiplexing technique to combine two functions in a single kinoform. One of the functions is, simply, f,(x, y ) in eq. (2.39); the second is f 2 ( x , y ) =e ~ ~ v ) - ed (x,~ Y)I ,

-

eJ*P(X,

Y)

[cos O(x, y ) -j sin O(x, y ) ] .

(2.40)

It is clear that A ( x , y)eJqP(x,y) is equal to O.5{f1(x,y ) + f 2 ( x , y)}. These two functions are combined in the kinoform by means of the delay sampling technique discussed in 0 2.2.3. The sampled function f ( x , y) used to make the kinoform is

Because f i ( x , y) and f2(x, y ) have only phase variation, f ( x , y) has constant amplitude and is suitable for recording by using the kinoform technique. In the reconstruction process, when f(x, y) is passed through a low-pass filter with impulse response, g ( x , Y ) = (sin ~x/d,)(d,/.rrx)(sin ~ y / d , , ) ( d , , / ~ y ) ,

(2.42)

the output from the filter will be

The output, fo(x, y), is the desired complex wave field. The delay sampling technique used in 52.2.3 not only combines four functions in a hologram, but also generates the proper phase shifts for each one in the reconstruction by means of a bandpass filter. However, the delay sampling technique used here simply combines two functions for recording in the kinoform. The phase shift due to the delay in the sampling is not used in the reconstruction. Figure 13 shows such a kinoform and the image reconstructed from it. Note that the noise from the second term in the function f i ( x , y) is not

111, § 21


151

Fig. 13 The phase variation shown in (a) is generated by using the parity sequence method which can add amplitude variation to the reconstructed wave. The reconstructed image is shown in (b). (Courtesy of Chu and Fienup.)

present in the low-frequency region where the reconstructed image is located. However, the bright spot in the center of the reconstruction indicates that there is a phase mismatch in the kinoform. If the function f2(x, y ) used in making the kinoform is f2(x, y ) = -el*(%Y)-e(X. Y ) (2.44) the correct wavefront will occur in the first diffracted order similar to Lee's hologram discussed in Q 2.2.3. In this case, the phase recorded by the delay sampling is used to cancel the second term in the function fl(x, y). Another phase function suggested by CHUand FIENUP[1974] is of the form f(x, y)=f1(x, y>+f2(x-L, Y ) .

(2.45)

The functions fi(x, y) and f2(x, y) are given in eqs. (2.39) and (2.40). The parameter L is the extent of the function fl(x, y') along the x-direction. It is used to make the functions fi(x, y) and f2(x, y) nonoverlapping in recording the kinoform. If F,(u, u ) and F,(u, u ) are the Fourier transforms of eJ'4(X,Y) cos O(x, y) and e"P("3Y)sin O(x, y), the Fourier transform of f(x, y) in eq. (2.45) can be written as F ( u , u ) = F,(u, u ) cos 2 r L u + F,(u, u ) sin 27~Lu.

(2.46)

Therefore, when the function F(u, u ) is sampled at u = n/L, its value is equal to F,( u, v), the Fourier transform of the desired complex wavefront. This technique is useful in making Fourier transform type kinoforms. In

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[III, 5 2

general this method is not as attractive as the first one discussed because of the difficulty in separating the functions F,(u,v) and Fi(u,v) in the reconstruction process.

2.4. COMPUTER-GENERATED INTERFEROGRAMS

2.4.1. Generation of binary holograms In many applications, the wavefronts to be recorded in the holograms have only phase variations. When these wavefronts are recorded as image and LOHMANN holograms, they are similar to interferograms (BRYNGDAHL [1968]). If a wavefront with phase variation only is recorded as an off-axis reference beam hologram, the amplitude transmittance of the hologram is t(x, y ) = 0.5{1 +COS

[ ~ T C Z X - P(X, y)]}.

(2.47)

In eq. (2.47) the function t(x, y ) has its maximum values at locations where 2rrax - q(x, y ) = 2mn,

(2.48)

and its minimum values at locations where n 2 ~ ax q(x, y) = 2 ~ ( +$).

(2.49)

Either eq. (2.48) or eq. (2.49) defines the location of the fringes in the hologram. The contrast of the fringes in the hologram can be enhanced by using the nonlinearity of the photographic film in the recording process to produce a binary fringe pattern. The fact that binary interferograms can be obtained by using the nonlinearity of photographic film suggests that binary computer-generated holograms can also be obtained by passing the function cos [27rax - q(x, y)] through a nonlinear limiter simulated by the computer during the computation. The desired nonlinear operation on the sinusoidal signal is shown in Fig. 14(a). For any input, the output of the limiter is either 0 or 1. The bias, cos Tq, is added to the input signal to control the width of the fringes in the binary hologram. The relationship between the output and the input of this nonlinear limiter can be analyzed by assuming that the input function is cos ~ T which Z produces at the output of the limiter the periodic function shown in Fig. 14(b). The width of the rectangular pulses in the figure is equal to q. Hence, the

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MAK"G COMPUTER-GENERATED HOLOGRAMS

153

(b)

Fig. 14. Nonlinear limiter for generating binary holograms

output function h ( z ) can be expanded as a Fourier series:

(2.50) The output of the limiter with cos [27rax - q ( x , y)] as input is obtained x cp(x, y ) for 2rr.Z and q(x, y ) for q into eq. (2.50) by substituting 2 ~ a to get

(2.51) In eq. (2.51) it is assumed that the bias function in Fig. 14(a) is a space-variant function. By selecting the function q(x, y) in such a way that A(x, y) =sin rrq(x, y), the m = -1 term in eq. (2.51) produces the wavefront A(x, y)eJq(x, y ) . Therefore, binary holograms can record both the amplitude and phase information of a wavefront without resorting to approximations such as those used in the detour phase hologram. For wavefronts that have constant amplitude, the parameter q can be used to determine the diffraction efficiency of the hologram in the reconstruction. When q is 0.5, all the even terms in eq. (2.51), except the m = 0 term, will disappear. This particular value of q allows the hologram to have a diffraction efficiency of 10% at its first diffracted order. Still

151


[III, 6 2

higher diffraction efficiency (40%) can be achieved by bleaching the hologram to convert it to a phase relief hologram. From Fig. 14(a) the hologram function h(x, y ) is equal to 1 if cos[2rrax-cp(x, y)]>cos 7rq.

(2.52)

This can also be written -qrr

< 27rax - q ( x , y) + 27rn < q7r,

(2.53)

where n is an integer. Equation (2.53) is the most general equation for making binary computer-generated holograms. When the wavefronts have only phase variation, it is easier to plot holograms with narrow fringes. This means that the value of q should be set to 0 for making a hologram with narrow fringes. For q = 0 eq. (2.53) becomes

2rrax - q ( x , y) = 2rrn.

(2.54)

This simplified equation is used below to discuss the procedure for making this type of binary hologram.

2.4.2. Considerations in making binary holograms The first consideration in making the hologram is the selection of the carrier frequency, a. Because there are many diffracted waves reconstructed from the binary hologram, it is important to use a sufficiently high carrier frequency to separate the first order diffracted wave from the higher-order waves.. From eq. (2.5 l), the spatial frequencies of the different diffracted orders in the x-direction are given by

%(X, y > = m b -(1/27r)dq(x, Y ) / W ,

(2.55)

where m indicates the diffracted orders. Similarly, the spatial frequencies in the y-direction for the diffracted waves are given by u,(x, Y)

= - (m/27r)fwix,YYdY.

(2.56)

It can be seen from eqs. (2.55) and (2.56) that the spatial frequency bandwidth along both directions increases linearly with m. The higherorder diffracted wave occupies a larger region in the frequency plane than the lower order diffracted wave. Because the spatial frequencies along the y-direction are independent of a, they will have no influence on the selection of the camer frequency a.

111,s 21

MAKING COMPLlTER-GENERATED HOLOGRAMS

15s

Suppose that the spatial frequencies along the x-direction are bounded between - U , and U,. To avoid overlapping between the first- and second-order diffracted waves in the frequency plane, the carrier frequency, a,must satisfy a

+ u,u2+2u,.

In eq. (2.57), cy + U, is the highest spatial frequency in the first-order diffracted wave and 2 a - 2 U , is the lowest spatial frequency from the second-order diffracted wave. Equation (2.57) is simply the mathematical statement for nonoverlapping between the first- and second-order diffracted waves. This condition also guarantees that the spatial frequencies from the higher-order waves will not extend to the spatial frequency region of the first-order diffracted wave. It is easy to show that the value of a determined by eq. (2.57) also satisfies the following inequality ff

+ u,< m(ff - U,),

(2.58)

where the righthand side of the inequality is just the lowest spatial frequency for the mth-order diffracted wave. Many wavefronts of interest have even symmetry about the y axis. This gives rise to symmetric distribution of the spatial frequencies of the wavefront about the u axis in the frequency plane. Therefore, the bounds on the spatial frequencies designated previously as UI and U, are the same and can be set equal to half of the bandwidth, U, of the wavefront along the x-direction. For these wavefronts, the condition given in eq. (2.56) for the carrier frequency a becomes a > 1.5 U.

(2.59)

This carrier frequency is three times higher than that needed for a Burch hologram. This is the price of recording the hologram in a binary format. When q = 1/2, the second diffracted order disappears because of the coefficient in eq. (2.51). This sets the third-order diffracted wave next to the first-order diffracted wave. This permits the condition on a to be somewhat relaxed, as can be seen in the following inequality a >33u,+ UJ.

(2.60)

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[ 111, (i 2

For a symmetric wavefront where U , = U,= U/2, eq. (2.60) becomes a>

u.

(2.61)

It is more time-consuming to make binary holograms with q = 0 . 5 (fringe widths equal to half of the fringe spacing). However, it does allow better utilization of the bandwidth of the hologram and produces a hologram with higher diffraction efficiency. The selection of the carrier frequency according to eq. (2.59) or eq. (2.61) also helps to reduce the complexities in locating the fringes in the hologram. Since the fringes are found by solving eq. (2.54) for some integer n, this is similar to finding the constant-value contour for an arbitrary function. But with the term 27mx on the left-hand side of eq. (2.541, the function is a monotonically increasing function in x. This property of the function on the left-hand side of the eq. (2.54), now designated by g(x, y), can be demonstrated by proving that

g(x1, y)-g(x*, Y)>O

(2.62)

for xl>x,. The left-hand side of eq. (2.62) is equal to

When x1- x2= A is small but positive, eq. (2.63) can be written as

g(xi, Y > - ~ ( x ,~, ) = 2 m A [ a- ( ~ / ~ T ) ~ c Py)laxI. (x,

(2.64)

By virtue of selection of the carrier frequency, the right-hand side of eq. (2.64) is always positive and this proves the inequality in eq. (2.62). The condition given in eq. (2.62) restricts the fringes, with the larger fringe index n, always on the right-hand side of the fringes with the lower index. As a result, once a fringe in the hologram is found, its location can be used as the starting point for searching for the next fringe. The inequality in eq. (2.62) also guarantees that the fringes in the hologram plane will not form closed loops. After selection of the carrier frequency, a, the next step in generating the computer hologram is to solve eq. (2.54) for the locations of the fringes. In general it is difficult to find an analytical expression that gives the location of y in terms of the coordinate x and the fringe index n. Therefore, the fringe locations are found by substituting the two coordinates successively into eq. (2.54) and testing to see whether they satisfy the equation. The spacing between the discrete points along the xdirection is given by T / M where T is l / a , the grating period of the

HI,§ 21


157

hologram, and M is an integer. The accuracy in locating the fringe is determine by TIM. The spacing between the points along the y-direction is equal to d,, = 1IV. V is the spatial bandwidth of the wavefront in the y-direction. A t these discrete locations where 3: = k,T/M and y = k,,dy eq. (2.54) is equal to

2~k,lM-cp(k,T/M, k,,d,,) = = 2 ~ n .

(2.65)

Multiplying both sides of eq. (2.65) by M / ~ Treduces the equation to

k , -(M/2rr)cp(kXT/M,k,d,)

= nM.

(2.66)

If modulo M is applied to eq. (2.66), the right-hand side of eq. (2.66) will be 0 because it is proportional to M. Thus, eq. (2.66) is simplified to , =O. Mod,{kx - ( M / ~ T ) v O ( ~ , T I Mk,,dy)}

(2.67)

By the use of residue arithmetic the fringe index II has been removed from eq. (2.66). Now instead of finding the pairs [k,, k,,]that can satisfy eq. (2.66) for a given value of n, it is only necessary to find the ones that make the residue of the left-hand side of eq. (2.66) equal to 0. This simplifies the computer programming involved in finding the fringes. There are two ways to plot binary holograms. Both require only a small number of memory locations for storing the computed data related to the hologram. In one method, a line segment with length d, is drawn at location ( k x ,k,,)where the condition in eq. (2.67) is met. In this case there is no need for storing computed data in the computer memory. Each point will be plotted as it is found. There is also no need to know the fringe index for the fringe points, as was pointed out earlier. However, this method requires longer plotting time because the pen has to move up and down for each line segment plotted. To solve this problem the successive points found along the y-directions are stored in the memory. The fringe index for each of these points is then calculated by substituting its location back into eq. (2.66). All the points with the same fringe index are then connected to form one fringe in the hologram. Once the first fringe has been found and plotted, the computer program will look for fringe points on the right-hand side of the previous fringe because of the property in eq. (2.62). This procedure is repeated until the whole hologram is plotted. In using this method, only one fringe line has to be in the computer memory at a time. The number of storage locations needed is equal to the space bandwidth product of the wavefront along the y direction.

I5X


[III, 0 2

2.4.3. Examples In this section, four different holograms are used to illustrate some of the properties of computer-generated interferograms. Some of the phase variations recorded in the holograms can be found in optical components such as a Fresnel lens, axicon, etc. For example, the spherical wavefronts in the first example can be obtained from a zone plate, or a lens. The conical wavefront in the second example is that of an axicon. The third hologram is equivalent to a Schmidt plate (LINFOOT [19S8]) used to correct the spherical aberrations in a spherical mirror. The last example is somewhat unconventional. The phase variation is linear in the azimuthal direction. This example demonstrates that a computer-generated interferogram can indeed produce phase variation that would be difficult to obtain with other fabrication techniques. A. Spherical wavefronts A spherical wavefront has phase variation given by

(2.68) where r 2 = x 2 + y 2 . The focal distance of the wavefront is F, and the wavelength of the laser illumination is A. The phase variation of a thin lens can be approximated by the phase variation in eq. (2.68). This wavefront has spatial frequencies along the x-direction and y-direction given by (2.69) The spatial frequencies of this wavefront are dependent on the location of the wavefront in a linear way. The maximum frequencies, as can be seen in eq. (2.69), occur at the boundary of the wavefront. Therefore, the spatial frequencies of the wavefront along the x-direction are bounded between -D/(2hF) and D/(2hF), where D is the lateral extent of the wavefront. Suppose that a hologram with a diameter of S m m is made to record the spherical wavefront with a focal distance of 8 m . The bandwidth of this wavefront along both axes is about 10 lines/mm €or h = 632.8 nm. To satisfy eq. (2.59), the carrier frequency, a,for the hologram is selected as 20 lines/mm. The average number of fringes in the hologram is given by N+= Da

= 100.

(2.70)

111, § 21


159

The maximum phase deviation of the wavefront at the edge of this hologram measured in 27r units (or number of wavelengths) can be obtained from eq. (2.68) and is equal to (2.71) In eq. (2.71) the relationship a = 2D/(AF) has been used. As will be found in other examples as well the maximum phase variation in the hologram is linearly proportional to the number of fringes in the hologram. The constant of proportionality between the number of fringes and the maximum phase deviation is strongly dependent on the maximum gradient (spatial frequency) in the wavefront. Figure 15(a) shows the computer-generated interferogram of the spherical wavefront. The original size of the hologram is 25.4 cm x 25.4 cm. It is plotted on a Hewlett-Packard 7202A graphic plotter. Plotting time and computation time is a total of about 20 minutes. The plot is photoreduced to about 5 mm o n the side onto Kodak 649F film. The Fourier transform of the hologram is shown in Fig. 15(b). The spectra of three different diffracted waves on either side of the optical axis can be observed. Because the spatial frequencies of this wavefront are linearly dependent on the coordinates of the wavefront, the spectrum of the wavefront takes on the shape of the hologram. The first two orders of the diffracted waves are separated as expected from the selection of the carrier frequency. The phase magnification in the higher order diffracted waves makes the spatial frequency bandwidth of the higher order difbacted wave larger. The bandwidth of the second-order diffracted wave is doubled along both coordinates so that the spatial frequencies occupy an area four times that of the first-order wave. The hologram in Fig. 15(a) is an image hologram of the phase variation in eq. (2.68). To show that the reconstructed wave from the hologram is indeed a spherical wave, we obtain an interferogram of the reconstructed wave. The computer-generated hologram is put at plane CGH in the optical system in Fig. 16. The regular grating in the optical system serves as a beam splitter to provide two illumination beams for the hologram. The frequency of the regular grating is matched to that of the hologram. With the two plane waves incident on the hologram many diffracted waves are reconstructed from the hologram. But only two of the first-order conjugate waves reconstructed are propagated along the optical axis. The

160


[III, 5 2

Fig. 15. (a) Computer-generated hologram of a spherical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

III,8 21

Grating

MAKING COMPUTER-GENERATED HOL.OGRAMS

L,

M,

CGH

L,

L,

M*

161

L,

I

Fig. 16. Optical system for obtaining the interferogram in Fig. 15(c)

second aperture mask in the optical system lets these two waves pass to form the interferogram at plane I. The intensity variation of the interference pattern is given by I(x, y)=c0sZq(x, y)=;{l+coS2q(x, y)}.

(2.72)

The interferogram shows twice the amount of the phase variation recorded in the hologram. The fringes in the interferogram are the constant phase contours of the wavefront. The interferogram in Fig. 15(c) is similar to a Fresnel zone plate. Therefore, this method can indeed generate holograms of wavefronts which are given to the computer only as mathematical descriptions.

B. Conical wavefronts A plane wave after passing through an axicon (MCLEOD[1954]) has a cone-shape wavefront. The phase variation of such a wavefront can be described by q ( x , y) = 2rrrlro.

(2.73)

r, determines the gradients of the wavefront along the radial direction. The bandwidth U of this wavefront along the x-direction is equal to 2/r0. Using a carrier frequency which is twice the bandwidth of the wavefront gives the following relationship between r, and a : (Y

= 4/ro.

(2.74)

The hologram shown in Fig. 17(a) is made by using a = 20 lineslmm to give an average of 100 fringes in a 5 mm-wide hologram. The maximum phase deviation of the conical wavefront recorded in this hologram is N,

= $D/r,,= iNf =

12.25.

(2.75)

The constant slope of the conical wavefront allows more phase variation to be recorded in this hologram than in the previous hologram, even

162


[III, $ 2

though the number of fringes in the holograms are the same. Since the slope of the wavefront along the radial direction is constant, the Fourier transforms of the diffracted waves from the hologram shown in Fig. 17(b) are circles. The interferogram of this wavefront (see Fig. 17(c)) consists of equally spaced concentric circles. C. Aspheric wavefronts One of the phase variations for an aspheric wavefront is given by q ( x , y ) = - ( g . r r l A F ) ( ~ ~ + y * ) + ( 2 ~ / 8 A F ~ ) ( x * + y ~ ) (2.76) ~,

where g = 1/16(D/F)2.F is the focal length of the spherical mirror having this amount of spherical aberration and D is the diameter of the mirror. The factor g is chosen so that the corrected wavefront will have the same phase variation as the spherical wave at the center and the circumference of the mirror. That is, q ( x , y) is zero at both r = 0 and r = D/2. This hologram can be used to correct the spherical aberration of a mirror with focal length F. The hologram shown in Fig. 18(a) is made with parameters D = 10 mm, F = 75 mm and A = 632.8 nm. The carrier frequency for this hologram is again 20 lines/mm to give 100 fringes in the hologram. The Fourier transform of the wavefront shown in Fig. 18(b) is similar to the focused spot of a lens with the same amount of spherical aberration. Because the phase varies as r4, the spatial frequencies at the four corners of the hologram are very high. These frequencies produce the star-shape in the Fourier transform. The phase variation of the reconstructed wavefront is shown in the interferogram in Fig. 18(c). The maximum phase variation at r = 0124 is

N,= 1.17.

(2.77)

Therefore, the same bandwidth that is used to record the spherical wave and the conical wave can only record a few wavelength variations in the corrector plate because of the large gradient of the wavefront at the circumference of the hologram. D. Helical wavefronts A helical wavefront has linear phase variation along the azimuthal direction: q ( x , Y ) = 27T%/%O,

where % =tan-' y l x .

(2.78)

111, § 21


163

Fig. 17. (a) Computer-generated hologram of a conical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

164

Fig. 18. (a) Computer-generated hologram of an aspheric wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

111, I21

MAKING COMPUTER-GENERATEDHOLOGRAMS

165

Fig. 19. (a) Computer-generated hologram of a helical wavefront. (b) The Fraunhofer diffraction pattern of the hologram in (a). (c) The interference pattern of the reconstructed wave from the hologram with a plane reference beam.

166

COMPUTER-GENERATED HOLOGRAMS TECHNIQUES AND APPLICATIONS

[III, (i 2

Phase variation of this type has not been made with conventional fabrication technique. The spatial frequency of this wavefront along the x -direction is -sin 0 (2.79) u,(r, 8)=-.

re,

The spatial frequency is inversely proportional to the radius r. Obviously, a fixed carrier frequency cannot satisfy eq. (2.59) for all radii. To get around this problem a small region in the center of the hologram is not plotted with fringes. The radius of that region must be larger than 1 . 5 ~ ~ 0 , so that the carrier frequency can satisfy eq. (2.59). 6 cx = 20 lines/mm. The hologram in Fig. 19 is made with O0 = 2 ~ / 1 and The diameter of the hole in the center of the hologram after photoreduction is about 0.1mm. The hologram contains about 100 fringes. The Fourier transform of the hologram is shown in Fig. 19(b). The fine structures in the higher spatial frequencies come from the region near the origin of the hologram. The outer circumference of the hologram contains all the low spatial frequencies of the wavefront. The interferogram of this wavefront shown in Fig. 19(c) consists of spokes along the radial direction. This interferogram is similar to the interferogram obtained from the kinoform in Fig. 11. 2.4.4, Generalization of computer-generated interferograms We have discussed how a space variant bias function coupled with a nonlinear limiter can produce a binary hologram that can modify the amplitude as well as the phase of a wavefront passing through the hologram. The amplitude of the diffracted wave from the hologram is recorded by the widths of the fringes in the hologram and the phase is recorded in the positions of the fringes. Both pieces of information are recorded along the x coordinate. To better utilize the two-dimensional characteristics of the hologram, it is better to record the amplitude information along the y -direction instead of the x-direction. A method for producing such a binary hologram is outlined in Fig. 20. In the lower branch of the figure, the phase information is still encoded in the same way as shown in Fig. 14(a) with the exception that the bias in Fig. 20 is now a constant q. In the upper branch a sinusoidal function with frequency y is used as a carrier and the amplitude variation A(x, y ) takes the place of the function q ( x , y ) . From previous discussions the output

111, § 21


167

hkv1 COS~TA(X,YI

(x)

=

qk"

h,bw 1

=

h,(x, y ) from the upper branch is

The first term in eq. (2.80) is the amplitude variation A(x, y). The output from the lower branch in Fig. 20 is the same as eq. (2.51) with q a constant. The function h(x, y) is the product of h,(x, y) and h,(x, y). By multiplying the right-hand side of eq. (2.80) with the right-hand side of eq. (2.51), one of the terms in the product h(x, y) with n = 0 and m = -1 is equal to A(x, y)exp{j[2rrax+cp(x, y)]}. In this generalization of the computer-generated hologram, the amplitude is recorded in a way similar to the halftone technique used in printing. The amplitude variation is controlled by the dot size or, in this case, the length of the line segments in the interferogram. When the amplitude of the wavefront is approximately constant inside each period of the sinusoidal carrier sin 2rryy and is equal to the sampled value of the wavefront at the center of each period, this generalized method is the same as the method used by Brown and Lohmann in their improved detour phase technique. In this regard, their improved method can indeed solve the many problems associated with the approximations in their original holograms.

168


[III, 5 3

D 3. Quantizations in Computer-Generated Holograms Before making a computer-generated hologram, the function representing the amplitude transmittance of the hologram or the complex wavefront itself has to be digitized by the computer. The digitization involves first sampling the function at regular intervals and then quantizing each of the sampled values into a finite number of levels. The sampling process, if carried out at a sufficiently high rate, will incur no loss of information to the original function. However, when the sampled values are quantized, irreducible errors are added to the original function. The errors which are due to quantization are often regarded as additive noise. In computer-generated holograms,the quantization noise limits the accuracy of the wavefront reconstructed from the hologram. It also reduces the detectability of digital data or images stored in the computergenerated holograms (POWERS and GOODMAN [ 19751). The term quantization has also been used to mean the conversion of a continuous function into a discrete valued function (GOODMAN and SILVESTRI [1970], DALLAS [1971a and b], DALLAS and LOHMANN 119721).A continuous function will become a discrete-valued function if it is passed through a nonlinear limiter with an input-output characteristic such as that shown in Fig. 2 1. The distinction between a quantized function and a discrete valued function is illustrated in Fig. 22. The digitized function is shown as the solid step function which is obtained by first sampling the continuous curve at the center of the sampling cell and then assigning its sampled values into one of ten levels. The transitions in the digitized function occurs at x = n. On the other hand, if the same curve is passed through the nonlinear limiter in Fig. 21, the resulting discrete function is the dotted step function in Fig. 22. The transition of the discrete valued function does not occur at x = n. Instead it occurs whenever cp(x)= 2 ~ n / 1 0 where , n is an integrer. It can be seen that the discrete-valued Z

Fig. 21. Input-utput

characteristic of a limiter for generating discrete valued functions

111, 0 31

QUANTIZATIONS IN COMPUTER-GENERATED HOLOGRAMS

169

Fig. 22. The solid step function is a quantized function of the continuous curve. The dotted step function is a discrete valued function derived from the continuous function by using the limiter in Fig. 21.

function generated by passing through the limiter approximates the original function better. This special interpretation of quantization, as it turns out, is useful for studying kinoforms made by depositing a finite number of thin films on a substrate (D’AuRIA, HUIGINARD, ROYand Spnz [1972]). As will be shown in the following section, the discrete phase kinoform has the same properties as the continuous-phase kinoform.

3.1. DISCRETE PHASE KINOFORMS

Discrete-phase kinoforms are made from a phase function which has been passed through the limiter with the input-output relationship shown in Fig. 21. The step size in the staircase function is generally equal to l/Na, where N, is the total number of quantization levels. As N, approaches infinity, the quantization limiter becomes the limiter shown in Fig. 12(a). Therefore, the procedure used in 0 2.3.2 to derive the output function of a nonlinear limiter is applicable here. From Fig. 21 it is clear that the output 2’ of the limiter is a periodic function in Z, and so is the function g(Z)=exp (j27rZ’). The function g(Z) can be expanded into a Fourier series:

I70


[III, 5 3

where

The summation in the Fourier coefficient c, is equal to 1 when (1- m ) / N , is equal to an integer and 0 for other values of m. As a result, c, can be written as c, =

where (l-m)/N,=

sin n ( n - lIiV,) n ( n - l/NJ '

(3.3)

a. Thus, the function g ( Z ) is equal to

The wavefront of a discrete phase kinoform of the phase function q ( x , y ) is obtained by substituting q ( x , y) for 27rZ in eq. (3.4):

The n = 0 term in the wavefront of the discrete-phase kinoform is equal to e j p k y ) , which is the same as in the continuous-phase kinoform. Because of the presence of additional diffracted waves in eq. ( 3 . 9 , the light efficiency of the discrete phase kinoform cannot be as high as the continuous-phase kinoform. Moreover, when the higher order waves from the kinoform cannot be neglected from eq. (3.9, a linear phase term should be added to q ( x , y ) so that the higher order waves from the kinoform will not propagate in the same direction as the 0th-order wave. As N, approaches infinity, all the Fourier coefficients c,, except c ~ , become zero. The discrete-phase kinoform becomes identical to the continuous-phase kinoform. Phase mismatch can happen in discrete phase kinoforms, especially when the kinoform is designed for use in one wavelength and is actually

III,g 31

QUANTIZATIONS IN COMPUTER-GENERATED HOLOGRAMS

171

used at a different wavelength. Or, the thin film layers that make up the phase variation in the kinoform do not add up to produce the correct amount of phase shift on the incident laser illumination. When this happens, the Fourier coefficients c, in eq. (3.2) should be modified by replacing ( 1 - m ) by ( l + p - m ) : 1 sin mT/Na 1 c-=N, 1 m T / ~ a

=-

1

Na

-eJ2n(l+6-tn)

-ej2dI+P-m)JNA

exp ( j ~ ( lp+- m ) ( l - l/Na)} X

sin(rnT/Na) sin . r r ( l + p - m ) . rnr/Na v (1 + P - m ) / N ,

sin

(3.6)

It can be shown that the coefficient ch approaches the coefficient given in eq. (2.31) as N, becomes infinite. ct, is again reduced to the coefficients in eq. (3.3) when p = 0.

3.2. QUANTIZATION NOISE

The effect of quantization on a function can be modelled as an additive noise n(x, y ) with a probability density function given by (3.7)

The amplitude of the noise is uniformly distributed between *l/(2Na). The severity of the effect of the quantization on the wavefront reconstructed from a quantized computer-generated hologram can be measured in terms of the mean square errors in the reconstructed wavefront. For example, the amplitude transmittance of an off-axis reference beam hologram after quantization can be written as t'(x,

Y) = t(x, Y ) + n(x, Y ) .

(3.8)

The mean square error in the reconstruction (attributed to quantization) is simply equal to MSE = E[n2(x, y)] = 1 / ( 1 2 N 3 .

(3.9)

E[. . .] in eq. (3.9) indicates the statistical average of the function inside the bracket. For this type of computer generated hologram the mean square error is independent of t ( x , y). However, when the amplitude and

172


[III, 5 3

phase of the wavefront are recorded separately, as in the detour phase holograms or the modified kinoforms, the wavefront reconstructed from a hologram in which quantization has taken place is given by f(x, Y ) = [ A ( ~ ,) + n , ( x~)Iexp{j[cp(x, , y)+n2(x,

v>lI.

(3.10)

nl(x, y) and n,(x, y ) are, respectively, the quantization noise in the amplitude and phase of the wavefront. The mean square error in the reconstructed wave is MSE = E[(A(x, y)eJwp(x, y’ - { A(x, Y ) + n ,(x,Y )I exp {j[cp(x, Y) + n 2 k Y )1>1’1

=E[ni(x, Y ) ’ + ~ A ( x ,y){A(x, ~>+fin,(x, Y ) I {-COS ~ nz(x3 Y)II. (3.11) The probability density function pn(a) of the random noise, n,(x, y), is given in eq. (3.7). The phase quantization noise, n,(x, y), is also assumed to be uniformly distributed with the density function (3.12) where N, is the number of phase quantization levels. The mean square error MSE in eq. (3.11) then becomes

MSE= 1/(12N3+2A2(x, y)[l-(N,/~) sin (TIN,)].

(3.13)

The mean square error in the quantized wavefront has two parts. One is due entirely to the amplitude quantization. Its value is equal to that in the off-axis reference beam hologram. The second part is due to the phase quantization alone. The effect of phase quantization is coupled with the amplitude variation of the wavefront and is largest when A(x, y ) = 1. As N , becomes infinite, the second term in eq. (3.13) becomes zero. The mean square error becomes indentical to that of the off-axis reference beam hologram.

3.3. PHASE ERROR FROM QUANTIZATION

In many applications, only phase information is recorded in computergenerated holograms. In those cases, it is useful to determine the effect of quantization on the accuracy of the phase variation reconstructed from the hologram. When the phase variation is recorded as a computer generated interferogram as discussed in 0 2.4, the inaccuracy in the

111, § 41

173

APPLICATIONS IN COMPUTER-GENERATED HOLOGRAMS

position of the fringes because of quantization error creates a wavefront with phase variation given by (PYX,

Y) = d x , Y ) + n,(x, Y ) .

(3.14)

The root mean square phase error is

RMSE= JE[n,(x, y ) ' ] = .rr/(,Np3t).

(3.15)

The result in eq. (3.15) indicates that with Np=30 the root mean square phase error in the reconstructed wavefront is less than 1/100 of a wavelength. Although the off -axis reference beam hologram has only amplitude quantization, the phase reconstructed from the hologram is also affected by the quantization noise. Suppose that the off-axis reference beam hologram is just a sinusoidal grating. The amplitude transmittance of the grating after quantization is

t'(x, y) =$[1+sin 27rax]+ n(x, y).

(3.16)

In the absence of quantization noise, the transmittance of the grating is equal to 1 / 2 at x = n / 2 a . With the quantization noise, the location where the transmittance is 1 / 2 occurs at sin 2 ~ a =x- 2 n ( x , y ) or

2rax

= m.rr+sin-'

2n(x, y ) .

(3.17)

For small n(x, y ) the phase error is approximately given by 2 n ( x , y). The RMSE phase noise in this case is equal to l/(&N,). The phase error in an off-axis reference beam hologram is less affected by the amplitude quantization. However, note that this result is obtained by assuming that the sampled data are recorded without jitter. Jitter in the sampling position in the off-axis reference beam hologram will produce phase error whose RMS fluctuation will be determined by an equation similar to eq. (3.15).

P

4. Applications of Computer-Generated Holograms

4.1. 3-D IMAGE DISPLAY W M COMPUTER-GENERATED HOLOGRAMS

The feasibility of a particular technique for making computergenerated holograms is often demonstrated experimentally by using the

171


[III, 5 4

technique to make a Fourier transform hologram of a simple object. This, indirectly, demonstrates the usefulness of computer-generated holograms as a medium for displaying images that may not physically exist. In § 2, we have discussed the many techniques for recording wavefronts in computer-generated holograms. These methods can also record the wavefronts from a three-dimensional solid object in computer-generated holograms. However, because the exact calculation of the wavefront of a 3-D object at the hologram plane is rather complicated, if not impossible, this causes some difficulties in realization of such a computer-generated hologram. This rather difficult problem has been solved by two methods which we will discuss here. One way to simplify the computation of the wavefront of a 3-D object is to assume that the object consists of many independent scatterers (WATERS[1968], LESEM,HIRSCHand JORDAN[1969], BROWNand LOHMANN [1969]). Each scatterer is considered as a point source with a parabolic wavefront at the hologram plane. Figure 23 illustrates the optical system used for the calculation of the wavefront at the hologram plane. The 3-D object is located at the front focal plane of lens L. The hologram is assumed to be at the back focal plane. For an object point at (x, y, 2,) the wavefront at the hologram plane with the usual parabolic approximation is equal to

where S(x, y, 2 , ) is the amplitude of the point source and f is the focal length of lens L. When there is more than one point at a distance z, from the lens, the collective wavefronts at the hologram from these points is given by

(4.2)

'-2

--

f -

Fig. 23. Optical system used in computing the wavefront of a 3-D object.

111,s 41


175

The summation in eq. (4.2) is the discrete two-dimensional Fourier transform of the points at a distance z,, from the lens. The total wavefront at the hologram for the entire 3-D object is the superposition of the functions w(u, TI, zn): (4.3) The summation in eq. (4.3) is carried out for all the planes that intersect the object. W(u,u) is a paraxial approximation of the wavefront of the object at the hologram plane. For If- z,,I I2+IQd~)I2=2 for

1x1 = 1,

(4.48)

210


[III, 3 4

we can show by deduction that

for 1x1 = 1. Hence,

IPk(X)lS(2N)f.

(4.50)

The inequality in eq. (4.46) is obtained by substituting eJnefor x in eq. (4.50). From this inequality we can further show that I

I n

5 (2N);.

(4.51)

The left-hand side of eq. (4.51) is just the cosine transform of an all 1 sequence. Therefore, the inequality in eq. (4.51) gives us the upper bound o n the amplitude of the cosine transform. With this upper bound, it is clear that the value of t ( x ) will not exceed the range [0,1] if the value of can be set equal to 1/(2N)f. In the optical memory application, the number of bits in the data sequence is always equal to 2k where the bound in eq. (4.51) is applicable. For other values of N,Rudin showed that the upper bound is equal to (5N)f. To compare Rudin's phase sequence with other phase coding schemes, graphs of the cosine transform of the four phase coding methods are calculated and shown plotted in Fig. 50. The number N in the cosine transform is equal to 8. The phase codes used in obtaining the curves are (in order): (a) deterministic binary code: (b) random binary code: (c) Schroeder code:

cpn/.rr = 0, 0, 0, 1 , 0 , 0 , 1 , 0

(d) random polyphase code:

cp,/.rr = .96, .72, .26, .32, 1.14,1.84,1.12,

cp"l.rr

= 0, 1 , 1 , 1 , 0 , 1, 1 , o

cpn/.rr = 0, .125, .5,1.125,2,3.125,4.5,

6.125 0.6. As can be seen in all the cases except for the random polyphase code, the maximum values of the cosine transforms are bounded by 4 which is in agreement with the bound derived by Rudin. The binary phase code (deterministic or random) produces a cosine transform that has even symmetry about the origin. This permits a 50% reduction in computation time for the cosine transform. Polyphase codes do not have this advantage.

111, § 41


21 1

4

2 0

-2 -4

t

Fig. 50. The cosine transforms of an all 1 sequence with four different phase codings: (a) deterministic binary phase code, (b) random binary phase code, (c) deterministic polyphase code, (d) random polyphase code.

Although the selection of the phase code has centered on the cosine transform for making one-dimensional holograms, the fluctuation in the cosine transform is related to the amplitude of the Fourier transform of the same digital data. Therefore, the results derived in this section can also be applied to other types of Fourier transform holograms.

4.5. LASER BEAM SCANNING

Laser beam scanning devices such as the galvanometer mirror scanner, acoustooptical beam deflectors or polygonal mirror scanners have now been widely used in electronic systems for facsimile reproduction, document reading, display, pattern generation for IC circuitry or computer

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output printers. Besides these well developed laser beam scanning devices, a new type of mechanical scanner using a holographically recorded grating to deflect laser beams has been investigated (see BRYNGDAHL and LEE [19761 and their references therein). Holographic grating scanners have properties similar to both the polygonal mirror scanner and the acoustooptical beam deflectors. Like the polygonal mirror scanner, the grating scanner requires mechanical motions to change the deflection of the laser beam. However, it deflects the laser beam by means of diffraction rather than reflection. The holographic grating scanner is capable of scanning over large angles and with high resolution. In comparison with other types of mechanical scanners, holographic scanners have these advantages: (1) The holographic grating scanner can be used in transmission. This has the advantage that the grating scanner can, therefore, be used in a prescan mode. This permits the focusing lens following the scanner to produce a flat scan line at the back focal plane of the lens. (2) The holographic grating scanner can be made in such a way that the wobbling of the scanner has a very small effect on the scan line. (3) The scan angle in a holographic grating scanner is independent of the number of hologram facets on the scanner. This permits construction of a multifaceted scanner that can scan over a larger cone angle than an equivalent polygonal mirror scanner of the same size. (4) Holographic gratings can be recorded on the circumference of a disc to make a high speed rotating disc scanner. (5) The holographic grating scanner can be used without any focusing lens in the optical system. In this section two methods for making holographic grating scanners will be discussed. In both of these methods the computer-generated hologram plays an important part in construction of the laser scanner.

4.5.1. Computer-generated holograms for laser beam scanning

There are two ways to use a holographic grating to scan a laser beam. One way is to scan the laser beam by rotating a constant frequency grating (see Fig. 51). As the grating rotates, the focused spot at the back focal plane of the lens generates a circular scan line. The radius of the circular scan is proportional to the spatial frequency of the grating and the focal length of the lens. An arc from this circular scan is used as the

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A

Fig. 51. A laser beam is scanned by rotating a constant frequency grating, G

scan line. To obtain a straight scan line, the document or recording surface must conform to the curvature of the circular scan. The circular scan can be corrected by additional optical components (BRAMLEY [1973], WYANTr197.51). Because the spatial frequency of the grating is constant across the grating, partially illuminating the grating will have no effect on the location of the scan spot. Therefore, the scan spot is invariant to the location in the grating illuminated by the laser beam. A different method of scanning the laser beam is to use a grating which has linear spatial frequency variation across the grating. When this grating is partially illuminated by a laser beam, the frequency variation in the different parts of the grating can change the direction of the diffracted laser beam. This is the major distinction between the constant-frequency grating scanner and the variable-frequency grating scanner. An example of a variable-frequencv scanner is shown in Fig. 52. The grating is X

Fig. 52. A drum scanner with holographic grating wrapped around its circumference.

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mounted on the circumference of a drum. The rotation of the drum moves the different parts of the grating across the laser beam. In practice, there is usually more than one grating on the circumference of the drum. The scan line from this drum scanner does not have the curvature problem of the constant-frequency grating scanner. The variable-frequency grating for scanning the laser beam can be made as a computer-generated interferogram with the grating equation: 27$y

+ q ( x , y) = 2 m .

(4.52)

The carrier frequency in the computer-generated interferogram is along the y-direction. The phase q ( x , y) is chosen so that the laser beam will be scanned along the x-direction. The spatial frequency of the grating made according to eq. (4.52) will be %(X,

Y)

= (1/2n)Mx, y)ldx.

(4.53)

In raster scanning, the spatial frequency of the grating must be a linear function in x. Hence, the phase function, q ( x , y), in eq. (4.53) is the solution to the following differential equation: a q ( ~ ,ypax = ~ ~ X I W A X .

(4.54)

The parameters w and Ax represent the hologram width and the incremental displacement of the grating needed to deflect the laser beam to the next resolvable position along the scan line. A function satisfying eq. (4.54) is d x , Y ) = (.rrx2/wAx)+d y ) .

(4.55)

g(y) is the constant of integration when eq. (4.54) is integrated to obtain the solution in eq. (4.55). Two different forms of g(y) can be used in making the grating for scanning the laser beam: (4.56)

The computer-generated holograms of q ( x , y) with two different forms of g(y) is shown in Fig. 53. When gl(y) is used, the spacing between the fringes along the y-direction is constant as shown in Fig. 53(b). This hologram is an off-axis cylindrical zone plate. When g,(y) is used to make the hologram in Fig. 53(a), the resulting hologram is just an off-axis Fresnel zone plate. In either hologram, the spatial frequency along x is a linear function of x.

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Fig. 53. The holograms in (a) and (b) can produce a linear scan line by moving across a laser beam. (a) is obtained from an off-axis section of a Fresnel zone plate. (b) is a cylindrical zone plate with carrier frequency normal to the phase variation.

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Suppose that the hologram in Fig. 53(a) is illuminated by a collimated beam at x = x f . The phase variation of one of the diffracted beams is equal to

cp(x - X I ,

y) = - T { ( x -x’)’+ y2}/(wAx)-27Tpy T(X2 --

+ y’)

w Ax

- 2mpy

2TXXI TXf2 +-w Ax w Ax’

(4.57)

The first term in the phase function produces a spherical wavefront that converges to a point at a distance wAx/A from the hologram. Since this quadratic phase function is independent of X I , it can be compensated for by using a diverging beam originated from a point source at a distance wAx/A from the hologram. The second term in eq. (4.57) causes a tilt of the diffracted beam along the y-direction. The third term which is linear is x, and x’ is responsible for the continuous deflection of the laser beam when the grating is moved across it. The deflection angle as a function of x’ is

Since the diameter of the laser beam can be equal to w, the angular resolution of the diffracted beam is A 6 = A / w . From eq. (4.58) if the grating is moved from x f to x f + Ax, the angle of the diffracted beam will be changed by he. This affirms that the parameter A x in the phase function q ( x , y) is the incremental distance that the grating must travel to address the next resolution position. For a grating with length L, the number of resolvable spots along a scan line is given by N

=L/Ax.

(4.59)

Some experimental results demonstrating the scanning capability of the linear frequency grating scanner are shown in Figs. 54-56. The computergenerated hologram used in the experiment is similar to the hologram in Fig. 53(b). The hologram was plotted on a Calcomp plotter and was 120cm in length and 20cm in width. The parameter Ax is equal to 3/8 cm. There are a total of 640 fringes in the hologram. From eq. (4.59) the number of resolution elements that can be addressed by this grating is 320. Because this grating is made using gl(y) in eq. (4.59), the grating has focusing power only in the x-direction. When this grating is used in the drum scanner in Fig. 52, the laser beam along the y-direction will focus at

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the back focal plane of the lens. But, the laser beam along x will focus at * A ~ / W A Xfrom the back focal plane of the lens shown. F is the focal length of the lens in Fig. 52. Without any cylindrical component added to the scanner, a line rather than a well-focused spot will be scanned in the frequency plane. Because the grating has a constant frequency along y-direction, any movement of the grating along y will have no effect on the scan line. This is the advantage of this kind of astigmatic grating. Figure 54 illustrates the focusing properties of the grating in Fig. 53(b). When the area marked a in Fig. 53(b) is illuminated, the diffraction pattern at the back focal plane of the lens is as shown in Fig. 54(a). The focused spot in the center in Fig. 54(a) is from the undiffracted beam of the grating. The lines on either side of the center spot are from the diffracted waves of the hologram. These waves are focused along the y-direction at the back focal plane of the lens. The length of the line along the x-direction is caused by the quadratic phase variation in the hologram. One of the diffracted waves from the hologram will focus at a

Fig. 54. Recording made with the rotating drum scanner using the hologram in Fig. S3(b). (a) is the Fraunhofer diffraction pattern of the hologram when it is illuminated in the area marked a in Fig. S3(b). (b) is taken at a short distance from the back focal plane of the lens. It shows the astigmatic focus of the diffracted wave from the hologram. The center disk is due to the undiffracted laser beam. (c) shows the correction of the astigmatism in the hologram with a cylindrical lens inserted just behind the hologram.

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small distance from the back focal plane as shown in Fig. 54(b). However, at that plane the diffracted wave along the y-direction gets out of focus. Figure 54(c) shows correction of the astigmatism by a cylindrical lens. The dependence of the scan spot on the different parts of the grating is demonstrated in Fig. 55. Figures 55(a) through 55(c) show the positions of the scan spot at the back focal plane when the holograms are illuminated at areas marked “a”, “b” and “c”. When the drum rotates, the scan line produced by the grating is shown in Fig. 55(d). To test the resolution of the scanner, the laser beam is passed through a modulator which turns the intensity of the laser beam on and off at selected rates. The modulated scan line is then recorded on film. Figure 56 shows the resolution of the scanner at 100 spots/scan, 200 spots/scan and 300 spots/scan. The capacity of the scanner is as predicted by eq. (4.59).

Fig. 5 5 . (a) to (c) show the positions of the scan spots when the areas marked a. b, and c o f the hologram in Fig. 53(b) are illuminated; (d) shows the complete scan line when the drum rotates.

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Fig. 56. Shows resolution of a feasibility scanner. Recording made with the rotating drum scanner using the hologram in Fig. 53(b): (a) 100 spots/scan; (b) 200 spotslscan; (c) 300 spots/scan,

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4.5.2. Interferometric grating scanner with aberration correction The number of resolution elements that can be scanned by a computergenerated hologram scanner is limited by the recording system used to make the hologram. For instance, the Calcomp plotter in the experiment described in the last section was a 28 mm (11in.) plotter with positional accuracy of 0.25 mm (0.01 in.). The period of the highest frequency in the experimental grating is about 1.25 mm which is only five pen positions of the plotter. It is difficult to increase the spatial frequency of the computer-generated hologram further without causing other type errors. The limitation of the Calcomp plotter can be circumvented by going to a laser recorder which uses a focused laser beam to write the computergenerated hologram directly on film. Even then, to make a computergenerated hologram scan over a cone angle exceeding +30", the laser recorder will require a precision similar to that of a ruling engine for making diffraction gratings. In this section, an alternative method for making holograms for scanning laser beams will be discussed. This method also illustrates a potentially important application of computergenerated holograms in correcting aberration in holographic optical elements. We have shown that the phase variation needed to produce a linear (4.60)

where f = wAlA. As written in eq. (4.60), f is the focal length of the spherical wavefront. The phase q ( x , y) in eq. (4.60) is equal to 2 m at radius r,, = ( 2 ~ n f ) d . (4.61) The radii {r"} are equal to the radii of the zones in a Fresnel zone plate. Therefore, the problem of making a computer-generated hologram to scan over a large cone angle is equivalent to that of making an off-axis zone plate with a large number of zones. Besides using plotters or laser recorders to make zone plates, Fresnel zone plates can be made by recording the interference pattern between a divergent wavefront from a point source and a collimated reference wave (HORMAN and CHAU[1967], CHAMPAGNE [1968], CHAU[1969]). The divergent wave recorded in the hologram has a phase variation of (4.62)

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The constant phase term, 27rflA, is included in eq. (4.62) so that (p2(x, y) = 0 at x = y = 0. The radius of the constant phase of (p2(x,y ) is equal to rl,= (n2h2+2nAf):.

(4.63)

The radii {rln) of the interferometric zone plate are different from those of the Fresnel zone plate. Only when f > >nh/4 and the difference between r,, and r; is small can the point source hologram approximate a Fresnel zone plate. When the interferometric zone plate is not used in an optical system similar to its construction geometry, the diffracted wave from an off-axis part of the grating will have aberrations. This is the case when the interferometric zone plate is used in the drum scanner in Fig. 52. The presence of off-axis aberration in the interferometric zone plate can be shown by expanding the phase function (p2(x, y) about an off-axis point (xo, Yo): V2(X - xo7 Y - Yo) = (27r/h){(1/2f)[(x-x,)’+(~

- Y ~ ) ’ I - ( ~ / ~ ~ ~ ) I I ( X - X ~ ) ~ + ( Y.-.I~ ~ ) ~ I ’ + .

= (p2(x, Y)+2.rrl~){-(1/f)(xox

+YYd

+ ( ~ / S ~ ’ > [ - ~ ( X ~ + Y ~Y) (YX~X) + ~~ + ( . K X ” + ~ ~., .J}.~ ] +(4.64) .

In eq. (4.64) the first term is again the spherical wavefront. The second term which is linear in x and y will change the direction of the diffracted beam. The remaining terms in the equation are the third-order aberration of the interferometric zone plate. By letting x = p cos 4 and y = p sin d ~ , the third-order phase error in the interferometric zone plate can be written as @’ = (27r/h){- (p3/2f3)(xOcos 4 + yo sin 4) + (p’/2f3)(xi cos2 4 + y i sin’ 4 + 2x,y,

cos 4 sin 4)).

(4.65)

The two terms in @3 are the coma and the astigmatism. These aberrations occur because the playback geometry of the hologram is not the same as its constructional geometry. Although the interferometric zone plate is recorded by using a large cone from a divergent wavefront and thus has a small f-number, the actual f-number of the zone plate used in scanning the laser beam is typically about 10. A t this f-number, the coma has less than one wavelength variation within the illuminated part of the zone plate. Hence, they have no noticable effect on the scan spot. The

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most serious aberration in the diffracted wave from the interferometric zone plate is caused by the astigmatism which makes the wavefront in the x-direction to focus at a distance

(4.66) from the zone plate. Whereas, with the laser beam centered about yo = 0, the wavefront along y will always focus at f, =f. As a result, there are two focal planes for the diffracted wave emerging from an off-axis part of the interferometric zone plate. The separation between the two focal planes is a function of the scan angle 8 because sin 8 = x,/f. The following experiment will demonstrate how this off -axis astigmatism affects the focused spot of the scanning laser beam. An interferometric zone plate is recorded in the setup shown in Fig. 57.

Fig. 57. Side view of an interferometer for making an IZP. The mirror, M, and the beam splitter, B, are used to combine the collimated reference beam with the divergent wavefront from the point source 0 for recording the IZP. The IZP recorded in this setup is narrow and has a width, W, along the y-direction. (b) Top view of the interferometer. The length of the IZP along x is equal to L.

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The optical system is basically an interferometer. The mirror, M, and the beam splitter, B, combine a collimated beam with the divergent wave from a point source, 0.The distance between the film plane and the point source is f. The mirror and beam splitter arrangement in the interferometer provides a reference beam angle, $, that can be as small as a few degrees. The interferometric zone plate recorded has width w. The top view of the interferometer is shown in Fig. 57(b). It shows that the cone angle from the point source is 8 and the length of the zone plate is L. The cone angle that can be scanned by this grating is equal to the cone angle 8 of the divergent beam. For the experiment, a zone plate was recorded using.f=37.5 mm, L = 18 mm and w = 3 mm. The cone angle 8 of the divergent beam from the point source was about 24 degrees. The reference beam angle $ was about 5 degrees. The interferometric zone plate was recorded on Kodak 649F plate. In reconstruction, the interferometric zone plate was illuminated by a divergent beam whose diameter on the zone plate was about 3mm. The distance of the reconstruction point source was such that one of the diffracted waves from the zone plate focused at a distance of 40cm from the zone plate. Photographs of the magnified spots at different scan angles are shown in Fig. %(a). It can be seen that the spot becomes a line at large scan angles because of the off-axis astigmatism. The angular range in which the scan spots are diffraction limited is small. For comparison the diameter of the on-axis spot in Fig. %(a) is about 100 pm. The off-axis aberration in the interferometric zone plate can be corrected by a corrector plate. Since the interferometric zone plate is recorded in an interferometer, the correction o f the aberration can be incorporated in the zone plate during the recording process. The phase variation needed to correct the aberration can be obtained from a computer-generated hologram. In the following we will discuss some of the considerations in using computer-generated holograms to correct aberrations in the interferometric zone plate. As pointed out in eq (4.60), the ideal phase variation for scanning a laser beam is cpl(x, y). This phase function has the property that the focusing power is independent of the part of the grating illuminated. Therefore, the phase variation needed to correct the aberration in an interferometric zone plate is given by the difference between (p2(x, y ) and PO,(X, Y):

Fig. 58. The magnified spots in (a) illustrate the off-axis astigmatism in the drum scanner. The focal power along the x-direction is strongly dependent on the scan angle, especially at large angles. As a reference, the diameter of the on-axis spot in (a) and (b) is about 100 pm.The scan angle hetween the two adjacent scan spots shown is about 0.15". (b) shows the spots from a corrected IZP. Most of the astigmatism in (a) has been removed by the correction recorded in the IZP.

e

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Equation (4.67) is derived by expanding cp2(x, y ) in a Taylor series about the origin. The terms 6(x, y) are the third- and higher-order terms in the expansion of (p2(x,y). In comparison to the first term in eq. (4.67), 6(x, y ) is generally small and can be neglected in making the corrector plate. The maximum phase deviation in A(p,(x, y) occurs at the boundary of the function. Its value depends on the focal length, f, and the f-number o f the zone plate. In the experiment just described, the maximum phase deviation is about 25h. The phase variation, Acp,(x, y), is only one of the many phase variations that can be used to correct the aberrations in the zone plate scanner. Another one that can correct the aberrations and at the same time minimize the phase deviation in the corrector plate is

A’(P~(x, y ) = - (d16Af)(L/f)’(x2 + y’)

+ (2r/8Af3)(x2+ Y’)~. (4.68)

The first term in A(p2(x,y) will change the focal length of the corrected interferometric zone plate by a small amount. ‘The addition of that phase term makes the phase function A(p2(x,y) vanish at x = y = 0 and x = L/2 and y = 0. The maximum value of hq2(x, y ) occurs at x = L/& and is 1/4 of that in Acpl(x, y). The aberration in the interferometric zone plate is corrected by the second term in A(p2(x,y). A computer-generated hologram of Acp2(x, y) is shown in Fig. 59(a). The carrier frequency of the hologram is along the y-direction, because most of the phase variation of A(pz(x,y) is along the x-direction. The interferogram of the wavefront reconstructed from this hologram is shown in Fig. 59(b). The interferometer in Fig. 60 is modified from the

(b) Fig. 59. Computer-generated hologram used as corrector plate. For clearness, the hologram contains fewer fringes than the one used in the experiment. The interferograrn (b) is obtained by combining an inline collimated reference beam with the wavefront reconstructed from the hologram in (a). The fringes show the phase variation added to the IZP.

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Fig. 60. Side view of the modified interferometer for making 1ZP's with aberration correction. The phase variation from the hologram corrector plate is imaged onto the film plane by the telecentric system. Because the computer-generated hologram is a diffracted optical element, the mask in the back focal plane of the first lens is used to select one of the diffracted waves from the hologram.

interferometer in Fig. 57 to include the computer-generated corrector plate in the making of the interferometric zone plate. The diffracted wave from the computer-generated hologram replaces the collimated beam used in the previous interferometer. As shown in Fig. 60, the computergenerated hologram is illuminated by a collimated beam and is imaged one-to-one by the telecentric lens system to the film plane. A mask in the back focal plane of lens L1 selects only one of the diffracted waves from the hologram to form the interference pattern on the film plane. The intensity variation of the interference pattern on the film plane is proportional to

The first term in eq. (4.69) is from the computer-generated hologram; the second term comes from the point source. The reference angle is 4.When Aq2(x, y) in eq. (4.68) is substituted into eq. (4.69), the function I(x, y ) becomes

(4.70) This interference pattern will produce an interferometric zone plate identical to a Fresnel zone plate.

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The photographs in Fig. 58(b) show the magnified scan spots from an aberration corrected, interferometric zone plate having the same parameters as the uncorrected zone plate discussed previously. A significant improvement in the focused spots at large scan angles can be seen in Fig. 58(b). This method of using a computer-generated hologram as a corrector plate is not only useful in making gratings for scanning laser beams, but is also useful for correcting the aberrations in other types of holographic optical elements. Up to now, the aberrations inherent in the holographic elements could only be minimized by the recording geometry or by using multiple elements. Using computer-generated holograms as corrector plates in recording holographical optical elements can help to eliminate certain aberrations in the holographic optical elements.

0 5. SummaryandComments We have discussed in some detail the many methods for making computer-generated holograms. Since each of the methods discussed is unique, it is difficult to specify the best method for making computergenerated holograms. When a laser recorder is available, Burch’s method of making off-axis reference beam holograms is the one to use. His method takes advantage of the uniform sampling in a raster scanning system and permits maximum use of the spatial bandwidth of the recording system. On the other hand, if a binary hologram is desired, the computer-generated interferogram or the improved detour phase method should be used. Kinoforms with their various extensions offer better light efficiency than any other methods. Kinoforms do not require a carrier frequency to record the complex wavefront; however, it is better to add a linear phase term to the kinoform so that if the kinoform has phase mismatch, the linear phase term can separate the different diffracted waves from the kinoform. In recent years there has been considerable interest in the effect of quantization on computer-generated holograms. In 8 3, we looked at two types of quantization. Quantization in assigning the sampled value of a function into one of a finite number of levels, introduces irrecoverable errors to the function. These errors are often random in nature. On the other hand, the term quantization has also been used to mean the conversion of a continuous function into a discrete-valued function. In

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this case, it has been shown that under certain conditions the orginal function can be recovered intact from the discrete-valued functions. Five different applications of computer-generated holograms have been discussed. These are by no means the only applications of computergenerated holograms. For example, we have not discussed the use of computer-generated holograms in long optical wavelengths for laser [19731, SWEENEY, STEVENSON, CAMPBELL machining (ENGELand HERZIGER and SHAFFER[1976]), because more work is expected from this area. In all the applications discussed, except for the 3-D display and data storage applications, computer-generated holograms can be considered as optical elements. The only difference between computer-generated holograms and other optical components such as lenses or mirrors etc., is that computer-generated holograms are diffractive optical elements. Their usefulness is limited to narrow band and spatially coherent light sources. The computer-generated holograms are most useful in supplementing other types of optical elements rather than replacing them. This is especially evident in the laser-beam scanning application. The computergenerated hologram can be made to scan a laser beam. But, the combination of an interferometric hologram and computer-generated hologram is a more practical approach to making laser scanners for scanning over large cone angles. We want to emphasize again the importance of copying computer-generated holograms in an interferometer. By doing so, we make an optical hologram which has the same wavefront as the original computer-generated hologram, possibly with additional phase variation. Development of the computer-generated hologram is strongly motivated by the need to synthesize spatial filters for coherent optical data processing. However, the filtering done by the coherent optical systems is mainly limited to linear filtering. Recently, it has been shown that with computer-generated holograms the optical system can be converted into a [19741, CASASENI and KRAUS[ 19761, space-variant system (BRYNGDAHL CASASENTand SZCZUTKOWSKI [1976]). However, more work must be done to improve the quality of this kind of nonlinear optical transformation. Image processing for many applications will invariably be dominated by digital methods. The computer-generated holograms have their greatest potential in the area of interferometry. They have been shown to be useful in supplementing existing methods of optical testing. Laser beam scanning is another promising area for computer-generated holograms. Holographic grating scanners are in many ways better than other mechanical mirror

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scanners. Complicated scan patterns, in addition to the raster scan, can be obtained by using holographic gratings. With other types of scanners these patterns will require the use of two different beam deflectors. Laser scanners have other optical components in them, some of which can be combined into the holographic grating to produce a more compact scanning system. In this chapter we have not discussed recording materials or the diffraction efficiency of computer-generated holograms because the considerations in selecting recording materials for an optically recorded hologram could be similarly applied to computer-generated holograms. A good review on the recording materials for holography can be found in the book by SMITH [1976].

References* AKAHORI, H.. 1973, Appl. Opt. 12, 2336. ALLEBACH, J. P. and B. LIU, 1976, Appl. Opt. 14, 3062. ANDERSON, G. B. and T. S. HUANG,1969, Spring Joint Computer Conf., AFIPS Conf. Proc., Vol. 34, pp. 173-185. ANDERSON, D. R., 1961, Proc. IRE 49, 357. ANDREW, H.C., A. G. TESCHERand R. P. KRUGER,1972, IEEE Spectrum, Vol. 9, no. 7. p. 20. BECKER,H. and W. J. DALLAS,1975, Opt Comm. 15, 50. BESTE,D. C. and E. N. LEITH,1966, IEEE Trans. Aerosp. Electron. Syst. AES-2, 376. BIRCH,K. G. and F. J. GREEN,1972, J. Phys. D: Apply. I’hys. 5, 1982. BRAMLEY,A,, 1973, U. S. Patent 3,721, 486. BRAUNECKER, B. and A. W. LOHMANN, 1974, Opt. Comm. 11, 141. BROWN,B. R. and A. W. LOHMANN, 1966, Appl. Opt. 5, 967. BROWN, B. R., A. W. LOHMANN and D. P. PARIS,1966, Opt. Acta 13, 377. BROWN,B. R. and A. W. LOHMANN, 1969, IBM J. Res. Develop. 13, 160. BRYNGDAHL, 0.and A. W. LOHMANN, 1968, J. Opt. SOC.Am. 58, 141. BRYNGDAHL, O., 1973, J. Opt. Soc. Am. 63, 1098. BRYNGDAHL, O.,1974a, Opt. Comm. 10, 164. BRYNGDAHL, O.,1974b, J. Opt. Soc. Am. 64, 1092. BRYNGDAHL, 0. and W-H. LEE, 1974, J. Opt. SOC.Am. 64, 1606. BRYNGDAHI, 0.and W-H. LEE, 1975, J. Opt. SOC.Am. 65, 1124(a). BRYNGDAHL, 0. 1975, Opt. Comm. 15, 237. O.,1975, Opt. Eng. 14, 426. BRYNGDAHL, BRYNGDAHL, 0.and W-H. LEE, 1976, Appl. Opt. 15, 183. BURCH,J. J., 1967, Proc. IEEE 55, 599. BURCKHARDT, C. B., 1970, Appl. Opt. 9, 1949. CMBEIL, K., G. W. WECKSUNG and C. R. MANSFIELD, 1974a Opt. Eng. 13, 175. CAMPBELL, K., G. W. WECKSLJNG and C. R. MANSFIEID,1974b, Proc. S.P.I.E. 48, 69.

* Some of the references are not quoted in the main text. They are included here to provide a reasonably complete bibliography on the subject.

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COMPUTER-GENERATED HOLOGRAMS: TECHNIOUES AND APPLICATIONS

rrrr

CARTER,W. H., 1968, Proc. IEEE 56, 96. CASASENT, D. and M. KRAUS, 1976, Opt. Comm. 19,212. CASASENT,D. and C. SZCZUTKOWSKI, 1976, Comm. 19, 217. CATHEYJr., W. T., 1968, Optik 27, 317. CATHEYJr., W. T., 1970, Appl. Opt. 9, 1478. CHAMPAGNE, E., 1968, Appl. Opt. 7, 381. CHAU,H. M., 1969, Appl. Opt. 8, 1209. CHAVEL, P. and J. P. HUGONIN, 1976, J. Opt. SOC.Am. 66,989. CHU,D. C. and J. W. GOODMAN, 1972, Appl. Opt. 11, 1716. CHU,D. C., J. R. FIENLPand J. W. GOODMAN, 1973, Appl. Opt. 12, 1386. CHU,D. C. and J. R. FIENW, 1974, Opt. Eng. 13, 189. CHU,D. C., 1974, Proc. S.P.I.E. 48, 81. CLAIR,J.-J., 1972, Opt. Comm. 6,135. CLITRONA, L. J., E. N. LEITH,L. J. PORCELLO and W. E. VNIAN,1966, Proc. IEEE 54, 1026. DALLAS, W. J., 1971a, Appl. Opt. 10, 673. DALLAS, W. J., 1971b, Appl. Opt. 10, 674. DALLAS, W. J. and A. W. LOHMA", 1972a, Appl. Opt. 11, 192. DALLAS, W. J., and A. W. LOHMANN, 1972b, Opt. Comm. 5, 78. DALLAS, W. J., Y. ICHIOKAand A. W. LOHMA", 1972, J. Opt. SOC.Am. 62, 737(A). DALLAS,W. J., 1973a, Appl. Opt. 12, 1179. DALLAS, W. J., 1973b, Opt. Comm. 8, 340. DALLAS, W. J., 1974, Appl. Opt. 13, 2274. DAMMANN, H., 1969, Phys. Lett. 29A, 301. DAMMANN, H., 1970, Optik 31, 95. D'AURIA,L., J. P. HUIGINARD, A. M. ROYand E. SPITZ, 1972, Opt. Comm. 5, 232. DYSON,J., 1958, Proc. Roy. SOC.A m , 93. ENGEL,A. and G. HERZIGER, 1973, Appl. Opt. 12, 471. FAULDE,M., A. F. FERCHER,R. TORGEand R. N. WILSON,1973, Opt. Comm. 4, 363. FERCHER, A. F. and M. KRIESE,1972, Optik 35, 168. FERCHER, A. F., P. SKALICKY and J. TOMISER,1973, Opt. and laser Technology, April, p. 75. FERCHER, A. F., 1976, Optica Acta 23, 347. FIENW,J. R., 1974, Proc. S.P.I.E. 48, 101. FIENLP,J.R. and J. W. GOODMAN, 1974, Nouv. Rev. Optique 5, 269. FILLMORE, G. L., 1972, Appl. Opt. 11, 2193. FLEURET,J., 1974, Nouv. Rev. Optique 5, 219. FRIESEM,A. A. and D. PERI, 1976, Opt. Comm. 19,382. GABEL,R. A. and B. LIU, 1970, App. Opt. 9, 1180. GABEL,R. A., 1975, Appl. Opt. 14, 2252. GALLAGHER Jr., N. C. and B. Lru, 1973, Appl. Opt. 12, 2328. GALLAGHER Jr., N. C. 1974, Proc. S.P.I.E. 48, 87. GALLAGHER Jr., N. C. and B. Lru, 1975, Optik 42, 65. GALLAGHER Jr., N. C., 1976, IEEE Trans. Info. Th. IT-22, 622. GOODMAN, J. W., 1968, Introduction to Fourier Optics (McGraw-Hill) Chapter 7. GOODMAN, J. W. and A. M. SILVESTRI, 1970, IBM J. Res. Develop. 14, 478. HANSCHE, B. D., 1976, Proc. S.P.I.E. 74, 70. HASKELL, R. E. and B. C. CULVER,1972, Appl. Opt. 11,2712. HASKELL,R. E., 1973, J. Opt. SOC.Am. 63, 504. HASKELL,R. E. and P. TAMURA, 1974, Proc. S.P.I.E. 52, 43. HASKELL, R. E., 1975, Opt. Eng. 14,195.

1111

REFERENCES

23 1

HELSTROM,C. W., 1967, J. Opt. SOC.Am. 57, 297. HORMAN, M. H. and H. M. CHAU,1967, Appl. Opt. 6, 317. HUANG,T.S. and B. PRASADA,1966, MIT/RLE Quar. Prog. Rep. 81, 199. HUANG,T.S., 1971, Proc. IEEE 59, 1335. HUGONIN J. P. and P. CHAVEL,1975, Opt. Comm. 16, 342. Y.and A. W. LOHMANN, 1972, Appl. Opt. 11, 2597. ICHIOKA, ICHIOKA, Y., M. IZUMIand T. SUZUKI,1969, Appl. Opt. 8, 2461. ICHIOKA,Y.,M. IZUMIand T. SUZUKI,1971, Appl. Opt. 10, 403. KATYL,R. H., 1972, Appl. Opt. 11, 198. KEETON,S. C., 1968, Proc. IEEE 56, 325. KETTERER, G., 1972, Optik 36, 565. KING.M. C., A. M. NOLLand D. H. BERRY,1970, Appl. Opt. 9, 471. KIRMISCH, D., 1970, J. Opt. SOC.Am. 60, 15. KIRK,J. P. and A. L. JONES,1971, J. Opt. SOC.Am. 61, 1023. KOZMA,A. and D. L. KELLY,1965, Appl. Opt. 4, 387. KOZMA,A,, W-H. LEEand P. J. PETERS, 1971, IEEE/OSA Conf. on Laser Engineering and Aplications. KOZMA,A,, E. N. LEITH and N. G. ~ ~ A S S E1972, Y , Appl. Opt. 11, 1766. KOZMA,A., 1973, in: Topical Meeting on Optical Storage of Digital Data, Digest of Technical Papers (Optical Society of America, March 1973). LEE, W-H., 1970a, Appl. Opt. 9, 639. LEE, W-H., 1970b, Pattern Recog., 2, 127. LEE, W-H. and M. 0. GREER,1974, Appl. Opt. 13, 925. LEE, W-H., 1974, Appl. Opt. 13, 1677. 1974, Opt. Comm. 12, 382. LEE, W-H. and 0. BRYNGDAHL, LEE, W-H., 1974, Proc. S.P.I.E. 48,77. LEE, W-H., 1975a, J. Opt. Soc. Am. 65, 518. LEE, W-H., 1975b, Appl. Opt. 14, 2217. LEE, W-H., 1975c, Appl. Opt. 14, 2447. LEE, W-H., 1977, Appl. Opt. 16, 1392. LEITH,E. N. and J. UPATNIEKS, 1962, J. Opt. Soc. Am. S2, 1123. 1968, Appl. Opt. 7, 539. LEITH,E. N. and A. L. INGALLS, LEITH,E. N., 1968, IEEE Trans. Aerosp. Electron. Syst. AES-4, 879. LEITH,E. N., 1971, Proc. IEEE 59, 1305. LEITH,E. N., 1976, in: Advances in Holography, Vol. 2, ed. N. H. Farhat (Marcel Dekker Inc.). and J. A. JORDAN Jr., 1967a, Roc. Symp. Modern Optics (New LESEM,L. B., P. M. HIR~CH York, Polytechnic Institute of Brooklyn). Jr., 1967b, Proc. 1967 Fall Joint Comp. Conf., LESEM,L. B., P. M. HIRSCHand J. A. JORDAN Vol. 31 (Washington D.C., Thompson Books) pp. 41-47. Jr., 1968, Commun. ACM 11, 661. LESEM,L. B., P. M. HIRSCHand J. A. JORDAN and J. A. JORDAN Jr., 1969, IBM J. Res Develop. 13, 150. LESEM,L. B., P. M. HIRSCH LESEM, L. B., P. M. HIRSCHand J. A. JORDAN Jr., 1970, Opt. Spectra 4, 18. LINFOOT,E. H., 1958, Recent Advances in Optics (Oxford) p. 176. LIU, B. and N. C. GALLEGHERJr., 1974, Appl. Opt. 13, 2470. A.W., D. P. PARISand H. W. WERLICH,1967, Appl. Opt. 6, 1139. LOHMANN, LQHMANN, A. W. and D. P. PARIS, 1967, Appl. Opt. 6, 1567, 1739. LOHMANN, A. W. and D. P. PARIS,1968, Appl. Opt. 7, 651. LOHMANN, A. W., 1973, NEREM Record, Part 2, p. 148. S. and P. CHAVEL, 1974, Appl. Opt. 13, 718. LQWENTHAL,

231

COMPUTER-GENERATF.D HOLOGRAMS: TECHNIQUES A N D AI'PI lCAl ION\

[III

MACGOVERN, A. J . and J . C. WYANT,1971, Appl. Opt. 10, 619. D.,1973, Opt. Comm. 8, 76. MACQUIGG, MACQUIGG, D., 1974, Proc. S.P.I.E. 48,96. and G. W. WECKSUNG, 1974, Proc. S.P.I.E. 52, 88. MANSFIELD, C. R., K. CAMPBELL MCLEOD,J. H. 1954, J. Opt SOC.Am. 44, 592. Mt:Yr.K. A. J . and R . HIC'KIING, 1967, J . Opt. Soc. Am. 57, 1388. MIYAMOTO, K., 1961, J. Opt. SOC.Am. 51, 17. MIYAMOTO, K., 1961, J . Opt. SOC.Am. 51, 21. K. and T. ASAKURA, 1976, Opt. Comm. 17, 273. NAGASHIMA, NAIDU,P. S. 1975, Opt. Comm. 15, 361. PASTOR,J., 1969, Appl. Opt. 8, 525. Jr., 1970, IBM J . Res. Develop. PATAU,J. C., L. B. LESEM,P. M. HIRSCHand J. A. JORDAN 14, 485. POWERS,R. S. and J. W. GOODMAN, 1975, Appl. Opt. 14, 1690. RANSOM, P. L., 1972, Appl. Opt. 11, 2554. RANSOM, P. L. and R. M. SINGLETON, 1974, Appl. Opt. 13, 717. RANSOM, P. L. and R. F. HENTON,1974, Appl. Opt. 13, 2765. RANSOM, P. L., 1975, Appl. Opt. 14, 563. RUDIN,W., 1959, Proc. Am. Math. Soc. 10, 855. SCHLLZ, G. and J. SCHWIDER, 1976, in: Progress in Optics, Vol. 13, ed. E. Wolf (NorthHolland, Amsterdam). 1976, Opt. Acta 23, 229. SIROHI,R. S., H. BLUMEand K.-J. ROSENBRUCH, SMITH,H., 1975, Principles of Holography (second edition, Wiley). T. C., 1974, Opt. Eng. 13, 219. STRAND, SWEENEY, D.W.. W. H. STEVENSON, D. K. CAMPBELLand G. SHAFFEK,1976. Appl. Opt. 15, 2959. TAKAHASHI, T., K. KONNO and M. I(AwAi, 1974, Proc. ICO Conf. Opt. Methods in Sci. and Ind. Meas., Tokyo, p. 247. TAKAHASHI, T., K. KONNO, M. KAWAIand M. ISSHIKI,1976, Appl. Opt. 15, 546. J., 1963, in: Progress in Optics, Vol. 2, ed. E. Wolf (North-Holland, TSUJIUCHI. Amsterdam) pp. 131-180. VANDER LUGT,A., 1964, IEEE Trans. Inform. Theory IT-10,139. WATERS,J. P., 1966, Appl. Phys. Lett. 9, 405. WATERS,J. P., 1968, J. Opt. SOC.Am. 58, 1284. WATERS,J. P. and F. MICHAEL,1969, Appl. Opt. 8, 714. T. R. and R. P. WILLIAMS, 1976, Opt. Acta 23, 237. WELBERRY, WYANTJ. C., 1974, in: Proc. 9th Intern. Congress of ICO, eds. B. J. Thompson and R. R. Shannon (National Academy of Science) pp. 643-664. J. C., 1975, Appl. Opt. 14, 1057. WYANT, WYANT,J. C. and V. P. BENNETT,1972, Appl. Opt. 11, 2833. WYANT, J. C. and P. K. O'NEILL,1974, Appl. Opt. 13, 2762. YATAGAI, T., 1974, Opt. Comm. 12, 43. YATAGAI, T., 1976, Appl. Opt. 15, 2722. YONEZAWA, S., Y . h m o , S. KASAI and A. MAEKAWA, 1971, Japan. J. Appl. Phys. 10, 1279. YOWEZAWA, S., 1972, Opt. Comm. 19, 370. YOUNG,M.,1972, J. Opt. SOC.Am. 62, 972.

E. WOLF,PROGRESS I N OPTICS XVI @ NORTH-HOLLAND 1978

IV

SPECKLE INTERFEROMETRY BY

A. E. ENNOS National Physical Laboratory, Teddingion, Middlesex, England

CONTENTS PAGE

3 1. INTRODUCTION . . . . . . . . . . . . . . . . 235 $ 2 . LASER SPECKLE CHARACTERISTICS

. . . . . . . 236

$ 3 . INTERFERENCE OF LASER SPECKLE . . . . . . . 240

5 4 . DIRECT OBSERVATION SPECKLE INTERFEROMETRY . . . . . . . . . . . . . . . . . 243

5 5 . INTERFEROMETRY BASED UPON SPECKLE CORRELATION

. . . . . . . . . . . . . . . . 246

$ 6 . SPECIAL PURPOSE CORRELATION INTERFEROME-

TERS . . . . . . . . . . . . . . . . . . . . .

249

$ 7 . ELECTRONIC SPECKLE PATTERN INTERFEROMETRY . . . . . . . . . . . . . . . . . . . 259

9 8. INTERFERENCE

EFFECTS WITH RECORDED SPECKLE PATTERNS . . . . . . . . . . . . . . 266

5 9 . SPECKLE PHOTOGRAPHY

. . . . . . . . . . . 268

§ 10. FURTHER TECHNIQUES OF SPECKLE PHOTOG-

RAPHY . . . . . . . . . . . . . . . . . . . .

272

0 11. APPLICATIONS OF SPECKLE PHOTOGRAPHY IN METROLOGY . . . . . . . . . . . . . . . . . 278

0 12. ‘WHITE LIGHT’ SPECKLE PHOTOGRAPHY

. . . . 281

5 13. MEASUREMENT OF SURFACE ROUGHNESS BY SPECKLE INTERFEROMETRY . . . . . . . . . . 283 0 14. CONCLUSIONS

. . . . . . . . . . . . . . . . 7-85

REFERENCES . . . . . . . . . . . . . . . . . . . 286

§

1. Introduction

The speckle effect in optics is the generation of a random intensity pattern formed when coherent light is scattered by a rough surface, or is diffused by a medium containing scattering centres. Although such speckle effects can be observed under special conditions when using light of limited coherence, it was with the advent of the laser in 1960 that the and GORDON [ 19621 speckle phenomenon came into prominence. RIGDEN first described the high-contrast, grainy appearance of a coherently lit surface, having a roughness depth greater than the wavelength of the light, and correctly ascribed its origin to the interaction of the scattered waves. With the subsequent widespread use of coherent light in optical systems, however, laser speckle was seen to set severe limits to the system’s performance, principally because the scale of the speckle pattern was approximately equal to the theoretical resolution limit of the system; consequently, the main research efforts were directed towards the reduction of laser speckle. Only at a later stage was it realised that the speckle effect could be used as an information carrier characterising the scattering medium, and that it might be used to study both the scatterer itself and any change taking place in its position or form. Speckle interferometry is the generic name given to the wide variety of experimental techniques used to extract this information. It must be mentioned, however, that the term “speckle interferometry” has also come to be applied to the technique first developed by LABEYRIE [ 19701 for obtaining diff raction-limited resolution of a stellar object despite the presence of the turbulent atmosphere through which a telescope has to view it. Although this technique bears a strong relationship to the coherent light methods to be described it will not be dealt with here; Labeyrie has in fact reviewed the subject in Volume XIV of this Series. T o a large extent the developments in speckle interferometry have followed on logically from the prior studies of holographic interferometry. These showed that measurements of interferometric sensitivity, previously restricted to smooth regular surfaces, could be extended to surfaces 235

236

SPECKLE INTERFEROMETRY

[IV. $ 2

of any shape or quality, with application to analysing their deformation and vibration behaviour. However, there are a number of fundamental drawbacks to the use of interference holography in many practical situations; firstly, it requires the recording of very high-resolution detail (the hologram): this means the use of photographic plates of relatively low sensitivity, which imposes a time delay in their processing. Secondly, and more fundamentally, the holographic method stores far more information than is necessary for performing differential interferometry, since none of the detail pertaining to the shape or reflectivity of the surface is needed. Thirdly, although the output interference patterns from a hologram interferometer will, in theory at least, give complete information on displacement in three dimensions, it is difficult to use them in practice to extract the desired information, e.g. the distortion, in the presence of unwanted information, such as the rigid body motion. For this reason research has been directed towards methods not requiring the intermediate step of holography, and which yield directly only the measurement information that is desired. True speckle interferometry is concerned with the interaction of the complex wavefronts arising from light that has suffered scattering, and this will be covered in the first part of this review. There are, however, many interesting optical effects that take place when speckle patterns are recorded photographically, o n plate or film, and used as diffracting structures; the second part of this article is concerned with these. Firstly, it is necessary to give an outline of the nature of the speckle pattern, and to define some of its basic characteristics.

8 2. Laser Speckle Characteristics Laser speckle takes the form of a high contrast granular pattern, generated whenever coherent light is scattered either from a rough surface or by a translucent diffusing medium. A typical pattern is shown in Fig. 1. The phenomenon is caused by interference of the individually scattered wavelets, each having a random relative phase, which diverge from every scattering point. T h e speckle field is not localised in space, but fills the coniplete volume occupied by the interfering waves, and its detailed nature depends not only upon the configuration of the scattering medium, but upon the angular extent over which the scattered waves are

IV, 0 21

LASER SPECKLE CHARACTERISTI(’S

237

Fig. 1. A laser speckle pattern

received. It is convenient when discussing speckle interferometry to differentiate between two types of speckle pattern, depending upon whether the interfering waves emanate directly from the scatterer, or whether they have first passed through an optical imaging system. (Fundamentally there is no difference between the two types.)

2.1. OBJECTIVE SPECKLE PATTERN

A n objective speckle pattern is formed in the whole of the space in front o f an optically rough surface illuminated by coherent light, or in the region behind a transparent diffuser. A flat screen AB held some distance from the scatterer will have a section of the three-dimensional interference pattern projected on to it (Fig. 2a). If the area of the scatterer is increased in size, not only will the detailed nature of the pattern change, due to the additional wavelets taking part in the interference process, but the size of the individual speckles will decrease, since the angular subtense is increased. The average “diameter” (a,) of a speckle, defined as the average distance between two bright speckles, is related to the wavelength of the light A, and to the angle subtended by the scattering

238


7--

1

[IV,

a2

_

L

Fig. 2. Formation of (a) objective speckle, (b) subjective speckle

area at the screen. For a circular illuminated patch of diameter 0,distant L from the screen, the approximate formula (a,)- 1.2hWD

(1)

applies.

2.2. SUBJECTIVE SPECKLE PATTERN

If waves from the scattering medium are imaged by an optical system (Fig. 2b) the resultant speckle pattern is termed “subjective”, the reason for this being that this type of speckle is seen by the human eye when viewing a coherently lit surface. The size of the speckles is now dependent

IV,5 21

LASER SPECKLE CHARACIERISTICS

235)

upon the solid angle subtended by the exit pupil of the imaging system at the image. In this case (us)= 0.6A/N.A.,

(2)

where (a,)is the subjective speckle size, and N.A. is the numerical aperture of the system. For small apertures, the speckle becomes large, and vice-versa. Since imaging systems are widely used in speckle interferometry, it is useful to relate the speckle size to F, the aperture ratio (flnumber) of the imaging lens, and to the magnification M with which the scattering surface is imaged,

(u,)=1.2(1+M)hF.

(3)

A further concept useful when considering the interference of speckle fields is to imagine the subjective speckle pattern as being the image of a similar pattern, of different scale, lying in the object plane. This pattern will have speckles of average diameter (&)= 1.2(1+M)AF/M.

(4)

The brightness distribution among the speckles in a speckle field is independent of the detailed configuration of the surface of the scatterer (provided that it generates scattered wavelets having fully randomised phase differences) but it does depend upon whether or not the scattered wavelets are fully capable of mutual interference. This is achieved only if the illumination is coherent and the scattered waves are polarised in the same plane, in which case the speckle pattern is said to be fully deueloped. The probability density function of the intensity distribution of these speckles follows the exponential relationship

P U ) = ( l / Z O ) exp (-l/Io),

(5)

where p ( I ) is the probability that a particular speckle has an intensity between I and ( I + dl). This relationship is shown in Fig. 3, curve 1, and it predicts that the most probable brightness of a speckle is zero, i.e. there are more completely dark speckles than those of any other brightness. A scatterer that depolarises the light completely will generate a pattern that can be considered to be the incoherent addition of two speckle fields polarised mutually at right angles to one another. The brightness distribution now follows the relationship

p ( I ) = (4I/I3exp (-21/10),

(6)

240

[IV, 3 3


I

I

0.5

I 1.0

I

-

I

1.5

I

2 .o

I

2.5

Fig. 3. Probability density functions of the brightness distribution of speckle fields. 1) Single speckle field, 2) incoherent combination of two speckle fields, 3) coherent combination of speckle field and uniform field.

shown as curve 2, Fig. 3. No completely dark speckles are present in this case, and the most probable brightness is not zero, but takes a value of about one half the average brightness. In considering the interference of speckle fields it will be assumed that the scatterer produces random local phase differences of at least 277, a property held by all but highly-specular surfaces. When the surface roughness is less than a wavelength deep, speckle contrast is reduced (FUJIand ASAKURA [1974]).

0 3. Interference of Laser Speckle The light forming a speckle pattern is coherent and is thus capable of optical interference with any light field formed with radiation of the same exact wavelength. In the general case this interference effect will completely modify both the size and the brightness of the resultant combination pattern. There are, however, special cases where these modifications

IV, § 31

INTERFERENCE OF LASER SPECKLE

2-11

are strictly limited to the extent that the interference effects can be used in a practical way for measurement purposes. One can distinguish two such conditions: (i) The case where a speckle pattern formed by a scatterer interacts with a uniform wave directed along the axes of the speckle-forming waves (Fig. 4a). An example of this might be that of a Michelson interferometer in which one of the mirrors has been replaced by a scattering surface. (ii) The case where two speckle patterns, derived from scatterers having the same angular aperture and directed along the same common axis, are superimposed (Fig. 4b). The Michelson interferometer with both mirrors replaced by a scattering surface, typifies this condition. Althoughstrictly a branch of (ii), a third case of special importance can be distinguished: (iii) Interference between two speckle fields derived from identical copies of the same scatterer. Interference effects based on this principle have been known for many years (e.g. Quetelet's rings, or the scatter plate interferometer of BURCH[1953]), but of special interest in the context of speckle interferometry are the phenomena associated with the interference of light scattered from two photographically recorded speckle patterns. The developments in speckle photography, outlined in $0 8, 9 and 10, come into this category. Interference of laser speckle depends on whether the fields are combined coherently or incoherently. Treating the above cases separately, we have: (i)a. Coherent combination of a speckle field with a uniform field. Both the size and brightness distribution of the combination speckle field will be different from those of the original speckle field. The size is approximately doubled, since the maximum angle a 1between rays generating the uniform field and those from the extremities of the scatterer (Fig. 4a) is one half of a", the maximum angle between rays from the two extremities of the scatterer. The statistical brightness distribution is also changed, to an extent that depends upon the relative intensities of the speckle and uniform fields. BURCH[ 19701 has computed the distribution curves for different combinations, and an example of one o f these is shown as curve 3, Fig. 3. (The case where the average brightness of the speckle field equals that of the uniform field.) (i)b. Incoherent combination of speckle and uniform field. Superimposing these two fields leads to a speckle pattern of the same average size, but with a brightness distribution showing reduced contrast, since it is

142

[IV.


Scotterinq S urfoce

s3

Reference

Imaqe

Source

Laser illumination

image

Fig. 4. Optical interference of (a) speckle field with uniform field, (b) two different speckle fields.

evident that there will be no regions having a brightness less than that of the uniform field. (ii)a. Coherent combination of two speckle fields. When two speckle fields derived from coaxial scatterers interfere together the average size of the resultant pattern will remain the same as that of either speckle field, and the brightness distribution of speckles within it will also be the same

IV, 8 41

DIRECT OBSERVATION SPECKLE INTERFEROMETRY

243

as for either of the two interfering fields. Only the detailed configuration of the combined pattern will differ from that of either of the components. (ii)b. Incoherent combination of two speckle patterns. In this case also the speckle size of the combination will not differ from that of its two components, but the brightness distribution will be modified. When the two average brightnesses are equal, the distribution curve is that of Fig. 3, curve 2 (the same as that for a depolarising scatterer). When the two speckle patterns are partially coherent with one another, the distribution curve lies between curves 1 and 2, as shown by BURCH[1971].

8 4. Direct Observation Speckle Interferometry By utilising the interference properties (i)a and (i)b of the speckle and TAYLOR pattern fields, it was shown by ARCHBOLD, BURCH,ENNOS [1969] that the effects could be observed directly if the speckle size was made large enough. This was done by viewing a scattering surface through an optical system having a small exit pupil (about i m m ) . The speckle field was combined with a smooth reference wave produced by internal reflection within the system. A more convenient instrument was later described by STETSON [1970], and is shown diagrammatically in Fig. 5. Basically this is a telescope, through which one observes the scattering surface, with an internal beam-splitter for feeding in a reference beam directly from the laser source. This reference beam, providing the uniform field, is split off externally from the main illuminating beam and is diverged, by means of a lens, from a point that is optically conjugate with the entrance pupil of the telescope. Condition (i) of 0 3 is thus satisfied. An adjustable iris diaphragm, for restricting the entrance pupil, is mounted in front of the telescope together with a polariser that transmits Reference beam

Rotating poloroid

Adjustable aperture

Wedge beam splitter

Fig. 5. Optical system of visual speckle interferometer (after K. A. Stetson).

244


“V. % 4

only light of the same polarisation as that of t h e reference beam. A second aperture, at the exit pupil of the telescope, accepts only one of the two beams reflected by the wedged beam-splitter. In using the interferometer, the illuminated surface is viewed through the telescope and the intensity of the reference beam is adjusted to be two or three times as bright as the average brightness of the speckle field. The combined field then appears as in Fig. 6a. In this condition each individual speckle is sensitive to a phase change of the light in either beam. This could, for example, be caused by a movement of the illuminated surface in the line-of-sight direction, or a change in refractive index of the medium through which the interfering beams pass. If the surface moves steadily towards or away from the observer, the speckles will vary in brightness in a cyclic fashion, from bright to dark to bright again for every half wavelength of the displacement. However, each combination speckle starts its brightness fluctuation at a random point on the cycle, since the phases of individual speckles are randomly distributed, so that the whole speckle pattern appears to “twinkle” for a slow steady surface movement. In a corresponding way, the field will twinkle if refractive index changes due to air currents are occurring. The direct observation speckle interferometer is a useful adjunct to any system for recording holograms, since it can detect instability of the object being recorded, or of any of the optical components. It is a simple matter to include the instrument in the system, by providing an additional beam-splitter for its reference beam. Observation through the instrument will show whether any movement detrimental to the recording of the hologram is taking place; this is particularly valuable when materials that creep, such as wood or plastics, are being studied. The most important use of the interferometer, however, is in its application to the direct study of surface vibration, again in conjunction with holography. If a surface is vibrating such that its amplitude of travel in the direction of the observer is a quarter of a wavelength or more, the high-contrast speckle will be blurred out, but where the surface is stationary, i.e. at the nodes, the speckles remain sharply defined. Fig. 6 b shows the appearance of a vibrating plate as viewed through a speckle interferometer; the nodal areas can easily be picked out. It is thus possible to find the resonance frequencies of a vibrating body by direct observation, and to determine the best means of exciting them individually. If more information is required, such as the amplitude distribution of the vibration, a timeaveraged hologram can then be recorded under the same conditions.

DIRECT OBSERVATION SPECKLE INTERFEROMETRY

M

c ffi

.c

E 2

P

c

0

-

246

SPECKLE IWERFEROMETRY

[IV. 4 5

§ 5. Interferometry Based upon Speckle Correlation

It will be apparent from the previous section that simple interference of speckle fields does not lead to quantitative read-out of phase differences, in t h e manner in which a conventional interferometer operates, i.e. by generation of fringes contouring regions of equal difference in phase. On a small scale, however, each speckle behaves as a unique element of a “conventional” fringe, and it can be shown that by means of correlation processes the information relating to phase change can be usefully extracted. Suppose two “subjective” speckle fields F,(x, y ) and F2(x, y) are superimposed in a speckle interferometer, as previously illustrated in Fig. 4b. These will combine coherently to form a third speckle field F3(x, y). If the scatterer giving rise to one of the speckle patterns moves in the direction of its normal, the combined speckle field will change from F3(x, y)” to F3(x, y)&,where 6 is the phase change at all parts of the field due to the displacement. When S=2n.rr ( n being an integer) every speckle will have undergone a complete number of brightness change cycles and returned to its original brightness, independent of what its original phase was. The speckle pattern will then have become recorrelated with itself. On the other hand, when 6 = ( 2 n + 1)m, maximum decorrelation of the pattern with its original condition will have taken place. If a means can be found of distinguishing between correlated and uncorrelated speckle patterns, the interferometer can be used to measure quantitatively the surface motion, or to analyse a phase change brought about in some other way. The following methods of performing the correlation process have been developed: (i) Photographic mask correlation. LEENDERTZ [1970a, b] first described this method of speckle correlation. A photograph of the combined ~ recorded on a photographic plate of adequate speckle pattern F3(x, Y ) is resolution, and after processing it the plate is returned to its holder so as to take up exactly its original position. Since bright speckles now fall upon their own black silver images when the speckle pattern is correlated with F,(x, y)”, zero light will be transmitted. When the pattern is uncorrelated, however, some light does get transmitted. A variation of 6 over the image area will then result in a corresponding variation of the transmitted light, and speckle correlation fringes are formed. These are much grainier in appearance than two-beam interference fringes, and in this instance are of very low intensity, since most of the light is absorbed by the correlating

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INTERFEROMETRY BASED UPON SPECKLE CORRELATION

217

mask. When the mask is in place, however, the interferometer can follow any phase change ‘live’, as in the case of a conventional instrument. (ii) Photographic subtraction. An alternative photographic method, also due to Leendertz, gives a permanent record of the fringe pattern. The patterns F,(x, Y ) and ~ F3(x, Y ) ~are recorded on two separate photographic plates A and B. A positive contact print of B is made o n another plate C , and the patterns on A and C are then carefully placed together in register. Speckle correlation fringes will be formed by transmission through the pair, in a manner similar to (i) above. An additional advantage of this method is that the fringe pattern can be made bright by illuminating the plates with a concentrated beam; correct registration of the two images may, however, be difficult if the speckle size is small. (iii) Double exposure recording on a non-linear detector. In contrast with hologram interferometry, where double exposure recording and subsequent simultaneous reconstruction of the two images lead to the addition of image amplitudes, the double exposure of two speckle patterns will only add the intensities of the patterns, and since correlated and uncorrelated areas of speckle have the same average intensity, no differentiation between them is possible. However, if a non-linear recording material is used, they may be distinguished for the following reasons (ARCHBOLD, BURCHand ENNOS[1970]). Consider one speckle in the output image, formed by superposition of two component speckles having amplitudes a , and a2 and phase difference E . The resultant intensity I is given by

I=a:+a~+2a,a,cos~.

(7)

If a double exposure recording is made (a) where E remains the same or changes by 2 n r , the total intensity recorded will be

I , = 2( a:

+ a; + 2 a , a, cos E ) .

(8 )

On the other hand (b) if the phase changes by ( 2 n + l ) n between recordings, the total intensity recorded will be = af

+ a: + 2 a , a , cos E + a f + a; + 2 a

a2 cos [ ( 2 n + 1)n + E ]

or

z2 = 2 ( 4 + a$).

(9)

Case (b) is therefore equivalent to the addition of two uncorrelated speckles, since only amplitudes, and not phases, are involved. The speckle patterns corresponding to (a) and (b) will have different intensity distributions, those of curves 1 and 2 in Fig. 3. If therefore they are recorded on

748

LIV.


B

5

a photographic material having a non-linear characteristic (e.g. one of high contrast), different densities of developed silver will result. The pattern with distribution 1 (with a large number of dark speckles) will yield a recorded image having greater optical transmission than the pattern with distribution 2 (which has fewer dark speckles). Speckle correlation fringes with a visibility of up to 40% can be formed in this way if the photographic exposure is adjusted to give a high density negative. (iv) Double exposure recording with image displacement (ARCHBOLD, BURCH and ENNOS [1970], BUTTERS and LEENDERTZ [1971a]). It will be shown in § 8 that if a speckle pattern is recorded on a film or plate, and the negative subsequently examined in a diffractometer, light will be diffracted into a circular “halo” surrounding the directly transmitted beam. If a double exposure recording of two similar patterns, slightly displaced with respect to one another, is made, the diffraction halo will be modulated by a set of parallel equi-spaced fringes similar to those obtained by Young in his classical interference experiment. Speckle correlation can be performed using this phenomenon, since the Young’s fringes will be generated only if the two displaced patterns are identical; otherwise the halo only is produced. Areas of image where the pattern is correlated can be identified by using a spatial filtering system of the type shown in Fig. 7. The double exposure recording (with image displacement) is re-imaged by a lens having a double-slot aperture that passes only the 3=+ order (dark) Young’s fringes generated by the doubled pattern. Correlated areas therefore appear dark in the filtered image, and uncorrelated areas appear bright. The fringe contrast obtainable when using this method is good. A modified version of the spatial filtering [1974a], dispenses with the system, described by JONES and LEENDERTZ double slots. Instead, the negative is illuminated by an off-axis beam of white light from a quartz-iodine lamp, directed at an angle corresponding to that of the first-order minimum of diffraction. Provided that the speckle field translation is sufficiently small, the angular extent of this

Film

recording

Spatial filter

FIItered image

Fig. 7. Optical filtering system far correlating speckle patterns.

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SPECIAL PURPOSE CORRELATION INTERFEROMETERS

249

diffraction order will be large and relatively insensitive to the wavelength of the light used. .(v) Electronic subtraction. An entirely different approach to the problem of speckle correlation was pioneered by BUTTERSand LEENDERTZ [1971b], using television techniques. As will be described more fully in § 6, the speckle pattern is imaged on to the sensitive surface of a television pick-up tube and the electrical signals from it are amplified and displayed on a monitor screen. Provided that the speckle size projected on to the T V tube is large enough to be resolved, the variations of speckle brightness over the field are faithfully reproduced. Subtraction of two speckle fields can now be carried out in the following way: the output signal corresponding to continuous scanning of the first speckle field is amplified and recorded on magnetic tape or magnetic disc, each frame of the recording occupying 1/25 second of storage time. The second speckle field is now scanned, and the amplified T V signal fed into an electronic mixing unit, into which the first speckle field signal is simultaneously played back from its recording, after electronically inverting it. If good frame synchronisation is maintained, the resultant electronic signal will correspond to the subtracted pattern, i.e. it will display speckles where the two patterns were uncorrelated. Further electronic processing can be used to convert the presence or absence of a speckle signal into a variation of monitor brightness, and in this way high contrast correlation fringes are obtained.

0 6. Special Purpose Correlation Interferometers Speckle correlation interferometers offer a greater flexibility in their application to the measurement of surface displacement than d o conventional or hologram interferometers. Special systems that have been devised fall into the following categories.

6.1, OUT-OF-PLANE DISPLACEMENT INTERFEROMETERS 8

These may be exemplified by a Michelson interferometer with scattering “mirrors”, or by the optical system of the visual speckle interferometer with means for performing speckle correlation. If the surface under examination is illuminated by the laser beam at angle 8 to the normal,

250

[IV.

SPECKLE INlERFEROMETRY

s; 6

correlation fringes will be obtained whenever the out-of-plane displacement d, takes on values given by dz= (1+ cos 0) nh.

(10)

The interferometer therefore has a sensitivity similar to that of a hologram interferometer used under comparable illuminating conditions. N o particular advantage is gained by using the speckle instrument rather than the hologram interferometer unless the imaging processing stage can be speeded up, as for example by employing electronic speckle interferometry.

6.2. IN-PLANE DISPLACEMENT INTERFEROMETERS

Measurement of the displacement vector lying in the plane of a surface has important application in engineering, since it leads to a non-contact method of measuring the strain. Developments in hologram interferometry failed to solve the problem of independently measuring this parameter. LEENDERTZ [1970a, b] showed that this could, however, be done employing speckle techniques, by the expedient of using the surface under investigation as both the scattering elements of the equivalent “Michelson-type” interferometer. To do this, the surface is illuminated by two beams of coherent light, making equal angles 8 to the surface normal, and it is imaged by a lens on to the speckle pattern detector (Fig. 8). The image speckle field can now be thought of as the result of interfering

Photographic Plate

‘i

Y

- 2

\

Fig. 8. Double illumination speckle interferometer for measuring in-plane displacement (after Leendertz).

IV, P 61


2s 1

together the two independent speckle fields generated by each illuminating beam acting alone. If the surface now moves a distance d, in the outof-plane direction, both these interfering beams suffer a path difference d,(l+cos 8), and there is no change in the combined speckle field. Displacement in the y-direction also produces no change in path length. However, if the surface moves a distance d, in its own plane in the x-direction, one interfering beam will increase in path length by d, sin 8, while the other will decrease by the same amount, and a total path difference of 2dx sin 8 results. The image speckle pattern will thus be changed, such that it becomes re-correlated with itself whenever 2dx sin 8 = nA.

(11)

Speckle correlation contours obtained in this way are a measure of the resolved part of the motion in the x-direction in the surface, independent of other components of motion and of the imaging properties of the lens. The spacing of the fringes will be A/2 sin 8, so that the sensitivity of measurement can be varied by changing the angle of incidence of the illuminating beams. Typically, for 8 = 60°, the sensitivity is 0.87 wavelengths per fringe; little increase in sensitivity is achieved by using larger angles of incidence, and for most practical purposes 4.5" is most often used. The bi-directional illumination can be obtained by dividing the original laser beam with a semi-reflecting beam-splitter, but it is often more convenient to mount a plane mirror perpendicularly to the object surface and to illuminate both object and mirror by a wide collimated and ENNOS [1974]). This diagram shows a beam, as in Fig. 9 (ARCHBOLD

Ti I tinq

w-

C35mm amera

Glass Plate

Fig. 9. Practical system for measuring in-plane displacement of a plastic pipe joint.

752


[IV. 6 6

photographic correlation interferometer of type 4(iv) for measuring the in-plane strain o n the surface of a polyethylene pipe joint, produced when the pipe was pressurised internally. The tilting glass plate mounted in front of the camera acts as an optical micrometer for obtaining a lateral image shift necessary for the subsequent optical filtering process. Fig. 1 0 is a typical pattern of speckle correlation fringes obtained in this way. By differentiating the fringe spacing of the pattern with respect to directions x and y, both the linear strain adxiax and the apparent shear strain adJay can be obtained in absolute units. The sign of the strain (whether tensile or compressive) cannot be ascertained from the pattern alone. As with any form of interferometry, directional information can be gained only by “live” observation of the fringe movement, or by introducing a known overall object movement with which the unknown displacement can be compared. The pattern of Fig. 10 does, however, show from the change in slope of the fringes that a double strain inversion has taken place along the axis of the pipe joint. Analysis of the complete strain field over a surface necessitates measuring both displacement vector components d, and d, at every point. JONES and LEENDERTZ [1974b] went some way towards doing this by adding a

Fig. 10. Speckle correlation fringes on the surface of a plastic pipe joint, caused by the surface stretching as the internal pressure is increased.

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third illuminating beam R, to the system, lying in a plane normal to that defined by the two original illuminating beams R, and R,. By recording correlation fringes on one part of the object surface using R, and R,, and fringes using R, and R, on an adjacent area, the complete strain tensor could be derived along the line of demarcation of the two areas. This is particularly useful for measuring the Young’s modulus and Poisson’s ratio of a bar of material subjected to pure bending forces. JONES [1976] has subsequently developed an interferometer in which double illumination in both the xz and yz planes can be readily alternated without disturbing the object, so that two contour patterns corresponding to d, and d, are recorded almost simultaneously. An alternative method of measuring the in-plane displacement of a [1972]. surface, independent of the normal component, is due to DUFFY Instead of using dual beam illumination, the surface is effectively viewed in two directions by splitting the imaging cone of rays into two parts by means of twin apertures placed symmetrically over the imaging lens (Fig. 11). Under these conditions each aperture generates a comparatively large “subjective” speckle pattern, and these are combined coherently in the image plane to give each speckle a grating-like modulation which can be recorded on suitably fine-grain film. Displacement of the surface in the line-of-sight direction, or in the y-direction, will affect the phase of the light accepted by each aperture in a similar manner, and the modulation pattern remains unchanged. However, for a displacement d, in the x-direction, the relative phase change causes the periodic pattern to be displaced, parallel to itself, by an amount equal to a complete number o f

xL

Y

Fig. 11. Double-aperture speckle interferometer for measuring in-plane displacement (after Duffy).

-. 754


[IV. 6 6

grid spacings whenever d, = nAI2 sin 4, where 4 is the semi-angle subtended by the apertures at the object surface. When the surface displacement d, = ( n +4)A/2 sin 4, the grid pattern will become out-of-phase with the original pattern, so that if a double exposure is recorded, the periodic structure will then vanish. Areas of the film over which these two conditions operate can be distinguished by spatial filtering; contours of the x-displacement are generated by illuminating the film with collimated light and viewing it in the direction of the strongly diffracted beam generated by the periodic structure. A high contrast pattern is obtained in this way. The technique is less sensitive than the Leendertz method, since 4 must necessarily be limited by the size of the lens pupil, and the image brightness will also be low due to the small size of the apertures. In addition, object tilt will impose limitations on the technique (see below). 6.3. SPECKLE SHEARING INTERFEROMETERS

Shearing interferometers, by definition, compare the phase of a wavefront with that of the same wavefront displaced by a small amount laterally. In conventional interferometry they are employed to determine the curvature of the wave, usually for the purpose of measuring lens or mirror aberrations. The technique is also used in experimental mechanics to study the flexural deformation of beams and plates under load, but these have to be prepared with initially flat, specularly reflecting surfaces (LIGTENBERG [ 19541). Speckle shearing interferometers allow one to make similar types of measurements on scattering (unpolished) materials. The importance of this to the engineer lies in the fact that the bending moment B of a plate or shell under flexural strain is related to the change in curvature p of the surface, occurring along a direction s, by the equation

B = - dp/ds.

(12)

Shearing of the imaged speckle field generated by a scattering surface can be effected in a number of ways. The original system, due to [1973] employed an optical element, mounted in LEENDERTZ and BUTTERS front of the imaging lens, consisting of a beam-splitter and two plane mirrors (Fig. 12a). The mirrors are mounted at an angle to one another of slightly greater than 90°, so that by reflection, two laterally displaced images are formed. The image shift 6x is proportional to this out-ofsquareness. At a particular point P in the image plane, the speckle pattern


255

a Image plane

F(x,y) + F ( X + ~ X , Y )

Fig. 12. Speckle shearing interferometer systems due to (a) Leendertz and Butters, (b) Hung, Rowlands and Daniel.

will be formed by the coherent combination of two speckle wavefronts F(x, y ) and F(x + 6x, y), arising from two entirely different areas of the object surface, q ( x , y ) and q ( x + 6 x , y). If the surface is now deformed such that the region q ( x , y ) becomes displaced by distances (u, TI, w) in the three orthogonal directions, and the region q ( x + 6x, y) is displaced by distances ( u + 6u, TI + 6u, w + 6w),the modified speckle wavefront arriving

256

SPECKLE INTEREFEROMETRY

[IV, § 6

at P will differ in phase from its former value by an amount d @ given by

where 8 is the angle of illumination, awlax is the surface slope change that has taken place and aulax is the component of linear strain, both measured in the x-direction. If a correlation of the speckle pattern fields before and after deformation is performed by one of the methods described in Q 4, a pattern of fringes with a spacing corresponding to d@/2n7r7will be obtained. These fringes are contours of constant slope awlax in the x-direction, provided that 8 = 0 (normal incidence illumination). The component of slope in the y-direction can similarly be obtained by shearing the image in the y-direction. HUNG[1974] suggested that the surface strain duldx might also be determined by performing two similar shearing experiments, but with different values of 8. Solving for the two unknowns aulax and awlax in eq. (13) then yields the strain. The accuracy of measurement using this method is, however, limited. Other designs of shearing interferometer have been described. HUNG and TAYLOR [1973] used Duffy's double aperture imaging system of Fig. 11, but with small optical micrometer plates mounted in front of each aperture. Rotating either of these parallel-sided glass plates about an axis normal to the beam causes a lateral shift of the corresponding image. The double exposure photographs obtained with this system are analysed in the same way as for the Duffy technique used to measure in-plane ROWLANDS and DANIEL [19751 showed that shearing displacement. HUNG, could also be obtained with a double aperture system simply by defocusing it, as shown in Fig. 12b. Using the notation of that figure, the apparent shear 6x of the object is given by 6x = RD/SM, where M is the magnification of the system and S is the out-of-focus distance. Information relating to the x-direction and the y-direction slope change can be obtained when four symmetrically disposed apertures are used instead of two. If this is done, the intermodulation of the light from adjacent apertures also produces a shearing effect related to directions at 45" to the x and y axes, and this information can be used to determine the sense of the slope change. HARIHARAN El9751 described a simplified form of speckle shearing interferometer in which the shearing is accomplished by mounting a pair of identical diffraction gratings, face to face, in front of the camera lens used to record the image of the test object. The camera is focused on a

IV, 8 61


257

first order diffracted image, and shearing is effected by rotating the gratings in their own plane so that the rulings are inclined to one another by a small angle. Another method of obtaining shear, using a compact optical element, was described by DEBRUS [1977]. In this case a Savart polariscope, placed between two crossed polarisers, was mounted in front of the camera lens. The image shear is related to the thickness of the Savart plate. 6.4. SPECKLE INTERFEROMETRY FOR THREE-DIMENSIONAL CONTOURING

Among the interferometric techniques developed using holography, it was demonstrated that surface contours could readily be generated over [19671, SHIOTKE, any three-dimensional object (HILDEBRAND and HAINES TSURUTA and ITOH [1968]). This was achieved by recording a double exposure hologram with a change of one of the system parameters between exposures, such as the wavelength of the light, the optical refractivity of the medium surrounding the object, or the direction of the illumination. In the same way speckle interferometers may be used to perform a similar function (BUTTERSand LEENDERTZ [1974]). For example, suppose that the non-flat, scattering surface is inserted in one arm of an interferometer of the Michelson type, having a plane mirror in the reference arm, and that it is illuminated with coherent light of wavelength A,. A sensitive speckle pattern will be generated in the output image. Now if the wavelength is changed to h2, the image speckle will, in the general case, become de-correlated with its former self, since for different parts of the non-flat surface, the path difference A between object and reference beam varies, and there will be a differential phase change for the two wavelengths. However, correlation will recur at regular intervals whenever A = m,A, = m2A2, where both m, and m2 are integers. The path difference dA between successive positions for which these conditions are satisfied is thus given by

+

dA = m,A, = ( m , l ) A 2 or dA = h l A 2 l ( h ~ - A ~ ) = h l A 2 I A A .

(14)

A speckle interferometer of this type, with means for speckle correlation, will generate contour fringes over the surface having a contour interval AlA2/2AA (since the light has to travel twice the distance of the

258


[IV, I 6

depth change). If an argon laser is used as the illuminant, several pairs of wavelengths are available to give different contour intervals. For example, with the 488.0 nm and 514.5 nm lines, contours of 4.7 p,m spacing are generated; with the 488.0 nm and 486.5 nm lines the intervals are 14.2 p,m. To obtain good visibility fringes, it is, of course, necessary that the roughness of the surface, i.e. its high-frequency depth variation, should be small in comparison with the depth interval (see 0 13). Two-wavelength contouring of non-specularly reflecting surfaces may be carried out with any interferometer system having the necessary speckle correlation facilities. The majority of research in this field, however, has used the ESP1 (Electronic Speckle Pattern Interferometer) system described in the rlext section, which enables rapid correlation to be effected. Apart from this factor, speckle contouring using two wavelengths has the advantage over contouring by holographic means that it employs an imaging system; the image position does not therefore change with change in wavelength. With holographic systems, on the other hand, the need to use an off-axis reference beam causes the reconstructed image to move when the wavelength is altered, unless additional means for preventing this are taken. The simple speckle interferometer system for generating contours used as illustration here (and, indeed, holographic systems as well) measure true depth contours, since the object surface is compared with a plane wavefront. This is useful for defining the shape of nearly flat surfaces using contour intervals of a few micrometers. However, speckle pattern contouring holds out the possibility that surfaces of a much more complicated nature might be compared with nominally similar shapes, since one is free to generate the shape of the reference wave. This is a potentially more attractive goal from the engineering metrologist’s standpoint. For example, a precision component of complicated shape, such as a turbine blade, might be compared with a “master” blade during the final finishing and LEENDERTZ [1974] first proved the operations. To this end BUTTERS feasibility of comparing surfaces of revolution (such as spheres and cylinders) in a speckle interferometer. The surface was illuminated by a wave having comparable curvature to that of the surface, and the light scattered back along the same path was then compared with a reference wave of similar curvature. This interferometer is analogous to the Twyman-Green instru.ment used for testing curved mirrors, but it has the advantage that the speckle system can test the mirror blanks in their ground but unpolished state, with reduced sensitivity.

IV, § 71

ELECTRONIC SPECKLE P A T E R N INTERFEROMETRY

259

The comparison of more complicated shapes by speckle interferometry poses greater problems. The ideal way of doing this would be to put the “master” shape and the part-finished component in the two arms of an interferometer, but unfortunately the fringe contrast one obtains in this way is very poor if both surfaces are optically “rough”. The reason for this is two-fold: (i) the appearance of a speckle pattern, when viewed in the direction corresponding to that of specular reflection, does not change appreciably if the wavelength is altered, but it does so rapidly when the viewing angle departs from the specular direction; (ii) even when specular conditions of illuminating and viewing are observed, correlation between speckle patterns becomes worse with increase in depth difference as the surfaces become rougher. The degree of correlation depends upon the ratio of the depth of roughness to the contour interval defined by the two wavelengths (WYKES[1977]). When two dissimilar components are to be compared, the compounding of this effect by both object surfaces will still further lower the fringe visibility in the output of the interferometer. The problem can be overcome by designing an interferometer in which the component to be inspected (having a rough surface) is mounted in one arm and illuminated in such a way that the instrument “sees” only the light that is scattered from it in the specular direction. A smooth reference wave can then be used in the other arm. DENBY,QUINTANILLA and B U ~ E R[1976] S showed that the first proviso could be achieved by constructing a holographic illuminator for the test object from a polished “master” shape. The hologram was recorded using an illuminating beam similar in divergence to that accepted by the interferometer, so that when a real image from the hologram is “played back” on to the test object (by reversing the reference beam), every part of its surface is illuminated at the correct angle.

8 7. Electronic Speckle Pattern Interferometry Electronic Speckle Pattern Interferometry (ESPI) has developed in response to the need for making direct “on-line” interferometric measurements or inspection, especially for non-destructive testing. The earlier developments in holographic interferometry marked the first stage in applying high-sensitivity optical techniques to problems of measuring deformation, vibration, etc., in engineering, but by the nature of the holographic process it is necessary to use a light receptor that can respond

260


[IV, s 7

to very high spatial frequencies. This almost invariably means using a photographic film or plate, and the time delay involved in developing it renders the process too slow for industrial inspection applications, except for special high-cost items, or where it is used as an exploratory tool. There are processes of electro-photography, such as that first developed and MEIER[1966], that are almost instantanefor holography by URBACH ous but these often gain the advantage of immediacy at the expense of speed and resolution when compared to photography. The possibility that television techniques might be used to record holographic information is, for all practical purposes, precluded, since television camera tubes have a very limited resolution. In the case of a speckle interferometer the information is carried in the speckle pattern, and not in high frequency detail whose scale depends upon the off-axis angle of the reference beam. Since the size of the speckle depends only upon the aperture of the imaging lens used (as explained in 5 2.2), it can be made as large as is required by reducing the aperture sufficiently. Television techniques can thus be used. A standard 600 line vidicon pick-up tube has, typically, a limiting resolution of 500 elements per line, and to match the speckle pattern to this size, the imaging lens must be stopped down to at least f/16: even so, some averaging of the signal over more than one speckle may take place, giving some reduction in speckle contrast. However, provided that adequate resolution is obtained, speckle correlation may be performed electronically on the signals picked up by the TV tube, in the manner described in 9 5(v). The advantages of using a television system are: (1) a real-time display of the fringe pattern is obtained on a bright, large-sized monitor screen that can be readily viewed by more than one person, (2) no lengthy photographic processing is required; (3) the TV signal can be electronically processed and stored if necessary; if the signal is digitised, it can be fed directly to a computer for further analysis; (4) speckle correlation can be performed almost instantaneously by combining image storage with electronic subtraction. Against these advantages must be set the following drawbacks: (1) high levels of laser illumination of the object under study must be employed, due to the necessity for using a low aperture imaging system; (2) the complexity of the system means high capital cost, and perhaps skilled maintenance may be required.

Developments of systems employing TV pick-up have largely been carried out at Loughborough University, England, by Professor Butters and his group, using the in-line reference beam methods already outlined. MACOVSKI, RAMSEYand SCHAEFER [197 13, of Stanford Research Institute, U.S.A., also describe a number of novel methods directed to the same goal, namely that of providing means for real-time measurement of surface distortion, contouring, etc. Although TV pick-up is employed, the systems described cannot strictly be called speckle interferometers, since off-axis reference beams are used; however, the same picture storage principle, for image subtraction, is employed. The optical system of the ESP1 is shown diagrammatically in Fig. 13. The object, illuminated by laser light, is imaged by means of a lens operating at low aperture, on to the face of a vidicon pick-up tube. The reference beam, whose source point lies effectively in the centre of the lens pupil, is introduced by means of a beam-splitter, which is wedged to eliminate reflections from the second surface. A thin wedge prism is also cemented to the face of the vidicon tube, to prevent low frequency fringe patterns being formed by reflection of the reference beam at the two surfaces of the window. Path lengths of signal and reference beams can be matched by translation of the roof reflector towards and away from the beam-splitter, so that laser light of limited coherence may be used. A vidicon works on the storage principle, such that light falling on any elementary area of the tube face is integrated over the period of one complete scan (1/25 second for European systems). This gives a high image sampling rate, but also sets a limit to the effective maximum

Object

camera

Fig. 13. The electronic speckle pattern interferometer system.

262


IIV, 9: 7

exposure time for the system. Consequently the object surface brightness must be great enough to give adequate signal/noise ratio in the vidicon G output during this time. To overcome this, PEDERSEN, L ~ K B E Rand FORRE[1974] used a silicon target vidicon to obtain higher sensitivity to the helium-neon wavelength, but this was gained at the expense of spatial resolution, resulting in poorer quality fringes. The vidicon, in addition to receiving light from the object, is also illuminated by the reference beam, whose brightness can be varied at will. Despite the fact that a strong reference beam would alleviate the problem of low signal, it was found by DENBY, QUINTANILLA and BUTTERS [19761 that optimum fringe visibility is obtained when the reference beam brightness is set at a value only slightly higher than the average speckle field brightness. The signal generated by the vidicon is amplified before being passed on to a TV monitor tube, where it is scanned in synchronism with the camera scan. It is usual to perform some form of electronic processing on the signal before it is displayed, in order to improve the visibility of the fringe pattern. This may take the form of a band-pass filter. The ESPI can be operated in a number of different modes.

7.1. ON-LINE OPERATION FOR VIBRATION ANALYSIS

Vibrating surfaces may be studied directly by use of the ESPI, with CW laser illumination of the object. In this case the output signal from the vidicon is first amplified and sent through a high-pass filter, which removes the effects of slow variations of surface brightness, but passes the higher speckle “frequencies”. (The scanning action of the camera system converts spatial frequency into temporal frequency.) The signal is then full-wave rectified before displaying it on a monitor. Provided that the vibration frequency is high enough for a sufficient number of cycles of vibration to be completed in 1/25 second, a time-averaged interference pattern of fringes will be obtained, bright fringes appearing where the speckle pattern has maintained good contrast and dark fringes appearing in other regions. It was shown by EKand MOLIN[1971] that this contrast C is related to the vibration amplitude a and to the ratio between the reference beam intensity and the object beam intensity y by the equation C = [1+2yJ~(4~alh)]f/(l+y)

(15)

where J, is the Bessel function of zero-order. This contrast will of course

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263

be modified by the electronic processing; in particular, the constant term can be subtracted. A typical ESPI vibration pattern, showing interference fringes on a flat metal plate vibrating at one of its resonances, is illustrated in Fig. 14a. Unlike the appearance of the field as seen through a visual speckle interferometer, which only indicates the nodal regions, the EPSI picture gives bright fringes for amplitudes of vibration corresponding to the maxima of the Bessel function (approximately whenever a = (n +$)h/4). In addition to giving this amplitude information, the electronic system is capable of following transient vibrations, provided that the vibration amplitude does not change too rapidly in comparison with the effective exposure time of 1/25 second. The output may then be recorded on video tape and played back later for detailed analysis. The ESPI vibration analyser is relatively insensitive to extraneous vibration, noise and the unwanted effects of air currents, due to the short exposure time of every frame. Fringes of greater clarity can be obtained if the illuminating beam is double pulsed at the vibration frequency, using an electro-optic modulator with a feedback link to the TV frame generator. By suitable

Fig. 14. Vibration patterns on a metal plate as viewed on the ESPI monitor; system operating in (a) time-averaged mode, (b) pulse-width modulation mode (courtesy of Prof. J . N. Butters, Loughborough University).

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synchronisation, any part of the vibration cycle can be examined by means of this pulse width modulation technique. Figure 14b shows the improved fringe pattern that results. A variation on the design of the vibration ESPI system described above, due to BIEDERMANN, EK and OSTLUND [1976], uses a lens with a double slot aperture instead of the usual circular iris; the reference beam source lies effectively at a point midway between the two slots. The advantage of this arrangement is that, with high-pass filtering of the electronic signal generated by the vidicon pick-up tube, only the interference effects between signal and reference beams are let through, but not the autocorrelation terms generated by interference between all the rays from a complete circular aperture. Up to 15 J2,-type fringes are observable on the monitor, using this method. For studying large amplitude vibrations that would give too great a number of fringes for resolution by the standard ESPI system, L ~ K B E R G and HaGMoEN [1976] modulated the phase of the reference beam in synchronism with the object vibration frequency, so that the apparent vibration amplitude was reduced. This method, analogous to the holographic technique used by ALEKSOFF [19711 for extending the vibration amplitude range, also allows one to determine the relative phases of vibration of different parts of the surface. 7.2. ON-LINE DISPLACEMENT MEASUREMENT

For displacement measurement, it is necessary to provide means for correlating two speckle fields; this can be performed electronically, as described previously in 0 4(v). A video tape-recorder was used in the earlier developments of this technique (BUTTERSand LEENDERTZ [197 l]), but the present practice is to use video disc storage, which allows more rapid access. A system using an electronic storage tube was demonstrated by the Eumig Company of Austria in 1973 but has not been described in the literature. More recently, LDKBERG,HOLJE and PEDERSEN[1976] described experiments with a Hughes Scan Converter Memory Tube, which stores a single frame and allows this to be read off continuously. However, storage time is limited if the image quality is not to be degraded. Whichever storage method is employed, the two signals, corresponding to the speckle pattern fields before and after object didplacement, are combined and the resultant difference signal sent through a high-pass

IV, § 71

ELECTRONIC SPECKLE PA'ITERN INTERFEROMETRY

265

filter and full-wave rectifier, as for the vibration instrument. A typical interference pattern, photographed from the T V monitor, is shown in Fig. 15. The fringes-relate to the distortion of a specimen of carbon-fibre reinforced plastic, when under axial tensile load. It was recorded using an ESPI system with two-beam illumination of the object, and the fringes are thus contours of equal displacement in the plane of the surface, resolved in the axial direction in this instance. Changes in the fringe pattern can be followed directly as the load is varied, and the whole sequence can be recorded on video tape for subsequent analysis. The ESPI with means for signal storage and subtraction can be used in conjunction with any of the forms of speckle interferometer previously described, and therefore has wide potential application as a tool for non-destructive testing (BUTTERS[1977]). A further extension of its powers is in the study of high speed events, by recording the speckle pattern generated with a pulsed laser as illurninant. HUGHES[1976] showed that, in principle, the pattern due to one short flash could be recorded on a single frame of the T V scan and subsequently processed in the same way as when using CW radiation. With a double pulse laser it is also possible to obtain a fringe pattern related to the surface movement between the pulses. Since the pulses occur in a time interval small

Fig. 15. Speckle correlation fringes on a carbon fibre-reinforced plastic bar under tensile loading, as viewed on the ESPI monitor, showing in-plane deformation (courtesy of Prof. J. B . Butters, Loughborough University).

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[IV, s; 8

compared with one frame period, the signal is integrated on the vidicon target, and special electronic techniques are necessary to extract the difference signal.

0 8. Intederence Effects with Recorded Speckle Patterns The systems described so far are truly interferometric in that they are concerned with the coherent interaction of speckle fields. However, considerable attention has also been paid to “interference” effects between recorded speckle fields, or more precisely, to the interference of light diffracted by the recordings. When a recording is made, the phase information of the individual speckle is lost, and it is the spatial relation between complete patterns that is of interest. It will be shown that displacements between similar patterns can be readily measured utilising the interference effects of their diffraction fields. The principle on which the techniques to be described is based follows [1968] on multiple exposure from the work of BURCHand TOKARSKI imaging of speckle patterns. Their experiment was to expose a photographic plate to an objective speckle pattern generated by illuminating a diffuser with laser light (Fig. 16a). The plate was first exposed for half the normal exposure time; it was then displaced a small distance in its own plane and a second exposure of similar duration was made. When the Fraunhofer diffraction spectrum of the processed photographic plate is formed by directing a narrow converging beam of monochromatic light through it, a pattern of straight parallel equi-spaced fringes is observed in the plane of focus. These are analogous to Young’s fringes. A simplified explanation is as follows. The two speckle pattern intensities recorded by the plate can be represented by D(x, y) and D(x + d,, y), where D is the speckle distribution, and the plate has moved a distance d, in the x-direction. The total intensity recorded is therefore

where 8 represents a convolution, and 6(x, y) is the delta function at point (x, y).

IV,8 81

INTERFERENCE EF'FEnS WITH RECORDED SPECKLE P A T E R N S

267

Laser P

Liqht

a

Diffuser

P h o t o q raphic Plate

Fig. 16. (a) Recording of double-exposure objective speckle pattern (Burch and Tokarski's experiment). (b) Young's fringe pattern obtained by diffraction at a double-exposure speckle photograph.

If the recording is made upon photographic material whose characteristics shows a linear relationship between the recorded intensity I and the resultant amplitude transmittance t of the processed plate, then t = A - b.1

where A and b are constants.

Substituting in (16), t = A - b{D@[6(x,

y ) + 6(x + d,, y)]).

268

SPECKLE INTEFWEROMETRY

[IV, B 9

The spectrum of this distribution is obtained by taking its Fourier transform

U(u,u ) = A . G ( u , u ) - b { o . [ l + e x p (ikvd,)]},

(17)

where k = 2rrh and u and v are angular co-ordinates. The first term of this distribution represents the directly transmitted light converging to a focus; 6 represents the spectrum of a single speckle pattern, known as the auto-correlation halo. Its angular extent depends upon the scattering properties of the diffuser and the angular size that it subtends at the photographic plate. The brightness of this halo is modulated by the [l+exp(ikvd,)] term, and this gives rise to the fringe pattern. The intensity distribution of the light in the fringes is thus

If= 101"cos2 (kudJ2).

(18)

A typical fringe pattern obtained in this way is shown in Fig. 16b. Interesting extensions of this simple experiment may be made by recording more than two speckle patterns on the same plate. If (N+1) exposures are given, with equal increments of displacement d, in the same direction each time, the intensity recorded will be the speckle distribution convoluted with a series of delta functions 6(x, y), G(x+ d,, y), 6(x +2d,, y), . . . 6(x + Nd,, y). The diffraction spectrum of the recording will now take the form

I ; = 1fi>1" {sin' [(N+ 1)kv&/2]/sin2 [kud,/2]}.

(19)

This formula is readily recognised as being similar to that of the diffraction spectrum generated by (N + 1) equi-spaced small apertures; between each pair of primary diffraction maxima, there will be (N- 1) subsidiary maxima. In principle, it is possible to simulate, by recording multiple exposure speckle patterns with a predetermined spatial relationship, the spectrum to be expected from a similarly disposed array of coherent point sources; furthermore, by varying the relative duration of each exposure, the intensity of these simulated point sources can be modified in any desired way.

8 9. Speckle Photography The procedure described above, used in reverse to analyse the way in which a scattering object moves, has a much wider practical application if

IV, § 91

SPECKLE PHOTOGRAPHY

269

the “subjective” speckle pattern from it is recorded, i.e. if the scatterer is imaged by a lens. In this case one will be able to differentiate between different regions of the scatterer; for example, for an object subjected to mechanical stress, the surface strain can then be measured. Techniques based upon recording of the image speckle pattern are usually referred to as “speckle photography”, and they have attracted considerable attention from workers in the coherent optics field, due to their simplicity and wide range of application (FINKand BUGER[1970], KOPF [1972], ARCHBOLD and ENNOS [1972a]). In its simplest form, speckle photography is used to measure the local displacements of a surface in a direction normal to the line of sight. The object is illuminated with a single beam of laser light from any convenient direction, and a double-exposure photograph of its focused image is recorded on suitably fine-grain film, before and after the unknown displacement takes place. The processed film is then analysed point-bypoint, by directing a narrow beam of light (conveniently from a laser) through it. The diffraction spectrum of Young’s fringes corresponding to the image point selected is then generated (Fig. 17). The direction of the surface displacement is indicated by the orientation of the parallel fringes (it will be orthogonal to them), and the magnitude of displacement d, of the image obtained from the fringe spacing, through the formula di sin p

= nh,

where /3 is the angle of diffraction of the nth bright fringe. The magnitude of the object surface displacement can then be calculated if the magnification of the system is known. This method of speckle photography for measuring displacement in the plane of a surface can be thought of as an extension of the interferometric

Fig. 17. Formation of fringes by diffraction at a double speckle pattern.

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[IV.

D9

speckle methods previously described in 0 5(ii). In the case of the interferometric techniques, it was a necessary condition that the object movement did not cause an image speckle to displace by more than its own diameter. In speckle photography, on the other hand, the speckle must displace by at least one diameter; it is thus inherently less sensitive than speckle interferometry, and the range of sensitivity can be extended downwards by demagnification of the object when photographing it. A reduction in sensitivity brings the following advantages: (1) The requirements for stability of the apparatus are far less severe than for interferometric arrangements. Since the configuration of the speckle pattern generated is independent of the phase of the light illuminating the object, movement of the laser source during exposure can be allowed to some extent. (2) Coherence requirements for the illuminating source can also be greatly relaxed, since it is necessary for the light to be coherent only over one “object” speckle diameter (2,). This means that even an un-Qswitched ruby laser can be used satisfactorily in the recording stage. (3) The technique is insensitive to line-of-sight ( 2 ) motion provided that this is not so large as to cause the speckle pattern to change and become de-correlated with its former self. The criterion to be met is that the change of focus of the image remains within the Rayleigh limit; in terms of object motion A Z , this condition is

A Z < f 2[( 1+ M)/MI2F’A. For demagnifying systems the allowable z motion can thus be considerable. (4) Analysis of the Young’s fringes from one photograph yields both the x and the y components of the displacement at every point of the object. In principle, an imaging lens of any quality may be used for speckle photography, since a speckle pattern is obtained whether lens aberrations are present or not. In practice this is true only when small translatory movements in the focal plane are to be measured, since complications arise when rotation of the object also takes place (see below). A good quality camera lens stopped down to f/4 is therefore normally employed, in conjunction with holographic film or plate (Agfa-Gevaert 10E75 for helium-neon light, 10E56 for the argon laser). Since the minimum speckle size to be recorded at f/4 is about 3 micrometres, faster film can

IV, § 91

27 I

SPECKLE PHOTOGRAPHY

be used to advantage; a good choice for the helium-neon wavelength is Kodak S0253, which, besides having greater speed, is less contrasty, thus relaxing the need for a very uniform illumination of the object. It is an advantage to use the largest possible camera aperture, since not only is less illumination needed, but the measurement sensitivity is thereby increased. This arises because the auto-correlation halo lOl2generated by the subjective speckle pattern extends over a circle of diameter proportional to 1/F; the greater the number of diffraction fringes within the halo, the greater the accuracy with which their spacing can be measured. Double-exposure speckle photographs may also be analysed by an optical filtering operation. The recorded negative is inserted in a system that re-images it, using only light that has been diffracted in a particular direction defined by a small aperture mounted off-axis in the plane of a transform lens (Fig. 18). Those parts of the negative in which the speckle displacement, resolved in the appropriate azimuth direction, is an integral multiple of h/sin p, will be bright, while those for which it takes halfintegral values, will remain dark. Contour fringes of the speckle displacement will thus be generated over the negative. Figures 19a and 19b show horizontal and vertical displacement contours on a loaded cantilever, obtained by filtering the same double exposure speckle photograph, with the filtering aperture offset in the corresponding azimuth directions. The sensitivity of this method of analysis to displacement measurement can be varied by changing the angle p, the greatest number of contours being obtained when the aperture is off-set to the edge of the auto-correlation halo. The contour intervals in Fig. 19 correspond to incremental displacements in the plane of the cantilever of 50 micrometres, the relatively low sensitivity being due to using a camera at a demagnification of five times.

D o u b l e Exposure Soeckle Photoqraph

Aperture I I

hltered lmaqe \ -4

I

Fig. 18. Optical filtering system for obtaining contours from a double-exposure speckle photograph.

2 17


rIv. s lo

Fig. 19. Filtered image of double exposure speckle photograph of a loaded cantilever, showing fringe contours of (a) equal vertical displacement, (b) equal horizontal displacement.

8 10. Further Techniques of Speckle Photography 10.1 SURFACE TILT MEASUREMENT

The speckle pattern formed by the scattering of laser light at a rough surface can be regarded as the coherent superposition of the diffraction spectra generated by a large number of fine gratings of random pitch and orientation, distributed over the surface. Using this simple concept one

IV,0 101

FURTHER TECHNIQUES OF SPECKLE PHOTOGRAPHY

21 3

may show that the effect of tilting the surface about an axis lying in its plane is to rotate the objective speckle pattern, as a whole, about the tilt axis, just as the spectrum formed by a diffraction grating will rotate when the grating is tilted. Geometrically, if the light is incident on the scattering surface at an angle i with respect to the surface normal, and the surface is rotated through a small angle S@ about an axis lying within its plane, then the speckle pattern generated in the direction making an angle r to the surface normal will appear to rotate through an angle SP, where 6 P = (1 + cos ilcos r ) S@.

(20)

For angles of incidence and scattering close to the normal, 8 9 = 2S@, i.e. the surface acts like a mirror to the speckle pattern. ARCHBOLD and ENNOS [1972a] showed that this property of the speckle pattern could be used to measure local tilting of the surface, by recording double exposure photographs with the object surface out of focus by a known amount. If the apparent displacement d‘ of the speckle is measured from the Young’s fringes generated by the doubled speckle recording, then the tilt angle 6@ will be given by 6@ = d‘12M SU,

(21)

where SU is the (small) defocus distance, and M‘ is the system magnification. De-focusing will of course blur the image slightly, and the tilt angle that is measured will be the value averaged over the circle of confusion recorded. For a body that rotates as a whole, T~ZIANI [1972a] showed that there are advantages in de-focusing to such a degree that the Fourier transform of its surface is recorded, i.e. the object and photographic plate are positioned at the two focal planes of the lens. Under these conditions any simultaneous lateral translation of the object produces no speckle movement, so that the system is sensitive to tilt only. It was subsequently shown by GREGORY [1976a, b] that the same translational independence could be obtained with a lesser degree of de-focusing, provided that certain conditions are met; these are (i) that the object surface is illuminated and photographed at nearly normal incidence, (ii) that the illuminating laser beam is divergent, and (iii) that the camera is focused on the plane containing the “image” of the laser source, “reflected” in the surface, regarding it as a mirror. GREGORY [1977] has developed this technique more fully as a method of detecting small surface deformations, and measuring surface profile changes.

275


[IV, a 10

Fig. 20. (a) Partial slope contours on a metal diaphragm distorting under internal pressure, obtained by out-of-focus speckle photography. (b) Out-of-plane displacement contours of the same diaphragm obtained by holographic interferometry.

IV, 0 101


275

Analysis of the de-focused speckle photographs can also be undertaken by the optical filtering method described in the previous section. The fringes generated in this way are partial slope contours, resolved in a direction corresponding to the direction of offset of the filtering aperture. Figure 20a illustrates such a pattern on a circular metal diaphragm clamped round its edge, and deformed by pneumatic pressure. The corresponding pattern oi’ displacement contours, obtained (with a similar pressure rise) by holographic interferometry, is shown in Fig. 20b. The disparity between the two photographs arises from the fact that the slope contours obtained by speckle photography refer to the component of slope resolved in the horizontal direction only, while the holographic fringes correspond to increments of displacement of A/2 in the normal direction. If a three-dimensional object is under investigation, the degree of de-focus will vary from point to point on its surface, and this complicates [1976] the analysis of the displacement and tilt that it undergoes. STETSON has, however, suggested that by the simultaneous recording of double exposure speckle photographs in a number of different image planes, the complete solution to the object deformation might be obtained. His theoretical treatment indicates that local surface strain, shear and rotation might be extracted from the fringe information.

10.2. ANALYSIS OF OBJECT MOTION BY TIME-AVERAGED SPECKLE PHOTOGRAPHY

It is well known that useful information about the behaviour of a moving surface may be obtained by recording its hologram while the object is in motion (the “time-averaged’’ method). In an analogous way, speckle photography can be employed to analyse continuous displacement or vibration of a surface, but in this case the sensitivity of measurement will be lower, and the sensitivity vector will lie in the plane of the surface, and not normal to it. If a “time-averaged’’ speckle photograph is recorded and subsequently analysed by forming its diffraction spectrum, the distribution of light to be expected in the Fourier plane may be predicted theoretically by modifying equation (16), in such a way that the two delta functions (representing two positions of a surface) are replaced by a position probability density function. For example ( F R A N c o “ ~ ~ ~ ~ ] ) , if the surface moves at uniform velocity in the x-direction during the

216


[IV, § 10

exposure, this expression will be a rectangular function, and the distribution of light in the diffraction spectrum will contain, instead of the cos2 term, the factor sinc2 (~rvdJA),where d, is the total displacement that has taken place. The brightness of the higher order fringes will thus fall off rapidly. Of greater interest is the case where the surface is vibrating in its own plane (TIZIANI[19711, ARCHBOLD and ENNOS [1972b]). Each speckle will then be smeared out in the direction of vibration, and its position probability density will take the form

w, = [(lwhere x , is the amplitude of vibration, assumed to be sinusoidal. The corresponding intensity distribution in the fringe pattern generated by diffraction at the speckle recording will now be

A considerable number of fringes will be visible in this pattern, since the amplitude of modulation of the Bessel function falls off far less rapidly than that of the sine' function. Motion in the plane of the surface does not necessarily follow a straight line path, but can contain components in both the x- and y-directions. The resulting speckle recording will have a diffraction spectrum characteristic of that motion, as ARCHBOLD and ENNOS [1975a] showed. If the x and y components are both sinusoidal, with frequencies related integrally to one another, the speckles will execute Lissajous’ figures, and the diffraction spectrum then becomes symmetrical about its centre, as seen in Fig. 21. Here the two orthogonal vibrations are in frequency ratio 2 : 1, and the speckles trace out parabolic paths. LOHMANN and WEIGELT [19751 extended this work to show that speckle photography can also be used to analyse the trajectory traced out by a surface moving in its own plane, rather than to infer it from the diffraction spectrum. To carry out the analysis, a double exposure recording is made: the first, a time-averaged one with the surface moving over its complete trajectory; the second, with it stationary, but displaced laterally so that the “centre of gravity” of the speckle pattern is translated a distance do.The total intensity recorded on the film is then

1 = D @ [W(x,y) + 6(x + do,y)l.

IV, § 10


271

Fig. 21. Diffraction pattern from a time-average speckle photograph of a surface vibrating in its own plane, with x and y sinusoidal vibrations in 2: I frequency ratio.

Following the same reasoning as in § 7, the intensity distribution I, in the spectrum of this recorded speckle pattern will be

W(x, y)+exp (-ikvd,)(2

kudo)+ W*(x, y) exp (-ikvd,)

+ I W(X, y)l2I, (24)

where W is the Fourier transform of W, and W* is the complex conjugate of W. The expression in the square brackets can be recognised as being the intensity distribution that would have been generated by forming the Fourier hologram of an object having a light intensity distribution W(x,y). Thus, if a second recording, this time of the spectrum from the original speckle photograph, is made on film, the motion trajectory may be extracted from it by treating the second recording as a Fourier hologram, and reconstructing the “image” from it in the usual way by illumination with a plane or divergent reference beam.

77s


[IV, 9: 11

8 11. Applications of Speckle Photography in Metrology By its simplicity and versatility, speckle photography lends itself to a wide variety of applications, and a number of variations of the technique have been developed to suit particular measurement problems. As an example of simple displacement mapping in an industrial environment, double exposure speckle photography was used to study the opening and closing of a crack that had developed in a welded joint of a large test [1975b]). machine, when it was subjected to load (ARCHBOLD and ENNOS A standard 35 mm camera with holographic film was rigidly attached to the machine, while the light source (a 15 mW helium-neon laser) was mounted on a separate tripod, and the beam diverged on to the crack by means of a lens. The speckle photographs were analysed to give the vector displacements of the image at points just above and just below the crack line, relative to the camera; subtracting these vectorially gives the and LUXMOORE [1974] crack opening at that point. In a similar way EVANS examined the very small displacements occurring in the region surrounding the tip of a crack in a piece of Araldite plastic, as the crack was growing. To obtain high sensitivity and resolution, the crack tip was magnified ten times, using a profile projector. Cracks in concrete have also been detected in the same way, by applying loads to the structure and recording double exposure photographs (DE BACKER[ 19751). Creep studies are also possible using speckle photography. The dimensional change, after quenching, of a single crystal specimen of aluminium only 10 mm long was followed (LUXMOORE, AMIN and EVANS [1974]), and the creep of polypropylene plastic strip specimens under tension has also been and ARCHBOLD [ 19761). investigated, using a magnifying system (ENNOS The strain behaviour of plastic materials is particularly suited to study by speckle photography, since the displacement that results from applying a stress is relatively large. Since it is possible to measure the vector displacement of every point on the surface undergoing stress by speckle photography, it follows that the strain field can be derived by differentiation, as when using a conventional moirC method. Although this was implicit in the original and JUANG [1976al demonpapers describing the technique, CHIANG strated that quantitative analysis of plane strain problems could be made, using both the point-by-point and the full field method. Similarly, KHETAN and CHIANC [ 19761 applied the out-of-focus technique to measure the two-dimensional bending of plates.

IV,P 111

APPLICATIONS OF SPECKLE PHOTOGRAPHY IN METROLOGY

219

T o obtain an adequate accuracy when using speckle photography it is necessary that the speckle displacement in the image be not too great; large rigid body displacement or rotation can reduce sensitivity. This problem, comnnn to all non-contacting methods of measurement, can be overcome if two separate records are made and subsequently compared. Ideally one should superimpose the two speckle photographs in exact register, but the practical difficulty of bringing them into the same plane means that in practice the recorded speckle patterns are separated in the line-of-sight direction, giving rise to curved transform fringes. KOPF [1973a] demonstrated this for two similar speckle fields, and ADAMSand MADDUX[ 1975al described a “sandwich” technique whereby the two speckle photographs are recorded on plates held in a special holder allowing the rigid body motions to be partially cancelled, and the residual displacement derived from the curved transform fringes. The measurement sensitivity of a system for speckle photography is directly dependent upon the magnification at which the recordings are made. A method of calibrating the system was described by CLOUD [1975], but this is only useful when the object under study performs simple translational motion. If a tilt is also present false values of displacement may be measured due to aberrations of the camera lens system. In particular, astigmatism and field curvature can lead to significant errors in the derived strain field. A number of variations of the basic technique of speckle photography are closely related to holography. For example, a focused-image hologram is really a speckle photograph with the coherent addition of an off-axis reference beam, and ADAMSand M A DD ~[1974] JX showed that if a double-exposure hologram of this type was recorded with a high signal beam to reference beam ratio, then it could be used both as hologram and speckle photograph. The reconstructed image, obtained by directing the reference beam on to the processed plate, will then give a fringe contour pattern related to the out-of-plane displacement components of object motion, and the Young’s fringes generated by diffraction from a small region of the hologram by a narrow light probe will give the in-plane displacement that has taken place. Alternatively, in-plane displacement contours, resolved in any desired direction, can be formed by optical filtering. This technique was used to investigate distortion of a metal plate in the region surrounding an interference-fit fastener, when the fastener [1974]). Another techdowel was driven home (ADAMSand GRIFFITHS nique developed by the same group (ADAMS and MADDUX [1975b]) analyses

280


[IV, 8 1 1

the displacements, as recorded in a double exposure hologram, by photographing the reconstructed image using a camera loaded with holographic film. When this photograph is examined with a light probe, Young’s type fringes related to the in-plane displacement are generated; the contrast of these is poorer than for a direct speckle photograph of the object, since, in the hologram reconstruction process, the two slightly displaced images are coherent with one another, and therefore interfere, whereas in speckle photography the recordings are separated in time, and thus cannot interact. A technique developed by BOONE[1976] also aims at solving the complete displacement field, and makes use of reflection (Lippmann) holograms. The photographic plate is attached to the distorting surface, and the illuminating beam sent through the plate; light scattered back forms -‘?lost a “contact” print. In an analogous way to the focused image hologram method described above, the reflection hologram can be analysed to give both out-of-plane and in-plane components of displacement. The strain field induced within a translucent material, such as Araldite, can also be measured by a variant of the speckle photography technique. A narrow laser beam is directed through the plastic in, say, the x -direction. Double exposure speckle photographs of the light scattered in the y-diriction and in the z-direction are then recorded. By analysing these, displacement values in all three coordinate directions may be [1976]). BARKER measured for points along the illuminated path (CHIANG and FOURNEY[1976] similarly demonstrated the technique, and also showed that if the displacement distances were less than one speckle diameter (&), the Leendertz interferometric technique (§ 5 (ii)), could be used. In this case the illuminating beam is directed through the scattering plastic in the form of a sheet of light, and reflected back along the same path with a mirror. The illuminated sheet is photographed from the side, and from the double exposure recordings one obtains speckle correlation fringe patterns with a spacing corresponding to a surface displacement of half a wavelength. These techniques all require a high intensity illuminating beam if the recording is to be made in a reasonably short time. For non-destructive testing applications under difficult environmental [1976a] showed that a non-Q-switched pulsed ruby conditions GREGORY laser could be used with advantage. By the defocused speckle photography technique, he was able to detect flaws in a large steel pressure vessel at an ambient temperature of 50°C. His system, sensitive only to surface tilt, indicated where the surface bulged locally when internal

IV, § 121

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281

pressure was applied to the vessel. A subsequent paper (GREGORY [1977]) gives some details of the way in which the tilt fringe patterns can be interpreted in practice. Speckle photography has been applied to the investigation of vibrations, in the case of the oscillation of the tuning fork of an electronic watch (TIZIANI [ 19721). From well-focused time-average recordings, the amplitude of vibration of the tines of the fork could be derived, and by recording the speckle pattern in the Fourier transform plane, the tilt component for various regions of its surface are obtained. The performance of the watch can be improved by re-designing the fork so as to minimise tilting, and speckle photography provides a means of monitoring the results. More recently, CHIANGand JUANG[1976b] have shown that by recording time-averaged speckle photographs slightly out of focus, and optically filtering the resultant vibration photograph in the manner described in § 8, partial anti-nodal patterns of the vibration mode are obtained. A partial anti-nodal pattern delineates points on the surface where the component of oscillatory bending along the azimuthal direction, defined by the filtering aperture, is zero. In themselves, these patterns provide only limited information on the vibration behaviour, but it is also possible to filter out the true anti-nodal regions by using an annular aperture instead of a circular aperture in the filter plane. It may be remarked that the surface at these anti-nodes is performing large amplitude oscillations in the direction of line-of-sight, but since there is no component o f bending, the out-of-focus speckle pattern does not change.

Q 12. “White Light” Speckle Photography Although this review is concerned principally with interferometry of speckle patterns generated by laser light, a number of associated techniques have been developed using “artificial” speckle patterns applied to [1975], BOONEand DE the surface under investigation (BURCHand FORNO BACKER[1976]). Since speckle photography as a means of displacement measurement is really a method of positional correlation of two random intensity patterns, it does not matter how these are generated, and spray-painting, retro-reflecting paint or even the rough texture of a concrete surface have been used to characterise the surface. White light illumination can then be employed, but it is necessary to use a good quality camera lens in order to obtain high resolution, and focusing is

282

SPECKLE MTERFEROMETRY

[IV, § 12

critical. Displacements of magnitude greater than 10 p m can be measured in this way. If the object under investigation is a flat translucent plate, contact printing can be adopted instead of imaging. (A similar procedure was used in the manufacture of 180” scatter plates for a scatter-plate [1970].) interferometer by HOUSTON A great improvement in the recording system for “white-light speck[1975]), by fitting the camera with an les” was made (BURCHand FORNO aperture stop in the form of four slots arranged like a square picture frame. This enhances the response of the lens for a particular spatial frequency determined by the slot separation. It also increases the depth of focus over that which would be obtained when using an equivalent circular aperture. The tuned frequency uo that the masked lens selects is given by uo = dfh,

(25)

where s is the slot separation, and f is the focal length of the lens. When using a Pentax 55 mm Takumar lens, it is possible to obtain a resolution of 300 lines/mm over the whole field, a value comparable to that of laser speckles generated by the same lens. Although this system of recording was developed principally for use with quasi-periodic grids applied to the surface under examination (the spacing of these being tuned to the slotted aperture lens), it also operates well if a high-contrast randomly spaced pattern is applied. A surface coated with retro-reflective paint, illuminated from a source close to the camera lens, is imaged by the slotted aperture lens as an array of fine grating patterns, in the manner that the Duffy technique images a laser speckle pattern (see 0 5(ii)). If a double exposure image is recorded on holographic film, is bleached, and is examined in an optical filtering system similar to Duffy’s, fringe patterns will be obtained relating to the surface movement lateral to the line of sight that has taken place between the exposures. FORNO [1975] showed that this type of white light speckle photography could be used for analysing the distortion of large sized objects, and also for vibration analysis and surface contouring. Figure 22 is a fringe pattern obtained by recording a double exposure white light photograph of a stainless steel sink bowl, coated with retro-reflective paint, when a load was applied between the exposures. Besides not requiring a laser, white light speckle photography has other advantages to offset the fact that the surface must be coated in some way. The principal one is that the system is insensitive to surface tilt, which has

IV, 5 131

MEASUREMENT OF SURFACE ROUGHNESS BY SPECKLE INTERFEROMETRY

283

Fig. 22. “White light” speckle fringe pattern on a metal sink coated with retro-reflective paint, vertical displacement contours spaced at 140 p,m (courtesy C. Forno).

been seen to complicate analysis of distortions when laser speckle photography is employed. It is also possible to cover a much larger area with a powerful light source; flash tube illumination is advantageous in this respect.

§ 13. Measurement of Surface Roughness by Speckle Interferometry

The detailed configuration of a laser speckle pattern is strongly dependent upon the microscopic nature of the surface irregularities at which the laser light is scattered. It also depends upon the surface reflectivity. Since all the types of speckle interferometry described here depend upon maintaining speckle correlation, it is evident that they will fail if the surface changes in any way during the course of a measurement. This property can, however, be used to detect when surface contamination,

283

[rv,w 13


corrosion or oxidation, or surface wear has taken place (KOPF[1973b]). A double exposure laser photograph is recorded with a small lateral translation of the surface, before and after the suspected degradation has occurred. On optically filtering the negative, only those areas that have been unaffected will be re-imaged, the disturbed regions remaining dark. Speckle de-correlation can also occur for other reasons, such as a change in the laser wavelength, or a change in its direction, and its extent will depend upon the depth of the surface roughness. Assuming that the distribution of the variations in height of the surface profile from the mean is Gaussian, the effect of a change in wavelength of Ah will be to cause complete de-correlation when

(5)2 A2/2AA,

(26)

where (5) is the mean square depth variation. TRIBILLON [1974] used this as a basis for a method of measuring surface roughness by speckle photography. The usual double exposures with object translation are given, but with a change in the wavelength between the exposures. The Young’s fringes generated by diffraction at the negative become less contrasty as the wavelength difference increases. When the recordings no longer give discernible fringes, eq. (26) can be applied to find the roughness. An argon laser is conveniently used to supply the coherent source of variable wavelength. The de-correlation occurring when the direction of illumination is altered can also be measured using the same photographic technique (LEGER,MATTHIEU and PERRIN[1975]). The theoretical variation of the visibility V of the transform fringes with change in illumination direction SO is given by V = exp { - [ ( 2 ~ ( 5 ) / h )sin 8 1

SO]’},

where O is the angle of the illuminating beam relative to the surface normal. In applying the technique, the far-field speckle pattern is recorded on film, without using a lens to image the surface. Due to the change in the illuminating angle, the speckle pattern will translate laterally by a distance h cos 6, where h is the distance between surface and film. Since this speckle shift may be large, it is necessary to compensate for it by moving the film between exposures by an amount calculated to leave a small residual speckle separation, sufficient to give suitably spaced Young’s fringes. A further method using speckle interferometry, developed by LEGERand

IV,§ 141

CONCLUSIONS

285

PERRIN [1976], aims to measure surface roughness over the range 1-30 kin, and to do it in real time. The surface is simultaneously illuminated by two beams, at angles O and (0 + SO) on one side of the normal, and the light scattered off at similar angles on the other side of the normal is combined in an interferometer, to give a fringe pattern. The visibility of these fringes is measured electronically by varying the relative path lengths of the two arms of this interferometer, and recording the modulation depth of the signal picked up by a photoelectric detector. Equation (27) can then be used to find the surface roughness depth. The instrument that was developed used a Michelson-type interferometer in both illuminating and viewing beams, with mirrors adjusted to give the same wavefront tilt. It must be noted that these methods for measuring surface roughness not only assume a Gaussian distribution of height profile, but also that the incident light is singly scattered by the surface. ARCHBOLD and ENNOS [1972a] showed that for surfaces that are strongly re-entrant, or those into which the sight penetrates to some degree, de-correlation of the speckle pattern will occur much more rapidly when a change in one of the illumination parameters is made. The maintenance of speckle correlation is, in fact, of paramount importance for all types of speckle interferometry, and this is a subject which has not as yet received much attention. Since imaging of the scattering surface is employed in practically every case, speckle interferometers will always be limited in their measurement capability by the amount of surface tilt that occurs, if only because the objective speckle pattern accepted by the imaging lens aperture will gradually change as tilt takes place. A theoretical treatment of de-correlation effects in the plane-strain interferometer has, however, been given by JONES and WYKES[1977].

0 14. Conclusions This review has concentrated upon the instrumental aspects of speckle interferometry and their applications, rather than a detailed theoretical treatment. It is seen that a considerable degree of ingenuity has had to be employed to overcome the inherent difficulties that occur when interference takes place between randomly distributed phase fields rather than between smoothly varying ones. Although the aim of the research has been to develop techniques useful to the engineer, it may well be that much of it is too complex to hav,e widespread application. However, the

286

SPECKLE INERFEROMETRY

[W

interest so engendered in the subject of the coherent speckle phenomenon is having repercussions in other branches of optics, and much theoretical work of a statistical nature, not referred to here, is presently being carried out. Developments are also being made in the use of speckle patterns for information processing (see, for example, FRANCON [1975]), where the pattern is employed as an optical carrier. Although this work bears a close relation to speckle photography, it also has not been included in the review.

References ADAMS,F. D. and W. I. GRIFFITHS,1974, Interference-fit fastener displacement measurement by speckle photography, Proc. A r m y Symposium on Solid Mechanics, AMMRC MS 74-8. ADAMS,F. D. and G. E. MADDUX,1974, Appl. Opt. 13, 219. ADAMS,F. D. and G. E. MADDUX,197Sa, Dual plate speckle photography, US Air Force Technical Memorandum, AFFDL-TR-75-92. , In-plane displacement measurement using ADAMS,F. D. and G. E. ~ ~ A D D U X1975b, speckle photographs of holographic images, US Air Force Technical Memorandum, AFFDL-TR-75-45. ALEKSOFF, C. C., 1971, Appl. Opt. 10, 1329. and P. A. TAYLOR,1969, Nature 222,263. ARCHBOLD, E., J. M. BURCH,A. E. ENNOS and A. E. ENNOS, 1970, Optica Acta 17, 883. ARCHBOLD,E., J. M. BUHCH ARCHBOLD, E. and A. E. ENNOS,1972a, Optica Acta 19, 253. ARCHBOLD,E. and A. E. ENNOS,1972b, Measurement by laser photography, Proc. Electro-Optics '72 Intern. Conf. (Brighton, England) p. 65. E. and A. E. ENNOS. 1974, J. Strain Analysis 9, 10. ARCHBOLD, ARCHBOLD, E. and A. E. ENNOS,1975a, Opt. Laser Technol. 7,17. ARCHBOLD, E. and A. E. ENNOS,197Sb, Non-Destructive Testing 8, 181. BARKER,D. B. and M. E., FOURNEY, 1976, Experimental Mechanics 16, 209. BIEDERMANN, K., L. EK and L. Os'rLuND, 1976, The Engineering Uses of Coherent Optics (Cambridge University Press, London) p. 219 BOONE,P. M., 1976, The Engineering Uses of Coherent Optics (Cambridge University Press, London) p. 81. BOONE,P. M. and L. C. DE BACKER,1976, Optik 44, 343. BURCH,J. M., 1953 Nature 171, 889. BURCH,J. M., 1970, Optical Instruments and Techniques (Oriel Press, Newcastle-uponTyne) p. 213. BURCH,J. M., 1971, S.P.I.E. Seminar Proc. 25, 149. BURCH,J. M. and C. FORNO,1975, Optical Engineering 14, 178. BURCH,J. M. and J. M. J. TOKARSKI,1968, Optica Acta 15, 101. BUTTERS,J. N., 1977, Opt. Laser Technol. 9, 117. BUTTERS,J. N. and J. A. LEENDERTZ,1971a, J. Phys. E. (Sci. Instrum.) 4, 277. BUTTERS,J. N. and J. A. LEENDERTZ,1971b, Measurement and Control 4, 349. BUTTERS, J. N. and J. A. LEENDERTZ, 1974, Component Inspection using Speckle Pattern, Proc. Electro-Optics '73 Intern. Conf. (Brighton, England) p. 43. CHIANG,F. P., 1976, The Engineering Uses of Coherent Optics (Cambridge University Press, London) p. 249.

IVI

REFERENCES

287

CHIANG, F. P. and R. M. JUANG,1976a, Appl. Opt. 15,2199. CHIANG, F. P. and R. M. JUANG,1976b, Optica Acta 23, 997. CLOUD,G., 1975, Appl. Opt. 14, 878. DEBRUS,S., 1977. Opt. Commun. 20, 257. and J. N. BUTTERS,1976, The Engineering Uses of DENBY,D., G. E. QUINTANILLA Coherent Optics (Cambridge University Press, London) p. 171. DE BACKER, L. C., 1975, Non-Destructive Testing 8, 17. DUFFY,D. E., 1972, Appl. Opt. 11, 1778. EK, L. E. and N.-E. MOLIN,1971, Opt. Commun. 2, 419. ENNOS,A. E. and E. ARCHBOLD, 1976, Plastics and Rubber: Materials and Applications 1, 116. EVANS,W. T. and A. R. LUXMOORE, 1974, Engineering Fracture Mechanics 6, 735. FINK,W. and Z. BUGER,1970, Angew. Physik 213, 176. FORNO,C., 1975, Opt. Laser Technol. 7 , 217. M., 1975, Laser Speckle and Related Phenomena (Springer-Verlag, Berlin) Ch. FRANCON, 5. FUJI, H. and T. ASAKURA,1974, Opt. Commun. 11, 35. GREGORY, D. A,, 1976a, The Engineering Uses of Coherent Optics (Cambridge University Press, London) p. 263. GREGORY, D. A,, 1976b, Opt. Laser Technol. 8, 201. GREGORY,D. A,, 1977, Opt. Laser Technol. 9, 17. P., 1975, Appl. Opt. 14, 2.563. HARIHARAN, B. P. and K. A. HAINES,1967, J. Opt. Soc. Am. 57, 155. HILDEBRAND, HOUSTON, J. B., 1970, Optical Spectra 4 (6), 32. HUGHES,R. G., 1976, The Engineering Uses of Coherent Optics (Cambridge University Press, London) p. 199. HUNG,Y. Y., 1974, Opt. Commun. 11, 132. and I. M. DANIEL. 197.5, Appl. Opt. 14,618. HUNG,Y. Y., R. E. ROWLANDS HUNG,Y. Y. and C. E. TAYLOR,1973, Proc. S.P.I.E. 41, 169. JONES,R., 1976, Opt. Laser Technol. 8, 215. 1974a, J. Phys. E. (Sci. Instrum.) 7 , 616. JONES,R. and J. A. LEENDERTZ, JONES,R. and J. A. LEENDERTZ, 1974b, J. Phys. E. (Sci. Instrum.) 7, 653. JONES,R. and C. WYKES,1977, Optica Acta 24, 533. KHETAN,R. P. and F. P. CHIANG,1976, Appl. Opt. 15, 2205. KOPF, U., 1972, Optik 35, 144. KOPF, U., 1973a, Opt. Commun. 4, 108. KOPF, U., 1973b, Siemens Forsch.-u.-Entwick1.-Ber.2, 277. LABEYRIE, A., 1970, Astronom. Astrophys. 6, 85. LEENDERTZ, J. A., 1970a, Optical Instruments and Techniques (Oriel Press, Newcastleupon-Tyne) p. 256. LEENDERTZ, J. A,, 1970b, J. Phys. E. (Sci. Instrum.) 3, 214. J. A. and J. N. BUTTERS, 1973, J. Phys. E. (Sci. Instrum) 6, 1107. LEENDERTZ, LBGER,D., E. MATTHIEUand J. C. PERRIN,1975, Appl. Opt. 14, 872. LBGER,D. and J. C. PERRIN,1976, J. Opt. SOC.Am. 66, 1210. LIGTENBERG, F. K., 1954, Proc. SESA 12, 83. LOHMANN, A. W. and G. P. WEIGELT,1975, Opt. Commun. 14, 252. LBKBERG, 0.J. and K . H0GMOEN, 1976, J. Phys. E. (sci. Instrum.) 9,847. LBKBERG, 0. J., 0. M. HOUE, and H. M. PEDERSEN, 1976, Opt. Laser Technol. 8, 17. LUXMOORE, A. R., F. A. A. AMINand W. T. EVANS,1974, J. Strain Analysis 9, 26. A., S. D. RAMSEYand L. F. SCHAEFER,1971, Appl. Opt. 10, 2722. MACOVSKI, PEDERSEN, H. M., 0. J. LDKBERG and B. M. FORRE,1974, Opt. Commun. 12,421.

288


RIGDEN,J. D. and E. I. GORDON,1962, Proc. IRE 50, 2367. N., T. TSURUTAand Y . ITOH,1968, Jap. J. Appl. Phys. 7, 904. SHIOTKE, STETSON, K. A., 1970, Opt. Laser. Techno]. 2, 179. STETSON,K. A,, 1976, J. Opt. SOC.Am. 66, 1267. TIZIANI,H., 1971, Optica Acta 18, 891. H., 1972a, Opt. Commun. 5, 271. TIZIANI, TIZIANI, H., 1972b, Optik 34, 442. TRIBILLON,G., 1974, Opt. Commun. 11, 172. URBACH, J. C. and R. W. MEIER, 1966, Appl. Opt. 5, 666. WYKES,C., 1977, Optica Acta 24, 517.


V

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATTERN RECOGNITION BY

DAVID CASASENT and DEMETRI PSALTIS Carnegie -Mellon University, Department of Electrical Engineering, Pittsburgh, P A 15213, U.S.A.

CONTENTS PAGE

. .

. . . . . .

. . . . . . 291

8

1. INTRODUCTION

D

2. SPACE-VARIANTOPTICALPROCESSING

.

.

. . . .

Q 3. SCALE-INVARIANT SPACE-VARIANT OPTICAL PROCESSOR . . . . . . . . . . . . . . . . 8 4. APPLICATIONS OF SCALE INVARIANT SYSTEMS

296

. 302

. .

314

3 5. ROTATIONAL INVARIANT SPACE-VARIANT SYSTEMS. . . . . . . . . . . . . . . . . . . 332 8 6. MULTIPLE INVARIANT SPACE-VARIANT PROCESSORS . . . . . . . . . . . . . . . . . 345

D

7. SUMMARY AND CONCLUSION.

. .

. .

.

.

. .

. 354

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . 355 REFERENCES . . . . .

. . . . . . .

.

. . . . . . 355

0 1. Introduction The classic operations possible in a coherent optical processor have been reviewed several times (VANDER LUGT[ 19741, GOODMAN [1977]). All of these basic operations rely on the Fourier transform property of a lens. This Fourier transform operation and the correlation and linear space invariant systems that result by cascading Fourier transform systems are the hallmarks of a coherent optical processor. However, the resultant systems (especially the optical correlators) have seen limited practical use. One reason for the extensive use of digital, rather than optical, processing methods in pattern recognition is the enormous flexibility and myriad of pattern recognition algorithms possible in a digital processing system. This rigidity and lack of flexibility of an optical processor and the limited number of operations achievable on such systems have greatly limited its practical applications (CASASENT and CAULFIELD [19781). One approach to these problems has been the development of hybrid optical/digital processors in which the best features of each processing method are combined (CASASENT and STERLING [1975], CASASENT [1978]). A more recent approach has involved the development of linear and non-linear space-invariant optical processors (SOC.Photo Opt. Instru. Engr., 1976). In this chapter, we are concerned with optical pattern recognition applications. Specifically, we consider practical pattern recognition problems in which differences exist between the input and reference functions to be correlated. The approach to be emphasized involves the development of space-variant optical pattern recognition systems that are invariant to various expected deformations that can arise between the input and reference functions to be correlated. The specific type of space-variant optical processor that we emphasize is realized by applying a coordinate transformation preprocessing operation to the input and reference data. Types of space-variant processors and applications of these systems, other than pattern recognition, are briefly summarized for completeness. Although the optical realization of these space-variant pattern recognition systems is emphasized (to retain 29 1

292

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATTERN RECOGNITION

[V, 5 1

the high speed and parallel processing advantages of an optical system), the methods to be presented can be realized by digital and electronic methods.

1 . I . OPTICAL PATTERN RECOGNITION

A brief review of the relevant, conventional, coherent optical processing, pattern recognition systems is included for completeness, to indicate the notation to be used, and to describe two of the optical correlator architectures that are frequently used in the experiments to be described in subsequent sections. A schematic diagram of the classic coherent optical pattern recognition system is shown in Fig. 1. A transparency with amplitude transmittance f(xo, yo) is placed in the input plane Po. Spherical lens L1 forms the two-dimensional optical Fourier transform: m

~(w,, my>= [ [ f ( x o . yo) exp [j(axxo+ayyo)~ dxo dyo

(1)

-m

of f(xo, yo) at plane PI. Capital letters are used to denote the Fourier transform of the corresponding lower case variables. 0, and wy are the spatial radian frequency coordinates of the transform plane. They are related to the space coordinates (xl, yl) of plane P, by W,

= 2~xl/AfL,

wY = 2 ~ ~ 1 / A f = ,

(2)

where fL is the focal length of lens L, and A is the wavelength of the laser light used. For pattern recognition, we record the complex conjugate G*(w,, w,,)

f(x

d

y )

0

L,

LP

Fig. 1. Schematic diagram of a frequency plane optical correlator.

v, 8 11

290

INTRODUCTION

of the Fourier transform G ( w x ,o,,) of the reference function g(x,, yo) at plane PI. This is a matched spatial filter and is formed holographically by placing g(x,, yo) at plane Po, recording at P1 the interference of its transform G ( w x , o y )(formed by L, at P,) and a plane wave reference beam U , = a exp (-j27rax1) of amplitude a at an off-axis angle 8 [where a =(sin 8 ) / h ] .The pattern recorded at P1 and the subsequent transmittance of P, is: t,(w,, o y ) = I G +U R ( 2 = ~ 2 + l G 1 2 + exp ~ G(+j27rax1)

+ aG* exp (-j27raxI).

(3)

The term of interest in the transmittance of P, is the last term in eq. (3). During correlation, the pattern described by eq. (3) is stored at P, and f ( x 0 , yo) is placed at Po. The light amplitude distribution leaving P, is Ftl. forms at P2 the Fourier transform of the product of these two Lens transforms. At P,, for the term of interest, we obtain: wx2,

Y2) =

a f 0 g * N X 2 - (YhfL),

(4)

where 0 denotes correlation. Thus, the desired correlation of the input function f and the reference function g appears at x2 = ahfL in plane P2 (where equal focal lengths are assumed for L, and L2). The convolution f * g of the two functions appears centered at x2 = -ahfL. This term and the other terms centered at the origin of P2 do not concern us. Because the correlation is realized by multiplication in the transform plane, this is referred to as a frequency plane correlator. The schematic diagram of a second correlator that we will use is shown in Fig. 2. In this system, the two functions f and g to be correlated are placed side-by-side in the input plane Po with a center-to-center separation 2b. In 1-D (for simplicity only), we can describe the transmittance of the input plane Po by

-1;:

-

‘READ LIGHT

Fig. 2. Schematic diagram of a joint transform optical correlator

294

DEFORMATION INVAFUANT, SPACE-VARIANT OPTICAL PATTERN RECOGNITION

[V, § 1

Lens L1 forms the magnitude squared of the Fourier transform of U, at P I . The subsequent amplitude transmittance of P, is then U , ( o )= IF(o)exp (job) + G(w) exp (-jub)I2 = IF12+IG12+FG*exp (jw2b)+F*G

exp (-jw2b).

(6)

With the transmittance of plane P1 given by eq. (6), we illuminate plane P, with a plane wave in reflection from beam splitter (BS). We show this wave as A, in Fig. 2. When real-time spatial light modulators are used at PI, different wavelengths of light are usually required for writing and [1977]). Hence the reading and readout is in reflection (CASASENT schematic of Fig. 2 is similar to the actual one that would be used. Because both functions to be correlated are placed side-by-side in the space domain input plane, this is referred to as a joint transform correlator. Both optical correlators in Figs. 1 and 2 can be converted to multichannel 1-D systems by replacing lenses L, and L, by cylindrical/ spherical lens pairs. In these systems, plane Po is imaged onto plane PI vertically and the transforms of the signals on each of N lines at Po are formed on N lines at PI. With N one-dimensional matched spatial filters recorded on N lines at PI, the output plane P, pattern consists of N correlations of the N input signals with the N reference signals recorded at PI. We will see several examples of the use of each of these correlations in subsequent sections.

1.2. NON-LINEAR AND SPACE VARIANT SYSTEMS

Conventional optical processors are linear, space-invariant systems in which the system’s impulse response (the output of the system with a point delta source as the input) does not change shape (but shifts across the output plane) as the input point source is moved across the input plane. A space-invariant system is thus shift-invariant and can be described by

g,

where the input and output coordinates are $. and the input and output functions are f , ( [ ) and fl(g), and O,({-[) is the system’s impulse response.

v, 5 11

INTRODUCTION

295

Space invariance has proven to be a severe limitation in optical processing, since it restricts the operations achievable to shift-invariant correlation and convolution. It has been clear that to increase the range of applications of optical processors and their practicality, space-invariant and/or non-linear optical systems are needed. Nonlinearities have been produced by preprocessing the input data with half -tone screens (DASHIEL and SAWCHUCK [1975]). By appropriate spatial filtering of this binary input signal, various nonlinear operations have been achieved such as: A/D conversion, level slicing, log operations, etc. Non-linear optical systems have also been developed using optical feedback (STALKER and LEE [1974]) applied to image enhancement. Optical systems capable of performing geometrical transformations on spatial functions have also been reported (BRYNGDAHL [1974]). In this latter case, computer generated holograms were used to convert the optical processor into a spacevariant system. Several researchers (ROBBINSand HUANG[19721, SAWCHUK [19741) realized that certain space-variant imaging systems could be described by a combination of coordinate transformations and spaceinvariant systems. These were demonstrated digitally. Volume holograms (WALKUPand HAGLER [1975]) are also being considered for use as space-variant optical elements. These systems mentioned above are only a sample of the methods that have been developed to extend the flexibility and repertoire of applications for an optical processor, However, these prior approaches were not applied to optical correlation and pattern recognition systems. In the following sections, we show how space-variant systems can be used to correlate deformed inputs that differ significantly from the reference function.

1.3. SCOPE OF C H A F E R

In 52, a general formulation is provided for the procedure whereby a coordinate transformation preprocessing step is applied to the input function and to the reference function such that these coordinate transformed functions can be used in a conventional space-invariant correlator. In § 3, we consider the specific space-variant system required to produce a scale-invariant correlator. The resultant transform that is produced in this case is a Mellin transform. The theoretical description of this operation, several methods of implementing it, the space bandwidth and

296

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATTERN

RECOGNITION

[V, 5 2

accuracy requirements of the resultant system, and several examples of optical and digital Mellin transforms are included. In $4, we consider several demonstrations of the use of the scaleinvariant transforms described in § 3 in pattern recognition and signal processing and several variations and extensions of them. Rotationally invariant space-variant pattern recognition systems are described in 0 5 . Several methods of implementation, the space bandwidth requirements, several applications, and a combined shift, scale and rotation invariant space-variant processor are presented. In 0 6 , we conclude with a discussion of multiple invariant systems using a method of extracting the phase from a complex wavefront.

§

2. Space-Variant Optical Processing (PSALTIS and CASASENT [1977])

In this section, we present a general formulation of the particular procedure to be used throughout this chapter for realization of an optical space-variant processor. As noted earlier, this procedure involves application of a coordinate transformation to the input function and the reference function. These new coordinate transformed functions are then used in a conventional space-invariant correlator. The resultant system is space-variant and yet uses conventional optical processors as those in Figs. 1 and 2, thereby retaining the advantages of the well-understood Fourier transform based systems that are the hallmark of coherent optics.

2.1. NOTATION

One-dimensional functions will be used to simplify the notation. Let us consider the linear shift-invariant system described by eq. (7). If the coordinate transformations x = h ( 5 ) and ,i? = h ( g ) are applied to eq. (7), we obtain

from which

v, 8 21

SPACE-VARIANT OPTICAL PROCESSING

291

TABLE 1 Space-variant system terms and parameters input spatial coordinate undistorted reference function distorted input function distortion applied to x distortion parameter output coordinate output of space variant system transformed input coordinate transformed output coordinate undistorted input to the space invariant system distorted input to the space invariant system impulse response of the space invariant system impulse response of the space variant system filter function of the space variant system

f,(O

f:(8 O,(&- 6)

o(a,x)

O,(P, x )

where

m

= Pl[h-'l,

f(x) = fl[h-'(x>l, O(i, X) = O,[h-'(i)- h-'(x)] [dh-'(x)ldx]. From this, we see that the coordinate transformation x = h(5) has converted the space-invariant system described by eq. (7) into a spacevariant system described by eq. (9), in which the space-variant impulse response O(i, x) depends both on the impulse response of the spaceinvariant system and the coordinate transformation. The terms and parameters to be used in describing these space-variant systems are summarized in Table 1 for easy reference. 2.2. SPACE-VARIANT CORRELATION

In Fig. 3a, we show the block diagram of a general space-variant correlator. The reference, or undeformed, function is f(x). We assume that the input function f(x) is degraded by the generalized deformation function g(x, a ) and becomes

f[g(x, a>]=fb') =?(XI.

(10)

x'= g(x, a ) describes the deformation as a function of both the input coordinate and a distortion parameter a. If we apply the coordinate transformation 5 = h-'(x) to both the reference f ( x ) function and the deformed input f(x) function, we obtain the coordinate transformed

298

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATI'ERN RECOGNITION

[v,5 2

COORDINATE

Fig. 3. General space variant optical correlators: (a) block diagram (PSALTIS and CASASENT [1977]), (b) schematic diagram.

functions f [ h ( 5 ) ]=f,(() and f ' [ h ( 5 ) ] = f ; ( 5 ) .If these new functions are to be correlated using a space-invariant system, they must be shifted versions of one another, i.e.

where is a constant dependent only on a. If 5 = h - ' ( x ) is a one-to-one correspondence function, then fl(5)and f;(c

max [h-'(x)]-min [h-'(x)] min [dh-'(x)/dx] Ax '

(20)

where c is a constant. We will determine the required space bandwidth N' (compared to N ) for each of the space variant systems to be considered (see 5 3.4, 5 4.4, and 0 5.3, (CASASENT and PsAurrs [ 1 9 7 7 ~d]). ,

302

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PA7TERN RECOGNITION

[v, 8 3

0 3. Scale-Invariant Space-Variant Optical Processor In this section, we consider (as the first specific space-variant optical processor and as the first specific distortion to be expected between the input function and the reference function) a scale-invariant optical processor and the correlation of two functions that differ in scale.

3.1. GENERAL ANALYSIS

For this case

where g(x,

a ) = ax

(22)

describes the distortion as a scale change in the input function. From eq. ( 16), we find the required coordinate transformation to be h - ' ( x ) = -a(d&/da) In x,

(23)

from which, we obtain

6 = h - l ( x ) =In x,

to= -In a.

(244

( 2 4 ~

As noted in 5 2, we form the Fourier transform of f(exp 5) where x = exp 5. This function is

I, T

Mtjw) =

Substituting

f(exp 5) exp t - j w t ) dt.

(25a)

6 =In x, we obtain M ( j w )= b'f(x) xJW-'dx.

(2Sb)

From this formulation, we recognize that the Fourier transform of f(exp 6 ) is the Mellin transform (SNEDDON [1972]) M ( j w ) of f ( x ) along the imaginary axis of the complex plane. All of the deformation correcting coordinate transformations do not yield specific transforms such as the Mellin transform, known by other names and studied previously.

v, 5 31

SCALE-INVARIANT SPACE-VARIANT OPTIC4L PROCESSOR

303

3.2. MELLIN TRANSFORM

Since the Mellin transform is better understood than the other spacevariant transforms we will consider, we include a brief discussion of it as it relates to this present application. This should provide insight into the idea of coordinate transformation processing and space-variant optical systems. The inverse Mellin transform is defined by

Since the Mellin transform of f(x) is the Fourier transform of f(exp c), w e can discuss the existence of the Mellin transform by stating that f(x) is a Mellin transformable function if, and only if, f(exp ,$) is a Fourier transformable function. Various properties of Mellin transforms follow in which we denote the Mellin transform of f(x) by M(jw). Property (a):

Lw

~oCDf(ax)xi"-ldx= f(x) (,x>j--l - a-jw

Lmf(x)xj-l

dx a

-

dx = a-'-M(jw),

(27)

where a is a constant. Equation (27) demonstrates that the Mellin transform converts a scale in the input into a phase factor. This is used in the Mellin transform scale-invariant correlator to be described in 9 4. Property (b):

where a is a constant. Equation (28) shows that an exponentiation of the input coordinate is equivalent to a scaling in Mellin space. This property is used in Q 4.4 in another example of the use of Mellin transforms to realize a different distortion invariant correlator.

303

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL P A T E R N RECOGNITION

[v,8 3

Property (c):

1,';~.

xu+'"-' dx = M(jo + a ) ,

f(x) xlWp1dx = [f(x)

(29)

where a is a constant. Equation (29) indicates that an amplitude weighting x a is equivalent to a shift in Mellin frequency space. This property is also used in a new distortion invariant correlator described in 5 4.4. The general integral equation for this scale-invariant system is found by substituting .$=In x and [ = l n P into eqs. (8) and (9) to yield

where O,(x) = Ol(ln x ) is the filter function. In a space-variant system, the filter function is not equivalent to the impulse response of the system O(f, x) (as is the case in space-invariant systems). In the case of scale invariance O ( i ,x) = (l/x)O,(f/x). Substituting f(ax) for f(x) in eq. (30) yields:

4,

oz

ZC

f(x) O , [ ( a ~ ) / x l( l l x ) dx =f(aP), (31)

f(ax> 0 2 ( i / x )(Ux) dx = 0

from which the scale invariance of this system follows since a scaling by a of the input coordinate x results in a scaling by a of the output coordinate i . This is analogous to the fact that a shift in the input coordinate of a space-invariant system results in a shift in the output coordinate. 3.3. SCALE-INVARIANT CORRELATION

To obtain the convolution theorem for a scale-invariant system, we form the Mellin transform of the integral equation describing such a system:

M [ [,exp' f(x) O,(i/x) dx]

= 6 x J U p 'f(x) dx

[:PJw-'

0,(2) d i

= M(jo) Mdjo),

(32)

where M(jw) and M,(jo) are the Mellin transforms of f(x) and 0 2 ( x ) , respectively. The inverse Mellin transform of eq. (32) yields

M-'[M(jw) M,(jo)] =

I:

x-' f(x) O,(f/x) dx.

(33)

SCALE-INVARIANT SPACE-VARIANT OPTICAL PROCESSOR

305

Equations (32) and (33) are analogous to the convolution theorem for the Fourier transform. For that reason, we refer to eq. (31) as the definition of the Mellin-type convolution o r scale-invariant convolution. We define the Mellin-type correlation or scale-invariant correlation of f(x> and OJx) by R ( x )= M-"[M(jw) M f ( j w ) ]=

f(x) O,(xi) ( l l x ) dx,

(34)

which describes a scale-invariant system. Substitution of x = exp 5 and R = exp ( yields

(35) which is the conventional correlation of f,(5) and O,(c). Substituting f(ax) for f(x) in eq. (35), the correlation output R (exp 8) becomes R[exp ({+ln a)] from which we see that a scale change by a in the input x coordinate simply shifts the output { coordinate by In a. The intensity of the output correlation pattern is thus not affected by a scale change and hence the peak intensity and signal-to-noise ratio of the output correlation are invariant to a scale difference between the input function and the reference function. In addition, the displacement of the output correlation peak (from its autocorrelation position) in the coordinate is proportional to the logarithm of the scale difference between the input function and the reference function. These properties make the Mellin correlator of immense practical use in pattern recognition and signal processing. Examples of such correlations will be presented in 3 4. The inverse Mellin transform is equivalent to a Fourier transform Consequently infollowed by the coordinate transformation x = exp stead of the inverse Mellin transform in eq. (34), we can form the Fourier transform of the product M ( j w ) @ ( j w ) and obtain R(exp& or the conventional correlation of f(exp 5) and 02(exp f ) , i.e.

(8.

I

m

FT[M(jw) @(Jw>l=

-a

f ( w

6) 02[exp (5+ 8 1 d 5 = R(exp 8. (36)

Scale-invariant correlation can be realized by either eq. (34) or eq. (36). When eq. (36) is used, the location of the peak of the auto-correlation function occurs at i = l n a rather than at 2 = a.

306

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PAlTERN RECOGNITION

[v,8 3

3.4. SPACE BANDWIDTH A N D ACCURACY REQUIREMENTS (CASASENT and PSALTIS [1977c])

In § 2.5, we noted that the space bandwidth product N' of the system required to represent and process f1(5)and f;(.$> is larger than the space bandwidth product N for the functions f(x> and f'(x). For the scaleinvariant processor using Mellin transforms, we consider an input function extending from 0 to X and ignore the data (of space bandwidth A4) i n the 0 5 x 1 6 region of the input. For this case, x max=X = N Ax, 5 max = f = max (In x) = In X = In ( N Ax), and 5 min = 5- = In 6 = In (A4 Ax). From eq. (19), we find

and from eq. (20),

c - 5 In(NAx)-ln(MAx) A5 1/N

">A=

=N

In (NIM).

Since (MIN) is the portion of the input omitted, (NIM) describes the accuracy of the operation. From eq. (38), we see that the space bandwidth N' of the coordinate transformed function increases with the space bandwidth N of the input function and that N' is a function of the accuracy ( N / M )desired. Since (NIM) also determines the allowed range of scale search that a given Mellin transform system can accommodate, N' TABLE2 Space-bandwidth requirements for the optical Mellin transform for various accuracies, scale search ranges, and input space-bandwidths (CASASENTand PSALTIS [ 1977c1)

100 MIN

N

Input spacebandwidth

500 500 500 500 1000 1000 1000 1000

o /'

Accuracy

1n/n 1Yo 2 Yo 2 Yn 1% 1% 2O h 2%

100 a

N'

Max. % scale change

Mellin spacebandwidth

100% 200% 100% 200% 100% 200% 100% 200%

2650 2850 2300 2500 5300 5700 4600 5000

V, 8 31


307

is thus also determined by the range of scale search desired, which in turn is directly related to the accuracy of the Mellin transform. In Table 2, we list the N' required for various input N for various accuracies MIN and percent ranges of scale change. 3.5. iMPLEMENTATION (USING NON-LINEAR SCANNING) (CASASENT and PSALTIS [1977d])

In the initial experimental demonstrations of the optical Mellin transform, an electronic raster scanned signal representative of the input image was formed by focusing a TV o n the input pattern. In the real-time version of this transform system, the video output from the TV system could be fed to the deflection system of the input real-time spatial and PSALTIS [1976a]). The schematic of Fig. 4a modulator (CASASENT log x

INPUT-

LASER-

.-

LIGHT-

INPUT READ LASER L I G H T

Fig. 4. Real-time realization of the Mellin transform: (a) using an electron beam addressed input spatial light modulator and (b) using an acousto-optic laser beam addressed input spatial light modulator (CASASENT and PSALTIS [1976a]).

308

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PATTERN

RECOGNITION

[V, 9 3

shows how this would be realized using an electron beam addressed input spatial light modulator. The corresponding schematic using an acoustooptic laser beam scanner and an optically addressed input spatial light modulator is shown in Fig. 4b. The scanner for the real-time device and the real-time spatial modulator itself must exhibit the resolution l/A[ and the space bandwidth N' of the coordinate transformed function. In practice, the systems of Fig. 4 produce a nonuniform amplitude modulation. The coordinate transformation 5 = h - l ( x ) should only relocate data values and not alter the value of the function at any point. However, because all spatial light modulator recording materials detect energy, the recorded intensity is proportional to the electron beam current and inversely proportional to the speed of the electron beam. With a nonlinear scan, the speed of the electron beam is not constant over the area of the input spatial light modulator. This results in a nonuniform amplitude modulation exp (&) that must be corrected by multiplying the amplitude of the video signal by l/t, where t, is the time corresponding to the horizontal deflection position [', of the electron beam. This difficulty can be avoided by non-linearly scanning the TV camera itself. The scanner for the real-time input spatial light modulator can be linear. However, because the video signal for the TV is now logarithmically deformed, its time bandwidth is larger (by a factor of 5 or so) and the amplifiers and deflection circuitry for the TV must now support this larger bandwidth and time-bandwidth product. For experimental demonstration purposes, the output deflection signals from the TV were logarithmically scaled. The availability of high bandwidth and high slew rate log amps and a high bandwidth camera would make the other scheme more attractive. However, these items were not available to us. In Q 5 , a modified camera scan system to realize a different coordinate transformation is described. In Figs. 5 a 4 , we show for two simple input functions f(x) and f;(x) = f(ax) (circles differing in scale by loo%), their log coordinate transformed versions fi([) and f i ( 5 ) (using log amplifiers in the TV's output deflection system as in Fig. 4), and the spatially produced Mellin transforms. Shading effects are apparent in these images due to the lack of amplitude compexation. In Figs. 5e-f the input and log coordinate scaled versions of the pattern produced by varying the sweep of the TV camera are shown. The desired log coordinate scaling of the scan was achieved simply by adding a resistor in parallel with the feedback capacitor in the

V, 8 31


309

Fig. 5. Optically produced Mellin transforms using log amplifiers in the output deflection signals from a TV. (a)-(b) inputs, (c)-(d) Mellin transforms. Optically produced Mellin transforms using modified T V scan circuit, (e) input, (f) log-coordinate transform.

3 10

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL PA’ITERN RECOGNITION

Fig. 5 (continued)

[v,5 3

V, § 31


31 1

integrator circuit of the deflection system. A TV without automatic gain control was used to achieve the linear displays of Fig. 5f of the modified video output signal.

3.6. IMPLEMENTATION (USING COMPUTER-GENERATED HOLOGRAMS)

The implementation schemes noted in Q 3.5 require sequentially scanned input spatial light modulators addressed by a scanning electron or laser beam. In many instances, it may be realistic to image the pattern onto the input noncoherent to coherent optical converter input spatial light modulator from a high quality TV monitor o r from a photographic transparency of the input data. In such instances a method of realizing the log coordinate conversion in parallel on 2-D data is desirable. This can be achieved by the use of computer generated holograms (BRYNGDAHL [1974], CASASENT and SZCZUTKOWSKI [1976a, 1976b1). One system in which this can be achieved is shown schematically in Fig. 6a. The input f(x, y) is placed in contact with the computer-generated hologram Ho in plane Po. Ho = exp [j4(x, y)] is a pure phase function. Lens L, forms the Fourier transform of fHoat plane PI, where the pattern is: m

Equation (39) can be solved using the approximate saddle point integration method (BORNand WOLF[1965]). This states that the major contribution to the value of fl(&q ) at (&, qa) comes from the point (xa, y,) for which the derivative of the total phase is zero, i.e:

The magnitude squared of the contribution of each saddle point is given by If1(5a,

where

qa)12= 14rTr2f2(xa, ~ J I ( 4 x x 4 y y- 42y)I,

(41)

3 12

DEFORMATION INVARIANT, SPACE-VARIANT OPTICAL P A T E R N RECOGNITION

[v,5 3

from which we see that the output light intensity at (ta, qa)is proportional to the light intensity at (xa,ya). Furthermore, the input location and the output location are related by eq. (40), and this relationship can be controlled by appropriately selecting the function 4(x, y). For the particular case, 5 =In x and q =In y, eq. (40) becomes,

The required phase function for H, is thus +(x, y) = x In x - x + y In y

-

y,

(43)

and eq. (41) becomes,

lfl(5, q)l2= 14.rr2xyf2(x,y)l,

(44)

which shows that an undesirable amplitude modulation occurs due to the coordinate transformation, and it has to be compensated for. Interestingly enough, this amplitude modulation is identical to the one that occurs in scanning. The computer-generated holograms used to create the function in eq. (43) were constructed using a method described earlier (LEE[1974]). A Calcomp computer plotter was used to draw a fringe at all points (x, y) that satisfy -$qsx/T+[+(x, y)/2m]+n s $ q ,

(45)

where 1/T is the average fringe spacing, n is an integer and q is a positive number from 0 to 1. When n is incremented, different fringes are drawn. A variation of q will vary the width of the fringes. When a phase only and PARIS hologram is desired, q is constant. It can be shown (LOHMANN [1967]) that when a transparent mask constructed in this manner is illuminated with a plane wave, the diffracted wavefronts can be approximated by exp [ M x , Y ) ] . A 62.5 cm X 62.5 cm pattern with a phase transfer function satisfying eq. (43) was prepared on the Calcomp plotter. This pattern was then photoreduced by a factor of 200 to 3 . 3 m m x 3 . 3 m m . This resultant hologram had center spatial frequency of 50 lines/mm and a 2 : l frequency range. This photoreduced transparency was placed in the optical arrangement of Fig. 6a. Figure 6 b shows the first orders in the output plane pattern at P, of Fig. 6a for a donut shaped input placed at Po. A magnified image of one of the first order terms containing the desired coordinate transformed version of the input object is shown in Fig. 6c.

v, 5 31


313

Fig. 6. Realization of the optical Mellin transforms using computer generated holograms: (a) system schematic, (b) first and sccond order output plane PI pattern, (c) enlargement of the first order image in (b), (CASASENT and SZ~JCZLTKOWSKI [1976a, b]).

Unfortunately this method carries its own problems. The implicit assumption in the saddle point integration method is that the function being integrated varies much slower than the total function in eq. (39). This imposes a severe restriction on the allowable bandwidth of the input function. In addition, the resolution of the input has to be at least an order of magnitude less than the resolution of the computer-generated hologram, a restriction imposed by the discrete nature of the hologram. Another disadvantage of this method is that the output in general carries a phase function. Consequently a Fourier transform lens placed behind plane PI of Fig. 6a cannot focus the Fourier transform of this coordinate transformed function (e.g. produce the Mellin transform of the input €unction). Thus, this coordinate transformed function must be power detected at plane PI prior to further processing. For these reasons, we choose to concentrate on the scanning methods implementing the Mellin and other space variant transforms.

3.7. IMPLEMENTATION (BY DIGITAL COMPUTER) [19751)

(HUANG,

RUSSELLand CHEN

The Mellin transform can also be implemented using a digital computer. In this case, the integration defined in eq. (25b) is performed digitally

3 14


[v,0 4

by replacing the integration by a summation. The one-dimensional Mellin transform then becomes N

M ( k Au)=

f(i Ax)(i Ax)JkA u - l Ax, i=l

where the resolution elements in the input and transform space are Ax and Au, respectively. Matrix multiplication can be used to realize a two-dimensional transform. If the input function f is an N X N matrix of the sampled input, we can form the Mellin transform as TjT, where T' is the transposed matrix and the transformation matrix T is exp (j Au In Ax)

4 exp ( j Au In 2hx)

...

.exp (jM Au In Ax)

4 exp ( J MAu In 2 Ax)

...

I

- exp (j

N

Au In NAx)

T=-1 Ax I ~

N

exp (jM Au In N A x )

The processing time required for such a digital realization is quite long since no fast computational algorithm exists. Exponential sampling of the input followed by a 2-D FFT is also possible but the required sampling space bandwidth and speed of the resultant system are not attractive. For completeness we include this method of implementation and show in Fig. 7 several examples of a 2 5 6 x 2 5 6 sample size digital Mellin transform. These calculations were repeated for 200% larger inputs; the resultant outputs were of course identical as expected. As noted earlier, we still concentrate on optically produced Mellin and other space variant transforms because of the parallel processing and real-time features of an optical processor.

0 4. Applications of Scale Invariant Systems Our primary applications of the scale invariant Mellin transform and of the other space variant systems to be described will be in pattern recognition. The effect of scale differences a between the input and reference functions is shown in Fig. 8, in which a 2% scale change is seen to cause a dramatic drop in SNR from 30 to 2dB.

v, 8 41

APPLICATIONS OF SCALE INVARIANT SYSTEMS

3 15

Fig. 7. Mellin transforms produced by digital computer: (a) square input, (b) triangular input (HUANG, RUSSELL and CHEN[1975]).

3 16

DEFORMATION INVARIANT. SPACE-VARIANT OPTICAL PATTERN RECOGNITION

0

I 2 S C A L E CH A N G E

I,

[V, 5 4

I,

Fig. 8. Effect of scale change between input and reference on the SNR of the output correlation (CASASENT and PSALTIS [1976b]).

4.1. PRIOR APPROACHES

Prior approaches to reducing the effect of a scale change have included the use of the converging beam correlator (VANDER LUGT[1966]) in which the input plane Po is placed a distance D behind the transform lens L,. Varying D varies the scale of the Fourier transform. Other approaches have involved the use of multiple matched spatial filters, one for [1967]). The first method each of the different input scales (BURCKHARDT is unattractive because the mechanical colineal motion required is not compatible with the real-time feature of optical processing. The second method suffers from practical synthesis problems and a loss in diffraction efficiency.

4.2. MELLIN TRANSFORM CORRELATOR

As noted in § 3.3, the scale invariant correlation of two functions can be realized using the system shown schematically in Fig. 9. The system is identical to that of Fig. 1 except that f;(exp 6, exp q ) rather than f'(x,?, y,,) is present at the input during correlation. During synthesis of the matched spatial filter, fi(exp 6, exp q ) is placed at Po and its Mellin transform hologram M;" is formed at PI by interfering M I (formed by L2 at P2) with a plane wave reference beam. With M;" stored at P,, the light amplitude

v, 5 41


Ll

PI

317

p2

L2

Fig. 9. Schematic diagram of a Mellin type scale invariant correlator (CASASENT and PSALTIS [1977a]).

distribution incident on P, is M i , the light amplitude distribution transmitted by P, is M I m and at P, we obtain the desired correlation f;(exp [, exp q ) @ fl(exp 6, exp q). From § 3.3, we see fl

0 f2

=

m[M;MTI

= Fr[M,MTa-'"] =

FT[M,M exp (jw 111a ) ]

= f l 0 f*a({+lna) l s(+j-Ina),

(46)

of,

from which we see that the cross-correlation f ; is the same as the autocorrelation f l 0 fl of fl and that this Mellin transform correlator is scale invariant. In addition, the location of the output correlation peak is proportional to the scale difference a between the input and reference functions.

4.3. SCALE INVARIANT PATTERN RECOGNITION (CASASENT and PSALTIS [1976b, 1977aI)

To demonstrate the scale invariance of this space variant correlator, the two-input functions of Figs. 10a and 10b (with a 100% scale difference between them) were prepared. The horizontal and vertical coordinates of the two functions were logarithmically scaled as described in Q 3.5. We denote the small square as the original undistorted input function f(x) (1-D notation is used for simplicity only) and the large square as f ' ( x ) = f(x'), where x r = g(x, a ) = ax with a = 2. The log coordinate transformed functions are fl(5) and f;([>, respectively. During filter synthesis, fl([) is placed at Po of Fig. 9 and the interference of M l ( w ) and a planewave reference beam UR=exp(-j2mrx,) (at an off-axis angle 8 where a = (sin B)/h) is recorded at plane PI. The term of interest in this pattern at P, and in the subsequent transmittance t , ( x , , yl) of PI is M ( w ) exp (-j27axl>.

3 18


[v,$ 4

Autoccrrelation

-//)kr;l Fig. 1 0 . Demonstration of an optical scale invariant Mellin transformation: (a)-(b) input and reference functions, (c) output cross-correlation, (d) cross-sectional scan of the output auto-correlation of the small square input function, (e) cross-sectional scan of the output cross-correlation pattern of (c) showing the scale invariance of the Mellin transform and the displacement of the location of the correlation peak proportional to the scale difference a between the input and reference functions (CASASENT and PSALTIS[1976b, 1977a1).

With this pattern recorded at P I and fI(5) in place at Po, the light amplitude transmitted through P, (for the term of interest) is MI* exp (-j27raxI). Lens L2 in Fig. 9 forms the Fourier transform of this product of two Mellin transforms. At Pz we obtain

Wi? .;r>=fl * S(&X,) 3

of1

= ( f l @ f l ) * S(i-x,-Inu).

(47)

This output plane pattern is shown in Fig. 1Oc. With f,([) at Po and the same filter described by *(o) at P,, the output plane P, pattern is the auto-correlation of f l. The cross-sectional scan through the correlation pattern of Fig. 1Oc is shown in Fig. 10e.


319

From this, we see that the amplitude of the cross-correlation is the same as that of the auto-correlation as predicted by eq. (46) and that the location of the cross-correlation peak is displaced from the x2 = x, reference position for the auto-correlation peak in Fig. 10d by an amount proportional to a as predicted from eq. (46). The theoretical shift should be 1.035 an which agrees within experimental error with the measured 1cm shift. 4.4. EXPONENTIATED COORDINATE DISTORTIONS (CASASENTand PSALTIS [ 1977b])

A distortion that can also be compensated for by coordinate transformations similar to those used in the scale invariant system of 9 3 and 9 4.3 is an exponentiated coordinate distortion. For this case, the undistorted input reference function is again f(x) and the deformed function is

fYx) = f(x) = W), where the deformation is x’= g(x, a ) = x a .

(48)

This type of deformation can arise due to non-linear scanning, aberrations in an imaging system (WELFORD[1974]), Doppler shifts due to nonuniform target motion (KELLYand WISHNER [1965]), o r imaging from curved surfaces (BRYNGDAHL [1974]). To determine the coordinate transformation 5 = h-l(x) required for this system, we substitute d g ( x , a)/& = a x a - ’ and d g ( x , a ) / d a = x a In x in eq..(16) to obtain

h-’(x) = Setting -d[o(a)lda = l / a and transformation to be

to= -In

(49)

a, we find the required coordinate

5 = h - l ( x ) =In (In x).

(50)

The transformation described in eq. (50) converts the distortion described as an exponentiation of the input coordinates into a shift in the transformed [ space, since x o = [h(t)3”= [exp (exp t)la= h(S- t o ) .

Thus, a system in which the coordinate transformation described by eq. (50) is applied to the input function followed by a conventional spaceinvariant system will be invariant to the deformation function described

320


[V, 9: 4

by eq. (48). This system can also be viewed as a combination of a logarithmic coordinate transformation followed by the scale-invariant system described in 5 4.3. A discussion of these two alternate implementations of this transformation sheds considerable light on the issue of the accuracy of a space-variant correlator. We refer to these two implementations as: (a) an input coordinate transformation eq. (50) followed by a space-invariant correlator, and (b) a scale-invariant correlator operating on the magnitudes of the Mellin transforms of the input functions. Implementation by method (a) involves two successive logarithmic coordinate transformations, whereas implementation by method (b) involves a single logarithmic transformation followed by the formation of the Fourier transform of the coordinate transformed functions. In this latter implementation the magnitude of the Mellin transforms of the two functions are formed. These are then used as inputs to the conventional scale invariant correlator of 5 4.3. The integral describing either of these systems is:

where f(x) is the input function, f(2) is the output function, and O,(ln i/ln x)/(x In x) is the system’s impulse response. From this expression, we see that the domain of the input function is restricted to the region 1 < x < M, since the input function is usually defined over the entire positive axis. This implies that the portion of the function contained in the region 0 < x 5 1 will be neglected in this implementation. In terms of the two implementation methods noted above, method (b) will require formation of the magnitude of the Mellin transform of the input functions. This will result in loss of the phase information present in the Mellin transform of the function. In method (a), the 0 < x 5 1 region of the full input function extending from O < x < a will map into the negative region of the logarithmically scaled function. Thus data present in this 0 < x 5 1 region of the input function will be lost in method (a), since one cannot form the logarithm of a negative number. The specific application used will determine which of these implementation methods is best . It is easy to show by extension of the remarks in 3 3.4, that the space bandwidth product N” required for this system is

N” = N In ( N / M )In [(NIM)In (NIM)],

(52)

where N is the space bandwidth product of the input function, and M is

321


the number of resolution elements in the 0 < x 5 1 region that are deleted when the logarithmic coordinate transformation is performed. As stated earlier, our primary application is to perform correlations invariant to the distortion described by eq. (48). For this case, the filter O,(lnx) in eq. (51) is the complex conjugate transform of the input function. Experimental confirmation of a space-invariant correlator invariant to the deformation described by eq. (48) is shown in Fig. 11. The reference function and the distorted input function used are shown in Figs. l l a and

I

b

a

h

e

Fig. 11. Demonstration on an optical space-variant correlator invariant to an input coordinate exponentiation deformation: (a)-(b) reference and input functions, (c) output crosscorrelation, (d)-(e) cross-sectional scans of the auto- and cross-correlations of the functions in (a) and (b) showing the deformation invariance of the system and the shift in the location and of the correlation peak proportional to the deformation parameter a (CASASENT PSALTIS [1977b]).

322

DEFORMATION INVARIANT, SPACE-VARIANT OFTICAL

PATTERN RECOGNITION

[V, 3 4

l l b . A one-dimensional grating was chosen for f(x) for simplicity. The distorted input function is described by f(x) = f(x“) and shown in Fig. 1lb. For demonstration purposes, we used f’(x) = f(x’.’). The horizontal components of both functions were logarithmically scaled (in one dimension only). This coordinate exponentiation converts the exponentiated coordinate distortion into a scale change by a in the new function. A second logarithmic coordinate transformation was then performed. This converts the scaling (produced by the first transformation) into a shift. This process on the deformed coordinate x’ = xa can be described by the coordinate transformation

x = h(6)= exp (exp 8)

(53)

xa = I[h(5>1”= Cexp (exp 5)Y,

(54)

applied to xa. This yields

which can be manipulated as follows xa =exp ( a expc)=exp[exp (c+ln a ) ] ,

(55)

where the first version demonstrates the conversion of an exponentiation in the input coordinates into a scale change and the second version demonstrates the conversion of a scale change in the coordinates into a shift. We can then write eq. ( 5 5 ) as

x u = h(5-ln a ) = h(6-t0),

(56)

where the shift toin the 5 space representation of the distorted function is In a or, as required by eq. ( l l ) ,

f ( 0 = f(S-

50).

(57)

The two log coordinate transformed versions fl(6) and fl(&) of the distorted input function and the undistorted reference function were formed as indicated above. With fl(6) placed at plane Po of Fig. 9, a holographic spatial filter FT of it was formed at plane P1. With fi(5)at the input, the autocorrelation fi 0 fi appears at plane Pz. The cross-sectional scan of this autocorrelation pattern is shown in Fig. l l d . With fi(5) in place at plane Po, the cross correlation f; 0 fi appears at plane P2. This output optical correlation plane pattern is shown in Fig. l l c and the cross-sectional scan across this pattern is shown in Fig. lle. Comparing Figs. l l d and l l e we see that there is no SNR loss between the two correlations and that the location of the cross correlation peak is shifted

v, I 41


323

from the location of the autocorrelation peak by an amount proportional to the deformation factor a as predicted by theory.

4.5. DOPPLER SIGNAL PROCESSING (CASASENT and PSALTIS [I 976d], CASASENT and KRAUS[1976])

In this section we consider the application of scale-invariant, spacevariant correlations to the case of certain one-dimensional signals. The most obvious application of this lies in Doppler signal processing (CASASENT and PSALTIS [1976d], CASASENT and K R A U[1976a]). ~ In this application, a similar procedure to that used in two-dimensional pattern recognition applications can be employed. Before embarking on a description of this application, several remarks on Doppler signals are needed. The term “Doppler shifted” is misleading, since the Doppler effect actually scales the frequency of the transmitted signal rather than shifting the signal. However, when the carrier frequency of the transmitted signal is high and the velocity of the source is much lower than the propagation velocity of the wave in the medium, the Doppler effect can be approximated as a shift in the frequency. Hence we will use the term Doppler shift to describe the Doppler effect, when we consider a wave traveling through a medium at a velocity uo. The source is assumed to move in the direction of propagation with a velocity us. The frequency detected by a stationary observer is then given by (LEVI [19681) I

=-

w

1-vslvo’ where w is the transmitted frequency and w ‘ is the received o r observed frequency. When the propagation of electromagnetic waves in space is considered, relativistic considerations modify eq. (58) to yield

where o: is the relativistic Doppler frequency and uo = c is the velocity of the electromagnetic or light wave in free space. For v, ~ - n y o ) + f n 2 ( x - x ( ) + x n , Y - ~ Y , ) ,

(61)

n=l

where n is the channel number, the vertical separation between channels is yo, and the center to center horizontal spacing between signals on a given channel is 2x,. The one-dimensional (horizontal) Fourier transform of this input pattern was recorded on film at plane P, of Fig. 2. This pattern was then illuminated with a plane wave and a second one-dimensional transform of this recording produced at plane P,. The output plane P2 pattern is then described by N

C

fnlo fn2

* 6 ( f - 2 x o - x n , 9 - ny,).

(62)

n=l

This pattern corresponds to the desired correlations fn 0f n 2 on n lines at i = 2 x 0 + x , and 9 = ny,,, where (a, 9 ) are the output plane coordinates. As shown in Fig. 12, all correlation peaks are of the same intensity and the relative location of the correlation peak on each line is proportional to the scale change between the corresponding pair of input signals and

Fig. 12. Output optical correlation plane pattern for a space-variant Doppler signal processor. The location of each of the nine correlation peaks corresponds to the relative Doppler difference between the two signals o n the corresponding line in the input (CASASEN~ and PSALTIS[1976d]).

326

DEFORMATION INVAKIANT. SPACE-VARIANT OVTICAL PATTERN RECOGNITION

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hence to the Doppler shift between these signals. This vividly demonstrates that a single channel scale-invariant optical processor is capable of performing Doppler signal processing over a wide range of the Doppler shift. This scheme can also obviously be realized using one-dimensional acousto-optic correlation techniques. At this point it is appropriate to stress the features of space-invariant processing as they apply to one dimensional signals. As noted earlier, a scale-invariant correlator is not tolerant of a shift in the input coordinates and therefore a precise form of two-dimensional input positioning is required. However, if the input is a one-dimensional signal, it is normally sampled at short intervals and the signal recorded at the same horizontal starting location. This alleviates the rigid input positioning requirements of a space-variant image processor and the associated phase loss problems noted earlier. 4.6. CORRELATION OF NON-VERTICAL IMAGERY (CASASENT and FURMAN [1977])

In this section, we consider the application of scale-invariant systems to the processing of non-vertical imagery. Non-vertical imagery results from an imaging system whose object planes are not parallel. Such a situation arises in numerous airborne reconnaissance and photographic applications. It is also the basic geometry used in photogrammetric applications and is typical of the type of aspect distortion to be expected in other realistic object identification applications. It will be shown below that the effect of non-vertical imaging is equivalent to a one-dimensional scaling of the image coordinates. Hence, it is possible to use a scale-invariant system to process such data. In Fig. 13, the geometry of the non-vertical imaging system to be used is defined. The Cartesian coordinate system o n the surface of the earth is defined by (xe, ye, z,) and the Cartesian coordinate system at the image plane is defined by (xi,yi, zi). For vertical imaging, the z axes coincide and x, and ye are parallel to xi and yi, i.e. (xi,

yi, zi) = (axe, aye, ze+ zo),

(63)

where a is a constant depending on the elevation of the imaging system and its focal length and zo is the height of the imaging system. The image plane system and the earth coordinate system can be related bv

v, I 41


VERTICAL IMAGING

T

327

Li

NON-VERTICAL

Fig. 13. Geometry of a non-vertical imaging system (CASASENT and FURMAN[1977]).

where 4 is the tilt angle of the imaging system with respect to the earth's vertical direction, and 8 is the orientation angle of the imaging system with respect to the object on earth. From eq. (64) it is seen that if the orientation of the imaging system were always the same with respect to the object, then 8 = 0 and eq. (64) becomes

Equation (65) shows that the effect of a tilt in the imaging system is equivalent to a scaling in the y i coordinate. In a practical situation 8 can have any value and cannot be restricted to 0, thus a complete non-vertical imaging system will require preprocessing of the data or use of a multiple invariant space-variant system (0 6). To transform this distorted image described by eq. (64), we multiply both sides of eq. (64) by the matrix TI, cos 8 -sin 8 T1= [sin 8 cos 8 This results in a new coordinate function with coordinates (xl, y l ) , where

328


[v,5 4

To correlate the vertical image described by f(x,, ye) = f and the nonvertical image f(xi, yi), the coordinate transformation T, described by eq. (66) is applied to f ( x i , y,). This produces the new coordinate transformed function f(xl, yl). A two-dimensional invariant correlator system can then be used to correlate the coordinate transformed function f ( x l , yl) and the undistorted function f(xe, ye). In the output correlation plane P,. the horizontal displacement of the correlation peak will be proportional to In a and the vertical displacement of the correlation peak will be proportional to In a +In (cos 4). Thus, the values of the unknown distorting parameters, specifically the scale change a and the tilt angle 4 of the system, can be determined from the location of the output correlation peak. The coordinate transformation described by the matrix TI is equivalent to a rotation of the input function by an angle -8 about the z axis. Thus, if 8 is known, T , can be implemented by simply orientating f(xi, yi) appropriately. If 8 is unknown, the function f(xi, yi) can be continuously rotated by 360" to search all possible angles 8.This can be accomplished by a mechanical rotator in the input plane or by a combination of a rotation invariant space-variant system (0 5) combined with this present space-variant system. As a simple demonstration of this new space-variant correlation process, the vertical input image of Fig. 14a was produced along with the non-vertical image of Fig. 14b. In this non-vertical image, the effective tilt angle 4 of the camera system was chosen to be 30" and a 20% scale variation ( a = 1.2) was used. The conventional frequency plane correlator of Fig. 1 was chosen for this application. A matched spatial filter of the vertical image f ( x e , ye) was produced at plane P, by methods previously described. The coordinate transformation T , described by eq. (66) was applied to the non-vertical image f(xi, yi) to produce the coordinate transformed function f(xi, yi). This function was placed in the input plane Po. The output correlation plane P, pattern is shown in Fig. 14c. The cross-sectional scans of the auto-correlation and cross-correlation pattern in the vertical direction are shown in Figs. 14d and 14e. These patterns show no signal-to-noise ratio loss and the displacement of the location of the correlation peak to be propo- tional to the distortion, i.e. In a + l n ( c o s O)=ln(l.2)+ln(0.82).

(68)

This once again confirms the ability of a space-variant system to process yet another type of deformation to be expected between an input and

APPLICATTONS OF SCALE INVARIANT SYSTEMS

329

Fig. 14. Space variant correlation of non-vertical imagery: (a) vertical image, (b) nonvertical image (note the difference in the angle between the runways in the two images), (c) output correlation plane pattern, (d)-(e) cross-sectional scans of the auto- and crosscorrelation peaks (CASASENT and FURMAN [1977]).

330


[v,5 4

Fig. 14 (Continued)

reference image. From these examples presented in the previous sections, it should be apparent that quite a number of deformations are expressible by coordinate transformations. In these types of applications, the use of space-variant correlators, such as the ones previously described, is of immense value.

V, 0 41


331

4.7. OTHER SPACE-VARIANT SYSTEMS

In the previous sections, various pattern recognition applications of space-variant processors have been considered. In this section, a general treatment of the use of the scale-invariant system to intentionally introduce certain desirable distortions or modifications into an input function will be provided and several examples included. The general transformation for such a system is repeated here for convenience,

Assuming f(x) = 6[ln (ax)], eq. (69) becomes

which shows that the output of the system with an input 6 function at x = l/a is simply a scaled version of the function O,(x). One application for a scale-invariant system besides pattern recognition and signal processing is the use of this system to intentionally introduce magnifications in an input function in a manner analogous to the use of a zoom lens system but without any moving parts. To automatically scale an input function, we place a phase grating with spatial frequency (In a)/27rAfL described by the function exp (jw In a ) = a’, (the Mellin transform of s[ln(ax)]) at the filter plane P, of the Mellin correlator of Fig. 9. The output of this system with an arbitrary function f(x), will be f ( a i ) as predicted by eq. (70). Another operation that may be realized in such a system involves the use of filtering in the Mellin frequency domain. High pass filtering tends to suppress those portions of the input function that lie close to the origin while enhancing the rest of the function. Low pass filtering has the opposite effect. Filtering in the Mellin frequency domain can be used to selectively enhance desired portions of the input function automatically. This operation is analogous to the physical movement of slits and arbitrarily shaped and weighted masks about the input space plane. However the scale-invariant filter implementation is more flexible and automatic and, in addition, does not require any moving parts.

332

DEFORMATION INVARIANT, SPACE-VARIANT O ~ I C A LPATTERN RECOGNITION

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A final example of the general uses of scale-invariant systems is space variant image restoration (ROBBINS and HUANC[19721). To show this, we consider a system whose response is described by eq. (69). The optical inversion of such a system or the recovery (restoration) of f(x) can be achieved by forming the Mellin transform, I&@), of the degraded image f ( 2 ) ,dividing this by the Mellin transform M&o) of the system’s spaceinvariant response Oz(x) and inverse Mellin transforming this quotient. The Mellin transform of eq. (69) is f i ( j w ) = Wjo) Mo(jw),

(71)

where M(jo) is the Mellin transform of f(x). We can write f(x) as:

This sequence of operations can be realized optically by recording l/Mo(jo) holographically in the Mellin frequency plane and placing f(expg) in the input plane of Fig. 9. The optical wave incident o n P, is k(jo),the optical wave transmitted through P, is M(jo)/M,,(jw) and the output plane will thus contain f(exp 5). An output coordinate transformation x = exp 5 will produce the desired undistorted function f(x). Systems that can be described by eq. (69) are spherical imaging systems with coma aberrations and cylindrical lens systems with a tilt in the system’s axis (ROBBINS [19701). Thus, the distortions produced by such systems can be optically removed and the original image restored using the scale-invariant system described above.

0 5. Rotational Invariant Space-Variant Systems 5.1. INTRODUCTION

In this section, we consider the use of coordinate transformations and optical transform techniques to realize a rotationally invariant optical processor. This type of distortion (a rotational misalignment of the input function and the reference function) occurs quite frequently and is another major practical source of error that a conventional Fourier transform based optical pattern recognition system cannot tolerate. To demonstrate the dependence of the signal-to-noise ratio of a conventional

v, s 51

ROTATIONAL INVARIANT SPACE-VARIANT SYSTEMS

0

10

20

30

333

40

DEGREES OF ROTATION

Fig. 15. Effect of rotational misalignment between the input and reference functions on the signal-to-noise ratio for an aerial image (CAsAsEwr and PSALTIS[1976c]).

space-invariant optical correlator on the angle of rotational alignment between the input and the reference function, we show in Fig. 15 a plot obtained from a high resolution aerial image. From this figure it is seen that a rotational misalignment of only 3" causes a 27 dB loss in signal-tonoise ratio. Various solutions to the rotational invariant problem that have been considered up to now include the use of multiple reference functions (LAMACCHIA and VINCELEITE [1968]) and the use of a mechanical rotator in the input plane to perform a complete orientational search (VANDER LUGT[1966]). Both of these appproaches are not attractive for reasons similar to those advanced in 5 4.1. The multiple filter system suffers from synthesis difficulties and a diffraction efficiency loss, whereas the mechanical system is not conducive with the real-time operation possible in an optical processor. We thus considered coordinate transformation processing as a method whereby total rotational invariance as well as real time performance of an optical pattern recognition system could be realized.

5.2. POLAR TRANSFORMATION (CASASENT and PSALTIS[ 1976~1)

The coordinate transformation required to achieve rotational invariance can be shown to be a polar transformation. The rotation of a two-dimensional function about the origin is described by the coordinate

334

DEFORMATION WARIANT, SPACE-VARIANT OPTICAL PATTERN RECOGNITION

[v,5 5

transformation

where

In this application, we again consider the unrotated function to be f ( x , y ) and the rotated distorted function to be f'(x, y ) , the distorted coordinate transformations are now described by g, = g x ( x , y, 8,) and g, = g,(x, y, O0), where the distortion parameter is now 8, (the angle of rotation between the input and the reference function). From eq. (74), we see that each rotated coordinate (g,, g,) depends on both input coordinates (x, y). Thus, two coordinate transformations, x = h,(& q), and y = h,,(,<s, q ) that depend on both transformed coordinates, must be used. These must satisfy eq. (17) for the distortion described by eq. (74) in order that the rotation 8, be converted to a shift in this new transformed space. Using eq. (74), (17a) becomes

This partial differential has a solution

and is also satisfied for all g, and g, for which

where R is any positive real constant. The second general coordinate transformation r ) ) must satisfy eq. (17c). For this case this requirement can be expressed as

h(&

-g,ahblagx

+ g,ah;lag, =dqo(Qld&.

(78)

Since the rotation present in the input function has already been translated into a shift in the [ coordinate, the r ) coordinate must not be affected by the rotation as discussed in 9 2. Consequently qo must be independent of 8, and aqold8, = 0. Equation (78) is satisfied by


335

From this, we see that eq. (77) states that eq. (76) is the solution to the differential equation of eq. (75) for any positive real value of q = hb. The inverse transformation of the pair of required transformations is given by

These expressions describe the relationship between a Cartesian (x, y ) and a polar (q,5) coordinate system. Since the notation (r, 0) is widely accepted for polar coordinates, it will be adopted for the remainder of this section instead of the more general coordinate transformed space notation (q,() previously used. Substituting eq. (80) into eq. (14), the factor 5” is found to be 5”(0,) = f%+n.rr,

(81)

where n is any integer. From the analysis of previous sections, we realize that a rotationally invariant system can be realized by a polar coordinate transformation followed by a space-invariant correlator. A rotation of the input to such a system will result only in a shift in one of the output coordinates and will not affect the shape of the output pattern nor its signal-to-noise ratio. 5.3. SPACE BANDWIDTH PRODUCT REQUIREMENTS (CASASENTand PSALTIS [1977dI)

As expected, the space bandwidth requirements for a polar coordinate transformed function are expected to be larger than for the corresponding Cartesian coordinate representation. It is thus important to consider this issue before various methods of implementation of the transform are considered. For this analysis we will assume the parameters of Table 3 . TABLE3 Parameters used in the space bandwidth analysis of a polar coordinate transformed function Parameter Ax

N2 NAx A% A1 ”2

Interpretation Input resolution Input space bandwidth product max (x)-min (x) Resolution in the 0 coordinate Resolution in the r coordinate Space bandwidth product of the transformed function

336


[V, $ 5

For this transformation, we find r,,, = NAx/&, 0,,, = 4 tan-' (N/2).In addition we know that Ar = Ax and A8 =&IN rad. The size of a resolution element in the r and 8 directions can be approximated by l / A x and N/&, from which the space bandwidth products in the two directions are

N: = N / h and

NA = [4N/&] tan-' (NI2). The product of N: and N i is N". The space bandwidth product of the transformed image is thus

N"

= NiNA = 2 I P

tan-' (N/2)= r N 2 .

(83)

5.4. IMPLEMENTATION

From eq. (83), we see that the space bandwidth product of a polar coordinate transformed function is approximately three times larger than the corresponding Cartesian coordinate representation of this function. Of the various methods noted in 0 3 for performing coordinate transformations on a spatial function, the method in which the internal sweep circuitry of a closed circuit TV system is modified is the most applicable for this particular polar transformation. A system of this type consists of a modified TV camera and a scanning real-time spatial light modulator as shown in Fig. 4. The deflection signals x and y from the camera are fed to two electronic circuits which produce two new deflection signals and tan-' ( y / x ) for the real-time spatial light modulator. These signals are used to deflect the scanning electron beam addressed or acousto-optic addressed input device in synchronization with the video signal from the camera. This video signal is used to vary the intensity of the writing electron beam or laser beam. The resultant image recorded on the input light modulator is thus the desired polar coordinate transformed version of the original input image. To achieve the optimum system in which the input image is scanned non-linearly in such a manner that a linear recording of the video signal on the input spatial light modulator is possible, a polar camera is needed. To realize this, the camera's electron gun is deflected by two sinusoidal signals 90" out of phase. This causes the electron gun to trace a circle in the input plane, with the radius of the circle proportional to the amplitude of the sinusoids. The entire input plane can be traced in this manner by

V, § 51


331

varying the amplitude of the sinusoids. An alternate system would be possible if arc tangent analog modules capable of operation at TV bandwidths of 5 megahertz are available. Since such analog modules were not available, a modified polar coordinate scanning television was designed and fabricated. This system is used in all of the examples included in this section. Digital image processing systems have also frequently been used to perform polar coordinate transformation (ROBBINS [19701). The time required to digitally transform the coordinates of an input function are typically four to eight times the time required to digitally form the Fourier transform of this pattern. Neither the speed of the input digital coordinate transformation system, nor the speed of the digital Fourier transform system are comparable to the rates obtainable using optical processing techniques and analog circuitry. For these reasons, only analog implementations were considered. The implementation of a coordinate transformation using computer generated holograms was noted in § 3.6. This method is not useful in this case because the coordinates in the required polar coordinate transformation are coupled.

5.5. ROTATION INVARIANT PATTERN RECOGNITION (CASASENT and KRAUS [19781)

To demonstrate the use of the polar coordinate transformation to produce a rotation invariant space-variant processor, the following experiment was performed. The modified polar camera discussed previously was fabricated. The airplane shown in Fig. 16a was used as the input image. The television camera was focused on this image and the resultant signal from the camera linearly displayed on a monitor and the screen of the monitor photographed. A typical polar coordinate version of this input image is shown in Fig. 16b. In a real-time implementation the output of the television monitor could be focused directly onto a realtime optically addressable device or the output video signal fed to an electron beam or laser beam scanner for recording on the input device. As the input image of the airplane was rotated through a full 180" as shown in the top sequence of images in Fig. 17 the polar transform version of these signals was produced on the polar camera. A matched spatial filter of the zero degree orientation version of the input image was

338


[v,3 5

Fig. 16. Airplane (a) and its polar coordinate transformed version (b) obtained using the and KRAUS[1978]). polar camera (CASASENT

recorded at plane PI of Fig. 9 and used as the matched spatial filter for all subsequent correlations. Each new image was placed in the input plane Fig. 9. The resultant output plane patterns at plane P2 of Fig. 9 are shown in Fig. 17b for the various orientations of the input image. As noted earlier, the location of the output correlation peak shifts vertically by an amount proportional to the rotational angle between the input function and the reference function. The cross-sectional scans of these correlation peaks are shown in Fig. 17c. As expected, they indicate only a modest

v, 5 51


339

Fig. 17. Rotation invariant pattern recognition using a space variant optical correlator: (a) input images, (b) output correlation plane pattern, (c) cross-sectional scans of the crosscorrelations (CASASENT and KRAUS [1978]).

loss in signal-to-noise ratio despite a full 180" rotation of the input. The loss in signal-to-noise ratio was due to non-linearities in the output television monitor used, not in non-linearities or non-uniformities in the polar camera system itself. This system is quite useful in applications in which the input object is centered by Gimbal trackers or similar devices.

5.6. ROTATION AND SCALE INVARIANT PATTERN RECOGNITION (CASASENT and PSALTIS[1976c])

Since a rotation is always associated with a two-dimensional system and a rotation is described by only one parameter, the second coordinate r of the system described in 5 5.5 can be used to accommodate a different distortion. In this section we consider the first of several multiple invariant space-variant correlators. Indeed, a rotation of the input only effects the 8 coordinate of the transformed image, whereas a scaling in the input coordinates only effects the r coordinate in the transformed

340

DEFORMATION VARIANT,

SPACE-VARIANT OPTICAL PATTERN

RECOGNITION

[V,

55

imzge. This makes the realization of a scale-invariant and rotationinvariant system possible. In this section, we discuss the theory and experimental demonstration of this type of space-variant correlator. The rotation-invariant system described above is also shift-invariant. The Fourier transform (which is rotation- and scale-invariant but is shift-variant) can be applied to the input function to produce an overall system invariant to all three distortions (rotation, scale, and shift) in the input coordinates. The first step in the synthesis of such a triple invariant system is the formation of the magnitude of the Fourier transform IF(w,, w,)l of the input function f(x, y ) . This initial transformation eliminates the effects of any shifts in the input image and automatically centers the resultant transformed light distribution on the optical axis. To understand how the combined rotation-invariant and scale-invariant correlator can be fabricated, let us consider what happens when the input function f(x, y) is rotated. This causes its Fourier transform F(w,, a,,) to be rotated by the same angle. Accordingly, a scale change in f ( x , y ) by a will scale its Fourier transform F(w,, a,)by lla. The effects of a rotation change and scale change in the resultant light distribution can be separated by forming the polar transformation of the Fourier transform F of the input function f. This converts the input from (ax,a,)coordinates to (r, 8 ) coordinates. A scale change in F by a does not effect the 0 coordinate in this transformed pattern, but merely scales the r coordinate to r' = ar. A two-dimensional scaling of the input function is thus reduced to a scaling in only one dimension (the r coordinate) in this transformed function F(r, 8). If the r coordinate is logarithmically transformed to p = In r, the scaling will then be transformed into a shift in the p coordinate proportional to In a. This system is, of course, restricted to scalings in which the scale factor change is equal along both axes. We now consider the effects of a rotation of the input function f(x, y ) by an angle 8,, in detail. As noted above, the rotation does not effect the r coordinate of the transformed function, but it will shift the 0 coordinate of this transformed function by an amount proportional to 0" as predicted by eq. (81). The effects on the 8 coordinate are best seen with reference to Fig. 18. To see this effect, we partition the input function F(w,,w,) into two sections F,(wx, a,)and &(ax,a,),where F2 is the segment of F that subtends the angle 0" as shown in Fig. 18a. The polar coordinate transformation of this function is shown in Fig. 18b. The Fl(r, 0) section of F(r, 8) extends from 0 to 27r-00 in the 8 direction, while F2(r,0) occupies the 27r - 8" to 27r region of the 8 axis.

v, 5 51


B

A

AW

34 1

Y

b wx

C

Fig. 18. Effects of a rotation by 0, in the input (q, m y ) space on the image in (r, 0) space: (a) input function, (b) polar transformation of (a), (c) rotated input function, (d) polar and PSALTIS[1976c]). transformation of (c),(CASASENT

A version of Fig. 18a, rotated clockwise by Oo, is shown in Fig. 18c. For convenience, the sections of this rotated function are denoted by FI and F;. The polar coordinate transformation of this F' function is shown in Fig. 18d. The effect of a rotation by 0, is seen to be an upward shift in F , by O0 and a downward shift in F2 by 2 m - 0 0 . Thus, whereas this transformation has converted a rotation in the input into a shift in the transformed space, the shift is not the same for all parts of the function. To see the effects of this different shift for different parts of the input transformed function, we refer to the block diagram of Fig. 19. The steps involved in forming a position, rotation, and scale-invariant correlator are indicated here. The original input function is f(x, y); the magnitude of its

342


[V, 0 5

Fig. 19. Block diagram of a position, scale, and rotation invariant, space-variant correlator (CASASENT and PSALTIS [1976cl).

Fourier transform IF1 is formed. This Fourier transform pattern is then transformed to polar coordinates to produce F(r, O), and the r axis of this transformed function is then logarithmically scaled to produce the new function f(exp p, 8). We denote the input coordinate transformed function as F(exp P, 8 ) = Fi(exp P, 8 ) + F Z ( ~P, X0). P

(84)

With this function placed in the input plane Po of the correlator in Fig. 20, the Fourier transform of eq. (84) is formed by lens L, at plane PI of Fig. 20. Its Fourier transform is denoted by W w , , we) =M,(w,, % ) + ~ z ( w pw, , ) .

(85)

A holographic matched spatial filter denoted by M* is formed and placed in position at plane P,. We denote the scaled and rotated distorted input function by F'(exp p, 6 ) . This function is placed at the input plane Po of the system in Fig. 20 during correlation. The pattern incident on plane P, REF

Fig. 20. Schematic diagram of a scale and rotation invariant, space-variant, optical correlator.

V, § 51


343

is then the Fourier transform of F', which is given by M'(w,,

0,)

= M,(w,, we>exp {-j(w,

In a + w,Oo)}

+ M2(wp,w,) exp {-j[w,

In a - w , ( 2 ~ - 0,)J). (86)

The light distribution emerging from plane PI is then the product W M ' , which we write as

M'M* = W M , exp {-j(w, In a + w,80)) + @M2 exp {-j[w,

In a - ~ ~ ( 2 7-1O,)]}. .

(87)

The Fourier transform of eq. (87) is then formed by lens L, at plane P,. This pattern consists of two terms: (a) the cross-correlation F,(exp p, 0) 0 F(exp p, 0) located at p'=ln a and 0'= O0, and (b) the cross-correlation F,(exp p, 0) 0 F(exp p, 0) at p' =In a and 0' = - 2 7 ~+ O0, where the coordinates of the output Fourier transform plane are denoted as ( p ' , 0'). If the intensities of these two cross-correlation peaks are summed, the resultant intensity will equal that of the auto-correlation peak of the function f(exp p, 0). As shown, the SNR of the output correlation is effectively invariant to a rotation o r scale change in the input function. Furthermore, the location of the output correlation peak along the p' axis is proportional to the scale change a between the input and reference function and the location along the 8' axis is proportional to the rotational change 0" between the input and reference functions. Therefore, two functions differing in both scale and rotation can be correlated with no loss in signal-to-noise ratio, and from the location of the correlation peak and the distortion parameters (here the scale factor a and the rotation angle 0,) can be determined. The positional information concerning the input is lost in forming the magnitude of the Fourier transform in the initial step shown in Fig. 19. This causes additional loss of phase information contained in the transform of the input function. We will show later in Q 6 that it is possible to recover this lost information and produce a complete invariant system. The results of an experimental demonstration of this scale and rotationinvariant correlation are shown in Fig. 21. The two input objects used are shown in Figs. 21a and 21b. They differ in scale by 100 percent and are rotated from one another by a full 180 degrees. The polar transformed versions of these functions logarithmically scaled in r, were prepared. The Mellin transform type hologram @(up,w,) of the undistorted function was formed at plane PI of Fig. 20. With F(exp p, 0) in place at plane Po,

344


[v,5 5

Fig. 21. Rotation and scale invariant pattern recognition using a space variant optical correlator: (a)-(b) input images, (c) output optical correlation plane pattern, (d)-(e) crossand PSALTIS [1976c]). sectional scans of the auto- and cross-correlations (CASASENT

the auto-correlation of F(exp p, 8) was formed at plane Pz. A crosssectional scan of this auto-correlation pattern is shown in Fig. 21d. With F'(exp p, 8) placed at plane Po the cross-correlation of the functions occurs at plane P,. It contains two peaks rather than one as noted above. The cross-correlation scan of this cross-correlation pattern in the 8' shows two cross-correlation peaks whose sum equals that of the auto-correlation peak. These peaks are separated by 2.rr and shifted from the 8' of the auto-correlation peak by the 0, = .rr rotation present between the input and reference function. The cross-sectional scans in the p' direction (using a scanning slit) show that the p' coordinate of the correlation peak is

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MULTIPLE INVARIANT SPACE-VARIANT PROCESSORS

345

related to the scale change between the input and the reference objects and that there is no loss in signal-to-noise ratio between the autocorrelation and cross-correlation peaks. These results indicate that it is possible to correlate two objects that differ in both scale and rotation with no loss in signal-to-noise ratio, and that it is possible to determine the unknown scale and rotational differences between the input function and the reference function from the location of the output correlation peak.

§ 6. Multiple Invariant Space-Variant Processors

(CASASENT and PSALTIS [1978])

6.1. INTRODUCTION

In previous sections, we have presented various versions of spacevariant, distortion-invariant optical processors using coordinate transformations. In several of these instances loss of information contained in the input data resulted when multiple invariances or special transformations were needed. In this final section we consider an approach whereby multiple invariance can be obtained with no loss of data information contained in the input function. The need for such systems arises because, using coordinate transformations, the number of distortion parameters is restricted to the number of dimensions of the system (two for the case of a two-dimensional processor). One very important situation in which this arises is the case of a shift in the input coordinates. In general, a shift is described by n parameters where n is the number of dimensions of the processing space (since the shift along one axis can occur independent of the other axis). Therefore, if shift invariance is required, all of the dimensions of the processing system are used, leaving no available axes to which other distortion parameters can be assigned. Multiple invariance, in the context of this section, is understood to mean invariance to more than one distortion parameter per axis of the processor. To achieve multiple invariance, the input must be preprocessed before it is coordinate transformed. This processing should reduce the number of distorting parameters to n, without the loss of any useful information. Coordinate transformation processing as previously described can then be applied to provide invariance with respect to the remaining parameters.

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In § 6.2, various approaches to multiple invariant space-variant processing are discussed. From the previous sections, it is apparent that extracting and retaining the phase information present in a complex wave front is the vital key step required to realize a multiple invariant processor. A novel approach to this problem is described in 0 6.3. In § 6.4, we discuss multiple invariant processors using this phase extraction technique. Experimental verification as well as system block diagrams are included.

6.2. APPROACHES TO MULTIPLE INVARIANT SPACE-VARIANT PROCESSING

There are basically two approaches to achieve multiple invariance. These consist of either scanning through all of the values of the additional distorting parameter, or eliminating the additional parameters by filtering in the transformed plane. To visualize this difference, we consider the following function

fYx) = fC&>

a,

2

41,

(88)

where f is the distorting function, f is the undistorted function, and g(x, a,, as) is the deforming transformation which depends on two deformation parameters a , and a2. For simplicity and for convenience we have chosen without any loss of generality, a one-dimensional function with two distorting parameters. The first approach to the construction of the system that is invariant to both a , and a, would be to construct a number of parallel systems invariant to one of the parameters, for instance a * ,with each one of these parallel systems corresponding to one value of the second parameter u2. If these different systems cover all possible values of a2, the output of at least one of them will be the same for all values of a , and a2 for all input functions related b y eq. (88). Practical systems of this kind are the multiple hologram systems, conventional multi-channel Doppler signal processors and the mechanical movement systems described in previous sections. In practice, these systems are either parallel or serial. Parallel systems consist of many subsystems operating simultaneously. In this case, each subsystem corresponds to a specific value of a2 and is invariant to a,. Obviously, as the range of a , and the accuracy requirements of the system increases, the number of subsystems required increases dramatically. This results in parallel processors which require very large processing space, thus rendering them impractical. Serial systems consist of a

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347

single processor invariant to a,. The system is then continuously altered to correspond to different values of a2 with the input reprocessed each time the system is altered. These systems correspond to the mechanical movement systems noted earlier. They are in general impractical for use in real time applications. The second approach to multiple invariant processing uses the fact that a shift in the input coordinates is transformed into a linear phase factor in Fourier transform space. If this linear phase factor can be eliminated, and the non-linear phase factors containing the phase information of the function can be retained, then any shifts present in the input coordinates will also be eliminated. If the linear phase factor can be retained, then positional information on the input data present can be retained for further use. The first step in the realization of a practical multiple invariant processor is thus the conversion of one of the distorting parameters in the input into a shift by use of coordinate transformations. The desired coordinate transformation x = h ( 5 ) must satisfy

where &(al) is a constant dependent only on the parameter a,. The transformation x = h(6) is determined by eq. (16). From eq. (16) and the one-dimensional function with two deformation parameters described in eq. (88), we see that since h-'(x) depends on 6g/6ai and since

SglGa, f 6glSUJ,

(90)

each of the distortion parameters requires a different coordinate transformation h-'(x). It thus follows that the maximum number of deformation parameters that coordinate transformation processing can accommodate, equals the number of dimensions of the processing system. Substituting eq. (89) into eq. (88) we obtain

f ( x > =fCs{~-'(S-5o),a,}l=P'(5-50,~2)-

(91)

where F(o)is the The Fourier transform of eq. (91) is exp (-joto)F(o), Fourier transform of f"(& az). Equation (91) clearly shows that the dependence of the function f ( x ) on the parameter a, is described by the shift = eo(al). From the Fourier transform of this expression, we see and hence the a , dependence of f ( x ) , appears as a linear phase that factor in Fourier space. Therefore, if this factor can be extracted from the frequency space representation, the total dependence of the input on a ,

e0 co,

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[v,5 6

can be eliminated as a multiple invariance realized. This problem therefore reduces to extracting a linear phase factor from a complex wave front. We consider a system whereby this can be achieved in the next section.

6.3. PHASE EXTRACTION FROM A COMPLEX WAVE FRONT (PSALTISand CASASENT [1978])

It is very tempting to remove the linear phase factor in the Fourier transform of the input function by recording the magnitude of the Fourier transform, However, this also removes the phase associated with the function itself as well as the linear phase term. The schematic diagram of a system capable of extracting either the linear o r non-linear portion of the phase from a complex wavefront is shown in Fig. 22. For the specific example to be considered, we concern ourselves with the function f ( x ) whose Fourier transform is F ( w ) = IF(w)l exp [j+(w)]. A one-dimensional example is used for simplicity only. The first step in this system is to form the interference of the Fourier transform F(o).We can achieve this either by interferring the Fourier transform itself with a plane wave reference beam as in Fig. 1 or by positioning the input function f(x - xo)+ f ( - x + xo) in the input plane as in Fig. 2. Both of these systems are well known to record the effective hologram of the input function which thus contains both the amplitude and phase distribution of F ( w ) . The present problem is the extraction of the phase from this complex wave front. We use the joint transform-like input at plane PI of Fig. 22. The pattern detected by the television camera is thus

u,( w ) = 2 IF(W)(*[1 + cos(2wx, - 2 4 ( w ) ) ] .

P

(92)

L.

Fig. 22. Schematic block diagram of a system capable of extracting the phase from a complex amplitude-modulated optical wavefront (PSALTISand CASASENT [ 19771).

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349

We desire to extract the non-linear phase term c$(w), (the phase of the Fourier transform of f), and the linear phase term wxo (proportional to the location of the function in the input plane). To achieve this, we first pass the video output of the TV described by eq. (92) through a non-linear limiter with a threshold level R. The output of this limiter is then

where q = q ( w )= cos-1

[C(W)]

n =0, + l ,* 2 , . . . C(W) =

R 2 IF(w>l

(94)

-1

3

4

xlm)

usUcEEN1

Fig. 24. Theoretical versus experimental values of shift determined using the system of Fig. 22 (CASASENT and PSALTIS[1978]).

inputs was increased by increasing x1 and the intensity in figures similar to Fig. 23c was measured. From these data, experimental and theoretical values of x1 were obtained. A plot of these results is shown in Fig. 24. The departure of the two curves in Fig. 24 is due to the degraded performance of the lens at large aperture and high space bandwidth products caused as the separation between the inputs was increased. The ability of this system to extract the phase 4 ( w ) associated with the Fourier transform of the input function has also been demonstrated (CASASENT and PSALTIS [1978]). This phase extraction system forms the basis for the multiple invariant space variant processor we now discuss.

6.4. MULTIPLE INVARIANT OPTICAL PROCESSOR (CASASENT and PSALTIS [1978])

For simplicity, we consider the one dimensional distorting function described by eq. (98), where

and the use of the phase extraction system described in § 6.3 to achieve multiple invariance. We consider the undistorted reference function f(x) and the deformed input function f ( x ) = f(x’), where the distortion is described by eqs. (89) and (99). The object of this multiple-invariant, space-variant system is to correlate these two functions with no loss in SNR and to determine the two unknown distortion parameters a , and a2. The block diagram of the multiple-invariant correlator is given in Fig. 25.

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Fig. 25. Block diagram of a multiple-invariant, space-variant optical correlator (CASASENI and PSALTIS [1978]).

A coordinate transformation x = h , ( e , ) is first applied to f'(x). The 'transformation 5, = h;'(x) effects only the distortion parameter a , . Its purpose is to convert the effect of a , into a shift by tol in the 5, coordinate, i.e.

g[h,(5,), a,, a21 = g,(t, - 501, a2L

( 100)

tolis a constant dependent only on a , . The Fourier transform of fl(tl - to,, az) is F ( w ) exp (juto1).Using the linear phase extraction scheme of § 6.3, we can determine tO1 and hence the unknown distortion

where

parameter a , as shown in the right-hand side of Fig. 25. The 50,(a,) value is then used to shift f , ( ~ l - ~ o la,) , to produce f,(tl,a2)as shown in Fig. 25. The inverse transform = h;'(x) is applied to this function to produce

el

f { g , [ h ; l ( x ) , a33 =f[g(x, a,>],

(101)

which is now independent of a , . We have thus reduced the number of distortion parameters to the number of dimensions of the optical system

354

DEFORMATION INVARIANT. SPACE-VARIANT

OPTICAL PATERN

RECOGNITION

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(one for the simple one-dimensional case considered here). Conventional space variant correlation techniques can now be used to determine u2 and to produce an output correlation invariant to the distortion parameter a2. This is accomplished by applying the coordinate transformation .& = h;'(x) to both f ( x ) and f[g(x,a,)]. The transformation & is chosen to satisfy eq. (16). The two new coordinate transformed functions f(&) and f(& - to2) are then used as the inputs to a conventional space invariant is a constant dependent only on u2. The output correlator. Note that tO2 of this correlator will then be invariant to the multiple distortion described by eqs. (89) and (99) and the location of the output correlation peak will be proportional to Eo2 and hence to u2. The desired distortion invariant correlation has thus been realized and the unknown distortion parameters a, and a2 have been determined. This approach can easily be extended to distortions of higher order.

In the previous section, we have presented a summary of all spacevariant optical correlation work using coordinance transformation processing. The principle purpose of this effort has been to achieve optical correlation with no loss in SNR even in the presence of distortion differences between the input and the reference functions. We believe that this vital approach to optical pattern recognition is needed if optical correlators are to see extensive use in real practical applications and if they are to supplant sophisticated digital pattern recognition systems with their extensive algorithms but slow speed high cost. This presentation has proceeded from a general formulation in which general expressions for the required coordinate transformation were derived €or the case of the general distorting functions (82). The most well-known of the space-variant correlations is the Mellin transform system (0 3). Its implementation using computer generated holograms and non-linear scanning has been described and demonstrated. Its use in pattern recognition on scaled imagery ( 5 4.3), aberrated imagery ( 5 4.4), Doppler signal processing (0 4.5), and non-vertical imagery (8 4.6) have been demonstrated and explained. The development of a rotationinvariant space-variant correlator and its combination with a Mellin transformed system to produce a scale-invariant rotation-invariant optical

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pattern recognition system has also been explained and demonstrated and its implementation described (§ 5). Of the many facets of this work included in this chapter, the detection scheme described in 0 6.3 in which the linear and non-linear phase of a complex amplitude modulated wave front can be determined has far reaching applications in many optical data processing systems. This scheme relieves the last road block in the theory and realization of viable space-variant distortion-invariant optical pattern recognition system and makes possible the realization of the multiple invariant system described in 0 6.4.

Acknowledgements The authors wish to thank the Office of Naval Research (Contract NR350-Oll), the Air Force Office of Scientific Research, Air Systems Command (Grant AFOSR-75-285 1) and the Ballistic Missile Advanced Technology Center (Grant DASG60-77-C-0034) for support of various phases and applications of this work.

References BORN,M. and E. WOLF,1965, Principles of Optics (Pergarnon, New York, third ed.) p. 753. O., 1974, J. Opt. SOC.Amer. 64, 1092. BRYNGDAHL, BURCKHARDT, C. B., 1967, Appl. Opt. 6, 1359. CASASENT, D., 1977, Proc. lEEE65, 143. CASASENT, D., 1978, Optical Information Processing, ed. S. H. Lee (Springer-Verlag, Heidelberg) Chap. 6. CASASENT, D. and H. J. CAULFTELD, 1978, Applications of Optical Data Processing, eds. D. Casasent and H. J. Caulfield (Springer-Verlag, Heidelberg). D. and A. FURMAN, 1977, Appl. Opt. 16, 1955. CASASENT, CASASENT, D. and M. KRAUS,1976, Opt. Commun. 19, 212. CASASENT, D. and M. KRAUS,1978, Appl. Opt. 17, 1559. CASASENT, D. and D. PSALTIS,1976a, Opt. En@. 15, 258. CASASENT, D. and D. PSALTIS, 1976b, Opt. Cornrnun. 17, 59. CASASENT, D. and D. PSALTIS,1976c, Appl. Opt. 15, 1795. CASASENT, D. and D. PSALTIS,1976d, Appl. Opt. 15, 2015. CASASENT, D. and D. PSALTIS,1977a, Proc. IEEE 65, 77. CASASENT, D. and D. PSALTIS,1977b, Opt. Comrnun. 21, 307. CASASENT, D. and D. PSALTIS, 1977c, Appl. Opt. 16, 1472. CASASENT, D. and D. PSALTIS, 1977d, Opt. Cornrnun. 23, 2 0 0 CASASENT, D. and D. PSALTIS,1978, Appl. Opt. 17,6.55.

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CASASENT, D., D. PSALTISand M. KRAUS,1976, Elec. Opt. Sys. Des. Conf., Anaheim (Ind. and Sci. Conf. Mgment Inc.). CASASENT, D. and W. STERLING, 1975, IEEE TC-24, 337. CASASENT, D. and C. SZUCZUTKOWSKI, 1976a, Proc. SOC.Photo. Instru. Engr. 83, 91. CASASENT, D. and C. SZUCZUTKOWSKI, 1976b, Opt. Commun. 19, 217. DASHIEL, S. and A. SAWCHUK, 1975, Elec. Opt. Sys. Des. Conf., Anaheim (Ind. and Sci. Conf. Mgment Inc.). J. W., 1977, Proc. IEEE 65, 29. GOODMAN, HUANG,G., F. RUSSELL and W. CHEN, 1975, EIA Symposium, College Park. KELLY,E. and R. WISHNER, 1965, IEEE MIL-9, 56. LAMACCHIA, T. and C. VINCELETIE, 1968, Appl. Opt. 7,1857. LEE, W. H., 1974, Appl. Opt. 13, 1677. LEVI, L., 1968, Appl. Opt.: A Guide to Modern Optical Systems Design (Wiley, New York). LOHMANN, A. W. and D. P. PARIS,1967, Appl. Opt. 9, 1567. PSALTIS, D. and D. CASASENT, 1977, Appl. Opt. 16,2288. PSALTIS,D. and D. CASASENT, 1978, Appl. Opt. 17, 1136. ROBBINS, G. M., 1970, PhD Thesis, The Inversion of Linear Shift Variant Imaging Systems, Massachussetts Inst. of Technology. ROBBINS, G. M. and T. S. HUANG,1972, Proc. IEEE 60, 860. SAWCHUK, A., 1974, J. Opt. SOC.h e r . 64, 138. I. H., 1972, The Use of Integral Transforms (McGraw-Hill, New York). SNEDDON, SOC.Photo Intru. Engr.. 1976. Vol. 83, ed. D. Casasent. K. T. and S. H. LEE, 1974, J. Opt. Soc. Amer. 64, 564. STALKER VANDER LUGT,A., 1966, Appl. Opt. 5, 1760. VANDER LUGT,A,, 1974, Proc. IEEE 62, 1300. J. and M. 0. HAGLER,1975, Elec. Opt. Sys. Des. Conf., Anaheim. WALKUP, WELFORD, W., 1974. Aberrations of the Symmetrical Optical System (Academic Press, London).


VI

LIGHT EMISSION FROM HIGH-CURRENT SURFACE-SPARK DISCHARGES BY

R. E. BEVERLY I11 Battelle, Columbus Laboratories, 505 King Aoenue, Columbus, Ohio 43201, U.S.A.

CONTENTS PAGE

$ 1. INTRODUCTION

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

$ 2. CHANNEL DEVELOPMENT AND GASDYNAMICS

359

.

362

5 3 . RADIATIVE PROPERTIES . . . . . . . . . . . .

370

$ 4. APPLICATIONS

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

389

$ 5 . CONCLUDING REMARKS . . . . . . . . . . . .

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

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ACKNOWLEDGEMENTS

8 1. Introduction 1.1. INTRODUCTION

Surface sparks, also called “creeping”, “guided”, “sliding”, or “gliding” sparks, have fascinated experimentalists for some time. Although many geometrical configurations have been investigated, the distinguishing characteristic of a surface spark is the presence of a dielectric solid adjacent to the discharge channel. This surface, which bounds the discharge plasma in at least one dimension, affects the gas-dynamic and radiative properties of the spark such that notable differences exist between surface discharges and unconfined (free or open) discharges. This latter statement must be qualified by noting the dependence of these physical mechanisms on the spark current; very weak discharges ‘will not vaporize material from the surface and will deviate little from free sparks. For this reason, this review will be concerned with high-current surface sparks and their optical properties. The luminosity induced by static-charge effects or low-energy discharges over surfaces, such as Lichtenberg patterns, is excluded from this review. The reader will notice an intentional stress on applications, particularly laser-related applications, reflecting the author’s own interests. Another major application area, light sources for high-speed photography and cinematography, is discussed in conjunction with the historical perspective presented in this section and will not be reviewed further. The interested reader is referred to the review of FRUNGEL [1965] which is largely concerned with this topic. 1.2. HISTORY

Accounts of early and historically significant work are given in the review by FRUNGEL [1965], including the application of a surface spark for schlieren photography published in 1867 by A. Toepler. The first investigation of the detailed electrical and optical characteristics of “creeping” 359

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[VI, $ 1

discharges, however, was published by FUNFER[1949]. By using rare gases such as Ar, Kr and Xe to obtain increased light intensity over sparks in air, he obtained optical efficiencies of about 20% and found that the intensity increases with gas pressure. Since the luminosity is proportional to the spark length and long-path discharges can be initiated over surfaces at considerably lower voltages, these light sources found numerous applications in scientific cinematography and high-speed photography (SCHARD I N and FUNFER [ 1952a, b]). A common device of this period, the “Defatron” (FAYOLLE and NASIJN [ 1953]), consisted of an insulating rod or tube with coaxial electrodes at both ends. An internal trigger rod running the length of the discharge gap served to initiate the spark along the insulating surface between the two main electrodes. The Defatron was plagued with several problems; the spark was highly nonreproducible and partial occultation often occurred when the spark wound itself around the rod. Long, straight spark lengths were difficult to adapt to optical reflectors, and the fragile insulating rod easily broke as the result of erosion or shock. A device similar to the Defatron, developed by EDGERTON, TREDWELL and COOPER [1962] for short-duration (0.01-0.1 psec) reflectedlight photography, was operated in several gas atmospheres in addition to air, including O,, N,, CO,, H, and Ar. To avoid several of the difficulties inherent to the Defatron, LUY and SCHADE[1954, 19561 employed a planar substrate of porous ceramic impregnated with electrolyte solution. Since one end of the substrate was immersed in a reservoir, vaporized electrolyte was continuously replenished permitting repetitively-pulsed operation as a source for frontlight photography at rates up to 10 000 frames per second. Sources were investigated in air, Xe, and Ar atmospheres and produced light emission having a duration of about 1psec. The ability to initiate discharges across gaps 10-20 times that possible with a free spark made it possible to deposit in excess of 90% of the capacitively-stored energy into the surface-spark channel (see 0 3.1 for a full discussion). TAWIL[1957] developed a “guided” spark a lm g an S-shaped channel in synthetic resin (ethoxyline) with triggering accomplished by an embedded control electrode. Interestingly, these resins were found to be slightly phosphorescent. To approximate a point source, MODEN,REECEand POOLEY[1963] investigated various sources operating in air which employed a discharge over a semiconductor surface (TiO) between coaxial electrodes. For a given width of the discharge annulus, the spark resistance was found to decrease with increasing capacitor voltage, and the

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36 1

INTRODUCTION

current waveform was successively overdamped, critically damped, and finally became oscillatory. The peak intensity occurred at that voltage which produced critical damping, and both parameters increased with increasing annulus width. A peak intensity of more than 20Mcd was obtained for a source having a 19-mm annulus width driven by a low-inductance capacitor bank ( C = 4.6 pF, V, = 5 kV and L 10 nH).

-

1.3. NOMENCLATURE

The simplest configuration, shown schematically in Fig. 1, consists of a planar ceramic substrate with tight-fitting electrodes separated by a predetermined gap. The composition of the electrodes and substrate strongly influence radiative properties. For prolonged life, tungsten is frequently employed as the electrode material. In experiments conducted in the author’s laboratory, alteration of the substrate composition is relied upon as the primary method for altering optical properties to suit a given application. The surrounding gas composition and pressure strongly affect the emission characteristics of surface sparks, and experiments have been conducted under conditions ranging from hard vacuum to pressures of many atmospheres. A wide variety of pure gases and gas mixtures have been investigated. A complete description of a given experiment requires specification of the parameters given in Table 1. Many unaccountable differences in supposedly identical experiments are attributable to subtle yet important circuit considerations as discussed in 0 3.1. Other secondary parameters Plasma Channel Ceramic Substrate

Fig. 1. Planar surface-spark geometry and spectrometer image positions for the spectroscopic studies of BEVERLY, BARNES,MOELLER and WONC[1977].

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TABLE1 Surface-spark discharge parameters Electrical: Capacitance of the energy-storage capacitor, C Circuit inductance, L Charging voltage, V, Electrical circuit efficiency, ‘ T ) ~ , (see 5 3.1) Mechanical: Substrate composition Electrode composition Source geometry Surface gap, ?i Environmental: Gas composition Gas pressure, P

include the stored electrical energy E , = $Vf, and the discharge “hardness” or “strength”, VJL. This latter parameter imposes an upper limit for the initial rate of change of the current with time and largely determines the initial rate of energy input to the discharge channel. Many radiative and gas-dynamic processes are understandably sensitive functions of the discharge parameters.

§

2. Channel Development and Gasdynamics

2.1. BREAKDOWN MECHANISMS

Surface breakdown is defined as the development of a conducting channel between two electrodes resting on a dielectric surface. I n the presence of a surrounding gas atmosphere, three development phases are discernible - the streamer, spark and surface discharge stages. With the arrival of a voltage pulse V at the cathode, a sharply inhomogeneous electric field having a large normal component is present because of an interelectrode surface capacitance C,. The normal field component En is related to the surface charge density a, by En= 4

7 = ~ T V~ C ,~.

(1)

The ensuing impact ionization of gas near the surface of the dielectric produces a weak predischarge streamer traveling along the surface toward

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363

the anode. Upon completion of the discharge circuit, current flow in the conducting channel produces a cylindrical plasma column just above the surface having properties analogous to those for a free or unconfined spark. Radiation emitted by this plasma heats and vaporizes surface constituents which modify the gas-dynamic and radiative behavior. The breakdown then has progressed into a fully-developed surface-spark discharge. KRASYUK,LIPATOVand PASHININ [1976] optically measured the predischarge streamer velocity, up, across a BaTiO, ceramic which possesses a large dielectric permittivity ( E = 1600). By precisely timing their discharge using a laser-triggered spark gap, they found a value of up= (1.44) x l o 7 cm/sec for air, V, = 20-50 kV and t = 1 2 cm. In general, the streamer velocity depends upon the discharge voltage, the interelectrode capacitance, and the gas composition and pressure. For a cathode[1971] estimated directed surface spark across glass in air, ZOLEDZIOWSKI a charge carrier density at the streamer tip of 9.7 x lo1* cmP3 and a linear density of 5 x lo9 cm-l. Since the predischarge streamer completes the discharge circuit, the incipient breakdown voltage Vi is also expected to depend on similar parameters. For a given substrate material and specified gas environment,

via c,*,

(2)

where the exponent a is usually less than unity. For example, BERTSEV, DASHUK and LYSAKOVSKII [1963] found the following empirical relation for polyethylene in air:

vi=54c;0.4-

10,

0.1 5 c,5 10,

(3)

where Vi is in kV and C, is in pF/cm2. On the whole, however, there is a paucity of breakdown voltage data in the published literature for various ceramics of interest in rare-gas atmospheres, and appreciable scatter in the data is to be expected because of differencesin the surface condition. Depending on the electrode-substrate geometry, C, may be difficult to measure or calculate. For the practical case of a dielectric bonded to a rear-surface ground plane as described in § 4.2, Fig. 24, however, the surface capacitance is simply C,

=dd,

(4)

where d is the thickness of the dielectric sheet. Hence, lower breakdown

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voltages will be obtained for thin substrates having a large dielectric permittivity. Characteristic breakdown voltages for surface sparks are much lower than €or free sparks, and discharge paths typically a factor of 10-20 times VANYUKOV and longer can be driven with the same value of V,. ANDREEV, DANIEL’ [1966] found that Vi increased only from 8 kV to 13 kV as t was increased from 2 to 8 cm for discharges across TiOz substrates in A r or Xe gas at pressures of several atmospheres. Figures 2 and 3 give static breakdown data for discharges between stainless electrodes mounted on A120, and BN substrates in various gas atmospheres. Figure 2 shows that the value of VJPe approaches a lower asymptotic value with increasing pressure which depends only on the substrate composition and the gas. This high-pressure breakdown field is shown in Fig. 3 as a function of e only for the A1,03 substrate and Ar and He gases. Hence, for P>>1atm, and t>>1cm, there is a decided tendency for the breakdown field to be only weakly dependent on the substrate material and surface gap. Gases which strongly attach electrons and store excess energy in vibrational modes (e.g., SF,), however, give breakdown voltages substantially higher than for the rare gas. For vacuum surface discharges, DASHUK, KICHAYEVA and YARYSHEVA [1967] observed no streamer or spark stages during the breakdown

0

2

4 6 8 Pressure,otm

1

0

Fig. 2 . Static VJPP for Ar, He, N, and CO, gases as a function of P for e = 0.25 cm, and A1,0, (open symbols) and BN (solid symbols) substrates (BEVERLY,BARNES, MOELLERand WONG[1976]).

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CHANNEL DEVE-LOPMENT AND GASDYNAMICS

0 0

4 6 0 Surface Gap, mm

2

365

10

Fig. 3. High-pressure static breakdown field for Ar and H e gases as a function of A1,0, substrate (BEVERLY, BARNES, MOELLERand WONG[1976]).

e for an

process and postulated that breakdown occurs as the result of cascade ionization of the gas layer adsorbed onto the surface. KALYATSKII and KASSIROV [1964] have shown that Vi depends upon the dielectric material, the surface finish, the field inhomogeneity especially at the cathode, and even the voltage pulse duration. Deterioration of the surface results from (1) ionic bombardment, (2) chemical attack from atmospheric constituents or from plasma reaction products, and (3) vaporization due to radiative heating. For high-current surface sparks, erosion due to radiative vaporization of the substrate material ultimately limits the practical life expectancy of the source and is of great concern in particular applications. The dominant effect of this mechanism was demonstrated in the experiment of VANYUKOV and [1969] in which the optical emission from a surface spark across DANIEL’ TiO, in a Xe atmosphere was projected by a cylindrical mirror onto an identical substrate. After 50 flashes, the two surfaces exhibited identical degrees of degradation. Many plastics and some ceramics form a carbonized conducting path under the influence of surf ace discharges. Historically, many experimentalists have simply drawn a pencil mark between the two electrodes insuring that the first few discharges followed the desired path. A few plastics, such as polytetrafluoroethylene and polychlorotrifluoroethylene, evaporate cleanly when used as substrates (MASON[1959], § 5 ) . Many ceramics undergo reduction and/or metallization reactions in the surface layer. Sputtered material from the electrodes which is transported into the discharge channel can partially condense onto the substrate following

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the discharge. These deterioration mechanisms can greatly alter the value of V,. “Fresh” ceramic substrates will usually require a larger breakdown potential than a “broken-in’’ substrate whose surface has been partially metallized. Little data exist in the literature on erosion rates. As a rule of thumb, materials having higher vaporization temperatures and larger latent heats of vaporization will yield better erosion resistance. The erosion rate, defined as the mass loss per unit discharge path length per unit time of discharge duration, was measured by DANIEL’ [1968] to be 2.5 g/sec-mm for a TiO, substrate in air with V J L = 0.9 X 10” V/H. At the instant of maximum luminance, DaniC1’ found that the vaporization rate was proportional to the rate of change of the discharge current and, hence, V J L .

2 . 2 . CHANNEL EXPANSION AND SUBSTRATE VAPORIZATION

Many of the interesting radiative characteristics of surface sparks are directly attributable to the mechanisms of channel expansion and substrate vaporization. Considerable research has been devoted to the gasdynamic behavior of surface-discharge plasmas, especially by Soviet researchers, and the particular traits of surface sparks which differ from those of free sparks can be qualitatively described. After channel breakdown, radiation from the highly ionized high-temperature cylindrical plasma column is partially intercepted by the substrate which heats and evolves constituent species into the discharge. Until this occurs, the physical processes of the discharge are the same as for a free spark. Because of surface vaporization, the conductivity of the plasma channel becomes largest near the surface, and the channel quickly becomes asymmetric with higher expansion velocities parallel to the surface rather than perpendicular. The plasma can be described as being “ribbonlike” rather than cylindrical. As the plasma is drawn to the substrate and as the plasma temperature rises, increased vaporization leads to high electron particle densities in a thin layer adjacent to the surface. Due to the smaller cross-sectional area of surface sparks, the current density and limiting brightness temperature are higher than attainable with free sparks; in addition, surface sparks are characterized by a higher resistance per unit length and a shorter optical pulse width than the equivalent free spark. Most diagnostic investigations of channel development have employed

VI, i 21

CHANNEL DEVELOPMENT A N D CASI>YNAMICS

361

streak and framing photography (DANIEL'[ 19651, ANDREEV, VANYUKOV and DANIEL,[1966], and VANYUKOV and DANIEL, [1967a, 1967b, 1969]), although recent work used spectroscopic techniques (BEVERLY [ 19771). The degree of channel spreading and the extent of channel asymmetry are related to the (i) substrate composition, (ii) the gas environment, (iii) the magnitude and rate of energy deposition into the discharge channel, and with a weaker dependence, (iv) the gas pressure. Concerning (i), experiments performed by VANYUKOV and DANIEV[1967b] were used in support of conclusions which related a large degree of channel spreading to a low ionization potential of the metallic constituent(s) of the ceramic substrate. Using bulk ceramics with and without various dielectric coatings, they found that spreading is greater in the case of a discharge across BaO than for TiO, or S O z , and attributed this to the low ionization potential of barium (5.21 eV) when compared with titanium (6.82 eV) or silicon (8.15eV). Analysis of the data presented by BEVERLY[1977] revealed no similar trend, and, in fact, the substrate giving the largest channel width (ZnO) contained the element with the highest ionization potential of those investigated (Zn, 9.39 eV). I proposed an alternate explanation which satisfactorily resolves both results. Metallization of a surface layer occurs for most of the ceramics, and the rate of substrate vaporization and the degree of channel spreading are related to the surface vapor pressure. Intense radiation intercepted by the surface heats and vaporizes this metallized layer, and the spatial extent of the vaporization process is a function of the local vapor pressure. A high value gives a large rate of substrate species evolution from surface zones well removed from the discharge axis. These neutral species are quickly ionized causing current to flow in the outer zones; therefore, this mechanism receives positive feedback for channel growth until other constraints, such as energy deposition and radiation losses, limit the spreading. This mechanism is also consistent with the observations of VANYUKOV and DANIEL. [1967b] which show no dependence of the luminous channel expansion on dielectric properties or polarizabilities of the ceramics. With regard to (ii) and (iv), the dependence of the channel gasdynamics on the atmospheric composition and pressure is due to the dependence of vaporization processes on the spectral emission coefficients of the initial plasma channel. The presence of gases which dissociate or attach electrons in the discharge plasma will also alter the gas-dynamic behavior (see Fig. 12, 0 3.2). Gases having lower ionization potentials and larger emission coefficients in the ultraviolet, such as Xe and Ar, produce distinctive

368

LIGHT EMISSION FROM HIGH-CURRENI‘ SURFACE-SPARK DISCHARGES

0

2

4

6

[VI, 3 2

8

Time, psec (C)

(d 1

Fig. 4. Luminous channel expansion of surface-spark discharges across TiO, measured from streak camera records: (a) He, (b) air at ambient pressure, (c) Xe, and (d) Xe with V, = 15 kV. Other parameters are C = 94 nF, L = 2.2pH, V, = 25 kV and t = 40 mm unless otherwise noted (VANYUKOV and DANIEL‘[1967a]).

“ribbonlike” channels. Finally, in connection with (iii), the gas-dynamic response of surface sparks is related to electrical parameters which affect the magnitude and rate with which capacitively-stored energy is delivered to the surface-discharge channel. The electrical efficiency qel, as defined in 0 3.1, and the discharge strength V J L strongly influence channel and expansion velocities. Excerpted results from the work of VANYUKOV DANIEL‘[1967a] are given in Fig. 4 showing the luminous channel expansion for various gases and values of P, V, and L.

2.3. PARTICLE DENSITY DISTRIBUTION AND CHANNEL CONDUCTIVITY

The density distribution of gas along a section of a surface-discharge channel in air was investigated by DANIEL’ [1968] using a Mach-Zehnder interferometer. A short-duration free spark, whose firing was adjustable at different times into the evolution of the surface discharge, served as the illumination source of the interferometer and gave exposure times of 0.4 p sec. The spectral region recorded by the interferogram was selected by an optical filter (A = 5800 A, Ah = 80 A). By assuming that the density distribution has cylindrical symmetry with an effective axis located beneath the surface of the substrate to account for “flattening” of the

v1, $!

?I

CHANNEL DEVELOPMENT A N D GASDYNAMICS

369

discharge channel, an approximate solution of Abel’s integral was found by a numerical method yielding the index of refraction as a function of radius. The electron density n, is related to the real component of the index of refraction n by the relation ( n < 1 ) n 2 =1 - ( ~ , , / ~ ) ~ = 1 - 3 . 1 8 ~ 1 O ~ n , / ~ ~ ,

(5)

where wpe and w are the electron plasma and probe light angular frequencies, respectively. The density distribution p in the shock wavefront, where the temperature does not exceed 2000-3000°K and the gas is only weakly ionized, can be obtained from the empirical relation ( n > 1)

n - 1= K p ,

(6)

where K is the Gladstone-Dale constant. Examples of electron and gas density distributions from this work are given in Fig. 5. The wavefront positions are replotted in Fig. 6 for the same experiment along with the luminous channel positions obtained from streak photographs and the current waveform. Interestingly, the zone of strong ionization initially occupies a volume smaller than the luminous channel. After 2 psec, this zone and the luminous channel coincide while the shock wavefront perpendicular to the surface rapidly separates from the luminous channel at a velocity of the order of 1 km/sec. The plasma conductivity and electron density are related by the wellknown expressions (+

= J/E = evdn,/E,

(7)

where J is the current density, E is the field strength, e is the electronic charge and vd is the electron drift velocity. Because of larger values of n, 4

3 ; c

W

>

3 >

BaT'03"' P 1 Zno\

3

1 0 20 30 40 50 Stared Energy ( & ) , J

60

Fig. 17. Radiated energy in the soft uv (2500-2900 A) from surface-spark discharges across various substrates in Ar with P = 10 atm, C = 1 cm, C = 82.8 nF and L = 0.45pH (BEVERLY, BARNES,MOELLER and WONG[:1977]).

388


2 5

6

‘OL O L

0

[VI, 5 3

Free Spark

c W

,

I

10 20 30 40 50 Stored Energy ( E s ) , J

60

0

10 2 0 30 40 50 Stored Energy ( E s ) , J

60

Fig. 18. Ultraviolet energy density and time-integrated effective channel width, defined as the equivalent width of a square envelope having the same peak intensity as the luminous BARNES, channel (measured at 2750&, for the same conditions as in Fig. 17 (BEVERLY, MOELLERand WONG[1977]).

the data in Figs. 17 and 18. For reference, data obtained from a free spark under identical conditions are also shown in these figures. Selection of the substrate material is a function of the desired radiative properties and the intended application of the light source. Metal-oxide ceramics are often chosen as a good compromise between maximum vaporization to affect spectral modifications and maximum erosion resistance to give the desired life expectancy, In general, the silicates have melting temperatures which are too low and which can lead to severe erosion and short life expectancies. Carbides and nitrides are oftentimes difficult to fabricate and are expensive. Further selection criteria are based on certain properties of the constituents; namely, the (i) ionization potential, (ii) vaporization temperature, (iii) known characteristics in ordinary spark spectra and (iv) stoichiometric relationship. Since the surfaces of many oxide ceramics undergo metallization, the metallic constituents are chosen on the basis of their vaporization temperature and ionization potential so that significant concentrations of metal ions enter

VI, $41

APPLICATIONS

389

the discharge, are excited by electron impact, and radiate via line transition series in the wavelength region of interest. Maximization of the metal/oxygen stoichiometry is desired when practical since abundant 0 line transitions are not generally observed in the visible and soft-uv spectral regions from typical discharges. This may mean that the bulk substrate is a semiconductor, but no operational difficulties have been encountered with such sources as evidenced by MODEN,REECEand POOLEY [1963].

0 4. Applications 4.1 SPECTROSCOPIC SOURCES

In the study of atomic spectra, the vacuum spark is a convenient source of far uv radiation originating from a high-temperature plasma of vaporized electrode species. Under high-vacuum conditions, the large potentials required for breakdown of short gaps result in large electric fields producing significant shifts and broadening in line transitions due to the Stark effect. With a vacuum surface spark, however, less violent discharges can permit easier line identifications without such complications and with greater discharge stability. These features were important in the [1955]) and Pr I11 (SUGAR study of the spectra of C I11 (BOCKASTEN [19631). In the first application of a vacuum surface spark to far uv spectroscopy, VODARand ASTOIN[1950] employed a commercial carbon rod resistor fitted with copper ring electrodes. Gaps of several cm were easily discharged with V, = 30 kV. Spectra obtained were similar to those obtained with the Millikan spark except for the appearance of carbon lines. Later work (ASTOIN[1952] and ASTOINand VODAR[1953]) was devoted to spectroscopic and spectrophotometric investigations employing other electrode metals (Al, Be, Fe, Ni, U, W) in addition to copper. These initial investigations were followed by an active and continuing effort devoted to uv source development at the Laboratoire des Haute Pressions, Centre National de la Recherche Scientifique, Meudon, Bellevue, France. Early work by ROMAND and BALLOFFET [1955] and BALLOFFET and ROMAND [1955, 19561 concentrated on optimization of the source geometry and identification of highly-ionized electrode, substrate and impurity line species. Transitions from multiply-charged ions, such as As V, C IV, CI VII, Ge IV, N V, 0 VI, P V, S VI, Se VI and Si IV, were

390


[VI, 8 4

2

Fig. 19. Source configurations for vuv spectroscopy using surface sparks (ROMANDand BALLOFFET [1955]).

excited and identified. Four source geometries are shown in Fig. 19; best results were obtained from configurations 1 and 4. A review of vuv and VODAR sources, including these surface sparks, is given by ROMAND [1956]. Surface discharges across alumina or zirconia produced far less substrate erosion, and hence, a greater source life than discharges across glass. Alumina gave the least number of line transitions attributable to vaporized substrate species. Intense vuv emission was produced when organic polymers were used as substrates (ROMANDand BALLOFFET [1957a]) due to the rich carbon spectra. Using vacuum surface sparks or surface-spark triggered vacuum sparks, researchers at the Laboratoire des Hautes Pressions also developed techniques for the spectrochemical analysis of impurities at low concentrations (ROMANDand BALLOFFET [1957b], ROMAND, BALLOFFET and VODAR[1959], and BALLOFFET [1960]). Also worthy of mention is the work of DAMANY, RONCIN and DAMANYASTOIN [1966], ESTEVA,BON,SCHWOB and ROMAND [1970], and BOURSEY and DAMANY [1974] in which an auxiliary surface spark is incorporated into the design of a vacuum spark, such as the BRV source (BALLOFFET, ROMAND and VODAR[196 l]), permitting accurate and reproducible triggering of the main spark. A similar modification was made in the design of an end-viewing Lyman capillary by MORLAIS and ROBIN[1964]. Endand side-viewing capillary discharges in vacuum have been developed which give intense continuum emission from the visible into the soft

VI, 8 41

391

APPLICATIONS

X-ray region (see, for example, BOGEN,CONRADS, GATTIand KOHLHAAS [1968], ROMAND, BALLOFFET and VODAR[1955], FITZPATRICK, HUBBARD and THALER [1950]). A powerful vuv light source was recently developed by LIE,BOGENand HINTZ[ 197 11 for the production of spatially-homogeneous, low-density plasmas of large volume (40 cm diameter x 100 cm length) by photoionization of gases at low particle concentrations. Their source, a vacuum surface discharge over solid Xe, produced a high-temperature, highdensity plasma having a large emitting area (=1.6 cm') and sufficiently long emission duration ( t p = 5 Ksec). A schematic diagram of their arrangement is shown in Fig. 20. The two electrons have a width of 2 c m and a separation of 1 cm. A liquid-air cooled insulator between the electrodes develops a layer of frozen Xe from the surrounding gas. Although the vapor pressure of Xe at 77 "K is about 1 mTorr, it was necessary to evacuate the discharge environment to a pressure lower than 0.4mTorr once the solid layer had formed to insure a reliable surface discharge. The electrical circuit consisted of a slower evaporating bank and a main discharge bank. The main bank is fired after a delay of 10 Ksec, such that the second discharge commences when the current from the evaporating bank is a maximum. Spectroscopic measurements of the absolute intensity were performed using a calibrated focusing mirror and Seya-Namioka monochromator. Intensity plots are shown in Fig. 21 for two different main bank energies, but due to destruction of the Ge/ZnS coating on the focusing mirror by the intense vuv radiation, detailed measurements showing the line spectral features were not performed. The intensity as a function of wavelength shows a dependence which is much stronger than would be expected from a pure bremsstrahlung source for which IA 0: llh2. On going from 3000 A to 750 A, I, Cooled Insulator

Evaporating Bank

Main

Fig. 20. Electrical circuit for the production of a vacuum surface discharge over solid xenon (LIE,BOGENand HINTZ[1971]).

392


I500 2000 Wavelength,

2500

[VI, 5 4

3000

Fig. 21. Intensity as a function of wavelength for the vuv source of LIE, BOGENand HINTZ [1971].

should increase by a factor of 16, whereas an experimental factor of 100 was observed. This can be explained in part by additional contributions from recombination continua and line radiation and, at least for the high-capacitance discharge, by reabsorption at longer wavelengths. The light intensity reaches a maximum simultaneously with the current, and the energy radiated into the 450 A to 950 A wavelength band over 4rr sr was estimated by extrapolation to be 750 J, representing an electrical-tooptical conversion efficiency of 10%. The source was used to photoionize a low-pressure gaseous Xe cell with the electron density measured by a microwave interferometer. This measurement inferred a radiated energy value over the same band of 1.5 kJ, representing a conversion efficiency of about 20%.

4.2 LASER UV PREIONIZATION

4.2.1. Spark uv preionization of pulsed CO, lasers Ultraviolet preionization of large-volume, high-pressure glow discharges has been the subject of intensive research as a simple and reliable

VI, 9 41

APPLICATIOUS

393

method which allows a higher arc-free energy loading into the discharge volume when compared with unconditioned discharges. Although this technique has had the greatest impact on the development of compact, high-power, pulsed CO, lasers, uv preionization is currently being exploited in the development of other electrically-excited gas lasers employing transverse discharges. Considerable effort had been devoted to understanding the uv photon generation, transport, and photoionization processes in CO, laser gas mixtures (SEGUIN,MCKENand TULIP[1976, 19771, MCKEN,SEGUINand TULIP[1976], and BABCOCK, LIBERMAN and PARTLOW [1976]). In typical gas mixtures at atmospheric pressure, CO, controls the photon transmission for wavelengths ,

(3.16)

0

(3.17)

x Jot dtlIot1dt2i([D‘”(t,), D‘O)(t,)],)sin o(t,- t,),

(3.18)

and

Since qndis the energy lost by the molecules due to the initial excitation of the field, we may regard it as induced emission, and since is the energy lost by the molecules independently of the initial excitation of the field, we may regard it as spontaneous emission. In order to simplify the discussion and concentrate only on the aspects essential for present purposes, we deal first with the special case for which (D“’(t))= 0,

(3.20a)

( Y ) = 0.

(3.20b)

or One can then write

x (Ai([D“)(t,), D(O)(f2)lP) sin w ( t , - fz) -({D(’)(t1),D(O)(tZ)}) cos o(t1-

424

[VII, 5 3

SEMICLASSICAL RADIATION THEORY

Consider a situation in which all the molecules are in the ground state. (For classical interpretation, this means that the molecular energy has its lowest value.) If the molecules respond to the field (or absorb energy from the field) in second order, which we assume, then &in,, must be negative. The expression for &ind leads, therefore, to the inequality

6'

dt,['

dtzi(0)[D'O'(tl), D'"(t2)],)0)sin w ( t l - t2)> 0,

(3.22)

where 10) is the ground state both quantum-mechanically and classically. It is clear that ([D'"(tl), D(o'(t2)]p) does not vanish in either formalism when the molecules are in the ground state. The result is different, however, with respect to the expression ({D(o'(tl),D(')(t2)}).Quantummechanically, this expression cannot vanish in the ground state, since D'O'( t ) does not commute with the molecular Hamiltonian (otherwise it would not oscillate). On the other hand, D'O'(t) and {D("'(tl), D'"'(t,)} must vanish classically, since internal motion ceases when a classical system is at its lowest energy. It is useful to have an explicit quantummechanical evaluation of the ground-state expectation values considered above in terms of the matrix elements of D(0). Noting that (3.23) ojE'(t)== D j k eXp (iWjkt), where D j k = D;E'(O),hwjk = Ej - &, with the Ei's being the molecular

energy levels, we obtain, with obvious labeling,

(ol(oco'(t,),D ' O ) ( t J ) l o ) q m

= 2 C IDok)' COS W k O ( t l -

t2).

(3.25)

k

Classically, we merely write i(ol[Dco'(tl), ~ ' o ' ( t ~ ) l , l O ~#c 0, ~

(3.26)

(Ol{D'"'(tl),D'"'(tz)llo>c,= 0.

(3.27)

Let us consider the expression €or spontaneous emission, with the molecules in the ground state, using all four possible types of description, as follows: (i) Both the field and the molecules are described quantummechanically. Utilizing the previously used approximation of retaining only resonant terms in the integrand, we obtain cancellation of the two

VII, § 31

SIhPLE EXAMPLE OF FIELD-ATOMS INTERACTION

425

terms in the integrand of (3.21), so that Espont

= 0.

(3.28)

(ii) Both the field and the molecules are described classically. In this case each term in the integrand of (3.21) vanishes, and Espont

= 0.

(3.29)

(iii) The field is described quantum-mechanically and the atoms are destribed classically. The first term of the integrand in (3.21) is now positive, according to the inequality (3.22), and the second term vanishes, so that (3.30) One notes that cSpont, in this instance, is exactly equal to &ind if we replace U’(O)U(O) by 1/2 in Bind. (iv) The field is described classically and the molecules are described quantum-mechanically. The first term of the integrand in (3.21) vanishes, and the second term produces a positive contribution to the right-hand side, so that Espont’o. (3.31) Since spontaneous emission is independent of the field energy, we apply these results to the case where the field is also in its ground state. In descriptions (i) and (ii), the results are entirely reasonable, since molecules in the ground state cannot emit energy. In descriptions (iii) and (iv), however, the conclusion contradicts the assumption; if the molecules and the field are both in their state of lowest energy, the molecules should not be able to emit or absorb energy. This internal inconsistency may be explained by the statement that, formally, the zero-point oscillation of the quantum-mechanical system is “seen” by the classical system as motion which can do work. Thus, in description (iii) the classical molecules absorb energy from the field as though it were induced absorption from a field with energy of quantum number 1/2. Likewise, in description (iv), the quantum-mechanical molecule appears to be doing work through its zero point oscillation on the classical field. ((Ol{D(”)( tl), D‘’’( t2)}10)may be regarded as a measure of the molecular zero-point oscillation.) Under certain conditions, the expression for cSpontis meaningful for all possible descriptions. From (3.19), these conditions are, clearly, those for which the term containing A is negligible compared to the other terms, in

426


[VII,

D4

which case the formalism used for the field is irrelevant. For (D'"'(t ) ) = 0 or ( y) = 0, such a condition is given by

where the subscript "max" indicates the maximum value during a cycle of tl-t2. This condition may be described as the existence of a large (essentially classical) molecular amplitude of oscillation. It is satisfied, for instance, by a harmonic oscillator in a high energy state (MESSIAH [1961]). If we extend our consideration to the case (D("(t))# 0, ( y ) # 0, the N 2 term will be dominant in E ~ for ~sufficiently ~ ~ large ~ N., Physically this case may be described as that of a large number of molecules oscillating in phase and producing a large (essentially classical) oscillating dipole moment. In other words, the method of description of the field in the expression for spontaneous emission is irrelevant when the oscillating dipole moment is essentially classical. We turn, next, to induced emission. In principle, both &(l) and &ind may be regarded as induced emission? since both quantities depend on the initial field; in usual practice, however, one has &ind in mind when referring to induced emission, and assumes that either ( y ) or (D'"'(t))is negligible. It is seen that neither qndnor E ( ' ) depends on the formalism used to describe the field, but only on the numbers that we associate with either at(0)a(O) or with q")(t).The result for qndwill be identical in both formalisms if, for instance, we say, quantum-mechanically, that the field is in an energy state corresponding to a given energy eigenvalue, or, classically, that the energy of the field has this same value. An identity of results also exists if the prescription of the field is such as to yield a statistical description of the energy, provided that this statistical description is the same in the classical and the quantum-mechanical prescriptions of the field. § 4.

Discussion of the Semiclassical Theories

We return to the consideration of the semiclassical theories described in 3 2 . 4.1. SEMICLASSICAL THEORY I

Classical treatment of the field and quantum-mechanical treatment of the atomic system is the proper analytical method of studying the atomic

VII, § 41

DISCUSSION OF THE SEMICLASSICAL THEORIES

427

behavior, from a quantum-mechanical viewpoint, when the field is prescribed, that is, the behavior of the field is specified without reference to the atoms. If one deals with the case of a strong field where the reaction of the atoms is negligible, then a classical prescription of the field is clearly justified. The field assumes the role of a (time dependent) parameter in either Schrodinger’s or Heisenberg’s equations of motion, and the problem is a computational one only, soluble, in principle, to all orders of perturbation theory. If one deals with the case of a weak field, a classical prescription of the field is justified under certain conditions. Suppose that the question posed does not refer to the effect of the atoms on the field, and the expression for the answer depends on one field variable (the energy, say). Then as seen in § 3, an appropriate classical description of the field will yield the same result as a quantum-mechanical description. An illustration of the first case is the behavior of atoms in a strong laser field. An illustration of the second case is induced emission or absorption up to second order in perturbation theory. (Another illustration, photoelectric detection, will be discussed later.) Although the word “photon” may be used in connection with these phenomena, it should, from a strict point of view, be interpreted as referring to the atom rather than the field. Thus, when one speaks of the absorption of a photon-or a number of photons, in multiphoton phenomena - the statement should be understood as referring to the energy of the atom only, since prescription of the field (independently of the atomic behavior) is inconsistent with calculation - or consideration - of atomic effects on the field. Taking account of the reaction-or the action-of atoms on the field is, in general, logically impossible in SCT I, since quantummechanical atoms radiate a field that must be described, in general, quantum-mechanically. This follows from the fact that the inhomogeneous solution of Maxwell’s equations (which, in operator form, are the Heisenberg equations of motion for the electromagnetic field) is an expression containing the quantum-mechanical matter variables. Furthermore, as has been shown in § 3, a classical field “sees” the zero-point motion of quantum-mechanically described atoms as motion which can do work, a fact that leads to absurd results under certain conditions. Spontaneous emission is describable in SCT I only when the oscillating dipole moment is sufficiently large (in a sense described quantitatively in § 3 ) to be essentially classical. One may enquire if a prescription of the field (during the time of interaction with the atoms) can be given in quantum-mechanical terms.

-128


[VII, § 4

Why cannot the field be prescribed “quantum mechanically” with the statement that it is initially in a given quantum state and its subsequent behavior is unaffected by the atoms? Such a prescription implies that any (ideal) measurement performed on the field -which, presumably, could be carried out by means of the atoms - will also leave it undisturbed. This implication is inconsistent with the uncertainty principle. Thus, if the field is prescribed independently of the atoms during the interaction process, the prescription must be considered to be essentially classical. It need not necessarily be deterministic, however; the prescription may be given in a statistical form. 4.1.1. Photoelectric detection The use of SCT I in photelectric detection has been the object of some criticism which is based on the point of view that one is dealing with a “photon field”, and “there is ultimately no substitute for the quantum [19631). It is instructive to discuss theory in describing quanta” (GLAUBER this subject in some detail. Although there exist several ways of looking at the phenomenon of photoelectric emission (KELLEY and KLEINER [1964], GLAUBER [1964], MANDEL, SUDARSHAN and WOLF[1964], MANDEL and WOLF[1965,1966], SENITZKY [1968b], LAMBand SCULLY [1969]), it is hardly necessary to point out that what one observes experimentally are electrons, and not photons. The fundamental expressions in photoelectric detection should, therefore, refer to the excitation (or ionization) of the atoms rather than to the disappearance of photons. We can use our simple model of § 3 for this purpose, letting the atoms in the cavity be the “detector”. If we consider the emission of a photoelectron as the excitation of an atom to an energy in the neighborhood of Eo + hw, where E, is the ground state energy, then the photoelectric current is given, from (3.18), by

I=

1 d hw d t

Eind

J

= ~ ( y ’ > a ~ ( o ) a ( o dt,i([D“’(t), )

~ ‘ “ ’ ( t ~ ) sin ] , ) w ( t - r,).

(4.1)

0

Assuming a spread in the excited atomic energy levels with a level density r ( E , + h w ) , we obtain, approximately, I = .rrNhy?ai(0)a(O)r(E,,+hw)lD12,

(4.2)

VJI, § 41

DISCUSSION O F T H E SEMICLASSICAL THEORIES

429

where 10)2 is the average of I D O k ) * for Ek near E , + h o . This is the photoelectric current during the time t for which second-order perturbation theory is valid. Setting w = (I/N)t,

(4.3)

and noting that the expectation value in (4.1) came from the averaging procedure implicit in the law of large numbers, we must interpret w as the probability per atom for the emission of a photoelectron during the time 1.

The above theory describes the effect of the field on atoms in a lossless cavity. This theory has been developed further by taking into account cavity losses and the presence of a source that drives the field (SENITZKY [1967, 1968133). The source is assumed to be negligibly affected by the detector, and the spectral width of the field is assumed to be less than that of the atoms. The result for the photoelectric current is formally similar to (4.1), except that the operator at(0)a(O)is now replaced by the c-number a * ( t ) a ( t )which describes the energy of the field generated by the source. (The insensitivity of the source to the detector makes the source appear [1965, 1966, 1967]).) 'We can ''classical'' to the detector (SENITZKY therefore write dw/dt = const. x a*(t)a(t).

(4.4)

In other words, the emission probability per atom per unit time at time t is proportional to the field energy at time t, described classically. One can proceed, using this expression together with the assumption that the atoms emit independently of each other [which is implicit in the factor N of (4.1), and is the result of the use of lowsst order perturbation theory] to calculate the various statistical properties of the photocurrent as a function of the statistical properties of the field, now viewed as a (classical) random process (MANDEL[1958, 19591, MANDELand WOLF [19651). Thus, if the reaction of the photoelectrons on the field is ignored, SCT I is valid, from a quantum mechanical viewpoint, in the analyses of photoelectric phenomena. It should be emphasized that this result follows naturally from the fact that we address ourselves to the question concerning the emission of photoelectrons (MANDEL,SUDARSHAN and WOLF [1964], SENITZKY [1968b], LAMBand SCULLY[1969]) rather than to the

430


[VII, ij 4

question concerning the disappearance of photons (GLAUBER[ 1963, 19641). As is well known, the form of the question, or the choice of variable with respect to which the question is posed, plays a significant role in quantum theory. 4.2. SEMICLASSICAL THEORY I1

In contrast to SCTI, where the mutual interaction between the field and the molecules cannot be described generally in a self-consistent manner, S CT I I provides a prescription for doing so. As mentioned earlier, SCT I1 calls for replacement of the matter variables in Maxwell’s equations by expectation values derived from Schrodinger’s equation. It is useful to outline the method of derivation of these expectation values for the case of an atomic system coupled to a classical field through its dipole moment. Consider, for the sake of simplicity, only a finite number of discrete levels of the atomic system to be of relevance in the interaction. We describe the state of the atom in the Schrodinger picture by (4.5)

Iq(t)) =Ctct (t)Iqt)>

where the lq,)’s satisfy the relationship

HJl%)= ELbA

(4.6)

H , being the Hamiltonian of the free atom. The dipole moment operator associated with the ij pair of levels can be expressed as a linear superposition of the operators

9:; =4(1cp,)(cp,l ’

’:I’

= -ii(l

+lcpCX@,l) ‘ P I >(qZ

I- 1

q I

9:;’= i(lqj)(qj I - Iqz ) ( q t zf

J.

1)

>(qJ

(4.7)

I),

The total Hamiltonian for the atom in SCT I1 is Ha,= Ho + H’,

(4.8)

with

where F:j’’’) is, most generally, a linear superposition of (classical) elec-

VII, 0 41

DISCUSSION OF TH!3 SEMICLASSICAL ‘THEORIES

431

tromagnetic field components that contains appropriate coupling constants. Schrodinger’s equation of motion for the atom can be written (SCHIFF [1949])

with the summation being taken over j only. The expectation value of the dipole moment is given by (9:;))

=+(CTCi

+ C*Cj)

(gf)) = -+i(cTci - ci* ci)

(4.10)

($3:;)) = &CTCj - C * C i ) .

According to SCTII, the dipole moment of the atom that yields directly the polarization vector in Maxwell’s equations (polarization being dipole moment per unit volume) is to be considered a linear superposition of these quantities. Before we discuss the degree of validity of SCTII from a quantum mechanical viewpoint, certain features of SCT I1 that may be regarded as unsatisfactory from a conceptual viewpoint should be pointed out. Firstly, the connection between the classical field equations and the quantummechanical matter equations is made through an ad hoc prescription, namely, “let the matter variables in Maxwell’s equations be replaced by their expectation values”. This prescription does not arise from a formal theory based on conventional laws of physics. Secondly, one can regard this prescription as containing a degree of arbitrariness. Why should the matter variables in Maxwell’s equations be replaced by their expectation values before the equations are solved rather than after the equations are solved? It is clear that the results will, in general, be different. For instance, the solution for radiated field energy from a dipole oscillator into free space depends on the square of the dipole moment (see, for instance, SCHIFF[1949]), while Maxwell’s equations contain the first power of the dipole moment, and as is well known, the expectation value of the square is not necessarily equal to the square of the expectation value. More generally, if one wants to replace the operator 0 by a number, any one of the (generally different) possibilities (On)’”’, n = 1, 2, . . . , is available, and there exists no a priori reason for choosing one rather than another.

432


[VII, § 4

SCT I1 ignores the statistics inherent in a purely quantum-mechanical description. Quantum mechanics is applicable, in principle, not only to microscopic systems, where the macroscopic experimental result is obtained by an averaging process, but also to macroscopic systems. Consider, for instance, the radiation by an excited macroscopic harmonic oscillator. Now, there is nothing to prevent us, in principle, from describing the quantum-mechanical state of this harmonic oscillator (if it is suitably prepared) as an energy state, no matter how high its energy may be. In this case, the expectation value of the matter variables in Maxwell’s equations are zero (since the variables are linear in the harmonic oscillator coordinates) and the radiated field (or energy) obtained is zero! The reason for this incorrect result is evident. An energy state describes a harmonic oscillator only statistically, and, in the limit of high quantum numbers, the energy state describes an ensemble of classical oscillators with precise amplitude and random phase (MESSIAH [1961]). An ensemble average over the coordinates of the oscillator before one calculates the radiated field (or its energy) will therefore yield nothing meaningful. When does SCT I1 provide a useful formalism for the analysis of the interaction between matter and the field? The above discussion serves to motivate part of the answer. If the matter variables occurring in Maxwell’s equations are macroscopic, and have a relatively small quantummechanical dispersion (so that the effect of the purely quantummechanical statistics is insignificant), then the replacement of the matter variables by their expectation values may be regarded as a good approximation. It is instructive to analyze a simple, idealized, example in order to illustrate this point. Consider N identical, but distinguishable, two-level systems with parallel dipole moments. Let the energy states of the ith two-level system be labeled by 11,) and 12,), and the respective energy levels by E l and E,. The total dipole moment along the direction of polarization is given by N

D

=

2 di,

(4.11)

i = l

where di is the dipole-moment operator of the ith two level system. We consider, for simplicity, the case in which di has only off-diagonal matrix elements. Using dimensionless units, we can then set d = 9 y d for each two-level system, so that

di =4(11i)(2iI+ l2i)(1il).

(4.12)

VII, § 41

DISCUSSION OF THE SEMICLASSICAL THEORIES

433

Let the total system be in the state (of the Schliidingerpicture)

n N

PI)=

[c,(t)lll)+ C2(Wl)1?

(4.13)

1=1

where (c112+lc212=1, and t is short compared to the time during which coupling produces significant changes in the atomic state. (This type of state has been referred to as a coherent state, or a Bloch state, in the literature (SENITZKY [1958], ARECCHI,COURTENS, GILMOREand THOMAS [1972]).) Then the expectation value of the total dipole moment is given by ( D )=+N(kTk,e-'"'+ k:kle""'), (4.14) where w = ( E 2 - E l ) / h and k, = c,(O). We also obtain, for this state (SENITZKY [1977])

( D 2 )= $ N ( N - l)(kyk$e-2'"'

+ k;2k:e21u1t

+21k1121k2)2)+$N.(4.15)

One notes that the term gf order N 2 in the expression for ( D 2 ) is precisely (D)', and that ( ( D 2 ) - ( 0 ) 2 ) / ( D 2N-l. )~ (4.16) Thus, for a large number of two-level systems described by the state /TI), the statistical spread in the dipole moment introduced by the quantum mechanics is negligible, and replacement of the dipole moment (or polarization) by its expectation value in Maxwell's equations is a good approximation. Consider now the situation in which the two-level systems are described by the state (4.17) where P is the operator that permutes the atomic indices so as to produce different product states, and the summation is to be taken over all such permutations, of which there are N ! [ N - r)!r!]-'. One may describe as an energy state that is symmetric with respect to the interchange of any pair of atoms. (In the literature it has been described as a symmetric Dicke state (SENITZKY [1958], ARECCHI, COURTENS, GILMORE and THOMAS [1972]).) Its energy eigenvalue is rE,+(N- r)E2.For this state, we have (SENITZKY [1977]) PIIIDlTd= 0, (4.18) (~IllD21'P~J = $ [ r ( N - r) +*N].

(4.19)

434


[VII. 5 5

One sees that the (quantum-mechanical) statistics associated with are such that the dispersion of the total dipole moment is of the same order of magnitude as the dipole moment itself. The replacement of the dipole moment of two-level systems in the state I*I ) by its expectation value in Maxwell’s equation is, therefore, unjustified, and will yield qualitatively incorrect results. It is interesting to note that for r = Nlk,12 the term of order N 2 in the time average of ( ~ I ~ D 2 ~ is *equal I ) to the term of order N 2 in (UI,(D’l*II), so that under certain conditions, substitution for the classical 0’ of the quantum-mechanical (0’)is a more meaningful procedure than substitution of (0)’.

4.3. SEMICLASSICAL THEORY I11

SCT 111 is closely related to SCT 11. In fact, it is a formulation of SCT I1 that constitutes a self-consistent classical dynamical theory without any ad hoc prescriptions. SCT I11 achieves the same result as SCT I1 by describing the field classically and by introducing a classical atomic model. An appropriate definition of dipole moment makes it equivalent to the corresponding quantum-mechanical expectation value. In order to demonstrate how SCT I11 produces this result, we refer to the “canonical” equations of motion (2.5) for the atomic variables a, and a: introduced in the definition of SCT 111. Explicit differentiation of the Hamiltonian yields, as the atomic equations of motion, equations for the a,’s that are identical to those for the c,’s, eqs. (4.9), in SCTII. Furthermore, the expresion for the dipole moment in terms of the d!;”’s of SCTIII is exactly the same as the expression for the expectation value of the dipole moment in terms of the (9;;”’)’s of SCT 11. Thus, the “dipole moment” of SCTIII is identical to the “expectation value of the dipole moment” of SCT 11, since, as pointed out in 0 2.3, the corresponding initial values are the same. From a conceptual viewpoint, SCTIII has the advantage of avoiding the ad hoc prescription connecting the classical description of the field with the quantum-mechanical description of the atoms.

9 5. Classical Limit of Quantum-Mechanical Radiation Theory If SCT is viewed as an approximation based on quantum theory, it should be possible to derive its formulation directly from quantum theory

VII,

5 51

CLASSICAL LIMIT OF QUANTUM-MECHANICAL RADIATION THEORY

435

for those conditions for which it is a valid approximation. We do so in the present section, following SENITZKY [ 1974, 19771. In addition to furnishing insight into the validity of SCT, this derivation yields a theory that contains new features.

5.1. BOSON-SECOND-QUANTIZATION FORMALISM

We begin by considering a number of identical atoms that are characterized by their spectrum and associated dipole-moment strengths. (The dipole moments may be either electric or magnetic.) Let the relevant spectrum consist of energy levels hw,, i = 1 , 2 , . . . n. Although the atoms are, in principle, distinguishable, we assume that they couple to the electromagnetic field identically, so that, as far as their effect on the field is concerned, they are indistinguishable. A boson second quantization (BSQ) formalism will be introduced to describe the state of these atoms. The space in which these states are described is spanned by the orthonorma1 basis vectors Ir, . . . r , . . . rn),

(5.1)

where the ri’s are non-negative integers. The fundamental operators, in terms of which all dynamical variables are constructed, are ui(t)and a t ( t ) (we use the Heisenberg picture, unless stated otherwise), such that

ai(0)lrl . . . ri . . . r,,) = r!’*Irl . . . ri .- 1 . . . rn), u!(0)lrl

. . . ri . . . r,) = (ri + I)”*Irl . . . ri + 1 . . . r,).

(5.2)

(5.3)

The operator commutation relationships are

(5.4) with all other equal-time commutators vanishing. The Hamiltonian which describes the collection of atoms is given by

One sees immediately that ai and a f are the usual annihilation and creation operators associated with a harmonic oscillator of frequency mi. The dipole moment of the entire collection is a linear superposition of the

436


[VII, 8 5

operators

For brevity, the d$,!’”’s, rn = 1 , 2 , 3 , will be referred to as the dipole moment components associated with the frequency lwi - mil. The coupling between the atoms and the electromagnetic field - assumed to be described quantum-mechanically, at this point - is given most generally by the interaction Hamiltonian (5.7) where the fiy)’~are linear superpositions of the components of the electromagnetic field multiplied by appropriate coupling constants. For the total Hamiltonian, we have

H

= Ho

+ H’+ Hf,

(5.8)

Hf being the field Hamiltonian (which we do not need in explicit form). For later discussion, it is convenient to introduce the “reduced” variables A,(t), Ar(t), defined by the relationships a j ( t )= Ai(t)epiwl‘,

a:(t) = A;(t)elw1’.

(5.9)

One verifies easily that the reduced variables are constants in the absence of coupling to the field, and vary slowly (compared to the natural atomic oscillations) for sufficiently weak coupling, the only kind of coupling to be considered. The basis vectors l r l . . . r,) are, clearly, eigenvectors of the occupation number operators ni(0),defined by

n. = ata. = A . 1

t

(5.10)

and satisfy the eigenvalue equation, ni(0)lvi... r n ) = r i l r i . . . r,).

(5.11)

They are also eigenstates of Ho, and are, therefore, energy states of the free (uncoupled) collection of atoms. It can be shown (SENITZKY [1974]) that, with the above interaction Hamiltonian, the operator Zini is a

VII, 8 51


437

constant of motion. All the energy states are eigenstates of this operator,

(c 1

ni Ir, . . . r,,) = Nlr, . . . rn),

(5.12)

where N

=

Cri, I

and those corresponding to a given N form a subspace which is invariant under the transformation of the total Hamiltonian. We describe the collection of atoms under consideration by a state (the initial state, if the Schrodinger picture is used) which lies in the subspace for which N is the total number of atoms. We obtain, thus, a BSQ formalism in which the atoms may be considered to be the bosom. From a first-quantization viewpoint, this means that the states used are fully symmetric with respect to the atoms, and from a physical viewpoint, this means that a situation of maximum atomic cooperation (or complete similarity of atomic behavior) is being considered. The operator n, now corresponds to the number of atoms (regardless of their identity) found in the ith (one-atom) energy state by an experiment designed to yield such a result, and the fact that Zini is a constant of motion merely means that the total number of atoms is invariant under interaction with the field. In addition to the set of energy states labeled by the integers rl, . . . r,,, it is useful to define a set of “coherent” states labeled by the complex constants c l r .. . c, (which we normalise by 1 1ciI2= 1 and designate collectively by {c}) defined as follows:

. . c : [ r l . . . r,,),

(5.13)

where the parenthetical superscript ( N ) in the summation indicates that the summation is to be taken over all values of rl, r,, . . . r,, for which

C ri = N. I

A t present, we merely note the properties (5.14) Certain other properties and the significance of the coherent states will be

438


[VII, P 5

discussed later. (Coherent states have been discussed in various contexts by SENITZKY [1958], ARECCHI,COURTENS, GILMOREand THOMAS[1972], GILMORE [1972], ARECCHI, GILMORE and KIM [1973], GILMORE,BOWDEN and NARDUCCI [1975], SENITZKY [1977].) 5.2. CLASSICAL LIMIT (SEMICLASSICAL THEORY IV) AND RELATIONSHIP TO SEMICLASSICAL THEORIES I1 AND I11

As is well known, a boson formalism is particularly suitable for passage to the classical limit. (One might remark that this is the reason, from a quantum-mechanical viewpoint, for the success of the classical theory of the electromagnetic field in contrast with the classical theory of the atom.) It should be noted that our collection of atoms has been described in the BSQ formalism by y1 harmonic oscillators, the operators a, and a: being the complex amplitudes

a, = 2-l”(qJ

+ ip,),

a: = 2-”2 (qJ-ipJ)?

(5.15)

where q, and p, are the dimensionless coordinate and momentum of the jth oscillator, satisfying the commutation relationship [q,, Pa1 = i,

(5.16)

and in terms of which the Hamiltonian of the jth oscillator is expressed by H, =+hw,(q;+p;)= hw,(n,+i).

(5.17)

Now, the conditions under which a harmonic oscillator may be regarded as classical for all purposes are well known. The traditional description of these conditions is the Correspondence Principle, which presents them as t h e limit of high quantum numbers. We obtain these conditions in quantitative form for the present case in the following argument. The essential difference between the quantum-mechanical description and the classical description of the harmonic oscillator is the fact that a, and a! are (noncommuting) q-numbers rather than (commuting) cnumbers. If, however, the numbers associated with a,a: and a:a, are approximately the same, that is, if [a,, a:] is relatively negligible, then a, and a: may be treated as c-numbers and the description becomes classical. This is indeed the case when the numbers associated with the operator n, are large compared to unity. What are these numbers? If our system is in an energy state, then, clearly, the only number associated with n, is its eigenvalue r,. If the system is not in an eigenstate of n,,then there appears to be more than one number that is associated with n,.For

VII, 5 51

CLASSICAL LIMITOF QUANTUM-MECHANICAL RADIATION THEORY

439

instance, a coherent state consists of terms which belong to eigenvalues of ni that range from 0 to N. These terms are weighted by their coefficients, however, and a reasonable estimate of the significance of the various eigenvalues of ni is provided by the quantity (q). We will use this quantity as a measure of the numbers associated with ni.Our criterion for treating the oscillator classically is, therefore, the following: if ( n , )>> 1, a, and a: may be treated as c-numbers, and the ith harmonic oscillator may be regarded as classical. In order that the entire system may be regarded as classical, this criterion must be satisfied by all the oscillators:

( n , ) >> 1,

i = 1, . . . a.

(5.18)

It follows immediately, of course, that the condition N >. 1,

(5.19)

must also be satisfied. For clarity, the symbol tilde will be used henceforth over a variable where it is necessary to indicate that it is being treated classically (that is, as a c-number). When the inequalities (5.18) are satisfied, Ho can be regarded as the classical Hamiltonian of the free (uncoupled) collection of atoms, expressed either in terms of the complex variables ii,, 6: or in terms of the (dimensionless) coordinates and momenta iji and pi,and is the same as the Hamiltonian of n classical harmonic oscillators. (It should be noted that h appears in Ho only for dimensional reasons.) The time variation of any variable is obtained from the Hamiltonian by the usual Poisson-bracket relationship. The Poisson bracket expressed in terms of partial derivatives of ij, and pi must be written so as to take into account the fact that these are dimensionless coordinates. Thus, for any variable X ( t ) , in the present notation, the equation of motion is (5.20) if X is considered to be a function of the Q’s and

ax aH

aH

ax

AS,or (5.21)

if X is considered to be a function of the Gi’sand 5:’s. The equations for iij and ii? become (5.22)

440


[VII, 5 5

which, for the free collection of atoms, lead to %+ = /pelO,f, 6J = AJ e-’“,‘, J J

(5.23) are now classical constants, obeying the inequality where AJ and lAJl*>> 1. The dipole-moment components are, in this case, classical quantities, two of which, d!:) and d::), oscillate with the frequency Iw, - wJI. It will be shown later that the electromagnetic field interacting with a classical atomic system may be treated classically, so that the fI;”s in the interaction Hamiltonian (5.7), as well as the variables of the field Hamiltonian, may now be regarded as classical variables. The classical limit of the BSQ formalism gives us, therefore, a self-consistent classical radiation theory for a number of n-level atoms interacting with the electromagnetic field. We refer to this theory as SCTIV. Comparison of SCTIV with SCTIII shows that the equations of motion for the dynamical variables (GI,67, and the field variables) are identical in both theories. It might therefore appear, at first glance, that we have derived the equivalent of SCT I11 (and also SCT 11, for which SCTIII constitutes a Hamiltonian formalism) by means of the above limiting procedure, provided, of course, that the conditions for this procedure are satisfied by SCT 111. Further inspection reveals, however, that SCT IV is more general than SCT 111, the reason being the specification of initial conditions. The initial conditions in SCT I11 are specified by identification of the initial values of the variables with the corresponding initial quantum-mechanical expectation values. In SCT IV, however, the initial values are automatically determined by the limiting procedure in a manner to be described below, since passage to the classical limit, which converts q-numbers into c-numbers, does not allow an arbitrary prescription of initial values. These must come from the quantum-mechanical specification of initial values, that is, from the quantum state of the system. Now, description of a system by means of quantum states is statistical. It is important to note that some statistical properties do not disappear, generally, even in the classical limit. Consider, for instance, the energy state (rl . . . r,,). According to the previous discussion, the system described by this state may be treated classically (at least initially, if it is interacting with other systems, and for all time if it is free) provided

r,;.>l,

t = 1 , . . . n,

(5.24)

a condition we assume. The relationships

. . . r,,lai(rl. . . r,)= 0 , t ( I I . . . rnlaiailrl.. . r,)= r,, (rl

(5.25)

VII, 8 51

CLASSICAL LIMITOF QUANTUM-MECHANICAL

RADIATION THEORY

441

express the fact that the average of 6 is zero, and the average of (ii(' is ri. The description must be interpreted as being a (classical) statistical description of 6 ; in other words, ii must be regarded as a random variable. Thus, in the classical limit, the quantum-mechanical variables become classical random variables, with their initial values specified by probability distributions - or some other method of statistical description - determined by the quantum state which describes the system (the initial quantum state, if the Schrodinger picture is used). The connection between the statistical description and the quantum states will be discussed later. Presently, we merely note that this is not the conventional SCT approach. As mentioned previously, in both SCTII and SCT 111, the classical variables are equated to the respective quantummechanical expectation values. In those cases where the quantum state is reasonably deterministic (that is, the statistical spread of the dipole moment is relatively negligible) SCT I1 and SCT I11 are essentially equivalent to SCTIV, if the classical-limit conditions are met. However, in those cases where the quantum state is not sufficiently deterministic, SCT I1 and SCT 111- as pointed out in 4 4 - cannot be used, while SCT IV remains applicable. In order to complete the formulation of SCTIV, we must set up a method of prescribing the initial classicaI variables iii(0) and 6T(O) (from which all other initial variables can be constructed) from information furnished by the (initial) quantum state. A random variable may be described completely by a probability distribution; it may also be described completely by an appropriate infinite set of moments. The two methods of description are related, of course (KATZ[1967]). Our method of description will consist of specifying a finite number of moments, which makes the description somewhat incomplete. For sufficiently large values of the (ni)'s, however, the moments that can be specified are sufficient to make this limitation insignificant. The variables Gj(O) and ZT(0) are identical with /$(O) and Ay(O), and if the collection of atoms is free (uncoupled to the electromagnetic field), the time argument can be omitted in the reduced variables, since they are constant. For simplicity of notation, we consider such a free collection. Let it be described quantum A statistical description of the variables Aj mechanically by the state (9). is furnished by the specification of the moments (5.26)

for all integral vi and wi, where the average is understood to be taken

442


[VII, 0 5

over an ensemble of atomic collections (that is, an ensemble of N-atom systems) associated with the statistical description of the collection. Such an ensemble is also involved in the quantum-mechanical description of the collection of atoms, or in the interpretation of the quantum state of the system. It is most reasonable, therefore, to set each moment equal to the corresponding quantum-mechanical moment with respect to the state under consideration. In general, there are many “corresponding” quantum-mechanical moments, since all ordering arrangements of the classical variables constitute the same moment, while different ordering arrangements of the quantum-mechanical variables may yield different values. In the present instance of the classical limit, however, we are dealing with a situation in which the commutators are considered relatively negligible, so that all ordering arrangements of the quantummechanical variables yield approximately equivalent results. Implicit in this approximation is the assumption u j + wj,

(5.36)

where 6, has a uniform probability distribution P ( e j )between 0 and 27r,

(5.37)

p(eJ = ( 2 7 ~ - l ,

and all the Ai’s are independent random variables, the joint probability, P(&, 62,. . 0,) being

.

p(el, e, . . . en) = p(e,)p(e2) . . . p(e,). For a coherent state the probability distribution for Aj = N1”cieie,

Aj may

(5.38) be given by

(5.39)

where 8 has a uniform probability distribution between 0 and 27~,

p(e)=(27r)-1,

(5.40)

but in contrast to the case of the energy state, 8 is the same for all Aj’s. The A,’s are thus dependent random variables; if the phase is specified for any Aj, it is determined for all Aj’s. From the above distributions, we can obtain distributions for all dynamical variables. It is easy to see why (d:;))av and (d$f))av vanish for energy states and do not vanish for coherent states. The phase of the dipole moment (at any one time) is uniformly distributed over the ensemble of N-atom systems associated with the statistical description of the energy state, while it is identical for all members of the ensemble associated with the coherent state. If we take rj = NIcj(’,

(5.41)

the amplitude of the oscillating components of the dipole moment associated with any one frequency has the same magnitude in both ensembles. It is seen that SCT I 1 and SCT I11 are restricted forms of a classical limit theory that are applicable to those quantum-mechanical descriptions that become entirely deterministic in the classical limit, but inapplicable to those quantum-mechanical descriptions that become statistical (in the sense of non-deterministic) in the classical limit. The coherent states are an example of the former, and the energy states are an example of the latter. In the discussion of the classical limit of the BSQ formalism, we began with a quantum-mechanical description of both atoms and field, and

VII, 8 51


445

examined the conditions under which the description of the atoms becomes essentially classical. What can be said about the field under the same conditions? The solution of the equations of motion for the fieldMaxwell’s equations - indicates that classical sources produce a classical field, given by the inhomogeneous part of the solution, and containing source variables only. Thus, in a quantum-mechanical description of the field, only the homogeneous solution can remain quantum-mechanical when the sources become classical. If the homogeneous solution contains a part that is due to external sources, this part is prescribed, and is, therefore, also classical, in accordance with our analysis of prescribed behavior in the discussion of SCTI. The only remaining part of the solution is the zero-point field. We have seen in the example of 0 3 that in order to have a meaningful radiation theory involving the interaction between the field and classical systems, we must assume that the classical systems do not “see” the zero-point field, since, to such systems, the zero-point oscillation appears as oscillation which can do work. Thus, if our interest lies in the atoms and in the field with which they interact or which they generate, then a classical description of the atoms validates a classical description of the field. 5.3. ADDITIONAL REMARKS

It should be recognized that the condition for the validity of SCTIV, namely, that all the (nj)’s be sufficiently large, involves quantities that change with time, since the (nj)’s are time dependent when the atoms interact with the field. It may, therefore, happen that the condition will be satisfied at certain times and not satisfied at other times, for a given system of atoms and field. Such cases are well known, and have been investigated by a combination of quantum theory and SCT, the former being used during the time when the latter is inapplicable (SENITZKY [1972]). On the other hand, if the time during which SCT is invalid is sufficiently short compared to the time during which the system undergoes a significant change according to quantum theory, then one may reasonably expect SCT to be a good approximation throughout. The time during which the system undergoes a significant change depends on the state of the system, of course. If the atoms are in the ground state, that is (al)= N, (n,)= 0 , its 1, and the field is in the ground state, then the system undergoes no change at all. This is obviously a trivial case, and its method of description is inconsequential. One could

446


[VII, (i 5

claim, in this instance, that the system can be described classically, for N >> 1, since all levels other than the first may be considered irrelevant, and the “relevant spectrum” consists of one level for which the requirements for a classical description are formally met. Consider, now, the case in which one of the higher atomic levels is populated, the lower ones are empty, and the field is again in the ground state. In this case, the lower levels are relevant, since they participate in spontaneous emission (assuming that the transitions are allowed), and the requirement for the validity of SCTIV is not met. If, however, with an upper atomic level occupied and the lower ones empty, the field is not in the ground state, and is sufficiently strong so that its action on the atoms will fill up the lower levels much faster than spontaneous emission, then the time during which SCT is invalid is sufficiently short to be ignored. This example also serves to illustrate the fact that the “relevant spectrum” used to define the classical atomic model depends not only on the real atom but also on the interaction under consideration, and must be chosen accordingly. Since SCT I1 and SCT 111 are special cases of SCT IV, the conditions for the validity of SCT IV also apply to SCT I1 and SCT III. However, in addition to the requirement that all the (n,)’s be large compared to unity, the requirement that the collection of atoms under consideration be in a fully symmetric state (in a first quantization formalism), implicit in the fact that SCTIV was derived from a BSQ formalism, must be stated explicitly as a requirement for the validity of SCTII and SCTIII. The physical significance of the symmetrization requirement may be understood, intuitively, as a requirement that all the atoms behave similarly (SENITZKY [1974, 19771). One may, of course, divide a number of atoms into separate collections, and treat each collection as bosons of a different kind. For instance, in the semiclassical laser analysis by LAMB[1964], a number of atoms distributed among two levels is divided into two collections, each one of which consists of atoms that are in the same one-atom energy state. SCTII and SCTIII must also satisfy the additional requirements - which is responsible for the fact that they are special cases of SCT IV - that the quantum-mechanical description of the dipole moment be deterministic (within the limits of quantum-mechanics), that is, that the quantum-mechanical dispersion in the dipole moment must be relatively small. This requirement does not apply to SCT IV. Needless to say, the validity conditions do not permit the application of SCT 11-IV to a single atom, an application that has been suggested in the case of

VII]

REFERENCES

447

SCTIII (JAYNES [1973]) as part of a proposal to make it a fundamental microscopic theory. The above discussion concerning the requirement for the validity of SCT I1 and SCT 111 needs an additional comment for completeness. Logically, it has been shown only that these “requirements” are sufficient conditions, from a quantum-mechanical viewpoint. In other words, we found conditions under which the theories are valid. Are these, however, necessary conditions? In order to answer this question, we consider SCTIII only, since this may be taken to be the formal dynamical theory that yields SCT 11. Let us begin with SCT 111 in its Hamiltonian form as a classical theory for describing atoms interacting with the electromagnetic field, and convert it into a quantum-mechanical theory. Using the conventional methods of passage from a classical Hamiltonian description to a quantum-mechanical Hamiltonian description, we obtain precisely the present BSQ formalism. One sees that this formalism is the quantummechanical version of SCTIII. Thus, there can exist no other quantummechanical formalism the classical version of which is SCT 111. In other words, if SCTIII is to be derived as the classical limit of a quantummechanical description, this description must be that of the BSQ formalism. The quantum-mechanical requirements for the validity of SCT 111 are, therefore, necessary and sufficient. Strictly speaking SCT 111 and SCT IV are not semiclassical theories at all, but completely classical theories. However, the atomic model in these theories is of a schematic nature, and is derived as the classical limit of a quantum-mechanical description. (Note that, in a sense, this is the reverse of the usual procedure followed in obtaining a quantum-mechanical model.) This classical model is, therefore, not divorced from its quantummechanical origin, and may be considered to be only “semiclassical”. References ARECCHI, F. T., E. COURTENS, R. GILMORE and H. THOMAS, 1972, Phys. Rev. A6, 2211. ARECCHI, F. T., R. GILMORE and D. M. KIM, 1973, Lett. Nuovo Cim. 6, 219. BOYER,J. H., 1969, Phys. Rev. 182, 1318, 1374. BOYER,J. H., 1970, Phys. Rev. D1, 1526, 2257. CRISP,M. D. and E. T. JAYNES, 1969, Phys. Rev. 179, 1253. EBERLY,J . H., 1976, in: Physics of Quantum Electronics, Vol. IV, eds. S. Jacobs, M. Sargeant and M. 0. Scully (Addison-Wesley, Reading, Mass.) p. 421. GILMORE, R., 1972, Ann. Phys. (N.Y.) 74, 391. GILMORE, R., C. M. BOWDEN and L. M. NARDUCCI,1975, Phys. Rev. A 12, 1019.

448

SEMICLASSICAL. RADIATION THEORY

[VII

GLAUBER, R. J., 1963, Phys. Rev. Letters 10, 84. GLAUBER, R. J., 1964, in: Quantum Optics and Electronics, eds. C. de Witt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York) p. 65. HAKEN,H., 1970, Handbuch der Physik, Vol. XXV/2c (Springer-Verlag, Berlin). HEITLER,W., 1954, The Quantum Theory of Radiation (3rd ed., Oxford University Press, London) ch. I. JAYNS, E. T., 1973, in: Coherence and Quantum Optics, eds. L. Mandel and E. Wolf (Plenum Press, New York) p. 35. KATZ,A., 1967, Principles of Statistical Mechanics (W. H. Freeman and Co., San Francisco and London) ch. 4. KELLEY,P. L. and W. H. KLEINER,1964, Phys. Rev. 136, A 316. LAMBJr., W. E., 1964, Phys. Rev. 134, A 1429. LAMBJr., W. E. and M. 0. SCULLY,1969, in: Polarization: Matiere et Rayonnement (Presses Universitaires de France, Paris) p. 363. MANDEL,L., 1958, Proc. Phys. SOC.(London) 72, 1037. MANDEL,L., 1959, Proc. Phys. SOC.(London) 74, 233. MANDEL,L., 1976, in: Progress in Optics, Vol. XIII, ed. E. Wolf (North-Holland Publishing Co., Amsterdam) p. 27. and E. WOLF,1964, Proc. Phys. SOC.(London) 84,435. MANDEL,L., E. C. G. SUDARSHAN MANDEL,L. and E. WOLF, 1965, Rev. Mod. Phys. 37, 231. MANDEL,L. and E. WOLF, 1966, Phys. Rev. 149, 1033. MARSHALL,J. W., 1963, Proc. Phys. SOC.(London) A276, 475. MARSHALL, J. W., 1965, I1 Nuovo Cim. 38, 206. MESSIAH,Albert, 1961, Quantum Mechanics (North-Holland Publishing Co., Amsterdam) ch. XII, Ex. 4. MILONNI,P. W., 1976, Phys. Reports 25, 1. ScHiw, L. I., 1949, Quantum Mechanics (McGraw-Hill Book Co., New York) p. 240, p. 190. SENITZKY, I. R., 1958, Phys. Rev. 111, 3. SENITZKY, I. R., 1965, Phys. Rev. Letters 15, 233. SENITZKY, I. R., 1966, Phys. Rev. Letters 16, 619. SENITZKY, I. R., 1967, Phys. Rev. 155, 1387. SENITZKY, I. R., 1968a, Phys. Rev. Letters 20, 1062. SENITZKY, I. R., 1968b, Phys. Rev. 174, 1588. SENITZKY, I. R., 1972, Phys. Rev. A 6, 1175. SENITZKY,I. R., 1974, Phys. Rev. A 10, 1868. SENITZKY, I. R., 1977, Phys. Rev. A 15, 284. STROUDJr., C. R. and E. T. JAYNES, 1970, Phys. Rev. A 1, 106.

AUTHOR INDEX A h m o ~C., I., 90, 92, 98, 113, 116 ~ O U A F-MA R GU IN, L., 45, 67 ADAMS,F. D., 279, 286 AKAHORI,H., 122, 207, 229 ALCOCK, A. J., 396, 398, 410 ALDRIDGEIII, J. P., 4, 22, 65 ALEKSOFF, C. C., 264, 286 ALLEBACH, J. P., 122, 229 ALPHONSE,G. A,, 70, 115 AMSARTZUMIAN,R. V., 4, 15, 18, 20, 22, 25, 27, 28, 29, 31-39, 49, 50, 52, 56, 57, 61, 65, 66, 68 AMIN,F. A. A,, 278, 287 ANDERSON, D. R., 207, 229 ANDERSON, G. B., 122, 229 ANDERSON, R. W., 19, 67 ANDREEV, S. I., 364, 367, 372, 373, 374, 381, 382, 383, 397, 398, 399, 408 ANDREEV, S. V., 59, 66 H. C., 184, 229 ANDREW, ANTONOV, A. V., 403, 408 ANTONOV, V. S., 59, 60, 66 APATIN,V. M., 28, 29, 56, 57, 65 APOSTO, D., 115 ARCHBOLD, E., 243, 247, 248, 251, 269, 273, 276, 278, 285, 286, 287 ARECCHI, F. T., 433, 438, 447 ARNOLDI,D., 44, 45, 66 ASAKURA, T., 232, 240, 287 ASTOIN,N., 389, 408, 411 AUMONT,R., 374, 381, 385, 408

B BA~COCK, R. V., 393, 408 BADER,T. R., 115 BADGER,R. M., 43, 66 BAGRATASHVILI, V. N., 40, 41, 66 BAKER,H. J., 400, 408

BALKIN,V. I., 30, 43, 66 BALKANSKY, M., 102, 103, 104, 116 BALLOFFET,G., 374, 382, 384, 385, 389, 390, 391, 408, 410 BARANOV, V. Yu., 40, 41, 66 BARANOVSKI, A. P., 30, 66 BARKER,D. B., 280, 286 BARNES, R. H., 361, 364, 365, 374, 375, 377-380, 382, 386, 387, 388, 402, 408 BARNET.M. E., 115 BASHKIN, A. S., 404, 408 BASOV,N. G., 16, 45, 66, 403, 404, 408 BAZHIN,N. M., 29, 67 BAZHUTIN, S. A,, 30, 67 BECKER,H., 229 BEKOV,G. I., 25, 49, 58, 65, 67 BELENOV,E. M., 16, 45, 66, 67 BELOUSOVA, I. M., 397, 408 BENNETT,J. J., 115 BENNEXT, V. P., 124, 126, 195, 232 BERRY,D. H., 124, 175, 231 BERTOLOTIT, M., 115 BERTSEV, V. V., 363, 408 BESTE,D. C., 229 BEVEIUY 111, R. E., 361, 364, 365, 367, 374, 375, 377-380, 382, 385-388, 402, 408 BmERMAN, L. M., 376, 409 BIEDERMANN, K., 264, 286 BILLINGS, B. H., 42, 67, 101, 116 BIRCH,K. G., 126, 229 BIRELY,J. H. 4, 22, 65 BLOEMBERGEN, N., 35, 38, 67 BLUME,H., 115, 118, 126, 232 BOCKASTEN, K., 389, 409 BOGEN,P., 384, 391, 392, 409, 410 BON,M., 390, 409 BOONE,P. M., 280, 281, 286 B o ~ t s o v V. , M., 394, 409 BORN,M., 311, 355

449

450

AUTHOR WDEX

BOROVICH, B. L., 403, 409 E., 390, 409 BOURSEY, BOWDEN, C. M., 438, 447 BOWEN,E. J., 13, 67 BOYER,J. H., 418, 447 A,, 213, 229 BRAMLEY, J. I., 21, 67 BRAUMAN, BRAUNECKER, B., 229 BROADBEN, D. K., 100, 116 BROUSSAUD, G., 110, 116 BROWN,B. R., 82, 86, 116, 122, 124, 125, 174, 178, 229 BRYNGDAHL, O., 114, 126, 144, 152, 195, 199, 212, 228, 229, 231, 295, 311, 319, 355 BUGER,Z., 269, 287 BURCH,J. J., 123, 125, 136, 137, 229 BURCH,J. M., 241,243,247,248,266, 281, 282, 286 BURCKHARDT, C. B., 84, 116, 123, 143, 229, 316, 355 R., 398, 409 BURNHAM, BUTTERS,J. N., 248, 249, 254, 257, 258, 259, 262, 264, 265, 286, 287 C CMANNES, F., 378, 409 A., 30, 66 CABELLO, E., 91, 103, 104, 105, 116 CAMATINI, CAMPBELL, D. K., 228, 232 K., 126, 141, 229, 232 CAMPBELL, CAMUS,J., 91, 116 CANTRELL, C. D., 4, 22, 38, 65, 67 CARLSON, L. R.,25, 49, 69 CARNEY,J. A., 97, 116 CARTER, W. H., 229 CARTWWGHT, D. C., 4, 22, 65 CASASENT, D., 228, 229, 230, 291, 294, 296, 298, 299, 301, 306, 307, 311, 313, 316-319, 321, 323, 325, 326, 327, 329, 333, 335, 337, 338, 339, 341, 342, 344, 348, 350-353, 355, 356 CATHEYJr., W. T., 230 H. J., 291, 355 CAULFIELD, CEGLIO,N. M., 396, 409 CHAMPAGNE, E., 220, 230 J., 378, 409 CHAPELLE, CHAU,H. M., 220, 230 CHAVEL,P., 182, 230, 231 V. P., 56, 68 CHEBOTAYEV,

CHEKALIN, N. V., 18, 20, 28, 31, 32, 33, 39, 40, 65, 66, 67 L. L., 397, 408 CHELNOKOV, CHEN,W., 313, 315, 356 CHIANG,F. P., 278, 280, 281, 286, 287 CHU, D. C., 123, 124, 125, 149, 151. 230 A,, 92, 116 CHUTJIAN, CLAIR,J. J., 89-92, 98, 113, 116, 230 CLARK,L. B., 52, 67 CLARK,J. H., 30, 66, 67 CLARK,M. D., 45, 69 CLARK,R., 91, 116 P. J. B., 111, 116 CLAFUUCOATS, CLOUD,G., 279, 287 D., B., 97, 116 COBL~Z M. J., 35, 67 COGGIOLA, COLLIER,R. J., 92, 116 P., 110, 116 COMPAAN-KRAMER, CONRADS, H., 384, 391, 409 COOPERJr., K. W., 360, 409 W. D. M., 116 CORNISH, A,, 77, 116 COUDERC, E., 433, 438, 447 COURTENS, CRILLO,A., 111, 112, 116 CRISP,M. D., 417, 447 CROCE,P., 78, 111, 117 CLLVER, B. C., 230 L. J., 78, 111, 116, 186, 230 CUTRONA, CZERNY,P., 101, 116 D

DALLAS, W. J., 89, 116, 122, 168, 207, 229, 230 DAMANY, H., 390, 409 DAMANFASTON, N., 390, 409 H., 230 DAMMANN, E. V., 364-370, 372, 373, 374, DANI~L, 381, 382, 383, 385, 387, 398, 399, 408, 409, 411 DANIEL,I. M., 256, 287 DASHIEL,S., 295, 356 DASHUK,P. N., 363, 364, 397, 406, 408, 409 DATTA,S., 19, 42, 67, 68 D'AURIA,L., 90, 117, 169, 230 DE BACKER, L. C., 278,281, 286, 287 DEBRUS,S., 257, 287 DEMIDOV, M. I., 383, 410 DENBY,D., 259, 262, 287 DENES, L. J., 394, 395, 396, 409, 410

AUTHOR INDEX

DEPALMA, J. J., 114 DEWEY,H. J., 43, 68 DJEU, N., 398, 409 DJIDJOEV, M. s., 16, 67 J. A., 77, 116 DOBROWOLSKY, V. S., 20, 31, 33, 39, 40,65, 67 DOUIKOV, DONDES,S., 42, 67, 68 H., 45, 67 DUBOST. DUBRIN, J. W., 15, 23, 69 T. W., 25, 49, 67 DUCAS, DUFFY,D. E., 253, 287 DUNN,O., 42, 67 DUPOISOT, H., 90, 116 J., 230 DYSON,

45 1

FORRE, B. M., 262, 287 FORSEN, H. K., 23, 24, 67 M. E., 280, 286 FOURNEY, M., 275, 286, 287 FRANCON, G. F., 115 FRAZIER, R. R., 25, 49, 67 FREEMAN. J., 98, 116 FRWLICH, FREUND, S. M., 45, 67 A. A., 101, 116, 230 FRIESEM, F R ~ G E F. L ,B. A., 359, 384, 409 FUJI,H., 240, 287 E., 360, 374, 409, 410 F~TNF-ER, A,, 326 327, 329, 355 FURMAN, N. P., 18, 36, 37, 39, 49, 50, 56, FURZIKOV, 57, 65, 66

E EBERLY, J. H., 417, 447 H. E., 360, 409 EDGERTON, EK, L., 264, 286 EK, L. E., 262, 287 ELTON.R. C., 405, 411 EMMONY, D. C., 77, 116 A., 205, 228, 230 ENGEL, A. E., 243,247,248,251,269,273, ENNOS. 276, 278, 285, 286, 287 ESEVA,J.-M., 390, 409 W. T., 278, 287 EVANS, T. U., 61, 69 EVLASHOVA,

F FATEEV, V. A,, 91, 117 FAULDE, M., 230 P., 360, 409 FAYOLLE, A,, 114 FEDOTOWSKY, N. B., 45, 66, 67 FERAPOIWOV, FERCHER, A. F., 126, 230 A,, 115 FERRARI, FIENW,J. R., 95, 96, 116, 123, 124, 125, 149, 151, 230 E. P., 49, 50, 65 FILIPPOV, FILLMORE, G. L., 230 FINK,W., 269, 287 FIRESTER,A. H., 90, 114, 116 FITZPATRICK, J. A,, 391, 409 J., 111, 112, 116 FLAMAND, J., 230 FLEURET, FOLEY,R. J., 23, 69 FORNO,C., 281, 282, 286, 287

G GABEL,R. A,, 122, 230 I. I., 396, 409 GALAKTIONOV, GALATIN. J. D., 86, 116 GALE,M. T., 110, 116 Jr., N. C., 122, 207, 230, 231 GALLAGHER GAITI,G., 384, 391, 409 GAVIOLA. E., 76, 117 L. K., 45, 66 GAVRILINA, GAY,P. F., 115 T. F., 115 GAYLORD, E. V., 402, 403, 410, 411 GEORGE, GIBERT,R., 44, 67 R., 433, 438, 447 GILMORE, GIRARD, F., 91, 116 R. J., 415, 428, 430, 448 GLAUBER, K. S., 16, 67 GOCHELASHVILI, J. W., 89, 95, 96, 116, 122, 123, GOODMAN, 125, 149, 168, 178, 230, 232, 291, 356 GORDON, E. I., 235, 288 GORELOV, V. Yu., 396, 409 Yu. A., 18, 20, 28, 31-37, 39, GOROKHOV, 49, 50, 65, 66 GORYACHKIN, D. A., 396, 409 GRANT,E. R., 35, 67 A., 97, 116 GRAUBE. GREEN,F. J., 126, 229 GREER,M. O., 126, 231 GREGORY, D. A,, 273, 280, 281, 287 GIUFFT~HS, W. I., 279, 286 A. V., 409 GRIGOR'EV, P. G., 404, 406, 408 GRIGOR'EV, GRIGOR'EV, S. L., 49, 50, 65

452

AUTHOR INDEX

GROSS, R. W. F., 13, 69, 399, 400, 409 GUNNING, H. E., 42, 43, 67, 69 GUFTA,S. D., 115 GUSINOW, M. A., 400, 401, 409

HUNG,Y. Y., 256, 287 HURST,G. S., 57, 67 I

H HAAS,Y., 30, 66, 67 HAGLER,M. O., 288, 295, 356 HAHN,E. L., 52, 69 HAINES,K. A., 257, 287 HAKEN,H., 416, 417,448 HALLSWORTH, R. S., 20, 31, 35, 67 B. D., 230 HANSCHE, P., 256, 287 HARIHARAN, P., 42, 67, 68 HARTECK, HARTLEY, H., 13, 67, 69 HASKELL,R. E., 84, 115, 116, 135, 230 HAYAT,G. S.. 111, 112, 116 HEITLER.W., 418, 448 HELMER,M. E., 114 C. W., 184, 230 HELSTROM, HENTON,R. F., 232 HERSCHELL, J. F. W., 101, 116 G., 205, 228, 230 HERZIGER, R., 82, 85, 117, 232 HICKLING, HILDEBRAND, B. P., 257, 287 HINTZ,E., 384, 391, 392, 410 HIRSH,P. M., 87, 88, 89, 117, 123, 125, 143, 174, 231, 232 W. J., 42, 67 HITCHCOCK, H. 0.. 114 HOADLEY, R. M., 16, 31, 68 HOCHSTRASSER, D. M., 114 HOFFMAN, HDGMOEN, K., 264, 287 HOUE. 0. M., 264, 287 T. M., 86, 116 HOLLADAY, M. H., 220, 230 HORMAN, HOUSTON, J. B., 282, 287 P. L., 30, 66, 67 HOUSTON, HRBECK, G . W., 110, 116 HSIA.J., 398, 409 HUANG, G., 313, 315, 356 T. S., 82, 117, 122, 123, 125, 138, HUANG, 229, 231, 295, 332, 356 HUBBARD, J. C., 391, 409 HUGHES,R. G., 265, 287 HUGONIN, J. P., 230, 231 HUIGNARD, J. P., 90, 106. 108, 110, 117, 169, 230

ICHIOKA, Y., 86, 114, 117, 126, 205, 230, 23 1 INGALLS, A. L., 186, 231 INNES, K. K., 31, 67 V. M., 396, 409 IRTUGANOV, V. A., 16, 45, 66 ISAKOV, ISENOR,N. R., 20, 31, 32, 35, 67 ISSHIKI,M., 232 ITOH,Y., 257, 288 ITZKAN,I., 15, 23, 67 L. N., 22, 49, 57, 67 IVANOV, IZAWA,Y., 28, 69 IZUMI,M., 86, 117, 118, 231

J JACOB,J. H., 398, 409 JAMES, E. A.. 114 JANES, G . S.. 15. 22, 23, 24. 67, 68 JAYNES, E. T., 415, 417, 418, 447, 448 JENSEN,R. J., 32, 69 JOHNSON, S. A., 25, 49, 69 JONES,A. L., 123, 125, 148. 231 JONES,R., 248, 252, 253, 285, 287 JORDAN Jr., J. A,, 87, 88, 89, 117, 123, 125, 143, 174, 231, 232 JUANG,R. M.. 278, 281, 286. 287

K KAISER,W., 51, 52, 68 V. P., 15, 22, 66, 396, 409 KALININ,

KALYATSKII,I. I., 365, 409 KAMRUKOV,A. S., 4 0 0 , 4 0 1 , 4 0 9 , 4 1 0 Y., 232 KANDO, KARL,R. R., 31. 67 KARLOV,N. V., 16, 67, 397, 408 KARNAUKHOV, v. A,, 55, 67 KASAI,S., 232 G. N., 400, 401, 409, 410 KASHNIKOV, G. M., 365, 409 KASSIROV, KATULIN, V. A., 403, 408, 409 KATYL,R. H., 231 KATZ,A,, 441, 442, 448 KAUFMANN, K., 44, 45, 66 KAWAI,M., 126, 232

AUTHOR INDEX

KAZAKOV,S. A,, 40, 41, 66 KEETON,S. C., 231 KELLER,R. A,, 43, 68 KELLEY. P. L., 428, 448 KELLY,D. L., 82, 111, 117, 231 KELLY,E., 319, 356 KETTERER,G., 231 R. P., 278, 287 KHETAN, KHOKHLOV,R. V., 16, 67 G. S., 364, 409 KICHAYEVA, KIM, D. M., 438, 447 KING, D. S., 16, 31, 68 KING, M. C., 124, 175, 231 KING, T. A,, 400, 408 KIRK,J. P., 123, 125, 148, 231 KIRMISCH,D., 231 KISELEV,A. V., 16, 67 KLEINER, W. H., 428, 448 KLEPPNER,D., 25, 49, 67, 68 KLINE,L. E., 394, 395, 396, 409, 410 KMETZ,A. R., 103, 117 KNOPP, K., 110, 116 KNYAZEV,I. N., 59, 60, 61, 66, 68, 69 KOHLHAAS,W., 384, 391, 409 H., 64, 69 KOHLHAGE, KOLOMIISKY, Yu. R., 39, 40, 41, 66, 67 KOLPAKOVA, I. V., 382, 383, 411 KONNO, K., 126, 232 KOPF, U., 269, 279, 284, 287 KOROL'KOV,K. S., 403, 408 KORONKEV~CH, V. P., 91, 117 Koz~ov,N. P., 400, 401, 409, 410 KOZMA, A., 82, 111, 117, 126, 187, 205, 206, 231 KRASWK, I. K., 363, 410 KRAUS,M., 228, 229, 323, 337, 338, 339, 355, 356 KRIESE,M., 126, 230 KRIKORIAN,O., 23, 69 R. P., 184, 229 KRUGER, KRUPKE,W. F., 402, 403, 410, 411 KUHN,W., 13, 68 KUNG,A. H., 30, 66 KURTZ,C. N., 114 KUZ'MIN,G. P., 397, 408 KWOK,M. A,, 13, 69

L LABEYRIE, A,, 235, 287 M., 111, 112, 116 LACROIX,

453

LADIK,J., 53, 54, 68 T., 333, 356 LAMACCHIA, LAMBJr., W. E., 416, 417, 428, 429, 446, 448 LAMOTTE, M., 43, 68 D. M., 38, 67 LARSEN, A,, 51, 52, 68 LAUBEREAU, J. D., 396, 409 LAWRENCE, LEE,S. H., 111, 117, 288, 295, 356 LEE,T. N., 405, 411 LEE,W. E., 84, 117 LEE, W-H., 123-126, 133, 139, 199, 212, 229, 231, 288, 312, 356 LEE,Y. T., 35, 67 J. A,, 246, 248, 249,250,252, LEENDERTZ, 254, 257, 258, 264, 286, 287 LEGAY, F., 45, 67 LEGER,D., 284, 285, 287 LEHOVEC. K., 114 LEITH,E. N., 111, 117, 121, 186, 187, 229, 230, 231 LEMA~RE, G., 76, 117 LENKOVA, G. A., 91, 117 LEONE,S. R., 30, 68 LEONOV, Yu. S., 16, 66 LEOPOLD, K. E., 396, 398, 410 LESEM,L. B., 87, 88, 89, 117, 123, 125, 143, 174, 231, 232 V. S., 4, 14-18, 20, 22, 24, 25, LETOKHOV, 27-41, 43, 47-52, 54-61, 63, 65-69 LEVI,L., 323, 356 LEVIN,L., 15, 23, 67 LEVIN,M., 22, 69 LEVY,R. H., 15, 22, 23, 24, 67, 68 I., 393, 408 LIBERMAN, LE, Y. T., 384, 391, 392, 410 F. K., 254, 287 LIGTENBERG, L m , R. C., 396, 410 LINDSAY, L. M., 115 LINFOOT,E. H., 158, 231 LPATOV,N. I., 363, 410 LITMAN, M. C., 25, 49, 67, 68 Lru, B., 122, 207, 229, 230, 231 Lm, D., 42, 68 Lnm, G., 42, 68 LOHMAN, V. N., 39, 40, 67 LOHMANN, A. W., 82, 86, 89, 111, 116, 117, 122, 124, 125, 126, 152, 168, 174, 178, 205, 229, 230, 231, 276, 287, 312, 356 L0KBERG, 0. J., 262, 264,287

454

AUTHOR INDEX

LOKHMAN, V. N., 33, 67 J. C., 101, 117 LOULERGUE, LOWDIN,P. O., 54, 69 S., 182, 231 LOWENTHAL, LUXMOORE, A. R , 278, 287 LUY. H., 360, 410 LYGIN,V. J., 16, 67 LYMAN.J. L., 32, 69 G. G.. 363, 408 LYSAKOVSKII,

M MACGOVERN,A. J., 126, 195, 231 A., 261, 287 MACOVSKI, MACQUIGG,D., 232 MADDUX,G. E., 279. 286 A., 232 MAEKAWA, MAKAROV,A. A., 15. 18, 20, 27, 28, 3037. 39, 49, 5 0 , 52, 56, 57, 65, 67, 68 MAKAROV, G. N.. 28, 65, 66 V. A,, 400,401,409,410 MALASHCHENKO, Yu. A , , 49, 50, 65 MALININ, MANUEL,L., 415, 428, 429, 448 J. A,, 398, 409 MANGANO, MANSFIELD,C. R., 126, 141, 229, 232 Y. J., 45, 69 MANUCCIA, A. S., 86, 117 MARATHAY, MARECHAL,A., 78, 111, 117 MARIOGE,J. P., 75, 117 MARKIN,E. P., 16, 45, 66, 67 MARKOV,S. N., 406, 409 MARLING.J. B., 30, 69 MARSHALL,J. W., 418. 448 MARTIN,H., 13, 68 MASON.J. H., 365, 410 MASSEY,N. G., 187, 231 MATEKUNAS, F., 1 15 MATTHIEU,E., 284, 287 MATTHIJSSE,P., 115 MAY, C. A,, 25, 49, 69 MAYER.S. W., 13, 69 MCCALL,S. L., 52, 69 MCKEN,D. C., 393, 396, 410, 411 MCLELLAN,J. H., 115 MCLEOD.J. H., 161, 232 MEIER, R. W., 260, 288 V., 20, 31, 35, 67 MERCHANT, MERTON,T. R., 13, 67, 69 MESSIAH,A., 426, 432, 448

MEYER,A. J.. 82. 85. 117, 232 MICHAEL,F.. 232 MICHERON,F., 106, 108, 110, 117 G. V.. 403, 408 MIKHAIL.OV. I. A,, 91, 117 MIKHAL'TOOVA. MILES, P., 115 MILONNI,P. W.. 41.5. 448 MISHIN,V. I., 22. 24, 25, 29, 30, 43, 48. 19, 56, 57, 58, 65-68 MIYAMOTO, K.. 87, 117. 144, 232 MODEN,J. C.. 360. 389, 410 MOELLER,C. E.. 361, 364. 365, 374, 375, 377-380, 382, 386, 387, 388, 402, 408 MOLIN,N.-E., 262, 287 MOLIN,Yu. N., 29, 67 MOILET, 2.. 10 1. 116 MON, K. K.. 1 15 MOORE, C. B., 1,15, 20, 22, 27, 29, 30, 66-69 MORIZET,J., 90, 116 MORLAIS,M., 390, 410 Moos, H. W., 13, 69 MOVSHEV,V. G., 59, 60, 61, 66, 68. 69 S., 13, 69 MROZOWSKI, MUELLER,P. F., 111, 117 MULLER, E. W., 62, 6 9 MURRAY,J. R., 403, 411

N

NAGASHIMA,K.. 232 NAIDU,P. S., 122. 232 NAMIOT,N. A,. 16, 67 NARDUCCI,L. M.. 438, 447 NASLIN,P., 360. 409 NAYFEH, M. H., 57, 67 I.. 22, 69 NEBENZAHL. NESTERIKHIN,Yu. E., 104, 117 NETEMIN,V. N., 403, 408 NIKIFOROV,S. M., 397, 408 NIKOLAEV,F. A,. 403, 408 NISHIDA,N., 97, 117 NIZ'EV, V. G.. 40, 41, 66 NOGUCHI,N., 28, 69 NOLL, A. M.. 124, 17.5, 231 NORMAN,G. E., 376, 409 NOSACH,0. Yu.. 403, 408, 409 NOSACH,V. Yu., 403, 408, 409

AUTHOR INDEX

0 OGURTSOVA, N. N., 383, 410 O’LEARY,T. J., 21, 67 O’NEILL,E. L., 78, 111, 117 O’NEILL,P. K., 232 A. N., 16,45,66.67,404,408 ORAEVSKY, ORLOV.A. N., 16, 67 ORLOV. V. K., 400, 401, 409, 410 Osrpov. A. J., 16, 67 L., 264, 286 OSTLUND,

P PAISNER, J. A,, 25, 49, 69 PALMER,A. J., 394, 410 PANCHENKO, V. J., 16, 67 PARIS,D. P., 111, 117, 124. 178, 229, 231 312, 356 PARTLOW, w. D., 393, 408 PASHININ, P. P., 363, 410 0. I., 396, 409 PASHKOV, PASTOR,J., 126, 232 PATAU,J. C., 87, 88, 89, 117, 232 M. J., 394, 410 PECHERSKY, PEDERSEN. H. M., 262, 264, 287 PERI,D., 230 PERRIN,J. C., 284, 285, 287 PERTEL,R., 42, 69 PETERS,P. J., 126, 231 0. G., 15, 23, 69 PETERSON, PETROV,A. L., 403, 408 PETROV,R. P., 16, 67 PETROV,Yu. N., 16, 67 PIKE.C. T., 15, 23, 67 V. D., 40, 41, 66 PISMENNY, A. G., 52, 66 PLATOVA, I. V., 383, 396, 409, 410 PODMOSHENSKN, POLIKANOV, S. M., 55, 67 PONDER,A. O., 13, 67 POOLEY,S., 360, 389, 410 L. J., 186, 230 PORCELLO, 0. E., 404, 408 PORODINKOV, POTAPOV, v. K., 61, 69 F. X., 398, 409 POWELL, POWERS, R. S., 168, 232 PRASADA,B., 123, 125, 138, 231 A. M., 16, 67, 397, 408 PROKHOROV, Yu. S., 400, 401, 410 PROTASOV, B. I., 16, 67 PROVOTOROV, D., 296, 298, 299, 301, 306, 307, PSALTIS,

455

316-319, 321, 323, 325, 333, 335, 339, 331, 342, 344, 348, 350-353, 355, 356 PUECH,C., 90, 117 PURETZKII, A. A., 15, 18, 22, 24, 28, 3237, 39, 48, 49, 50, 52, 56, 57, 65, 66, 68

Q QUINTANILLA, G. E., 259, 262, 287

R RABIDEAU, S. W., 28, 69 RAMSEY. S. D., 261, 287 RANSOM, P. L., 82, 86, 117, 232 REECE,CI. W., 360, 389, 410 REHM,G. G., 45, 69 REILLY,J., 30, 66 V. J., 91, 117 REMESNIK, G. O., 111, 117 REYNOLDS, RICH,J. W., 45, 69 M. C., 20, 31, 32, 35, 67, RICHARDSON, 396, 410 RIGDEN,J. D., 235, 288 RITIER, J. J., 43, 45, 67, 68 G. M., 295, 332, 337, 356 ROBBINS, ROBIN,S., 390, 410 S., 28, 69 ROCKWOOD, J., 389, 390, 391, 408, 409, 410 ROMAND. V. I., 16, 45, 66, 67 ROMANENKO, RONCIN,J.-Y., 390, 409 ROSENBRUCH, K.-J., 126, 232 ROWLANDS, R. E., 256, 287 ROWLANDS, R. E., 256, 287 ROY,A. M., 90, 117, 169, 230 Rozmov, V. B., 401, 410 RUDIN,W., 207, 208, 232 D., 111, 117 RUDOLPH, F., 313, 315, 356 RUSSELL, E. A,, 18, 20, 28, 31, 32, 39, 40, RYABOV, 41, 65--68 S SARJEAN-T, w. J., 398, 410 SATOV,Yu. A,, 394, 409 SAwcnuK, A., 295, 356 SCHADE,R., 360, 410 SCHAEFER,L. F., 261, 287 SCHAFER, F. P., 43, 69 H., 360, 410 SCHARDIN, A. L., 13, 21, 67, 69 SCHAWLOW,

456

AUTHOR INDEX

SCHIFF, L. I., 416, 431, 448 S C H L ~ D., R , 376, 378, 411 S ~ LG., ,111, 117 L. G., 76, 77, 117, 205, 232 SCHULZ, P. A., 35, 67 SCHULTZ, J., 205, 232 SCHWIDER, SCHWOB, J.-L., 390, 409 SCUDIERI, F., 115 M. O., 428, 429, 448 SCULLY, H. J., 393, 396, 410, 411 SEGUIN, SEILMEIER,A., 52, 68 V. A., 30, 43, 66, 67, 68 SEMCHISHEN, SE-Y, I. R., 418, 428, 429, 433, 435, 436, 438, 445, 446, 448 SERAPHIN, B. o., 102, 117 SHAFFER,G., 228, 232 R. P., 75, 117 SHANNON, A. V., 403, 408 SHELOBOLIN, S m , Y. R., 35, 67 SHIBANOV, A. N., 33, 67 K., 56, 69 SHIMODA, SHIOTKE,N., 257, 288 S m o v , V. L., 406, 409 SIWROV,A. N., 397, 408 A. J., 115 SIEVERS, SWAST, W. T., 403, 411 SILVESTRI, A. M., 89, 116, 122, 168, 230 R. M., 232 SINGLETON, SIROHI, R. S., 115, 118, 126, 232 P., 230 SKALICKY, G. I., 29, 67 SKUBNEVSKAYA, SMITH,H. M., 97, 117, 229, 232 B. B. 15, 23, 69 SNAVELY, I. H., 302, 356 SNEDDON, SOLARZ, R. W., 25, 49, 69 V. A., 396,409 SOLOV'EV, N. I., 29, 67 SOROKIN, D. L., 13, 69 SPENCER, S P ~E.,, 90,106, 108, 110,117,169,230 K. T., 295, 356 STALKER, A. I., 40, 41, 66 STARODUBTZEV, STARTSEV, A. V., 403, 409 STERLING, w., 291, 356 STETSON,K. A. 243, 275, 288 W. H., 115, 228, 232 STEVENSON, STOILOV, YU., 403, 409 STRAND, J. C., 232

STRAUSZ,0. P., 42, 43, 67 STROKE, G. W., 92, 104, 117 STRONG, J., 76, 117

STROUD Jr., C. R., 417, 419, 448 M., 43, 69 STRUKE, Su, S. F., 115 V. V., 394, 409 SUDAKOV, E. C. G., 428, 429, 448 SUDARSHAN, Aa. S., 35, 67 SUDBO, SUGAR, J., 389, 411 SUSI,H., 53, 54, 69 SUZUKI, T., 86, 117, 231 SWEENY, D. W., 115, 228, 232 SWINGLE, J. C., 403, 411 SZCZU~

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