PROGRESS IN OPTICS VOLUME XVII
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,
Sydney, Australia
W. T. WELFORD,
London, England
P R O G R E S S I N OPTICS VOLUME XVII
EDITED BY
E. WOLF Unicersity of Rochesfer. N . Y.. U.S.A
Con frihrcfors
R. DANDLIKER, E. GIACOBINO. B. CAGNAC M. SCHUBERT, B. WILHELMI W. J. TANGO, R. Q. TWISS, A. L. MIKAELIAN
1980
NORTH-HOLLAND PUBLISHING COMPANYAMSTERDAM. NEW YORK . OXFORD
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CONTENTS OF VOLUME 1 ( 1 9 6 1 ) I. I1.
THEMODERN DEVELOPMENT OF HAMILTOMAN Omcs. R . J . PEGIS . . . 1-29 WAVE O m c s AND GEOMETRICALOPTICS IN OFTICAL DESIGN. K. MIYAMOTO. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1-66 I11. THE INTENSITY DISTRB~ONAND TOTALILLLIMINATION OF ABERRATIONFREEDIFFRACTION IMAGES. R . BARAKAT . . . . . . . . . . . . . . 67-108 IV . LIGHTANDINFORMATION. D . GABOR . . . . . . . . . . . . . . . 109-153 V . ON BASICANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN O ~ C AAND L ELECTRONIC INFORMATION. H. WOLTER. . . . . . . . . . . . . . . 155-210 COLOR.H . KUBOTA . . . . . . . . . . . . . . . . 21 1-251 VI . INTERFERENCE VII . DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES. A . FIORENTINI . . . 253-288 VIII . MODERN ALIGNMENT DEVICES.A . C. S . VAN HEEL . . . . . . . . . . 289-329
CONTENTS OF VOLUME I1 (1963) I. I1. I11.
rv. V. VI .
RULING.TESTINGAND USEOF OFTICALGRATINGS FOR HIGH-RESOLUTION 1-72 SPECTROSCOPY.G. w . STROKE . . . . . . . . . . . . . . . . . . THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS.J . M. BURCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-108 DIFFUSION TROUGH NON-UNIFORM MEDIA. R . G . GIOVANELLI. . . . . 109-129 OF OFTICAL IMAGESBY COMPENSATION OF ABERRATIONS AND CORRECTION BY SPATIAL FREQUENCY FILTERING. J . TSUJIUCHI. . . . . . . . . . . 131-180 OF LIGHT BEAMS.L. MANDEL . . . . . . . . . . . . FLUCTUATIONS 181-248 METHODSFOR DETERMINING OFTICALPARAMETERS OF THINFILMS.F . ABELES ............................ 249-288
CONTENTS OF VOLUME I11 (1964) I. I1. 111.
. . . . . . . . THEELEMENTSOF RADIATIVE TRANSFER. F . KOTTL.ER APODISATION.P . JACQUINOT AND B . ROIZEN-DOSSIER . . . . . . . . COHERENCE. H . GAMO . . . . . . . MATRIX TREATMENTOF PARTIAL
1-28 29-186 187-332
CONTENTS OF VOLUME IV (1965) I. I1. 111. IV . V. VI . VII .
HIGHERORDERABERRATION THEORY. J . FOCKE . . . . . . . . . . APPLICATIONS OF SHEARING INTERFEROMETRY. 0. BRYNGDAHL . . . . OF OFTICAL GLASSES.K. KINOSITA . . . . . SURFACEDETERIORATION O ~ C ACONSTANTS L OF THINFILMS.P . ROUARD AND P . BOUSQLJET. . . THEMIYAMOTO-WOLF DIFFRACTION WAVE.A . RUBINOWICZ. . . . . ABERRATION THEORYOF GRATINGS AND GRATINGMOUNTINGS. W. T. WELFORD . . . . . . . . . . . . . . . . . . . . . . . . . . . DIFFRACTION AT A BLACKSCREEN.PART I: KIRCHHOFFT THEORY.F . KOTTLER . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-36 37-83 85-143 145-1 97 199-240 241-280 281-314
CONTENTS OF VOLUME V (1966) I. I1. I11.
OFTICAL PUMVTMG. C . COHEN-TANNOUDJI AND A . KASTLER . . . . . . NON-LINEAR Omcs. P. S . PERSHAN . . . . . . . . . . . . . . . . TWO-BEAM INTERFEROMETRY. W. H . STEEL . . . . . . . . . . . . .
1-81 83-144 145-197
w.
INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFERFUNCTIONS,K. MURATA . . . . . . . . . . . . . . . . . . . . . . . . . . . V. LIGHT F~FLECTION FROM m s OF CONTINUOUSLY VARYINGR E F R A m INDEX,R. JACOBSSON ...... . . . . .. .. . .. . , . . VI. X-RAY C R Y ~ ~ A L - S T R UDETERMINATION ~~~RE AS A BRANCH OF PHYSICAL Omcs, H. LIPSONANDC.A. TAYLOR . . , . . , . . . . . . . . . VII. THE WAVEOF A MOVINGCLASSICALELECTRON,J. Prcm . . . . . . .
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199-245 247-286 287-350 35 1-370
C O N T E N T S OF V O L U M E V I ( 1 9 6 7 )
. . 1-52 ROUGHSURFACES, P.BECKMANN . . . . . . 53-69 MEASUREMENT OF THE SECOND ORDER DEGREE OF COHERENCE, M. FRANCON AND S. WICK . . . . . . . . . . . . . . . . . . . . 71-104 Iv. DESIGNOF ZOOMLENSES,K. Y m . . . . . . . . . . . . . . . 105-170 V. SOMEAPPLICATIONS OFLASERSTO INTERFEROMETRY, D. R. HERRIOTC . 171-209 STUDIESOF INTENSITYFLUCTUATIONS IN LASERS,J. A. VI. E~ERIMENTAL ARMSTRONG AND A. w. SMITH . . . . . . . . . . . . . . . . . 21 1-257 VII. FOURIERSPECTROSCOPY, G. A. VANASSE AND H:SAKAI . . . . . . . . 259-330 VIII. DIFFRACTIONAT A BLACKSCREEN,PART11: ELECTROMAGNETIC THEORY, F.KOTTLER , . . . . . . . . . . . . , . . . . . . . . . , . . 331-377 I.
RECENT ADVANCES IN HOLOGRAPHY, E. N. L m AND J. UPATNEKS
11. 111.
SCATTERING OF LIGHTBY
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C O N T E N T S OF V O L U M E V I I ( 1 9 6 9 ) MULTIPLE-BEAMINTERFERENCE AND NATURALMODES IN OPEN RESONATORS, G . KOPPELMAN. . . . . . . . . . . . . . . . . . . . . 1-66 11. METHODS OF SYNTHESIS FOR DIELECTRIC MULTILAYERFILTERS,E. DELANO m R . J. PEGIS . . . . . . . . . . , . . . . . . . . . . . . . . 67-137 111. ECHOESATOFTCALFREQUENCIES, I. D. ABELLA . . . . . . . . . . 139-168 IV. IMAGEFORMATION wrm PARTIALLY COHERENTLIGHT,B. J. THOMPSON . 169-230 V. QUASI-CLASSICAL THEORY OF LASERRADIATION, A. L. MIKAELIAN AND M. L. TER-MIKAELIAN . . . . . . . . . . . . . . . . . . . . . . . 231-297 VI. THE PHOTOGRAPHIC IMAGE,S. OOUE . . . . . . . . . . . . . . . 299-358 VII. hIERACTION OF VERY INTENSE LIGHTWITH 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 ) SYNTHETIC-APERATURE Omcs, J. W. GOODMAN . . . . . . . . . . 1-50 TIIE O m c a PERFORMANCE OFTHE HUMAN EYE, G . A. FRY . . . . . 51-131 LIGHTBEATING SPECTROSCOPY, H. Z. CWINS AND H. L. SWINNEY . . 133-200 Iv. MULTILAYERANTIREFLECTION COATINGS, A. MUSSETAND A. THELEN . 20 1-237 V. STATISTICAL PROPERTIES OF LASER LIGHT,H. RISKEN . . . . . . . . . 239-294 OF SOURCE-SIZECOMPENSATION IN INTERFERENCE VI. COHERENCETHEORY T. YAMAMOTO . , . . . . . . . . . . . . . . . . . 295-341 MICROSCOPY, VII. VISIONIN COMMUNICATION, L. LEVI . . . . . . . . . . . . . . . . 343-372 VIII. THEORY OF PHOTOELECTRON COUNTING, C. L. MEHTA . . . . . . . . 373-440
I.
11. 111.
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 LENGTHMEASUREMENTS.
A.L. BLOOM
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1-30
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11. ~COSECONDLASERPULSES,A. J. DEMARIA . , , .... . ... J. W. 111. OPTICAL PROPAGATION THROUGH THE TURBULENT ATMOSPHERE, STROHFJEHN . . . . . . . . .... . . .. . . . . . IV. SYNTHESIS OF OPTICAL BIREFRINGENT NETWORKS, E. 0. AMMA" . . . V. MODE LOCKING IN GASLASERS, L. ALLENAND D. G. C. JONES . . . . L W~THSPATIAL DISPERSION, V. M. AGRANOVICH AND V. L. VI. C ~ Y W AOPTICS GINZBURG . . . . . . . . . . . . . . . . . . . . . . . . . VII. APPLICATIONSOF OPTICAL METHODS IN THE DIFFRACTION THEORYOF ELASTICWAVES,K. GNIADEK ANDJ.Pmxmwcz . . . . . . . . . . VIII. EVALUATION, DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL SIGNALS, BASEDON USE OF THE PROLATE FUNCTIONS, B. R.-EN ..
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... .
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31-71 73-122 123-177 179-234 235-280 281-310 311-407
CONTENTS OF VOLUME X ( 1 9 7 2 ) I. 11. 111.
BANDWIDTH COMPRESSION OF OPTICAL ~ A G E ST., S. HUANG THE USE OF IMAGE TUBES AS SHU~-I'ERS, R. SMlTH
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TOOLSOF THEORETICAL Q u m OPTICS,M. 0. SCUUYAND K. G. WHITNEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . FOR ASTRONOMICAL TELESCOPES, C. G . WYNNE . . Iv. FIELDCORRECTORS OPTICALABSORFTION STRENGTH OF DEFECISIN INSULATORS, D. Y. SMITH V. ANDD.L. D m R . . . . . . . . . . . . . . . . . . . . . . . AND DEFLECTION, E. K.S m G . . . VI. ELASTOOPTIC LIGHTMODULATION THEORY,C. W. HELSTROM . . . . . . . . . . VII. QUANTUM DETECTION
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1-44 45-87 89-135 137- 164 165-228 229-288 289-369
CONTENTS OF VOLUME X I ( 1 9 7 3 ) EQUATION METHODS M Q U A N T U OPnCS, M G . s. AGARWAL . . RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . ...,... . . . . . .... . ..... . 111. INTERACTIONOF LIGHTANDACOUSTICSURFACEWAVES, E. G . LEAN . . rv. EVANESCENTWAVESIN OPTICALIMAGING,0.BRYNGDAHL. . . . , . OF ELECTRON PROBESUSINGA FIELDEMISSION SOURCE,A. V. PRODUCTION v.cREwE . . . . . . . . . . . . . . . . . . . . . . . . . . . THEORY OF BEAMMODEPROPAGATION, J. A. ARNAUD . VI. HAMILTOMAN INDEXLENSES,E. W. MARCHAND . . . . . . . . . . . . VII. GRADIENT I. 11.
MASTER
1-76 77-122 123-166 167-221 223-246 247-304 305-337
CONTENTS OF VOLUME XI1 (1974) SELF-FOCUSING, SELF-TRAPPING, AND SELF-PHASE MODULATION OF LASER BEAMS,0. SVELTO . . . . . . . . . . . . . . . . . . . . . . . 1-51 TRANSPARENCY, R. E. SLUSHER . . . . . . . . . . . 53-100 11. SELF-INDUCED TECHNIQUES M SPECTROMETRY, M. HARWIT,J. A. DECKER 111. MODULATION JR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101-162 OF LIGHTWITH MONOMOLECULAR DYE LAYERS,K. H. IV. INTERACTION DREXHAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . 163-232 CONCEPT AND COHERENCE IN ATOMIC EMISSION, V. THE PHASE TRANSITION R.GRAHAM . . . . . . . . . . . . . . . . . . . . . . . . . . 233-286 SPECTROSCOPY, S. BASHKIN. . . . . . . . . . . . . . . 287-344 VI. BEAM-FOIL
I.
CONTENTS OF VOLUME XI11 (1976) I.
ONTHE VALIDITY OF KIRCHHOFF'SLAWOF HEATF~DIATIONFOR A BODY IN A NONEQUILIBRIUMENVIRONMENT, H. P. BALTES . . . . . . . . .
1-2s
THE CASE FORAND AGAINST SEMICLASSICALRADIATIONTHEORY, L. MANDEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-68 ~ SUBJEXXIVE SPHERICALABERRATION MEASUREMENTSOF 111. O B J E AND THE HUMANEYE,W. M. ROSENBLUM, J. L. CHRISTENSEN . . . . . . 69-91 Iv. INTERFEROMETRIC TESTINGOF SMOOTH SURFACES, G. SCHULZ, J. SCHWIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . 93-167 OF LASERBEAMSIN PLASMAS AND SEMICONDUCTORS, M. S . V. SELFFOCUSING SODHA,A. K. GHATAK, V. K. TRPATHI . . . . . . . . . . . . . . 169-265 AND ISOPLANATISM, W. T. WEWORD . . . . . . . . . . 267-292 VI . APLANATISM 11.
C O N T E N T S OF V O L U M E X I V ( 1 9 7 7 ) 1-46 THESTATISTICSOF SPECKLEP A ~ R N SJ. ,C. DAINTY . . . . . . . . . 47-87 HIGH-RESOLUTION "IQUES IN OPTICALASTRONOMY, A. LABEYRIE . 111. RELAXATION PHENOMENA IN --EARTH LUMINESCENE, L. A. RISEBERG, 89-IS9 M. J. WEBER . . . . . . . . . . . . . . . . . . . . . . . . . . L S-R, M. A. DUGUAY. . . . . . . 161-193 Iv. THEULTRAFASTO ~ I C AKERR V. HOLOGRAPHIC DIFFRACTION GRATINGS, G. SCHMAHL, D. RUDOLPH . . 195-244 P. J. VERNIER . . . . . . . . . . . . . . . . . . 245-325 VI. PHOTOEMISSION, L WAVEGUIDES-A REVIEW.P. J. B. CLARRICOATS . . . 327-402 VII. O ~ C AFIBRE
I. 11.
C O N T E N T S OF V O L U M E XV ( 1 9 7 7 ) PARAMETRIC h L I F I C A T I O N AND OSCILLATION, w . 1-75 BRUNNER,H.PAUL . . . . . . , , . , . . . . . . . . . . . . 77-137 OFTHIN METALFILMS,P. ROUARD,A. MEESSEN . 11. OPTICAL PROPERTIES 111. PROJECTION-TYPE HOLOGRAPHY, T. OKOSHI. . . . . . . . . . . . . 139-1 85 TECHNIQUES OF RADIO ASTRONOMY, T. W. COLE . . . 187-244 Iv. QUASI-OPTTCAL V. FOUNDATIONS OF THE MACROSCOPIC ELECTROMAGNETIC THEORY OF DIELECTRIC MEDIA,J. VANKRANENDONK,J. E. SPE . . . . . . . . . 245-350
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THEORY OF OPnCAL
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C O N T E N T S OF V O L U M E X V I ( 1 9 7 8 ) LASER SELECTIVE PHOTOPHYSICS AND PHOTOCHEMISTRY, V. S. 1-69 LETOKHOV. . . . . . . . . . . . . . . . . . . , . , 11. RECENTADVANCES I N PHASEPROFILES GENERATION, J. J. CLAIR,C. 1. 71-117 ABITBOL. . . . . . . . . . . . . . . . . . . . . . . HOLOGRAMS: TECHNIQUES AND APPLICATIONS, 111. COMPUTER-GENERATED W-H.LEE . . . . . . . . . . . . . . . , . . . . . . 119-232 INTERFEROMETRY, A. E. ENNOS . . . . . . . . . . . 233-288 IV. SPECKLE INVARIANT, SPACE-VARIANT OPTICALPATTERN RECOGNIV. DEFORMATION TION,D. CASASENT, D. PSALTIS. . . . . . . . . . . . . . . 289-356 FROM HIGH-CURRENT SURFACE-SPARK DISCHARGES, VI. LIGHT EMISSION 357-41 1 R. E. BEVERLY111 . . . . . . . . . . . . . . . . . . . RADIATION THEORYWITHINA QUANTUM-MECHANICAL VII. SEMICLASSICAL 418-448 FRAMEWORK, I. R. SENITZKY.. . . . . . . . . . . . . . I.
PREFACE It was planned to publish this volume in 1979, but for technical reasons that were beyond the control of the Publisher and the Editor, this did not prove to be feasible. It is my hope that even with the slight delay, this volume will be welcomed by our readers. It contains five review articles dealing with topics that cover a broad range of subjects, to which optical scientists have devoted a good deal of attention in recent years. Department of Physics and Astronomy University of Rochester Rochester, N . Y. 14627 January 1980
EMILWOLF
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CONTENTS I . HETERODYNE HOLOGRAPHIC INTERFEROMETRY by R . DANDLIKER(BADEN.SWITZERLAND)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . INTRODUCTION 2 . INTERFEROMETRY WITH DIFFUSELY SCATTERING SURFACES . . . . . . . . . . 2.1 Coherent image of a rough surface . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Holographic interferometry of objects with a rough surface 2.3 Heterodyne interferometry of rough surfaces . . . . . . . . . . . . . . 3 . TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Recording and reconstructed images 3.2 Alignment and wavelength sensitivity . . . . . . . . . . . . . . . . . 3.3 Effect of misalignment on the interference patterns . . . . . . . . . . . 3.4 Nonlinear cross-talk . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Overlapping reconstructions . . . . . . . . . . . . . . . . . . . . . 4 . EXPERIMENTAL REALIZATION AND RESULTS . . . . . . . . . . . . . . . . 4 . I Some general considerations . . . . . . . . . . . . . . . . . . . . . 4.2 Experimental verification . . . . . . . . . . . . . . . . . . . . . . 4.3 Advanced experimental arrangement . . . . . . . . . . . . . . . . . 4.4 Accuracy and reproducibility . . . . . . . . . . . . . . . . . . . . . 5. APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Double exposure holographic interferometry . . . . . . . . . . . . . . 5.2 Real-time holographic interferometry . . . . . . . . . . . . . . . . . 5.3 Vibration analysis . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Depth contouring . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Measurement of mechanical strain and stress . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 7 7 10 13 18 19 25 28 33 40 45 45 46 50 54 60 61 63 64 67 69 82
I1. DOPPLER-FREE MULTIPHOTON SPECTROSCOPY and B . CAGNAC(PARIS) by E . GIACOBINO
1. INTRODUCTTON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . CALCULATION OF INTENSITIES. LINESHAPESAND LIGHT-SHIFIS . . . . . . . . 2.1 Two-photon transition probability and two-photon operator . . . . . . . 2.1.1 Introduction of the two-photon operator . . . . . . . . . . . . . 2.1.2 Expression of the transition probability in terms of the oscillator strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 The case of non-monochromatic fields . . . . . . . . . . . . . . 2.1.4 Comparison with stepwise excitation in the density matrix formalism . 2.2 Application of the irreducible tensorial set formalism to Q&, . . . . . . 2.2.1 Expansion of the two-photon operator QEIE2 o n the irreducible tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Selection rules . . . . . . . . . . . . . . . . . . . . . . . .
87 89 91 91 94 95 98 101 101 102
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2.2.3 Line intensities. The case of hyperfine components . . . . . . . . 2.2.4 Density matrix of the excited state . . . . . . . . . . . . . . . . 2.3 Light shift of the two-photon resonance . . . . . . . . . . . . . . . . 2.3.1 Calculation of the light-shift . . . . . . . . . . . . . . . . . . 2.3.2 Order of magnitude . . . . . . . . . . . . . . . . . . . . . . 2.4 Three-photon and multiphoton transitions . . . . . . . . . . . . . . . 2.4.1 Three-photon transition probability and generalization to n-photon transition . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Light-shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Background due to first-order Doppler effect . . . . . . . . . . . 2.5.2 Second order Doppler effect . . . . . . . . . . . . . . . . . . 2.5.3 Effect of transit time . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Effect of collisions . . . . . . . . . . . . . . . . . . . . . . . 3 TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME . . . . . . . . . . . 3.1 Experimental set-up for two-photon spectroscopy . . . . . . . . . . . . 3.1.1 Dye laser and wavelength control . . . . . . . . . . . . . . . . 3.1.2 Set-up for observation of the transitions . . . . . . . . . . . . . 3.2 Doppler-free two-photon experiments in sodium . . . . . . . . . . . . 3.3 Review of Doppler-free two-photon spectroscopy experiments . . . . . . 3.3.1 Fine and hyperfine structures; Zeeman structure . . . . . . . . . 3.3.2 Isotopic shifts . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Study of molecular structures . . . . . . . . . . . . . . . . . . 3.4 Study of Rydberg states using ion detection . . . . . . . . . . . . . . 3.5 Hyperfine components of a two-photon line . . . . . . . . . . . . . . 3.6 Studies of collisional effects . . . . . . . . . . . . . . . . . . . . . 3.7 Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. TRANSIENT PROCESSESINVOLVING TWO-PHOTON EXCITATION. . . . . . . . . 4.1 Resonant excitation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Off-resonance excitation . . . . . . . . . . . . . . . . . . . . . . . 4.3 Ramsey’s fringes in Doppler-free two-photon resonances . . . . . . . . 4.3.1 Principle of the experiment . . . . . . . . . . . . . . . . . . . 4.3.2 Experimental realization . . . . . . . . . . . . . . . . . . . . 5 . THREE-PHOTON DOPPLER-FREE TRANSITIONS IN SODIUM. . . . . . . . . . . 6. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX I: REMARKS ON THE CHOICE OF GAUGE . . . . . . . . . . . . . . . APPENDIX 11: CALCULATION OF THE ENERGY ABSORBED BY ONEATOM . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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104 106 107 107 108 110 110 112 113 113 115 117 119 120 121 121 123 124 127 127 128 128 128 130 132 135 136 136 139 142 143 146 148 151 152 154 158
111. THE MUTUAL DEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR OPTICAL PROCESSES by M. SCHUBERT and B . WILHELMI(JENA.G.D.R.) 1 . INTRODUCTION ............................. PROPERTIES OF LIGHT . . . . . . . . . . . . . . . . . . . . 2. COHERENCE 2.1 Correlation functions and definition of coherence . . . . . . . . . . . . 2.2 States and measurable quantities of the field . . . . . . . . . . . . . . 2.3 Measurement of statistical properties of light . . . . . . . . . . . . . . ........................ 3. ONE-PHOTONPROCESES 3.1 Change of the coherence properties of light by spontaneous emission from the atomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
165 165 166 171 179 184 184
xiii
CONTENTS
3.2 Change of the coherence properties of light by nonequilibrium atomic . . . systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . MULTI-PHOTON ABSORITION . . . . . . . . . . . . . . . . . . . . 4.1 Transition probabilities . . . . . . . . . . . . . . . . . . . . ... 4.2 Alteration of the field . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Propagation problems . . . . . . . . . . . . . . . . . . . . . . . . 5 . TWO-PHOTON EMISSION AND TWO-PHOTONLASING PROCESS . . . . . . 6. PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION. . . . . . 6.1 General aspects of the problem . . . . . . . . . . . . . . . . . 6.2 Effectswith neglected depletion of pump fields . . . . . . . . . . . . . 6.3 General correlation behavior . . . . . . . . . . . . . . . . .. . . 7 . STIMULATED RAMANSCATTERING . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186 196 196 203 210 215 217 218 221 225 229 232
IV . MICHELSON STELLAR INTERFEROMETRY by W . J . TANGO (SYDNEY.AUSTRALIA) and R . Q. miss (LONDON. ENGLAND)
1 . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . STELLAR INTERFEROMETER. . . . . . . . . . . . 2 . THEMODERNMICHELSON OF THE MICHEUONSTELLAR I~TERFEROMETER . . . . . . 3 . THEBASICTHEORY 3.1 The quasi-monochromatic theory . . . . . . . . . . . . . . . . . . . 3.2 The effect of a finite bandwidth . . . . . . . . . . . . . . . . . . . . 3.2.1 Atmospheric dispersion . . . . . . . . . . . . . . . . . . . . 3.3 Measurementof the fringevisibility by photoncounting . . . . . . . . . 4 . THE E m & OF ATMOSPHERIC TURBULENCE ON A MICHELSON STELLAR INTERFEROMETER
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . .........
4.1 Large scale phase fluctuations . . . . . . . . . . . 4.2 Temporal fluctuationsof the wave amplitude . . . . 4.2.1 Temporal phase fluctuations . . . . . . . . 4.2.2 Irradiance fluctuations . . . . . . . . . . . . 4.3 Spatial fluctuations of the wave amplitude . . . . . 4.3.1 Removal of wavefront tilts . . . . . . . . . 4.3.2 Measurement of the coherence loss . . . . . 4.4 The signal to noise ratio . . . . . . . . . . . . . 5 . THETILTCORRECTING SERVOSYSTEM . . . . . . . . 5.1 Atmospheric dispersion . . . . . . . . . . . . . 6 . SLJMMARY AND DISCUSSION . . . . . . . . . . . . . COUNTING STATISTICS . . . . . APPENDIX A: THEPHOTON APPENDIX B: THEANGLEOF ARRIVAL SPECTRUM. . . . . ACKNOWLEDGEMENT. . . . . . . . . . . . . . . . . REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .
241 243 247 247 250 252 253 255 256 257 257 258 259 261 262 263 264 268 268 270 273 276 276
V . SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION by A . L . MIKAELIAN (Moscow)
PREFACE. by A . M . PROKHOROV . . . . . . . . . . . . . . . . . . . 1. INTRODUCITON .......................... 1.1 State of the field . . . . . . . . . . . . . . . . . . . . . . . 1.2 Historical review . . . . . . . . . . . . . . . . . . . . . . . 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
...
...
... ...
...
281 283 283 283 286
xiv
CONTENTS
2 . FOCUSING INHOMOGENEOUS MEDIAw m CENTRAL SYMMETRY . . . . . . . . 2.1 Maxwell lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Luneburg lens . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . FOCUSING LAMINATED INHOMOGENEOUS CYLINDRICALMEDIUM . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Self-focusing cylindrical waveguide . . . . . . . . . . . . . . . . . . 3.3 Mikaelian lens . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Wave propagation in a self-focusing medium . . . . . . . . . . . . . . 3.5 Image tfansmission in a self-focusing medium . . . . . . . . . . . . . 4 . FLATLAMINATED INHOMOGENEOUS FOCUSING MEDIA . . . . . . . . . . . . 4.1 Flat self-focusing waveguide . . . . . . . . . . . . . . . . . . . . . 4.2 Parabolic waveguide . . . . . . . . . . . . . . . . . . . . . . . . 5 . EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 . MEWODSFOR THE CALCULATION OF INHOMOGENEOUS FOCUSING MEDIA . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Method based on integration of Euler’s equations . . . . . . . . . . . . 6.3 Method based on the principle of inhomogeneous media similarity . . . . 7 . QUASI-REGULAR CYLJNDRICALINHOMOGENEOUS MEDIA . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Self-focusing waveguides (SELFOCS) . . . . . . . . . . . . . . . . . . 7.3 Telescopic waveguides ....................... 7.4 Irregular waveguides with variable refractive index . . . . . . . . . . . 8. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
288 288 291 297 297 297 300 302 306 311 311 318 321 324 324 325 330 332 332 333 337 340 342 343
AUTHOR INDEX . . . . . . . . . . . . . SUBJECT INDEX . . . . . . . . . . . . . CUMULATIVE INDEX - VOLUMES I-XVII
347
............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
355 359
E. WOLF, PROGRESS IN OPTICS XVII @ NORTH-HOLLAND 1980
I
HETERODYNE HOLOGRAPHIC INTERFEROMETRY BY
RENE
DANDLIKER"
Brown Boveri Research Centre, CH-5405Baden, Switzerland
* Now
at Institut de Microtechnique de PUniversitB, CH- 2000 Newhiitel, Switzerland.
CONTENTS PAGE
§ 1. INTRODUCTION
. . . . . . . . . . . . . . . .
0 2 . INTERFEROMETRY WITH DIFFUSELY SCATTERING SURFACES . . . . . . . . . . . . . . . . . .
3 7
0 3 . TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY . . . . . . . . . . . . . . . 18 9 4 . EXPERIMENTAL REALIZATION AND RESULTS . . . 45 §
5 . APPLICATIONS
REFERENCES .
. . . . . . . . . . . . . . . .
60
. . . . . . . . . . . . . . . . . . 82
8 1. Introduction Interferometry is an old and very powerful technique to measure the deviation between two wavefields with a sensitivity of a fraction of a wavelength. Coherent light V(x, t ) of high optical quality, mostly plane or spherical waves, probes the object by reflection or transmission. This results in a wavefield
V’(x, t ) = a(x)ei+‘x’V(x, t )
(1.1)
with slightly distorted phase +(x) and amplitude a ( x ) . It is mainly the distribution of the phase +(x, y) in a plane (x, y) near the object, which carries the wanted information. That information is extracted by superposition of V ( x ,t ) and V’(x, t) with the help of some kind of a beam splitter and looking at the resulting intensity
I ( x , y)=IV+V’12=IV12[(1+a2)+2a(x,y)cos+(x, y)l
(1.2)
in the image of the considered plane near the object. The result is an interference fringe pattern, where the phase information +(x, y ) is transformed by the cosine-function into an intensity distribution. As long as the local amplitude variations a(x, y ) are small compared with the fringe separation the phase information can be quantitatively extracted from this fringe pattern. In holographic interferometry at least one of the wavefields to be compared interferometrically is stored in a hologram. The hologram is usually recorded experimentally with the help of a reference wave. But computer-generated holograms constructed theoretically may also be used to supply special, previously not existing wavefields. In most cases the hologram itself also acts as the beamsplitter for the superposition of the two wavefields. Holographic interferometry has some unique properties which makes it superior to classical interferometry: (1) Wavefields from the same object, but under different conditions and at different times, can be compared. This is an essential prerequisite to compare interferometrically different states of solid objects with opaque, diffusely 3
4
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 1
scattering surfaces. (2) Transient wavefields can be frozen instantaneously by short exposure times, e.g. by pulsed or modulated laser light sources, and afterwards compared under stationary conditions. (3) Time averaged wavefields can be recorded and afterwards compared. This results in reduction of noise and increase of accuracy and stability. However, quantitative information on the interference phase +(x) can only be obtained reliably from the maxima and minima of the interference fringes, corresponding to multiples of 180" or T for the phase. Any interpolation between the fringes is difficult and not very accurate. Heterodyne holographic interferometry is an opto-electronic technique which overcomes this limitation and allows to determine the interference phase at any position within the fringe pattern with an accuracy of better than 0.4" or 1/1000 of a fringe. The basic idea of heterodyne interferometry is to introduce a small frequency shift between the optical frequencies of the two interfering light fields. This results in an intensity modulation at the beat frequency of the two light fields for any given point of the interference pattern. The optical phase difference is converted into the phase of the beat frequency. The two light fields are then described by their complex amplitudes
where a1,2are the real amplitudes, +1.2 the phases, and the optical frequencies. A photodetector placed at the point P(x) in the superposition of these two light fields sees the time dependent intensity
Comparison with eq. (1.2) shows that the interference phase +(x) = +l(x) - c$~(x),i.e. the optical phase difference between the two light fields, appears as the phase of the intensity modulation at the beat frequency 0 = w1 - w2. As long as R is small enough to be resolved by a photodetector, this modulation can be separated by an electronic filter centered at R and the phase can be measured electronically with respect to a reference signal at the same frequency. The interference phase can be measured essentially independent of the amplitude of the modulated signal and
1, 0
11
5
INTRODUCI'ION
Q
I
t
= rnn
771
X i = frinqe positions,
671
xn= positions of interest, tpn = measured phase
I
58 471
37 'Pn
271
71 0
-x Fig. 1.1. Comparison of fringe counting and interference phase measurement.
therefore also independent of the amplitudes al(x) and u2(x) of the interfering light fields. The accuracy for the electronic phase measurement can easily be better than 1" or 2 ~ / 4 0 0 . It is instructive to compare heterodyne interferometry and classical fringe counting interferometry with the help of Fig. 1.1, which shows the interference phase 4(x) as a function of the position x in an arbitrary direction across the fringe pattern. The positions Xi of the dark and bright fringes are determined by the maxima and minima of the cosinefunction in eq. (1.2). This means 4i = for the corresponding values of the interference phase, where m is an integer. Therefore only a limited number of samples of 4(w), equidistant in phase rather than in space, can be obtained. The accuracy of the determined fringe positions Xi depends strongly on the slope of $(x) and on the resolution of the intensity detection within the fringe pattern, because of the fact that the intensity is stationary at the maxima and minima of the cosine-function. Interpolation between the fringes is not very reliable since any intermediate value of the intensity in the fringe pattern depends both on phase 2nd average intensity, which is in general not constant across the image. The heterodyne interferometry overcomes this limitation, because phase and amplitude of the interference term can be separated electronically and the fringes travel across the image so that sensitivity and accuracy are the same at any position. As indicated in Fig. 1.1one may select any arbitrary position x, or even a set of equidistant positions, and determine the corresponding value 4, of the interference phase.
6
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 1
Heterodyne interferometry was first described and experimentally realized for a conventional two-beam interferometer by CRANE[19691. The optical heterodyne concept was also used by DENTINO and BARNES [1970] to measure the complex amplitude in a coherent optical image. In connection with early work on holographic television systems ENLOE, JAKES and RUBINSTELN [19681 suggested already a hologram heterodyne scanner to save spatial bandwidth by using a temporal carrier frequency, produced by a reference beam with frequency offset, instead of the high spatial carrier frequency introduced by the angular offset of the reference beam in conventional holography. The heterodyne technique can be applied, with some restriction, to nearly all known kinds of holographic interferometry. For this purpose it is only necessary that the two wavefields to be compared interferometrically can be reconstructed with slightly different optical frequencies. This can be accomplished in real-time holographic interferometry by a frequency offset between the reference beam, reconstructing the first wavefield from the hologram, and the object illumination, generating the second wavefield. In case of double exposure holography, however, the two holographically recorded wavefields have to be stored independently, so that they can be reconstructed with different optical frequencies. This is most conveniently achieved by using two different reference waves. The problems associated with two-reference-beam holographic interferometry and its application to diffusely scattering objects will be discussed in § 2 and § 3. Heterodyne holographic interferometry using two-reference-beam recording was first reported by BALLAFCD [197 11 for transparent objects and by DANDLIKER, INEICHEN and MOPIER [1973, 19741 for displacement measurement of diffusely scattering objects. The heterodyne technique is not applicable to time-average holographic interferometry of vibrations, since the average wavefield shows already frozen interference fringes in the hologram recording. Nevertheless, heterodyne holographic interferometry can be applied to vibration analysis by combining two-reference-beam holography with stroboscopic illumination. After all the list of heterodyne holographic interferometry includes: (1) real time holographic interferometry with high temporal resolution and accuracy, (2) double exposure holography of transparent or diffusely scattering objects, (3) high speed holographic interferometry using pulsed lasers for the recording, (4)holographic interferometry with stroboscopic illumination for vibration analysis, (5) holographic depth contouring using either dual illumination source or dual wavelength recording.
1, I21
INTERFEROMETRY WITH DIFFUSELY SCAlTERING SURFACES
I
7
2. Interterometry with Diffusely Scattering Surfaces
2.1 COHERENT IMAGE O F A ROUGH SURFACE
It is well known that laser light scattered at a rough surface has a granular structure. The statistical properties of the intensity in the image of a diffuse object have been investigated by LOWENTHAL and ARSENAULT [ 19701. The surface roughness is described by a complex reflection coefficient p ( x ) with statistical properties. The usual hypothesis is made that the complex amplitude of the scattered light field is stationary and Gaussian with independent real and imaginary parts having zero mean and the same variance. It is further assumed that the coherent imaging system, shown in Fig. 2.1, can be described by a space invariant amplitude impulse response function h(x), which is true in most practical optical systems. Using appropriate coordinates x, = (x,, yo) and xI = (xI,y,) in the object and image plane, respectively, the analytic signal V(xI) in the image plane takes the form
(2.1) where M = dJd, is the magnification and O(x,) is the light from the object smoothed with respect to the surface roughness (cf. e.g. GOODMAN [1968]). Therefore O(X,) describes the macroscopic shape of the object surface. The impulse response function is related to the pupil or
Fig. 2.1. Coherent imaging of the object plane 0 through the aperture P and the lens L to the image plane I.
8
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
aperture function P(x,) by the Fourier transformation
XI - Mx,)
= [l/fip(Ad,)l
I
dZx,P(xd exp { - ikx,[(x,/d,) - (xo/do)ll,
(2.2) where k = 2rr/h is the wavenumber of the light. Following eq. (2.2) the impulse response function is normalized, i.e. J lhI2 d2xI= 1, and
I
4 = lP(xp)lzdzxp
(2.3)
is the effective area of the pupil. From eq. (2.2) and the similarity theorem for Fourier transformations one gets the relation
(Ax), = M ( A x ) , = d,A/(Ax),
(2.4)
between the width (Ax), of the pupil P(x,) and the diffraction limited resolution (Ax), in image or (Ax), on the object. The resulting intensity in the image I(X1)
= V(X1) V*(X,)
shows speckles due to the random variations of the phase of the reflectivity p. To evaluate the statistical properties of V(x,) and I ( x , ) ensemble averages over a large number of diffusors p(x,) for the same average surface O(x,) are calculated in the usual manner. For this purpose the correlation functions of p(x,) have to be known. Since the correlation peak of the surface roughness is very narrow compared with the resolution (Ax), of the image system, the correlation functions may be approximated by Dirac delta functions (DANDLIKER and MOTTIER [1971]). The twofold and fourfold correlation functions are then given by (P(X)P*(X’N
(2.6a)
= S(x-x’),
( p ( ~ ) p * ( x ’ ) p ( ~ ” ) p * ( ~=“ ’~) () x - x ’ )S(X“-X”’)+
S(X-X‘”)
S(X’ -x“).
(2.6b)
Using eqs. (2. l), (2.5) and (2.6) the autocorrelation functions of the image amplitude and the image intensity are found to be
(V(x,)V*(x;))=
I
d’x,
lOl” ( x , ) h ( x , - M x , ) h * ( ~ ; - M x , ) ,
(k)I(xf))= ( I ) ( X I ) ( m : ) + I(V(X1) V*(x;))l2.
(2.7a) (2.7b)
1, 821
INTERFEROMETRY WITH DIFFUSELY SCATITRING SURFACES
9
As already stated by LOWENTHAL and ARSENAULT [1970] the mean intensity in the image plane
r
as obtained from eq. (2.7) for xI=x;, is seen to be equivalent to an incoherent imaging of the object given by the intensity distribution lblz(x,) = d(x,)d*(x,). Since the object intensity (x,) can be assumed to change slowly with respect to the resolution of the imaging system, the shape of the autocorrelation function of the amplitude, as given in eq. (2.7a), is essentially determined by the autocorrelation function
lol’
of the impulse response function h(x,). Therefore the so-called speckle size, i.e. the correlation length of amplitude or intensity in the image, is equal to the width of ch(xI), which is the diffraction limited resolution (AX), of the imaging system. Using the Fourier transform relation of eq. (2.2) the autocorrelation Ch(x1) is directly related to the pupil function P(x,) through Ch(xJ= (l/Ap)
J dZXpIP(xp)lzexp {ikxIxP/dIl,
(2.10)
where d, is the distance between lens and image plane (Fig. 2.1). The speckles in the image of a coherently illuminated rough surface introduce substantial intensity noise, which is not present undex incoherent illumination. For a review of the statistical properties of laser speckle patterns see e.g. GOODMAN [1975]. The influence of speckle noise on the accuracy of intensity measurements in the image plane is characterized by the relative average variations (AIz)/(ID)2 of the detected intensity IDand depends on the ratio of the detector size and the speckle size. The detected variations are reduced by the average number N of independent speckles or correlation cells within the detector area. The resulting noise is given in a good approximation by
+
= 1/(1 N ) .
(2.11)
10
[I, 8 2
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
The number N of speckles within the detector area AD can be calculated from the autocorrelation function c h ( x I ) , given in eqs. (2.9) and (2.10). For uniformly transmitting pupil P(x,) and uniformly sensitive detector area A,, the speckle number N in case of N >> 1 is found to be
(2.12) For practical purposes the area A, of the pupil is obtained from the F-number and the focal length f of the lens through the relation A,
= nf2/4F2.
(2.13)
If the detector size A, is smaller than the speckle size one gets N = 0 and eq. (2.11) yields the original speckle contrast of the image intensity, which can be obtained from eqs. (2.7) with x I = r ; and is known to be (A12)/(I)”= ((12)-(1)2)/(1)2 = 1. Equations (2.11) and (2.12) make clear, that a reasonably accurate intensity measurement in the coherent image of a di hs e l y scattering object can only be obtained at reduced spatial resolution, well below the diffraction limit of the imaging system.
2.2. HOLOGRAPHIC INTERFEROMETRY OF OBJECTS WITH A ROUGH
SURFACE
The coherent superposition of the light fields from two diffusely scattering object surfaces o , p , and 0 2 p 2 does result in a new light field V b I )
= VdXI)
=
I
+ VZ(X1)
d 2 ~ , [ d ~ ( ~ , ) ~+~~2(x,)p2(x,)lh(x,(~,) Mx,),
(2.14)
which shows also speckles. In spite of the fact that the two coherent light fields VI and V, always interfere locally or microscopically, generating a new speckle field, it is not evident that also interference fringes, i.e. macroscopic interference, can be observed. To distinguish between microscopic interference, leading to speckle noise, and macroscopic interference, leading to fringes in the image, one has to investigate the averaged intensity ( W*)or more specifically the averaged interference term (V,(XI)
e(Xd>
I, 521
INTERFEROMETRY WITH DIFFUSELY SCATTERING SURFACES
11
For non vanishing average interference eq. (2.15) requires that the two random phase distributions pl(xo) and p2(x;) are correlated, i.e. (pI(xo)p~(x&)) # 0, at least for two mutual positions x, and x;. This means that the two rough surfaces have to be microscopically identical and that the same polarization of the scattered light should be selected, since the statistical properties of the light from most kinds of surface roughness are polarization dependent. Moreover, the speckles depend strongly on the observation aperture. It was shown by D ~ D L I K E MAROM R, and MOTTIER [ 19721 that therefore interference fringes are not visible if two completely different apertures for the observation of the fields V1 and V, are used. This implies for practical applications that only two states of the same solid object with a rough surface under identical illumination, observation and polarization conditions can be compared interferometrically with each other. And this is only possible with the help of holographic interferometry, because the two wavefields V1 and V, cannot originally exist at the same time. At least one wave field has to be stored in a hologram to be compared afterwards with the second wave field. The basic setup for holographic interferometry of objects with diffusely scattering surface is shown in Fig. 2.2. The observed part of the object surface is supposed to be in or very close to the object plane (xo, yo) of the imaging system consisting of the lens L with the pupil P. The hologram H just acts as the intermediate storage medium for the observed wavefields and can be located anywhere between the object and the lens, but a position close to the lens gives the least limitation of the field of view. The object is illuminated from the point source Q and the
Fig. 2.2. Holographic interferometry of diffusely scattering surfaces. Illumination source Q, object point P with displacement u, observation point A, hologram H.
12
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 5 2
mean direction of observation is determined by the center A of the lens aperture. The states of the object surface to be compared interferometrically differ by a slight displacement u(x,). The second wave field is therefore following eq. (2.1)
I
Vz(xI)= d 2 x , b ( ~ o + u ) ~ ( x o + u ) h ( ~ I - M r , ) . (2.16)
The displacement of the rough surface has two essentially different effects on the field Vz(x,) in the image plane: First, a change of the phase due to the change of the optical path length from the source Q to the lens aperture when the point P is displaced by u to its new position. From this interference phase
4 = u(k, -kA)= 2kuE
(2.17)
the displacement component (uE)in the direction of the sensitivity vector E, can be determined (SOLLID [1969]). As shown in Fig. 2.2 k, and k, are the wave vectors of the illuminating and the observed light, respectively. Second, the speckle field in the image plane ( x I , yI) is shifted by uI= Mu, due to the transverse displacement u, of the rough surface in the object plane (xo, yo). This effect is related to the so-called fringe localization (VELZEL[1970]) and results in a reduction of the fringe contrast. Additional phase change and transverse shift of the wave field may be caused in two-reference-beam holographic interferometry by misalignment or change of the wavelength (see § 3). Anyhow, the second wave field is then described by V2(xI)= V,(xI+uI)ei*(xJ
As long as the variations of the object intensity 16/’(x) and the interference phase C#J(X) are resolved by the imaging system one gets from eqs. (2.1), (2.9) and (2.18) for the average interference term
I
( V , G ) = lblze-i4(x~)d2x,h(xI-Mx,)h*(xI+uI-Mx,) =
1Olze-i+(xJC,, (XI).
(2.19)
The contribution of the interference term depends essentially on the autocorrelation C, of the impulse response function. Following eq. (2.9)
I, $21
13
INTERFEROMETRY WITH DIFFOSELY SCATERING SURFACES
Fig. 2.3. Reduction of the fringe contrast y due to the transverse displacement w1 of the speckle patterns in the image plane in terms of the diffraction limited resolution (Ax),= hdI/D for a circular aperture of diameter D.
the autocorrelation c h defines also the speckle size in the image. Therefore eq. (2.19) means that the interference fringes are only visible as long as the mutual shift of the speckle patterns is smaller than the speckle size. The actual fringe contrast compared with the maximum possible fringe contrast is given by y(u1) = IW, ~
1.
(2.20)
~ 1 / ~ ~ 1 ~ 1 1= 2 I ~C ~ h (1~ ~, ) 2l ~1 2 ~ ~ ~
For well designed imaging systems the pupil function P(xp)is symmetric. Therefore c h is, following eq. (2.10), a real valued function and does not contribute any additional phase to the interference term in eq. (2.19). For the most common case of a circular aperture P(xp)of diameter D the autocorrelation c h and the fringe contrast y are found from eqs. (2.10) and (2.20) to be given by the well known Airy function Y ( U J = 2J,(.srDu,/hd,)/(.srDu,/Ad,),
(2.21)
where uI is the transverse mutual shift of the speckle patterns in the image, dI is the distance from the lens to the image plane, and J1 is the first order Bessel function. The reduction of fringe contrast with increasing transverse shift uI in terms of the diffraction limited optical resolution AX)^ = hd,/D in the image is shown in Fig. 2.3. 2.3. HETERODYNE INTERFEROMETRY OF ROUGH SURFACES
In the case of heterodyne interferometry of rough surfaces the statistical properties of the speckle fields and their mutual correlation, discussed
14
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
11, 5 2
in the previous section, have to be considered. Therefore the complex amplitude of the two interfering light fields given in eqs. (1.3) for smooth wavefields of transparent or reflecting objects have to be replaced by V,(x, t ) = al(x)u(x)eiolt
v,(x, t ) = a,(x)[yu(x)e'""+
(1- y2)fw(x)]eiw2'.
(2.22)
Both V, and V, are similar speckle fields but at the two slightly different optical frequencies w , and w,, respectively. The statistical aspects are brought in by the two Gaussian distributed, complex variables u ( x ) and w(x), which are assumed to be normalized and uncorrelated. The statistical properties of u ( x ) and w(x) are derived from eqs. (2.6) and (2.7). They can be summarized as follows:
The complex amplitudes at different positions x and x' are only correlated within one speckle, the size of which is equal to the width of the autocorrelation C, of the impulse response h(x,), as defined in eqs. (2.9) or (2.10). Using eqs. (2.22) and (2.23) it is verified that a, and a2 are the average real amplitudes of V, and V,, i.e. (lVllz)= a:, (IV,l") = a;. The fact that V, is assumed to consist of two uncorrelated parts takes into account that the two speckle fields may be mutually shifted and exhibit only reduced macroscopic interference with fringe contrast y, as described by eqs. (2.19) and (2.20). Indeed, the average interference obtained from eq. (2.22) is only ( V, = y[a,a, exp i(0t - +)I. The time dependent intensity within one speckle is therefore
e)
I ( x , t)=aS (uI2+a: lyve'Q+(1-y2)~w)2 +2a,az Re{[yuu*ei~+(1-y2)fuw*]ei"'}.
(2.24)
The ac-signal at the beat frequency 0 = w , - w , consists of two essentially different contributions, viz. the correlated part yuu* and the uncorrelated part (1- y2):uw*. The uncorrelated part has arbitrary phase and thus zero average amplitude. The corresponding detector signal is therefore composed of a true signal and a spurious signal of arbitrary phase. The spurious signal introduces a statistical phase error 6 4 which may limit the overall accuracy of the measurement. The error can be reduced by averaging over several speckles, using a detector of area A, which covers N speckles with uncorrelated complex amplitudes, i.e. uncorrelated u,
1, 821
INTERFEROMETRY WITH DIFFUSELY SCAlTERING SURFACES
15
Fig. 2.4. Phase error 84 due to the superposition of correlated signals i, and uncorrelated signals i,.
and wn. The complex amplitude of the detector signal is then given by N
iN
=
C
N
in= i,
n=l
C [y.+,u:ei+ + (1 - y2)fv,,w:1eia* = + i,, 1,’
(2.25)
n=l
where the average signal amplitude i, = 2a,a2(AD/N)of one speckle area (ADIN) is introduced. The correlated and the uncorrelated contributions to the detector signal iN are i, and i, respectively. As shown in Fig. 2.4 the correlated contributions add up coherently whereas the uncorrelated ones yield a relatively small signal of random phase 4 as a result of their random walk behavior. For small phase errors one gets from Fig. 2.4
(64’) = (liul’ sin2 W(lic12) =~(ILI~)/(I~~I~),
(2.26)
since q!~and li,l are uncorrelated and (sin2 +) = for random phase. Using eq. (2.25) and the statistical properties of v, and wn summarized in eqs. (2.23) one finds
(li,,12)
= i:(l-
y2)
1
= i,2(1- y2)~(lu1’)()wI’)
n. m
=(:i
1- y2)N.
(2.27b)
The resulting phase error is thus
84 = [( 1- y2)/2y2(N + l)]k
(2.28)
The numerical evaluation of eq. (2.28) is shown in Fig. 2.5 for N = 1, N = 200 and N = 2 X lo4. The number N of speckles within the detector area A, can be calculated with eq. (2.12), knowing aperture and
16
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, § 2
Fig. 2.5. Phase error &$ versus fringe contrast y for different numbers of speckles N within the detector area.
geometry of the imaging system. It is seen from eq. (2.28) that the phase error 84 is zero for unity fringe contrast y(u,) = 1 independent of the number N of speckles within the detector area. If the speckle fields are slightly displaced, and therefore y < 1, the phase error can be reduced by averaging over a number N of speckles. For the most common case of a circular aperture P(xp)the fringe contrast is given in eq. (2.21) and Fig. 2.3. The increase of phase error 8 4 with increasing displacement uI in terms of the optical resolution (AX),= Ad,/D in the image is shown in Fig. 2.6 for a single speckle ( N = 1) and for averaging over N = 200 and N = 2 X lo4 speckles, corresponding to detector diameters of about 15(Ax), and 15O(Ax),, respectively. In Fig. 2.7 the required speckle number N to get a phase error of less than 84 = 0.3" is given for different transverse mutual shift u, of the speckle patterns. It is seen that even for a transverse shift of u,=O.SAd,/D, which is half the optical resolution, N = 2 x lo4 speckles are sufficient to measure the phase with an accuracy of better than Sb, = 0.3". It is instructive to compare heterodyne interferometry and classical fringe intensity detection with respect to the accuracy of interference phase measurement. First of all it should be pointed out that in case of intensity detection the interference phase can only be reliably deduced from the positions of the fringe maxima and the fringe minima, because any intermediate value of the intensity depends on both phase and average intensity, which is in general not constant across the image. The
I, $21
INTERFEROhtBTRY WITH DIFFUSELY SCA7TERING SURFACES
17
Fig. 2.6. Phase error S ~ versus I transverse displacement uI in terms of diffraction limited resolution AX)^ = AdJD for different numbers of speckles N within the detector area.
heterodyne interferometry overcomes this limitation, as discussed in 9 1 and illustrated in Fig. 1.1. Moreover, in the interference pattern of diffusely scattering objects the intensity measurements are disturbed by the speckle noise, as described in D 2.1. The intensity fluctuations in a speckle field are equal to the average intensity itself and therefore no meaningful intensity measurement can be made at all with a detector size smaller or equal to the speckle size. This means that N >> 1 is required for intensity measurements. Since the variation of intensity is quadratic with phase in the maxima and minima
Fig. 2.7. Minimum number of speckles N required for a phase error less than 84 = 0.3" versus transverse displacement uI, in terms of (Ax), = Ad,/D.
18
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 3
of the cosine fringe function one gets for the phase error
(w), = (2 A I , , / I , ~ * = &(N+
1)-li4
( N " 11,
(2.29)
where eq. (2.11) has been used for the value of the speckle noise. A detector size corresponding to N = 2 X lo4, which is sufficient for 6 4 = 0.3" in heterodyne interferometry even for considerable transverse speckle shift (see Fig. 2.7), yields only (W), = 7" or 1/50 of a fringe. The required speckle numbers N for higher accuracy are prohibitive, because of the fourth power dependence in eq. (2.29). In heterodyne interferometry the intensity noise of the speckles appears only as a variation of the amplitude of the beat signal, from one observation point to another, which does not directly influence the phase measurement. To keep these local amplitude variations below 20%, which is sufficient to avoid a substantial reduction of the electronic signal to noise ratio, a speckle number of N = 2 5 within the detector area is already sufficient. Larger speckle numbers may be required to reduce .an eventual phase error 64 in case of reduced fringe contrast y due to a mutual shift of the interfering speckle patterns.
§
3. Two-Reference-Beam Holographic Interferometry
Double exposure holographic interferometry is the most common and convenient kind of holographic interferometry. It is therefore very important to find a solution to use the heterodyne technique together with double exposure holographic interferometry. This is possible if the two wavefields are stored independently in the hologram, so that during reconstruction the different frequencies for the two interfering light fields can be introduced by using two reference waves of different frequencies. The most convenient realization is to use two different reference waves for that purpose (BALLARD [1971], DANDLIKER, INEICHENand MOTTIER [1973, 19741). Multiple-reference-beam holography was already proposed and applied by DE and SEVIGNY [1967a, 1967bl for inspecting phase objects, by LOHMANN [1965], BRYNGDAHL [1967] and FOURNAY, WAGGONER and MATE[1968] for holographic recording of polarization, or by BALLARD[1968], TSURUTA, SHIOTAKEand ITOH[1968] and KERSCH [ 19711 as means for introducing flexibility into conventional double exposed holograms. Indeed, if each image has its own reconstruction beam, one has access to each image separately, as well as to their mutual
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TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
19
interference pattern. A general analysis of multi-reference-beam holography was already given by POLITCH, SHAMIR and BEN-URI [1971], POLITCH and BEN-URI[1973], and SURGET [1974]. The properties of tworeference-beam holography and their implications to quantitative interferometry were analyzed by DANDLIKER, MAROMand MOTTIER[1976]. Special attention was given to the multiplicity of the reconstructed images, guidelines to separate the useful from the disturbing reconstructions, and the influence of misalignment on the fringe pattern in the image. 3. I . RECORDING AND RECONSTRUCTED IMAGES
The setup for recording a double exposure hologram utilizing two reference beams R, and R, with the corresponding objects 0, and 0, (0, being a distorted version of 0,) is shown in Fig. 3.1. The processed photographic plate will have an amplitude transmittance T equal to = T~ -
0 ( R u1 -tR uy -k R z u2+ R , uz),
(3.1)
where U1 and U, are the complex amplitudes of the light fields corresponding to 0, and 0, at the recording photographic plate, and T,, p are constants. Upon illumination of the hologram with the two reference beams R , and R, eight reconstructions result from the product of R, + R z with T. As shown in Fig. 3.2, two pairs of reconstructions, viz. R,RyO,, R,R;O, and R,R, Oy, RIR,Oz, will be in exact register giving rise to interference. Object
-
\
.Photographic Plate
Fig. 3.1. Setup for the recording of double exposure holograms with two reference beams. R,and 0, are used for the first exposure, while R, and 0, are used for the second one.
20
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, § 3
Fig. 3.2. Reconstructed images in case of two-reference-beam holographic interferometry. Reconstruction with R , only (A), R , and R, simultaneously (B), and R, only (C). (DANDLIKER, MAROM and MOTTIER[1976].)
1, 8 31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
21
The images R,RyO,, R,R,*O, and R,R,Oy, R2R20,*are the primary and conjugate self-reconstructions, respectively, of the two independent holograms with their own reference waves. In addition, R,RyO,, R,RzO, and R,R, OT, R, R 2 0 S are the primary and conjugate crossreconstructions, respectively, of the two holograms with the wrong reference waves. The primary reconstructions consist of terms proportional to the original wave fields 0, or O,, while the conjugate ones consist of terms proportional to the conjugate wave fields 0: or 0:. The locations of the reconstructed images depend on the mutual position of the reference waves and the object during recording. To enable better visualization of the various image positions, the direction and the Gaussian focus distance of the various reconstructions of one object point will be discussed in the following. The reference waves and the object waves are assumed to be spherical and observed through a small aperture, which is determined by the hologram plate. Therefore the wave fronts in the hologram plane can be described by a propagation vector k and a radius of curvature p , which is equal to the distance of the corresponding point sources to the hologram plane. The directions of the reconstructed waves are calculated and visualized with the help of the k-space where each wave is represented by its wave-vector k, as shown in Fig. 3.3. The wavevectors are k, for the recorded point, k, for the recording reference wave, k, for the reconstructing reference wave, k, for the primary, and k, for the conjugate reconstruction. The directions of the reconstructions are found from the plane wave approximation of the complex amplitudes of the reconstructed waves in the hologram plane. One gets for the primary reconstruction
6dxH>= exp i(wt -kpxH) = exp i[wt - (k,- k, + k , ) ~ H l
(3.2a)
and for the conjugated reconstruction
OC(xH) = exp i(wt-kkcxH)= exp i[wt-(k,+k,-k,)x,],
(3.2b)
where x H = ( x , y) is in the hologram plane. The eqs. (3.2) show that the projections m of the wavevectors k in the hologram plane have to match. This means for the two reconstructions mp = m, - mR+ mor
m, = m, + m R - m,.
(3.3)
In addition the magnitudes of the wavevectors of the reconstructing reference lk,l and the reconstructed wave (kplor lk,l have to be equal. The
22
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 3
qp tk’
I
Fig. 3.3. Wavevectors in k-space for hologram recording and reconstruction: references k, = k,, object k,, primary reconstruction k,, conjugate reconstruction k,.
third component k, of either k, or k, is therefore obtained from
k,
=
(lkl”- lmp.c12)~.
(3.4)
This equation has only a real solution if Im(5 Ik,l, which means that the reconstructed wave does only propagate in space if ( m l slkrl, otherwise it becomes an evanescent wave. The configuration in the k-space for identical recording and reconstructing reference waves (kR= k,) is sketched in Fig. 3.3. The primary reconstruction is identical with the recorded wave (kp=k,) and the projection m, of the conjugate wave is constructed following eq. (3.3). The direction of propagation of the various reconstructed waves for a two-reference-beam hologram are constructed and visualized with the help of a sphere of radius k in the k-space where each propagation vector
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TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
23
is represented by a point on the sphere, For the example sketched in Fig. 3.4 the object vectors k,, and k,,, have been chosen along the z direction, i.e. normal to the hologram plane. The optimum choice of the reference source to avoid disturbing overlapping of the different reconstructions can be deduced from Fig. 3.4. Symmetrically positioned reference beams with respect to the object location are not recommended, since for this case the two pairs of interfering images will be reconstructed along the same propagation direction. The best choice is to have both reference sources on the same side of the object with a mutual separation as smail as possible. This means that the angular spacing between the two reference waves should be just larger than the angular size of the object in the corresponding direction, so that the reconstructions R,RTO,, R,R:O2 and
Fig. 3.4. Propagation directions associated with various image terms are marked as points on the surface of the k-sphere. The wave vectors of the object and the reference waves are k,, k , and k,, respectively ( D ~ D L I K E MAROM R , and MO’ITIER[1976]).
24
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 5 3
R2R,0T, R,R,O: carrying interference information are mutually separated and do not overlap with R2RT0, nor R1R;02. The wave fronts of the reconstructed waves are in general not spherical but astigmatic, even in the paraxial approximation (INEICHEN, KOGELSCHATZ and DANDLIKER [1973]). However, a Gaussian focus distance pp and pc for the primary and the conjugate reconstruction of the point source can be defined and calculated following CHAMPAGNE [1967]. One finds
where po, pR and pr are the radii of curvature of the object wave, the recording reference, and the reconstructing wave, respectively. It is seen from eq. (3.5) that for similar radii of the reference waves (pr=pR) the primary reconstructions are always virtual and at the same distance behind the hologram as the original object (pp= po), whereas the conjugate reconstructions can either be real (p,
O). In the special case of po=pr=pR the conjugate image appears at the same distance po behind the hologram as the primary image. This arrangement was used to record the hologram reconstructed in Fig. 3.2. For reasons of aberrations it is recommended to work exclusively with the interfering pair of primary, self-reconstructed waves R,R?O, and R,R?O,. Two-reference-beam holographic interferometry is used in the following way to apply the heterodyne method to double exposure holographic interferometry (see Fig. 3.1): The first object state 0, is recorded using beam R, as a reference. A stop is placed in beam R, so that it does not illuminate the hologram plate. The second object state 0, is recorded in the same manner, except that the beam R, is used as a reference and beam R, is stopped. All light fields during recording have the same optical frequency 0,. After processing the hologram is reconstructed with geometrically identical reference waves R , and R,, but of slightly different optical frequencies o1 and w,, respectively. The reconstructed wave fields 0, and 0, have the same frequencies as their respective reference waves. This meets precisely the conditions necessary for heterodyne interferometry between these two reconstructed wave fields. The frequency difference 0 = w , - w2 has to be small enough to be resolved by photodetectors (0/27r< 100 h4Hz). The relative frequency change 0 / w , , , is therefore smaller than 2 x lo-’ for visible light. The resulting changes
1, $31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
25
in the wave propagation are thus very much below any optical resolution and can be completely ignored.
3.2. ALIGNMENT AND WAVELENGTH SENSITIVITY
Two-reference-beam holographic interferometry is expected to be sensitive to repositioning errors and wavelength changes because the propagation of the two reconstructed wave fields to interfere is differently affected by these alterations. DANDLIKER, MAROMand MOTTIER[1976] have shown that misalignment of both hologram plate o r reference waves causes mainly a linear phase deviation between the reconstructed wavefields across the hologram. In the following the analysis will be carried out for collimated reference waves. This describes the essential aspects, even for the more general case of spherical waves. Using eq. (3.2) the primary reconstructions and of the recorded wave fields 0, and 0, with the altered reference waves R: and R: respectively, are given by
ol
o2
where Ul(xH) and UZ(xH)are the complex amplitudes of the wave fields 0, and O2 in the hologram plane xH,respectively. Both the wavevectors k,,, and the hologram position xH may have changed with respect to the hologram recording. For small changes Akl,2= k;,2- kl,2 and AxH= xh - xH from recording to reconstruction the resulting additional phase difference $ between the reconstructed waves 0, and is found from eq. (3.6) to be
o2
$(xH)
= $1 - $2
(k, - kz) AX,
+ (Akl -Ak2)XH-
(3.7)
The change of the hologram coordinates can be described by
A x , = ~ + ( wXXH),
(3.8)
where t is the translation vector and w = (Aa, A@, A y) is the rotation vector for small rotations ha, A@, Ay around the x, y, z-axes, respectively. It is seen from eqs. (3.7) and (3.8) that a pure translation of the hologram causes only a constant phase shift, whereas a rotation introduces a linear phase deviation across the hologram plane. Since a constant phase shift
26
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, B 3
a)
Fig. 3.5. Reconstruction conditions in two-reference-beam holography after: (a) change of the mutual angle 26 by 2 AS or (b) change of the wavelength corresponding to Ak.
can be ignored the only relevant contributions in eq. (3.7) are +(TH)=[(~~-~~)xw]xH+(A~~-A~~)xH.
(3.9)
Comparison of the first and the second term in eq. (3.9) suggests that changes A&, and Akz of the reference waves could be compensated by a rotation of the hologram. This is only partially true, because the vector product ( k , - k,) x w is always perpendicular to the difference vector A k = ( k , - k2) so that the component of (Akl - Ak2) parallel to Ak can not be compensated for. From the geometrical interpretation of this component in Fig. 3.5 one finds that it is either due to a change of the mutual angle 26 between the two reference waves or due to a change of the wavelength, i.e. (k:l= (k;l# Ik,l= lk21, or a combination thereof. The first term in eq. (3.9) shows that the phase deviation caused by a hologram rotation depends only on the difference A k = ( k , - k2) of the two wavevectors and does vanish for identical reference waves, which is the case in classical double exposure holography. The maximum slope of the corresponding phase deviation is obtained from eq. (3.9) as \grad +,.,l=[(Aky A y - A k , Ap)’+(AkZ A a - A k , Ay)2]$,
(3.10)
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TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
21
where the z-axis is assumed to be normal to the hologram surface and Aa, AP, A? are the angular rotations around the x, y, z-axis, respectively. With a partially symmetric choice of the reference waves one can obtain Ak, = 0, so that the rotation A? around the normal to the hologram plate remains as the only essential contribution to the phase deviation. A change 2A6 of the mutual angle 26 between the reference waves produces following Fig. 3.5a a phase deviation 48
(xH) = 2 k cos 6 ___ ( k 1 - k 2 ) ~ H A 6 = A 6ctanS (k,-k2)xH. Ikl - k2l
(3.11)
From Fig. 3.5b one gets for a change of the wavelength
where Ak = lk;.,I- Ik,.,J = (2r/Af)-(2r/A) is the change of the length of the wavevectors due to the different wavelengths A and A ’ used for recording and reconstruction, respectively. Therefore the wavelength sensitivity decreases for decreasing difference between the two reference waves R , and R , and does even vanish for identical reference waves, which is the case in classical double exposure holography. Misalignment of hologram and reference waves manifests itself by the appearance of fringes in the hologram plane, which are easily recognized looking at the hologram plate under the reconstruction condition. A pattern of parallel fringes appears if the object has not changed between the two exposures and if plane reference waves are used. In the practical cases of partially deformed objects and slightly diverging reference waves the fringes deviate somewhat from straight lines, but they still show a preferential direction. Inspecting eqs. (3.9) and (3.11) the fringe direction can be related to the different types of misalignment. - Rotation around an axis in the hologram plane (Aa, AP) yields fringes parallel to that axis, and only if Akz = ( k , , - k , , ) f O , - Rotation around the hologram normal (Ay) yields fringes perpendicular to the projection of Ak = k, - k, on the hologram plane, - Change of the mutual angle of the reference waves (AS) yields fringes parallel to the projection of Ak = k , - k, on the hologram plane. One may take advantage of these properties to readjust systematically the reconstruction setup for minimum misalignment. In any case the alignment requirements are considerably less stringent than in real-time holography. If the same experimental setup is used for recording and
28
[I,
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
(i 3
reconstruction, only a slight readjustment of the rotational position of the hologram plate is necessary to get minimum or zero fringes across the hologram and, as will be shown in § 3.3, at the same time maximum fringe contrast in the image.
3.3. EFFECT OF MISALIGNMENT ON THE INTERFERENCE PATTERNS
For interferometric studies of object deformations one looks at the interference pattern as it appears in the image of the object. It is therefore important to examine the effect of hologram or reference wave misalignment on the formation of this interference pattern. For this purpose one has to analyze the effect of misalignment on the formation of the coherent image. As shown in § 3.2 misalignment causes mainly an additional linear phase deviation ICIbH)
(3.13)
= kPXH
between the reconstructed wave fields across the hologram plane xH. The effect of these linear phase deviations on the image of the object, obtained by a lens in the reconstructed wave field, can be simulated by a wedge positioned in the hologram plane affecting only one of the reconstructions, as sketched in Fig. 3.6. The resulting effect on the fringe MAROMand pattern in the image plane x4 was analyzed by DANDLIKER, M O ~ I E[1976] R as a function of the distance b between hologram and imaging lens. As will be shown, a linear phase distortion $(x2) = kpx2 in the x2(hologram)-plane is equivalent to two phase distortions, one in the xl(object)-plane and the other in the x3-(pupil)-plane. The misalignment effect is arbitrarily assigned to the 0, reconstruction. The propagation of the wave field 0, from the object- to the pupil-plane is therefore, in the Fresnel approximation, described by
I
Ul(x3)= d2x1d2x2O1(xl)exp
- x3I 2b
1x2
where k p is the gradient of the linear phase distortion in the hologram plane. The exponent in eq. (3.14) can be rewritten as
I, Q 31
29
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
H
L
I
L
I
B) Fig. 3.6. Schematic arrangement for evaluating misalignment fringe effects on the interference pattern. (A) Setup for inspecting the interference of 0, and 0, in the image plane I. The image is formed by the lens L. 0, and 0, are reconstructed from the hologram H by their respective reference waves R , and R,. (B) Equivalent setup to (A), where the misalignment of R , is substituted by a wedge D , in the hologram plane H acting on the R image 0, only but not on the image 0, (DANDLIKER, MAROMand M O ~ I E[1976J).
to perform the integration over x2. The result is U,(x3)=
ik j d2X,O,(X,) exp (m {ab lpI2-2bpx1
+(XI
-x312-2apx,}).
(3.16) Compared with undistorted propagation (p = 0) one gets a constant phase shift proportional to IpI2 which can be ignored, a linear phase distortion 4,(xi)=-kpx,b/(a+b)
(3.17a)
across the object, and a linear phase distortion
4p(xJ = -kpx,a/(a + b )
(3.17b)
across the pupil of the lens. Following eq. (2.2) the pupil function P(x,) and the impulse response function h(x4) are related by a Fourier transformation and therefore the &(x3) results in an additional lateral shift of the image by u4 = c a p / ( a+ b ) = Map, (3.18) where M = c/(a+ b ) is the magnification of the imaging system.
30
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, § 3
For the discussion of the influence of these two distortions on the fringe pattern one has to distinguish between holographic interferometry with smooth wavefields from transparent or reflecting objects and holographic interferometry with speckled wavefields from diffusely scattering objects. For smooth wavefields both distortions will alter the fringe pattern; &,(xJ as an additive phase across the object and its image and u4 as a shearing of the two interfering wavefields. The fringe contrast is not reduced. For speckled wavefields, however, only the phase distortion 4"(xl) changes the interference pattern, whereas the lateral shift u4 does only reduce the fringe contrast (see § 2.2). Fortunately the phase distortion &(xl) can be nearly eliminated by placing the lens as close as possible to the hologram, so that b/(a+ b)<<1 in eq. (3.17a). Under this condition the lateral shift in the image plane is approximately u4=d1p, where d , = c is the image distance from the lens. The corresponding reduction of the fringe contrast y can be obtained for the most common case of a circular lens aperture of diameter D from eq. (2.21) or Fig. 2.3, using uID/AdI= pD/A. From the definition of p , given in eq. (3.13), it is seen that the phase variation across the pupil is $ = k p D = 2 ~ p D / Aand therefore pD/A is equal to the number of misalignment fringes within the lens aperture. Figure 2.3 indicates y = 0 for pD/A = 1.2. This means that the fringe contrast is nearly reduced to zero for one fringe across the lens aperture. Since the misalignment fringes are easily observed on the hologram plane, which is favorably located just behind the lens, the criterion of having less than one fringe within the lens aperture is very helpful to judge the realignment accuracy in tworeference-beam holographic interferometry. The effect of the lateral displacement of the speckle fields on the accuracy of measuring the interference phase by heterodyne interferometry is discussed in § 2.3. The resulting phase error &#J can be calculated with eq. (2.28) knowing the fringe contrast y and the number of speckles N within the detector aperture. The influence of misalignment on the interference pattern displayed in the image of the object, as described by eqs. (3.17) and (3.18), is experimentally verified. Some results for different pupil diameter D and different distances b of the lens from the hologram are shown in Fig. 3.7. The object was the same bent cantilever as in Fig. 3.2 located at a distance of a = 1.3 m from the hologram. All photographs of the reconstruction were taken with an f = 360 mm objective. The first picture, in Fig. 3.7 shows the interference pattern under ideal
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TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
31
Fig. 3.7. Influence of misalignment of the hologram o n the interference pattern demonR, and MOTHER[1976]. The distance between object and strated by D ~ D L I K EMAROM hologram is a = 1.3 m and the focal length of the imaging lens is f = 360 mm for all pictures. pD/A is the number of misalignment fringes across the aperture of the lens with diameter D.
conditions, i.e. no misalignment fringes on the hologram (p = O), taken with an aperture of fl9 close to the hologram ( b = 0.05 m). The next three pictures in Fig. 3.7 taken with the objective at the same position ( b = 0.05 m), demonstrate the influence of misalignment on the fringe contrast for different pupil diameter D. The misalignment yielded about one fringe
32
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, I 3
per 10 mm on the hologram which means p/A = 0.1 mm-'. The interference fringes are only visible for sufficiently small apertures. The number of misalignment fringes across the pupil is given by pD/A. The fringe distribution is exactly the same as for p = 0, but the speckle size increases with reduced apertures. The last picture in Fig. 3.7 is taken with the objective at a distance of b = 0.84 m from the hologram with an aperture of f/45.The interference fringes are clearly seen to be distorted by the misalignment and fringes appear also on parts of the object which have not changed between the two exposures. The analysis of two-reference-beam holographic interferometry has shown that the realignment is facilitated by the appearance of clearly visible fringes in the hologram plane. The complete elimination of these fringes indicates perfect alignment. However, perfect alignment is not necessary for the formation of the genuine interference fringe pattern in the image, since it is not altered and has enough contrast if the pupil of the observation optics is close to the hologram and there is less than one alignment fringe across the aperture. As illustration the alignment requirements for a typical two-reference-beam setup with a field of view of about 10" without overlapping of the different reconstructed images will be calculated. The reference waves are given by k , = k(sin 20°, 0, cos 20") and k , = k(sin 30°, 0, cos 30"), where k = 24A. Following eqs. (3.9), (3.11)-(3.13) the linear phase shifts for the different alignment errors are described by the vectors pw = [-O.O7Ap, (0.07Aa + 0.16Ay), 01, pa = (-1.8A60. O), and ph = (-0.16Ak/k, 0. O), where A a . Ap, Ay are rotations around the x, y, z-axis of the hologram, respectively, A6 is the change of the mutual angle 26 of the reference waves, and Ak/k=AA/A is the relative wavelength change. Allowing a reduction of the fringe contrast to y = 0.7 one gets from eq. (2.21) or Fig. 2.3 uID/Ad,= pD/A = 0.5, which means one half alignment fringe across the pupil. Assuming a pupil diameter of D = 30 mm and a wavelength of A = 633 nm (He-Ne-laser) one finds p = lo-'. The corresponding maximum tolerable alignment errors are Act = A@ = 1 . 4 ~ Ay = 0.6 x lop4, A 6 = 0.5 x lop5 and AA/A = 0 . 6 ~ The angular adjustment requirements can readily be fulfilled with standard optical precision mounts. Important wavelength changes occur in case of high-speed hologram recording with a pulsed laser, e.g. ruby laser at A = 694 nm, and subsequent reconstruction with a continuous laser, e.g. He-Ne laser at A' = 633 nm. Following eq. (3.12) the corresponding change of the wavenumber Ak/k=0.09 would require that the mutual angle of the
I, 5 31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
33
reference waves is less than 26 = to get y = 0.7 for D = 30 mm. Such small angular separations can be realized but the cross-reconstructions do completely overlap with the self-reconstructions. It will be shown in 0 3.5 that for a large enough number N of detected speckles interference phase measurements by heterodyne interferometry of better than 64 = 1" are still possible.
3.4. NONLINEAR CROSS-TALK
Nonlinear hologram recording leads to cross-talk in two-referencebeam holographic interferometry. This cross-talk generates spurious interference fringes and introduces an error in heterodyne holographic interferometry. It was shown by DANDLIKER and INEICHEN [1976] that the cross-talk depends essentially on the change of the recorded scene between the two exposures and that it can be eliminated by clearing the scene from any visible parts which remain unchanged in position. The basic setup for two-reference-beam holographic interferometry is shown in Fig. 3.1. In the Fresnel approximation the object wave in the hologram plane is described by
J
U(XJ = d2x O(X)exp {(ik/2d,)
IX -xHI2},
(3.19)
where O(x) is the complex amplitude at the object, k is the wavenumber, and do is the distance between the object and the hologram. The total recorded energy E(xH)during the two exposures T, and T2 is given by E(xH)=
lU1+R112+T2
(3.20)
1U2+R212,
where V1,2 are the two object waves to be recorded and R1,2the corresponding reference beams. Taking nonlinear recording into account, the amplitude transmission 7(xH)of the developed hologram is related to the energy E ( x H ) by dxH)= T,)+ P , ( E -Eo) + pz(E-Eo)'+
-,
(3.21)
where Eo = T , IR,12+ T2IR2l2is introduced as the bias energy. The reconstruction R17(xH)of that hologram with the reference-beam R , alone generates a variety of wave fields, as discussed in 0 3.1, some of them propagating like the original object waves 0, and 02, or U , and V2
34
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
respectively. The corresponding complex amplitude gram plane reads
O I ( ~ H= )lR112T , { p ,UI i2TiP2U1
[I.
P3
U,(X,) in the holo-
-t2T2P2 UIfiu,}.
(3.22)
The statistical properties of the light scattered from a rough surface 5 2.1, can be used to simplify eq. (3.22). Using eq. (2.6a), viz. (p(x)p*(x’))=ti(x-x’), one sees that the terms U,Ut V , contain only the following significant contributions: O(x) = O(x)p(r), as described in
where O,,(X>is the value of O,(x) smoothed with respect to surface roughness. Introducing the mutual intensity
eq. (3.22) can be written as
This equation shows clearly the nonlinear cross-talk, i.e. a contribution of the object wave U2 in the reconstruction of the hologram with the reference wave R , alone. This contribution is proportional to the nonlinearity parameter p2 of the hologram recording and the mutual intensity 112.
In a similar manner one obtains from eqs. (3.20) and (3.21) a reconstructed wave fi,(x,) propagating like the reference beam R,, namely
This means that reconstructing the nonlinearly recorded hologram with the first reference beam R1 alone generates the second reference beam R2 proportional to the nonlinearity parameter p2 and the mutual intensity 112.
I, 5 31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
35
The validity of the foregoing treatment of nonlinear cross-talk can be verified experimentally by some qualitative tests of conclusions and predictions from eqs. (3.24)-(3.26). The mutual intensity TI, depends essentially on the deformation of the object between the two exposures. Since 02(x) is assumed to be a slightly distortzd version of O,(x) the difference between 0, and 0, is given by a phase function +(x) and I,, is obtained from eq. (3.24) as (3.27) Therefore TI, is maximum for a constant phase shift across the object, e.g. no deformation at all, and I,, = 0 for many fringes over the object. A set of experiments was performed by DANDLIKER and INEICHEN [1976] to show the dependence of nonlinear cross-talk on I,,.For this purpose an object O(x) consisting of two parts A and B of nearly equal size was used. The following three experiments were performed: a) No change of the object (0,= 0, = A +B), corresponding to TI, = f, b) deformation of the upper part B of the object only (0,= A + B, 0, = A + B’), corresponding to I,, =if,c) deformation of both parts (0,= A + B, 0, = A‘ + B’), corresponding to I,, = 0. Then the holograms were reconstructed with the reference wave R , and the following experimental observations were made in agreement with eqs. (3.25) and (3.26): (a) No cross-talk fringes on the object (no deformation), clearly visible nonlinear reconstruction of reference R,. (b) Cross-talk fringes, corresponding to the deformation, on the upper part B of the object (Fig. 3.8a), clearly visible nonlinear reconstruction of reference R,. (c) No cross-talk fringes on the object (Fig. 3.8b), hardly visible nonlinear reconstruction of reference R,. Equation (3.24) indicates that the nonlinear cross-talk of the object 0, is independent of the presence of the reference R, during the hologram recording. This fact has also been verified by the observation that the spurious cross-talk fringes in case b) appear also if R, is blocked during the second hologram exposure. This effect demonstrates, that it is the unchanged part A of the object which acts as a diffuse reference (ARSENAULT [1971], COLLIER, BURCKHARDT and LIN[1971a]) for the rest of the object in nonlinear hologram recording and reconstruction. Additional experiments were performed to obtain more quantitative information on this nonlinear cross-talk, the contrast of the spurious
36
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 5 3
Fig. 3.8. Verification of the dependence of the nonlinear cross-talk on the mutual intensity f,, by D ~ D L I K Eand R INEICHEN [1976]. (a) Object part A (lower part) not changed between the two exposures (0,= A + B , 0, = A+B’). (The fringe contrast is enhanced by the photographic process.) (b) Both object parts A and B changed between the two exposures (0,= A + B , O,=A’+B’, i 1 2 = O ) .
fringes and their influence on the heterodyne holographic interferometry. From eq. (3.25) the contrast K of the cross-talk fringes is found to be
(3.28) where the average intensity f = I, = f,, and the normalized mutual intensity, or the complex degree of coherence, y12=f1,/fhave been introduced. For eq. (3.28) it has been assumed that the nonlinear terms are small compared to the linear ones, i.e. PI >> T1,2f12f,,,,,.Equation (3.28)
1, 831
37
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
suggests to introduce the dimensionless parameter ff
= T2T IP2IPII
(3.29)
for the relative magnitude of the nonlinearity. Experimentally a can be determined by comparison of the reconstruction intensity I of the object and the intensity Id2\”of the nonlinear reconstruction of the second reference beam R2. The main contribution to is the linear part (PI >> T,,2P2fn,) in eq. (3.25). Therefore, the relative nonlinearity can be determined through
oIl2 loll2
(3.30) where IU112/101(2 and IRz12/lR212are the ratios of the recorded and the reconstructed intensities of the object 0, and reference beam R2, respectively. These ratios can be measured easily. Since (yl2I2=1 in case a) where the object has not changed a can be calculated directly from eq. (3.30). For partial changes of the object, eq. (3.30) can also be used to determine experimentally the degree of coherence Iy121which is the most important parameter for the fringe contrast K as given in eq. (3.28). The result for the three cases of the object O = A + B are shown in Table 3.1. From the independently measured intensities of the parts A and B, respectively, in the hologram plane the mutual intensity y12(theor) can be calculated. From case a) with y l z= 1 the relative nonlinearity is calculated With this value the degree of coherence y12and the fringe as a = 8.8 x contrast K are calculated from the measured a 2 Iy1212,using eqs. (3.28) and (3.30). TABLE3.1 Experimental determination of nonlinear cross-talk and calculated fringe contrast by DANDLIKER and INEICHEN[19761. y ,,(theor) is calculated from the measured intensities due to objects A and B, respectively, in the hologram plane. a* IyI2l2is determined by eq. (3.30) from the measured intensity ratios. The fringe contrast K is calculated from a’ ly1212with a =8.8X as obtained from case (a). O,=A+B (a) 0, = A + B y,,(theor) lY121*
1 .0 7.8 x lo-’
Y12
1.o
K = 14ay12(
3.5%
O,=A+B (’) 0 2 = A + B ’
0.59 2.3 x lo-’ 0.54 1.9%
0,= A + B O,=A‘+B‘ 0.0 2.0 x 10-7 0.05 0.2%
38
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 3
These results were obtained with the following experimental parameters: single frequency Ar laser (A = 514 nm) with 500 mW as light source, spatial carrier frequency 700 mm-I for R, and 890 mm-' for R2, ratio between object and reference wave U,,2: R,,2= 1:3, Afga-Gevaert Millimask (neg. Type) as holographic plates, exposure time TI= T2= l s so that a density of D = 1.1 was obtained after about 2.5 min. development with D-19. As seen from Table 3.1 the agreement between y12(theor) and yI2 is within 5% of the maximum value y12= 1. But even in case c) with total deformation of the object the experimentally determined cross-talk does not vanish. This means that still some other parts besides the object are present in the holographically recorded scene and remain unchanged between the two exposures. The corresponding fringe contrast, however, is at least one order of magnitude smaller than the maximum possible value determined by the nonlinearity of the hologram recording. In heterodyne holographic interferometry the spurious fringes generated by the nonlinear cross-talk may introduce an error in the measured interference phase. Following eq. (3.25) the reconstructed object fields contain nonlinear cross-talk terms and are therefore altered to become (3.31) where the cross-talk coefficients are given by
r12
=I :, are complex The coefficients P i k are in general complex, since and also 0 , and P2 in eq. (3.21), describing the amplitude transmission of the hologram, may be complex. However, the nonlinearities are assumed to be small, which means I&I<< 1. For heterodyne interferometry the wavefields 0, and 6, are reconstructed with slightly different frequencies w l and w2. as shown in eq. (1.3). One gets therefore for the interference term at the beat frequency 0
1, 531
TWO-REFERENCE-BEAMHOLOGRAPHIC INTERFEROMETRY
39
Taking into account that 0, = 0,exp i4(x) and IPi,I<< 1, eq. (3.33)can be modified to see the phase distortion A+(x) due to the nonlinear crosstalk, at least in a linear approximation. The result is
[1+(P11+PT2)+(Pl2+~Tl)exPi4(x)lexPi[~~-4(x)l = exp i[Ot - +(x) + A ~ ( x ) = ] [1 + i A+(x)l exp i[Rt - 4 (x)].
(3.34)
This shows that A ~ ( x ) contains two contributions, viz. A+(X) = A ~ , + S ~ ( X )where , A&, is constant and S ~ ( Xdepends ) on the position x. These contributions are obtained from the imaginary part of the first term in eq. (3.34) as (3.35) The constant phase shift A & , can be ignored but S&(x) introduces a systematic error in the measured interference phase, which depends on the true value +(x) in the manner given in eq. (3.35). A correction for this effect is hardly possible, since the values of the complex factor (PI,+ PJ; can not be determined reliably from hologram recording and development procedure. Therefore one can only estimate the upper limit 84 of 64(x) for the worst case, i.e. (pI2+P2*1)= 2 lplzl= 2 Ip211.Using eqs. (3.32) and (3.35) one gets 84 =IP12+PT115 4 I(TP2 I P,)&,I=
K
= 4a
IY12L
(3.36)
where equal exposure times T , = T2= T are assumed. For practical purposes the relation of 84 to the contrast K of the residual fringes, defined in eq. (3.28), or to the product of relative nonlinearity a! and degree of coherence IyI21 is important. This product can be determined experimentally as described by eq. (3.30). Examples of experimental results are given in Table 3.1, yielding (Y = 8.8 X The corresponding phase errors are 8 4 5 1" for Jyl,l = 0.5 and 84 (0.1" for (yI21= 0.05, which was the lowest experimentally realized value of cross-talk. The conclusions are that nonlinear hologram recording leads to crosstalk in two-reference-beam holographic interferometry. This cross-talk depends essentially on the mutual intensity TI, = d , ( x ) O ~ ( xd2x ) between the two objects. The nonlinear cross-talk can be drastically reduced by taking care that the holographically recorded scene does not contain any visible parts which are unchanged between the two exposures, i.e.
40
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I. § 3
keeping lflzl as low as possible. The nonlinear cross-talk can be measured by monitoring the reconstructed intensity of the second reference beam if the hologram is only illuminated with the first reference beam and vice versa. In heterodyne holographic interferometry the spurious fringes generated by the nonlinear cross-talk introduce a phase error of 84, as given in eq. (3.36). Experimental results (see § 4) show clearly that this phase error can be kept below the resolution of 0.2" in heterodyne holographic interferometry, if the recorded scene is carefully cleared from spurious reference sources. A more extensive discussion of these effects was recently published by KATZand MAROM[1979].
3.5. OVERLAPPING RECONSTRUCTIONS
As shown in § 3.1 the angular separation of the two reference waves has to be large enough so that the undesired cross-reconstructions R,RTO,, R,R;02 do not overlap with the self-reconstructions R,RyO,, R,R$O,, carrying the interference information. On the other hand, however, there are several reasons to keep the separation of the two reference waves as small as possible: First of all the discussions in § 3.2 show that the repositioning requirements for the hologram decrease essentially for smaller differences k,-k, between the wavevectors of the two reference beams. The optical setup with the reference sources close together can be much more compact and is therefore easier to handle and more stable. The use of recording materials with restricted spatial frequency response, such as photothermoplastics which have a band pass characteristic (see e.g. URBACH [1977]) limits the useful range of angles between reference beams and object waves. Finally the application to highspeed holography would require a very small mutual angle of the two reference beams to become insensitive to the substantial wavelength change between recording with a pulsed laser and reconstruction with a continuous-wave laser (see P 3.3). It has been shown by DANDLIKER [1977al that under certain conditions overlapping of the cross-reconstructions can be tolerated for heterodyne holographic interferometry without undue loss of accuracy for the interference phase measurement. As long as the lateral offset of the crossreconstructions R,RTOI, R,R$02 is larger than the speckle size, the statistical phases of the speckle patterns of the desired and undesired images are not correlated and therefore their superposition does not
I, $31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
41
produce any macroscopic interference fringes. This means that the corresponding beat signals will have arbitrary phase shifts and their contributions to the signal of a photodetector with an area larger than the speckle size only add up incoherently (see § 2.3). Hence the restrictions on the separation of the reference beams are relaxed and the reference sources may be placed very close together, which makes two-reference-beam holographic interferometry nearly as compact and uncritical as classical double exposure holography. It should be noted, however, that this is only true for holographic interferometry of diffusely scattering objects, since the phase of smooth wavefields remains correlated even for large lateral shifts and thus the contribution of the undesired overlapping cross-reconstructions can not be suppressed by spatial averaging. The minimum required separation of the two reference beams R, and R, for a lateral shift of the images large enough for decorrelation of the speckle patterns can be calculated following the lines used in 0 3.2 and 8 3.3 to evaluate the influence of misalignment on the fringe pattern in the image. Following eq. (3.6) the complex amplitudes of the four primary reconstructions in the hologram plane xH are given by
R I R IU1= IR1I2 U~(XH>, R2R?U1= IR21 lRll UI(xH) exp N k 1- k2)xd3, (3.37) R2R?U2=IR212
UZ(XH),
R1RfU2=IR11 IR2I
U2(XH)
exp {-i(kl-kZ)xd*
This means that the two cross-reconstructions R,RTU, and R,RfU, have additional linear phase shifts $ % I ( ~ H )=
-$12(x~) = (ki- ~ z ) X H = kPxH
(3.38)
of opposite sign across the hologram plane compared with the selfreconstructions RIRTUI and R,RTUz. The effect of such a linear phase shift on the image formed by a lens behind the hologram (Fig. 3.6) is discussed in § 3.3. From eq. (3.18) one can calculate the resulting lateral shift u4 in the image plane. If the lens is placed closely behind the hologram, i.e. b
(3.39)
where d, = c is the distance between lens and image plane and 28 is the mutual angle of the two reference waves (Fig. 3.5). As shown in eq. (2.19) of P2, the average interference between two identical but mutually shifted speckle patterns is given by the autocorrelation function C,(u,)of
42
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 3
the impulse response function, defined in eqs. (2.9) and (2.10). The minimum shift to suppress the interference of the self-reconstructions with the cross-reconstructions by spatial averaging is therefore obtained from the requirement Ch(u1) = 0. For the most common case of a circular aperture of diameter D the decorrelation of the speckle patterns is given by eq. (2.12) and Fig. 2.3. From Fig. 2.3 one finds uID/AdI>1.2 to get lch(uI)\ =y(u,)=o. Using eq. (3.39) the minimum angular separation of the two reference waves is therefore
26 = 2 sin 6 > 1.2(A/D).
(3.40)
This is equivalent to the statement that the two reference sources have to be resolved by imaging lens, since A/D is the diffraction limited angular resolution. As discussed in § 3.3 and illustrated by Fig. 3.7 this means also that the two reference sources must produce more than one interference fringe across the lens aperture. Assuming again a pupil diameter of D = 30 mm and a wavelength of A = 633 nm, one gets from eq. (3.40) a minimum angular separation of 26 > 2.5 X lo-'= 5". This is smaller than the required value of 26 = calculated in § 3.3 for the same optical parameters to allow even a wavelength change from ruby laser (A = 694 nm) to He-Ne-laser (A = 633 nm) without undue loss of fringe contrast and phase accuracy. For a symmetrical setup of the reference waves, given by k, = ( k x ,k sin 6, k , ) and k, = (k, - k sin 6, k z ) ,the sensitivity to hologram misalignment is found from eqs. (3.9) or (3.10) to be ]grad I+,, = k p = 2 k A y sin 6. The number of misalignment fringes across the lens aperture is therefore pD/A = 2 Ay sin 6 D/A and does only depend critically on the rotation A y around t h e hologram normal. Even for an angular separation of 26 = 3 x lW4, ten times larger than required by eq. (3.40), this rotational alignment has only to be accurate within Ay = 2" for pD/h 5 0 . 5 and following Fig. 2.3 for a speckle correlation or fringe contrast of better than y = 0.7. The two interfering light waves in the image plane corresponding to eq. (2.22), are now
(3.41) where 0, and 0, are the speckled wavefields in the image plane corresponding to the object waves 0, and 02,respectively, and u is the
I,
§
31
TWO-REFERENCE-BEAM HOLOGRAPHIC INTERFEROMETRY
43
lateral shift of the cross-reconstructions as obtained from eqs. (3.37)3.39). The average interference term is therefore
where 4 is the interference phase and R = w , - w 2 . For this result it has been assumed that the lateral shift u is larger than the speckle size and that the two self-reconstructed fields O,(x) and O,(x) may also be slightly shifted, due to transverse object displacement, yielding a reduced fringe contrast y as shown in eq. (2.19). Using again the statistical and correlation properties of the envolved reconstructions one gets the average intensity
(3.43) The final result for the effective fringe contrast 7 in case of overlapping of the cross-reconstructions with the self-reconstructions is found from eqs. (3.42) and (3.43) to be
This means that the fringe contrast is reduced due to the overlapping by an additional factor of yo= 1/2. The corresponding reduction of phase accuracy 84 as a function of the number N of speckles within the detector area A, can be calculated with eq. (2.28) or read from Fig. 2.5. The resulting phase error due to overlapping only (y = 1,y = yo = 0.5) is found to be 84 = [3/2(N+ l)];,which is plotted in Fig. 3.9. It is seen that N = 200 is sufficient for an accuracy of 84 = 5", whereas N = 2 x lo4 is necessary for 8 4 = 0.5". It should be pointed out that 8 4 = 0.5" corresponds to 1/500 of a fringe and that N = 2 x lo4 is obtained already with a detector diameter which is only about 100 times the speckle size. Heterodyne holographic interferometry with overlapping reconstructions was investigated experimentally (DANDLIKER [1977a]) in a setup with a detector diameter of 1.3 mm, i.e. A,= 1.33 mm2, an imaging lens of focal length f = 300 mm and apertures between f/9 and f/64, a distance to the image plane of d, = 427 mm, and A = 5 14 nm. The angular difference
44
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 3
Fig. 3.9. Phase error 84 versus number of speckles N within the detector area in case of overlapping reconstruction-,. The experimental results for different F-numbers follow quite well the (1+ N)-:-law (dashed line) but are smaller than the theoretically predicted values (solid line), (DANDLIKER[1977a]).
26 between the directions of the two reference waves was about 5 x which is 4 times the diffraction limited resolution of the lens at f/64. For this small separation of the reference beams the most critical repositioning around the axis normal to the hologram plate is calculated using eq. (3.10) as A? = 1"for the maximum aperture fl9. With this type of setup two-reference-beam holographic interferometry is nearly as compact and uncritical as classical double exposure holography. The average phase error 8 4 was determined experimentally from the standard deviation of 40 independent measurements taken at different positions in the reconstructed image of an object which had not changed between the two exposures. The number N of speckles within the detector area A, was calculated for the different lens apertures using eqs. (2.12) and (2.13). The experimental results are shown in Fig. 3.9 together with the theoretical predictions. The measured phase variations 8 4 follow quite well the expected &-law but surprisingly they are considerably smaller than calculated. This would indicate that the contribution of the uncorrelated cross-reconstruction is smaller than expected, most probably due to a better averaging. Whether the effective number N of speckles (correlation cells) within the detector area is really larger than calculated by eqs. (2.12) and (2.13) has not been checked yet by complementary experiments and remains thus open for discussion.
EXPERIMENTAL REALIZATION AND RESULTS
45
0 4. Experimental Realization and Results 4.1. SOME GENERAL CONSIDERATIONS
Besides the standard holographic technique and equipment one needs for heterodyne holographic interferometry also methods to generate the desired frequency offset 0,to detect the modulated signals, and to measure their phases accurately. For small frequency offsets (0/277 < 1 kHz) a mechanically rotating h/2-plate and subsequent polarizing elements can be used as in the early experiments of heterodyne interferometry by CRANE[1969]. For the sake of stability, accuracy, and measuring speed, however, larger frequency offsets (0/277 = 100 kHz) are advisable. This can be realized adequately only with either rotating radial gratings (STEVENSON [1970]) or acousto-optical modulators (DIXON [1970], SITTIG [1972]). The disadvantage of the rotating radial grating is the fixed intensity ratios between the different object and reference beams, given by the diffraction efficiency of the grating, which does not allow to optimize the light economy for both recording and reconstruction independently. Moreover, most rotating radial gratings show residual amplitude modulation due to grating imperfections which may disturb the phase measurement eventually. Both methods will be covered in the following description of experiments. For the phase measurement one needs a signal to act as a reference. Therefore at least two photodetectors are placed in the image of the object under investigation. One detector may be at a fixed position while the other scans the image, or both detectors may be movable at a fixed mutual separation. The latter measures rather the fringe density or the gradient of the interference phase function than the interference phase itself. Nevertheless it has the advantage to be less sensitive to slight variations of the position of the detectors, as long as their mutual separation is kept constant. Moreover it yields directly local derivatives of the interference phase, which are proportional to the strains in case of the measurements of mechanical deformations. The integral interference phase can be calculated very accurately by summation of the increments. Photomultiplier tubes rather than semiconductor photodiodes should be used as photodetectors because of their nearly noise free and phase distortion free high gain. The disadvantage of photomultiplier tubes to be bulky can be overcome by using fibre optics to collect the light in the image plane.
46
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, § 4
All electronic amplifiers and filters in the signal paths should be designed carefully to avoid phase distortion which could reduce the accuracy of the phase measurements. Especially narrow band filters for noise reduction should be avoided. A bandwidth of somewhat less than half the modulation frequency 0 is advisable to cut down the dccomponent and harmonics. The phase is measured either by a phase sensitive detector and a calibrated, variable phase shifter or more conveniently by a zero-crossing phasemeter. Both kinds of instruments are commercially available with a resolution down to 0.1" for the phase. The holograms may be recorded on any kind of photosensitive material commonly used in holography. It should be noted however, that there has to be enough light intensity in the reconstructed image to yield a reasonably high signal-to-noise ratio for the electronic phase measurement. Therefore it is advisable to look for optimum diffraction efficiency of the hologram, e.g. by properly choosing the recording conditions and the development process for amplitude holograms or by making use of phase holograms, such as bleached photographic emulsions or photothermoplastic materials. The influence of hologram efficiency, laser power, object size, etc. on the signal-to-noise ratio and the accuracy of the interference phase measurement will be discussed in Q 4.4. In case of two-reference-beam hologram recording for double exposure heterodyne interferometry the optical setup has to fulfill the requirements for nonoverlapping cross-reconstructions and repositioning stability as described in § 3.1, § 3.2 and 03.3, unless the close reference arrangement presented in § 3.5 is used. The recorded scene should be carefully cleaned from spurious reference light sources, such as parts o r objects which remain unchanged between the two exposures, to avoid the effects of nonlinear cross-talk described in § 3.4.
4.2. EXPERIMENTAL VERIFICATION
Experimental results of heterodyne holographic interferometry were first reported by BALLARD [1971] for the investigation of transparent phase objects. DANDLIKER, INEICHEN and M O ~ ~ I E[1973, R 19741 verified the capability of heterodyne holographic interferometry by using it to determine quantitatively the bending of a cantilever under load and comparing the results with theory. These early experiments were performed with a rotating radial grating as frequency shifter and with a
I, $41
47
EXPERIMENTAL REALIZATION AND RESULTS
0 RATING
FORCE
OBJECT
Fig. 4.1. Experimental setup for heterodyne holographic interferometry of a bent cantilever INEICHENand MOTTIER[1974]. The two reference waves R, and R, are after DANDLIKER, generated by diffraction at a grating, which will be rotating during reconstruction to introduce the frequency offset R = 2 Am. D, and D, are photodetectors at the points P, and P,, respectively, in the image plane of the object.
readily available phase sensitive detector and the calibrated phase shifter of its reference unit to measure the interference phase. The experimental arrangement is shown schematically in Fig. 4.1. The beam of a 500 mW single frequency Ar-laser (A = 514 nm) is split in three beams by diffraction at a rotatable radial grating. The zero order beam illuminates the object. The two first order beams are used as the two different reference beams R, and R,. During the recording of the hologram, the grating stands still and all light fields have the same frequency 0,. The first position 0, of the object is recorded with R1only, i.e. R, is blocked. The second position O,, due to a different load, is recorded with reference R2 only, i.e. R, is blocked now. The processed hologram reconstructs the two wavefields corresponding to the object position 0, and 0, independently if it is illuminated with the respective reference waves R , and R2. A lens close to the hologram plate forms the images of the two states 0, and 0, which appear superposed in the image to produce interference fringes. The frequency difference R for the heterodyne detection is introduced by rotating the radial grating at constant speed. The rotating grating shifts the frequency of the first order beams by +Am and -Am, respectively (STEVENSON [1970]). The frequency difference R = 2 AOJ was in this experiment L?/27r = 80 kHz. Two different photodetectors D, and D, are placed at different image points P, and P,.
48
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, (i 4
Fig. 4.2. Schematic representation of the interference fringes on the cantilever and the corresponding beat frequency signals obtained from two photodetectors D, and D,. D, is at a fixed position. The relative phase of the signals gives the interference phase at different positions of D, within the fringe pattern (DANDLIKER, INEICHEN and MOTTIER[1974]).
As shown in Fig. 4.2 the interference phase 4 ( x ) was obtained by scanning the image of the object with the photodetector D1 and measuring the phase of the beat frequency with respect to the phase obtained from the second detector D, at a fixed reference point. The signals obtained from the moving interference pattern are shown in Fig. 4.2 for three positions, corresponding to 4 = 0", 90", 180", respectively. The phase difference of the two detector signals was measured by adjusting the calibrated phase shifter in the reference signal path to get minimum reading, corresponding to 90" phase difference, at the phase sensitive detector. The phase reading was reproducible within &#I = 0.3" at any position. This corresponds to less than lop3of the fringe separation. The
I, 541
EXPERIMENTAL. REALIZATION AND RESULTS
49
phase measurements were taken at intervals of A x = 3 m m along the cantilever and the corresponding normal displacement u of the object surface was calculated as u ( x ) = 4(x)/2k = 4(x)A/47r,
(4.1)
which follows from eq. (2.17) by assuming that both the illumination k, and the observation k, are nearly normal to the object surface. From the corresponding values u, = u(x,,) the local bending of the cantilever is numerically calculated through
u"(x)
+ U,_~)/~(AX)'.
= dZU/dX2= (~,,+2-2~,,
(4.2)
As a result one gets values of the local bending at intervals of Ax = 3 mm along the cantilever. The experimental results are compared in Fig. 4.3 with theory. In the case of pure bending the second derivative d2u/dx2of the normal displacement is proportional to the bending moment M(x): d2U/dX2= M(x)/EJ = K(1- x)/EJ,
(4.3)
where K is the force at distance 1 = 123 mm from the clamped end which is at x = 0, E is the modulus of elasticity, and J is the moment of inertia of the cantilever cross-section. The expected error Su" for the bending can be calculated from the accuracy Su of the displacement measurement. Assuming statistically independent measurement errors for the three points u,+', u,, u , - ~ ,one gets
SU"=
& 6~/4(Ax)'=0.6 6u/(Ax)'.
(4.4)
Fig. 4.3. Experimental results for the bending of a cantilever clamped at x = 0 and loaded at x = 123 mm. Comparison of the second derivatives d2u,/dx2 of the normal displacement u,(x) with theory indicates an accuracy for the interference phase measurement of 84 = 0.3",corresponding to of a fringe (DANDLIKER, INEICHENand MOTTIER[ 1 9 7 4 ] ) .
50
[I, 8 4
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
From the claimed accuracy of &#J = 0.3” for the phase measurement one gets, following eq. (4.l), an accuracy af Su = S4/2k = 2 X pm for the normal displacement and with eq. (4.4) Su”= 0.15 x lop4mp’for the bending. This fits quite well with the results shown in Fig. 4.3 and proves therefore the expected performance of heterodyne holographic interferometry. 4.3. ADVANCED EXPERIMENTAL ARRANGEMENT
A somewhat more advanced setup for heterodyne holographic interferometry, shown in Fig, 4.4, has been used by DANDLIKER and INEICHEN [1977] for quantitative strain measurement. In this case the frequency shift of 0/27r = 100 kHz is realized by two commercially available acousto-optical modulators MI and M,. The intensity ratios between object illumination and reference waves R , and R , can be adjusted independently by the two variable beam splitters S1 and S2 to get first optimum recording conditions and afterwards maximum power in the
I
I
Fig. 4.4. Advanced setup for heterodyne holographic interferometry, after D ~ D L I K E and R INEICHEN[1977], with two acousto-optical modulators M behind the beam splitters S and the array of three detectors D in the image plane (& q).
I, a41
EXPERIMENTAL REALIZATION AND RESULTS
51
Fig. 4.5. Arrangement of two acousto-optical modulators in cascade to generate the frequency offset for heterodyne holographic interferometry. All combinations of diffracted beams are separated and the desired frequency R = R , -a2can be selected by the corresponding opening in the screen.
reconstructing reference waves. The use of two standard 40MHz acousto-optical modulators in cascade to produce zero frequency shift for the recording and 100 kHz for the reconstruction of the two-referencebeam hologram is explained with the help of Fig. 4.5. The second modulator M, is mounted just behind the first one MI, but rotated by 90" around the light beam. This is necessary to get all the possible combinations of diffracted beams completely separated in a plane at some distance from the two modulators. The modulators are driven with the two frequencies 0,and R,, respectively, which are both within the operation bandwidth centered at 40 MHz. The first order diffractions are shifted in frequency by either +R1,, or -R1,,, depending on whether the diffraction is in the direction of the travelling acoustic wave or opposite to the acoustic wave. The diffraction angles are typically 5 mrad, which is about 15 times the diffraction limited divergence of a 1 mm diameter laser beam. The resulting total frequency shifts of the different combinations of diffracted beams are indicated in Fig. 4.5. For the desired purpose the beam with the net shift R = 0, - 0, is selected by an appropriate mask. During recording both modulators are driven with R1/257= R,/21r = 40 MHz so that R = 0. During reconstruction the first modulator is driven with R1/257= 40.1 MHz and the second one with f12/257= 40 MHz, which yields the desired difference frequency R / 2 r = (0, - 0, )/ 2a = 100 kHz for the heterodyne evaluation. By proper adjustment of the angular position of the modulators to meet the Bragg condition one gets more than 80% of the optical input power in the selected, twice diffracted
52
H F E R O D Y N E HOLOGRAPHIC INTERFEROMETRY
beam. Switching from recording conditions (0, = 0,) to reconstruction conditions (0, # 0,) has negligible effect on the propagation direction of the diffracted beams, since the 100 kHz change of the 40 MHz acoustic wave changes the diffraction angle only by 0.01 mrad, which is much smaller than the divergence of the laser beam. The two frequencies of 40 MHz and 40.1 MHz can be either generated by two quartz-stabilized oscillators or by an appropriate frequency-synthesizer. During recording both modulators should be driven from the same frequency source to insure that 0,and 0, are strictly identical and the relative phase of reference and object light at the hologram plate remains essentially stable during the exposure time. For the heterodyne measurement any crosstalk of the two frequencies 0,and 0,should be carefully prevented to avoid spurious amplitude modulation of the reference wave at the beat frequency 0 =a,-0,.MASSIEand NELSON[1978] reported recently investigations of the beam quality of acousto-optical frequency shifters for heterodyne interferometry. For heterodyne holographic interferometry, however, the beam quality is not crucial, since for both, recording and reconstruction, the same beams are used and only the phase change is important. The holograms are recorded on standard hologram plates either processed as amplitude holograms or bleached to get phase holograms with higher diffraction efficiency. The plate holder is of the same type as used for real time holography so that the hologram can be replaced after development quite accurately. As discussed in § 3.2 the most critical readjustment parameter is the rotation around the plate normal. The holograms should be developed very carefully to get high diffraction efficiency, low noise due to scattering, and uniform thickness of the emulsion. An appropriate photographic lens is placed just behind the hologram to form an image of the object in the plane of observation. An array of three detectors, as shown in Fig. 4.4, is used to scan the image. The two differences At+ and A,,+ in the orthogonal directions 8 and q are measured rather than the interference phase +(& q ) itself. The latter can be easily and accurately calculated by summation of the measured differences along a given path. The detector array is realized by the ends of three fiber bundles of about 1mm diameter, rigidly mounted at a separation of A t = A q = 3 mm, which feed the light to the three photomultiplier tubes D , , DZ,D,. The electronic equipment for automatic measurement and data acquisition devised by MASTNERand MASEK[1979] is shown in Fig. 4.6. The
Fig. 4.6. Electronic equipment for heterodyne holographic interferometry, used by DANDLIKER and INEICHEN 119771and devised by MASTNERand MASEK[1979]. From top to bottom: X-Y control unit for the detector position in the image plane (6, q), dual phasemeter measuring A& and A,,+, regulated power supplies for the photomultipliers and input filters, tape puncher for the data output, acousto-optical modulator driver.
54
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 4
center box contains the regulated power supplies for the photomultiplier tubes and filters of B = 10kHz bandwidth for the signals at 0/27r= 100 kHz. The signal amplitudes are kept constant independent of the intensity variations across the image by a feedback control of the photomultiplier voltage. The phase differences Ae4 and A,,+ are measured with two zero-crossing phasemeters, which interpolate the phase angle to 0.1" and count also the multiples of 360°, which corresponds to the fringe number. The detector array is mounted on a stepmotor driven stage to scan the image. The programmed scanning pattern is executed by the X-Y control unit (top of Fig. 4.6). For any measured position the 6, q-coordinates as well as the interference phases are digitally recorded on punch tape to be available for further data processing. The measuring time for one position, including displacement and data punch, is only a few seconds. At the bottom of Fig. 4.6 the driver for the two acoustooptical modulators (40 h4Hz and 40.1 MHz) is recognized. The described electronic system for heterodyne holographic interferometry is essentially computer compatible. With appropriate interface units on-line data processing and computer controlled interferogram scanning and evaluation can be realized. An example will be given in $5.5. 4.4. ACCURACY AND REPRODUCIBILITY
The overall accuracy and reproducibility of heterodyne holographic interferometry depend mainly on the specification of the phasemeter, the signal-to-noise ratio of thedetector signals and the mechanical stability of the opticalsetup. The performanceof thesystem described in P 4.3withrespect to these parameters was investigated by INEICHEN, DANDLIKER and MASTNER [1977]. In the following the accuracy of the phase measurement will be discussed first and afterwards the results of an experimental test of accuracy and reproducibility for the measurement of mechanical deformation will be reported. The relative phase of the two detector signals is measured with a zero-crossing phase meter. The stability and accuracy of t h e phase meter is electronically tested to be better than *0.05" for clean input signals at 100 kHz. The phase meter requires for proper operation, i.e. to avoid multiple zero crossings, a signal-to-noise ratio (SNR) of at least 20 dB and a noise bandwidth of less than the signal frequency. The noise introduces a phase error 84 due to the fluctuations of the zero crossings. This phase error is found to be 64 = (
s ~ ~ ) - =f (sNR)-~(~/T)-:, ~-f
(4.5)
I, 841
EXPERIMENTAL REALIZATION AND RESULTS
55
where M = T / T is the number of zero crossings observed during the integration time 7 of the phase meter. This means that a single measurement (M = 1) with SNR = 20 dB yields 6 4 = 6". This is reduced to 64 = 0.06" for T = 100 ms and a frequency of 100 kHz ( T = 10 ps). The SNR of the detector signals can be estimated from the holographic setup, the hologram efficiency, and the laser power. It is assumed, that photomultiplier tubes are used as detectors and that the light intensity is large enough to get shot-noise limited detection. In this case the signal-to-noise ratio is found to be
SNR = h 1 2 q ~ q ~ A ~ / 2 A , h V B ,
(4.6)
where P = power of the reconstructingreference beams, rn = observed fringe contrast, q H = hologram efficiency, q D = quantum efficiency of the detector, AD = detector area, A&=area of the illuminated object surface in the image, hv = photon energy, B = detection bandwidth. The power P of the reference beams is defined as the power falling on that area of the hologram, which contributes to the reconstructed image. The observed fringe contrast is equal to the depth of modulation of the detector output and is given by
where ImaX and Imin are the maximum and minimum intensity of the fringe, respectively. This observed fringe contrast is always smaller than the fringe contrast y or 7 defined in eqs. (2.20)and (3.44), since unequal amplitudes of the two interfering light fields and any scattered or ambient light causes an additional background intensity and reduces therefore the depth of modulation to rn < y. The parameters for the experiments described in the following were typically P = 20 mW, rn = 0.4, qH = 0.25, qD= 0.1, A,, = m2, A, = 0.1 m2, hv = 3 . 9 lo-" ~ Ws (A = 514 nm), and B = 10 kHz. The SNR was about 40 dB, which corresponds to the value estimated by eq. (4.6). The effective power of the reconstructing reference beams P is often surprisingly small, since the spot size of the reference beams are chosen larger than the hologram plate, to get uniform illumination, and the hologram is again larger than the useful area for the reconstruction, which is determined by the aperture of the lens behind the hologram. In most cases the light economy can be considerably improved by better matching the diameter of the reference beams and the maximum lens aperture. The ratio between the detector area A, = lop7m2 and the object area of A, = 0.1 m2corresponds to a resolution of lo6 points in the image of the object.
56
[I, 5 4
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
Besides the phase error due to amplitude noise, as shown in eq. (4.9, additional phase fluctuation may occur in the signal. These are mainly caused by optical instabilities, e.g. in the path length of the two reference waves, and by mechanical instabilities of the position of detector and reconstructed image. Both effects are reduced by averaging over the integration time T of the phase meter. The overall accuracy of the phase measurement, including these phase fluctuations, was determined experimentally. A double exposure, two-reference-beam hologram of an object, which has not been deformed between the two exposures, was recorded. The phase difference between the signals of two detectors at a constant separation of 3 m m was measured at different positions in the object image. The result is shown in Fig. 4.7. The vertical bars indicate the digital resolution of +0.05" for the phase meter reading. The measured phases show statistical variations with 6 4 = 0.22" around an average value of 4 = 1.05". This means that the interference phase can be measured with an accuracy of 64= 0.22". corresponding to an interpolation of 0.6 x of a fringe. The accuracy and reproducibility of the measurement of mechanical deformation have been experimentally investigated for the case of a turbine blade under static torsional load. The tested blade is about 1.2 m long. The torsional load is realized by the couple of two symmetric weights at the tip of the blade. The holograms were recorded at 1m
lP 1.5
1
t
0.5
0
I
I D
C
I
I
I
z
I
Fig. 4.7. Accuracy &$ of the interference phase measurement determined from the measured phase along an undeformed object (INEICHEN,DANDLIKERand MASTNER[1977]).
I. 841
EXPERIMENTAL REALlZATlON AND RESULTS
57
distance from the blade. The reconstruction was imaged with a magnification of 0.39 by a f/9 aperture objective ( f = 300 mm), located closely behind the hologram. The detector array used in the image plane is shown in Fig. 4.8a. The ends of two circular fibre bundles of 1 mm diameter are covered with 0.1 mm wide slits, parallel to the blade axis. The 5 mm separation of the slits corresponds to a distance of DObi = 12.8mm on the object. The fibre bundles feed the light to two photomultiplier tubes. Using eqs. (2.12) and (2.13) the number of speckles within the detector area of A D = 0.1 mm2 is calculated as N = 2 x lo3. This allows following Fig. 2.7 a transverse shift of the speckle pattern in the image plane up to u,=O.2 (AX), =0.2 (Ad,/D) = 1.3 p m for a phase error of less than 84 = 0.3". The fringe pattern caused by the torsion of the blade is also shown in Fig. 4.8b. The fringe density varies from zero to a maximum value of 8 fringeslmm on the object. The measured phase difference C#J between the slits is proportional to the twist 8(z) of the blade around its axis. Following eq. (4.1) the corresponding relation reads e(z)
= au/ax = 4
~ 1 4 AX, 7~
(4.8)
where u is the normal component of the surface displacement, A is the wavelength, and A x = DObj is the separation of the detector slits on the object. The differential change d8/dz of the tilt along the z-axis of the blade is called rate of twist or torsion. In a first approximation dO/dz is related to the torsional moment MT by d 8/d z
= MT/GIT,
(4.9)
where GI, is the torsional stiffness, given by the modulus of elasticity in shear G and a geometry factor IT of the cross-section (see e.g. TIMOSHENKO and GOODIER [1970]). On the other hand d0/dz is experimentally obtained from the difference of the phase measurements A ~ ( z , ) =+(zn+J-4(zn-,) along the blade axis: d0/dz = A 4 A 1 4 A ~ x Az,
(4.10)
where A z = z , + ~ - z , - ~ . The relative error of the torsion due to the measurement error S 4 of the interference phase is found to be S(d0/dz)/(dt3/dz) = f i S ~ / A $ J
(4.11)
and depends also on the spatial resolution Ax, Az. Experimental results for d8/dz are shown in Fig. 4.9, as circles. The spatial resolution on the
58
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 5 4
Fig. 4.8. Quantitative determination of the twist of a turbine blade under torsional load (INEICHEN, DANDLIKER and ~ ~ A S T N E [1977]). R (a) Detector array for the measurement of the phase in the fringe pattern. (b) Fringe pattern to be evaluated.
1, $41
EXPERIMENTAL REALIZATION AND RESULTS
59
Fig. 4.9. Measured torsion dO/dz of a turbine blade compared with theory (INEICHEN, DANDLIKER and MASTNER [1977]).
object is Ax = A2/2 = 12.8 mm. There is good agreement with theory, shown as X . The estimated absolute accuracy of about *5'/0 for the torsion is rather limited by the uncertainty of the mechanical and geometrical parameters, such as load, magnification factor, detector slit separation, than by the phase measurement itself. This becomes also clear from the following results for the reproducibility of the measurement of d9/dz. Three independent double exposure experiments have been performed. The mechanical and optical setup for these experiments was the same. The three double exposure holograms, corresponding to three independent mechanical deformations of the blade, are evaluated and compared. The differences in the values obtained for the torsion at each position z along the blade axis are calculated with a digital resolution of dO/dz = lop9mrr-'. According to eqs. (4.10) and (4.11) this just corresponds to a phase error of 84 = 0.3". From Fig. 4.9 the torsion at z = 50 mm is found to be dO/dz = 1.4X lO-'mm-', which yields according to eq. (4.10) a measured phase difference of only A 4 = 64" and according to eq. (4.11) with 84 = 0.3" a relative error of G(dO/dz)/(dO/dz)= 0.65%. The relative deviations of the three measurements, viz. 1353, 1354, 1355, are shown in Fig. 4.10. It is seen that all except two values, at z = 5 0 m m and z = 225 mm, are within the expected limits. This means that also the mechanical conditions, e.g. the load, are reproducible enough in this experiment to use the full accuracy of heterodyne holographic interferometry.
60
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 8 5
Fig. 4.10. Reproducibility of the measurement of the torsion for three independently recorded and evaluated holograms (1353, 1354, 1355). The relative deviations of the individual measurements are shown for each position along the blade axis z (INEICHEN, DANDLIKER and MASTNER [1977]).
8 5. Applications It has been shown and experimentally verified that heterodyne holography has the following outstanding properties: - fringe interpolation to better than of a fringe (64 = *0.3"), - measurement with the same accuracy at any desired position in the image, therefore high spatial resolution (more than 100X 100 points), - independent of brightness variations across the image, - inherently direction sensitive, i.e. increase and decrease of interference phase can be distinguished, - computer readable output both for position and phase easily obtained (allows on-line data-processing), - inherently less sensitive to speckle noise than fringe intensity measurement. For these reasons heterodyne holographic interferometry is considered to be a very powerful technique to collect data for quantitative evaluation of holographically recorded interference patterns. This is of great importance for quantitative reconstruction of three-dimensional refractive index distributions in transparent objects and quantitative measurement of surface displacement and strain ( $ 5 . 5 ) of solid objects. It will be shown
I, $ 5 1
APPLICATTONS
61
in the following that the heterodyne technique can be applied to nearly all known kinds of holographic interferometry, except for time average holograms. For a comprehensive review of holography, holographic interferometry and its applications the reader is referred to corresponding textbooks (e.g. COLLIER, BURCKHARDT and LIN [1971b], BUTTERS [1971], ERF[1974], VEST [19791). It should be noted, however, that heterodyne holographic interferometry or modification thereof may also be realized, at somewhat reduced accuracy, with less sophisticated equipment than described in 0 4.3. Even if the interference phase is measured only with 6" resolution it is still an essential improvement over classical fringe counting techniques, since it allows to interpolate the fringe pattern at any desired position to 1/50 of a fringe. Some of these alternative methods were reported by LAN= and SCHL-R [1977] or SOMMARCREN [1977], who use incremental steps of phase shift rather than frequency offset, which corresponds to a continuous linear phase shift. 5.1. DOUBLE EXPOSURE HOLOGRAPHIC INTERFEROMETRY
The experimental realization of double exposure heterodyne holographic interferometry is described extensively in § 4.2 and § 4.3. It makes essentially use of two-reference-beam hologram recording, which is discussed in § 3. The accuracy and reproducibility of the experimental results reported in § 4.4 indicate that heterodyne interferometry in connection with double exposure holography can develop its full capability of fringe interpolation accuracy and spatial resolution. For the problems associated with two-reference-beam holographic interferometry (§ 3) one should clearly distinguish between the investigation of diffusely scattered wave fields from rough surfaces (0 2) and of smooth wave fields from transparent phase objects or specularly reflecting surfaces. Following P 3.2 and § 3.3 any change of alignment or wavelength between recording and reconstruction introduces essentially a transverse shift or shearing of the interfering light fields in the reconstructed image. In case of speckled wave fields from rough surfaces the fringe pattern and the measured interference phase are not changed if the image lens is placed closely behind the hologram, but the fringe contrast is reduced and eventually completely lost so that no fringes can be seen and no phase measurement can be made any longer (D3.3). In case of smooth wavefields the transverse shift does not reduce the fringe contrast, but
62
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I, 5 5
may change the fringe pattern due to the well known effect of shearing in classical interferometry. The concept of two closely spaced reference sources to reduce alignment and wavelength sensitivity of two-referencebeam holographic interferometry, described in 4.3.5, can only be applied to speckled wave fields from rough surfaces, since the suppression of the contribution of the undesired overlapping reconstructions relies essentially on the decorrelation of speckles of the slightly shifted, undesired reconstructions. But on the other hand the alignment conditions are less severe for smooth wavefields, because transverse displacements can be tolerated as long as the corresponding change of the fringe pattern due to shearing can be accepted. Some additional comments are necessary regarding high speed double exposure holography using pulsed lasers for the recording. First of all one has to switch the two reference beams between the exposures. This can be accomplished by mechanical or fast electro-mechanical shutters for pulse separations larger than a few milliseconds and by acousto-optical shutters (Bragg-cells) or electro-optical shutters (Pockels-cells, Kerr-cells) for shorter pulse separations. The problem of the wavelength change from recording with a pulsed laser to reconstruction with a continuous laser has been discussed already in 4 3.3 and 8 3.5. As pulsed lasers most frequently either ruby-lasers at A = 694.3 nm or frequency-doubled Ndlasers at A = 532.4 nm are employed. A straightforward but somewhat expensive solution is the reconstruction of the hologram with a continuous dye-laser, tuned to the same wavelength as used for the recording. As seen from the numerical example given in 4 3.3 the required wavelength reproduction is about AA/A = 0.3 x which means AA = lo-’ nm or Av = 10 GHz for the reproducibility and stability of the tunable dye-laser. A cheaper and perhaps more suitable solution is to use a readily available continuous laser with a wavelength as close as possible to the recording wavelength and to compensate for the wavelength change by readjusting the mutual angle 26 between the reference waves, as indicated by eqs. (3.11) and (3.12), or to use closely spaced reference sources in case of speckled wave fields (8 3.5). A list of available gas-laser lines for both ruby-laser and frequency doubled Ndlaser and the corresponding relative change Ak/k=AA/A of the wavelength is given in Table 5.1. The wavelength mismatch can be made as small as Ak/k = 3 X lop2 or even Ak/k = 3 x respectively. The resulting transverse shift or shear is tolerable in most cases of holographic interferometry with smooth wave fields.
I, 9 51
63
APPLICATIONS
TABLE 5.1 Available laser lines for the reconstruction of two-reference-beam holograms recorded with pulsed lasers. Recording
Aklk
Reconstruction
frequency doubled Nd-YAG laser A = 532.4 nm
Ar-laser Ar-laser Kr-laser
A = 5 14.5 nm A = 528.7 nm A = 530.9 nm
3.36X 6.95 x 2.82 x 10-3
ruby laser A = 694.3 nm
He-Ne-laser Kr-laser
A = 632.8 nm A = 676.4 nm
8.86 X 2.58 X lo-*
5.2. REAL-TIME HOLOGRAPHIC INTERFEROMETRY
In real-time holographic interferometry only one wave field has to be stored holographically, therefore there is no need for two-reference-beam recording. In this case the frequency offset has to be introduced between the reconstructing reference beam and the object illumination. Two possible setups with acousto-optical modulators are sketched in Fig. 5.1. The two modulators can be either placed in cascade in the reference beam, similar as shown in Figs. 4.4 and 4.5, or one in the reference beam and the other one in the object illuminating beam. The latter version may
Obi. c
* Obj.
Fig. 5.1. Possible arrangement for the two acousto-optical modulators in real-time heterodyne holographic interferometry. (a) Modulators in cascade as shown in Fig. 4.5. (b) One modulator in each beam, using the same order of diffraction.
64
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
[I,
B5
be somewhat more convenient, since each beam passes only through one modulator and the different diffraction orders are easier to separate. As described in 54.3 the modulators are driven at identical frequencies R , = R2 for the hologram recording and at different frequencies 0 ,= R , - R, for the heterodyne evaluation. The interference phase measurement is expected to be less accurate than in the case of double exposure heterodyne holographic interferometry, since the object wave fields show considerable phask variations due to object movements, air turbulence, etc. But the real-time arrangement has the advantage that the temporal behavior of the interference phase 4(t) can be determined quantitatively for a fixed point or with additional detectors at several points simultaneously. The temporal resolution is limited first by the integration time of the phasemeter and finally by the difference frequency R, employed. The detector signals are of the form i(t) = a ( t ) cos [R,t+(b(r)l, (5.1) where a ( t ) is the signal amplitude, which may also be time dependent. Phase measurements by zero-crossing phasemeters or similar techniques are possible if d 4 / d t < < R , and the amplitude modulation a ( t ) can be suppressed by an amplitude limiter. The temporal phase resolution is given by
8& = .r(d(b/dt)= M(2~/Q,)(d4/dt), (5.2) where T is the integration time and M = T / T > ~is the number of observed zero-crossings during the integration time. Following eq. (4.5) the measuring accuracy for a given signal-to-noise ratio depends also on M . The optimum integration time can therefore be found by comparing of eq. (5.2) with 84 of eq. (4.5). From phasemeters with an analog output one gets a direct, continuous display of the time dependent interference phase either on an oscilloscope or on a plotter. The accuracy of the phase measurements depends besides the electronic equipment essentially on the mechanical stability of the optical setup and the amount of statistical phase fluctuations due to air turbulence in the optical paths. 5.3. VIBRATION ANALYSIS
Time-average holography, which is most commonly used for holographic vibration analysis, can not be combined with heterodyne evaluation, since the intereference fringes are already frozen in the hologram.
I, 6 51
APPLICATIONS
65
Synchronized double-pulse holography can be employed in the same manner as pulsed double exposure holography described in § 5.1. Realtime holography can be used following the explications given in § 5.2. From eq. (2.17) one finds for the time dependent interference phase
4(t)= 2 k ~ ~ ( t ) = 2 k ~ ( t ) ,
(5.3)
where it is assumed for the approximation that the displacement vector u ( t ) and the sensitive vector E are nearly parallel. As seen from eq. (5.1) the detector output is a phase modulated signal with carrier frequency 0,. The phase modulation is also equivalent to a frequency modulation following the relation
0 ( t )= R,+d4ld t 0,,+2k(duldt), (5.4) where R ( t ) is the instantaneous frequency of the signal and 0, is the beat-frequency introduced by the heterodyne technique. Analog techniques to determine the frequency modulation d4ldt are well known as FM-demodulators, such as used in FM-receivers. As seen from eq. (5.4) one measures in this manner rather the velocity duldt of the vibrating surface point than the displacement u(f) itself. This makes also clear that the method can be looked upon as a velocity measurement based on the Doppler frequency shift of an optical wave reflected or scattered at a moving target. A well developed application of this technique is the laser Doppler velocimetry used in flow research. For this purpose special wide band FM-demodulators, such as tracking receivers or fast frequency counters, have been devised (see e.g. WATRASIEWICZ and RUDD[ 19761, DURST,MELLINCand WHITELAW[1976]). Such systems can be used favorably for real-time heterodyne holographic vibration analysis. The so-called sampling FM-demodulator reported by ITEN and DANDLIKER [ 19721 is a very flexible instrument for rapidly changing instantaneous frequencies n(t),as in case of large vibration amplitudes u,, and high vibration frequencies f. In case of sinusoidal vibration the instantaneous frequency is obtained from eq. (5.4) as
0 ( t ) = 0,,+ 47rfku,, cos (27rft) = 0 ,+ A 0 cos (27rff),
(5.5)
where A 0 is the bandwidth and A0/0', the modulation index of the frequency modulated signal. It is seen from eq. (5.5) that the bandwidth
depends on both the vibration amplitude u, and the vibration frequency f.
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The numerical example for u,/h = 1 and f = 1 kHz yields a bandwidth of AO/27r = 12.6 kHz and indicates that higher difference frequencies than only O,/21r = 100 kHz as reported in § 4.2 and § 4.3 are needed in case of vibration analysis at high frequencies and large amplitudes. Real-time heterodyne holography allows to investigate the temporal behavior of vibrations, but only at one or a few points simultaneously, depending on the number of detectors employed. Moreover, the accuracy is significantly reduced due to the necessarily fast response of the phase or frequency detection. To use the full potential of accuracy of heterodyne holographic interferometry it is therefore necessary to store the information of the vibration amplitude holographically by some kind of double exposure recording and to perform the evaluation under stationary conditions afterwards. This can be accomplished by combination of stroboscopy and two-reference-beam holography. A corresponding setup, using three acousto-optical modulators, is sketched in Fig. 5.2a. Each of the two reference beams R, and RZ,as well as the object illuminating beam pass through one of the modulators, which are adjusted to give maximum output in the first order diffracted beam. During recording they are driven at identical frequencies 0,= O2= O3and used as amplitude modulators or fast shutters to generate the stroboscopic illumination. During reconstruction only the reference waves are used and their modulators are driven at slightly different frequencies 0, and O2 to generate the difference frequency O = 0 ,-02 for the heterodyne detection in a similar manner as described in $4.3. The pulse trains of the driving power fed to the three modulators are shown in Fig. 5.2b together with the vibration amplitude, which has to be detected at one point of the object for synchronization purpose. The object is illuminated during an appropriate time in iis two extreme positions of positive and negative vibration amplitude, as e.g. described by ALEKSOFF [1974]. The reference beams R 1 and R, are alternatively switched on at the same intervals, so that one reference beam records only the positive and the other one only the negative extreme position of the vibration. The result of this type of stroboscopic recording is a two-reference-beam hologram with independent recording of the two extreme positions of the vibrating object. This hologram can be evaluated by the heterodyne method in the same manner and with the same accuracy as described in 0 4 for double exposure holograms. Standard acousto-optical modulators at 40 MHz driving frequency can be used straightforward as amplitude modulators or fast shutters with pulse durations down to 1 ps or even shorter.
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Fig. 5.2. Combination of stroboscopy and heterodyne technique for holographic vibration analysis, using three acousto-optical modulators. (a) Each beam generated by the beam splitters S, and S, passes through one modulator M. During recording they act as amplitude = 0, = 0,. During modulators or fast shutters, driven at identical carrier-frequencies 0, reconstruction M, and M, are used to introduce the difference frequency 0 = 0,- 0, for the heterodyne evaluation. (b) Vibration amplitude u(t) and corresponding rf-power pulses PI, P,, P, applied to the modulators MI, M,, M, to record a stroboscopic two-referencebeam hologram.
5.4. DEPTH CONTOURING
Several methods of holographic and MoirC surface contouring are known. The reader is referred to the corresponding literature, which may be found in the review by VARNER[1974]. Although heterodyne techniques can also be applied to MoirC surface contouring, as reported by INDEBETOUW [1978] and PERRINand THOMAS [1979], the following discussion will be restricted to holographic methods. For holographic depth contouring essentially two different holographic recordings of the same object are made under different conditions of illumination, e.g. using different wavelengths of the laser source, different index of refraction of an immersion liquid, or different directions of illumination. If each of the two holograms is recorded independently by its own reference wave in a
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two-reference-beam setup (0 3), the heterodyne evaluation can be applied in the manner described in Q 4. The main advantages of heterodyne holographic depth contouring over the conventional fringe counting are inferred from the discussion of Fig. 1.1, namely - continuous plots of the profile along any desired path can be obtained with a resolution and accuracy of about of one depth contour fringe, - increase and decrease of depth are unambiguously detectable, - quantitative and computer readable values of the depth can be determined at any desired position of the object surface. Preliminary experimental results were already reported. by DANDLIKER, INEICHEN and MOITIER[19741. They recorded the object consecutively with two different wavelengths, or rather output frequencies within one line from an Ar-laser, using a two-reference-beam setup as described in 0 3.2. The two frequencies were readily obtained by changing the angular position of the mode selection etalon of the single frequency Ar-laser, which allows to tune the output frequency within the bandwidth of about 9 GHz of the green Ar-laser line centered at A = 5 14 nm. The heterodyne evaluation was made with a frequency offset between the two reference beams of 80 kHz, generated by a rotating radial grating as reported in Q 3.2. Similar to eq. (2.17), but rather assuming a change of the length of the wavevectors from k, to k, between the exposures than a displacement u of the object surface, one gets for the interference phase corresponding to the depth contours 4(xs)=2(kl- ~ , ) E . x , ~ ~ T ( A v / c ) z ,
(5.7)
where x, are the coordinates of the object surface in space, E is the sensitivity vector as defined in eq. (2.17), c is the speed of light, Av is the frequency difference of the light used for the two exposures, and z is the depth of the object in the direction of E if k, and k, are nearly parallel. The frequency shift in the reported experiment was Av = 6.7 GHz and yields following eq. (5.7) as separation of the depth contour fringes c/2Av = 22.4 mm, which seems to be quite coarse. But since the phase 4(xs) can be measured by the heterodyne technique with an accuracy as high as 84 = 0.3", the resolution for the depth of the object surface is already better than 8z = 0.02 mm. The experimental results shown in Fig. 5.3 were obtained by measuring the phase with a moderate reproducibility of only 84 = 6", which means about 8z = 0.4 mm for the accuracy of
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Fig. 5.3. Preliminary results of heterodyne holographic depth contouring reported by DANDLIKER, INEICHENand MOTITER [1974]. Two wavelengths separated by Av = 6.7 GHz within the 514nm line of an Ar-laser were used for the recording. The profile z(x) shown corresponds to less than one depth contour fringe (1$<360") which is 22.4mm. The resolution in both directions x and z is better than 0.4mm, even for a phase accuracy of only 84 = 6".
the surface profile. Note that the whole profile plotted in Fig. 5.3 corresponds to less than one depth contour fringe.
5.5. MEASUREMENT OF MECHANICAL STRAIN AND STRESS
Holographic interferometry applied to deformations of solid objects generates in general fringes which are contour lines of the surface displacement in the direction of the sensitivity vector. In many practical applications, however, it is rather the differential change of the surface displacement, i.e. strain, tilt, bending, torsion, than the displacement itself which is of primary interest. It is not surprising, therefore, that considerable effort has been expended in finding methods to determine strain from, the fringes of holographic interferometry. For a review of these methods the reader is referred to SCHUMANN and DUBAS[1979]. Several authors have given more or less general theoretical treatments of the relation between surface strain and observable interference fringes. Among them are SCHUMANN [1973], DUBASand SCHUMANN [19741, DANDLIKER, ELIASSON, INEICHENand MOTTIER [ 19761, STETSON r197.51, PRYPUTNIEWICZ and STETSON [1976], and DANDLIKER [1977a]. A common result of these treatments is the fact that the surface strain is essentially related to the local derivatives of the fringe pattern, or more precisely the gradient of
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the interference phase grad 4(x), which is identified with the so-called [ 19751. Obviously heterodyne holofringe-vector introduced by STETSON graphic interferometry is a very powerful tool to determine local derivatives of the interference phase, as shown in P 4.2 and P 4.3, thanks to its capability of accurate fringe interpolation. Therefore straightforward numerical differentiation of the measured interference phase can be used successfully to calculate surface strain, tilt, bending and torsion. The necessary theoretical relations between optics, geometry, and mechanics will be worked out in the following along the lines used by DANDLIKER [1977a, 1977bl and DANDLIKER and ELIASSON [1979]. The displacement of the object surface is completely described by a vector u(x, y, z ) , where the coordinates x, y and z are restricted to the surface of the object, defined by a function z = f(x, y),
or F(x, y, z ) = z -f(x, y ) = 0.
(5.8)
If it is supposed that the derivatives of the displacement are small compared with unity, the vector gradient gradu may be separated additively in a symmetric part &ik, the strain, and a skew-symmetric part a i k , the rotation: grad u = au,/axk = &ik
+
aik,
For linear elastic deformation of an isotropic material the relations between stress uikand strain &ik are given by Hooke’s law (5.10)
where E is the modulus of elasticity and v is Poisson’s ratio (& = 1 for i = k and 6 i k = O for i f k). For the discussion of the relations between surface displacement and surface strain it is appropriate to select the x, y, z coordinate system for the object point P so that the z-axis is parallel to the surface normal n and the x- and y-axes are parallel to the tangential plane (Fig. 5.4). From the observed surface displacement, however, it is not possible to determine grad u = ahlax, completely, because only the variations of u(x, y, z ) along the surface z = f ( x , y) are accessible. This means, expressed in the special coordinate system shown in Fig. 5.4, that only the six components au,/ax, can be obtained, where here and the following greek indices (e.g.
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Fig. 5.4. Object coordinates (x. y. 2) at point P: z has the direction of the surface normal, u is the surface displacement, s is in the tangential plane.
a,p), are used for in-plane components (u,, u,) and in-plane coordinates (x, y ) . The three remaining components ay/az have to be determined
from additional relations. These relations are determined uniquely by the mechanical boundary conditions at the object surface. They read Tk
(5.11)
= niUik,
stating that the external forces Tk are in equilibrium with the stress components normal to the surface. (In eq. (5.11) and in the following the convention of summation over repeatedly occurring indices is used, i.e. sum over i = x, y, 2, or a = x, y in case of greek indices.) In most practical cases the surface under observation is free of external forces (Tk= 0). This implies cr,, = 0, i.e. all normal stress components are zero. In case of homogeneous material this leads with eqs. (5.9) and (5.10) to the following relations for the strain components &ik and the components o k of the rotation vector n: Ex, = au,/ax,
& ,,
= 0,
E,,
E,, = 0,
a,= nzy= au,/ay,
= au,/ay, Ey,
v E,,
=.gau,iax
= --
1-V
+ au,/ay),
(Ex,
+ E,,),
(5.12)
oy= n,, = -au,/ax,
0,= a,, = +(a~,,/ax -au,lay). Bending and torsion are given by the change of curvature of the surface. The original curvature of the object surface is described by a
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tensor K , ~=
(5.13)
a2f/ax, ax,,
where z = f(x, y ) defines the surface following .eq. (5.8) and x, are the orthogonal coordinates (x, y ) in the tangential plane at the point P (Fig. 5.4). The change of curvature AK,, is obtained from comparison of the corresponding coefficients of the quadratic form approximating the surface around the point P before and after the deformation described by u(x). Restricting oneself to linear contributions in E,,, since E,, << 1, one gets finally the relations
AK,,
= a2u,/ax, ax, - (K,,E,,
+ E,,K,,).
(5.14)
The change of curvature is seen to consist of two different contributions, namely the variation of the surface tilt d2u,/dx, ax, and the influence of the surface strain E,,, which reduces the apparent curvature proportional to its original value K , ~ . In the case of dominant bending and torsion the contributions of the second terms in eq. (5.14) can be neglected, since the products of the curvature K,, and the strain E,, will be small. The bending (change of curvature) A K ~and the torsion (rate of twist) dL?Jds for an arbitrary direction s on the surface are obtained from A K , ~ by the relations
AK, = AK,. cos2 8 + 2 A K , ~cos 8 +AK,, sin2 8, dOs/ds = A K , ~cos2 8 + ~ ( A K - AK,,) ~ ~ sin2 8,
(5.15)
where 8 is the angle between the x-axis and the direction s (Fig. 5.4). After that, all relations between surface displacement and the mechanically relevant deformations are established. From eqs. (5.12) and (5.14) it is seen that the in-plane strains E,, are obtained from the first derivatives of the in-plane displacements u,, u,, whereas the change of curvature AK,, is mainly determined by the second derivatives of the out-of-plane displacement u,. Next, the relations for the calculation of the surface displacement vector and its derivatives from the observable interference phase in the image of the object surface will be established. The considered geometry of the setup for the holographic interferometry is shown in Fig. 5.5. It is assumed that the same imaging system is used for all the holographic interferograms with different sensitive vectors En. This can be realized using either different illumination sources Q, or different observation directions, e.g. different portions of the hologram or of the lens aperture.
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Fig. 5.5. Setup for holographic interferometry to determine the surface strain of arbitrarily shaped objects: image system (6, q. c), object system (x, y, z ) . illumination sources Q,. sensitivity vectors E,, (6, q ) .
It has also been suggested to use entirely different views of the object to get the sensitivity vectors E,,, but this is not recommended because of the different projections of the object formed in the images which makes proper position measurements and coordination of individual points on the surface quite difficult and inaccurate. The actual observation point P'((', q') in the.image plane is simply related to the point P(5, q ) in the conjugate object plant by the magnification. Therefore (6, q,5 ) are used as image coordinates. The object coordinate system (x, y, z ) for the corresponding point on the surface is mainly determined by the surface normal n, as shown in Fig. 5.4. The interference phase measured in the image is given by eq. (2.17), which reads
4n(t,
v),
(5.16)
where ko and k, are the wave vectors of the illumination and the observed light, respectively, and Un(&q ) are the apparent components of the displacement u in the directions of the sensitivity vectors E,,((, q). If the displacement vector is represented by its components uL(& q ) in the image coordinate system (6, q,t),eq. (5.16) can be written as
Un(&q ) = e n k ( &
q)u;(& q),
(5.17)
where enk are the components of the En(.$,q ) in the image system. Using
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HETERODYNE HOLOGRAPHIC INTERFEROMETRY
three linearly independent sensitivity vectors eq. (5.17) can be inverted to calculate the displacement compments U;(& q ) = f k n ( &
q)un(& q ) ~ with
fmnenk
= amk.
(5.18)
The elements fkn(& q ) of the inverse sensitivity matrix can be calculated from the recording and imaging geometry which determines the sensitivity vectors E n ( [ ,q). The derivatives of the displacement u;(& q ) in the image system are directly obtained from eq. (5.18), taking into account that also the inverse sensitivity f k , ( & , q ) depends on the position in the image plane. In general the contributions of the differential change of the sensitivity vectors En(&q ) cannot be neglected, as pointed out by BIJLand JONES[1974] ELIASSON, INEICHEN and MOTTIER [1976]. At least an and DANDLIKER, estimation of these contributions is recommended in each special case. Therefore the derivatives in the image system read
More than three sensitivity vectors lead to an overdetermined equation system for u;(& q). One can take advantage of that to eliminate the zero fringe ambiguity or to improve the accuracy by least-squares solutions of eqs. (5.18) and (5.19). The terms aU,,/a&x in eq. (5.19) are recognized as the components of the so-called fringe-vectors, introduced by STETSON [ 19751. The last step is to transform the displacement and its derivatives from the image system to the object system. For each point P the relation between the image system (&, q,1) and the object system (x, y, z ) can be described for that purpose approximately by a general rotation in space xm
= Rmk6k,
(5.20)
where R,, is orthonormal, i.e. RmkRnk = a,,. This approximation is valid as long as the distance between object surface and object plane (6, q ) is small compared with the viewing distance. The rotation matrix R,, is mainly determined by the direction of the surface normal
n(P)=(n,, n,,, nc),
with nz+n;+n,= 1,
(5.21)
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in the image system. If the orientation of the x-axis in the tangential plane is chosen so that it corresponds to the optical projection of the [-axis, i.e. in the ([, 5 ) plane, the rotation matrix is explicitly given by nJN R m k = -n,n,,lN
[
n,
0 N n,
-n,lN -n,,nJN n,
1
,
(5.22)
with N = += The inverse rotation matrix RLA is simply obtained from R;: = Rmkrsince Rmkis orthonormal. Using eq. (5.20) the components of the displacement u in the object system are obtained by y,
2) = RmkU;(& q).
(5.23)
For the transformation of the derivatives, however, it has to be considered that the observed displacement u(x, y, z ) is restricted to the surface defined by eq. (5.8), which means &(x) = &{x, y, z = f(x, y)}. Therefore one gets aU,/aX,
a2umlax, ax,
= Rmk(dU;/a&)R~~ = Rmk(a2uL/a&a&)R;iRiA
(5.24a)
+ R,,(au;la&)R;~K,,.
(5.24b)
Remember that greek indices (a,p, y, 6) refer only to in-plane coordinates (x, y ) or (6, q), respectively. As the final result one finds that the surface strain &ik and the rotation flik are obtained from the measured interference phase by consecutively applying eqs. (5.16), (5.18),(5.19a), (5.24a) and (5.12). For the calculation of bending and torsion (change of curvature AK,,) eqs. (5.19b), (5.24b) and (5.14) have to be used additionally. The sensitivity of the described method for the measurement of surface strain E , ~ using heterodyne holographic interferometry is estimated for the special case of three sensitivity vectors symmetrically distributed around the I-axis at the angle p and a simple rotation of the object system around the &axis by the angle 8. It is assumed that the interference phase is measured with an accuracy of 84 = 0.4" at equidistant points separated by A[ = A q = 3 mm, e.g. with a detector array as described in 0 4.3, and that the derivatives aU,,la& are calculated numerically using only the values from next neighbours. The estimated sensitivity 8~ for the different strain components is displayed in Fig. 5.6 as a function of the angle p for two values of the surface inclination, viz.
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I
I
4
1
10-
€
5
8".
10.40
2 360" A € = Aq = 3mm
2
A
=
514nm
4
5
2
\ 10-
I
(
50
10
1
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20
5'
I
10"
1
20'
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Fig. 5.6. Sensitivity SE for the in-plane strain components E , ~after DANDLIKER [1977a]. p is the angle between the sensitivity vectors En and the [-axis, 0 is the tilt of the surface normal, A[=Aq is the spatial resolution, SU is the measuring error of the displacement corresponding to 0.4"phase accuracy.
8 = 0" and 8 = 45". The sensitivity 8~ of the strain measurement depends
mainly on the angle p between the sensitivity vectors En and the C-axis and is nearly the same for all strain components and surface inclinations 8. A sensitivity of 6e = 1 pstrain is already obtained for p = 5", which can even be realized by observing the object through different portions of the same hologram. Therefore one could also try to combine the scanning reconstruction beam technique, as described by FOSSATI, and SONA[1974] and EK and BIEDERMANN [1977, 19781 with BELLANI heterodyne holographic interferometry. Further investigations concerning the accuracy of strain measurements based on numerical simulations with a computer program developed for the strain evaluation following the described lines are reported by DANDLIKER and ELIASSON [1979]. The results of Fig. 5.6 are confirmed for a more general setup with p=lO". Moreover it was found that the
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determination of the strains is much less sensitive to inaccurate measurement of the geometrical parameters of the optical setup than the determination of the displacement vector itself, which was recently analyzed by NOBISand VEST[1978]. Relative errors of 10% in the measured separation of illumination sources, object and imaging lens (Fig. 5.5) cause maximum deviations of 10% or even less in the calculated strain components. In case of pure bending and torsion the change of curvature AK,, defined in eq. (5.14) can be determined approximately using a single sensitive vector E , as already demonstrated by DANDLIKER, ELIASSON, INEICHEN and MOTTIER[1976]. Pure bending and torsion is characterized by the fact that a neutral line with zero strain exists inside the structure and that the strain increases linearly from the neutral line to the surface. Usually these structures are relatively thin, so that the surface is close to the neutral line and therefore the surface strain E,, is small compared with the surface tilt, i.e. E,, << Ox,0,. Moreover the differential change of the surface strain along the surface can be assumed to be negligibly small, which means that the line-element ds remains nearly constant. These assumptions allow to determine the change of curvature AK,, from only one measured component U = (uE)of the displacement u in the direction of the sensitive vector E . Neglecting the second term in eq. (5.14) and making appropriate use of the above mentioned assumptions the following approximation is found (DANDLIKER [1977a, 1977bl): 1 a2u 1 au AK,, = -,K e, e, ax, ax, axy ~
+p
(5.25) where e, = ex,y and e, are the components of the sensitivity vector E in the object coordinate system (x, y, 2). The derivatives of U in the object system (x, y ) are obtained from the corresponding derivatives in the image system (& q), similarly as shown in eqs. (5.24), by
(5.26)
For practical purposes it is convenient to use an experimental setup with the illumination source Q (Fig. 5.5) close to the imaging lens, i.e.
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HETERODYNE HOLOGRAPHIC INTERFEROMETRY
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illumination and observation directions are nearly the same. To reduce the contributions of the differential change of the sensitivity vector, the distance do between lens and object should be large compared with the transverse extension of the objects (do/&,ax,d,,/qma,> 5). Under these experimental conditions and for a rotation of tne object system around the &axis by the angle 8 (Fig. 5.6) one obtains from eqs. (5.25) and (5.26) the following simple expressions:
AK,,
= (a2U/a(2)/cos 8
AK,,, = (8’ U/d[ d q ) - (a U/dt) tan 8/do
(5.27)
A K ~ ,=, (d’U/dq”) cos 8 -2(dU/dq) sin 8/do. The two terms containing the original curvature K,@ in eqs. (5.25) and (5.26) compensate each other as long as the sensitivity and the imaging direction are close enough. The accuracy ~ ( A K of ) this method can be estimated from eqs. (5.27) for the same parameters as used to calculate 8~ (Fig. 5.6), viz. 6 4 = 0.4”, A t = A q = 3 mm, A = 5 14 nm. For 8 = 0 one gets ~ ( A K=) 5 X lo-’ m-’, which corresponds to a bending induced strain of e g = 5 X lo-’= 0.5 pstrain on a surface at 10 mm from the neutral. A number of examples using heterodyne holographic interferometry and eqs. (5.25) or (5.27) for investigations of mechanical objects subjected to pure bending and torsion were reported by DANDLIKER, ELIASSON, INEICHEN and MOTTIER [1976] and DANDLIKER and INEICHEN [1977]. Equations (5.27) turned out to be very useful for fast evaluation of experiments and to give quite accurate results for simple object geometries, as in the case of pure torsion of a turbine blade shown in Fig. 4.9. The validity of the approximations used to derive eq. (5.25) was checked for the case of a one-dimensional, curved object with the help of a so-called Bourdon tube. This is a circularly bent tube of elliptical crosssection, closed at one end. Bourdon tubes are commonly used as pressure gauges. By applying variable air pressure inside the tube its surface will be subject to a known displacement. The tube always remains circularly bent, but with a different radius of curvature. This corresponds to a constant change of curvature AK. The original radius of curvature was about 105 mm and the diameter of the cross-section along the major axis of the ellipse was 18mm. The experimental setup was similar to the one shown in Fig. 4.1. The distance do between the imaging lens and the object was about 1m. A typical interference pattern as observed in the image plane is shown in Fig. 5.7a. The experimental results for the change of curvature AK along the surface path s are plotted in Fig. 5.7b.
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Fig. 5.7. Measurement of pure bending of a one-dimensional, curved object (Bourdon tube) using a single sensitivity vector E reported by DANDLIKER, ELIASSON, INEICHEN and MO~TIER [1976]. (a) Interference pattern observed in the image of the Bourdon tube due to a change of the internal pressure. (b) Change of curvature AK (bending) of the Bourdon tube along its curved surface coordinate s compared to the expected constant value of AK = 1.5 x m-'.
The error bars indicate the expected accuracy of each individual measurement due to an error of 64 = 0.2" for the measurement of the interference phase. The average behavior of the results compares quite well with the theoretical predictions of constant change of curvature. The considerable local deviations from the average value can be easily attributed to variable cross-section or wall thickness of the tube. In Fig. 5.8 additional experimental results are shown for a turbine blade under static bending load. The experimental setup was the same as shown in Fig. 4.4. The bending along the blade axis in Fig. 5.8a shows good agreement with mechanical theory. Note that the spatial resolution of the measurement is about 3 m m and that local variations of the bending stiffness can be detected. The change of curvature A K of ~ the cross-sections normal to the axis have been measured in two independent experiments. The results in Fig. 5.8b show the excellent reproducibility of the measurements within the expected limits. Finally, a nearly fully automated test facility for turbine blades under static load is presented in Fig. 5.9 as an example of a
Fig. 5.8. Experimental results for pure bending of a turbine blade under static load (DANDLIKER and INEICHEN[1977]). (a) Bending A K ~along the blade axis z. The local deviations around z = 200 mm appear reproducibly at different positions within the blade ~ tilt R, of the cross-section. The triangles and circles profile. (b) Change of curvature A K and represent two independent experiments.
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Fig. 5.9. Automated test facility for turbine blades under static load using heterodyne holographic interferometry. The holograms are recorded o n photothermoplastic film and the evaluation of the measurements is performed on-line with a mini-computer (Brown Boveri Research Centre, Baden, Switzerland).
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[I, 8 5
very elaborate industrial application of heterodyne holographic interferometry. Cameras with photothermoplastic film are employed (MORAW [1976], PITLAK[1978], INEICHEN and MASTNER [1979]) to achieve fast and dry in-situ recording and processing of holograms with high diffraction efficiency. The entire recording procedure, including change of loading, switching reference and object beams, exposure and development, is automatically executed. A mini-computer controls the heterodyne evaluation of the fringe pattern and performs on-line processing of the measurements. Essential parts of this article are contained in the inaugural dissertation of the author at the Swiss Federal Institute of Technology, Zurich, Switzerland (DANDLIKER [1977b]).
References ALEKSOFF, C. C., 1974. in: Holographic Nondestructive Testing, ed. R. K. Erf (Academic Press, New York) pp. 247-263. ARSENAULT, H., 1971, Opt. Comm. 4, 267-270. G. S., 1968, J. Appl. Phys. 39, 4846-4848. BALLARD, BALLARD, G.S., 1971, in: Proc. Conf. Holography and Optical Filtering (NASA SP-.299) pp. 83-92. BIJL,D. and R. JONES,1974, Optica Acta 21, 105-118. 0.. 1967, J. Opt. SOC.Am. 57, 545-546. BRYNGDAHL, BUITERS,J. N., 1971, Holography and its Technology (Peter Peregrinus Ltd., London). CHAMPAGNE, E. B., 1967, J. Opt. Soc. Am. 57, 51-55. and L. H. LIN,1971a. Optical Holography (Academic COLLIER, R. J., Ch. B. BURCKHARDT Press, New York) pp. 409-414. COLLIER, R. J., Ch. B. BURCKHARDT and L. H. LIN,1971b, Optical Holography (Academic Press, New York). CRANE,R., 1969, Appl. Optics 8, 538-542. DANDLIKER, R., 1977a, in: Applications of Holography and Optical Data Processing, eds. E. Marom, A. A. Friesem and E. Wiener-Avnear (Pergamon Press, Oxford) pp. 169-181. DANDLIKER, R., 1977b, Strain and Stress Analysis Through Heterodyne Holographic Interferometry, Habilitationsschrift, ETH Zurich (Library Swiss Federal Institute of Technology, Zurich, Switzerland). DANDLIKER, R., B. ELIASSON, B. INEICHEN and F. M. MOTTIER,1976, in: The Engineering Uses of Coherent Optics, ed. E. R. Robertson (Cambridge Univ. Press, Cambridge) pp. 99-117. D ~ D L I K E R. R , and B. ELIASSON. 1979, Exp. Mech. 19, 93-101. DANDLIKER, R., B. INEICHEN and F. M. MOTTIER,1973, Opt. Comm. 9, 412-416. DANDLIKER, R., B. INEICHENand F. M. MO~TIER, 1974, in: Roc. Intern. Computing Conf. (IEEE, New York, Catalog No. 74CH0862-3C) pp. 69-72. DANDLIKER, R. and B. INEICHEN, 1976, Opt. Comm. 19, 365-369. R. and B. INEICHEN, 1977, in: SPIE vol. 99, Third European Electro-Optics DANDLIKER, Conf. 1976 (SOC.Photo-Optical Instr. Eng., Washington) pp. 90-98.
I1
REFERENCES
83
DANDLIKER,R., E. MAROM and F. M. MOT~IER, 1972, Opt. Comm. 6, 368-371. DANDLIKER,R., E. MAROM and F. M. MOITIER, 1976, J. Opt. Soc. Am. 66, 23-30. DANDLIKER,R. and F. M. MCYITIER,1971, J. Appl. Math. Phys. 22, 369-380. DE, M. and L. SEVIGNY,1967a, Appl. Phys. Lett. 10, 78-79. DE, M. and L. SBVIGNY,1967b. Appl. Optics 10, 1665-1671. DENTINO, M. J. and C. W. BARNES, 1970, J. Opt. Soc. Am. 60, 420-421. DIXON,R. W., 1970, IEEE Trans. Electron Devices ED-17, 229-235. DUBAS,M. and W. SCHUMA", 1974, Optica Acta 21, 547-562. DURST,F., A. MELLINGand J. H. WHITELAW, 1976, Principles and Practice of LaserDoppler Anemometry (Academic Press, London) pp. 156-271. EK,L. and K. BJ-EDERMANN, 1977, Appl. Optics 16, 2535-2542. EK, L. and K. BIEDERMA", 1978, Appl. Optics 17, 1727-1732. ENLOE, L. H., W. C. JAKESand C. B. RUBINSTEM,1968, Bell System Tech. J. 47, 1875-1882. ERF, R. K., 1974, Holographic Nondestructive Testing (Academic Press, New York). FOSSATIBELLANI, V. and A. SONA,1974, Appl. Optics 13, 1337-1341. FOURNAY, M. E., A. P. WAGGONER and K. V. MATE, 1968, J. Opt. Soc. Am. 58,701-702. GOODMAN, J. W., 1968, Introduction to Fourier Optics (McGraw-Hill, New York) pp. 102-105. GOODMAN, J. W., 1975, in: Laser Speckle and Related Phenomena, ed. J. C. Dainty (Springer, Heidelberg) pp. 46-5 1. INDEBETOUW, G., 1978, Appl. Optics 17, 2930-2933. INEICHEN, B., R. DANDLJKERand J. MASTNER,1977, in: Applications of Holography and Optical Data Processing, eds. E. Marom, A. A. Friesem and E. Wiener-Avnear (Pergamon Press, Oxford) pp. 207-212. INEICHEN, B., U. KOGELSCHATZ and R. DANDLIKER,1973, Appl. Optics 12, 2554-2556. INEICHEN, B. and J. MASTNER,1979, to be published. 1972, Proc. IEEE 60, 1470-1475. ITEN.P. D. and R. DANDLIKER. KATZ,J. and E. MAROM,1979, J. Opt. SOC. Am. 69,696-705. KERSCH, L. A,, 1971, Mater, Evaluation, 29, 125-140. LANZL, F. andM. SCHL~~TER, 1977,in: SPIEvol. 136,lst EuropeanCongressonOpticsApplied to Metrology (SOC.Photo-Optical Instr. Eng., Washington) pp. 166-171. LOHMA", A. W., 1965, Appl. Optics 4, 1667-1668. LOWENTHAL, S. and H. ARSENAULT, 1970, J. Opt. SOC.Am. 60, 1478-1483. MASSIE, N. A. and D. NELSON,1978, Optics Lett. 3, 46-47. MASTNER,J. and V. MASEK,1979, to be published. MORAW,R., 1976, in: Laser '75: Opto-Electronics Conference Proceedings, ed. W. Waidelich (IPC Science and Technology Press, Guildford, U.K.) pp. 179-180. NOBIS,D. and C. M. VEST, 1978, Appl. Optics 17, 2198-2204. PERRIN,J. C. and A. THOMAS, 1979, Appl. Optics 18,563-574. PITLAK,R. T., 1978, Electro-Optical Systems Design 10, July, 46-47. PO~S~CH, J., J. SHAMIR,and J. BEN-URI, 1971, Optics and Laser Technology 3, 226-228. POLITCH,J. and J. BEN-URI,1973, Optik 38, 368-386. PRWUTNIEWICZ, R. and K. A. STETSON,1976, Appl. Optics 15,725-728. SCHUMANN, W., 1973, Exp. Mech. 13, 225-231. SCHUMA",W. and M. DUBAS,1979, Holographic Interferometry (Springer, Heidelberg). S ~ G E., K., 1972, in: Progress in Optics, Vol. X, ed. E. Wolf (North-Holland, Amsterdam) pp. 231-288. SOLLID, J. E., 1969. Appl. Optics 8, 1587-1595. SOMMARGREN, G. E., 1977, Appl. Optics 16, 1736-1741.
84
HETERODYNE HOLOGRAPHIC INTERFEROMETRY
STETSON,K. A., 1975, Appl. Optics 14, 2256-2259. STEVENSON, W. H., 1970, Appl. Optics 9, 649-652. SURGET,J., 1974, Nouv. Rev. Optique 5, 201-217. S. P. and J. N. GOODIER,1970, Theory of Elasticity (3rd ed., McGraw-Hill, TIMOSHENKO, New York) pp. 291-299. and Y. ITOH, 1968, Jap. J. Appl. Phys. 7, 1092-1100. TSURUTA,T., N. SHIOTAKE URBACH,J. C., 1977, in: Holographic Recording Materials, ed. H. M. Smith (Springer, Heidelberg) pp. 161-206. VARNER, J. R., 1974, in: Holographic Nondestructive Testing, ed. R. K. Erf (Academic Press, New York) pp. 105-147. VELZEL,C. H. F., 1970, J. Opt. SOC.Am. 60, 419420. VEST, C. M. 1979, Holographic Interferometry (John Wiley & Sons, New York). WATRASIEWICZ, B. M. and M. J. RUDD,1976, Laser Doppler Measurements (Butterworths, London).
E. WOLF. PROGRESS IN OPTICS XVII @ NORTH-HOLLAND 1980
I1
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY BY
E. GIACOBINO and B. CAGNAC Laboratoire de Spectroscopie Hertzienne de L.E.N.S., Uniuersiti Pierre and Marie Curie, Paris, France
CONTENTS PAGE
. . . . . . . . . . . . . . .
87
2. CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFTS . . . . . . . . . . . . . . . .
89
3. TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME. . . . . . . . . . . . . . . . . . . .
120
§ 1 . INTRODUCTION §
§
Q 4. TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXCITATION. . . . . . . . . . . . . . . . . 136 Q 5. THREE-PHOTON DOPPLER-FREE TRANSITIONS IN SODIUM. . . . . . . . . . . . . . . . . . . 148
0 6. CONCLUSIONS
. . . . . . . . . . . . . . . .
APPENDIX I: REMARKS O N THE CHOICE OF GAUGE . . . . . . . . . . . . . . . . . . . . APPENDIX 11: CALCULATION OF THE ENERGY ABSORBED BY ONE ATOM. . . . . . . . . . REFERENCES
151 152
. .
154
. . . . . . . . . . . . . . . . . .
158
5 1. Introduction The laser techniques that have contributed to the revival of high resolution spectroscopy rely on various specific properties of lasers: temporal coherence and monochromaticity, spatial coherence, and the possibility of producing very short pulses. Among them, the nonlinear spectroscopy methods make use of the large intensity available in a narrow spectral range. As these nonlinear methods permit, in principle, the elimination of Doppler broadening, linewidths of the order of the natural width can be obtained if the laser is sufficiently monochromatic. We deal here with Doppler-free two-photon (or multiphoton) spectroscopy, that is to say, with the elimination of Doppler broadening in transitions where the atoms absorb two or several photons. The possibility of an atom absorbing two photons has been known for a long time (GOEPPERT-MAYER [1931]). It assumes that the .energy of the photon ho is equal to half of the energy difference ho, between the ground state E, and an excited state E, (Fig. la):
E , - E, = hog,= 2ho. Due to the necessity of intense radiation sources, most of the first two-photon, or multiphoton, experiments were done in situations where the problem of Doppler shift did not arise: either in the radio-frequency range (HUGHES and GRAEINER [1950a,b]; BROSSEL, CAGNAC and KASTJER [1953, 19541; WINTER [1959]) or on transitions towards wide bands of molecules or solids (BONCH-BRUEVICH and KHODOVOI[19651) and towards ionization continua (AGOSTINI, BARJOT, MAINFRAY,MANUS and THIBAUT [ 19701). The experiments concerning two-photon absorption between narrow atomic energy levels required tunable lasers and the first one was performed by ABELLA [1962]. He used a thermally tuned ruby laser to excite the 6S;-9Dg transition in caesium. A review presenting numerous [1972]. experiments of this type can be found in the paper by WORLOCK The idea of suppressing Doppler broadening on a multiphoton transition only appeared in the early seventies (VASILENKO, CHEBOTAYEV and SHISHAYEV [ 19701; CAGNAC, GRYNBERG and BIRABEN [1973]), approximately at the same time as the appearance of dye lasers. Indeed, owing to 87
88
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
atom
eT--Ee laser
I,
I
7 @;
'
0
I
9 L--Eg
cnygy
a standing wave +-
laboratory frame
t
laser
ir
[II, 5 1
levels
cal
atom frame
'o(lt\kk)
(b,
Fig. 1. Principle of two-photon transitions: (a) Energy diagram; (b) Doppler broadening cancellation.
the very wide tuning range of these lasers, the experiments were not limited to accidental coincidence between a fixed, or almost fixed, frequency of a gas or solid laser and half of an atomic frequency, but the method could be considered as a very general spectroscopic method. The basic idea of this method is extremely simple (Fig. lb). Suppose that an atom or a molecule moving with velocity u (of component v, on the laser beam direction Ox) can absorb two photons in making a transition from state g to state e, separated by the energy difference hw,,; the atom is submitted to a standing electromagnetic wave of angular frequency w , which is produced by interference of two counterpropagating waves, as the laser beam is reflected on itself by a mirror. In its rest frame the atom sees two Doppler shifted frequencies corresponding to the two oppositely travelling waves: (1 + V J C ) and (1-v,/c), so that the resonance condition for the absorption of one photon of each travelling wave is: w (1+ u,/c) + w (1 - v J c ) = 2 0 = w,,, (1.1) which is independent of velocity. All the atoms will undergo the twophoton transition for the same value of the laser frequency w , and the width of the absorption line will be the natural one. The generalization to Doppler-free multiphoton transitions is then straightforward. Consider an atom of velocity u interacting with several plane waves with angular frequencies w, and wave vectors k, ( k , = w , / c ) . The first-order Doppler shift on the n-photon transition is 1 k, * u. If
k,= 0 ,
(1.2)
there is no first-order Doppler shift. One immediately finds again the above result for a two-photon transition; the wave vectors k , and k, must verify k , = -k2, which means one must have a standing wave. It must be pointed out that eq. (1.2) is a momentum conservation equation; the total
11.8 21
CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIITS
89
momentum of the photons being equal to zero, there is no momentum change of the atom and consequently no recoil effect. Nevertheless one cannot cancel the second-order Doppler shift whose magnitude is wg,u2/2c2,which is negligible in most cases, except possibly in metrology. In this review we exclude the case of experiments with a resonant intermediate state (the three-level problem or stepwise excitation). Many experiments have been done, of course, with two successive resonant one-photon absorptions. These experiments, which require two light sources at different wavelengths, also permit the Doppler broadening to be cancelled by the selection of velocity classes among the atoms. The Doppler-free signal observed in these stepwise experiments is obtained only from the few atoms belonging to a particular velocity class, whereas in the “true” multiphoton absorption, above-described, all the atoms contribute together to the Doppler-free signal. The type of problems encountered in the three-level experiments have been widely discussed elsewhere (see for instance TOSCHEK [19731 or CHEBOTAYEV [1976]). We see that the problems are close to those encountered in saturated absorption, and that is the reason why we shall restrict the scope of this review to true multiphoton transitions without any resonant intermediate state; we will include only a brief comparison with the case of three-level problem (3 2.1.4). We begin this review with an important theoretical section (§ 2) in which the introduction of the two-photon transition operator permits us to derive line intensities and selection rules and in which problems of light-shifts and lineshapes are discussed. We hope to furnish a convenient tool for the physicists who aim at using the two-photon method. But detailed reading of this theoretical section is not absolutely necessary for the understanding of the following experimental sections, if one only looks for a general idea of the possibilities of the method. In this experimental part, we describe first the experiments performed in a “stationary” regime, which are the more numerous (§ 3). Then we present a few studies of transient phenomena involving two-photon excitation (§4); and lastly a brief section 5 presents the case of three-photon transitions.
P
2. Calculation of intensities, lineshapes and light-shifts
The two-photon transition process can be interpreted in the following way: supposing that the atom has three energy levels (Fig. 2), the initial
90
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
Natural width
oscillator strength fgr
processes
[II, 0 2
h re
flu Laser
process
Fig. 2. Energy diagram showing the notations used in the text.
(g) and final (e) levels of the two-photon transition and an intermediate level ( r ) (whose energy is ho,,); the atom can undergo a transition between states g and r by absorbing one photon Ao; but, as we have supposed there is no intermediate resonant state, the energy defect h(w - ogr)corresponding to the one-photon detuning is large and the atom can only spend a very short time in this relay level r, time of the order of l/(o- ogr), to satisfy the uncertainty principle. Consequently no quantity has time enough to evolve in this intermediate level before a second photon is absorbed and this fact enables us to introduce a two-photon transition operator QS, the form of which is quite simple. In 0 2.1, after calculating the transition probability with monochromatic waves, we deal with the more general case of non-monochromatic fields. We then rapidly discuss the connection with the formalism which is usually introduced to deal with the three-level problem. In Q 2.2 from the tensorial decomposition of the operator QSwe derive the selection rules particular to the two-photon transitions and the
11, P 21
CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFTS
91
relative intensities of the components of a two-photon line. This formalism permits us to easily calculate the components of the density matrix in the final state as well as in the initial state of the transition. In 0 2.3 we study the light-shift of the two photon transition and in 0 2.4 we consider the case of three, or multiphoton, transitions and compare their transition probabilities and light-shifts to the case of two-photon transitions. In 0 2.5 we study the question of lineshape in more detail, with particular attention to the problem of the transit time of the atom in the laser beam. The formalism used in this theoretical section will depend on the various problems we are dealing with. The time-dependent perturbation theory is better suited to calculate the lineshape of the transient experiments, but the comparison with the three-level problem is easier done in the density matrix formalism. On the other hand, the formalism of the “dressed” atom allows a very simple treatment of the light-shifts. 2.1. TWO-PHOTON TRANSITION PROBABETIT AND TWO-PHOTON OPERATOR
2.1.1. Introduction of the two-photon operator The calculations of the two-photon transition probability are well known and can be found in many references. Nevertheless in this section we perform again this calculation from the beginning in order to justify the introduction of the two-photon operator and the cancellation of the terms involving the atomic velocity. This two-photon operator has been introduced by CAGNAC, GRYNBERG and BIRABEN [1973] using the formalism of the dressed atom in the rest frame of the atom. Here, we introduce it using the time-dependent perturbation theory in the laboratory frame. The notations are those used at the beginning of this chapter and illustrated in Fig. 2, but we may now have many other r relay levels not necessarily located between g and e. We shall attribute the energy zero to the ground level g. Electric dipole transitions are allowed between lg) and some states Ir) and between Ir) and le), but not between lg) and le) because of parity. The hamiltonian of the atom interacting with the electromagnetic field is
H=Hl+V. (2.1) H1 is the hamiltonian of the free atom, including relaxation due to spontaneous emission. It is possible, indeed, to describe the coupling of the excited state with the empty modes of the electromagnetic field by the
92
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 8 2
addition of an imaginary part to the hamiltonian Ho of the free atom (COHEN-TANNOUDJI [19681)
H1= Ho - ihU2.
(2.2)
Ho is the hamiltonian in the absence of spontaneous emission and r is an operator acting only on states le) and Ir), such that for any a state of radiative decay rate r,:
H , la)= h(ogm-+ir,)
la).
V is the interaction between the radiation field E and the atom. In the electric dipole approximation (see Appendix I) this term can be written: V=-D*E. (2.3) D is the dipole momentum operator of the atom. The electric fields propagating in each direction with the same frequency can be written E l = &,Elexp {i(k,r - of)}+ETETexp {-i(klr - of)}, E, = e2E2exp {i(k,r - of)}+E:EZ exp {-i(k2r - of)},
(2.4)
following the convention of HEITLER [1954], where the first term with e-'"I will be effective for absorption, whereas the second imaginary conjugate term will be effective for emission. E, are the polarization vectors, E, the amplitudes, ki the wave vectors (i = 1,2). We restrict the calculation to the condition of Doppler-free experiments where k,+k, = 0. Let us set
D,, = D
-
E~
(2.5)
and suppose that the atom first absorbs a photon of wave 1, then a photon of wave 2; the state vector at time f of an atom initially in the ground state can be calculated using the second-order time-dependent perturbation theory:
Setting u = T - 7' and using the condition k, = -k, we obtain
x exp {i(w -k, u - H,/h)u}D,,Ig).
(2.7)
11, 8 21
93
CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFTS
It is important to notice that the velocity u remains only in the coefficient of u, but not in the coefficient of T. Performing the integration in (2.7) we obtain!
As we have supposed that all levels Ir) are far off-resonance for the one-photon transition, we can neglect the relaxation rate and the Doppler shift k , u in comparison with the energy defect in the intermediate level:
r
-
lAw,I=lw-w,,I>>r
and
>>k, - u ,
(2.9)
and we can do the approximation:
QElE2 is the two-photon transition operator.
We must then add to (2.8) the contribution of the path where the atom first absorbs a photon of wave 2, then a photon of wave 1. In the expression of the state vector $ ( t ) , the symmetric two-photon transition operator then appears
Qz,,,
a:,,,= i[Q,,,,+Q,,,,].
(2.11)
As a consequence of the condition (2.9), the atomic velocity u has disappeared from the expression (2.8) and the result of the calculation will be independent of velocity, i.e. without Doppler broadeuing. We must now perform the remaining integration of (2.8),project on the final state (el and deduce the transition probability from g to e per unit time which is also the lifetime of the ground state due to the two-photon transition
r:’,
(2.12) where PI = 2&,,ScE: and P2 = 2&,ScE: are the powers of waves 1 and 2 (do not forget that the amplitudes of the classical electric fields are 2E1 and 2E2). S is the section of the beam, reis the lifetime of the excited state, Sw = w - w,/2 is the detuning of the two-photon transition. We verify in (2.12) that all the atoms together contribute to the absorption whatever their velocities may be. The transition probability shows a
94
[XI, P 2
DOPPLER-FREE MULTTPHOTON SPECTROSCOPY
Lorentzian shape centered on the two-photon resonance 2 0 = use;the width of the line is the width reof the excited state. On the other hand, the interest of the two-photon operator will be demonstrated in 0 2.2. 2.1.2. Expression of the transition probability in terms of the oscillator strengths Developing the two-photon operator, we obtain: PlP2
I
r b : l = , , , m ,
(el D e , Ir)(rl o e ,
Is>+ (el
oe,
Ir>(rlDe,Id
bur X
re
4 aW2+ r : / 4 ’
(2.13)
where we have introduced the detuning of the one-photon transition Am, = 0 - wgr. As is well known we find a transition probability proportional to the power square and inversely proportional to the square of the detuning of the one-photon transition. Formula (2.13) obviously shows that interference can occur between the contributions of different relay levels. Nevertheless, in many cases only one intermediate level gives a dominant contribution. In this last case, to evaluate the transition probability it is interesting to introduce the oscillator strengths, which are tabulated:
(2.14) We now have to take the level degeneracy into account. At resonance and when both waves have equal powers P1= Pz = P, one finds
where r, = q2/4reomcz= 2.8 X cm is the classical radius of the electron, A,, = 2 r c / o g , and A,, = 2rc/o,, are the wavelengths of the onephoton transitions involved in the calculation and fgr and f,, are their oscillator strengths. The Clebsch-Gordan coefficients can be calculated
11, § 21
CALCULATION O F INTENSITIES. LINESHAPES AND LIGHT-SHIFB
95
from the quantum numbers J, rn of the atomic levels and from the index q which characterizes the polarization of the light. (q = +1,0, -1, respectively, for light waves of circular u+,linear T or circular u- polarization.) Formula (2.15) permits a numerical calculation of the transition probability and can be compared with experiment (see 0 3.2). It is worth noting that very intense powers P are not necessary to obtain significant transition probabilities. Formula (2.15) is valid only if the spectral width of the laser AWL is small compared to the natural width re of the excited state. In the opposite case, a more detailed analysis in frequency is necessary, and we give it in the next section. We shall show that in the case of an incoherent laser line of width A y much larger than re,one can obtain an order of magnitude of the transition probability by substituting l / A o L in place of l/reat the beginning of formula (2.15). 2.1.3. The case of non-monochromatic fields
To deal with the case of the two-photon absorption in a nonmonochromatic field, it is interesting to develop a slightly more sophisticated calculation involving evaluation of the energy absorbed by the atom. This leads to a very compact expression for the lineshape, which will be used further. This treatment has been described by BIRABEN [1977]. In its referential the atom is submitted to an electric field E(t), of polarization E . Let E ( o ) be the Fourier transform of E ( t ) E ( t )= E
urn
dwE(o)e-'"'+
E*
urn
doE*(w)e+'"'.
(2.16)
E ( w ) is a function centered in oL,of width Awb The following results are derived in the case of linear polarkation for the two oppositely traveling waves; then the expression of the Fourier transform is simpler. If we set $(a)= E ( o ) o > 0, zS(w) = E*(-o) o <0, then E(t )= E %'(w)e-'"'do. It is supposed that the atom is irradiated during a time A T out of which the electric field is zero. This is, in particular, the case of an atom of velocity IJ which passes through a laser beam. As is usually done in the lineshape theories, the method consists of determining the state It,!r(t)) of the system submitted to the perturbation, then calculating the induced dipole (D(t))=(t,!r(t)( D Ia,h(t)), and deducing the energy absorbed by the
96
[II, 5 2
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
atom, which is the work of the electric field on the induced dipole
(2.17) Let us point out that W can only be finite if the interaction has a finite duration AT. The detailed calculation is presented in Appendix 11. The two-photon transition operator defined by (2.11) is transformed in the interaction representation to Q:e, and the result is finally given by
The quantity A(R) depends only on the electromagnetic field +m
I
+m
'iE(w)%(0 - w ) d o =-!-
2T
E(t)2epi"' dt.
(2.19)
-m
This result is quite similar to the one obtained in the case of the one-photon absorption (see for example SOBELMAN [1972])
The correlation function of the electric dipole momentum is replaced for the two-photon absorption by the correlation function of the twophoton transition operator. The electric field is replaced by the electric field squared. When the electromagnetic field is the only perturbation applied on the atom, the calculation of the correlation function is simple and the lineshape is given by
It is worth pointing out that A ( 0 ) depends on E(t)* and not on the product of the electric field at two different times E ( t , ) * E(t2).This comes from the fact that the atom can only stay a very short time in the relay level, and the absorptions of the two photons are practically simultaneous. The function A ( 0 ) involves the relative phases of the various components of the field. This can be illustrated by two very simple examples: (i) We suppose that the electromagnetic field is the superposition of 2N monochromatic, equidistant modes of frequencies w, and amplitudes En. The modes are set out symmetrically on both sides of the resonance, and the distance between modes is large compared to the natural width re.
11,s 21
CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHImS
97
There is a resonance when the atom absorbs a photon in each of the modes symmetric with respect to w,/2. Replacing the integrals by sums, the absorbed power is proportional to:
We suppose that the amplitudes En are all equal to E. If the relative phases are random, this leads to N
A =4
JEnEZN-,,12-4NE4. n=l
If all the phases are the same (the case of the mode-locked laser), there is a constructive interference A
= 4(NE2)’= 4N’E4.
The physical meaning is simple: when the phases of the modes are random, the electric field is almost constant in time. On the contrary, if all the modes have the same phase, E ( t ) is zero most of the time and very intense during short times. The two-photon absorption being a non-linear process, the signal is larger in the second case. This effect has been MAINFRAY, MANUS demonstrated in multiphoton ionisation by LECOMFTE, and SANCHEZ [1974]. (ii) Many experiments use incoherent laser lines with spectral width AwL much larger than the natural width reof the excited state. In such cases, as in formula (2.20), the Lorentzian function can be replaced by a 6 distribution. On the other hand IA(fl)l’ can be evaluated by remarking that if the width of the laser line is AmL, the correlation time of the electric field is l/AwL (see Appendix 11). Thus, we calculate the absorbed power
and the transition probability is then
We can compare this expression with the transition probability (2.12) taken at resonance. Apart from numerical factors we find that l/Aw, replaces 1/re.The formalism of this section will be useful again in sections 2.5.3 and 2.5.4 in order to obtain detailed lineshapes.
98
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DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
2.1.4. Comparison with stepwise excitation in the density matrix formalism In the case where only one intermediate state gives a dominant contribution, one could now try to compare the transition probability in two-photon excitation and in stepwise excitation, but the comparison is not easy in the formalism developed in the preceding section. It is more understandable when one calculates the density matrix of the excited state. In this last formalism, the population of the excited state pee appears at fourth order with respect to the electric field (TOSCHEK [1973]). The processes can be schematized by diagrams showing the different elements of the density matrix created after each interaction with the electric field. Careful investigation of the possible diagrams shows that only three of them must be kept: (a)
+
Pgr
+
prr + Pre + Pee,
(a')
P B + ~ pgr + ~ r r
(b)
p g g+ pgr+ Pge
+
Per
+
+
Pee,
he -+ Pee.
In addition, one must consider the complex conjugates of these diagrams. Each arrow represents one interaction with the electric field. Paths (a) and (a') can be read in the following way: from the population of the ground state pa, one produces a coherence pgr (with a resonant energy denominator for o =up, in the atom rest frame). A population in the relay level r is then produced, followed by a coherence &e (or per), and finally a population in the excited state. For an atom with velocity 0, the calculation of this process (associated with its complex conjugate) using the density matrix master equation leads to the following expression of pee (see GIACOBINO [1976])
(2.21) Pee(a')
-- rrre
E"fgrfre
Re
1
1
rgr -i(ogr - o1+ kl - u ) rre+ i ( q e- W , + k,
v)
(2.22) where rr,and rgrare the relaxation rates of the optical coherences pgr and pre. In this section, contrary to the rest of the paper, we allow the two fields to have different frequencies o1 and o,, but they have the same amplitude E.
II,O 21
99
CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFTS
In path (b) no population is created in the relay level; the transition is made through the excitation of the coherence pge which oscillates at the frequency w , + 0,; this coherence presents a resonant denominator for o1+02= wge in the atom rest frame. One obtains pee(b)
1 -re Re [rgr - i(mgr- o1+ k , E4fgrfre
u)
1.
1 I-‘, - i[uge- (ol + 0,) + ( k ,+ k2) u ] r,,- i ( q e - 0, + k, u ) (2.23) For the sake of simplicity, we have considered here only the “cascade” array where level r is located between e and g. In (2.23) it clearly appears a “two-photon resonance’’ given by the condition X
1
*u=O.
0~~-(01+02)+(kl+kZ)
(2.24)
If we set for the one-photon detunings A ~ = c o ~ - w ~ , , A2=02-0,,, the resonance condition (2.24) can be written (A + A,) = (kl + k2)* u. (2.25) Depending on the values of the frequencies o1 and 0, and of the detunings A, and A,, three cases must be considered: (i) If k , = -k, (which implies o1= 0,) the resonance condition (2.25) is independent of velocity, and for A , = - A l , we find again the “true” Doppler-free two-photon resonance we study in this paper. Thus, the origin of the resonance is in the excitation of coherence pae;in particular the width of the resonance will be that of the optical coherence rge. As the first and third denominators in (2.23) are generally very far from resonance when (2.25) is fulfilled, the real parts and k o terms can be neglected and the matrix element pee is proportional to l / A 2 (A = A l = A2)Let us now consider again paths (a) and (a’) which involve population in the intermediate state p,,. It can easily be verified that when none of one-photon transitions is at resonance, the contributions of path (a) and path (a’) almost completely cancel one another resulting in a term proportional to l / A 4 . Thus the contribution of the diagrams (a) and (a’) involving creation of population in the intermediate state can be completely neglected in the study of Doppler-free two-photon processes. (ii) Let us now suppose that the Doppler-free condition is no longer fulfilled: kl# -k2, and that the one-photon transition is not on resonance when the two-photon transition is near resonance, i.e. the Doppler width AD is much smaller than A, and A, (AD<< A l , A,) Considering first
100
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DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
the contribution of path (b), the real part and the k u terms of the first and third denominators of (2.23) are negligible and the second denominator gives a resonance when condition (2.25) is satisfied. If A, is swept, A , being fixed, the center of the line (obtained for zero velocity in (2.25)) is given by A2 = -A,. Integration over the velocities then shows a linewidth due to first order Doppler broadening which is of the order of AD Ik, +k,l/k. If k , is not too different from -k2, this width can be much narrower than the Doppler width. Experiments have been done in such conditions by BJORKHOLM and LIAO[1974]. Compared with the “true” Doppler-free two-photon resonance ( 0 ,= 02) an important factor on the linewidth is lost, but the transition probability can be increased by several orders of magnitude when decreasing the detunings A, and A,. Considering the contribution of path (a) and path (a’), we see that the same considerations as in the Doppler-free case (i) permit us to neglect it. (iii) If now the detunings A , , and A, are of the order of the Doppler width A, (AD-A,, A2), the two-photon resonance condition in path (b) is still given by (2.25). Integration over velocities would give the same resonance shape as in the preceding case (ii), if a particular role were not played by resonant velocity classes: the first and third denominators can now be made resonant if the velocity u verifies A I = kl u and A, = k, u. Let us suppose that the wave-vectors k , and k, are collinear, and as before, that A , is fixed while A, is scanned. The resonant velocity class for the first denominator is centered on v, verifying u, = A,/k,, and when A, is scanned the third denominator becomes resonant for A, = k, v,; that is, A 2 = A l k 2 / k , (let us point out that this value of A, also makes the second denominator resonant). Thus, in addition to a first Dopplerbroadened resonance centered in A, = -Al, as in (ii), one must observe a narrow resonance centered in A, = A , k,/k, and with a width of the order of the natural widths of the involved levels. Moreover, one must now consider the contribution of path (a) and (a’) which obviously also give a resonance centered in A, = A , k2/kl, due to the velocity class v,. This signal has been experimentally observed by BJORKHOLM and LIAO [19761. The complete theory, involving integration over velocities, is clearly much too complex to be approached here, and it is out of the scope of this paper to give anything other than a very brief outline of the subject; indeed, the three-level problem has been extensively studied during the last ten years and one can refer, for instance, to FELDand [1970] and to the review paper of JAVAN[1969], HANSCHand TOSCHEK TOSCHEK [1973] or CHEBOTAEV [1976].
-
-
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101
2.2. APPLICATION OF THE IRREDUCIBLE TENSORIAL SET
FORMALISM TO Q:,*,
The principal interest in the two-photon operator introduced in § 2.1.1 is the possibility of simplifying many calculations by applying the formalism of irreducible tensorial sets. This application has been done first by CAGNAC, GRYNBERG and BIRABEN [1973] and later developed by GRYNBERG [1976], GRYNBERG, BIRABEN, GIACOBINO and CAGNAC [1976, 19771, FLUSBERG, MOSSBERG and HARTMA"[1976b]. As we shall see, it permits us to obtain selection rules and relative line intensities. 2.2.1. Expansion of the two-photon operator Q:,,, on the irreducible tensors Definition (2.11) shows that the two-photon transition operator symmetrical in the exchange of fields 1 and 2,
a:,,,is
This property holds only when both fields have the same frequency. In the case of two different frequencies for the oppositely travelling waves, many of the following properties are no longer valid. The rank of a:,,, cannot be larger than 2 since it is the product of two vectorial operators and one scalar operator. On the other hand, because of the symmetry, one can easily verify that there is no tensor of rank 1 in the expansion of QfIe2. Thus a:,,, can only involve a scalar operator and a quadrupolar operator. Depending on the polarization of the light, both components or only one of them may occur. Let us calculate the standard components Q," of Q:,,,, which are formed from the tensorial product of twice the -hu), vectorial operator D and the scalar operator l/(Hc, (2.27) where D, is the standard component q of the electric dipole moment. For any pair of polarizations e l and 2 , the operator a:,,, can be written
Q:,,,
=
1
&,>a,".
(2.28)
k.q
If the polarizations are standard polarizations designated by q 1 and q2,
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DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 5 2
comparison of (2.27) and (2.28) shows that:
a:(%, 92) = b[(k 4 I 1 1 41 42)+(k 4 I 1 1 q 2 41)l.
(2.29)
One easily verifies that af, is always zero, as mentioned above. The calculation of the coefficient aqk(el,eZ), in the case of non-standard polarization, can be found in GRYNBERG, BIRABEN, GIACOBINO and CAGNAC [1977].
2.2.2. Selection rules (i) Between J levels without nuclear spin, the two-photon transition operator being the sum of an operator of rank 0 and an operator of rank 2, the Wigner-Eckart theorem leads to the following selection rules for the transition between two levels a and f l : IJ, -5,152
I
J, = 0 +J, = 1
J, = 1 +J, = 0
forbidden.
(2.30)
The second of these selection rules may seem surprising since stepwise excitation is allowed between two such levels. The physical meaning of this property is shown in Fig. 3. First the excitation from a J = 0 level is
Fig. 3. Energy diagram showing the magnetic substates in the case of a two-photon transition from J = O to J = 1. We can take into account the components of various polarizations in the laser field and compare: (a) stepwise excitation where there is a predominant resonant path; (b) “true” two-photon excitation where two symmetrical paths give a destructive interference.
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CALCULATION OF INTENSITIES, LINESHAPES AND UGHT-SHIFIS
103
possible only if the field contains both u+ and u- polarizations (for example, linear u polarization for both counter-propagating field, or one field ' u polarized and the other u- polarized). Then, the relay levels in the two-photon excitation are the magnetic substates m, = *l of the r level, as indicated in Fig. 3. With two equal frequencies there is always a completely destructive interference between these two paths, whereas in the case of stepwise excitation with two different frequencies one path has a dominant contribution. (ii) When there is a nuclear spin Z, the preceding selection rules should apply to the total angular momentum F = Z + J , for the operator l / ( H o - h o ) is only invariant in a rotation of electronic and nuclear variables together. So we have
F,
= 0 + F, = 1
and F, = 1 + F, = 0 forbidden.
(iii) If the hyperjine structure in the intermediate state is small compared to the one-photon detuning A q , the eigenvalue of the operator l/(Hohw) is almost the same for the different F values in the relay level r. If we now introduce the projection operator P(r) on the relay level subspace, we see that P ( r ) [ l / ( H o -hw)]P(r)is practically invariant in the rotation of the electronic variables only and the preceding selection rules (2.30) on J are still valid. One can also understand this result as follows: as indicated above, the atom only spends a time of the order of l/Awr in the relay level; if this time is small enough, the electronic and nuclear momenta have not time enough to become coupled. Considering this case, we can derive more selection rules concerning the angular momentum F. Let us first point out that if we consider a transition between levels of different angular momentum J, # Je, only the quadrupolar part of the operator can yield the transition, and the selection rules on F are those of an ordinary quadrupolar transition. On the other hand, if J, =J,, both operators must be taken into account but if J, = 0 or 4, the transition is purely scalar, which implies that we have on F the following selection rules: A F = 0, Am, = 0. (iv) If now the fine structure of state r is small compared to h Ao,, the selection rules (2.30) apply to L and the above remarks about F can be made about J. In particular, consider the case of a two-photon transition
104
DOPPLER-FREE MULTIPHOTON SPECl'ROSCOPY
[II, 8 2
between two levels whose wave functions are pure states of LS coupling, the possible relay levels being in intermediate coupling. If the one-photon detuning Am, is small compared to the fine structure, the two-photon transition is possible between, for instance, triplet and singlet states. But if Amr is large, one has to take into account the contribution of all the levels of a relay configuration and the singlet-triplet transition is no longer allowed. As before, one can consider that L and S have not enough time to become coupled in the relay level, and thus S cannot be changed during the transition since the optical excitation only concerns L. Thus, two-photon intercombination transitions between a ground state and a metastable state will have an extremely small probability if these levels are well below the other excited levels. Obviously all this is valid for other coupling schemes, j j coupling or Racah coupling, for instance. In the rare gases (neon) the 2-photon transition between the metastable state (core jl =;) and the 4d' (core jl = $) excited states is possible because the relay state does not have a pure wave function in Racah coupling, and Aw, is small compared to the fine structure (BIRABEN, GIACOBINO and GRYNBERG [19751).
2.2.3. Line intensities. The case of hyperfine components Continuing with the calculation of the tensorial components of the two-photon operator, we now introduce the restriction of the operator to the transition g(JJ + e(Je),
a:,,,
(2.31) where P(e) and P(g) are the projectors on the subspaces e and g. As mentioned before even if is the sum of a scalar and of a quadrupomay be purely scalar or quadrupolar, depending on lar operator, egQE,E2 the 5 values. If the ground state is not polarized and in the absence of a magnetic field, the transition probability, averaged over the (25, + 1) magnetic substates, is proportional to the population created in the excited state. It is given at resonance by
a:,,,
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CALCULATTON OF INTENSITIES, LINESHAPES A N D LIGHT-SHIFTS
105
Using (2.28) and the orthogonality relations between the ClebschGordan coefficients, this probability is transformed into
If both e and g now present a hyperfine structure, (2.33) permits us to calculate the transition probability between the hyperfine levels F, and F,. But we must use the new statistical weight (2F,+1) in place of (23,+ 1) and the new reduced matrix element (JeIFel\Q k \IJ,IFe) in place of (Jell Q k llJ,). Under the hypothesis of 9 2.2.2 (hyperfine structure small compared to the one-photon detuning Am,). the properties of the irreducible tensorial operators permit us to express the new reduced matrix element as a function of (Jell Q k 113,) and of Wigner 63 coefficient. We thus obtain
rTFC
X
1(3e11O k113g)12 lu,k(e,,e2)I2. (2.34)
2k+l
In order to obtain the relative intensities of the various components F,+ F, of the two-photon absorption line, we must multiply the probabilities (2.34) by the populations of the lower levels F,. We thus calculate the intensities, which are proportional to:
(2.35) The results obtained from these formulas can now be discussed in various cases depending on the effective tensorial rank of the two-photon operator: (i) If “’QZ,,,
is scalar, (2.35) and (2.34) immediately give
- (2Fg+
IFnFc
l)&=Fn.
(2.36)
The intensity of each line is proportional to the degeneracy of the subspace F,.
106
DOPPLER-FREE MULTIPHOTON SPECI'ROSCOFT
[II, 5 2
(ii) If egQS,ez is purely quadrupolar, we obtain
(2.37) When the initial and final states have the same angular momentum J, = J, and J,>l, the rank of the two-photon operator depends only on the polarization of the light. For example, it is purely quadrupolar when both waves have the same circular polarization, but it is the sum of a scalar and of a quadrupolar operator when the light polarization is linear. contains both a scalar operator and a quadrupolar (iii) If egQS,e2 operator, we cannot derive the relative line intensities without knowing explicitly the reduced matrix elements (Jel l Qo I l J , ) and (Jell QZ IlJ,). From the definition of the Q$s one can calculate these reduced matrix elements as a fraction of the reduced matrix elements of the operator 0, using well-known relations between irreducible tensors:
If there is only one relay level r,
IFSF=is
then proportional to
If several relay levels must be taken into account, the above expression for IF,Fe is shown to be wrong and must be replaced by a much more complicated expression, including sums over levels r, and taking their respective detuning into account. It can easily be seen that interference effects may occur and that they may be quite different for the quadrupolar contribution and the scalar contribution. These effects are very sensitive to the oscillator strengths of the involved one-photon transitions and can thus provide an interesting test of the wave-functions of the relay levels (GRYNBERG, BIRABEN, GIACOBINO and CAGNAC [19771).
2.2.4. Density matrix of the excited state The expressions obtained up to this point have concerned the intensities of the two-photon absorption line, that is, the population created in
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CALCULATION OF INTENSITIES. LINESHAPES AND LIGHT-SHIFTS
107
the excited state. Actually, in many experiments, fluorescent light is examined, which can be sensitive to other tensorial quantities of the excited state. The standard components of the density matrix on a basis of irreducible tensorial operators can be calculated using the two-photon transition operator exactly as it is done for the one-photon transition (OMONT [1977]),
(2.40) using the coefficients "'4: which characterize the electromagnetic waves:
kk'4K 0- 1 ( - ) k ' - q ' ~ i (E eZ ) l a!$(el, , e2)(KQ 1 k k' q 4').
(2.41)
44'
In formula (2.40) we suppose that the ground state is not polarized. A more general formula can be found elsewhere (GRYNBERG [1976], GRYNBERG,BIRABEN, GIACOBINO and CAGNAC [1977]). From the density matrix, it is then possible to calculate the fluorescent intensities using well-known results (FANO and RACAH[1957], OMONT [1977]).
2.3. LIGHT SHIFT OF THE TWO-PHOTON RESONANCE
We have already noticed that very intense light power is not necessary to observe the two-photon transitions. Nevertheless, the required intensity PIS is of the order of 1kW/cm2 or more, much larger than in one-photon transitions. Such intensities are able to produce in the irradiated atoms important shifts of their energy levels, which depend on the one-photon detuning (COHEN-TANNOUDJI [19621). These light-shifts could prevent the application of the narrow Doppler-free two-photon lines to high resolution spectroscopy. Therefore, a detailed study of these light-shifts, which are also called dynamical Stark shifts, is necessary.
2.3.1. Calculation of the light-shift The calculation of the light-shift of both the ground and excited state e and g are more easily done in the formalism of the "dressed atom" (COHEN-TANNOUDJI [19681, COHEN-TANNOUDJI and DUPONT-ROC [19721). The second-order perturbation theory permits us to obtain a very simple
108
[II, 5 2
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
expression for the light shifts 60, and 60, of levels e and g (CAGNAC, GRYNBERG and BIRABEN [1973]),
[x
6 0 , =2&"h2SC ,
(el D
I[-'.
4 MI D - el le) wre + 0
(el D
- 4 I r k 1 D .e2le)
2&,h2Sc
ore
+ A [ E (el D 2&,h2Sc
I[-'+ 2&"h2SC ,
I
*
+0
e l Ir>(rl D 6're-w
(el D * e2Ir>(rlD "re--W
*
&T le)
-
& :
le)
1 1 I.
(2.43)
This derivation is only valid if the two-photon transition is far enough from saturation. In the ground state the light-shift arises from the "virtual" absorption and emission of photons 1 and 2; in the excited state, the first two terms correspond to the absorption and then the emission of photon 1 or photon 2, and their denominator is close to ( 3 0 - w , ) ; the last two terms correspond to stimulated emission, followed by absorption and their denominator is close to Aw,. The expression for 6w, contains only two terms because we have neglected the processes corresponding to the emission of one photon from the ground state, which are always antiresonant, but in the case of the excited level, we must take both processes into account, since level r can be either below or above level e.
2.3.2. Order of magnitude To give an order of magnitude of the light-shift we suppose that only one intermediate state gives a dominant contribution to the transition probability and in the same way to the shift. This is the case of the two-photon transitions 3 s + 4D and 3 s + 5s in sodium (see § 3.2). The 3 s ground state is principally coupled to the 3P level by the absorption of one photon. The 4D and 5 s states are principally coupled with the same
11,821
CALCULATION OF INTENSITIES. LINESHAPES A N D LIGHT-SHIFTS
109
3P level by stimulated emission. But the 5s and 4D states are also resonantly coupled with the states of the continuum by absorption of one photon. Nevertheless, it can be shown (AVAN,COHEN-TANNOUDJI, DUPONT-ROC and FABRE [19761) that the corresponding shift will be very small if the level of the continuum reached in such a process is far above the ionisation limit. Under these conditions, one can consider the sodium atom as a three-level system. This situation is also found in many atoms. Supposing that both counter-propagating beams are identical (same frequency, same power and same polarization), the expressions of the shifts are P (81 D E* Ir>(rl D e lg) (2.44) Sw, = E(,h2SC hwr P (el D e Ir)(rl D e* le) (2.45) 60, = EohZSC Awl Let us point out that Sw, and 60, have the same sign; if the oscillator strengths f,, and f,, of the one-photon transitions are the same, the two-photon transition is not shifted. It is interesting to compare the light-shifts 60, and 60, with the two-photon transition probability rz)(res) at resonance. Using (2.13), one has in this simplified case
-
-
Hence, 80,
- 6w, = :re- rE)(res).
(2.47)
Now we have supposed that the two-photon transition is far from saturation. Thus, we have: rg)(res)<< re.Therefore
awe- so, << ir:.
(2.48)
If the oscillator strengths are of the same order of magnitude, the two shifts So, and Sw, are also of the same order of magnitude and are small compared to the natural width; so the shift of the two-photon transition is negligible. But the two-photon transition shift is not negligible when the order of magnitude of the oscillator strengths are very different, as for instance in the case of the Rydberg states of alkalis: for the 4s-10D two-photon transition of potassium, the ratio of the oscillator strengths is of the order of lo4 and the light-shifts are in the same ratio:
s ~ , / s =~ fg,ifr, , = ,-, = lo4.
(2.49)
110
[11,9 2
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
Condition (2.48) must then be rewritten tio, << tio, <<
&re= sore,
(2.50)
which is much more restrictive than (2.48). But it can be pointed out that as far as measurements of structures in the excited states are concerned, the light-shift of the ground state is eliminated, for it is the same for all components of the line. As was pointed out above, all of 0 2.3 has been written under the hypothesis of two identical counter-propagating beams (same frequency and intensity). If this is not the case, the light-shift may be stronger. So we have shown that it is possible to adjust the experimental parameters to have negligible light-shift at the same time as significant transition probabilities, i.e., that it is possible to apply the Doppler-free two-photon transitions to high resolution spectroscopy.
2.4. THREE-PHOTON AND MULTIPHOTON TRANSITIONS
As seen in the introduction, the cancellation of Doppler broadening can be extended to multiphoton transitions with the condition of zero momentum (1.2); the experimental demonstration of this effect has been done by GRYNBERG, BIRABEN, BASSINI and CAGNAC [1976] (see 0 5). It is the reason why we generalize the evaluation of transition probabilities and light-shift to the case of multiphoton absorption. This evaluation shows that in this case the encountered problems are much more difficult. 2.4.1. Three-photon transition probability and generalization to n-photon transition The atom interacts with three waves of angular frequencies ol,w z , w3, verifying the zero-momentum condition (1.2). Using the third order perturbation theory, one obtains the three-photon transition probability r(3)
=,
3
where we have introduced the three-photon transition operator, Q'3' =
1D
(12.3) r'.r"
1
E~
-
1
Ir") -(rrrlD e2 Ir') -(r'l D hA d h Aa'
* E,,
(2.52)
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CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFIS
111
with the detunings A o ’ and A o ” for the one- and two-photon transitions towards levels rt and r”, respectively Am’= 01 - mar,;
AIM”=01 + o2- w ~ ~ , , .
(2.53)
We have to take the sum over the possible relay levels and over the order in which the atom absorbs the photons, but in many cases only a few terms will yield a significant contribution. We now give an order of magnitude of the probability for an atom of absorbing three photons of equal energies, supposing in addition that two particular relay levels r‘ and rtt give a predominant contribution. Using the same notations as in 02.1.2, (2.15), we introduce the oscillator strengths of the involved one-photon transitions. At resonance, we have:
where we have neglected the numerical factors which depend on the angular momenta and polarizations. It is interesting to compare this expression to those obtained for the two-photon transition probabilities from g to r“, pzi, and to one-photon transition probability from rrr to e, r$i,
(2.55) (2.56) Comparing the last three simplified expressions we see that the factor roAP/hS appears to the power equal to the number of photons involved in the transition. This factor has the dimension of a squared frequency, and is, in order of magnitude, the square of the Rabi frequency (for a transition whose oscillator strength is 1). For a power PIS = 1 W/mm2 and A = OSp., one finds roAP/hS = (3 x lo9 sec-’)’. The three-photon transition probability (2.54) can also be written:
-
rg)(res) r‘~?(res)r,?~(res)r,./Ao’”.
(2.57)
The three-photon absorption appears as a two-photon absorption from the ground state g to the relay level r”, followed by a one-photon transition between levels rrt and e. In order to obtain a significant
112
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
transition probability (i.e. the same order of magnitude for the light intensity has to be sufficient to ensure that: I':!:(res)
[II, 8 2
r(3) and r(')),
- Aw"211'r..
This means that the one-photon transition has to be saturated by several orders of magnitude. Although practically the observation of the threephoton transition requires high-power lasers, one can also choose three different laser frequencies so that the energy defects A u ' and A o " are small. Formula (2.57) can immediately be generalized to n-photon transition probability rg);one thus obtains a recurrent relation between r(n) and the ( n - 1)-photon transition probability Fn-l)
(2.58) where
r r ~ism the- natural ~ ~ lifetime of
Awn-, is
the (n-1)th relay level r(n-l) and the associated detuning Awn-, = w g r ~ n - ~(ln- - 1)w.
2.4.2. Light-shifts Expressions (2.51) and (2.52) for the three-photon transition probability rz) are valid if I'g) is small compared to the natural width of the excited level re.But contrary to the case of two-photon transitions, this last condition is not sufficient to evaluate the light-shifts. This comes from the fact that the transition probability and the shift do not appear at the same order of perturbation development. If we suppose that the detunings Aw' and A d ' , and the oscillator strengths of the involved one-photon transitions are of the same order of magnitude, and in addition, that the three electromagnetic waves have the same intensities and frequencies, the shift of the ground state is given by a formula similar to (2.44), and the relation between shift and transition probability must now be written (80,)~
- rg' re Aw. *
*
(2.59)
If we want the shift to be much smaller than the natural width, we must set
rgk re- relh,
(2.60)
which is much more restrictive than the corresponding condition for the two-photon transition, for the detuning Aw may be much larger than re.
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113
So the light-shift in three-photon transitions will ,only be negligible for very weak transition probabilities. 2.5. LINESHAPE
The essential features of the lineshape have already been discussed in OQ 2.1.1 and 2.1.3; we wish here to go further into details. 2.5.1. Background due to first-order Doppler eflecf We have shown in 0 2.1.1 that the Doppler-free two-photon absorption resonance has a Lorentzian shape whose width is the natural width of the excited state. But we have ignored the processes where one atom absorbs two photons propagating in the same direction. The total transition probability including such processes can easily be calculated in the same way as in $2.1.1
(k being the common modulus of the wave vectors k, and k2,and u, the projection of the atomic velocity on the laser beam direction). The first term gives the Doppler-free two-photon transition probability, the second corresponds to the absorption of two photons from wave 1, and the third corresponds to the absorption of two photons from wave 2. To obtain the absorption lineshape d ( 6 w ) , we have to average (2.61) over the velocity distribution f(v,) of the atoms
I, +m
d ( 6 w )=
du, Tg’(u,)f(v,).
(2.62)
114
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
Assuming a Maxwellian distribution
(2.63)
(k Boltzman constant, M atomic mass). The first term of (2.61) is obviously independent of velocity. For the two other terms, we suppose that the Doppler width is much larger than the natural width and in the integral we can identify the Lorentzian function to the Dirac distribution:
re
4(6w - ku,)2 + r:/4
~ 6 ( 6 0- kv,).
(2.64)
Performing the integration, we find the absorption total lineshape: 1 &(so)= 4s2h4&;c2
The absorption line is the superposition of a Gaussian shape whose width is 6wD = 2ku log 2 and a Lorentzian shape whose width is 6wN = re/2. This lineshape was first predicted by VASILENKO, CHEBOTAYEV and SHISHAYEV [1970]. The width of the Lorentzian curve is half the natural width because we have calculated the absorption as a function of 6w = w -w,/2, that is, for one exciting photon. It appears in (2.65) that when waves 1 and 2 have the same polarizations, the surface of the Lorentzian curve is twice that of the Gaussian curve: the probability of absorbing two photons propagating in opposite directions is four times the probability of absorbing two photons from the same wave. The physical meaning of this fact is the following: when we calculate the probability amplitude for absorbing two counter-propagating photons, we have to take the sum of the amplitudes corresponding to the processes “photon 1 then photon 2” and “photon 2 then photon 1”. Each of these amplitudes is equal to the probability amplitude of absorbing two photons from the same wave. When calculating the probabilities, we find the factor 4.
I I , 8 21
CALCULATION OF INTENSITIES. LINESHAPES AND LIGHT-SHIFTS
115
The areas of the two curves being of the same order of magnitude, their relative heights will depend on their respective widths. Currently the Doppler width A o D of the Gaussian curve is 100, or 1000 times larger than the natural width AwN of the Lorentzian curve, and the Gaussian curve will appear as very small background. Nevertheless, in practice, the relative importance of this background is often increased by small broadening phenomena (spectral width of the laser or collisiqns), which increase the width of the Lorentzian curve without practical effect on the Gaussian curve. Another phenomenon increases the relative importance of the Gaussian background: when one observes distinct components of a two-photon line inside the Doppler width, the corresponding Gaussian curves are superimposed. Such an example can be seen in Fig. 9 of 0 3; in high resolution experiments the slope of this Gaussian background must be taken into account. Let us point out that it is sometimes possible to eliminate the Doppler background by choosing different polarizations for waves 1 and 2, so that the absorption of two photons with the same polarization is forbidden by the selections rules (CAGNAC, GRYNBERG and BIRABEN[1973], BIRABEN, CAGNAC and GRYNBERG [1974a]).
2.5.2. Second order Doppler effect We go further into the problem of the two-photon transition by including the upper-order terms of relativistic mechanics, to evaluate their influence. Replacing the frequencies seen by the atom in the resonance condition (1-1), by their relativistic expression, one finds:
huge=
2ho
JGm'
(2.66)
In (2.66) we neglect the relativistic recoil effect of the atom. This term is of the order of (v2/c2)ho/Mc2,which is completely negligible in the optical domain. Up to second order (2.66) can be written
2 0 = uae(l- V2/2CZ).
(2.67)
The corrective term is of the order of
v2/2c2= qkT/Mc2. At 300°K for light elements ( M = 20) its value is approximately which is completely negligible in many experiments. But in some cases of
116
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 5 2
interest, it has to be taken into account. In these cases, it leads to a lineshape which is shifted and asymmetrical. Let us consider the case of the positronium (GRYNBERG [19761). The study of the 1s-2s transition of positronium would be of great interest because it permits a very precise measurement of the fine structure constant a.But the positronium atoms are produced with rather large velocities: v2/2c2= lo-' to lo-'; the linewidth is limited by the transit time through the beam and principally by the second order Doppler effect. The first term of (2.61) representing the Doppler-free absorption must be modified now according to (2.67) and averaged over the Maxwellian distribution (2.63). If we suppose that the mean velocify is large enough, we can identify the Lorentzian function with the Dirac distribution and we obtain the lineshape:
(2.68) with A = w,u2/4c2
= o,,kT/2Mc2.
(2.69)
The lineshape is shown in Fig. 4. The line is shifted by the quantity A and falls abruptly to zero for 60 = 0. This sharp edge is located exactly at the atomic resonance and this feature may be useful for spectroscopy; but it must be pointed out that this abrupt side would be somewhat smoothed if we had considered the width of the Lorentzian curve. On the other side, the transition probability decreases slowly as the detuning increases. ABSORPTION
I
Fig. 4. Two-photon lineshape taking into account the second order Doppler shift (other broadening effects are neglected).
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CALCULATTON OF INTENSITIES. LINESHAPES AND LIGHT-SHIRS
117
2.5.3. Efect of transit time The atoms in a vapor are often moving fast enough to spend a time in the electromagnetic beam, which can be as short as the lifetime, especially since the beam is usually focussed to improve the transition probability. This c a k e s an important modification in the lineshape. We present here the derivation given by Biraben in the formalism of 02.1.3 (see BIRABEN[1977], BIRABEN,BASSINIand CAGNAC [1979]). BORDE[1976] has obtained the same result in the density matrix formalism. If the laser is run in T E N o , single-mode operation, the electromagnetic field can be described by a Gaussian mode. The atoms are observed in the vicinity of the waist of the focussed Gaussian beam, on a length L which is small compared to the Rayleigh length zR (KOGELNIK and LI [1966]). In these conditions, the expression of the electric field is
E(x, Y, z,
t ) = Eo exp {-(x’+
.
y2)/w~1
x [exp {-i(wLt - kz)}+exp {-i(w,t
+ kz)}] +c.c.,
(2.70)
where wo is the beam radius at the waist. The method consists in first calculating the energy absorbed by an atom with a well-defined trajectory which passes through the beam, then averaging over all atoms to get the lineshape. Let us consider an atom following a straight line trajectory whose projection on a plane perpendicular to the beam axis is represented in Fig. 5. Let p be the minimum distance between the atomic trajectory and the beam axis, and u, and u, the radial and axial components of the velocity. We express the atomic coordinates in function of time (with time origin when the atom passes at the minimum distance p). We deduce the electric field seen by the atom: E ( t )= E, exp (-p’/wi) exp (-u:t2/wi) x [exp {-i(w,-
kv,)t}+exp {-i(w,+ ku,)t}] +c.c. (2.71)
/r
projection of the trajectory
Fig. 5 . Problem of transit time of the atoms through the laser beam.
118
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 5 2
The averaging calculations are simplified by the fact that the temporal variation of E ( t ) is independent of p. Hence, using (2.19),A(R) is given by 1 wo EE exp (-2p2/w;4)exp A(R) =
rVr
The total energy absorbed by the atom is then (formula 2.20)
[
+m
x
I,
re exp -(nz032]. 4v,z/w: (0- wge)z + 9re
(2.73)
For a given velocity, the absorption curve is a Voigt profile, the convolution of a Lorentzian curve and a Gaussian curve. We must now take the average over the radial velocities v, and the trajectories (characterized by p) to obtain the energy absorbed by unit time 9 . The energy absorbed by all the atoms which go through the beam during a time AT on a length L of the beam is +m
9 AT= N [
+m
dpb
2 dv, Lu, AT f(v,),W(v,,p).
N is the atomic density and f(v,) is the radial velocity distribution
Replacing f ( v , ) and W(v,, p) by their expressions (2.74) and (2.73), and integrating over p and v, one obtains:
where S = u / w o and C depends both on atomic parameters and on parameters describing the laser beam
(2.76) The absorption profile in function of oLis the convolution of a Lorentzian cwrue of width re with a double exponential curue of width S . The parameter S characterizes the broadening due to the transit time through the laser beam. This broadening is symmetrical and the two-photon
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CALCULATION OF INTENSITIES, LINESHAPES AND LIGHT-SHIFTS
119
absorption lines are not shifted by this effect. Such curves have been observed experimentally (see BASSINI [19771, BIRABEN, BASSINI and CAGNAC [1979]). Let us point out that the particular, sharp-pointed shape of the broadening is due to slow atoms, which remain in the beam for a long time and contribute much to the absorption and very little to the broadening. It can also be remarked that the calculation is not valid for atoms of very weak radial velocity ur, which do not have time enough to pass through the beam during AT. Nevertheless, it can be shown that the corresponding correction is negligible (see BIRABEN [19771).
2.5.4. Eflect of collisions It is well known that collisions affect the shape of spectral lines and we cannot here develop in detail the problems of collisions. We wish only to discuss one aspect, where the Doppler-free two-photon method appears as quite different from the saturated absorption method. We have underlined in 02.1.3 that the lineshape, with the help of the quantity A ( 0 ) , depends on E(t)’ and not on the product of the electric field at two different times; see formulas (2.19) and (2.20). The same remark is of some importance for the analysis of the two-photon lineshape in the presence of collisions which modify the atomic velocity. Let us first demonstrate that the lineshape does not depend on the trajectory. The trajectory is determined by a law r(t). At time t, in its reference frame the atom interacts with an electric field E ( t ) equal to (in the case of a monochromatic standing wave): E ( t )=e[EOexp {-im,t -ik
*
r(t)}+Eoexp {-im,t +ik * r(t)}]+c.c.
The value of A ( 0 ) is deduced using (2.19). We see that A ( 0 ) consists of three terms, and one of them does not depend on the trajectory. It is precisely that term which gives the Doppler-free two-photon line. It follows that the two-photon absorption lineshape is not sensitive to the velocity-changing collisions (contrary to the saturated absorption lineshape). Let us now notice that this result has the same range of validity as the previous demonstration which leads to formula (2.20). In particular, one must exclude the single photon absorption: the energy detuning h Am, has to be large compared to the spectral width of the electromagnetic field as
120
DOPPLER-FREE MULTlPHOTON SPECTROSCOPY
[II, § 3
it appears in the atomic frame. Because of the variation of the electromagnetic frequency (due to a change of velocity) which takes place in a time of the order of the collision time T ~ the , spectral width which has to be considered is l / ~We ~ can . thus neglect the velocity changing collisions as long as: A o , >> l / ~ ~ . This means that the time of the two-photon absorption l / A o , has to be short compared to a collision time T ~ Such . a condition excludes the case where a collision takes place between the absorption of two photons. In conclusion, the lineshape due to collisions will be Lorentzian, and it thus can be described by two quantities: the shift of the center and the broadening.
8 3. Two-Photon Experiments in a Stationary Regime We present in this section the main applications of the Doppler-free two-photon technique to spectroscopy, the study of collisional effects and metrology. We shall focus our interest on experiments performed either with C.W. or with pulsed lasers but where no transient effects are studied. These will be treated in § 4. The first experimental demonstrations of a Doppler-free two-photon transition in a standing wave were performed on sodium by BIRABEN, CAGNACand GRYNBERG [1974al and by LEVENSON and BLOEMBERGEN [ 19741. These two experiments used pulsed dye lasers. Simultaneously PFUTCHARD, Am and DUCAS [19741 observed two-photon transitions with a C.W. laser on an atomic beam. A short time later, HANSCH,HARVEY, MEISELand SCHAWLOW [ 19741 also observed the elimination of Doppler broadening with a C.W. laser. These experiments were followed by many others, a review of which we give in this chapter. We first describe the problems involved in the experimental set up (§ 3.1) and then the typical Doppler-free two-photon experiment performed on sodium (0 3.2). In 5 3.3 we give a review of spectroscopy experiments using a similar set-up. We then discuss two particular cases: the study of Rydberg states involving ion-detection (5 3.4), and the application of the tensorial formalism developed in 0 2.2.3 to identify the hyperfine components of a twophoton line (§ 3.5). In § 3.6, we describe collision effects studies and in 0 3.7, we deal with measurements of the absolute wavelength and the search for the two-photon optical frequency standard.
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121
3.1. EXPERIMENTAL SET-UP FOR TWO-PHOTON SPECTROSCOPY
3.1.1. Dye laser and wavelength control We first describe the C.W. dye laser; the pulsed dye laser will be presented further along. The experimental set-up in the E.N.S. laboratory (for the neon experiment, GIACOBINO, BIRABEN, GRYNBERG and CAGNAC [1977]) is shown in Fig. 6. The C.W. dye laser is pumped by an argon ion laser. In order to obtain good control of the laser frequency we use two servo-loops, as follows: (i) An internal Fabry-Perot etalon ensures single-mode operation of the laser; its thickness can be piezoelectrically varied and is locked to the length of the laser cavity. This is done by modulating the thickness of the internal etalon and using the corresponding modulation of the laser intensity as the error signal. (ii) The second servo-loop is used to control the frequency of the laser cavity and does not include any modulation. The length of the laser cavity, which can also be piezoelectrically varied, is locked to the side of the transmission peak of a very stable etalon which can be pressure scanned. This servo-loop also reduces the jitter of the laser frequency due to acoustical vibrations of the laser cavity. It is easy to reduce the frequency jitter of the laser to about 1 MHz, and some authors have reported even better performances (GROVE, Wu and EZEKIEL [19741, BARGER, WEST and ENGLISH [1975], STEINER, WALTHERand ZYGAN [19761).
CALIBRATION
Art
C SWEPT ETALON
RFGENE. RATOR
Fig. 6. Experimental set-up for Doppler-free two-photon spectroscopy.
122
DOPPLER-FREE MULTIPHOTON SPEDROSCOPY
[II, 8 3
As mentioned above, the laser frequency is scanned by scanning the pressure of the external etalon. A small part of the laser light passes through a second, very stable etalon which has a free spectral range of 75 MHz. The transmitted signal gives closely spaced transmission peaks which are used to calibrate the frequency axis. The maximum power we have obtained with such a laser is 200mW with rhodamine 6G. New devices involve a double Michelson interferometer instead of an internal AMINOF Fabry-Perot etalon to provide single-mode selection (PINARD, and LALOB [1978]). Another improvement is the use of ring cavities (SCHRODER, STEIN,FROLICH, FUGGER and WELLING [1977]). In both cases a power around 1 Watt can be expected in single-mode operation. Some researchers have also used multimode, C.W. lasers; two-photon resonances appear if 0,,/2 coincides either with one mode or with the mid-point between two modes. As several modes take part in the twophoton transition, the transition probability is enhanced by comparison to the case of a monomode laser, whose power would be that of one of the modes. The main drawback to this method is that each component of the structure of a line gives rise to a comb of fringes, and the measurement of the structure supposes prior knowledge of the line separation to an accuracy better than the fringe spacing. Nevertheless, it has been used by ROBERTSand FORTSON [1975] and ECKSTEIN,FERGUSON and HANSCH [1978]. The latter authors pointed out an interesting feature of the method if a mode-locked C.W.dye laser is used: if one has to double the laser frequency, as for the hydrogen 1s-2s two-photon transition, the efficiency of the second harmonic generation is considerably increased due to the pulse structure in the time domain. As the two-photon excitation rate depends on the square of the average ultraviolet power, the signal is expected to be enhanced by an important factor. Nevertheless, there are experiments which can be done only with pulsed lasers, either because the C.W.power is not sufficient or because the C.W. lasers do not work in the needed spectral region. The pulsed dye laser which has been used by most people is the nitrogen pumped dye laser built according to HANSCH’Sdesign [1972] (see also WALLENSTEIN and HANSCH [1974, 19751). The spectral width of such a laser is rather large essentially because of the shortness of the pulse (less than lOns), which limits the width of the Fourier transform. It is possible to decrease the spectral width by filtering the laser light with a passive Fabry-Perot cavity but this results in an important loss in intensity. In order to obtain a narrow spectral width, the easier solution is to use
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TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
123
longer pulses. LEVEVSON and EASLEY [1976] used a xenon pumped pulsed dye laser, the pulses from the xenon laser being longer than those from the nitrogen laser. An interesting possibility would be to use a flashlamppumped dye laser, whose pulses are around 1 p-long. But the flashlamppumped dye laser is very difficult to run in single-mode operation with passive etalons inside (as did GALE[1973]), and a promising device is to synchronise the flashlamp-pumped dye laser to a C.W. dye laser (TR~HIN, CAGNAC and GRYNBERG [1978]).
3.1.2. Set-up for obseruation of the transitions Before describing the experiment itself, we recall the method of focusing the light beam, which is usual in all processes of nonlinear optics. According to the formulas of 0 2.1, the two-photon transition probability is proportional to (PlS)', the square of the ratio of the power P of the laser beam over its section S. Decreasing the section S, the number of irradiated atoms is also decreased as S, but the transition probability increases as US', and the whole signal increases as 1/S. Nevertheless, the area S is limited downwards either by problems of saturation, or by the shortening of the transit time through the light beam (see § 2.5.3). We now understand why in Fig. 6 the light coming from the laser is focused with a lens into the experimental cell. The transmitted light is refocused at the same point using a spherical mirror whose center coincides with the focus of the lens. In some experiments, the energy density has been increased by placing the experimental cell in a spherical concentric Fabry-Perot resonator (BIRABEN [19771, GIACOBINO, BIRABEN, GRYNBERG and CAGNAC [1977]). The windows of the experimental cell must then be tilted at the Brewster angle to reduce the losses in the cavity. The length of this cavity is piezoelectrically locked to the laser frequency in order to maximize the energy density inside the cavity. To prevent interference between the return beam and the incident beam in the laser cavity and, thus, to cause mode-hopping, one uses an optical isolator between the laser and the experimental region. The optical isolator can be either a quarter wave plate or a Faraday rotator made of flint glass in a longitudinal field. In both cases the return beam has a polarization at right angles with the incoming beam and does not interfere with it. With the first device, the atoms are excited with circularly polarized light, in the second case with linearly polarized light. HANSCH,HARVEY,MEISELand SCHAWLOW [1974] use another device: they
124
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 5 3
reflect the laser beam on itself with a small mirror mounted on a vibrating loudspeaker, which rapidly modulates the phase of the return beam. The two-photon resonance is detected by collecting photons emitted from the excited state at a wavelength that is obviously different from that of the laser. The detection wavelength is selected with an interference filter or a monochromator, which allows complete elimination of the stray-light of the laser. If the two-photon transition starts from a metastable state excited in a discharge, as in the case of the neon experiments (BIRABEN, GIACOBINO and GRYNBERG [19751, GIACOBINO, BIRABEN, GRYNBERG and CAGNAC [1977]), it is useful to observe the two-photon excitation in the afterglow to avoid the light emitted by the atoms excited by the discharge. If the two-photon transition reaches a highly excited state, detection by ionisation is more efficient (see 0 3.4). 3.2. DOPPLER-FREE TWO-PHOTON EXPERIMENTS IN SODIUM
The first two-photon experiments mentioned above have been performed on sodium 3s-4D of 3s-5s transitions, schematized in the energy diagram of Fig. 7, because of several favorable factors: the needed wavelength lies in the central lasing range of rhodamine 6G; the corresponding one-photon detuning with the relay levels 3P are rather small ( A o J o 0.02), and the involved oscillator strengths are large ( f g r = 1,
-
1772M
1
1 3S%
Fig. 7. Energy diagram of sodium showing some two-photon transitions.
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TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
125
= 0.1). We can evaluate the two-photon transition probability in these conditions using formula (2.15) of 0 2.1.2. Assuming a linewidth of the order of 1OXsec-' and a laser power P = 5 0 m W on a section S 0.01 mm2 one obtains rg)- lo4sec-'. This roughly corresponds to the observed signals. As indicated in Fig. 7, the 3s; and 5s; levels are split into two hyperfine sublevels of angular momentum F = 1 and F = 2 (the nuclear spin of sodium is 3). As pointed out in P2.2.2, because the orbital angular momenta L of both levels of the transition are equal to 0, only A F = 0 transitions are allowed; so the spectrum is made up of two lines only, as can be seen on the upper curve of the experimental recording (Fig. 8), which shows the variation of the photoelectric current versus time, while the laser frequency is linearly scanned. On the lower curve, the simultaneous recording of the transmission peaks of a confocal Fabry-Perot etalon (length 25 cm) permits the scaling of the horizontal frequency axis. This recording was obtained with a 60 mW C.W. dye laser operating at 6022A, and a sodium pressure of lOP3torr (BIRABEN, CAGNACand GRYNBERC [1974d3). The resonance is detected by means of the fluorescent light emitted at 6 154 A and 6 160 A towards the 3P; and 3P; levels. The intensity of each component is proportional to the degeneracy (2F+ 1)of the level in agreement with theory (P 2.2.3). From the distance between
fre
Fig. 8. Recording of the two-photon fransition 3S-5s in sodium: (a) Current of the photomultiplier versus the laser frequency: (b) Frequency markers given by an external fixed [ 19761and BIRABEN [1977]). Fabry-Perot etalon (from GRYNBERC
126
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, I 3
the two peaks the hyperfine splitting of the 5s level can be deduced, for the hyperfine splitting of the ground state is very well known from radiofrequency measurements (KUSCH[1954]). BIRABEN[19771 found 155.1f 1 MHz in good agreement with earlier experiments. DUONG, LIBERMAN,PINARDand VIALLE[1974] using an atomic beam method found 1 5 9 k 6 M H z ; TSEKERIS,LIAO and GURA [1976] in a radiofrequency experiment obtained 155.2k0.4 MHz. These figures may suggest that radiofrequency experiments are more precise than two-photon experiments. In fact, there is no basic underlying reason for that, except that a very precise and very stable etalon is needed for such two-photon measurements, and it was not yet available at the time of the measurements. Among the early experiments, some were performed on the 3s-4D transition of sodium. The 4 D level is split into two fine structure sublevels 4D; and 4D; (Fig. 7), and the spectrum is made up of four lines, as can be seen in Fig. 9. In this recording a wider frequency range is represented and this allows us to see the small Gaussian background due to the absorption of two co-propagating photons. The hyperfine structure constants in both 4Dg and 4D3 levels are smaller than the natural width, and hyperfine sublevels cannot be separated. Nevertheless, by measuring the polarization rate of the re-emitted fluorescence after the two-photon excitation, BIRABEN and BEROFF[1978] could give a value of the hyperfine constants IA;l= 0.23 *O. 12 MHz, IA;l< 0.28 MHz (the natural width is
0
2 GHz atomic frequency
Fig. 9. Recording of the two-photon transition 3 S + 4 D in sodium. The small Doppler background and the four components a, b, c, d explained on the energy diagram of Fig. 7 should be noted (from BIRABEN, CAGNAC and GRYNBERG [1974c]).
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TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
127
3.2 MHz). It is worth noticing that Doppler-free two-photon excitation played a leading role in this experiment by Biraben and BCroff, because it permitted them to selectively populate only one of the two 4 D levels, although their separation is smaller than the Doppler width. This selective population was an absolute necessity for a precise interpretation of polarization measurements. Numerous other experiments have been performed on sodium. The fine structures have been measured in the 5D level (LEVENSON and SALOUR [1974J) and in 6D, 7D, 8D (SALOUR [1976J); the hyperfine structure of the 6s level has been measured by LEVENSON and SALOUR [1974]. The Zeeman effect has been studied on the 3s-4D transition by BIRABEN, CAGNAC and GRYNBERG [1974b and c] and on the 3s-5s transition by [19761 and by BLOEMBERGEN, LEVENSON and SALOUR[19741. GRYNBERG The Stark effect on the 3S-4D two-photon transition has been studied by HARVEY,HAWKINS, MEISELand SCHAWLOW [1975]. Very similar experiments on other elements and molecules are reviewed in the next section. 3.3. REVIEW OF DOPPLER-FREE TWO-PHOTON SPECTROSCOPY EXPERIMENTS
We give in this section a review of the main application of Doppler-free two-photon spectroscopy, in addition to those already mentioned in sodium. These applications are fine and hyperfine structures and Zeeman structures (§ 3.3.1), isotope shifts (5 3.3.2), and the study of molecular structures (§ 3.3.3). 3.3.1. Fine and hyperfine structures; Zeeman structure Fine splittings have been measured in potassium n D (8 5 n I19) by HARPERand LEVENSON [ 19761, n D (115 n 5 24) and n S (13 In I26) by HARPER,WHEATLEY and LEVENSON [1977], and in rubidium n D (11 c n 5 32) by KATO and STOICHEW[1976] using a conventional fluorescence detection technique. Measurements on higher Rydberg states using ion detection will be presented with more details below (Q3.4). Measurements have also been performed in neon 2ps4d configuration (GIACOBINO, BIRABEN, GRYNBERG and CAGNAC [1977], BIRABEN [1977]). Hyperfine structures have been measured in neon 2ps4d configuration (BIRABEN, GIACOBMO and GRYNBERG [19751, GIACOBINO, BIRABEN, GRYNBERG and CAGNAC [1977], BEROFF[1978]), in thallium 7P (FLUSBERG,
128
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 8 3
MOSSBERC and HARTMA”[ 1976a]), in barium 6P (MEISEL and JITSCHIN [19761). A study of the diamagnetic Zeeman shift has been performed on potassium Rydberg states 14-20s and 14-16D by HARPER and LEVENSON [1977]. Such types of experiments seem very promising, especially in the case of quasi-Landau levels in high magnetic fields. 3.3.2. Isotopic shifts With reference to the types of structures considered in the previous section, Doppler-free two-photon spectroscopy can be compared with other more conventional Doppler-free methods, such as radiofrequency or level-crossing methods. But for precise measurement of the isotopic shifts, the possible methods are not so numerous, and the Doppler-free two-photon spectroscopy has proven to be particularly useful. The following species have been studied: hydrogen and deuterium by LEE,WALLENSTEIN and HANSCH [1975], WIEMAN and HANSCH [1977], neon by BIRABEN, and BAUCHE [1976] and BEROFF [1978], rubidium GIACOBINO, GRYNBERC by ROBERTS and FORTSON [1975], KATOand STOPCHEW [1976], thallium by FLUSBERG, MOSSBERC and HARTMANN [ 1976bl.
3.3.3. Study of molecular structures Doppler-free two-photon excitation of molecules has also been performed: (i) between two vibrational states of a symmetric-top molecule, in the infrared range, on CH3F by BISCHEL, KELLYand RHODES [1975, 1976a1, KELLYand RHODES [1976b]; and on NH, by BISCHEL, (ii) between two electronic states in the optical range, on NO by GELBWACHS, JONES and WESSEL [1975], on C6H6 by WALLENSTEIN [1976], on Na, by WOERDMAN [1976], on CO and N2 by FILSETH,WALLENSTEIN and ZACHARIAS [1977], and on naphtalhne by CHEN,KHOO,STEENHOEK and YEUNC[1977].
3.4. STUDY OF RYDBERG STATES USING ION DETECTION
We go further into the details about the spectroscopy of Rydberg states because it is a new and promising direction of Doppler-free two-photon spectroscopy. But when the atom is excited in very high n quantum
11,s 31
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TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
OVEN
OR
TO FREOUENCY AND WAVELENGTH
Fig. 10. Experimental set-up for Doppler-free two-photon spectroscopy of Rydberg states, and STO~CHEFF [19771). using the thermoionic detection (from HARVEY
number levels, the re-emitted fluorescence is weak and distributed over many various wavelengths, and one has to find more efficient detection means. HARVEY and STO~CHEFF [1977] proposed the use of a modified type of thermoionic detector, already used by MARRand WHERRE~T [1972] and by POPESCU, PASCU,COLLINS, JOHNSONand POPESCU [1973] in studies of highly excited states. Since then, other authors have employed this technique to study highly excited n D states of caesium (18 In 5 2 3 ) (NIEMAX and WEBER[1978]). We describe here with more details the experiment of HARVEY and STOPCHEFF [1977] on rubidium. Their experimental set-up with the usual C.W.stabilized dye laser is shown in Fig. 10. The main improvement is the use of a therrnoionic detector, which may be thought of as a simple diode working in the space-charge regime. It consists of a cylindrical anode surrounding a wire cathode. The highly excited atoms produced by the laser excitation are ionized by thermal collisions, and their presence in the space-charge affects the flow current. The trapping of the ions greatly enhances the sensitivity of the device; almost 10Oo/o efficiency is achieved, with gains of the order of 10'. Since the polarizabilities of the states under investigation are large, an electrostatically shielded volume is made inside the detector: the anode is separated into two compartments by a nickel mesh, one of them containing the cathode tungsten-wire, the second one being free of the electric field. The laser is focused into the second compartment, and the excited atoms diffuse into the region containing the cathode. Owing to this device, no Stark effect was observed even at
130
II1,B 3
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
7.0r
4.0
I
1
3.2
3.4
1
3.6 I n (net,)
1
3.8
I
4.0
Fig. 11. Experimental results of Doppler-free spectroscopy in Rydberg states n2D of rubidium. The logarithm of the fine splitting A, is plotted versus the logarithm of the effective quantum number neff (from HARVEY and STOICHEW [1977]).
n = 85. Two-photon absorption spectra 52S + n2D of R b were recorded for n = 25 to n = 85. Figure 11 shows the fine structure splitting as a function of the effective quantum number. From these results on the Rydberg series, the authors could deduce the first terms of the expansion of the fine structure splitting in inverse powers of the effective quantum number neff Afs = An;: with A
= 10 800*
+
15 GHz and B = -84870rt 100 GHz.
3.5. HYPERFINE COMPONENTS OF A TWO-PHOTON LINE
Numerous experiments involving hyperfine structure measurements have been quoted in 0 3.3.1. We come again to this kind of experiment, in order to consider the problems concerning the identification of the components. We have shown in P2.2.3 that the relative intensities of the components of a two-photon line can be predicted, using the formalism of tensorial operators. This theory has been experimentally verified on *‘Ne by GRYNBERG, BIRABEN, GIACOBINO and CAGNAC [1976] and on thallium by FLUSBERG, MOSSBERG and HARTMANN [1976]. We describe here the experiment of Grynberg and collaborators. The energy levels involved in
11, 0 31
TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
LASER 5923A
/
I
131
\,
DISCHARGE
i 2P0
Fig. 12. Energy diagram of neon showing the two-photon transitions of interest.
the experiment are presented in Fig. 12. The metastable states are populated by a discharge in a cell which contains 0.6Torr neon. The twc-photon transitions are produced in the afterglow of the chopped discharge from the lower metastable state to the levels of 2ps4d and 2p55s configurations. The intermediate states of the 2p53p configuration constitute efficient relay levels in the two-photon process, and permit the experiment to be done with a C.W.laser. To bring about the superposition of scalar and quadrupolar spectra, linear polarized light has been used (if both counter-propagating waves have the same circular polarization, the spectrum is purely quadrupolar). Correct interpretation of the fluorescence spectrum should involve calculation of the whole density matrix including collision effects; nevertheless it appears that in the experimental conditions, the fluorescence intensities are proportional to the absorption intensities. The two-photon transition starts from the metastable 3s [g] J = 2 level; when the angular momentum of the two-photon excited state is different from J = 2 (cf. the excited state 4d' [g] J = 3), the spectrum appears to be purely quadrupolar, as shown in Fig. 13. Figure 14 shows that, on the contrary, for the excited 4d [$I J = 2 level, the spectrum appears to be the superposition of a scalar and of a
132
DOPPLER-FREE MULTIPHOTON SPEmROSCOPY
[II, 8 3
Fig. 13. Recording of the two-photon transition 3s [$I J = 2 + 4d' [$I J = 3 in isotope 21 of neon. The lower part of the figure shows the strengths of the hyperfine components GIACOBINO and calculated according to a quadrupolar absorption formula (from BIRABEN, [1975]). GRYNBERG
quadrupolar spectrum. The non-zero J, = 5 + J, = $ component shows evidence for the presence of the scalar spectrum, for this transition is forbidden in a quadrupolar excitation (due to the vanishing 65 Wigner coefficient
From the relative intensities of the line components, the ratio between the reduced matrix elements (Jell Q" 115,) and (Jell Q2115), can be deduced (see § 2.2.3), and hence information about the oscillator strengths of the involved one-photon transitions (eq. 2.38). In fact, an exact calculation, as done by GRYNBERG [1976], taking several relay levels into account is necessary to correctly interpret the spectrum. 3.6. STUDIES O F COLLISIONAL EFFECTS
In studying collisional effects two types of experiments can be performed: first, taking advantage of the fact that the line is not Doppler broadened, shifts and broadenings due to very low pressures of foreign gases can be measured. Second, it is possible to excite only one sublevel inside the Doppler width and observe the collisional transfer of population and coherence towards other sublevels. (i) As it was noticed earlier (§ 2.5.4) the two-photon absorption lineshape does not depend on velocity changing collisions. The lineshape is thus Lorentzian, and the influence of collisions on the lineshape can be described by two quantities: the shift of the center of the line and the
11, I31
TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
33
T'Z
133
li
Fig. 14. Recording of the two photon transition 3s [$]J = 2 4d J = 2 in isotope 21 of neon: (a) experimental recording; (b) theoretical intensities of the various hyperfine components (F,F,) for a quadrupolar absorption. From left to right: (2,;) (1. i) 1) 5) ($, +) (2.3) ($, 2) (2,;) (4,;) (f, f ) (2, (c) theoretical intensities of the hyperfine components (F,F,) for a scalar absorption. From left to right: ,(; 4) ($, i) (5, (f, f ) (from G R Y N B E R G , BIRABEN, GIACOBINO and CAGNAC [1977]). --.)
z);
(z, (z,
z)
broadening of the line. Shift and broadening measurements have been done on sodium 3s-4D and 3s-5s transitions by BIRABEN, CAGNAC and GRYNBERG [1975a], BIRABEN, CAGNAC, GIACOBINO and GRYNBERG [19771. The experimental set-up is similar to the classical one ( 5 3.1), but twophoton transitions are excited simultaneously by two parts of the same laser beam in two sodium cells, the one containing pure sodium, the other, sodium and a foreign gas. The fluorescence signals are simultaneously recorded when the laser wavelength is swept. Such recordings are shown in Fig. 15 for the 3s-4D and 3s-5s transition in the presence of 1Torr of neon. The broadenings are of the same order of magnitude in both cases, but the shifts have opposite signs. It can be verified on the experimental curves that the broadened lines remain Lorentzian-shaped as is predicted by theory (GRYNBERG [1976]). A systematic study of shifts and broadening have been done on these two two-photon transitions with the five noble gases. The results have been compared with calculation
134
[II, 8 3
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
3s-4D
17,
,
,
o
atomic iooMHz*frcqucncy
0
100MHz
.
Fig. 15. Simultaneous recordings of the two-photon lines of sodium in the presence of 1 Torr of neon (upper curve) and in a cell of pure sodium (lower curve): at left one component of the 3s -+ 4D transition, showing a red shift, at right one component of the 3 S + 5s transition, showing a blue shift (from BIRABEN, CAGNAC, GIACOBINO and G R Y N BERG [1977]).
using either van der Waals interaction potentials or PASCALE and VANDEPLANQUE [1974] computed potentials. Similar experiments have been performed by BOLTON, HARVEYand STOICHEFF [19771 on highly excited Rb atoms colliding with fundamental R b or with argon. (ii) Collisional transfer between fine structure sublevels have been extensively studied in the case of the sodium 3S-4D transition by BIRABEN, CAGNAC and GRYNBERG [1975b], BIRABEN, BEROFF, GIACOBINO and GRYNBERG [1978] (see also BIRABEN [1977]). The atoms are selectively excited by two-photon absorption either in the 4D; or 4D; sublevels, and the re-emitted fluorescence towards 3Pi and 3P; is analysed in intensity and polarization (see Fig. 7). Advantage is taken of the fact that the 4D;-3P; fluorescent line is forbidden, and that the 4D;-3P; line is five times more intense than the 4Dt-3P; line. This allows the determination of the transfer between the two 4 D sublevels by analyzing the light with a monochromator. Monitoring not only the intensity but also the polarization permits the study of the transfer of population and of coherence as a function of the foreign gas pressure. In fact, in order to analyze the intensities with precision, one also has to take into account the possibility of quenching. It appears that the 4F level is near enough to the 4D level for the two levels to be connected in a collision with a noble gas atom. The measured cross-sections for the different relaxation and transfer processes considered above have also been compared with the theoretical predictions given by the computation mentioned above. Especially at intermediate and long interatomic distances, this comparison provides
I t 8 31
TWO-PHOTON EXPERIMENTS IN A STATIONARY REGIME
135
information on the rare gas-sodium interaction potentials, whose contribution is dominant. 3.7. METROLOGY
Very little work in two-photon spectroscopy has been done on this subject until now. Nevertheless, it is probably one of the most promising applications of the method. Two types of experiments can be considered: the absolute measurement of energy levels, yielding new determination of a fundamental constant or the use of a two-photon transition between two very long-lived states to provide a new optical frequency standard. (i) An example of the first type of experiment is the 1S-2s two-photon excitation of hydrogen (HANSCH, LEE,WALLENSTEIN and WIEMAN [1975], WIEMAN and HANSCH[1977]). Due to the very long lifetime of the 2 s metastable state, the two-photon line can be extremely narrow. Furthermore, the 1S-2s frequency is directly related to the Rydberg constant (or to the Lamb-shift of the ground state if one supposes the Rydberg constant to be known). The two-photon transition is excited, using a C.W. monomode laser amplified by a three-stage pulsed dye amplifier and doubled in a lithium formate crystal near 2 4 3 0 A . The two-photon excitation is monitored by observing the collision-induced 2P-1s fluorescence at 1215 A. The C.W. dye laser output near 4 860 A allows simultaneous observation of the Balmer-P line by a polarization spectroscopy method (WIEMAN and HANSCH[1976]), which is an improvement of the classical saturated absorption spectroscopy. The determination of the 1s Lamb shift is performed by comparing the frequencies of the 1S-2s two-photon transition and the 2Pq-4Dg component of the Balmer-P line. After corrections for several experimental spurious shifts, the value of 8 159.2*29 MHz is found, which is in good agreement with theory (8 149.43*0.08 MHz). (ii) A two-photon optical frequency standard is actually being perfected by HALL,POULSEN, LEE and BERGQUIST [1978] in the case of the transition from the ( 6 ~4S; ) ~ground state of 209Bito the ( 6 ~*Pi ) ~metastable state: the main advantage of this system is the transition wavelength (6 030.5 A), its suitability for atomic beam studies and its long lifetime (4ms). Because of experimental problems, the major one coming from the very low transition probability ( the Na 3S-4D transition probability for the same laser intensity), only rather broad resonances (2.6 MHz full width) have yet been observed, but important future improvement is
136
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 0 4
expected. Another idea proposed by the same authors is to use twophoton transitions towards high n rubidium Rydberg states (30 s n s 50) whose lifetime is long without excessive polarisability shifts.
0 4. Transient Processes Involving Two-Photon Excitation Doppler free two-photon excitation is also a very useful tool to investigate transient effects. As is well known, the coherent field of the laser has permitted the generalization to the optical range of many transient experiments, which were only possible earlier in the radiofrequency range: free precession, nutation (TORREY[1949]), spin echoes (HAHN[1950]). The first generalization was the photon-echo method developed by KURNIT,ABELLAand HARTMANN [1964] (see also ABELLA [ 19691). But a new problem appears in these optical transients, due to the Doppler broadening: in the observation of transient optical coherences (BREWERand SHOEMAKER [1972], DUCLOY,LEITEand FELD[1977]) the Doppler dephasing results in destructive interference between different velocity classes, and shortens the observed decay. This difficulty is suppressed with Doppler-free two-photon excitation, which permits the excitation of .all the atoms exactly at resonance, or off-resonance with the same energy detuning, whatever their velocities may be. This paragraph is divided into three parts, the first one concerning resonant excitation and relaxation-time studies, the second one dealing with off -resonance excitation, and dependence of the observed ringing with the detuning and shape of the pulse. The third part will be devoted to a recently proposed and very attractive method, which consists in transposing the classical radiofrequency technique of Ramsey fringes in the optical domain.
4.1.
RESONANT EXCITATION
Pulsed two-photon excitation can be used to populate an excited state; it permits the observation of fluorescent decay or of superradiant decay (WALLENSTEIN [19761). But in such experiments, the two-photon excitation plays the same role as a stepwise excitation, and we will rather focus our interest on experiments which take advantage of the Doppler-free character to observe the two-photon optical coherence. The first experiments of two-photon optical free induction decay have
11, 8 41
TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXClTAllON
137
been realized by Lov [1977] and LIAO,BJORKHOLM and GOR~ON [1977]. This type of experiment has been improved by LIAO,ECONOMOU and FREEMAN [1977], who observed the evolution of the two-photon coherence by a delayed pulse technique. We describe here their experiments with more details. They use two identical counter-propagating, simultaneous laser pulses to produce a two-photon coherent state. The decay of this coherent state is monitored by a delayed probe pulse. In the same way as for the exciting pulses, the phase-matching condition is such that the probe pulse produces a backward emission whose amplitude is proportional to the coherence. As long as the atoms remain coherently excited, the probe will continue to generate a signal; measurements of the decrease of the backward-wave intensity with increasing probe delay gives the lifetime T2 of the optical coherence. The interesting point is that observation of this phenomen does not require narrow linewidth pulses, as the exciting (and probing) frequency does not need to be measured: (i) the finite bandwidth of the pump pulse only leads to a very slight Doppler dephasing. This dephasing occurs because the two-photon transition will not be excited by frequency components at exactly o=oge/2,but at combinations of frequencies w,/2 + S and o2 /, - 6. Because the frequencies are unequal, a residual Doppler width of 26 u/c is obtained (where .t, is the mean quadratic velocity). For a laser linewidth of AUJC = 0.1 cm-' this additional linewidth is only of a few kHz; (ii) the finite bandwidth of the probe gives a finite bandwidth to the generated signal but does not affect its intensity. Nevertheless the bandwidth of the laser should be kept reasonably narrow since the backward-wave signal can only be generated on the laser's coherence length c/Au,. The experiment has been performed on sodium and the pressurebroadening coefficients for the 3s-4D two-photon transition for collision with neon, argon and helium have been determined. The sodium atoms are pumped with the 5 ns, 15 kW pulses from a nitrogen-laser-pumped dye laser; the bandwidth is approximately 4 GHz. The output of the laser is divided into two beams of orthogonal polarization. One of them is reflected back on itself to pump the two-photon transition, and the second one is used to probe the two-photon coherence. As the backward re-emitted pulse has the same polarization as the probe pulse, it can easily be isolated from the pump pulse. Moreover, the probe pulse is not necessarily collinear with the pump pulse, and the signal can also be isolated spatially. The sodium vapor pressure is maintained at approximately 0.3Torr. Figure 16 shows the dependence of the signal intensity
138
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
r11, Q 4
1.0
0.1 v)
z
W
I-
E
0.01
A
a L ‘3 v)
0.001
0.0001 10 DELAY (nrec)
20
Fig. 16. Optical two-photon transients after a very short resonant excitation (in sodium 3s + 4D). The two-photon coherence is excited by two counterpropagatinglaser pulses and is monitored with a delayed probe which produces a backward emission: Intensity of the backward wave versus the delay (with 0.8 Torr of neon buffer gas) (from LIAO,ECONOMOIJ and FREEMAN [1977]).
on the probe delay when 0.8Torr of neon buffer gas is added. The pressure-broadening coefficients measured here at 400°C, agree with those of BIRABEN, CAGNAC, GIACOBINO and GRYNBERG [1977] (see 0 3.6). The experiments we have mentioned above by LIAO,BJORKHOLM and GORDON [1977] on the same Na transition, and by Lou [1977] on NH3 molecules in the infrared range are quite similar in their principle, except for the following features: (i) The excitation is done by two counter-propagating lasers at slightly different frequencies w, and w2, which gives a small residual Doppler broadening, but increases the transition probability and permits easier saturation of the two-photon transition. (ii) The Stark switching technique (BREWER and SHOEMAKER [1971]) is used to produce the transient process; the sudden application of an electric field shifts the studied energy level by the Stark effect and brings the two-photon transition widely off -resonance. Then the two-photon optical coherence begins a free precession with the shifted atomic frequency wRe, whereas the laser frequencies remain constant; a beat signal can be observed between the shifted atomic frequency w, and the “two-photon frequency” w1 + w2.
11, 0 41
TRANSIENT PROCESSES INVOLVING TWO-PHOTONEXCITATION
139
So in these experiments the laser beams successively play two different roles: first, they resonantly excite the optical coherence; second, after the Stark pulse, they probe the evolution of the optical coherence; the beam at frequency o1 generates the backward emission at the complementary frequency wge-ol.
4.2. OFF-RESONANCE EXCITATION
The important point in off-resonance excitation is that all the atoms can be excited with the same energy detuning. It is thus possible to study very precisely the off -resonant transient effects, particularly for weak detunings from resonance. A quite simple experiment has been performed by BASSINI,BIRABEN,CAGNAC and GRYNBERG [1977] on the 3s-4D twophoton transition of sodium. They observed the transient fluorescence signal when the light from a C.W. dye laser is suddenly switched off or on, for various fixed detunings of the laser frequency. The theory of these two-photon transients is identical to the theory of the one-photon transients (when the Doppler effect is ignored), provided that the energy defect do, with the relay level is large enough (see §2.2.1); the laser frequency detuning 8w, must be replaced by the actual energy detuning 6’0, which is twice as large (8’0 = 2w,- oge = 280,). The behaviour of the transient is quite different in the respective cases as the light is suddenly switched on or off: (i) The sudden irradiation of atoms causes the ringing corresponding to optical nutation. Very similar effects have been studied for a long time in the radiofrequency domain (TORREY[1949]). It is well known from elementary quantum mechanical calculation that in such cases it is possible to observe oscillations of Rabi frequency [( Weg/fI)2 + (8’0)~]1,where We, is the matrix element of the perturbation hamiltonian (due here to the laser electromagnetic field) which causes the transition between levels e and g. These oscillations decrease with a time constant equal to T2,the relaxation time of the optical coherences. In the experiment of Bassini et al., the frequency of the Rabi nutation is 8‘0 because the two-photon excitation is very far from saturation and thus W,,/A<<1/Tz-6’o. The preceding assertions are valid for zero rise time of the pulse or at least when this rise time is short compared to the Rabi frequency. Otherwise, the amplitude of the oscillations is reduced compared to the theoretical prediction given by a square shaped function. This can be
140
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
PI, 5 4
easily understood if one considers the Fourier transform of the pulse: when off -resonance electromagnetic radiation is turned on, the atomic resonance frequency wge is contained in the excitation Fourier spectrum if the edge of the pulse is steep enough compared to 1/6’w. Under that condition beats will be observed between the resonance excitation at the atomic frequency wle and the Raman excitation at 2w,, twice the laser frequency. If the rise time of the pulse is not short enough, the component at the atomic frequency w, is less intense and can even disappear, causing a decrease in the observed interference between resonant and Raman re-emission. (ii) If the light pulse is cut 08instantaneously, the emitted light simply decays following an exponential function with a time constant equal to T , which is the relaxation time of the level population. But here the imperfection of the shutter considerably influences the shape of the signal. There has been a large theoretical interest about these effects (see for example MUKAMEL and JORTNER[1975] and SZOKEand COURTENS [1975]) after the work of WILLIAMS, ROUSSEAU and DVORETSKI [1974] who observed different time-dependent decays between pure resonance fluorescence and pure Raman scattering after an exciting pulse, in a one-photon transition. Suppose that the light is not cut off instantaneously but with a time constant 8 shorter than TI.Then the decay of the scattered light depends on the energy detuning 6‘0: -if 6 ’ w c l/8, there is only one exponential curve with time constant T ,; - if 6’w >> l/8, the scattered light decreases in the same way as the light pulse; -in the intermediate cases (6’0 * 8 l), the decay is described by two successive curves, the first one follows the exciting pulse, the second one is an exponential curve with time constant TI. The relative amplitude of the two successive curves depends on the product 6’0 * 8. The experiment by Williams et al. was performed on a one-photon transition in iodine and the phenomena were blurred because of the average on the Doppler shifts of different velocity classes. In the twophoton experiment by Bassini et al. the energy detuning 6’w is well defined and can easily be compared with 8. The experiment was performed with a C.W.dye laser focussed into the sodium cell, and reflected back on itself to provide Doppler-free excitation. An acousto-optic modulator was used to deflect the light beam; the rise time of the system
-
11, P 41
TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXCITATION
141
Fig. 17. Optical two-photon transients observed by sudden irradiation, for various values of the two-photon energy detuning h 6'v (in sodium 3s 4D): number of spontaneously scattered photons versus the time (0start of the switching; pulse's rise time -1411s) (from BASSINI,BIRABEN, CAGNAC and GRYNBERC [ 19771).
(14 ns) was smaller than the lifetime (50 ns) of the 4D state. The repetition rate of the light pulses was of the order of lo6s-l. The number of fluorescence photons with C.W. excitation was small (lo3 s-' to lo6 s-' at resonance), and the probability of observing more than one photon during the observation time following each pulse (150 ns) was negligible. A time-to-amplitude converter measured the arrival time of each photon after the leading edge of the pulse. A multichannel analyser gave the plot of the number of photons versus the arrival time. Figure 17 shows the experimental curves obtained at the rising edge of the pulse for various values of the energy detuning 6'w = 21r 6'v. The oscillations at frequencies corresponding to 6'v can be observed. The amplitude and decay of these oscillations are in agreement with the expected values (BASSINI [19771). Figure 18 shows the decay of the fluorescent light at the trailing edge of the pulse (decay time of the modulator 8 = 22 ns). One clearly sees the progressive appearance of the sharp curve which follows the light pulse and the decrease of the amplitude of the slow T , exponential curve. The amplitudes of the two successive curves are almost equal for case (c) corresponding to the product 6'w * 8 = 21r 6'v * 8 = 1.5. (The ringing
142
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 5 4
Fig. 18. Optical two-photon transients observed by the cut-off of the light, for various values of the two-photon energy detuning h 6'v (in sodium 3s + 4D): number of spontaneously scattered photons versus the time (0start of the cut-off; 0 = 20 ns end of the cut-off BIRABEN, CAGNAC and GRYNBERG [1977]). time) (from BASSINI,
which appears on the last two curves is mostly due to the fact that the acousto-optic modulator does not deflect the whole beam but lets a part of it pass through the cell. The ringing thus corresponds to the sudden transition from a stationary Raman scattering regime to another one with a different intensity of the laser light.) 4.3. RAMSEY'S FRINGES IN DOPPLER-FREE TWO-PHOTON RESONANCES
When the two-photon transition takes place between two very narrow levels such as ground and metastable states (an important example being the 1s: + 2s; transition of H),the width of the Doppler-free two-photon line in the usual conditions is limited by the transit time of the atoms through the laser beam (see 52.5.3). To overcome this problem, BAKLANOV, CHEBOTAYEV and DUBETSKY [1976] have proposed the use of two spatially separated regions of interaction with the laser; the signal due to the atoms that have passed through the two regions at times separated by the interval T, exhibits interference fringes with a splitting 1/2T in the
11, 5 41
TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXCITATION
143
profile of the Doppler-free two-photon resonance. The experiment has been performed by CHEBOTAYEV, SHISHAYEV, YURSHINand VASILENKO [1978] on the 3S-4D two-photon transition in sodium. The same effect also appears if the two excitations are shifted in time rather than in space, that is, if the atom is illuminated by two successive coherent pulses, as demonstrated by SALOUR and COHEN-TANNOUDJI [1977] on the 3s-4D transition in sodium. The use of two time-delayed coherent laser pulses permits taking advantage of the high power of pulsed dye lasers, without loss in resolution: the linewidth is no longer limited by the inverse of the time duration 7 of the pulse but, as above mentioned, by the inverse of the time separation T between the two pulses. These experiments can easily be understood as a transposition of the original radiofrequency experi[1949] (see also RAMSEY [19563) to the optical domain. ment of RAMSEY The fact that the Doppler-free two-photon transition is produced in a standing wave, of definite phase in all points of space, is essential for this transposition. We will first recall the principle of the calculation of this effect and then describe Salour’s experiment with delayed pulse.
4.3.1. Principle of the experiment Consider two successive coherent pulses having the same phase obtained, for instance, by amplifying two portions of a sinusoidal wave emitted by a C.W. monomode laser of frequency o. The atoms are submitted to an electric field which can be written:
(4.1)
( E ( t )= 0
for any other time,
where T is the duration of the pulses, T is the time-delay between the first and the second pulse. Each pulse is reflected back on itself by a mirror placed near the cell containing the atoms so that the atoms are submitted to a standing wave pulse, whose phase is independent of z. As in 0 2.1.1, a straightforward perturbation calculation permits the obtaining of the two-photon transition probability amplitude b ( t ) between levels g and e at the end of the first and the second pulse. In a first step, the finite life-time of level e will be ignored.
144
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, § 4
In the interaction representation the transition probability amplitude becomes:
6(t)= b ( t ) exp (iw,,t)
(4.2)
and it is given by the equation
=O
for any other time,
where 0 1=
E2 h(el
QZ
lg)
(cf. eqs. (2.10) and (2.11)).
(4.4)
Integrating (4.3) between 0 and T , with the initial condition 6(0)= 0, one obtains b ( T ) at the end of the first pulse
b , = 6 ( ~=)
-"1
wge- 2 0
[exp {i(wge- 2 0 ) ~-)13.
(4.5)
lb,I2 is the transition probability after the first pulse sin (w, - 2 0 ) 7 / 2 (ogc - 2w)/2
SW
(4.6)
As expected, the curve representing lb1I2 as a function of 6w = w -lugeis
a diffraction curve centered on Sw = 0 with a width of the order of 27r/r (dotted curve in Fig. 19): the linewidth is inversely proportional to the duration of the pulse. According to (4.3), 6(t) does not change between r and T. After the AlU=w/T
P
Fig. 19. Principle of Ramsey's fringes: transition probability versus the laser frequency after one pulse (dotted line) and after two pulses separated by time detuning 60 = o - w,,/2, T (solid line).
11.8 41
TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXCITATION
145
second pulse, integration of (4.3) between T and T + T with the initial condition 6 ( T )= b , leads to
b 2 = g(Tt-7)= b,[l +exp {i(oge-20)T}],
(4.7)
and the transition probability after the second pulse is
[
lb2I2= 4 J b l2 , cos
2
= 4 JbIl2* COS* (6w * 7').
(4.8)
Interference fringes appear within the diffraction profile of lb1I2 (solid curve of Fig. 19). When the laser frequency is scanned, one observes fringes with a spacing Am = or Av = 1/2T.
So the resolution is improved by a factor ~ T / and T the central fringe is exactly centered on o = oge/2,which is important for spectroscopic applications. One can easily see that a phase variation between the two pulses would lead to a shift of the whole interference pattern within the diffraction curve. Thus, important phase fluctuations between the two pulses would wash out the interference fringes. This means that the two pulses must have a constant phase difference during the experiment. If this difference is not zero, the fringes are not centered on the atomic resonance. The constant phase condition also implies that the two pulses have to be Fourier limited; i.e. their coherence time must be longer than their duration. On the other hand, fluctuation of the time-delay T between the pulses will produce a decrease in the contrast of the lateral fringes since the spacing depends on T, but will alter neither the position of the central fringe nor its amplitude. One can now take into account the effect of the finite lifetime l/reof level e. If we suppose that the duration of the pulse is much shorter than this lifetime (T<<1/re), only eq. (4.7) has to be modified and 1 replaced by exp (-reT/2).As the two terms which interfere in the bracket (4.7) no longer have the same modulus, the contrast C of the fringes is decreased. One finds 2 exp (-TeT) C= (4.9) 1+exp (-Ten * So increasing T gives a better resolution but a poorer contrast. This comes from the fact that a smaller number of atoms live a time long enough to interact with the two pulses. In practice, it is difficult to use values of T much longer than the lifetime l/re, and the method is better
146
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
used to observe lines with natural width T of the pulses.
CII, 5 4
re,despite the too short duration
4.3.2. Experimental realization As mentioned before, the delay time T between the two successive light interactions can be obtained experimentally by two methods: (i) one uses the movement of the atoms, which pass through two different interaction regions separated by the distance I, as in the original Ramsey experiments. The delay T = l/v depends on the atomic velocity u but, as we have seen, the central fringe does not depend on the delay T (CHEBOTAEV, SHISHAEV, YURSHIN and VASILENKO [ 19781); (ii) one effectively produces two successive pulses separated by the decay time T (of the order of the lifetime), and those pulses irradiate the same region of a cell. We give a few more details about this method which was used by SALOUR and COHEN-TANNOUDJI [1977] because it raises some problems. The easiest method to produce time-delays of this order of magnitude is to use an optical delay line of length L = cT. But in this case the phase of the delayed pulse is modified by an amount cp = o T = kL. This dephasing depends on the laser frequency o, and will not be kept constant during the experiment, as we will sweep the frequency w. But we have seen that the constant phase is the essential condition for observation of Ramsey fringes. Moreover it is easy to show that this particular phase variation versus o leads to a complete vanishing of the fringes: in eq. (4.1) we must then replace the electric field during the second pulse ( T < t < T + T) by the new expression:
E ( t )= E exp {i(kz - ot)+ ikL} = E exp {i(kz -cot)+ ioT}.
(4.10)
Performing the second integration, as in the above calculation, we find:
b2 = b,[l +exp (iwgeT)].
(4.11)
The ratio b2/b, does not depend now on o,because the dephasing cp between the two pulses exactly compensates the dephasing between the optical coherence of the atom and the laser wave. The fringes of eq. (4.8) have completely disappeared. In order to observe the Ramsey fringes it is necessary to maintain constant the phase cp = wT between the two pulses by changing the delay T in inverse function of o,which is possible as the delay itself need not be
I I , 8 41
147
TRANSIENT PROCESSES INVOLVING TWO-PHOTON EXCITATION
constant. In order to observe centered Ramsey fringes we must keep Q = w T = n * 27r (n integer). This condition was realized in Salour's experiment in which a single mode C.W. dye laser, frequency stabilized to a pressure tuned etalon was scanned over the 3s-4D two-photon transition in sodium. The output of the C.W. laser was amplified in three stages of the dye amplifiers pumped by a nitrogen laser. The output of the third amplifier was split into two pulses and the second one delayed in a delay line of length L ; both parts were focussed at the same point of the cell. To control the length L of the delay line the fact that the C.W. laser beams and the two amplified pulses exactly follow the same paths was made use of: the piezoceramic pushing the return mirror of the delay line was controlled in such a manner that the interference between the direct and delayed C.W. beams gave a maximum intensity. Thus, the two pulses were phase-locked to zero during the entire experiment. Furthermore, the delayed beam, with its longer optical path, had a wavefront with a larger radius of curvature than that of the direct beam, and a high-quality lens was used to match the curvatures of the wavefronts and to avoid phase variations within the section of the beam. Figure 20 recorded by SALOUR
2Tz25ns C
L
0
- -
-c
I
I
"c
I
2
'
LASER FREQUENCY (GHd
Fig. 20. Ramsey's fringes in two-photon transition 3 S 4 4 D of sodium: (A)recording of the four components of the transition in a reference cell excited with a single pulse (same curve as Fig. 9 but the components are enlarged by the too short pulse duration T 8 ns). (B) and (D)recording in the sample cell excited with two time-delayed coherent pulses (delay, respectively, T = 12.5 ns and T = 16.5 ns). (C) checking of the technique permitting the elimination of the diffraction background in curves (B) and (D)(from SALOUR [1978]).
-
148
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, 8 5
[ 19781 shows the four well-known two-photon resonances of the 3s-4D transition of sodium (see section 3.1.2 and Fig. 9), observed with a single pulse (upper curve A) and with two pulses delayed by, respectively, 2 T = 25 ns and 2 T = 33 ns. In the last two cases the interfringes roughly vary as expected from (4.8). In the simplest experiments the fringes are superimposed to the broad lines of recording A with a poor contrast; but curves B and D have been recorded using a data accumulation technique which permits the substraction of the broad lines (curve C was a verification of the substraction technique).
0 5. Three-Photon Doppler-Free Transitions in Sodium As mentioned in D 1, the cancellation of the Doppler broadening is not limited to the two-photon transition and the method can, in theory, be extended to any n-photon transition (n >2 ). The interest of such an extension is obvious: using the two-photon absorption one can investigate states of the same parity as the ground state, whereas in an n-photon absorption (with n odd) it is possible to study states of the opposite parity. Nevertheless, a three-photon experiment is generally more difficult to realize than a two-photm experiment due to the smaller transition probability (see P 2.4.1). To avoid the use of high power lasers GRYNBERG, [19761, have performed an experiment BIRABEN, BASSINIand CAGNAC with small values of the one-photon detunings A@,, using the threephoton excitation of the 3P4 resonance level of sodium, two photons being absorbed and one photon emitted. The principle of the experiment is shown in Fig. 21. The three wave-vectors kl,k; and k, are disposed in
y
Ibl
iai
A
"
(C)
[
&F;2 l9OMHZ -F: 1
Z
/ \k2/'k'l
7 c1
kl,
-/-
\
- 4-
-
-F=
2
1772MHz
---FzI 3sVz
Fig. 21. Principle of Doppler-free three-photon experiment: (a) Spatial orientation of the wavevectors in order to eliminate the Doppler broadening in the three-photon transition. (b) Schematic mechanism of the three-photon transition. (c) Schematic energy-level diagram of sodium.
11, ! I51
THREE-PHOTON DOPPLER-FREE TRANSITIONS IN SODIUM
149
such a way that the total momentum exchanged between the atom and the photons is zero (see Fig. 21a):
hk, + hk; - hk2 = 0. Conservation of energy requires (see Fig. 21b):
E,- E,= hue, = hck, + hck; -hck,. In this case, the intermediate states involved in the three-photon process are e and g (due to the smallness of Au, other levels need not be taken into account in the calculation). The two ground and excited levels, 3s; and 3P;, are split into two hyperfine levels (see Fig. 21c), and we expect to observe four components of this three-photon resonance. In the experiment, the light beams come from two different lasers (see Fig. 22), so that hck, = hck: # hck,. By scanning the frequency of one laser, the frequency of the second being fixed, a Doppler-free threephoton spectrum can be observed. The resonance is detected by collecting photons spontaneously emitted from the 3Pt level. Figure 22 shows the experimental set-up. The two tunable light sources are C.W. dye lasers similar to those described in § 3.1.1. Each laser is locked to the side of the transmission peak of a pressure swept FabryPerot etalon. The laser powers are about 6 0 m W (laser 1) and 3 0 m W (laser 2), and the linewidth is the same for both lasers (about 7 MHz). The light coming from laser 1 is focused onto the sodium cell; then, using a set of mirrors and a lens, the transmitted light is refocused at the same point of the sodium cell, the angle between the two beams being equal to 120". The beam coming from laser 2 is parallel to the bisector of the preceding beams and is focused at the same point. To select the fluorescent light at the atomic frequency from the Rayleigh scattering at the laser frequency, a high resolution monochromator is used. 75MHz etalon Ar+ dye laser1 Ar
+
\
\
dye laser 2
Hp
IL
Fig. 22. Experimental set-up for the Doppler-free three-photon transition
150
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
I90MHz
1772MHz tb+-**-**-
'
[II, 8 5
+
Fig. 23. Experimental recording of the Doppler-free three-photon spectrum (following the scheme of Fig. 21). The laser 2 frequency is fixed, its value is A v = 2 ( A w / 2 ~ ) = 3 0 G H z , smaller than the sodium D , resonance line. The resonances are observed by scanning the laser 1 frequency. The separations between the lines are indicated for the atomic frequency and CACNAC [1976a]). (from GRYNBERC, BIRABEN, BASSIN]
Figure 23 shows the signal recorded when the frequency of laser 1 is scanned, while the frequency of laser 2 is fixed 30 GHz (or 1 cm-I) below the sodium resonance frequency (3St-3P;). The four peaks correspond to the transitions from the two hyperfine sublevels (F= 1 and F = 2) of the ground state to the two hyperfine sublevels (F'= 1 and F'= 2) of the excited state. When one of the three beams is interrupted, the resonances disappear. Even a slight modification of the relative position of the focus yields an abrupt decrease of the signal. The width of each resonance is about 60 MHz, which is much less than the Doppler width (2000 MHz) but is more than could be expected from the linewidth of the lasers and from the width of the excited level. The main sources of broadening seem to be: (i) a residual Doppler effect due to the fact that the angles between the laser beams may be slightly different from 60". (The residual Doppler effect due to the focussing is much smaller, of the order of 2MHz.) (ii) The light-shifts: in the case of two-photon transition, it has been shown in P 2.3.2 that it is usually possible to find experimental conditions for which the light shift is smaller than the natural width of the two-photon resonance line. But this statement is not true for the three-photon case (0 2.4.2). For instance, Fig. 24 shows recordings of the two low frequency components of the three-photon transition for a smaller one-photon detuning (10GHz) than in Fig. 23. These components are recorded for different values of the laser intensity (BIRABEN [1977]). Apart from an evident shift of the resonance, it appears that at higher intensity the
151
CONCLUSIONS
Av-~CHZ Pr 2 5 m W
0
2OOMHz
E'
Fig. 24. Recording of the F = 2 + F' = 1 and F = 2 + F' = 2 components of the 3S:-3Pt three-photon transition of sodium with a smaller detuning of laser 2: Av = 2(A0/2~r)= 10 GHz, and for different laser intensities. At higher power the resonances are shifted to higher frequencies. Abscissa: the atomic frequency (from BIRABEN [1977]).
three-photon resonance is broadened. There are two reasons for that: first, there are spatial nonuniformities of the laser intensity within the region of observation, and secondly, the light-shift changes from one atomic velocity group to another because the Doppler broadening is not very small compared to the energy gap Ao between the laser frequency and the sodium resonance frequency. This experiment demonstrates that the Doppler-free multiphoton spectroscopy is not restricted to two-photon transitions. There may be an interesting range of spectroscopic applications of Doppler-free threephoton spectroscopy. By absorption of three photons it should be possible to investigate very highly excited states whose parity is opposite that of the ground state. Nevertheless, it must be pointed out that most of these experiments need high-power lasers and one has to be careful with the light-shifts.
P 6.
Conclusions
The Doppler-free multiphoton spectroscopy is a new method, as the first experiments were done in 1974. Nevertheless, it has already had an important development, at least as far as two-photon spectroscopy is concerned. We have tried to give a general review of all the applications that have appeared up to now and we apologize to those whose experiments may have been forgotten. These applications have shown the
152
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, AQQ.I
possibility of obtaining a resolving limit of the order of the natural width of the atomic levels. It is not surprising that until now the most numerous experiments have concerned the measurement of small energy differences in atomic levels (fine and hyperfine structures, isotopic shifts). We must point out the particular usefulness of the method for the isotopic shifts, which cannot be measured by more classical Doppler-free methods, such as radiofrequency or level-crossing. The applications to molecular levels generally need higher power and are only in their infancy. In collision studies, the small natural width of the two-photon lines is made use of to measure shifts and broadenings; moreover, one can selectively populate one level among others within the Doppler width and measure population or polarization transfers. Transient experiments have been shown to be much simpler with two-photon excitation than with one-photon excitation because all the atoms see the same frequency whatever their velocities may be; one can expect interesting developments of the method. But, to our mind, the most promising application is in metrology either for the measurement of fundamental constants or for the realization of new optical frequency standards. Up to now the two-photon method has been much less developed in this field than other Doppler-free optical methods, such as saturated absorption or atomic beam methods, and owing to this fact, the limits of the two-photon method have not yet been reached. Concerning the generalization to Doppler-free multiphoton spectroscopy with more than two photons, care must be taken in talking about its future, because, even if the experiments have been shown to be feasible, the light shifts are very difficult to avoid. Although it is obviously impossible to predict the future extension of the Doppler-free multiphoton method, we think we have shown in this work the wide range of its actual and possible applications. Finally the authors would like to acknowledge the help provided by other members of their research team, D r FranGois Biraben and Dr Gilbert Grynberg. Appendix I: Remarks on the Choice of Gauge At the electric dipolar approximation both hamiltonians - D * E and A (q’/2m)A2 are equivalent since the former can be transformed into the latter by a change of gauge. Moreover (q2/2m)A’cannot -(q/m)p
+
11, APP.I1
REMARKS O N THE CHOICE OF GAUGE
153
couple different levels of the atom. So the calculation of the two-photon transition probability must give the same result with the hamiltonian - ( q / m ) p * A or -D E. Using the hamiltonian - ( q / m ) p A, one finds at resonance the following transition probability:
(We suppose here that both counter-propagating light waves are identical.) As this expression must be equal to (2.13) we have
This relation can also be shown directly (BUNKIN[1966J). The demonstration simply uses the commutation properties of the operators p and r with the atomic hamiltonian H,: [r,Ho]=ifi-P m
or
4 [D,Ho]=ih-pp, m
which gives for the components on the direction
E:
and one deduces:
(AI.3) can be rewritten:
In the same way, remarking that we are at resonance (o- ogr= or,- o), (AI.4) can be rewritten
Hence taking the half-sum of (AM) and (AI.6), the second summations give the matrix element (el DepE- peDe Ig), which is null. Then we change
154
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
[II, App. I1
p into D and we obtain
As wr,+wpr = 2w, we deduce the equality (A1.2), i.e. the equality of the transition probability in the two gauges. It must be noticed that this demonstration is only valid at resonance. Off -resonance, one would think that different results are obtained with the radiation gauge and the Coulomb gauge (KOBE[1978]). Indeed if one calculates the right significant physical quantity, the same result is obtained with the two gauges (YANG[1976], FORNEY, QUA-ITROPANI and BASSANI [1977], GRYNBERG and GIACOBINO [1979]). On the other hand, it must be pointed out that the contribution of each relay level r is not the same in one gauge as in the other. The result does not differ much if one quasi resonant relay-level gives a dominant contribution to the transition probability. When there is no quasiresonant intermediate level, it can be shown that the series converge much more rapidly in the gauge - D E than in the gauge A p. This has been done either by evaluating an upper limit of the correction to the single state approximation (BUNKIN [19661, GRYNBERG [19761) or by numerical computer calculation (BASSANI, FORNEYand QUA~TROPANI [1977]).
.
Appendix Ik Calculation of the Energy Absorbed by One Atom We justify here the formulas used in Q 2.1.3 in the case of nonmonochromatic fields (see BIRABEN [1977]). The electric field is supposed to be non-zero during a time A T = T2- TI, so the work of the electric field acting on the induced dipole is (see (2.17) and (2.16)): (AII. 1)
with m
E ( t )= e
dwE(w)e-'"'+e*
I
dwE*(w)e+iot.
(AII.2)
In the calculation of ( $ ( t ) l D I $ ( t ) ) we have to develop l $ ( t ) ) up to the third order. In order to avoid useless calculations we only keep the terms which will give a contribution to the integral (AII.l). These terms are: (a) the product of the first order term in the bra ($1 (or ket 19)) with the second order term in the ket (4) (or bra ($1). ( p ) the product of the zero
11, App. 111
ENERGY ABSORBED BY ONE ATOM
155
order term in the bra ($1 (or ket 14))with the third term in the ket I$) (or bra($[). It can be shown that the terms (a)and ( p ) give two equal contributions corresponding to the two steps of the whole process: absorption of one photon from state (8) to the intermediate state Ir); absorption of one photon from Ir) to the final excited state le) (see BIRABEN, BASSINI and CAGNAC [1979]). For the sake of simplicity we will here calculate only the terms (a)and we will multiply the obtained result by 2. So, we will only present here the calculation of I $ ( t ) ) up to the second order. This second order perturbation calculation is done in Schrodinger representation and not in the interaction representation, because it is easier when one uses the hamiltonian H , = Ha- il72 (see (2.2)) which represents the relaxation due to spontaneous emission, and which is not hermitic. We use the components 0.defined by (2.5) of the dipolar momentum operator; for simplicity we will suppose in this appendix that the polarization E is linear, and so real:
1
--
h2
6, 1;: dt2
+m
dt, I l d w 2 d ~ , E ( w ~ ) E ( w , ) e - ~ " ~ ' ~ e - ~ " ~ ' ~ 0
x e-iH,(r-t,)lh
t
Ih iH,r,/h
e 0, 18) +third order term. (AII.3) In this expression we take into account only the first term of (AII.2) with e-'"", which is the only coefficient efficient for absorption. On the other hand we must not forget that we have chosen the energy zero for the ground state 18) in such a way that e-iH1'/hlg) = Ig) for any t. Performing the integral over t3 is analogous to the transformation of (2.7) into (2.8): the relaxation time inside the hamiltonian HI reduces to zero the term corresponding to one of the two time-boundaries (if T 2 - TI>> l/I'=); and the approximation (2.9) permits the introduction of the two-photon operator (2.10):
m
0
+third order term.
(AII.4)
156
[II, App. I1
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
A straightforward algebra then yields the contribution of the (a)terms: I
m
TI
0
x [E*(w1)E(02)E(03)e+i(Oi1i-OZf2-01f2) (gl e+iH;(t-t,)lh
~~~-iH,(t-r~)lf' Q:, lg) + E("1)E*(w2)E*(w3)e-i(O'l,-w,t,-o,r,) (gl QZe x e+iH;(I-I,)/*Dee-iH,(I-I e, MI. )/hD
(AII.5)
Performing the integral over t1 permits us to introduce again the twophoton operator, and to obtain for the contribution of the (a)terms: ($(t)lDe
I$(r))=+i
m
[dT[[[dwld%dw3 0
x [E*( w ,)E( w2)E(~3)e-i('"'+w ?-+'
e+i(w2+o,)i
(81 Q:ee-iHIT'hQ:e18)
- E(wl)E* (w2)E*(w3)e+i(w2+03-01)t e-i(w,+o,)z
(AII.6)
(gl Q:ee+iH;T'hQfe lg)].
In this formula we have extended to infinity the upper boundary of the integral over T = t - t2, using the relaxation term in HI and assuming T2- Tl >> l/re.Now we must multiply by 2 the contribution (AII.6) of (a) terms. Using (AII.l) and (A11.3), the absorbed energy is given by d dt
W = 4 Re r d t p w 4 E * ( ~ ~ ) e + ~( $~( t4) (~D,-
I$(t)).
(AII.7)
We see that only the negative frequency part of (AII.6) gives a non-zero contribution. So +m
I
x E*(wl)E(w2)E(w3)E*(w4)
d7 ei("2+03)T (gl Qf,e-'HIT'*Qze Ig).
(AII.8) We suppose that the distribution E(w) is centered in wL with a rather
11, App. 111
157
ENERGY ABSORBED BY ONE ATOM
small width AwL; then w1-
0 2
-
w,+
0 3
- 0 1 = OL.
0 3 -
(AII.9)
WL,
and we can take On the other hand, as we have supposed that the electromagnetic field is zero before time T , and after time T,, we can take the integration over t between --oo and +m +m
I_,
eih
,+o,-o,-oJt
dt = 2~8(01+04-02-03).
(AII.lO)
We now introduce the variable R and the quantity B(R),
Formula (AII.8) takes the form: m
W = 8.rroL h
Re l l d R dR'B*(R)B(R') 8(R -a')
(AII. 12) Integrating over R', we obtain: 8rwL
W = -Re h
+m
+m
- ~ ) (AII.13) dR IB(R)121 dreiRT(g( Q ~ , ( 0 ) Q ~ e (Ig), 0
where QzJt) is the two-photon operator in the interaction representation (AII.14)
Q;Jt) = eiH,t/hQ:,e-iH:rlh
In the case of a linear polarization, the electric field, E ( t ) can be written E ( t )= E ( t )
-
+m
E =E
with 8(w)=E(w)
=E*(-o)
if
w>O;
if w < O .
1
(AII. 15)
158
DOPPLER-FREE MULTIPHOTON SPECTROSCOPY
If we set A ( 0 ) = lim8(w)8(R-w)dw
(AII.16)
--m
in the vicinity of 0 = 2w, which is the case of interest, we have A(0)=B ( 0 ) .
(AII. 17)
A ( 0 ) has a simple expression as a function of the electric field 1 A ( 0 )=2T
I
+m
E(f)'e-'"'
dt.
--m
The absorbed energy can finally be written W=-8~w~ Re
h
I, t m
d 0 )A(0))'jmdTeinT(g) Q ~ , ( O > ~ , ( - T >)g). (AII.18) 0
In the case of imaginary components (circular polarizations), the operators 0,and Qzeare no longer hermitian and we must replace some of them by their adjoint.
References ABELLA,I. D., 1962, Phys. Rev. Lett. 9, 453. ABELLA,I. D., 1969, in: Progress in Optics VII, ed. E. Wolf (North-Holland) p. 141. AGOSTINI, P., G. BARJOT,F. MAINFRAY, C. MANUSand J. THEBAUT, 1970, IEEE J. Quant. Elec. QE6, 782. A V ~P.,, C. COHEN-TANNOUDJI, J. DUPONT-ROC and C. FABRE,1976, J. Physique 37,993. BAKLANOV, E. V., V. P. CHEBOTAYEV and B. DUBESTKY, 1976, Appl. Phys. 11, 201. BARGER,R. L., J. B. WESTand T. C. ENGLISH,1975, Appl. Phys. 27, 31. BASSANI, F., J. J. FORNEY and A. QUATTROPANI, 1977, Phys. Rev. Lett. 39, 1070. BASSINI, M., 1977, Thbse de 3bme cycle Paris (unpublished). BASSINI, M., F. BIRABEN, B. CACNAC and G. GRYNBERC, 1977, Opt. Comm. 21, 263. BEROFF,K., 1978, Thbse de 3bme cycle Paris (unpublished). BIRABEN, F., 1977, Thbse Paris (unpublished). BIRABEN, F., M. BASSINIand B. CAGNAC, 1979, J. de Physique40.445. BIRABEN, F. and K. BEROFF,1978, Phys. Lett. 65A, 209. BIRABEN, F., K. BEROFF,E. GIACOBINO and G. GRYNBERG, 1978, J. de Physique 39, L-108. BIRABEN, F., B. CACNAC, E. GIACOBINO and G. GRYNBERC. 1977, J. Phys. B10, 2369. BIRABEN, F., B. CACNAC and G. GRYNBERG, 1974a, Phys. Rev. Lett. 32, 643. BIRABEN, F., B. CACNAC and G. GRYNBERC, 1974b. C.R. Acad. Sc. Paris 279B, 51. BIRABEN, F., B. CACNAC and G. GRYNBERG, 1974~.Phys. Lett. MA, 469. BIRABEN, F., B. CAGNAC and G. GRYNBERG, 1974d. Phys. Lett. 49A, 71. BIRABEN, F., B. CAGNAC and G. GRYNBERC, 1975a. J. Physique Lett. 36, L41. BIRABEN, F., B. CAGNAC and G. GRYNBERC. 1975b. C.R. Acad. Sc. Paris BOB, 235.
I11
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E. WOLF, PROGRESS IN OPTICS XVII @NORTH-HOLLAND 1980
I11
THE MUTUAL DEPENDENCE BETWEEN COHERENCE PROPERTIES OF LIGHT AND NONLINEAR OPTICAL PROCESSES BY
M. SCHUBERT and B. WILHELMI Department of Physics, Friedrich-Schiller-Uiuersity Jena, 69 Jena, G.D.R.
CONTENTS
PAGE
§
1 . INTRODUCTION
§
2 . COHERENCE PROPERTIES OF LIGHT .
§
3 . ONE-PHOTON PROCESSES
. . . . . . . . . . . . . . .
. . . . . .
165
. . . . . . . . . . .
184
5 4 . MULTI-PHOTON ABSORPTION §
§
165
. . . . . . . . .
5 . TWO-PHOTON EMISSION AND TWO-PHOTON LASING PROCESS . . . . . . . . . . .
. . .
6 . PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION . . . . . . . . . . . . . . .
.
196 215 217
0 7 . STIMULATED RAMAN SCATTERING . . . . . . . 229 REFERENCES
. . . . . . . . . . . . . . . . . .
232
# 1. Introduction This review is concerned with the mutual relationship between nonlinear optical effects and the coherence properties of light, a subject which seems to be gaining increasing importance, because of the recent development both of new high-power sources and of'sensitive and fast detection devices. If a medium is irradiated by light, changes of the state of the medium as well as of the statistical properties of light will be caused by nonlinear interactions. For instance, the probability of atomic transitions is influenced by the coherence of incident light; on the other hand, the parameters of the radiation field, such as the intensity correlation functions which, besides other effects, describe the bunching and antibunching of photons are altered due to nonlinear interactions. Moreover, correlation effects between several modes of the light field occur. In general, these mutual influences are essentially determined by the interaction Hamiltonian, the initial states of the atomic systems and the radiation field, the interaction time, and the propagation length. The quantum character of the atomic systems and of the radiation plays an important role and in many cases it has to be taken into account explicitly. We first present a concise compilation of relations concerning the coherence of light, by means of which the physical meaning of the classical and quantum-theoretical description is discussed in connection with experimental techniques and measuring procedures. Next, onephoton processes with nonlinear character are studied. We proceed with the treatment of several important multi-photon processes, such as multiphoton absorption, two-photon emission, optical parametric processes, and stimulated Raman scattering. In connection with the discussion of these effects, representative theoretical and experimental methods are outlined and employed.
8 2. Coherence Properties of Light In this paragraph quantities, definitions, and relations are compiled which are needed for the discussion in the subsequent sections; more 165
166
[III, cj 2
COHERENCE OF LIGHT AND NONLINEAR OWICAL PROCESSES
detailed information about the subject can be obtained from several books and papers, e.g. by GLAUBER[1963a, 1963b], KLAUDERand SUDARSHAN [1968], MANDELand WOLF[1970], PERINA[1972], LOUDON [1973], LOUISELL [1973], BERTOLOTTI [19741. An adequate description of the electromagnetic radiation requires the application of a fully quantumtheoretical formalism. However, many problems can be sufficiently precisely described by classical means which, in general, are associated with a simpler mathematical formulation. We will, therefore, deal with the classical as well as the quantum-theoretical description, including the physical relation between both of them.
2.1. CORRELATION FUNCTIONS AND DEFINITION OF COHERENCE
In the classical description the real electric field vector E(r, t ) can be decomposed into the sum of two complex conjugate parts E'-) and E'+), where E(-) is the so-called complex analytic signal which contains all the information about the field strength, E(r, t) =E'-)(r, t)+E'+'(r, t ) .
(1)
The quantity E'-)(r, t ) is connected with the negative frequency part of the temporal Fourier transform E(r, o)of the real field strength E(r, t ) by the relation E'-)(r, t ) = ( 1 / 2 ~ ) f !d~o E(r, o)exp [id]. The properties of the field can be described by s-fold joint probabilities dp,
= P,[E'-)(x,),
. . . ,E'-)(xi), . . . ,&)(x,)]
d2E'-'(xl)
* * *
d2E'-'(x,) (2)
for the signals &)(xi) at the space-time points xi, where xi consists of the spatial (ri) and time ( t i ) coordinate; P, represents the corresponding probability density. Correlation functions such as rl'm (XI,
. . . , X,)
=
I
n
I +m
dp,
E'+)(Xk) k=l
E'-)(xk), with
1 -k m
= S,
k=l+l
(3) can be defined as mean values of functions which consist of the factors E(+)(xk)and E'-)(xk). From definition (3) it is obvious that has the character of a tensor; however, a special notation may be omitted, because one-component fields are displayed in the following considerations.
111, § 21
167
COHERENCE PROPERTIES OF LIGHT
In the quantum-theoretical description the field vectors E, Jl-), become Hilbert space operators E, &), $+) which satisfy the relation
k(r, t ) = i F ) ( r , t ) + i?+)(r, t ) .
(4)
(Hilbert space operators will be denoted by a caret.) The operators &)(r, t ) and k(+)(r,t ) are Hermitian conjugates. The explicit display of the dependence on the time coordinate t is intended to indicate that these quantities are operators in the Heisenberg picture. The operator 2-)(r, t ) is related to the Fourier transform of E(r, t ) in the same way as it is given above for the classical quantities. The state of the radiation field is described by the density operator 6. The correlation functions are quantum-theoretical expectation values formed by means of the density operator: 1 +m
I.l'"(x,,
fl
. . . , x s ) = T r [ B kn=~l + ) ( X k ) k = l + l $-)(xk)},
l+m=S.
(5)
It should be noted that, in contrast to the classical case, the arrangement of the factors now exhibits a physical significance. Here the normally ordered product is used (all the operators are on the left-hand side of more general cases will also be discussed later on). all the operators k(-); In the classical case as well as in the quantum-theoretical case the following definition of coherence has been introduced from which, among others, the connection with stochastic influences can easily be seen (compare 0 2.2). A radiation field is coherent in Kth order, if the correlation function satisfies the factorization condition
&'
k=l
k=l+l
for all I, m 5 K. The function V(x) must be independent of the index k and has to be a solution of the Maxwell equations with the given boundary conditions; thus, V(x) corresponds to the complex analytic signal E?)(x). If K tends to infinity, the field is said to possess full coherence. An equivalent possibility for defining coherence rests on the application of the normalized correlation functions
168
[III,9 2
COHERENCE OF LIGHT AND NONLINEAR OPTlCAL PROCESSES
(GLAUBER [1963a]). If ly'*"( = 1 for all I, rn I K , then the field is coherent in the Kth order. In the case 1 = rn = 1 the quantity yl*'can be regarded as a degree of coherence, because the relation 0 5 \ y l q x l , xz)JI 1
(8)
holds. First order coherence exists if ly'.') = 1; full incoherence is characterized by y'*' = 0. If [ y ' * ' (lies within the interval (0, l),partial coherence exists. Concerning higher values of I, m, the case (y'.") > 1 may also occur, so that (y'.") cannot be regarded as a genuine degree of coherence. In contrast to this fact the quotient
(94 (MEHTA[1966, 19671; PERINAand PERINOVA [1965]) as well as the quotient ym.m(xl,. . . I x,)
-
rmyX1,.
[ r m . m ( ~ l ,.. ., xm, xm,
*
. . , x,)
. . ,x l ) p m ( & + l ,
-
* *
7 ~ 2 m 9 ~
2m9 * * *
9
xm+l)$
(9b) and SUDARSHAN [19681 Chapt. (introduced by Sudarshan, see KLAUDER 8, 5 2) can be regarded as a degree of coherence. In the classical case for all I, rn and all the arguments xk the relations ly'*") 5 1,
(y"'") 5 1
hold; in the quantum-theoretical case these relations hold if the field has [1967]) - compare 5 2.2. a nonnegative definite P-representation (MEHTA It should be emphasized that the statement on the coherence in the Kth order and of a high degree of coherence, respectively, depends on the choice of the s space-time points, which are the arguments of the field quantities in the correlation functions. With the help of equations which describe the space-time relations of the field quantities (in the classical case: the Maxwell equations and the wave equations for the vector potential and the electric field strength) the coherence properties in the different space-time regions can be evaluated. A representative example of the first order is the mutual coherence function r1*'(xl, x2) with x1 # x2. The connection with the explanation of
111.8 21
COHERENCE PROPERTIES OF LIGHT
169
Young's interference experiment will be considered first, thereby assuming simple geometrical conditions and stationarity of the field. The wave equation for the field strength yields the result that the total field strength E ( x ) at a point on the screen turns out to be a linear superposition of two terms representing the contributions from the two pinholes. The time dependence of each term is influenced by the propagation time between the pinhole and the point on the screen. This fact also holds for the complex analytic signal @(x) as well as for E'+'(x). By averaging the product E'+'(x)E'-'(x) with respect to the time, the intensity on the screen results. This quantity contains an additive term that is proportional to the real part of r1.'(x1, x2). Hence, r'*'(xl, x2), known as the mutual coherence function serves to explain the interference behavior; for simple geometrical conditions the modulus of the normalized correlation function ly'*'(xl,x2)l equals Michelson's visibility of the interference fringes. Using this result a certain space-time region with ly1.'(xl,x2)l = 1 can be determined which is said to be the so-called coherence volume. The connection of the mutual coherence function of a radiation source with the radiometric quantities has been investigated by WOLF[1978]. While serious difficulties appear with respect to the radiance and the radiant emittance, the radiant intensity (the rate at which energy is radiated by the source per unit solid angle) can be uniquely expressed in terms of the mutual coherence function at points in the source plane; the radiant intensity is proportional to the spatial Fourier transform of the crossspectral density function, which is a linear functional of r1*'(x1, x2) itself. This relation allows, besides other problems, the clarification of the relationship between the coherence behavior of light sources and the directionality of light beams. In deriving the radiant intensity the wave equation for the mutual coherence function was applied (BORNand WOLF [1975]). Passing from the classical treatment to an explanation on the quantum theoretical basis of these space-time phenomena, one may say the following: If the time dependence of the field operators in the Heisenberg picture only arises from the dynamical time dependence, then the spacetime relations between the operators are the same as they are between the corresponding classical quantities, e.g. the wave equation for the operator of the electric field strength
V2&,
1 a2 t ) -7 7 E(r, t ) = 0 . A
c at
170
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 82
Hence the solution of this linear differential equation under the condition of Young’s experiment leads to the result that the operators of the total field strength at a point on the screen consist additively of two field operators representing the contributions from the two pinholes. Further treatment of the problem leads to the application of the quantumtheoretical correlation function r1,’(xlrx,) for the explanation of the phenomenon. The operator for the field energy in a certain volume can be determined by integrating the energy density [tzOk2/2 + p0fi’/2] with respect to the spatial coordinates over the volume. A deeper insight into the significance of the coherence quantities can be gained by introducing operators that are associated directly with the existence of light particles in the quantized field. Here we will deal with a free electromagnetic field (whose coupling to atomic systems will be considered later on). Using standard methods, the expansion of the field operators in plane waves leads to the relations
where p is the mode index. A single mode is characterized by its frequency w,, a wave number q,, a unit polarization vector e,, the photon annihilation operator B,, and the creation operator 4: which is contained ) . normalization volume V (which is assumed to be a cube) may in ~ + The be chosen sufficiently large, so that any experimental situation with sufficiently closely spaced modes in the o-space and the q-space can be described. The Hamiltonian fi contains the photon number operators &, = of the different modes. The eigenstates of N, are the photon number states In,), with eigenvalues np. An operator, that represents the total photon number in a volume V’, with linear dimensions less than those of the normalization volume V, may be defined as the space integral Jv,d3rC(+)&) (MANDEL [1966]). The operators &) and &) are defined by
(it&,
111, 521
COHERENCE PROPERTIES OF LIGHT
171
A clear explanation of the properties of this total photon number operator with respect to the volume V' is possible, if the linear dimensions of V' are greater than the wave lengths of the occupied modes. The detailed investigation of the interaction of a radiation field with an ensemble of atoms, which is regarded as an ideal photon detector on the basis of the external photoelectric effect, led to the following result: provided that we deal with a point-like detector at the space point r, the correlation function P 1 ( x , x) is proportional to the counting rate of photoelectrons and it can, therefore, serve as a measure for the intensity at the space-time point x = (r, t ) . The function P 2 ( x 1 ,x2, x2, xl) is proportional to the joint probability for measuring the intensities r 1 * ' ( x 1 ,xl) at a point x1 and r1.'(xZ, x,) at a point x2, with ideal photon detectors (compare § 2.3); an extension is possible to the case of rfS1(xl,. . . ,xI, xl, . . . ,xI)with 1 > 2. For the same space points rl = r2 = r and different time values t , , t2, the expression yZ.'(x1, x2, x2, xl) yields important information about the statistical behavior of the radiation field. Under stationary conditions the expression y2*2(t,,f2, t2, t , ) depends on T = t2 - t , only; the quantity y 2 , * ( 7 ) is proportional to the conditional probability for detecting a photon at time t + T , if one photon had already been detected at time t. If this conditional probability exceeds the value of Poisson-distributed photons, then so-called bunching of the photons occurs. While the relation y 2 * 2 ( ~ ) >is1 associated with the bunching effect, an anti-bunching effect exists if y2.'(7) < 1 (compare § 2.3). 2.2. STATES AND MEASURABLE QUANTITIES OF THE FIELD
The representation of a pure or mixed state of the field can be given by means of eigenstates of certain dynamical variables and observables. We will first consider such quantities for one mode only; the corresponding mode index can then be omitted here. The consideration of a single mode allows us also to omit vector notation; let & be the field strength in the direction of the polarization vector e and let r be the space coordinate in the direction of the wave number vector q. It should be remarked that in this article we will always employ this simple scalar notation if a misunderstanding is not likely to arise. A very useful tool is the coherent state la) (GLAUBER [1963a, 1963bl; see also PERELOMOV [1977]); it can equivalently be defined by a) ii
la)= a
la) and b)
Ia)=e-1a12'2 a n ( n ! ) - fIn). n=O
(14)
172
[III, 5 2
COHERENCE OF LIGHT AND NONLINEAR OWICAL PROCESSES
Definition a) means that la) is a right-hand eigenstate of the (nonHermitian) annihilation operator. Equation (14) yields the following mean photon number and photon number distribution (which is a Poisson distribution) (alfila)=la12,
p(n; Ia))=I(n I a)J2=e-‘N’(&)”/n!.
(15)
With respect to the quadrature components h1=(6’+8)/2 and 6 2 = i (6’ - 6)/2 of the annihilation and creation operators, the mean square deviations are (a\(Ad,)’ ( c Y ) = ( ~ ( la)=;i. 1 (16) For general states the uncertainty principle leads to the inequality ((A61)2)((A62)2)2&. The coherent state is, therefore a special case of a minimum-uncertainty state (value of the uncertainty product With the help of coherent states la), the representation of an operator A?(&+, 6) can be given, and the following relation holds:
A).
In order to illustrate the connection between classical and quantumtheoretical description it is advantageous to use the eigenstates of the [1968]; SCHUBERT and VOGEL operator of the field strength (SCHUBERT [1978a]). The eigenstate IE(r))of the operator E ( r ) in the Schroedinger picture can be represented by means of coherent states as follows:
‘I
IE(r))= - d2a la)(27r)-+ 7r
E(r)(asf)*-Im2 (as’) 1
-- Im 2
1
with s’ = exp [iqr]. The eigenstates IE(r)) are orthogonal and satisfy the normalization and completeness relation. So-called hvo-photon coherent states have been introduced by YUEN [1975b, 19761 to describe coherent two-photon emission. With the help of the operator 9, which contains the operators 6+,6 and the complex
III,S21
173
COHERENCE PROPERTIES OF LIGHT
numbers p, u according to
the right-hand eigenstates Iy) of
9
can be obtained:
9 IY)=
Y IY).
The following representation of the two-photon coherent states can be obtained with the aid of coherent states:
U
U*
-- (a*I2+- y2+- a * y l 2P 2P pl
.
(20)
For the special case p + 1, v + O the states Iy) coincide with the coherent states. For IuI > O interesting differences appear as to counting statistics and the minimum-uncertainty state character. The mean-square deviations with respect to the quadrature components are ( Y I ( A ~ , ) ’ I Y ) = ~ I C L - U I ~ ,( Y I ( A ~ ~ ) ’ I Y ) =C~ L I + ~ ( ’ . (21)
Using these relations it can be shown that the states Iy) are minimumuncertainty states if p = gv, for a real number g. However, if 1< g <m, the value of (yI (Ahl)’ Iy) is less, or even much less, than This is a significant difference in comparison to the coherent states. The properties of a field state I+) may be illustrated by employing the probability density P(E(r); (4))= I(E(r) 1 +)I2 for the field strength having the value E at the space point r. In Fig. l a the quantity P is plotted versus E at a fixed point r, for photon number states In) and for a coherent state. In Figs. lb,c the mean value of the field strength and its r.m.s. deviation are plotted. For a photon number state the mean value is zero and the r.m.s. value J(hw/2~,V)(2n+ 1) is independent of r. For a coherent state the r-dependent expectation value is given by
a.
flu
(a1g(r)la)= J F V [ i a
exp [iqr]+(c.c.)l.
(22)
This expression represents a sine wave with the complex amplitude (hw/2~,V):(fip exp [i(arg a + r/2)]. The fluctuations are Gaussian with the constant r.m.s. value ( f i u / 2 ~V)t. , This value corresponds to the vacuum fluctuations [it should be noted that this feature does not apply to the two-photon coherent states (SCHUBERT and VOCEL[1978b]) - compare
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Fig. 1. Electric field strength. (a) distribution of the electric field strength for the two number states 10)and (3) and for a coherent state la).(b) Mean value ( E ) and (c) square root of the mean squared deviation (((A&)*))$ in dependence on the space coordinate r.
Q 5.1. We will now consider the relative deviations; because of the sinusoidal dependence of (E(r)) on r, we will relate them to the averaged value ((E(r))2)r. This leads to the equation ((AE(r))’)/((E(r))’)~ =1/2(i%.
(23)
The quotient on the left-hand side affords a comparison with the classical theory. Small relative quantum fluctuations involve the approach to the classical wave, with well-defined values of the field strength at the space-time points. Eq. (23) shows that for a coherent state the relative fluctuations tend to zero with increasing mean photon number. The coherent states) . 1 factorize the correlation functions rlSrn for arbitrary values of s = 1 + m,where V ( x )= i(hw/2~,,V ) f aexp [i(qr - w t ) ] ; this means that they possess the property of full coherence in the sense of the definition (6). In the classical description full coherence can be obtained, if the probability density P, [cf. eq. (2)] takes the form
111,5 21
COHERENCE PROPERTIES OF LIGHT
115
for all s-values. The quantities E(-)(xi)must be regarded as statistical variables,.whereas the V(xi) denote fixed values of the field. The existence of the &functions in eq. (24) implies that no fluctuations occur and the predicted values V(x,)are exactly attained. An arbitrary one-mode state 1,c, In) shows coherence in the first order, whereas higher order coherence depends on the behavior of the coefficients c,,; for instance if c, = 0 for all n > n’, then coherence in the order K > n’ does not exist.
So far we have dealt with pure states of the field. We will now discuss mixed states, which must be described by the density operator 6. We have seen that the coherent state la) corresponds most closely to a classical wave. It seems, therefore, useful to represent 6 as a superposition of the operators la)(aI. The so-called P-representation of 6 is defined by
I
fi = d2a P(a) [a)(a[.
(25)
In Fig. 2 the functions P(a) are explicitly given and illustrated for three important types of radiation fields. The function I belongs to ideal laser
Fig. 2. P-representation for important types of radiation fields.
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radiation with the complex amplitude factor a';I1 denotes the function associated with chaotic radiation, which can be completely derived from the requirement for maximum entropy under the condition of a fixed mean photon number (fi); a special case is thermal radiation for which (&)= (exp [AwlkTI- l)-' ( k being the Boltmann constant and T the temperature). The superposition of ideal laser radiation and of chaotic radiation (function 111) represents the radiation of a real laser. The weight function P(a) cannot be interpreted as a genuine probability distribution, because it is not positive definite. This fact shows resemblance to statements with respect to a more general quantumtheoretical problem, namely the impossibility of a general mapping of operators onto c-number functions with the property of genuine probability distributions (WIGNER [1971]; SRINIVAS and WOLF[1975]). In general, P(a) need not take the form of an ordinary point function, it may become a generalized function; so the P-representation of the number state In) is proportional to the nth derivative of the &function. Because of this property it is often advantageous to use more well-behaved functions for the description of the radiation field, for example the function PA(a)= 7 ~ - ' ( afil (a).The functions P and PA are connected by an integral relation. The expectation value of an antinormally ordered operator product can be expressed in terms of PA(a) in the same way as the expectation value of a normally ordered product can be expressed in terms of P(a) [cf. eq. (33)].It is useful to introduce also the functions K ( A ) = T r (fieA''e-'*')
and KA(h)=Tr{fie-A*'eA'")
(26)
known as the normally and antinormally ordered characteristic functions. K ( A ) and K ~ ( A )are the two-dimensional Fourier transforms of P(a) and PA(a), respectively. In contrast to P(a) and K ( A ) the functions PA(&) and K A ( A ) are always well-behaved functions with the property IPA(a)l5 1, I K ~ ( A ) I I 1. K ( A ) and K A ( A ) may be regarded as generating functions of the expectation values of operator products. K ( A ) generates the average of a normally ordered product:
a'
Tr {6(6+)'(6)"'} = -
am
c3A' d(-A*)"'
A =O
By differentiating K A ( A ) one obtains the average of the antinormally ordered product. The dependence of the density operator on time can be obtained by
111, B 21
COHERENCE PROPERTIES OF LIGHT
177
solving the following equation of motion for f i ( t ) :
Using this equation and the above noted relations between fi and P(a) or P A ( a ) , generalized Fokker-Planck equations for P(a, t) or P,(a, t ) may be derived. The time-dependent characteristic functions may be written in the form ~ ( ht),= T r {fi(0)eAd'(t)e-A*d(t) 1
and KA(A, t ) =
Tr {fi(0)e"~a~t~e"a""},
where h ( t ) and h + ( t ) are the photon annihilation and creation operators in the Heisenberg picture. For the description of multimode fields direct products of the singlemode eigenstates can be used, since the operators fi,, E,, d, commute among one another in different modes. For example, the multimode photon number state is given by
The so-called global coherent state
plays an important role; it is the right-hand eigenstate of the operator $-).
fi(-)(r, t ) I{a,}) = ~ ( - ) ( rt ,; {a,}) be}).
(30)
The eigenvalue
E(-)(r,t ; {a,})=
i(hw,l2eOV)fa, exp [i(q,r-w,t)] w
corresponds strictly to the complex analytic signal with the dimensionless amplitude factors {a,}. Global coherent states fulfil the factorization condition (6) for all I, m. In the multimode case eq. (25) is replaced by
S=
I
d%,}P({a,))
I{a,))
({q.JI,
(31)
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[HI, 5 2
where the function P depends on the set {a,} of the amplitude factors. In analogy, the functions PA, K , K~ may also be introduced for a multi-mode field. With the aid of P({a,}), the s-fold joint probability
can be obtained, from which the quantum-theoretical normally-ordered correlation functions can formally be derived, in the same manner as it is possible for the classical correlation functions (see eq. (3)); this means that the expression dp, in eq. (32) plays the same role as does the corresponding quantity of the classical description (see eq. (2)). This provides a justification of the classical procedure - at least in the case of normal ordering. In general, the classical description differs from the quantumtheoretical one by the operator character (with the possible noncommutability) of the dynamical variables. This fact also influences the description of coherence properties by means of correlation functions. U p to now we have discussed normally ordered correlation functions representing the expectation values of such measurable quantities, for which the fluctuations of the photon vacuum are of no importance. Correlation functions with arbitrarily ordered operators also are of importance for the explanation of certain experiments. Let the expression &[&'(Xk); &'+'(xk,)] be such an operator product. Then, in general, the following relation for its expectation value holds;
where (0)is the photon vacuum state. The quantity R is a certain part of the matrix element (01 6 10); R vanishes only in the case of normal ordering. For nonvanishing R, fluctuations connected with the photon vacuum come into play. In the case of strong fields (great numbers of
111, P 21
COHERENCE PROPERTIES OF LIGHT
179
photons) the ordering is of minor importance and the quantumtheoretical results agree with the classical ones.
2.3. MEASUREMENT OF STATISTICAL PROPERTIES OF LIGHT
To determine the statistical properties of light, photoelectric counting and correlation methods based on one-photon interactions in a photoelectric detector, as well as nonlinear optical techniques, are applied. While these techniques will be discussed when treating the respective interaction effects, we wish to summarize here some fundamentals of photoelectric measurements in which light releases electrons - so-called photoelectrons - at the cathode of a detector, e.g., of a photomultiplier. The individual photoelectrons generate electric pulses by amplification and are measured by special electronic devices. Basic investigations concerning photoelectron counting were carried out by MANDEL [1958, 19631, GLAUBER [1963a, 19641, MANDEL, SUDARSHAN and WOLF[1964], MANDELand WOLF [1965]. Reviews are given by GLAUBER [1972], ARECCHI and DEGIORGIO [19721, JAKEMAN [1974] and ARECCHI [19751, which also deal with the experimental technique. For the interpretation of photon-counting experiments it is necessary to connect the above introduced parameters describing the state of the radiation field with the measurable photodetection distributions. At first we will consider the photons within the normalization volume V at a given time t. The photon statistics will be characterized by the probability p(n) for the occurrence of n photons, or by the mean value of all the kth moments of the number operator fi = C, fi,. The corresponding expectation values can be calculated by means of the given density operator; using eq. (31) we have
with
and
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COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
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The parameter W = C , \ ( Y , ~is~ equal to the expectation value of the photon number operator in the state [{a,}). Next, we will compare eq. (34) with the semiclassically derived probability distribution for the photoelectron emission of a radiation detector. A volume V in front of the detector with an area A irradiated perpendicularly by the light under investigation is considered. If S ( t ’ ) is the number of incident photons per unit time, W and S will be related by
I
t+T
W=
dt’S(t’) with
T = VIAc.
The probability pe(n, T ) for the emission of n photoelectrons in the time interval (t, t + T) is given (with a photoefficiency q = 1) by the expression on the right-hand side of eq. (34) if P’(W)is identified with the classical distribution for W, caused by the stochastic character of the intensity S. A fully quantum-theoretical analysis has to take into account the quantum character of the absorption of the photons by the individual atoms that constitute the detector. (In this connection the importance, for the measurements of intensities, of the correlation functions defined by eq. ( 5 ) becomes evident.) Under certain simplifying conditions (broad-band and point-like detector, quasi-monochromatic wave, number of atoms NA>>photon number n, Tcccoherence time of the radiation 7,) the quantum-theoretical and semiclassical treatments give the same results. In terms of the generating function Q(A, T ) =
L-
dWP’(W)e-AW,
the photoelectron count distribution is given by
and the moments are expressible as
For real detectors the photoefficiency q is smaller than 1 and W has to be replaced by qW. The generating function Q is represented by use of P’(W), which is directly related with P({a,}). Since, as mentioned above, there are fixed relations between the functions P, K , PA, K A , the generating function Q can also be represented by means of the well-behaved functions PA and K A (PERINA,PERINOVA and HORAK[1973]).
I I I , 8 21
COHERENCE PROPERTlES OF LIGHT
181
Fig. 3. Scheme of a device for the discrimination of noise pulses and the shaping of photoelectron pulses (Tdis the dead time of the device).
In principle, the accuracy attainable in photoelectron counting is limited by the finite number of signal pulses as well as by noise pulses which are caused by fluctuation processes in the detector or by irradiation of the detector with noise photons or other particles, e.g., from the cosmic radiation. In general, the noise pulses are characterized by an amplitude distribution different from that of the signal pulses. For example, the electrons emitted at the dynodes due to the noise are less strongly amplified than the photoelectrons from the cathode. An amplitude discrimination is, therefore, helpful (Fig. 3). Using this method and employing suitable detectors, the counting rate without signal can be decreased below one count per second, whereas the probability of a single photoelectron to be detected is close to unity. Behind the discriminator the pulses can be formed and digitally analyzed. One has to take into account the fact that the dead time of the device is not only caused by the photon detector but also by the electronic equipment, especially by the pulse former. Typical values for the dead time Td are of the order of some lo-" s. The counting loss due to the dead time is given by (fi)T,/T under usual conditions, so under certain assumptions about the light statistics, the dead time effect can be corrected (see, e.g., ARECCHI and DEGIORGIO [1972], GUPTAand MEHTA[1973] and MIRZAEV and RAJAPOV [1978]). In comparing the calculated and the measured values one must examine whether all the above conditions concerning the detector are fulfilled; otherwise the experimental results have to be corrected. For example CANTRELLand FISLD[1973] calculated corrections for a detector with finite aperture; these were applied and tested by BARKand SMITH[1977]. The fulfilment of the condition T K T ,may become difficult, especially with radiation sources of short coherence time as, such as those employed
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COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
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in the early experiments of Hanbury Brown and Twiss. At a short measuring time T the accuracy is not high because of the small number of signal pulses, limited by the intensity of the light source or by the dead time of the receiving device. In principle, it is possible to compensate the loss of accuracy of a single measurement by making a great number of independent measurements. This, however, requires high stability of all experimental parameters. To determine the statistics of the photoelectrons and thus, according to the relations discussed above, the statistics of the light field, different methods for measuring the parameters of the photoelectron distribution in one signal channel or in several channels have been developed: Measurement of the number of photoelectrons in a given time interval: After passing the pulse former the signal pulses are transmitted to a counter which is gated between ti and ti+-,- and which measures the number of pulses n, within this time interval. Thereafter, the content of the nith channel of a multi-channel analyzer is increased by one digit. From a great number of independent measurements (with ti+l -ti >> 7,)the probability distribution of photo-counts p,( n, T), which is immediately related with light statistics, is obtained. With pe(n, T ) the moments or the factorial moments can be directly calculated. On the other hand, it is possible to measure a signal proportional to ni by analogous devices, e.g., by an integration of the charges with a capacitor. The device may be gated by an electronic switch behind the detector or by gating the sensitivity of the detector. These methods are also usefully applied when utilizing two-dimensional arrangements of light detectors, as for instance in connection with special vidicons. Measurement of the time interval between two photo-counts: Time intervals can be electronically measured with high accuracy, and one makes use of this possibility in determining the distance between two consecutive pulses i and i + 1 (or more generally between i and i + j with j = 1 , 2 , . . .). One pulse serves as the start signal of a clock, the other pulse gives the stop signal. These measurements lead to the joint probability of second order, p2(t, t + T ) . In an analogous manner joint probabilities of higher order can be measured. Measurement of the correlation between several signals : The correlation between signals in different channels can also be measured by digital and analogous methods. The resulting signal behind the correlator in a device of the Hanbury Brown and Twiss type (Fig. 4a) is given by ( n , nz) or by
111,s 21
183
COHERENCE PROPERTIES OF LIOHT
Digital correlator
I
I
Ai
fI
Dj
Electronic delay, tn
II PFi
C1
ii I
A,
j
I
D,
PF, 1
c2
t c,
n
t-tr, t-tD+T
-
Tlme to digital converter
t-
el Computer
Fig. 4. Measurement of correlation between two signals 1 and 2. (PD,., photo detector; A,.* amplifier; Dl,2 discriminator; PF,., pulse former; Cl.2 counter.) (a) Measurement of (n, . n,). (Signal 1 is delayed for a time t,. The counters are gated for a time T.) (b) Measurement of the statistical distribution of the time between a photo-electron count in channel 1 and the next count in channel 2.
a quantity proportional to this product, where n , and n2 are the numbers of photo-counts in the two detectors 1 and 2 during a time interval T. (As a special case the photo-counts from one and the same detector measured at different times may be correlated.) By averaging many independent measurements, the quantity ( n l ( t ) n 2 ( +7)) t (with 7 = tD) is obtained, which is directly related to the correlation function I'2*2(t, t + T, t + T , t ) (for T
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COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
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is small compared to unity. The correlation measurement then reduces to the registration of a coincidence of signals in both channels. An improved version of this method has been described by KIKKAWA, YOSHIMURO and SUZUKI [1973]. Correlations between two channels can also be determined by time measurements in which one pulse of the first channel serves as the start signal for the clock and the following pulse (or an arbitrarily chosen later pulse) of the other channel stops the clock (Fig. 4b) (DAVIDSON and MANDEL [1968]). This method has been applied, e.g., by KIMBLE, DAGENAIS and MANDEL [19771 to measure photon antibunching in resonance fluorescence (see P 3). Correlations of higher order can be determined by similar methods.
9 3. One-Photon Processes In preparation for a discussion of multi-photon processes as the main effects of nonlinear optics, we will consider some methods and results concerning one-photon interaction that will lead to nonlinearities only in the case of saturation. Under resonance conditions the coherence properties of light can be changed even by one-photon processes, because of the stochastic character of spontaneous emission and because of the dependence of the population of the atomic levels on the light intensity. In this area numerous different lines of experimental and theoretical investigations have been pursued. We can only outline here a few representative ones. As far as possible we will refer to publications that present summaries and reviews of such works.
3.1. CHANGE OF THE COHERENCE PROPERTIES OF LIGHT BY SPONTANEOUS EMISSION FROM THE ATOMIC SYSTEMS
Following the considerations of CARUSO-ITO [1975] we will discuss the interaction of light with a system of atoms that have an energy separation h o between the states 12) and [I),which coincides with the photon energy of the considered mode (Fig. 5 ) . The dipole approximation yields the interaction Hamiltonian -diI?(ri) in the Schroedinger picture, where di and &ri) are the dipole operator of the jth atom and the operator of the electric field at the position of the jth atom, respectively. Applying standard methods of quantum mechanics (see, e.g., SCHIFF[1955] and
xi
III,§ 31
185
ONE-PHOTON PROCESSES
Fig. 5. Scheme of the resonant transition 11)
-
12).
HAKEN[1970]), the operator di can be expressed by means of the flip operator of the transition 1 + 2 , which flips the atom j from the 'state 11) to the state 12), and the hermitian conjugate operator (I& which transforms the atom in the opposite way. This representation of the dipole moment is analogous to the representation of the operator of the electric field strength E by means of the creation and annihilation operators d and d' or the corresponding complex field operators &) and &+). Then, using the rotating wave approximation, the interaction Hamiltonian may be expressed in the form
A,
-1 i
x(l)(&l)j
&)(rj)+{h.c.},
(38)
where x'" is the effective matrix element of the transition between the states 11) and 12). This interaction Hamiltonian is inserted into the equation of motion of the density operator 6 of the total system of atoms and radiation. Assuming the validity of the irreversible approximation, which means that the thermal equilibrium is not disturbed by the radiation, a differential equation for the density operator of the field dF alone is derived, by taking the trace over all the atomic variables. The exact solution of this equation can instructively be described in the Prepresentation for bF. Before the interaction has been switched on at t = 0, the state of the field is described by means of the such-probability Pda) as
6dO) = dZa Pda) la) (a\.
(39)
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COHERENCE OF LIGHT AND NONLINEAR OFTICAL PROCESSES
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where
N2 (fi(t ) ) = [exp {2q"'t(N2- N , ) } - 11. N2 - Nl
q'l' is proportional to and to the line shape function of the transition. N,, N , are the population numbers in thermal equilibrium of level 2 and 1, respectively. Pstimrepresents the change due to the stimulated processes, namely absorption and emission. The amplitudes a of the initial distribution are multiplied by a time-dependent factor. This means that only a translation of P,(a) in the a-space takes place. One may, therefore, say that both the stimulated processes essentially preserve the coherence properties. Pchaorepresents the chaotic field caused by spontaneous emission, with the time-dependent mean photon number (fi(t)).
3.2. CHANGE OF THE COHERENCE PROPERTIES OF LIGHT BY NONEQUILIBRIUM ATOMIC SYSTEMS
If the properties of the atomic ensemble, for instance the population numbers of the energy levels, are influenced by the radiation field even the one-photon interaction processes will become nonlinear. Saturable absorbers : Under the simplifying conditions that the radiation can be treated classically and that the transversal and longitudinal relaxation times are small compared with the correlation time of the incident light, the following nonlinear dependence between the light intensity I behind the two-level absorber and the incident intensity I. is obtained:
( a is the cross section of the absorption process, q = t - z / v , t time, z coordinate in direction of propagation, v light velocity, 5 = z , N , number of active atoms per volume), (see, for example, HERRMANN, WIENECKE and
111, § 31
ONE-PHOTON PROCESSES
187
Fig. 6. Photoelectron distribution p(n) of transmitted laser light and chaotic light, with mean number of photo-electron counts of incident radiation (n)o = 10. (After BENDJABALLAH [ 19761.)
WILHELMI [1975]). By such a nonlinear filter without memory the probability distribution Po(Io) of the initial light intensity I", which is functionally related to the intensity I by I. = f(I),is changed into the distribution P(I)=P[f(I)](df(I)/dI) of the intensity I. High intensities are seen to be strongly preferred by this nonlinear filter. Starting with this transformation of probabilities and using representative initial distributions, e.g., for single mode laser radiation and thermal light, BENDJABALLAH [ 19761 calculated the statistics of photoelectron counts caused by filtered light in a photon detector (Fig. 6). The distribution law maintains its structure during the nonlinear process only for an initial Poisson distribution, whereby in the calculated example the mean number of photoelectrons decreases from 10 to 3.2. The calculation of the normalized factorial moments ( n ( n- 1) * * (n - k + l))/(n)' shows that, except for ideal coherent light, the intensity fluctuations increase strongly. Processes in lasers: The interactions within the active medium of one-photon lasers belong to the nonlinear one-photon processes. The statistics of emitted light determined by spontaneous and stimulated
-
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COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
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emission in the active medium, as well as by feed-back in the resonator, has been investigated very extensively because of the importance of lasers as light sources. Besides, the changes of the statistics in the region of the laser threshold are of fundamental physical interest (see, for example, HAKEN[1975]). The calculated statistics of photo-counts generated by single-mode lasers below, at and above threshold, are in good agreement with experimental results; reviews of investigations in this area were given by ARMSTRONG and SMITH[1967], HAKEN [1970, 19721, RISKEN [1970], SARGENT and SCULLY [ 19721, KLIMONTOVICH, KOVALEVand LANDA [1972], ARECCHI and DEGIORGIO [1972], KLIMONTOVICH [19741 and ARECCHI [1975]. Even as regards the time development of the coherence of light due to transient processes after switching on the laser, a good agreement between calculations and experiments has been found. In most investigations the intensity correlation functions of the second order have been measured. The analysis of higher order correlations is also of great importance because near laser threshold the field fluctuations are nonGaussian and, therefore, the behavior of third and higher order correlations cannot be inferred from the behavior of correlations of the second order. CORTI and DEGIORGIO [1975, 19761 investigated third-order intensity correlation of light from a single-mode intensity-stabilized He-Nelaser, utilizing a fast digital correlator to measure the joint probability of photoelectron counts. By use of these results and additional second-order measurements, the so-called intrinsic third-order correlation has been calculated, which represents that part of the total third-order correlation which is not predictable from the knowledge of correlations up to the second-order. In agreement with laser theory, this intrinsic third-order correlation differs significantly from zero only in the region of laser threshold. Lasers working in the multi-mode regime are also of great physical and practical importance, and, for this reason, many papers have been devoted to their coherence properties, which are substantially influenced by mode competition (see the literature on lasers cited above). Often the nonlinearity of the active medium stabilizes the overall intensity of the laser in the multi-mode regime, but at the same time the fluctuations of the single modes are increased. Therefore anticorrelation between several modes may arise (ARMSTRONG and SMITH [1967]). By experimental results, TEHRANI and MANDEL [ 19771 confirmed their own predictions that, if slightly different losses exist between the counter running modes in a ring laser, the intensity of the weaker mode remains at a constant value as
111, S 31
ONE-PHOTON PROCESSES
189
the pump parameter increases and the cross correlation of intensity fluctuations between both the modes tends to a constant negative value. Because of the great complexity of the laser processes in the multi-mode regime, only qualitative agreement between experiment and calculations can be expected to be found in most cases (see, for example, ARECCHI and RICCA[1977] and AOKI,ENDOand SAKUKAI [1977]). Passive mode-locking with saturable absorbers : Nonlinear filters of the types described above can be used as saturable absorbers for mode synchronization of lasers (see, for example, LETOKHOV [ 19681, KFUUKOV and LETOKHOV [1972], NEW[1972], GLENN [1975], BRADLEY [1977]). The initial probability distribution of the field amplitudes given by fluctuation processes is strongly changed due to the saturable absorber and single pulses, with well-determined parameters arising from noise. HERRMANN and WEIDNER [1979] described the action of the fluctuation processes by introducing Langevin forces into the density operator equation of the atomic systems. The properties of the radiation field are characterized by correlation functions, mean square fluctuations and conditional probabilities of field strength. Expectation values of all measurable parameters can be calculated in this way. Oil this basis HERRMANN, WEIDNERand WILHELMI [19791 treated the simultaneous influence of the saturable absorber and of variations of the inversion of the active medium on the statistical properties of the radiation during the different phases of the generation of ultrashort pulses. Calculations have been performed on fluctuations of the pulse parameters (intensity and pulse length, signal-tonoise ratio), as well as on the probability of the occurrence of satellite pulses and on the probability that the generation process is broken off before well-behaved single pulses are formed. The results are in agreeVOCLERand WILHELMI ment with experiments (BELKE,D. SCHUBERT, [ 19781). Coherent states of atomic systems : During strong interactions between the radiation field and the atomic system, the coherence properties of the field can be transferred to the atom, thereby markedly influencing the further behavior of the atom. This is recognized by considering the equation of motion for the atomic operator 6;:
Here the atom is regarded as a two-level system with the two levels being associated with the transition under consideration. The operator
190
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 8 3
6,'(t) is the slowly varying part of the ordinary flip operator 6,,(t)= 6 ' ( t ) = 6:(t) exp[io,,t], which flips the atom from the lower (1) to the upper (2) state (see, for example, PONATH and SCHUBERT [1976, 19771). The first and second terms on the right-hand side describe the fluctuation and phase destroying action of a dissipative system coupled with the atom, whereas the third term gives the interaction with the incident radiation field. (For a complete description of the atomic system the equation of motion of &(t)6,(t) has also to be investigated.) If h-' ~ d Z l ~ , ~ ~ 7(8, ' ; ' is the field strength of a coherent field, d,, is the dipole matrix element of the transition and T,, is the transversal relaxation time or phase decay time), intrinsic excitations of the atom will take place before the phase of the atomic state is disturbed by the dissipative system. Under these circumstances coherence properties of the light can be transferred to the atom. In general, the dependence of the correlators of the atomic variables on the field correlators has to be calculated. This transfer of coherence properties leads to effects such as self-induced transparency, optical free induction decay, optical nutation and optical echo. For reviews of this subject see, for example, ABELLA[1969], BACHMANN, SAUERand WALLIS[ 19721, SLUSHER [19741. POLUEKTOV, and ROJTBERG [1974], BULLOUGH, CANDREY, EILBECK and GIBBEN POPOV [1974] and ALLENand EBERLY [1975]. If several transitions of the atom take part in the interaction with the radiation field, a coherent coupling between these transitions will be possible, leading for instance, to beats in the emitted radiation (see, for example, HAROCHE and PAISNER [1974] and LETOKHOV and CHEBOTAEV [1974]). Resonance fluorescence : In the last years resonance fluorescence gained increasing interest with respect to the coherence properties of the emitted light. The theoretical aspects are discussed by MOLLOW[19691, HERRMANN, SUSSEand WELSCH [1973], EBERLY [1976], WALLSand CARMICHAEL [1976], KIMBLEand MANDEL [1976, 19771, AGARWAL [1976, 19771, SMIRNOV and TROSHIN [1977] and APANASEVICH and KILIN[1977]. In these papers the temporal and the spectral behavior of fluorescence has been investigated, the atom being excited by a laser wave. EBERLY [1976], AGARWAL [1977] and KIMBLEand MANDEL[ 19771 calculated the influence on the properties of emitted light of a finite line width of the exciting radiation and of a deviation of the midfrequency of the laser from exact resonance. Experimental investigations on laser-induced resonance fluorescence are in good quantitative agreement with these calculations (see, for example, SCHUDA, STROUD and HERCHER [1974], Wu,
111, 5 31
191
ONE-PHOTON PROCESSES
GROVEand EZEKIEL [1975], WALTHER [1975] and GIBBSand VENKATESAN [ 19761). The correlation between fluorescence photons emitted at different frequencies have been discussed by APANASEVICH and KILIN[19771. HERRMANN, SUSSEand WELSCH[1973] and KIMBLEand MANDEL [1977] investigated the transient behavior of fluorescence on the condition that the laser field acts on the atom from the time t = 0 at which the atom is in the lower state. With exact resonance and irradiation of the atom by a monochromatic laser wave of amplitude ELfor t 2 0, the mean value of the fluorescence intensity is seen to increase from the value of zero at t = 0 to a maximum at t = d-*/ldzlE,I1and then damped oscillations occur, with the Rabi frequency a'=(dz,ELIh-', before the atom attains the stationary value after a time of about 27r/a'.The population probability pi$) of the excited atomic state changes in an appropriate manner (Fig. 7). According to KIMBLE and MANDEL [1977] the field strength correlation function f sof2fluorescence light in this process can be expressed in the factorized form
ryt,t f 7 ,
f+7,
t)=r"'(t, t)ry(7,7),
(43)
where T,(T,T)is the mean intensity at time 7 after switching on the . correlation interaction with the atom in the lower state at ~ = 0 The function T,(T,7 ) which is proportional to pi$)(^), as shown by HERRMANN, [1973], vanishes at 7 = 0 and increases with T up to the SUSSEand WELSCH first maximum. According to eq. (43) the correlation function f.* shows the same dependence on 7. Thus, the single atom pumped by coherent radiation represents a light source which shows antibunching because, every time after the registration of a fluorescence photon the atom is
-
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 7. Probability pi;) to find the atom- in the excited state 2 as a function of the normalized time T = r/T,, with p = fi(2d,,ELT,,)-' = 0.1 and 0.4,during the laser pulse of constant field strength EL switched on at T=O (full line) and after switching of the pulse (broken lines). (After HERRMANN,suss^ and WELSCH [1973].)
192
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 5 3
found in the ground state with probability 1 and it cannot emit the next photon immediately but, with high probability, only after a time of the order of UO’.Antibunched radiation cannot, in principle, be described by means of classical physics as mentioned in 9 2, because fields in which the photon number is defined more sharply than in a coherent state cannot have a P-representation with a well-behaved P-function (cf. 9 2.2). Therefore the resonance fluorescence of a single atom can serve as a source of light with typical quantum properties. (Other possibilities for the generation of antibunched radiation will be discussed in 09 4-6.) The first experiment for demonstration of antibunching of fluorescence light was proposed and carried out by KIMBLE,DAGENAIS and MANDEL [1977] by means of a very interesting experimental technique (Fig. 8). The resonance fluorescence of the transition (3‘S;, F = 2, mF = 2) c,(3*P;, F = 3, m, = 3) of pre-pumped sodium atoms within an atomic beam was investigated. By calibration of the laser intensity, ratios of the Rabi frequency 0’and the Einstein transition coefficient of spontaneous emission between 2 and 2.5 were obtained. In these experiments the transversal relaxation time T,, was determined only by spontaneous emission, i.e., one has T,, = 2/A. By employing a beam splitter, the fluorescence light was divided into two beams of equal intensity, ejecting photoelectrons at the cathodes of two photondetectors. The pulses of both detectors were transferred to the start and stop entrances of a time interval measuring device, after appropriate processing (Fig. 4b). The number of events n ( ~ ) with a time between start and stop in the interval (7 T + A T ) were
---
Pre -pumped
Fig. 8. Scheme of the antibunching experiment of KIMBLE,DAGENAIS and MNDEL [19771.
111, I31
193
ONE-PHOTON PROCESSES
--
TJIO-~S
200
I
I
25
50
I
75
100
Fig. 9. Number of recorded stop pulses n(T) within a time interval of AT = 2 x 1O-'s at time T between start and stop. (After KIMBLE, DACENAIS and MANDEL[1977].)
stored (Fig. 9). The number of stop events l t ( ~ is ) seen to increase from its smallest value at T = 0 to a maximum at about 25 ns, as expected from theory. The quantitative analysis of the experiment is rather difficult because the fluctuation of the number of emitting atoms and the mutual coherence between fluorescence and background have to be taken into account. Owing to the fundamental importance of the experiment, the results were also discussed by many other authors, e.g., JAKEMAN,PIKE, PUSEYand VAUGHAM [ 19771, CARMICHAEL, DRUMMOND, MEYSTRE and WALLS[1978], VILLAEYS and FREED[1978] and PERINA [1978]. JAKEMAN, PIKE,PUSEYand VAUGHAM [19771 and CARMICHAEL, DRUMMOND, MEYSTRE and WALLS[1978] proved that the antibunching effect of an ensemble of Poisson distributed atoms can vanish even if a single atom gives ideal antibunching. The deviation of the experimental values of l t ( ~ from ) zero at T = 0 can be explained by photoelectron counts caused by the action of scattered light and of fluorescence light from an atom other than that giving the start count. The influence of coherent light-beating between fluorescence light and background depends on the spatial coherence of both light sources (cf. KIMBLE,DAGENAIS and MANDEL [1977], JAKEMAN, PIKE, PUSEY and VAUGHAM [1977] and CARMICHAEL, DRUMMOND, MEYSTRE and WALLS[1978]). To calculate the decrease of n ( T ) after the maximum, the finite occupation time of the atoms within the observed volume has to be taken into account (VILLAYES and FREED[1978]). Analyzing the experimental results of KIMBLE,DAGENAIS and MANDEL
194
COHERENCE OF LIGHT AND NONLWEAR OPTICAL PROCESSES
[III, § 3
[1977], whereby the deviation of the number of atoms from unity and the action of background are particularly considered, it may be concluded, in our view, that the radiation from a single atom really reveals antibunching with y’.z(0) < 1. Superradiance: The rate at which light is spontaneously emitted by a system of two-level atoms has been calculated first by DICKE[1954]. Considering the whole ensemble as a single quantum-mechanical system, the individual atoms were found by Dicke to cooperate under certain conditions, and to emit radiation at a rate much higher than expected for incoherent superposition of the emission processes. Such a system is said to be “superradiant”. A system of atoms in a superradiant state exhibits specific coherence properties. We consider a simple model of NA twolevel atoms contained in a region with dimensions small compared to the wavelength of light and interacting with a radiation field described by the Hamiltonian kF.The atoms are assumed not to take part in any translational motion and they are coupled with one another only by the radiation field. The Hamiltonian of the whole system is given by
A = hoR, -A(e,R,
+e&
+ fiF
(44)
(tto represents the energy difference between the two atomic levels, el, e2
are unit vectors characterizing the polarization of the atomic transition, A is the vector potential of the radiation field and the fii are given in terms of the spin energy operators R , of the single atoms by Ri R, with i = 1,2,3) (see, for example, FAJN[1972] or ALLENand EBERLY[1975]). A state of the NA-atom system can be characterized by the quantum numbers r and M, which are defined by the relations R’ Ir, M)= r ( r + 1) (r, M ) and k,Ir, M ) = M Ir, M ) , with IMI 5 r 5$NA.The quantum number M is equal to one half of the population number inversion. The intensity of the spontaneous emission of the system is given by I = (r+M)(r-M+l) (DICKE[1954]). If all atoms are in the excited state, they will radiate independently of one another and hence, the total intensity I a NA results, because of M = r =$NA.On the other hand, if M = 0 and r = f N A ,the relation I 0:$NA($NA+ 1)=$N: will hold. Solving the Heisenberg equation for the atomic- and the field operators by means of a perturbation calculation, the normalized correlation functions of the emitted radiation are found to be
=xi=,
IIL8 31
ONE-PHOTON PROCESSES
195
(see, for example, KARCZEWSKI [1975]). As regards the superradiant state with M = 0, r = f N A , one finds that
For large NA the modulus of the normalized correlation function approaches unity with high accuracy, irrespective of the time coordinates chosen. This means that the radiation field emitted by a superradiant system with large N A is coherent up to very high orders (m 5 0 . 1 NA). HASSAN and WALLS[ 19781 discussed the photon statistics in Cooperative resonance fluorescence with regard to its dependence on the strength of the pumping field. The phenomenon of bistability has been obtained: in the non-cooperative branch y2v2(7)= 1 + (yl*l(~)( + O(N,'), whereas ~ ' ~ ~ =1 ( +O(N,') 7 ) in the cooperative branch (Fig. 10). Light emitted by a Dicke-system of atoms excited at time t = 0 to the upper state by a short coherent pulse exhibits antibunching. The dependence of the normalized correlation functions ym*" on time has been calculated by CHROSTOWSKI and KARCZEWSKI [1971] and by KARCZEWSKI [1975]. Further investigations such as those concerning the development of coherence in space and time after preparing the atomic state by a laser pulse, as well as investigations involving several modes of the radiation field and atomic ensembles confined to large volumes and involving several level atoms were carried out (BIALYNICKA-BIRULA [ 19701, EBERLY and REHLER [1970], REHLERand EBERLY[1971], AGARWAL [1974], FRIEDBERG and HARTMANN [1974], ALLENand EBERLY [1975], AGARWAL
0
-
pump field strength
Fig. 10. Normalized correlation function y2.2 in dependence on the field strength of the pump radiation. (After HASSANand WALLS[1978].)
196
COHERENCE OF LIGHT AND NONLINEAR OkTICAL PROCESSES
[III, 84
and TRIVEDI [1976], EMEWANOV and KLIMONTOVICH [1976], RESSEYRE and TALLET[1977] and S E N ~ Z K[1977]). Y The equations of motion of the whole system of two-level atoms interacting with a one-mode field have been solved exactly (KUMARand MEHTA[1977]).
8 4. Multi-Photon Absorption Soon after the construction of the first lasers the simultaneous absorption of two or more photons from one or several modes of the radiation field was observed. This effect had already been predicted and calculated in its main features by M. GOPPERT-MAYER [1931]. In such a process the transition probabilities depend mainly on the statistics of incident light and it is thus possible to determine these statistical properties. On the other hand, multi-photon absorption can be made use of to change the statistical behavior of the electromagnetic field. Both aspects will be discussed in this section. In numerous investigations reference is made not only to transitions between bound electron states but also to transitions from bound to free electron states, i.e. to a multiphoton ionization, in which, because of the high detection sensitivity for free electrons, nonlinear effects of very high order (rn 2 10) could be observed. Basic theoretical investigations concerning multi-photon absorption have been performed, among others, by TEICHand WOLGA[ 19661, LAMBROPOULOS, KIKUCHI and OSBORN [19661, SHEN[1967], MOLLOW [1968], AGARWAL [1970], MCNEILand WALLS [1974], TORNAU and BACH[1974] and SIMAAN and LOUDON [1975].
4.1. TRANSITION PROBABILITIES
In principle, an analogous procedure as it is used in the one-photon case can be applied. Instead of using the basic interaction Hamiltonian A, -diE(ri) in a higher-order perturbation calculation, we start with the effective interaction Hamiltonian
=xi
fir=--I ~ ' ~ ~ ' ( ~ ~ , ) i [ ~ ' - '+{h.c.} (ri)]m
(46)
i
for a system of atoms or molecules subjected to multi-photon absorption (Fig. 11). Here x'"' is the matrix element for simultaneous absorption of
197
MULTI-PHOTON ABSORPTION
Fig. 11. Multi-photon transition between two levels of the atomic system.
rn photons and (i,,), is the flip operator of the transition 1 + 2 of the jth atom (see, for example, SCHIJBERT and WILHELMI [1978]). For simplicity the incident light is assumed to be a single-mode field. A t first we will consider only one atomic system with the following initial condition at t = 0: the atom is in its ground state and the radiation field is described by the density operator &(O). The probability p'"'(t) for the transition of the atom from state 11) into the excited state 12) at time t > 0, namely
with
T"-"(t;,. . . , t;,
t,,
. . . , t,)
= T r {&(O)E(+)(ti)
*
- E(-)(t,)
E(-'(t,)}
is calculated in a straightforward way using time-dependent perturbation theory. The function W A contains the characteristic quantities of the atom -the dipole matrix elements of all the atomic transitions and all the energy eigenvalues. Further details are given by SHEN[1967], MOLLOW [1968] and AGARWAL [1970]. It can be seen that p(,) depends on the normally ordered correlation function Trn,,.This correlation function, and also ptm)are considerably affected by the coherence properties of the field at t = 0. Assuming stationarity and sufficiently large t-values, p(,) becomes proportional to t. Hence, a time-independent transition rate can be introduced. In this case or for quasimonochromatic fields (linewidth Aw, of the radiation<<spectral width of the atomic transition AmA) one finds that
198
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
near resonance
I
rn ! ( ~ ? + 6 ) ~
d p(m) a ((&+)m6") dt
=
[III, 84
for chaotic light for coherent light
(48)
-
n(n - 1 ) . * ( n - rn + 1) for light in a photon number state In) with (6'6) = n.
Thus, the transition rate is proportional to the expectation value of a 2rn-fold product of photon creation and annihilation operators. The dependence of this expression on the intensity (6'6) strongly differs for chaotic, coherent and number-state light. The result obtained implies that the transition rate for thermal light is rn! times higher than that for fully coherent light, i.e. for ideal laser light. Concerning its intensity, chaotic radiation exhibits more pronounced deviations from the average value than does laser light. Since the positive deviations - the intensity peaks yield larger contributions to the nonlinear response than the negative deviations, the transition rate is enhanced by the factor rn! for chaotic light, providing one has the same mean intensity in both cases. CHROSTOWSKI and KARCZEWSKI [1977] analyzed the relations of the two-photon transition probability without special assumptions regarding the ratio AwR/Aw,. In Fig. 12 the result of these calculations for the ratio between the transition rates of chaotic and ideal laser light r,= [t-'p~'(t)]/[t-'p~)(f)] is given in dependence on the interaction time t, where the same first-order correlation function
ryt, t) =(lir)exp(-io,(t'-tt)-~AoR
It'-tl)
(49)
_ _ _ _ _ _ _ _ _ ----------__
-
Ao,t
Fig. 12. Dependence of r2=[t-'pz]/[t-'p(:)] on time t, for Po,<< AwR (l), A w A = A w R (2) and Am, >>Ao,(3). (After CHROSTOWSKI and KARCZEWSKI [19771 and KARCZEWSKI[1978].)
MULTI-PHOTON ABSORPTION
111, I41
199
has been assumed for both kinds of light. The ratio r2 is seen to have its maximum at Ao,t 1. For t + m and Ao,<< AmA the result is r2 = 2!, in agreement with the above theory; for t + m and AwR>>AwA the value r2 = 4 is obtained. The transition probabilities for multi-photon absorption and multiphoton ionization and, in particular, their dependence on the coherence of incident light can be considerably influenced by resonant intermediate levels. Investigations dealing with this problem have been carried out by ARMSTRONG, LAMBROPOULOS and RAHMAN[19761, MOSTOWSKI [19761, DIXIT and LAMBROPOULOS [1978], KNIGHT[1977], SCHUBERTand WILHELMI [1975] and WILHELMI [1976a,b]. The results depend on the ratio between the correlation time of the signal T , and the lifetime of the intermediate level Tlo,as well as on the ratio between signal intensity and saturation intensity. For example, the dependence of the two-photon transition probability on photon statistics disappears if the transition between the ground state and the intermediate level is entirely saturated (Fig. 13); under these circumstances the interaction process resembles a one-photon absorption at the upper transition. On the other hand, at light intensities far below saturation the resonant system may act like a two-photon absorber without intermediate resonance. Three-photon absorption, with resonant-enhanced transition probability, has been discussed by SCHUBERT and WILHELMI [1975]. They also considered the choice of suitable absorbers with short relaxation times in the condensed phase. KNIGHT [1977] took into account the fact that under very strong irradiation, ( a , , l > T ; ' , a,"is the cross-section of the transition between the ground and intermediate level, T,, is the transversal relaxation time or phase decay time of this transition), the behavior of the atomic systems
-
Fig. 13. Saturation of the lower transition.
200
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 64
cannot be described by simple rate equations for the population numbers of atomic levels. He obtained damped oscillations for the population probability of the intermediate level, namely a kind of Rabi oscillations. The dependence of the transition probability of m-photon ionization on the coherence of light, under influence of finer details in the atomic level scheme, especially of a coupling between the resonant intermediate state 11) and a nearby state ll'), has been discussed by DIXIT and LAMBROPOULOS [1978]. The coupling leads to ratios r, = [ t - ' p ~ ~ ~ ( r ) ] / [ t - ' p ' , " ' ( t )greater l than m ! , even at large interaction times t and with AwR<
- p?t;
n) = n ( n - I )
11 + q(n - 1>12 S2+An[l+q(n-1)]2'
where q is determined by the strength of the coupling between 11) and 11') and 6 and A are also given by the properties of the atomic system. Using the probability distributions of photon numbers for chaotic and fully coherent light, the transition probabilities of both kinds of light and thus the ratio rz can be calculated from eq. (50). Using values of the atomic parameters representative of the pair of fine structure levels Pi and P: in alkali atoms, Dixit and Lambropoulos calculated maximum values of r, of nearly 4,at mean photon numbers (A)- 100, whereas at small mean photon numbers (N) + 0 the common result of unsaturated two-photon absorption r2 = 2! is obtained. With (A)+ the result r, = 1 agrees with that of saturated absorption of only one intermediate resonant level. ARSLANBEKOV, DELONE,MASALOV, TOBIRASHKU and FAJNSHTEJN[ 19771 showed the ratio r, between the transition probabilities of m-photon ionization for chaotic and laser light to deviate, in accordance with experimental results, from m! at high field strength, even without the action of resonant intermediate levels. These deviations are caused by tunnel transitions of electrons, the probability of which increases with increasing field strength (KELDYSH [1964]). Therefore, the ratio r, is not only dependent on the order m of the ionization process, but also on the adiabaticity parameter y = TTun/TVibr where TTunand TVib represent the transition time of electron tunneling and the vibration period of the radiation field, respectively (Fig. 14). For large values of y the parameter r,,, tends to m!.Ionization experiments were performed with xenon, using Nd-laser light at field strengths of about 5 x lo9 V/m. This experimental
111,141
MULTI-PHOTON ABSORPTION
201
Fig. 14. Dependence of r,,, =[t-lp;i:’]/[t-’p‘i]’] on the parameter of adiabaticity y = TTLm/Tv,b for xenon ( m= 1 1 ) . (After ARSLANBEKOV,DELONE, MASALOV, TOBIRASHKU and FAJNSHTEJN [1977].)
condition corresponds to m = 11 and y = 5. Values of r,, were measured to be of the order of lo5, in agreement with theoretical predictions. In order to determine the ratio r,,,, the transition probabilities of multi-photon absorption and multi-photon ionization for chaotic and single-mode laser light were measured in several experiments carried out within the last years (see, for example, LECOMITE,MAINFRAY, MANUS and SANCHEZ [1975], KRASINSKI, CHUDZYNSKI, MAJEWSKI and GLODZ[ 19751, KRASINSKI and DINEV[1976], KRASINSKI and DINEV[1977] and by SMIRNOVA and TIKHONOV [1977]). In all these experiments lasers oscillating in several non-synchronized modes were used as “chaotic” light sources of high power. Since, even without special auxiliary synchronizing elements within the resonator, the laser modes may exhibit coupling effects caused by the polarization and inversion of the active medium, checking of the synchronization behavior is indispensable in any such experiment. For this purpose the determination of the probability distribution P ( I ) or of the intensity correlation functions of several orders is necessary. This was done, for example, by SANCHEZ [1975] and HEIN[1977]. One must also take into account that even for independent phases of the M oscillating laser modes, the ratio r,,, depends on the number M and tends to m! only as M 4 0 0 . In the case of independent phases and Gaussian distributed amplitudes LECOMPTE,MAINFRAY,MANUSand SANCHEZ[1975] and ARSLANBEKOV, DELONE,MASALOV, TOBIRASHKU and FAJNSHTE~N [ 19771
202
107
COHERENCE OF LIGHT AND NONLINEAR OITICAL PROCESSES
[III, 84
--
106 -
lo5 lok+fl
103
I
10
20
30
Fig. 15. Dependence of r,,, = [r-l&mA]/[t-lp'm'] SM on the number of effective laser modes M. (a) for xenon (m=11)and (b) for rhodamin 6G ( m = 2 ) . (After LECOMFE, MAINFRAY, MANUSand SANCHEZ [1975]and SMIRNOVA and TIKHONOV [1977].)
derived the following relation: r,(M)= rn!
M r n ( M - l)! (rn+M-l)!'
Thus, with the increasing order rn of the process, the demand for a large number M of independent modes becomes more and more important. Experiments carried out by means of lasers with partially synchronized modes can be often approximately be described by taking into account an effective value of M.Fig. 15 show the experimental results of Lecompte et al. for 11-photon ionization, as well as those of SMIRNOVA and TIKHONOV [19771 for two-photon absorption, illustrating the dependence on the effective number of independent modes. In addition to the limiting value r l l ( r n = l l , M+a, y + w ) = l l ! the value of r(m=11, M + m , y = 10) is plotted in Fig. 15(a), y having been determined by us from the experimental data of LECOMPTE, MAINFRAY, MANUS and SANCHEZ [1975]. We see that the intensity correlation functions of higher order can be determined from measurements of transition probabilities for multiphoton absorption and ionization under precisely defined experimental conditions. For this purpose the observation of multi-photon fluorescence- i.e. a one-photon emission after a multi-photon absorption - has proved to be suitable for applications. This method was first applied to measurements of the intensity correlation function of RENTZEPIS, SHAPIRO and WECHT[1967]. Inlaser light by GIORDMAINE, terpretations of such results were discussed by WEBER[19681, WEBERand
111, $41
MULTI-PHOTON ABSORPTION
203
Fig. 16. Schema of a triangular TPF arrangement (M, M , , M, are mirrors, C is a cuvette with TPF sample, Ca is the camera).
DANDLIKER [19681, KUSNETZOVA [ 19691 and were summarized by BRAD[1977]. Fig. 16 shows a triangular optical configuration often used in such experiments. (For increasing the accuracy and simplifying the analysis we use an optical multi-channel analyzer instead of the camera.) This arrangement allows the determination of the dependence of the correlation function f * * ( O , T, T,0) on T = zlv (v is the light velocity in the medium) even for unique pulse-like signals, with a time resolution of about s. Beside two-photon fluorescence, higher order processes such as three-photon fluorescence have been investigated, mainly for the purpose of elucidating the asymmetry of light pulses from the intensity correlation function r3.3. This problem was first examined by WEBERand DANDLIKER [ 19681; more recent investigations have been performed by HAMAL,DARICEK,KUBECEKand NOVOTNY[1972] and by BAUMAN [ 19771. LEY
4.2. ALTERATION OF THE FIELD
So far we have considered only the measurement of given correlation functions of the field by means of transitions in the atomic systems. We will now discuss the alteration of the field due to multi-photon absorption. In analogy to the one-photon case one can derive the equation of
204
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 8 4
motion
+
N2[
d" (d+)"&- 2(d+)mfiFdm + &d "(d+)"B (52)
for the density operator &(t) of the field, using the Hamiltonian of eq. (46). The interaction constant q(") is proportional to the absolute value of the squared matrix element for the rn-photon absorption x'"'; N , and N 2 represent the population of states 1 and 2. (The irreversible approximation has been applied, that is, N , and N 2 remain at their initial values because of fast relaxation processes.) Equation (52) is a rather complicated relation; exact solutions of it are not yet known under general conditions. Starting from eq. (52) a set of differential equations can be derived, for the probabilities, p,,(t), that n photons are in the field. The following elementary considerations pertaining to two-photon absorption also lead to such a system of equations for p,,(t). The change of p,, with time is caused by 4 processes: (a) decrease of the photon number from n to n - 2 or (b) from (n + 2 ) to n with the transition of an atom from state 1 to state 2, (c) increase of the photon number from n to ( n + 2 ) or (d) from (n-2) to n with the transition of an atom from 2 to 1 (Fig. 17). Taking into consideration these processes and assuming N, atoms in the lower level and N 2 atoms in the upper one the relation d -pn =-q1'2'~,.(n-l)p,,-q'2'N2(n+l)(n+2)pn dt
Fig. 17. Change of the probability p, by two-photon processes.
111,141
MULTI-PHOTON ABSORPTION
205
is obtained (SIMAAN and LOUDON[1975]). The following set of equations for the moments ( a k ) of the photon-number distribution can be immediately obtained from eq. (53):
etc. It is obvious that the change of p,, depends on pn+’ and that the change of ( n k ) depends on the higher moment ( a k + ’ ) . This leads to difficulties regarding the solution of the system. A first insight into the variation of the probabilities or the moments with time can be obtained by using the short-time approximation. If NAq”’t<<1 it is possible to develop the solutions in powers of t and to neglect terms nonlinear in t. This means that the desired quantities are obtained from eqs. ( 5 3 ) or (54) by inserting the initial values into the right hand side of these equations. One then obtains, for example,
Often, the number of atoms in the excited state can be neglected, whereby with N 2 = 0 and N, = NA only 2 of the 4 processes considered in eq. (53) remain. Under these circumstances one obtains, for example, the following expression for the normalized correlation function with identical space-time points:
where T = q(”NAt. Some results appropriate to various important initial probability distributions are given in Table 1. (In the table the pulsed coherent field is a light field with rectangular pulses of length TL, the time To between adjacent pulses and a mean photon number (no)T,/TL during the pulse time TL, with (no) being the time average of the mean photon number.) Already from these approximate solutions the trend which
206
-7
,
N
&
0
v E
c,
L8
l=-"lc'
u r
I
rv
h
f N
c,
U N
v
r"
CI
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
h
.5
-+
I
-I$ -5 CI
" n
4c-'
+ "A P L
r"
A I
+
v
v S
N / .
111.441
MULTI-PHOTON ABSORFTION
207
affects the coherence properties is obvious. In particular, starting from a coherent state, a decrease of the normalized correlation function below unity is obtained. This means that the radiation shows an antibunching effect after the action of two-photon absorption. Assuming all the atoms to be in the ground state the set of rate equations for the p n ( t ) can be solved rigorously, by making use of a generating function method, as shown by AGARWAL [1970]. From the generating function Q, given by eq. (36) he passed to the generating function
by replacing ( 1 - A ) equation
by A' and t by
T.
From eq. (57) the differential
-aF F = [ ~ - ( A a' ) ~ ] ~ F a7
is obtained from the rate equation system (53), on multiplying the nth equation by (A')" and summing all equations. From the solution, obtained by the technique of separation of variables, the factorial moments
as well as the photon distribution
can easily be calculated. Fig. 18 gives, for example, the P , ( T ) for initial number state beams. The results differ significantly according whether (no) is or is not even. The strong influence of the quantum character on the two-photon absorption at low photon numbers can be seen, for example, from the fact that after a long interaction time the mean number of photons tends to zero or to unity for an initial number state beam according whether (no) is even or odd. For the same reason the degree of second-order coherence tends to zero or infinity. To get a clear insight into the change of the statistical behavior of a light beam that is initially in a coherent state, SCHUBERT and WLHELMI[19751 calculated numerically the time-dependent reduced variance ((Afi)')/(fi) as given in Fig.
208
COHERENCE OF LIGHT AND NONLINEAR OSTICAL PROCESSES
[III, 84
Fig. 18. Probability distribution P,(T) for an initial number-state beam containing 10 and 11 photons, respectively. (After SIMAAN and LDUDON [1975].)
19, the curve (with (A),=,,= 10) shows that this quantity considerably decreases with increasing time for N,q(')t < 0.2. To explain the physical meaning we compare the calculated reduced variance with values of some fundamental distributions: the Bose-Einstein distribution describing the chaotic field (value 1 +(A),=,),the Poisson distribution describing the
Bose -Einsteindistribution (chaotic state) Poisson distribution (coherent state)
0-
number state In>
Fig. 19. Change of the reduced variance ((A&)*)/(&) of a field by two-photon absorption (initial conditions: coherent state of the field; (N),=()= 10). (After SCHUBERT and WILHELMI [ 19751.)
111, B 41
MULTI-PHOTON ABSORPTION
209
coherent state (value l), and the distribution for the number state (value 0). The curve lies between the values of the coherent and of the number state (an appreciable deviation of -0.4 from the value of the coherent state already occurs at t = 0.2[NAq'2']-'). Additional exact calculations lead to the value 4(1 +exp [-2(fi),=,]) for t 3 a. Values between zero and unity characterize states where the photons show antibunching. In principle, the antibunching behavior cannot be comprehended in terms of an unquantized electromagnetic field, as we already mentioned in P 3.2. This result can be interpreted as follows: two-photon absorption leads from states which have no photon correlation to states with an anticorrelation (antibunching) effect. The problem of antibunching obtainable with multi-photon absorbers was discussed by CHANDRA and PRAKASH [1970], TORNAUand BACH[1974], SIMAAN and LOUDON [1975, 19781, EVERY [1975], BANDILLA and RITZE[1976a,b] and PAUL,MOHRand BRUNNER [1976]. The general case of m-photon absorption was treated by PAUL, MOHRand BRUNNER [1976]. Starting from the rate equations for p,,(t). differential equations for the higher moments of photon numbers were set up and solved by them. In interaction processes of higher order the antibunching effect becomes far more pronounced compared with twophoton absorption. For instance, the normalized second order intensity correlation function is given by
and this expression is less than 1 for m 2 2. BANDILLA and RITZE[1976b] calculated the influence of a multi-photon absorption within the laser resonator on the statistics of the generated light. However, up to now there have not been any realistic proposals as to the experimental generation of field states of low mean photon number which will show a measurable antibunching effect by means of multiphoton absorption (because of the very low efficiency of this nonlinear process at low intensities). When applying common photon counting techniques, the accuracy of the ratio u =((A&)')/(&) will be given by experirnentlal conditions. This parameter u takes on, for example, the values of 0 and l/(fi) for a coherent state and for a number state, respectively. Therefore, the changes of the parameter u to be measured are also of the order of l/(fi). It will, therefore, be difficult to find anticorrelation effects at mean photon numbers higher than 100 in the coherence volume. Moreover, the effect of antibunching obtainable with
210
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 14
multiphoton absorbers is reduced if one takes into account partially coherent driving fields. CHATURVEDI, DRUMMOND and WALLS [1977] presented calculations concerning two-photon absorption in a cavity with an external driving field EL(t)= [ vo + v,(t)] exp [i@(f)], characterized by phase and amplitude fluctuations for which (vl(t)vl(t’)) = a exp [-b It - t’l] and ( y ( t ) y ( t ’ ) )= I%(?- t ’ ) , with y ( t ) = d@(t)/dt.The following results were obtained :
1 4a y’*’(O)= 1--+-+{terms
3 ( N ) 914
of order I‘/(NAq(*’), a’/v;,
l/(fi)’)
(62)
and
+
Y’*’(T) = 1 [y’,’(O)
- l]e-7’T~,
with
The antibunching at T = 0 is seen to have decreased. To reach steady state conditions the extraction rate of the photons from the cavity must be less than the l h R , since (A) is limited according to the above considerations: Large values of NAq(’)are required.
4.3. PROPAGATION PROBLEMS
So far we have assumed in our discussion a bound system for the radiation maintained by a cavity. For the treatment of propagation problems we must consider the change in the coherence properties as functions of the space and time coordinates. A rigorous formulation and solution of this problem is very difficult. It was shown by SHEN[1967] that the stationary propagation problem can be treated in analogy to the cavity problem by replacing the time coordinate t by z/v ( z propagation length, u velocity of light in the medium). With this method the dependence of the coherence properties of light (represented for instance by correlation functions) on space coordinates can be calculated. In this way, all results of this section are applicable to stationary space-dependent processes. The propagation problem is dealt with more thoroughly in 0 6.1 for parametric interactions of quantized fields. In the case of
111, $41
MULTI-PHOTON ABSORPTION
21 1
multi-photon absorption with its normally ordered interaction Hamiltonian a much simpler classical description can be employed at high intensities. This means that the expectation values of normally ordered operators, such as the photon numbers and quantum-theoretical correlation functions are replaced by the respective classical quantities. The classical treatment proved to be useful in many applications even under nonstationary conditions. In this approximation multi-photon absorbers represent nonlinear filters by which the coherence properties of a given incident signal are changed. If intermediate resonant states are absent or if relaxation from such intermediate states to the ground state are very fast compared with the correlation time of the signal one has a nonlinear filter without memory. Then the relation between the intensity at the entrance z = O of the filter Zo(t) and at the exit z of the n-photon absorber Z(t) is given by
( N , is the number of absorbing particles per volume, a'"' is the crosssection of the process.) For a given probability distribution of the intensity at the entrance the corresponding distribution P(Z/(I))can be calculated (see WEBER[ 19711, SCHUBERT and WILHELMI [1975] and WILHELMI [1976b]). Fluctuations will be decreased due to the greater efficiency of the nonlinear process at higher intensities. In principle, this stabilization effect is far more pronounced with nonlinear effects of higher order (assuming an equal decrease of the mean intensity). However, difficulties arise concerning real conditions. In order to avoid the destruction of the medium, the field strength of the incident light must be smaller than the internal atomic field strength and the destruction threshold. Taking into account this serious condition, the obtainable transition rates diminish with increasing order of the nonlinear effect; this implies a decreasing influence on the statistical properties as well. It is, therefore, of importance, as mentioned above, that the efficiency of n-photon absorbers can be considerably enlarged in the presence of real intermediate states. Suitable experimental conditions have been discussed by SCHUBERT and WILHELMI [1975]. Figure 20 gives the intensity distribution and the normalized intensity correlation y2*2(0)= (I')/(Z)' at the exit of a 2- and 3-photon absorber, assuming Gaussian light at the entrance. The intensity stabilization of the three-photon absorber appreciably exceeds that of the two-photon absorber, as is seen in Fig. 20a. Figure 21 compares the
212 1
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES ~
"
"
"
I
I
~
I
[III, 14
I
Fig. 20. Intensity distribution P(Z/(I)) (a), and intensity correlation (12)/(Z)' (b) at the exit of a m-photon absorber ( m = 2,3) for Gaussian light at the entrance. (After SCHUBERT and WILHELMI [1975].)
0
0.2
0.4
0.6
0.8
1.0
tQ
Fig. 21. Influence of multi-photon absorbers on a fluctuating signal (10 =normalized time, S=(I)/(Z,,)=0.27): (a) entrance signal 1,); (b), (c) signal Z at the exit of a two-photon absorber and three-photon absorber, respectively. (After WEBER[19711 and WILHELMI [ 1976bl.)
IIL8 41
MULTI-PHOTON ABSORPTION
213
influence of a two- and three-photon absorber of the same mean transmittance T=(I)/(Io)on a signal fluctuating in time, as determined by WEBER[1971] and WILHELMI [1976b]. As was already stated in b 2 , the normalized second-order intensity correlation function y2*2 at identical space-time points may be regarded as a measure of intensity fluctuations; the value 2 corresponds to Gaussian behavior, the value 1 corresponds to completely intensity stabilized light. (In the classical description a value of 1 is a lower limit, in contrast with the results of quantum theory, given above.) It is obvious that the same stabilization effect for m = 3 can be obtained at an appreciably higher transmission in comparison with m = 2 . In general, multi-photon absorbers with resonant intermediate levels act as nonlinear filters with memory, and the memory will be negligible only if the energy relaxation times T oof the intermediate levels are short in comparison with the correlation time of the signal 7,. Usually, the influence of nonlinear filters with memory on statistical signals is rather complicated (cf. ZADEH[1952] and NEMES[1972]). As an example SCHUBERT and WILHELMI [1979] gave an approximate calculation of the change of the second-order intensity correlation function of a Gaussian signal with mean frequency oo in a three-photon absorber with an intermediate resonance at 2hw, (see Fig. 22), for different ratios between the relaxation time TI, and the correlation time of the entrance signal T ~ Figure 23 gives the derivative of the normalized correlation function y2.2 with respect to the effective transmittance 3= (I)&) in dependence on TIo/7,.This derivative, which characterizes the stabilization effect of the filter, assumes the value of 4 at T I O /=~ 0c (i.e. for filters without memory) and decreases with increasing relaxation time. Only for TI0/7,C 0.35 is the three-photon absorber with memory more effective in stabilizing the
I -
Fig. 22. Three-photon absorber with intermediate resonance at El - E,, = 2ho.
.
214
COHERENCE OF LIGHT AND NONLINEAR OpIlCAL PROCESSES
-
I I I I
0.2
[III, 54
Go/r,
I
0.4
0.6
0.8
3
Fig. 23. Derivative of the degree of second-order coherence y2.’ with respect to the transmission 9 of a three-photon absorber kith intermediate resonance as a function of TI&,. The dashed line gives the value of ay2.2(T)/aTfor a two-photon absorber without and WILHELMI [ 19791.) intermediate resonances. (After SCHUBERT
light intensity than a two-photon absorber without intermediate resonances. The corresponding relaxation times of such filters have been measured by WILHELMI, HEUMANN and TRIEBEL [ 19761. Their influence on the generation process of pico-second pulses has been measured by HEUMANN, TRIEBEL, WILHELMI [19761. Multi-photon absorbers have been used to stabilize fluctuating laser fields employed in measurements of nonlinear susceptibilities (KLEINSCHMIDT, RENTSCH,D. SCHUBERT and WILHELMI [1974]). KRASINSKIand DINEV[1976, 19771 have measured the dependence of the normalized second-order intensity correlation function of a light beam passing through a two-photon absorber on the input power. The spectrum of a broad-band dye laser source was centered at 490nm. (This is the peak of the two-photon absorption spectrum of a -chloronaphthalene.) Even when more than 90 percent of the laser beam was absorbed, the decrease of the correlation function turned out to be only about 20%. This indeed represents a significant deviation from the theoretical value, for the decrease of the correlation function should be nearly 50 percent. This difference might be explained by taking into account the high power density (approximately 5 GW/cm*), which seems to be high enough for the excitation of other nonlinear processes, such as self-focusing, filamentation, and stimulated scattering. Analogous influences of other nonlinear
III,§ 51
TWO-PHOTON EMISSION AND TWO-PHOTON LASING PROCESS
215
processes in the measurement of two-photon absorption have been observed in our laboratory. Moreover, one-photon absorption from the primary excited level or from levels populated by secondary processes can falsify the results (see, for example, KLEINSCHMIDT, RENTSCH, TOT~LEBEN and WILHELMI [1974]). KARCZEWSKI [1978] proposed similar measurements but performed with a very sensitive detection device constructed by his group, which allows the observation of two-photon absorption at excitation intensities below W.
4 5. Two-photon Emission and Two-Photon Lasing Process In connection with the observation of spontaneous and enhanced two-photon emission (YATSIV, ROKNI and BASAK [1968]), the investigation of the stimulated two-photon emission and the two-photon lasing process became important. The two-photon laser is expected to be useful for generating giant laser pulses with ultrashort rise times and pulse widths of the order of nanoseconds. Another feature of the two-photon laser is the possibility of generating new laser frequencies. Furthermore, the pulse amplification by stimulated two-photon emission is of interest for pulse shortening into the ps-range. On the basis of the rate equations the dynamical behavior of the two-photon laser was studied for the pulse regime by SOROKIN and BRASLAU [1964] and GARWIN [1964], and the stationary regime was studied by YUEN [1975a]. SCHUBERT and WIEDERHOLD [19791 analyzed the starting stage of the lasing process from the very beginning of the photon generation and they took into account spontaneous processes and a nonvanishing population of the lower laser level. The characteristic radiation states of the two-photon laser are the two-photon coherent states already mentioned in 0 2.2 (YUEN[1975b, 19761). These states cannot be obtained from other radiation sources available.
If loss processes are neglected, the operator
fi = fi,, + f i A 2 + fiF+ f i 1 , +&I2
(65)
may be regarded as the basic Hamiltonian for the two-photon laser. We will consider the degenerate case; fiF= hwd+h is the Hamiltonian of the radiation mode from which two photons with the same frequency o are simultaneously emitted and absorbed. This effect is associated with a
216
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 8 5
transition between energy levels of atomic systems ("atoms") of species 1 with the free Hamiltonian fiAl.The corresponding field interaction Hamiltonian is fiIl;the contribution of an individual atom to fil, may be described by (66) Here h,y; is the matrix element for simultaneous emission of two photons; it is proportional to x'~'*in eq. (46); 6:; is a flip operator, which flips the upper atomic laser state lu) to the lower laser state 14'). In contrast to the one-photon laser, the two-photon laser cannot originate from spontaneous emission alone; initially a sufficiently strong field at the lasing frequency o must be present. We will, therefore, adopt the following model: From host atoms (species 2), contained in the same cavity as those of species 1, the initial radiation is obtained by a one-photon lasing process, which is represented by the terms fiA2 and fil, for the free atomic Hamiltonian and the field interaction Hamiltonian. Of course, a population inversion has to be generated for both species of atoms. Altogether, it can be assumed that the total density operator factorizes into an atomic and a field part. The equation of motion for the field density operator fiF alone is then given by d ih - fiF = [Ei',&], df (fiIl)ind
where the Hamiltonian the atomic variables:
fi'= hod+&-fi[x:
= -hx;66:'(&')'+(h.C.}.
fi' is derived from fi by taking the trace over all
Tr (fi16~~)[4']'+{h.c.}]-h[x; Tr (fi26$$4++(h.c.}].
(68) fil and fi2 are the parts of the atomic density operator that are associated with the atoms 1 and 2; fil and fi2 can be assumed to be independent. According to general rules of quantum mechanics, a time development operator to) can be associated with the Hamiltonian I?'.' YUEN[1976] has shown that under general assumptions about the expressions xl Tr (fi,@;) and x; Tr (fi26y2),the operator to) implies a transition from an ordinary coherent state at to to a two-photon coherent state at t > to. SCHUBERT and VOGEL[1978b] discussed the spatial behavior of the two-photon coherent states Iy). The probability distribution of the measurable field strength is Gaussian at every space point r. The expectation value (y I E ( r ) Iy) is a sine wave of the form A sin [rolc + 471, where the amplitude factor A and the phase Q depend on the complex numbers
o(t,
o(f,
111,861
PARAMETRIC AMPUFICATION AND FREQUENCY CONVERSION
211
v, y (cf. eq. (20)), and on the real number ( h w / 2 ~ , V ) i The . mean square deviation ( y I (A&r))’ Iy) depends on. the spatial coordinate r ; it can attain values below the corresponding value of the vacuum state (01 8’(r) 10) within certain r-intervals. This property significantly differs from that of ordinary coherent states. So far we have not taken into account dissipation mechanisms. With regard to such mechanisms the following statements are valid. Far above threshold the radiation state of the laser approaches a two-photon coherent state. Below this limiting case external perturbations, due to dissipation mechanisms, require the averaging of the expectation values by means of a classical random probability function, for which the twophoton coherent states serve as basis. The intensity correlation behavior can be deduced from the quantity y’*’(O). Its value for a state Iy) is given by p,
For small Iv1 and appropriate values of the phases cpv, ( P ~ , cpv the 0 )1 occurs. The minimum value of y’*’(O) canantibunching case ~ ’ ~ ’ (< not reach the value corresponding to the photon number .state (HIROTA and IKEHARA [1976]). On the other hand, under certain conditions for u, p, y, values y2.’(0)z 1 can be attained and even an enhanced bunching effect with y’*’(O) > 2. The same result for spontaneous two-photon emission has been obtained by MCNEILand WALLS[1975] on the basis of an energy-spin model for the atoms.
8 6. Parametric Amplification and Frequency Conversion In §§ 3, 4 and 5 we dealt with effects in which the alteration of the radiation field is associated with the transition between real levels of the atomic system. In contrast to these situations parametric amplification and frequency conversion may be considered under the assumption that only photon annihilation and creation take place, without any change of the population of the atomic states. The interaction between several modes of the radiation field (and under certain circumstances the interaction of the field with a reservoir) may lead to interesting coherence effects, e.g., anticorrelation or antibunching effects.
218
COHERENCE OF LIGHT AND NONLINEAR OF’TICAL PROCESSES
[III, 86
6.1. GENERAL ASPECTS OF THE PROBLEM
For the description of representative cases of parametric amplification and frequency conversion in the lowest order, the following effective interaction Hamiltonian can be applied (SHEN[ 19671):
The indices 1, 2, 3 characterize three radiation modes with frequencies ~ 1 ~ 2 and . 3 wave numbers ql. q2, q3. The generalized susceptibility tensor x can be completely determined by second-order susceptibility components, used in the classical theory of nonlinear optics. The explicit representation of the field operators h(+), $-)in the Schroedinger picture, given by eq. (1l), leads to the expression
A, = - f i x ’
[v-’
d3r exp{i(q3-ql -qz)r}]d;dld3+{h.c.},
(71)
where the interaction constant x‘ is proportional to xele2e3, V-f and to the three factors ( ~ 1 , 2 . 3 / ~ 1 , 2 , 3 ) f(cf. SCHUBERT and WILHELMI[1971, 19781). If the phase-matching condition is fulfilled (as it will be assumed in the subsequent discussion), the expression in the brackets takes on the value unity; otherwise its absolute value becomes smaller. Conservation of energy implies the relation w 3 = wz+wl. We will call w1 and w 2 the sub-frequencies and w 3 the sum-frequency. Considering the ordinary parametric amplification, we will identify the modes 1 and 2 with the signal and the idler mode, respectively, and the mode 3 with the pump mode. The first term on the right-hand side of eq. (71) then obviously corresponds to a process in which one pump photon decays into one signal photon and into one idler photon. If the signal mode equals the idler mode, this term describes degenerate parametric amplification. By the same arguments it can be shown that the second term in eq. (71) (with the product cilci2d:) corresponds to frequency up-conversion; if both the sub-modes become equal one has the case of second harmonic generation. A straightforward calculation based on A, leads to the following equations of motion for the annihilation and the creation operators in the
111, § 61
PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION
219
Heisenberg picture: d dt
- ii,(t) = -ioliil(t)+iX’ii3(t)iii(t),
d dt
- ii3(t)= -io3ii3(t)+ix’iil(t)ii,(t). A more general treatment must take into account dissipation mechanisms, for which the same methods as for the description of lasing processes may be utilized (HAKEN and WEIDLICH [1966], HAKEN [1966]). Assuming that each of the three modes p = 1, 2 , 3 interacts, in addition, with a dissipative system, the total interaction Hamiltonian can be expressed as a sum of fir and of the additional term
The quantities R, R + are the annihilation and creation operators of the three dissipative systems (reservoirs), which are assumed to be boson systems with a broad-band quasicontinuous spectrum (the index I denotes the number of the reservoir mode). The influence of the reservoirs on the modes can arise from external random fields or from stochastic perturbations of the medium. It should be noted that the special form of the interaction operator fiR ensures that the absorption of one reservoir quantum (energy E l ) is associated with the simultaneous emission of one photon (energy hw, = E l ) and vice versa. From the total interaction Hamiltonian fiI+fiR one obtains equations of motion which differ from eqs. (72): on the right-hand side the expression ( y , / 2 ) 6 , ( t ) + k p ( t )must be added to each of the modes p = 1, 2 , 3. y, represents a real damping constant and fi, a fluctuation force; these forces are assumed to be independent of each other. The nonvanishing correlators of the fluctuation forces have a delta function-like time-dependence; under certain approximation conditions (PONATH and SCHUBERT [19761) the strength of the correlators may be taken as independent of the field operators. It has been pointed out in P 2.2 that the problems of the temporal development can be formulated with the help of the equations of motion for the operators h,(t) or, equivalently, by means of the generalized
220
COHERENCE OF LlGHT AND NONLINEAR OFTICAL PROCESSES
[III,8 6
Fokker-Planck equations for the distribution PA({a,}, t) or its Fourier transform, the characteristic function K A ( ( A , } , t ) . In general, the solutions of the equations of notion and those of the corresponding Fokker-Planck equation cannot be obtained in a compact form. Recursion procedures which, in principle, lead to solutions for the characteristic and generating functions for any time can be applied (PERINOVA and PERINA[1978]). As will be shown below, special physical conditions or assumptions can lead to compact formulas. Important information may often be obtained by applying the so-called short-time approximation; this is based on the expansion of the characteristic and generating functions in powers of the interaction time t, where the interaction is assumed to be switched on at ?=0.The validity of the resulting expressions with a finite number of terms depends on the convergence behavior; the coefficients of the powers of t have a physical meaning and depend on the initial coherence conditions, on the intensity of the incident waves, on the interaction strength etc. AGRAWAL and MEHTA [1974] have evaluated the time dependence of the density operator under various initial conditions of the pump and signal waves. So far we have considered the cavity problem. Real experimental situations have mostly to be regarded as propagation problems. SHEN [1967] has pointed out, that for quasi-stationary propagation in a medium the calculation can be carried out in the same manner as in the cavity case, with interaction time t replaced by z / u ; here z is the propagation distance and u the velocity of the light in the medium. Eliminating the (fast) main time-dependence exp [-iw,t] from the operators d,(t), one obtains slowly varying operators E,, which, equivalently, can be given in dependence on the propagation length z. With this replacement the equations of motion (72) become differential equations with respect to the spatial coordinate z for the desired solutions E,(z) (the resulting equations correspond to the so-called generating equations of the classical theory of nonlinear optics). In analogy to the above mentioned short-time approximation, the equations for the operators 2,(z) can be solved by applying the “short-path” approximation. For special experimental situations the convergence behavior has been estimated (KOZIEROWSKI, TANAS and KIEUCH[1978]); in the case of harmonic generation the short-path approximation is applicable within that range where the intensity of the generated beam does not approach that of the incident beam. It should be noted that the coherence properties (e.g., bunching or antibunching) in
IIL961
221
PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION
the short-time approximation may qualitatively differ from that of the long-time limit (WALLSand TINDLE [1971, 19721). On the basis of the relation between spontaneous and stimulated multi-photon processes (especially parametric effects and frequencyconversion) KLYSHKO [1977] proposed new methods for the measurement of brightness and radiation temperature of incoherent fields.
6.2. EFFECTS WITH NEGLECTED DEPLETION OF PUMP FIELDS
Let us first deal with parametric amplification. According to representative experimental conditions it will be assumed that the pump wave (index 3) is realized by a laser well above threshold. Its depletion caused by the parametric process shall be negligible. In accordance with these assumptions, the annihilation operator d 3 ( t ) in the Heisenberg picture can be replaced by the c-number Icz,~ exp [icp,(t)] exp [-iqt]; here ( c zis, ~the constant amplitude of the laser and the function cp3(t) reflects the timedependence of the laser phase. The following equation of motion for the signal wave (index 1) is then obtained from eq. (72): d dt
- 4 ( t ) = -io,h,(t)+iu(t) exp [-io,t]h:(t),
(74)
with u ( t ) = Ix'I Ia31exp [icp,.] exp [iq,(t)] and d:(t) is the creation operator of the idler mode. Next consider the limiting case d ~ , / d t= 0; i.e. the situation when u(t)=const. Under these assumptions eq. (74) can be solved exactly. The solution for d , ( t ) yields the following expression for the expectation number of signal photons:
-
in the range t hlul-'. The expression in the inner brackets [. .] shows that coherence effects come into play. Assuming initially coherent states for the signal and idler wave, the value of the expression in the brackets depends on the relation between the phase cpv of v and the phases cpl, cpz. If one or both the waves have randomly distributed phases, the expression vanishes; the same result holds if the signal and idler waves are [19771). Furthermore, from dl(t) the normalized uncorrelated (SCHUBERT second-order correlation function y2.'(0) of the signal wave can be
222
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, I 6
evaluated for sufficiently large t-values. Assuming a coherent initial signal state and a vacuum state for the idler, this function approaches the value of unity for a large initial number of signal photons and the value of two for a small number of initial signal photons (a detailed calculation including the case (A,(O))= (fi2(0)) = 0 has been carried out by CHMELA [1977]). In the first case the resulting signal light is coherent, in the second case it is incoherent. It should be noted that TRUNG and SCH~~?TE [1977] have presented a method for calculating the density matrix under arbitrary initial conditions for the signal and the idler state, with the help of the unitary time-development operator. We will now discuss the case of a real laser whose phase q3 is assumed to depend on t statistically. The t h e derivative dq,/dt = w is interpreted as the random variable (CROSIGNANI, DI PORTO,GANIEL,SOLIMENO and YARIV[1972]). Additionally the quantum-theoretical expectation values must be averaged with respect to the random variable w . Under the assumptions of slow phase variations (r.m.s. deviation of w much less than 1 1 ) the gain rate 2 1 1 , as given in eq. (75) for an ideal laser, must be replaced by the expression 2 1 1 (1 -(w2),,/2 \u12). This means that the gain rate of a real laser decreases with the increasing second statistical moment (w2),, of the random variable. Let us now discuss the degenerate parametric amplification in which the signal and idler are identical; the operator dl then equals ci2. Degenerate parametric amplification shows quite a different statistical behavior from the nondegenerate one. STOLER[1974] has studied the conditions for observing photon antibunching with the help of degenerate parametric amplification. Here the same assumptions that lead to eq. (74) for the signal operator d , ( t ) are used. Further assumptions are an ideal laser pump (dq3/dt =0) and an initial coherent state for the signal. The parameter u is assumed to be a real constant, which is proportional to the interaction constant and to the pump amplitude. Using the equation of motion for d , ( t ) the reduced variance can be calculated; it takes the form
where A(ut) and B ( v t ) are nonnegative functions. The reduced variance is unity at t = 0. Antibunching occurs if the variance falls below unity. There is only one way for obtaining such a result, namely if sin (2q, q3)< 0. The temporal behavior of the variance is schematically illustrated
111, 8 61
PARAMEIWC AMPLIFICATION AND FREQUENCY CONVERSION
I
0
0.35
I
-
223
vt
InYlhW
Fig. 24. Degenerate parametric amplification: Reduced variance in dependence on time.
in Fig. 24 where the optimum phase relation sin (2q1- (p,) = -1 (equivalent to 2(p1- (p, = - d 2 ) is chosen and the initial number of signal photons exceeds 10. The curve shows the following features: Its minimum value (-0.75) is attained at u t x 0 . 3 5 and it becomes unity again at u t = In [2 (fil(0)):y, for larger t it remains greater than unity. From calculations of the higher factorial moments carried out by MISTA,PERINAand PERINOVA [1977] it must be concluded that the initial coherent state cannot be reached again with increasing time, although the reduced variance ([Afil]*)/(fil) becomes unity for a certain value t > 0.3%-’. An experimental realization of the process might proceed as follows: A strong cw-laser produces second harmonic light. Both the fundamental and the harmonic wave enter the material, where the harmonic wave serves as the pump and the fundamental wave as the initial signal wave. In order to describe the spatial behavior we must replace ut by the dimensionless product u’z, where z is the propagation distance. A suitable crystal seems to be barium sodium niobate with v ’ = 9 x 10-4(1,/W)4. With a I,-value of lo4 Wcrnp2 the optimum value for antibunching u’z ~ 0 . 3 5is reached at z = 4 cm. It should be noted that even smaller z-values might lead to a considerable antibunching effect (compare Fig. 24). The proposed antibunching observation has the advantage, in comparison with this effect in the case of the two-photon absorption, that the mean photon number may be quite large. However, it should be emphasized that the observability of antibunching depends crucially on the relation between the phases cpl and (p,. The occurrence of fluctuations of realistic radiation sources must be taken into consideration. A monitoring of the relative phases seems to be possible. The
-
224
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 86
theoretical analysis of the problem including loss mechanisms, with the PERINA and pump again treated classically, was given by MISTA,PERINOVA, BRAUNEROVA [19771. From the numerical results, the time-dependent reduced variance can be calculated; the corresponding curve shows the same features as that in Fig. 24 if the damping constant y ( y = y1= y2) is chosen to be zero. An increasing damping constant y leads to a decreasing antibunching effect; for sufficiently large y the antibunching effect completely vanishes. Frequency conversion with a real laser pump follows the above pattern concerning the parametric amplification. We will deal with an upconverted signal, with the frequency w3 = w1 + w2. It is to be noted that the third relation of eqs. (72) must now be considered instead of eq. (74); the index 3 characterizes the signal wave, whereas index 1 is associated with the pump wave u ( t ) exp [-h,t], with u ( t ) = Ix’1 lall exp [icp,.] exp [icpl(t)]. We again assume slow phase variations of the pump; this means that the r.m.s. value of dq,/dt is much less than 1111. In Fig. 25 the influence of the statistical properties of the pump on the averaged value of the signal photon number ((fi3(f)))av is schematically illustrated. The solid line represents the case of the fully coherent pump. A complete energy exchange between the signal and the idler wave occurs periodically. The dashed line represents the case of slowly varying phases; asymptotically, the exchange becomes negligible and the mean photon numbers of the signal and the idler become equal. Moreover, the problem was treated under the assumption that the correlator of the fluctuations of the pump amplitude factor v ( t ) is delta functionlike and that damping
0
4x
63t
Fig. 25. Frequency up-conversion: Influence of phase fluctuations dq,/dt = o. [(After CROSIGNANI, DIPORTO, GANIEL,SOLIMENO and YARN [ 19721.)
111,861
PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION
225
terms are introduced into the equation of motion (KRYSZEWSKI and CHROSTOWSKI [1977]). At certain values of the correlator strength and the damping constant the signal wave exhibits antibunching. More recently frequency up-conversion has been investigated in metal vapors. The atomic motion in the gaseous matter influences the photon statistics; this aspect was analyzed for a second-harmonic field in a single-mode cavity by NAYAK and MOHANTY [1977].
6.3. GENERAL CORRELATION BEHAVIOR
Based on the concept outlined in § 6.1 for the treatment of propagation problems, the quantum fluctuations of second-harmonic generation had been analyzed by KOZIEROWSKI and TANAS [1977]. The modes 1 and 2 become equal and represent the fundamental wave (frequency q), whereas the second-harmonic wave has the frequency w 3 . Introducing the slowly varying operators E instead of 8, the following equations can be derived from eqs. (72): d dz
- E,(z)= 2i~*E:(z)t~(z),
d dz
-E,(z)=i~E:(z).
(77)
It is assumed that the phase-matching condition is fulfilled. The interaction constant K is proportional to x’. In contrast to the discussion in § 6.2 we will now take into account the alteration of the fundamental wave; thus, the solution of eq. (77) will be obtained by expansion in powers of z . Let us assume that at z = 0 only photons of the fundamental wave exist, whereas (E3(0))is assumed to be zero. With these initial conditions the power series for E,(z) and & ( z ) can be calculated. From these expressions the correlation functions
indicates that the spatial and can be evaluated. The argument 0 in Y’*’ temporal coordinate of the 21 field operators do not differ from each other (6.§ 2.1). However, via the z-dependence of the 12- and the ?+-operators the correlation functions Y‘.’ (0) vary with the propagation length z. As regards the occurrence of a bunching or antibunching effect it is interesting to know the expressions A2r1.3(z)= fi:i(O; 2)[Fi:i(O;z)]’(cf. § 2.1). From the power series for E,(z) and &(z) the
226
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, 16
power series for A2r(z) can be derived. For the fundamental wave the following results are obtained:
Equation (78a) holds for the case of an incident coherent wave. As it can be seen from the sign, the fundamental wave in the short-path approximation shows an increasing antibunching behavior with increasing z. Equation (78b) holds for an incident chaotic wave. Of course, the wave shows a bunching effect at z = 0. However, with increasing z this effect diminishes. It should be emphasized that the coefficients of the powers of z depend on the operator character of E and E' (commutation rule [t,E+] = f), which involves the quantum properties of the physical system under discussion. The corresponding results for the second-harmonic wave are:
Equation (79a) refers to an incident coherent wave, eq. (79b) to a chaotic one. Taking into account that the number of second-harmonic photons is assumed to be zero at z = 0, eqs. (79) reflect qualitatively the same effect as has been found for the fundamental wave; with increasing z a tendency towards antibunching and diminishing of bunching, respectively, becomes apparent. With the same assumptions that lead to eqs. (72) and including damping terms and fluctuation forces, PERINOVA and PERINA [1978] formulated the corresponding Fokker-Planck equation for the characteristic function K*({&}, t ) . Using the short-time approximation, the solution up to the terms of the order t2 has been given explicitly. Now a physical insight into the problem will be provided by considering measurable quantities for
111, § 61
PARAMETRIC AMPLIFICATION AND FREQUENCY CONVERSION
227
special physical situations. Such measurable quantities are represented by
A2r,(t) =
(%) 2E0v
2
[(dL’(t)ii’,(t))- (iiL(t)ii,(t))”]
and
-(a;(t)h,
(t))(ii;w,*(o)l.
The short-time approximation up to the terms t2 leads to the following relatively simple expressions, if the losses are neglected. Provided that the initial modes are coherent states (a1),laz),( a 3 )one has
A’r,(t)
=
hw 2EoV
2 (x’I2Ia,l2 la3I2t2 for p = 1,2,
A2r3= 0,
(80a) (gob)
h2W,W2
(Arl(t))(Ar2(f)) = [{ix’*aTa;a3t+{c.c.)) 4&ov +Ix’I2[(1+2 bi12+2 I.z12)
(Ar,(f))(AI‘3(f))
hw,w, =r lx’lz la,(2la3I2t2 4&ov
Ia312-Iai12 Iaz121t21,
for p = 1,2.
(80~) (god)
Before discussing these relations we will recall that the indices 1 and 2 indicate the sub-frequencies, while the index 3 characterizes the sumfrequency. Equation (80a) shows a bunching effect of the sub-harmonics due to the interaction in case of a nonvanishing initial number of the sum-frequency photons. On the other hand, in this order the coherence properties remain unchanged for mode 3 (if one considers higher powers of t, results similar to those of eq. (79) are obtained). The first term on the right-hand side of eq. (80c) involves the possibility of the occurrence of an anticorrelation effect between both the sub-harmonic frequencies. The existence of this effect depends on the phase relation between the three complex amplitudes aI, a2,a3 and x’. We already showed in connection with eq. (75) that such phase relations play an important role for the explanation of coherence properties. Note that the right-hand side of eq. (80c) contains an anticorrelation term, even if Ia3l2vanishes. The interaction between one sub-harmonic and the sum-frequency wave leads to an anticorrelation effect (cf. eq. (80d)).
228
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, B 6
If a loss mechanism is taken into account, additional terms appear (PERINOVA and PERINA [1978]). We might mention a very interesting aspect, namely that the tendency towards anticorrelation can be increased by loss quantities. As an example let us consider the sum-frequency generation on the assumption that the mode 1 is coherent light and the mode 2 is chaotic light with mean photon number (&). It then follows that
An increasing photon number (&) is connected with an increasing anticorrelation effect in this order of the approximation. Qualitatively the same statement holds for subfrequency generation if \all = 0. Similar results have been obtained for higher order processes of harmonic and sub-harmonic generation which are connected with interaction Hamiltonians of the form [-ttx"ii:Ci~+{h.c.}], with 1 > 2 (PERINA, and KNESEL [1977]). The interaction of three waves generated PERINOVA by a coherent pump has been investigated on the basis of a parametric cascade process, by ILINSKY, KLYSHKO and PETNIKOVA [1975], with regard to the mutual coherence of these waves.
0
QOY
a08
0.42
s/m m
Fig. 26. Measured mutual coherence function l-y'.'(s)l as a function of the path difference s for different angles p (KDP). (After AKHMANOV, GOUAEV, TUNKIN and CHIRKIN [1975].)
111, 5 71
STIMULATED RAMAN SCATTERING
229
AKHMANOV, GOUAEV,TUNKIN and CHIRKIN [19751 have studied both theoretically and experimentally the spatial coherence properties of second-harmonic radiation. By measuring Michelson's visibility they determined the modulus of the normalized mutual coherence function ly17'(xl,xz)\ (cf. 0 2.1). Besides other results, they found a strong influence of the orientation of the crystal (KDP) on xz)l. The dependence of this quantity on the path difference s for different angles p with respect to the axis Y was measured (see Fig. 26). Experimental investigations of parametric generation of coherent light by means of a spatially non-coherent pump have been carried out by BABIN,BEUAEVA, BEUAEV and FREJDMAN [1976]; the ratio of the correlation lengths of the resonant and the non-resonant radiation determines the results.
J 7. Stimulated Raman Scattering A very instructive early example concerning the influence of coherence properties on nonlinear processes was given by FREEDHOFF [ 19671 more than 10 years ago. She showed that the positive generation rate for stimulated anti-Stokes scattering can be explained only using the assumption of the existence of sufficiently coherent fields. In that direction, for which the momentum conservation law is satisfied, a positive anti-Stokes gain only occurs if a certain inequality holds for the phases of the pump, the Stokes, and the anti-Stokes waves. This requirement can only be met by sufficiently coherent fields, with small phase fluctuations. As to the description of the stimulated scattering by elementary excitations of matter, the same procedure as in the case of parametric amplification can be used (SCHUBERT and WILHELMI [1974]). In the interaction Hamiltonian the idler mode has to be replaced by the mode of the elementary excitation of the medium:
A, = -hx:ciLci:ci,
+{h.c.}.
(82)
Here ci;, ci: are the creation operators of the elementary excitation mode and the Stokes mode of the field, and ci, is the annihilation operator of the pump mode. The relation up= w,+ wE is assumed to hold. If the elementary excitations of the medium can be treated as bosons, all the calculations remain unchanged. As an example, scattering by polaritons
230
COHERENCE OF LIGHT AND NONLINEAR OPTICAL PROCESSES
[III, § 7
will be discussed. The following conditions shall be assumed: two powerful coherent beams enter the crystal. Thus, the Heisenberg operators ci,(t), ci,(t) can be replaced by complex c-numbers with a constant modulus :
At r = O the elementary mode is assumed to be in the vacuum state. Under these conditions the time development of the elementary excitation is obtained as the coherent state la&)), with a E ( f=)i,y'apaet for t > 0; this means that from the very beginning of the two-beam interaction process, a state with full coherence occurs. From this result it has been concluded in some papers (see, for example, BIRAUD-LAVAL, CHARTIER and REINISCH [19713) that for this reason the classical description can be used immediately, instead of the more complicated quantumtheoretical formalism. Later on the problem of the transition from the quantum-theoretical to the classical description has been studied in detail by SCHUBERT and VOGEL[1978a], on the basis of a solution of the eigenvalue problem of the electric field operator (SCHUBERT [1968]). An equation of motion for the density operator in a representation employing the eigenstates of the electric field has been derived. Under conditions determined by the actual physical problem, the solution of this equation allows an unambiguous comparison between the classical quantities and relations with the corresponding quantum-theoretical quantities and measurable values. This comparison is of importance with respect to the temporal behavior of the coherence properties. In the case of scattering by polaritons, a physical insight into the problem can be gained by considering the fluctuations of the electric field. In Fig. 27 the normalized expectation value of the field fluctuations is plotted versus time. At the beginning, large values arise due to the unavoidable quantum fluctuations. With increasing time the relative fluctuations attain small values; hence, in this region the electric field associated with the elementary excitation may be regarded as a prescribed function of time and a classical description is then possible. It can be seen that the lower bound of the time interval, which allows a classical description, decreases with an increasing interaction constant and with increasing mean photon numbers in both the incident light beams. Statistical effects appearing in stimulated Raman scattering under nonstationary conditions, for instance the development of the Stokes and the
111, 571
STIMULATED RAMAN SCATlEFUNG
23 1
Fig. 27. Polariton scattering: Dependence of the normalized expectation value of the field fluctuation on time.
anti-Stokes signal from noise have been treated (see, for example, AKHMANOV and CHIRKIN [19711, AKHMANOV, DRABOVICH, SUKHORUKOV and SHEDNOVA [1972] and HERRMANN [1977]). In stimulated Raman scattering, photon number correlations within one mode or between several modes of the radiation field can be calculated in the short-time approximation in the same way as was discussed in treating multi-photon absorption and parametric effects (see, for example, WALLS[1970, 19731, LOUDON[1973], SIMAAN [1975] and PERINA [1978]). Using the interaction Hamiltonian of eq. (82) and assuming the excitation mode to be in a chaotic initial state, with a mean number of excitations (&, and also assuming the pump and Stokes radiation to be in coherent states with complex amplitudes a, and as,the following results are obtained:
Treating the interaction between laser mode, Stokes- and anti-Stokes modes simultaneously by using the interaction Hamiltonian
232
COHERENCE OF LIGHT AND NONLINEAR OFTICAL PROCESSES
[111
+
(where XA is the susceptibility of the anti-Stokes process with wA = wp wE), additional correlation effects can be obtained;
(cf. PERINA[1978]). That photon bunching increases with time is obvious for each single mode; anticorrelation effects may arise between several modes. This is in agreement with results obtained by S C H ~ E , TANZLER and TRUNC [1978]. As regards higher order Raman scattering-, the so-called hyper-Raman process, photon anticorrelation has been calculated by SZLACHETKA and KIELICH[1978] who employed the same method. ALTMANN and STREY [19771 and STREY[1978] investigated the scattered intensity in dependence on the statistical properties and the polarization of the incident light. As with multi-photon absorption an increasing amplification factor appears for incident incoherent waves. The development of nonstationary spontaneous and stimulated Raman scattering processes is substantially affected by the fluctuation correlators, arising from the relaxation processes in the medium. In addition to their dependence on the properties of the medium (longitudinal and transversal relaxation times, temperature of the reservoir), these correlators also depend on external forces, such as the field strengths of the incident beams. In the framework of this concept, nonstationary time- and spacedependent spectral densities had been incorporated into the correlation functions of the fields which influence the coherence behavior (PONATH and SCHUBERT [1976, 19771).
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E. WOLF, PROGRESS IN OPTICS XVII @ NORTH-HOLLAND 1980
IV
MICHELSON STELLAR INTERFEROMETRY BY
WILLIAM J. TANGO Chanerton Astronomy Department, School of Physics, Unioersity of Sydney, N.S.W. 2006, Australia
and R. Q. TWISS C/OOptics Section, The Blackett Laboratory, Prince Consort Road, London, SW7 ZBZ, England
CONTENTS PAGE
5 1. INTRODUCTION
. . . . . . . . . . . . . . . 241
5 2. THE MODERN MICHELSON STELLAR INTERFEROMETER §3
. . . . . . . . . . . . . . . . . 243
. THE BASIC THEORY OF THE MICHELSON STELLAR INTERFEROMETER . . . . . . . . . . . . . .
247
5 4 . THE EFFECTS OF ATMOSPHERIC TURBULENCE ON A MICHELSON STELLAR INTERFEROMETER.
. . . 255
5 5 . THE TILT CORRECTING SERVO SYSTEM . . . . . 264 5 6. SUMMARY AND DISCUSSION . . . . . . . . . . 268 APPENDIX A: THE PHOTON COUNTING STATISTICS . . . . . . . . . . .
. . . . . . 270
APPENDIX B: THE ANGLE OF ARRIVAL SPECTRUM . . . . . . . . . . .
. . . . . . 273
ACKNOWLEDGEMENT . REFERENCES
. . . . . . . . . . . . . . 276
. . . . . . . . . . . . . . . . . . 276
8 1. Introduction Although FIZEAU[18681 suggested more than one hundred years ago that interferometry could be used for measuring the angular sues and separations of astronomical objects, throughout most of this period the technique has found only limited employment. There are two major reasons for this: (a) until recently it has been very difficult to maintain the instrumental precision and stability needed to obtain interference reliably; and (b) atmospheric turbulence produces large and rapidly varying distortions of the arriving wavefront. These distortions give rise to the phenomenon of seeing, i.e., the degradation of image quality observed in conventional telescopes. MICHELSON avoided these difficulties in his first stellar interferometer [1891] by placing a mask with two small holes over the aperture of a conventional telescope. This method was subsequently improved upon by Michelson and others. FINSEN [1951, 19541 developed the “eyepiece” interferometer still further and used it for his monumental binary star work. FINSEN’S [1971] review gives a good account of the difficulties associated with visual interferometry. More recently WICKESand DICKE [1973, 19741 have also used this approach. Since the angular resolution of an interferometer is approximately AlD, where D is the baseline separating the two apertures, an interferometer constructed by masking a telescope has only a modest resolution (of the order of a few hundredths of an arcsecond). MICHELSON and PEASE[1921] built an interferometer with two separate apertures on a beam attached to the front of the telescope; in this way they increased the resolution and were able to resolve the supergiant a Orionis. Larger instruments were built but the difficulty in maintaining the required stability limited their usefulness. The effects of instrumental and atmospheric instabilities are particularly detrimental to amplitude interferometers, i.e., ones in which the light from two separated points on a wavefront is linearly added together and then “mixed” in a square law detector. The discovery of intensity 241
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MICHELSON STELLAR INTERFEROMETRY
[IV, 5 1
interferometry (HANBURY BROWN and Twiss [1956a]) offered a way around these obstacles, and the resolution of a Canis Majoris (Sirius) confirmed its astronomical potential (HANBURY BROWN and TWISS [1956b]). A large intensity interferometer was then built in Australia (HANBURY BROWN [1974]) which was capable of resolving stars with angular diameters less than 1xlOP3 arcseconds. This instrument was used to determine the angular diameters of 32 stars, with an accuracy of a few percent in some cases (HANBURY BROWN, DAVISand ALLEN[1974]). The intensity interferometer has one notable drawback when compared with an amplitude interferometer, namely, a poor signal to noise ratio (TWISS[19691). Thus the Australian intensity interferometer had a limiting magnitude of about +2 while using flux collectors with a combined area of 60m2. Although it is feasible to build larger interferometers having greater sensitivity, they would be enormous and cost would become a limiting factor. As a consequence amplitude interferometry was again considered and a number of proposed instruments were put forward (National Academy of Science [1967]). Over the past ten years stellar interferometry based on amplitude interference has taken two courses. The first, which will be the main concern of this paper, utilizes relatively small apertures. In this context “small” means of the order of Fried’s correlation length r,, which is usually taken to be about lOcm (FRIEDand MEVERS[1974]). It seems appropriate to retain the name “Michelson stellar interferometer” for instruments based on this principle, since they are analogous to Michelson’s original interferometers. At least two such stellar interferometers and TANGO have been built (CURRIE,KNAPPand LIEWER[1974]; TWISS [19771). The second approach to stellar interferometry is speckle interferometry, a technique developed by Labeyrie (LABEYRIE [19701; GEZARI, LABEYRIE and STACHNIK [ 19721). Speckle differs from Michelson interferometry in that quite large apertures (of the order of one meter) are used, and an essentially statistical procedure is used to recover information from the highly aberrated images that result when large apertures are employed. LABEYRIE [1976] has reviewed the various techniques of modern stellar interferometry, and more recently a colloquium on high resolution stellar interferometry was sponsored by the International Astronomical Union (DAVISand TANGO [1979]); the relative merits of the different approaches to the problem of high angular resolution astronomy were extensively discussed at this meeting.
IV, 5 21
THE MODERN MICHELSON STELLAR INTERFEROMETER
243
In this paper we wish to confine ourselves to the problems that face the designer of a long baseline Michelson stellar interferometer. As we have indicated the difficulties are twofold: instrumental stability and atmospheric turbulence. Advances in optical technology, and especially the development of stabilized lasers, have largely removed the first difficulty, and therefore we shall not consider this aspect in any detail. The effects of the atmosphere, however, must be looked at more closely. The problems caused by the turbulent atmosphere are made more acute by the astrophysicists' demand that the fringe visibility be measured with an accuracy of 1-2% (DAVIS[1979]). This requirement means that the overall signal to noise ratio and the potential sources of systematic errors in any proposed instrument must be carefully evaluated. As we shall see, the necessity of high precision will be the chief factor in determining the limiting magnitude of the stellar interferometer.
8 2. The Modem Michelson Stellar Interferometer There is general agreement on the basic features which would characterize any two-aperture, long baseline amplitude interferometer, and we briefly summarize them in this section. The instrument we shall describe is based largely on the interferometer of the Royal Observatory, Edinburgh (TANGO [1979b]), and the instrument now under construction by the University of Sydney (DAVIS [1979]). The basic plan of the interferometer is shown in Fig. 1. Two siderostats (i.e., plane mirrors which reflect the incident starlight in a constant direction), having effective circular apertures of diameter d and located at fixed stations separated by the baseline distance Do, direct the starlight into the interferometer via suitable transfer optics (there would normally be beam-reducing telescopes in the transfer optics in order to reduce the beams to a convenient size). In general the light from the star will arrive at the oblique angle 0,and it is apparent that in the absence of atmospheric effects there will be a relative delay in the arrival times of the wavefront at the two siderostats of At = c-'Do cos 0,while the projected baseline as viewed from the source is D = Do sin 0.Since the delay time between the two interfering wavefronts must be kept much smaller than the coherence time of the light (set by optical filters in the instrument), some form of path compensation must be used. As usually envisaged, the compensators would consist of variable optical path length sections in each channel: these mieht be. for examole. retroreflectors which can he
244
MICHELSON STELLAR INTERFEROMETRY
[IV, I 2
Fig. 1 . Section through a long baseline Michelson stellar interferometer. S , . 2 are the siderostats which direct the starlight into the main instrument, which is shown in Fig. 2. The baseline length is D , and D is the projected baseline.
precisely moved and positioned on tracks. Tbe required compensation is determined by 0,which - again ignoring the atmosphere - depends solely upon the coordinates of the saurce and the siderostats themselves. Auxiliary laser interferometers can be employed to accurately control the path compensation with errors of the order of a few parts in 10'' or better (HALL[1978]); as well they can be used to monitor the variation in the distance Do due to tides, seismic activity, etc. The design and construction of the siderostats, compensators, and transfer optics are essentially problems of precision optical engineering, and although challenging, should be within the capabilities of modern technology. Two points about the construction which may not be immediately obvious should be noted, however. Care must be taken to insure that the light passes through the instrument symmetrically, for otherwise the polarization states of the interfering waves could differ significantly. Additionally, because of the very long paths involved, diffraction effects inside the interferometer may cause small but not and TWISS [1974]). negligible errors (TANGO We turn now to an examination of the central station of the interferometer, which is shown schematically in Fig. 2. For simplicity, the optics for the various guidance and alignment systems are omitted. Light from the two arms of the interferometer is directed towards the main beamsplitter B,, by the mirrors W,, W2 and S,,S2. The beamsplitting cubes b, and b2 are polarizing beamsplitters which transmit the
IV, 8 21
THE MODERN MICHELSON STELLAR IhZRFEROMETER
245
Fig. 2. Layout of the main interferometer. Light enters from X , and XI. After passing through the path compensating system the beams are reflected by the “wobbler” mirrors W,,2 and the phase switching mirrors Sl,2. The beams interfere at the main beamsplitter B,,. Fl,2are filters and Dl,2are the photon counting detectors. The polarizing beamsplitters bI,* deflect one component of the light to the guide sensors G , , 2which control the wobblers.
parallel (p) component and reflect the perpendicularly polarized (s) component. The light is incident on Bo at nominally the Brewster angle, but the angles of incidence will fluctuate because of guidance errors, and more importantly because of atmospheric turbulence. These fluctuations are monitored by the angular position sensors G I and GZ,which are used to drive the “wobbler” mirrors W, and W2 by means of a suitable servo system. The mirrors could conveniently be driven by piezoelectric actuators over a range of about *20. Because of the unavoidable path length errors which will exist due to atmospheric effects, uncertainties in the computed value of the path compensation, and various internal path errors, one must limit the optical bandwidth in order to see any interference. The filters F, and F, are assumed to be ideal devices which pass light in a selected band around the operating wavelength A”.
246
MICHELSON STELLAR INTERFEROMETRY
[IV, I 2
The interfering beams fall on the photon counting detectors D, and D2, and their outputs are used to determine the fringe visibility. If one were to view the source from the position of one or the other detector, in general two images would be seen separated by the angular distance be, which represents the difference in the angles of incidence of the beams at the main beamsplitter. If now one were to focus not on the source but on the entrance pupils, one would see the two apertures slightly displaced from one another, and the mutually illuminated field would be crossed by characteristic “tilt” fringes of interference, of spatial period h,/Ae. In the usual way we define the fringe visibility or contrast to be Imax- Imin
= I,,,
+ Imin
where I,,, and Iminare respectively the intensities of the bright fringe maxima and the dark minima, respectively. In the absence of atmospheric and instrumental effects which conspire to reduce the apparent visibility, a knowledge of IyI for all possible lengths and orientations of the baseline allows us to determine the source distribution.* The actual method of determining the visibility from the observational data is a distinguishing feature of the various proposed stellar interferometers. The different techniques are often analogous to methods used in radio interferometry (BLUMin [1959] was apparently the first to note the applicability of radio methods to the entire electromagnetic spectrum), and the approach we follow is based on RYLE’S[1952] phase switched interferometer. If the tilt angle A0 is made zero, the two beams of interfering light will be uniformly illuminated, and the total light reaching the two detectors will be proportional to l+lylcos@ and 1-1yI cos @, where @ is a phase which includes among other terms one proportional to the path difference. Since it is nearly impossible to determine the path errors with an accuracy equal to a small fraction of a wavelength, we must regard the phase as an unknown and essentially random variable. The unwanted phase may be eliminated as follows. We divide the observing period into very many short sample periods. The length of these fundamental periods is chosen so that the variations in the phase
* Strictly, the visibility function JyI determines only the source autocorrelation function rather than the irradiance distribution of the source. There is evidence, however, that it may be possible to reconstruct the source from the visibility alone (FEINUP[1978]).
IV,$31
BASIC THEORY
247
during a single sample can be ignored (the period will be of the order of a few milliseconds). To optimize the signal to noise ratio, photon counting techniques must be used, and we let n, and n2 be the number of photons counted by the detectors D, and D2 during a single sample. We form q = (n, - n2)2, which is proportional to lylz cos2 @, and average it over many samples (-10’). We thus obtain a quantity proportional to Iy12cos2@,and if the phase is rapidly changing, this average reduces to IYI2/2. We may completely eliminate the phase by a simple technique. In alternate sample periods we displace the mirrors S , and S2 by piezoelectric transducers and thereby introduce a path difference of Ao/4. Because of the phase switching, the average of q will be proportional to IyI2 (cos2Gi +sin2 Qi+,),where @i is the phase during the ith sample. If the phase changes but slowly over two samples, q becomes independent of the phase. The above brief discussion of the measurement of the fringe visibility does not take into account either the photon counting statistics or the effects of atmospheric turbulence. These will be covered in the following sections.
8 3. The Basic Theory of the Michelson Stellar Interferometer 3.1. THE QUASI-MONOCHROMATIC THEORY
We summarize here the basic results of the theory of partially coherent light as applied to stellar interferometry. A rigorous justification of our rather simplified treatment may be found, for example, in BORNand WOLF’S[1975] text. It is convenient to divide the analysis into two parts: we first determine the distribution of the amplitude of the incident starlight on a plane P just outside the atmosphere, and then consider the propagation of the light from this plane to our detectors. Figure 3 illustrates the geometry. For convenience we choose as the z-axis the geometrical light ray of wavelength An from the center of the source to the midpoint of the baseline Do. Outside the atmosphere we take a plane P perpendicular to the z-axis. The x-axis is defined by the intersection of P with the plane containing the baseline and source. The x-z plane also defines a great circle on the celestial sphere, and we use 6
248
MICHELSON STELLAR INTERFEROMETRY
[IV, 5 3
Fig. 3. Definition of coordinate axes. D, is the interferometer, and u is the source on the sky. a-a denotes the top of the atmosphere, and P is a plane perpendicular to the plane containing the source and D,).
to measure distance along this circle from the center of the source. The q coordinate is measured along the orthogonal great circle passing through the source center. In the vicinity of the source it will be permissible to regard the curvilinear coordinates (6, q ) as Cartesian. The irradiance produced on P by a point 5 = (6, q ) on the source will be denoted by E ( & v ) , where v is the optical frequency. Because of the remoteness of the source, the amplitude of the electric vector of the light at P (in a given polarization state) is very simply determined, and is proportional to: V(& x, v, t ) =
exp (27rivt - ik&* x + iJI(&,t ) } ,
(3.1)
here k = 2 7 r c- l ~is the optical wavenumber and JI(& t ) is a phase which is independent of the observation point x. Consider now the rays passing through the point x which reach some point X’ on one of the detectors (say D,). In general because of projection
IV, 8 31
249
BASIC THEORY
effects rays reaching the detector through the two apertures will not arrive at the same point x’, so that the interferometer will possess shear. Although the practical effects of shear would be fairly small, it greatly complicates the mathematics. Accordingly we assume that the optics are designed in such a way as to remove shear (a simple way of avoiding the problem is to introduce an extra reflection in one arm only of the interferometer). One can then choose a coordinate system (x’, y’) at one of the detectors so that y f= y and depending upon which aperture the sky is viewed through x’ = x + D/2 or x’ = x - D/2, where D is the projected baseline Do sin 0. The complex amplitude at the detector D, will be given by V, (we omit the phase shifts at the beamsplitter):
V,(&x’, v, t ) = JE(s;E;jexp (2wivf +ik(&’+ qy’)+ +(f
t)}
x[exp{ik(ED/2+Zi& x, v, t ) + I , +D/2)1
+ exp {ik(-5D/2 + Z,(& x’, v, t ) + 1, - D/2)}1,
(3.2)
where Z, and Z, are the optical paths through the atmosphere, and 11, I, are the paths within the interferometer. The irradiance at the point xfis found by first integrating IV,Iz over the source and then averaging over many periods of the optical frequency. A difficulty arises here because the paths Z1 and Z , may depend on the source coordinates; that is, rays from different parts of the source may have different optical paths through the atmosphere. If Z1 and Z, are independent of the source coordinates, the object is said to lie within a single isoplanatic patch. Although the size of a patch is not well defined, it is generally accepted that it is about 1’’ (YOUNG [1974]). Since most objects that would be studied with a stellar interferometer are much smaller than this, it is permissible to assume isoplanatism. In this case the irradiance at the point x’ in the plane of the detector Di(i = 1,2) is given by the following:
Ei(x’, V , t ) = E J v ) ’ [ l + ( - ) ’ I Y ( V ) ~ C O S { ~A( Z V+) k+ AI}], ~
(3.3)
where AZ and AI = I, - I, - D are respectively the external path error difference and the internal error (including errors due to path compensation), and y ( v )= (yIexp {ia}is the complex degree of coherence given by the van Cittert-Zernike theorem:
250
MICHELSON STELLAR INTERFEROMETRY
",
83
In eqs. (3.3,4) E,(v) is the total irradiance from the source. It is often convenient to use the line source function
I(5)= j d r l E(5, rl)lE,
(3.5)
and write eq. (3.4) as y ( v ) = I d 5 exp {-ikD(}1(5)
.
(3.4')
To find the output signal from the photodetectors, eq. (3.3) must be integrated over the illuminated area. The path error A 2 will, because of turbulence, depend on x' (the irradiance as well as the phase will fluctuate because of atmospheric turbulence; both of these effects can be combined in A 2 if we regard this as a complex function), and consequently the output of the detectors will be affected by atmospheric turbulence. This will be examined in detail in 0 4. It should be pointed out, however, that if a multi-element detector array is used, one can avoid the integration which is necessary with a single large photocathode. It can be shown that this type of detector is independent of seeing (TANGO [1979a]), however it appears that large apertures are needed to obtain a good signal to noise ratio (this technique is essentially the same as the pupil plane speckle interferometer described by GREENAWAY and DAINTY[1978]). For the rest of this section we shall assume that the aperture diameters are so small that the output signals of the photodetectors are given by eq. (3.3) with x' = 0.
3.2. THE EFFECT OF A FINITE BANDWIDTH
If F(k)is the transmission of the optical filters F, and F,, expressed as a function of the wavenumber k, and if E(x) is its normalized Fourier transform, the signals from the detectors will be proportional to: os Ei=EO{l+(-)' I ~ ( D ) ~ c@}
(3.6)
where @ is the overall phase and T ( D ) is given by eq. (3.7):
I
y(D )= 2~ d5 P(0 5 - AZ - Al)l( 5).
(3.7)
A loss of coherence will result if the source dimensions are excessively large or if the path errors become too great. While the first of these conditions should not normally occur, path errors will always be present and may cause a serious loss of coherence.
IV, 5 31
BASIC THEORY
25 1
We shall assume that the function F ( k ) is rectangular, with a full width of Sk centered at k, (the actual response of a filter is complicated by diffraction effects; see TWISS and WELFORD [1973]). In this case one may easily show that for the loss in coherence to be less than 1% the bandwidth must satisfy the inequality
Sk < 1/(2SL),
(3.8)
where 6L is the total path error. Although one should be able to monitor the internal path lengths of the interferometer with very high precision, there will remain systematic errors which can be significant. These arise from the fact that the computed value of the path compensation depends on the relative angular orientation of the source and baseline through the angle a. Even for bright stars the positions are not normally known to better than about 0.2” (SCOTT[1963]), and accordingly if one is to rely on catalog coordinates to determine the compensation one must expect that the differential path errors will not be less than about 10-6Do. Expressing the baseline length in meters and the bandwidth B, in herz, one has the following constraint on the allowable bandwidth, if the coherence loss is to be less than 1%:
B , < 2.5 x
(3.9)
One can always use a larger bandwidth by systematically varying the path compensation and determining the position of maximum coherence. As in radio interferometry, such “hunting” for the white light fringe will certainly be necessary when the interferometer is set up in order to determine the absolute position of the instrument. It is clearly undesirable to do this on a nightly basis, and one would normally use a bandwidth satisfying (3.9). A more worrying problem is the random fluctuation in path length due to the atmosphere. There is abundant data at radio frequencies (ELSMORE and RYLE[1976]) that path differences due to fluctuations in the index of refraction can be as large as several millimeters when the receivers are separated by distances of the order of a kilometer. The fluctuations at optical wavelengths are apparently much smaller, and we show in section 4.1 that the r.m.s. path errors predicted by the conventional theory of turbulence are of the same order of magnitude as those arising from astrometric uncertainties. Nevertheless, the radio data, coupled with the fact that at long baselines the path errors will be caused by very large
252
MICHELSON STELLAR INTERFEROMETRY
[IV, 8 3
scale atmospheric motions not well described by normal turbulence theory, would suggest that the inequality (3.9) is a rather optimistic one. The effective bandwidth which determines the signal to noise ratio and hence the limiting sensitivity of the instrument can be readily increased by multi-band spectral analysis. We suppose that the starlight is dispersed into a line spectrum, and the detectors are replaced with linear arrays of photodetectors. The dispersion of the system and the size of each detector element are chosen so that the bandwidth of each spectral channel satisfies the constraint B, < c / ( ~ TAl). (3.10) We show in section 3.3 that the signal to noise ratio is directly proportional to the bandwidth. It follows that if N,, spectral channels are used, and the data from the different channels are handled separately, the signal to noise ratio will be improved by (N,,);. Consequently, the effective bandwidth Be, will be given by Be,= N&B,. This arrangement also allows a greater flexibility, since one can optimize the bandwidth B, for the observing conditions by coherently combining the signals from two or more adjacent channels. Furthermore, one will also have the ability to observe an object in several colors simultaneously; this would be very useful for elucidating the structure of the object.
3.2.1. Atmospheric dispersion
,
Atmospheric dispersion produces a small displacement of the achromatic fringe in a Michelson stellar interferometer because of the small differential air path AL which arises from the combined effects of atmospheric turbulence, the curvature of the earth and aay height difference between the two ends of the baseline. From the theory of the Rayleigh interferometer (BORNand WOLF [1975]) we know that this displacement can be corrected by a change in the internal path compensation equal to AL(n - 1 - A 3n/3A)A=b (3.11) where n(A) is the refractive index of the atmosphere. Now AL<
IV, 8 31
BASIC THEORY
253
(EDLEN [1953]), that the associated loss of correlation that would arise with a non-zero bandwidth will be less than 1% if
(3.12) where D is in metres and A(, has been taken equal to 500nm. Since in practice D >> 1, we see, by comparison with eq. (3.9), that the limitation on bandwidth imposed by air dispersion is always unimportant, even under extreme conditions, with the limitation imposed by astrometric errors, provided, of course, that the path compensation has been properly optimized. The effects of dispersion can be further minimised by means of dispersion correctors within the interferometer. These would consist of pairs of opposing wedges, which, when correctly adjusted, would provide the necessary dispersion to correct for the residual differential air path. Such a complication might be justified if the astrometric errors could be greatly reduced, thus allowing the possibility of a significant increase in bandwidth.
3.3. MEASUREMENT OF THE FRINGE VISIBILITY BY PHOTON COUNTING
The direct measurement of the degree of coherence using the photocurrents from the two detectors is not generally possible because of the rapid phase fluctuations. We must sample in a time short compared to the fluctuation time of the phase. The sample time will be determined by the strength of atmospheric turbulence, but will be of the order of 1 millisecond (see § 4), and consequently photon counting must be used to measure the signals. In this section we investigate the effect of the counting statistics on the measurement of the degree of coherence. The mathematical details have been relegated to Appendix A. It is a property of thermal sources of radiation that if the sample time is very much greater than the coherence time of the light the probability of emission from a photocathode is given accurately by Poisson statistics (MANDEL[1959]). However, in practice the actual events which are observed can deviate from this ideal, due to a variety of instrumental effects. In particular, dead time effects (BEDARD[1967]) or multiple pulsing (afterpulsing) can seriously modify the counting statistics (GETHNER and FLYNN[1975]). If the form of the actual distribution is known,
254
MICHELSON STELLAR INTERFEROMETRY
[IV, § 3
one can take this into account (see for example SALEH'S [1978] monograph on photoelectron statistics), but for simplicity we shall consider only Poisson statistics. It is important to realize that deviations from an assumed statistical distribution can seriously bias the data, and such deviations will be an important source of systematic error in the interf erome ter . Suppose now that there are M successive sample periods of duration 27 seconds, where typically M - 5 X lo4 and T 1ms, giving a total observing time of T = 2M7 lo2seconds. Each period will be divided into two equal subintervals; in the first of these no extra phase shift will be introduced; while in the second there is a 90" phase shift present. Let Ejkr be the irradiance incident upon the detector Di during the kth sample and rth subinterval (k = 1 , 2 , . . . ,M ;r =0, 1). For the present we assume that the phase is constant throughout the kth sample and write it as @k. We also assume that the variation in the wave amplitudes over the entrance apertures can be neglected. Then one has
-
-
Ejkr
= EJI
+ (-)'
IyI COS (@k
+r~/4)].
(3.13)
We measure the three quantities ill, ii2 and i j defined by the equations: (3.14a) (3.14b) where njk, is the number of counts produced by Dj during the ( k , r ) subinterval. In Appendix A it is shown that the unnormalized correlation C2,
?f
= ij
-(ti,
+ ii2),
(3.15)
is an unbiased statistic for (2N07)' ( y ( ' , where No = (ill + A2), i.e., it is the expected number of photon events per second for both channels together. When M is large, the normalized correlation, defined by eq. (3.16), ;?I
+
= 4 2 / ( ti,
ii2)2,
(3.16)
has an expectation just equal to (y12. A measure of the statistical uncertainty in the determination of IyI2 is given by the standard deviation of E;: cl,,12
= 2(No7)-'42(7/T)(1
+No7 ")'I2).
(3.17)
IV, § 41
THE EFFECTS OF ATMOSPHERIC TURBULENCE
255
The signal to noise ratio, S/N(Iyl2)= Iy12/~1y12,is (3.18) For bright sources the signal to noise ratio is proportional to (NOT);,as one might expect. When N07<<1, that is, when less than one photon per sample is registered, the signal to noise ratio is given by the following: S/N(IrI*) = (NoT)(7/8T)' IyI2.
(3.19)
We can readily compute the signal to noise ratio for an unresolved source using eq. (3.19). The photon counting rate is given by No = a ~ ( m f ~ / 4 ) f , B , , , / ( h s-', u)
(3.20)
where OL is the photodetector quantum efficiency, K is a loss factor which will include atmospheric extinction and instrumental absorption losses, f, is the absolute flux from the star (measured in Wm-' Hz-I) at the optical frequency u and Be,, is the effective optical bandwidth. It is assumed that half of the light is removed by the polarizing beamsplitters. We adopt the numerical values a =0.25, K =0.25, d =0.10m, and assume that we observe an early type star of visual magnitude m,, for which we take a typical flux equal to f v = 4.08 x 10-23-mJ2.5(OKEand SCHILD [1970]). We also take B , = 2.5 x 10" Hz and use 100 spectral channels, centered around ho = 500 nm, giving an effective bandwidth of 2.5 x 10l2Hz. Under these assumptions the signal to noise ratio can be found from the formula: (3.21) log,, (SIN)= 3.2+0.5 log,, T- mJ2.5. The limiting magnitude m,(limit) may be rather arbitrarily defined as the magnitude which gives a signal to noise ratio of 3 in an integration time of one hour. From eq. (3.19) one has that m,(limit)=+ll. If only a single spectral channel is used, this drops to +8.5. For routine observations a more practical limit is that a signal to noise of 3 be achieved in 100 seconds; this gives a limiting magnitude of about +9 for the multichannel instrument. 04. The Effects of Atmospheric Turbulence on a
Michelson Stellar Interferometer The analysis of D 3 was based on the assumption that the phase and amplitude variations of the wavefront across the apertures were
256
MICHELSON STELLAR INTERFEROMETRY
[IV, P 4
negligible. This will not generally be true, because a plane wave incident upon the atmosphere will undergo multiple scattering as it propagates through the turbulent layers of the medium. The stochastic properties of the fluctuations induced by this process are by now fairly well understood, largely owing to the work of Kolmogorov, Tatarski, Fried and others. It is probably true to say that we now have a fairly plausible theory of “seeing”, at least for good sites under clear air conditions. The effects of turbulence can be divided into three catagories: (1) path length fluctuations due to large scale turbulence; (2) temporal fluctuations in the mean phase and amplitude of the wavefront at each aperture; and (3) the spatial variation in the phase and amplitude across each aperture. We consider these three areas in the subsections that follow.
4.1. LARGE SCALE PHASE FLUCTUATIONS
We have seen that the large scale, long period variations in relative phase may produce significant path errors. These may be estimated as follows. The structure function (FRIED [1966]) defined by eq. (4.1),
9(DO) = 6.88(Do/ro)f,
(4.1)
is the total strength in the relative phase fluctuations of the wavefront measured at two points separated by Do. When Do is !arge, the phase fluctuations grow as D,$ but they are confined primarily to the very low frequency end of the temporal power spectrum. The corresponding r.m.s. path length error is
(AZ)= 0.42h(Do/r,,)~,
(4.2)
and if one takes ro =lOcm and h = 5 0 0 n m , it will be seen that the expected path errors will be of the order of 10-6Do. This result is only valid for D O C LOwhere , Lo is the outer scale length of the turbulence, which is a few meters near the ground. Although D,, will surely exceed the outer scale length, it is nevertheless probable that these results give at least the order of magnitude of the effect, since the predictions of eq. (4.2) are in fair agreement with the radio data (WESLEY [1976]).
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THE EFFECTS OF ATMOSPHERIC TURBULENCE
251
4.2. TEMPORAL FLUCTUATIONS OF THE WAVE AMPLITUDE
We look next at the temporal fluctuations of the complex wave amplitude at a single point in one or the other aperture. The characteristic time of these fluctuations is important, for it will set the basic sampling time of the interferometer. 4.2.1. Temporal phase fluctuations In eq. (3.13) we have assumed that the phase @ is constant during a single observing period. If this is not so, one must integrate the time varying irradiance over the sample period, and use the result to determine the average probability of photoemission during the sample. If this is done one finds that the observed correlation will be reduced by the factor q k (71, where Jtk - r / 2
where 6, is a suitably chosen mean phase. When the coherence loss factor is small, it is permissible to expand the cosine function in eq. (4.3), and one finds to second order that the loss factor is given by 7)k = 1-4 A 2 @ k , (4.4) where A2Gk is the mean squared fluctuation in the phase during the kth sample period. If we require that the loss due to this source be less than 1%,then the phase fluctuations must satisfy the inequality (4.5): A2@ < 0.02. (4.5) We estimate the phase fluctuations as follows. Let (A2@) be the expected value of the fluctuations. It follows from the elementary properties of spectral distributions that (4.6) where W,,(f) is the spectral power distribution for the fluctuations in phase measured at two points a distance D apart. Because the two apertures are widely separated, we can regard the fluctuations at apertures to be statistically uncorrelated, and we therefore
258
[IV, 9: 4
MICHELSON STELLAR INTERFEROMETRY
approximate WD,(f) by assuming that it is just twice W&), the spectrum for the fluctuations at a single point (as we indicated in 0 4.1, there are large scale fluctuations associated with widely separated apertures, but these are of very low frequency and consequently will not affect the behavior of the more rapid fluctuations discussed herej. TATARSKI [1971] gives an expression for W,(f) for the high frequency fluctuations in the “inertial subrange”, and one has that
WD,(f)=0.066k2vt’f-’
I‘
(4.7)
Ci(2) dZ,
where C: is the characteristic structure constant for fluctuations in the index of refraction, and u, is an effective transverse wind speed. The integral is taken along the path of propagation through the atmosphere. It is convenient to introduce the correlation length r,, defined by the integral
ro-; = 0.42k2J C’,(Z)d Z
(4.8)
(FRIED [1965]). The experimental value of r, is usually taken to be about 10 cm (FRIEDand MEVERS[1974]). Integration of eq. (4.6) then gives for the mean squared phase fluctuations as a function of the observing time the following:
(Az@) = 1.42(wI/r,$.
(4.9)
The inequality (4.5) thus becomes TU, < O.O77r,,.
(4.10)
Under normal observing conditions we would expect that the wind speed would be of the order of a few meters per second, and thus if the sample time T is chosen to be about 1 msec this inequality will be satisfied.*
4.2.2. Irradiance fluctuations Because of scintillation the mean optical power through each aperture will also fluctuate irregularly. Denoting by fl(t) and fz(t) the power received from aperture A, and AZ,respectively, we see that the power reaching the detector Di (i = 1,2) will be given by eq. (4.11): I i ( t ) = ~ { I l ( t ) + 1 2 ( f ) + 2 ( - ) i cos@(t)} ~lyl
* This constraint represents a worst case; for finite apertures A*@
is:
i = 1,2.
(4.11)
0.528(u,~/r,,)i(v,~/d):.
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THE EFFECTS OF ATMOSPHERIC TURBULENCE
259
If we integrate eq. (4.11) over the sample time r we obtain for the average power during the kth sample: Iik
= Ik(l+(-)iqI
cos @ k ) ,
(4.12)
where fk is a mean power and qI a loss factor due to irradiance fluctuations, given by (4.13) To evaluate the loss due to scintillation we assume that the loss is small and that the fluctuations in irradiance at the two apertures are uncorrelated. In this case the integrand of eq. (4.13) can be expanded in a power series of the variable S = ( I ( t ) - ( I ) ) / ( I ) , where I ( t ) is the power received upon one aperture. The mean squared fluctuations in 6 can be related to a power spectrum, and we find that the expected loss will be given by the expression (4.14) where W,(f) is the normalized power spectrum of scintillation and a: is the mean squared fluctuation in the irradiance. Using TATARSKI'S [ 19713 expression for WI(f), one finds that the loss due to scintillation is given by qI = 0.108(mf,,r)b~,
(4.15)
where fo is a characteristic cutoff frequency for the scintillation spectrum. From data given by Tatarski, fo is about 100 Hz and u: is approximately 0.08 for a 10 cm aperture, and consequently the contribution of scintillation to the loss of coherence is quite negligible.
4.3. SPATIAL FLUCTUATIONS OF THE WAVE AMPLITUDE
By choosing the sample time r short enough, the fluctuations in mean phase and irradiance will be effectively frozen, and will not affect the measured coherence. The wavefront will however be distorted due to turbulence, and we consider this next. If the optical path lengths are correctly compensated, the amplitude of the arriving wavefront in the aperture plane is given by V(&,x,t):
V(S,x, t ) = E (x, t ) V(&,x, 0.
(4.16)
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MICHELSON STELLAR INTERFEROMETRY
[IV, § 4
Here V(& x, t ) represents the unaberrated wave of eq. (3.1) and E(X, t ) is the atmospheric transfer function. This latter is frequently written as E
(x, t ) = exp {i[@(x,t)- ix(x, t)l),
(4.17)
where 0 is the phase aberration and x is the log amplitude function which is used to represent scintillation. Physically, one can regard E(X, t) as a combination of classical aberrations. The lowest order term represents fluctuations in the mean phase and irradiance across the apertures (the so called “piston” mode): higher terms correspond to wavefront tilts or image motion, defocussing, astigmatism, etc. The r.m.s. power in each aberration mode has been calculated by NOLL[1976]; approximately 80% of the total aberration is in the form of wavefront tilts. The analysis of Q 3 can be repeated using U in place of V, and it is found that the optical power reaching the detector Di during the (k, r) subinterval ( k = 1,2, . . . ,M ; r = 0 , l ) will be: Ijkr = &[Jk
-k
(-y
Re {irHky}],
(4.18)
Here E,, is, as before, the total source irradiance; Jk and Hk are defined by eqs. (4.19,20):
A (x) is the aperture function defined by: (4.21) and the integrals in eqs. (4.19,20) are taken over the entire plane. If we set r k = EoJk and q k =Hk/Jk,eq. (4.18) can be put into the “standard” form: Iikr=Tk[l+(-)j I q k l ‘I~lcos{@k +rr/4)].
(4.18’)
The factor q k is a coherence loss factor, and the measured correlation will be given by: (4.22)
IV, Ei 41
THE EFFECTS OF ATMOSPHERIC TURBULENCE
261
where the average loss factor is just the average value of IHkl’ over the total integrating period (see Appendix A). The loss term will involve the fourth order statistics of the atmospheric transfer function and the general expression is not particularly useful. We note two important cases, however. First, if the effects of the atmosphere are small, eq. (4.20) can be expanded in a power series in @ and x. One finds, to second order, that the loss of coherence will be given by -
(qI2=1 - ( A 2 @ 1 + A 2 @ 2 ) - ( A 2 ~ I +A2x2),
(4.23)
where the terms on the right represent the mean squared fluctuations in phase and log amplitude about their average values at each aperture. In deriving eq. (4.23) it is assumed that the fluctuations at the two apertures are independent. Using Noll’s results, which are based on the Kolmogorov model for turbulence, one can write a numerical expression for the loss: lqI2= 1-2.06(d/r,,);,(d/r,cc 1).
(4.24)
Eq. (4.24) implies that for a conventional Michelson stellar interferometer the aperture diameters must be very much smaller than r,, in order to have good fringe contrast. A second way of estimating the loss assumes that the turbulence obeys the Kolmogorov model and is stationary. We also assume that the turbulence at the two apertures is uncorrelated. In this case one has the following estimate for the coherence loss: (Iq )1’
I
= (7rd2/4)-’
d2uT(u)B2(u),
(4.25)
where T(u) is the optical transfer function of a single aperture and B(u) is the second order correlation function defined by eq. (4.26): (4.26) B (Irl) = (E (r + r ’ ) E *(r’)). FRIED[1966] has given formulas for B(u) and in Fig. 4 we show the expected loss in the observed degree of coherence as a function of (d/rO). It is interesting to note that the coherence is reduced by almost exactly 50% when d = ro. 4.3.1. Removal of wavefront tilts It is clear that if nothing is done to correct at least partially for the effects of atmospherically produced wavefront distortions, one would
262
[IV, 5 4
MICHELSON STELLAR INTERFEROMETRY
01
10
d/r,
Fig. 4. The loss of coherence in a Michelson stellar interferometer due to atmospheric turbulence. Curve (a) is the loss expected when no tilt compensation is used; curve (b) is the loss with a tilt correcting servo.
need to use unacceptably small apertures. The obvious first step is to remove the lowest order effect, namely wavefront tilt. In the prototype instrument we are discussing, the tilts in each half of the interferometer are monitored by the sensors G , and G , (Fig. 2) which drive the wobbler mirrors W, and W, through a suitable servo system. The contribution of the tilt term to the total fluctuations is 0.898(d/ro): (NOLL[1976]), so that the loss of coherence, in the small aperture limit, will be
m=1 -0.13(d/r0)3, (d/rO<<1).
(4.27)
In the more general case, one can again use Fried's results to determine the loss and the resulting loss factor is shown as curve (b) in Fig. 4. In this case the observed degree of coherence is reduced by 50% when d = 2.2r0.
4.3.2. Measurement of the coherence loss
If we seek to measure the coherence with an accuracy of around 1%, and if we have no direct information on the value of 1qI2, it will be necessary to use quite small apertures, Even with tilt correction, the aperture size would need to be much less than the worst case value of r,. On the other hand, if the loss is not too large (perhaps -0.5), it may be possible to accurately estimate its value and apply a correction in the subsequent data analysis. There are a number of ways of estimating the loss. Perhaps the simplest approach is to measure the Strehl definition of the images formed at the
IV, 8 41
THE EFFECTS OF ATMOSPHERIC TURBULENCE
263
guidance sensors G1.2.The Strehl definition is a common measure of image quality, which when normalized can be written in the following form:
I
9 = (7rd2/4)-’ d2uT(u)B(u),
(4.28)
so that it differs from the coherence loss factor only by the appearance of B(u) instead of B2(u) in the integral. Since the correlation function B(u) is exponential for a Kolmogorov model, in fact the Strehl definition and coherence loss will differ only by a scale factor, so that if 9 is used to estimate the instantaneous value of ro it should be possible to determine the coherence loss quite accurately. The actual measurement of the Strehl definition (or equivalent measures of image sharpness) depends on the details of the actual sensors used. Since the turbulence will be locally stationary, it should be possible to set up auxiliary equipment near each aperture to measure the turbulence (the strength of turbulence will not necessarily be the same at each aperture if they are separated by a distance much larger than the outer scale of turbulence). One attractive way of estimating the loss from such “seeing” monitors is to use a shearing interferometer (DAINTYand SCADDAN [1974, 19751; RODDIER [1976]). Such an instrument measures B(u) directly, and should allow an accurate estimation of the coherence loss. It should be noted that any active optical system cannot provide perfect correction, and furthermore it will act itself as a source of noise. These are important practical considerations which we shall examine in 0 5 . 4.4.
THE SIGNAL TO NOISE RATIO
We look very briefly at the effect the phase and irradiance aberrations have on the statistics and the noise characteristics of the interferometer. The details are given in Appendix A. If the loss factor is known, then the signal to noise ratio for the measurement of the coherence will be given by eq. (A.17) of the Appendix, which we repeat here:
17712
The signal to noise ratio is obviously reduced by the presence of the loss factor, in addition it is reduced by the term AzNoINo. This is the excess
264
MICHEISON STELLAR INTERFEROMETRY
[IV,
Fi 5
noise due to scintillation, and is equivalent to the factor a: of 0 4.2.2. This excess noise term is a consequence of the Poisson counting statistics. As we have already seen. this term is of the order of 0.1, and thus can be neglected for most purposes. There is another factor h3 which appears in eq. (4.29). This factor depends on the sixth order statistics of the atmospheric transfer function, and we expect that it will be of the same order of magnitude as the coherence loss factor. Apart from the lowering of the signal to noise ratio because of the coherence loss, the signal to noise ratio is not changed significantly by turbulence.
5 5. The Tilt Correcting Servo System Since the behavior of the active optical system which removes the wavefront tilts will have an important effect on the overall performance of the interferometer, it must be examined in some detail. We shall assume that the position sensing detectors can be represented as linear devices with outputs proportional to the angular displacements A a X and A% of the image; we can write for the signal
S = N , Aala,,.
(5.1)
where N , is the photon counting rate at the guider and a,,a parameter of the order of the size of the image. The signals, either analog or digital, are passed through servo amplifiers having the complex frequency response G(f), and are used to control the “wobbler” mirrors. The precise form of G(f) will depend on a number of factors, but will have the shape of a low pass filter. Because any real servo will have a finite bandwidth, there will be residual tilt fluctuations, and these will be given by
where WD,(f) is the temporal power spectrum for the total fluctuation in the angle of arrival of the wavefronts at two apertures separated by D. Because the turbulence at the two apertures will be uncorrelated, and because the fluctuations in the orthogonal x and y directions will also be uncorrelated, WDa(f)= 4 W,\(f). There has been some disagreement
IV, 8 51
265
THE TILT CORRECTING SERVO SYSTEM
about the form of this spectrum, and apparently DE WOLF[1973] was the first to give a correct derivation. This is discussed more fully by HOGGE and Bums [1976], who also give spectra for the higher order aberrations. We derive the spectrum in Appendix B, and it is shown in Fig. 5 . The spectrum exhibits a f-' power law for frequencies below a cutoff frequency fi,;above this it falls off rapidly as f-q. The cutoff is given by
f0 = UT/(Td),
(5.3)
where uT is an effective transverse wind speed. The total power in the fluctuations is given by a;, = 0.680(d/r,)~(h/d)2.
0.01
0.1
1.0
(5.4)
flf,
Fig. 5. The temporal power spectrum of wavefront tilt fluctuations. The two lines indicate f - j and f-' power laws. f,, is a scaled frequency (see text).
266
MICHELSON STELLAR INTERFEROMETRY
[IV, 5 5
In order to work through a numerical example, we shall take Gcf)= fl/if, where f l is the servo (3dB) cutoff frequency. One finds that the fractional power remaining, p = (A2a)/a;,, is given by the integral*
The phase fluctuation corresponding to a small angular term will be (,rr2/8)(d/A)’a,and from eq. (4.23) we can estimate the loss of coherence due to the residual tilts: q, = 1 -o.84(d/r,>;p(fl/fo).
(5.6)
The residual power when f, = fo is about 0.114, so that there will be approximately a 10% loss in the apparent coherence when the cutoff frequencies are equal. Assuming that the electronics contribute a negligible amount of noise, one can show that a detector having the response of eq. (5.1) will introduce an angular “dither” due to shot noise equal to (A’a,):
where BN is the noise bandwidth of the servo. In the present example, B, = (?r/2)f,. ‘From the point of view of minimizing noise, it is clearly desirable to make the image “size” a. as small as possible and to use a detection method which is efficient. One can show that the best possible value of a”,in the sense of the Rao-Cramer theorem, is h/(,rrd) (FARRELL [1966]). This limit can be obtained by an optimal filter (WALLNER [1978]). In practice, one can expect that the guiding will be less than optimal, both because of instrumental effects and because turbulence degrades the image quality, lowering the sharpness of the image. This latter effect is in general difficult to estimate, but in the case of a quadrant detector a simple result is obtained. In a quadrant (or “Hartmann”) type detector, the image is dissected by an optical knife-edge, and image motion is sensed by an unbalance in the signals from the two sides of the edge. A second edge at right angles to the first is used to provide a signal for the orthogonal direction. For this configuration one can show that (5.8) * T h e function A , ( z ) is the normalized Bessel function 2 J , ( z ) / z .
IV, P 51
267
THE TILT CORRECTING SERVO SYSTEM
where (k) is a factor between zero and one which allows for the degradation of image quality. It is given by
where T is the single aperture OTF and B the atmospheric correlation function defined in eq. (4.26). Note that this is a one-dimensional integral, and B ( u ) is the "short exposure" correlation function (i.e., with ti1t removed). Since the shot noise power rises linearly with the servo bandwidth, while the uncompensated tilt fluctuations fall approximately as f-*, there will be an optimum bandwidth which produces a minimum of servo noise. In Table 1 we give the percentage loss in coherence and the best servo cutoff frequency f l for stars of various magnitudes and for aperture diameters of d = r, and d = 1.6r".We have assumed in calculating these numbers that the wind speed is 5 mls, r, = 10 cm, and that the optical bandwidth of the guidance system is about 100 nm centered at 500 nm. The numbers in the table should be taken as a rough guide only to the expected losses due to the servo, but it seems clear that a significant loss can occur for stars fainter than about rn, = 6 when a 10 cm aperture is used. As with the case of wavefront distortion, it should be possible to estimate the servo losses. To first order, the error signal generated by the TABLE 1 Visibility loss due to uncompensated angle of arrival Ructuations and shot noise
10
5 6
7 8 9
16
5 6 7
a 9 10
1.4 2.5 4.5 7.8 13.2
120 80 55 35 25
0.54 1.o 1.8 3.3 5.8 10.0
120 90
65 45 30 20
268
MICHELSON STELLAR INTERFEROMETRY
[IV, I 6
guide sensors is directly proportional to the shot noise within the servo bandwidth and to the residual tilt fluctuations. If the noise power of the error signal is measured, allowing for scintillation, one obtains an estimate of the combined shot noise and tilt fluctuations. When calibrated, this data can be used to correct the observed coherence. This procedure was in fact used with the interferometer of the Royal Observatory, Edinburgh with very good results (TANGO [1979b]). An even better estimate of the coherence loss can be made if the variation in the detector sensitivity is taken into account (eq. (5.8)). 5.1 ATMOSPHERIC DISPERSION
The effect of dispersion on the performance of the tilt correcting servo can be quite serious. The problem arises principally because the optical responses of the two servo systems will not necessarily be matched, nor will their peak sensitivities coincide with the operating wavelength of the main interferometer. As a consequence one may have differential offsets present which will be a function of both the position and the spectral type of the source. Dispersion correctors will reduce these offsets to negligible values. A second effect of dispersion is that the phase aberrations introduced by turbulence are themselves wavelength dependent. WALLNER [19771 has discussed ways of minimizing this effect. However, even with correction for dispersion, one can expect that if the optical bandwidth of the guidance system is made too large, one will lose accuracy, since the tilt fluctuations are a function of wavelength, and will not be correlated over a sufficiently large bandwidth. In addition there are the practical difficulties associated with the limited bandwidth of the photodetectors, the transmission of the optics, etc., all of which tend to reduce the servo optical bandwidth. We have chosen a perhaps optimistic value of 100 nm; it is clear that more experimental data is needed before an optimal guiding system can be designed.
D
6. Summary and Discussion
The resolution and accuracy of a modern optical Michelson stellar interferometer are limited principally by atmospheric turbulence. ' l o reach 9th magnitude the maximum practicable baseline is probably about 100 m because of large scale long period atmospheric fluctuations
IV, 5 61
SUMMARY AND DISCUSSION
269
in optical path. Longer baselines would require narrower optical bandwidths with an associated loss in signal-to-noise ratio and therefore of limiting magnitude; e.g., a tenfold increase in baseline and hence resolving power entails a loss of 2.5 in stellar magnitude. Perhaps more important from the astronomical point of view is the loss in accuracy caused by the atmosphere. Since any errors reduce the apparent degree of coherence, the effect of turbulence is to make sources appear resolved when in fact they are not. Smaller errors can give rise to spurious limb darkening effects, etc. These errors arise from the high frequency, but relatively small amplitude, fluctuations in the wavefronts at each aperture; there is essentially no correlation between the fluctuations at the separate apertures. By a proper choice of sampling time and by the use of an active optical servo to minimize wavefront tilts, it is possible to reduce the coherence loss to around 20-30% (that is, the sum of the losses given by eq. (4.25) and Table 1) for a star of magnitude +9 and an aperture of 1Ocm. It is important to note that the absolute limiting magnitude, set by photon noise in the optical “correlator”, can be found from 0 3.3 and is much fainter. Thus the practical limiting magnitude will be set by the tilt compensating servo and is fixed by the amount of signal loss we are willing to accept. If the losses are not too large, it is feasible to estimate them by auxiliary observations, either through the same apertures or by means of “seeing” monitors placed near to the primary apertures. When corrections for these residual losses are made during the data analysis, it should be possible to estimate IyI with an accuracy of a few percent for stars brighter than about rn, = +9. Most of the instrumentation needed for a Michelson stellar interferometer has already been developed, either for the existing prototype interferometers or for other applications, so that there is no reason why an interferometer with a baseline of 100 m and a limiting magnitude of +9 could not be built. We conclude with a brief discussion of future developments which may significantly improve the performance of interferometers. (1) Active optics. We have considered only the simplest method for reducing the wavefront aberrations. The subject of phase correction by active control of the optics is under intensive study. A recent review has been given by HARDY [1978]. The feasibility of sharpening stellar images by real time phase correction has been demonstrated by BUFFINGTON, CRAWFORD, POLLAINE, ORTHand MULLER[1978]. If these more elaborate
270
MICHELSON STELLAR INTERFEROMETRY
[IV, APP.A
techniques are used with a Michelson stellar interferometer it may be possible to use larger apertures with no loss in accuracy. As Hardy points out, however, the signal to noise ratio of an adaptive optical system is limited by the photon flux through the smallest subaperture over which phase correction is desired. As this size will be no greater than ro a larger overall aperture size would not necessarily yield a correspondingly fainter limiting magnitude. On the other hand, the signal to noise ratio of the main “correlator” of the interferometer would certainly be improved. This would allow shorter integration times, which are desirable in order to minimize systematic errors which may arise in the photon counting equipment. As well, since more light is available it would be possible to get more information out of the interferometer, which could be used, for example, in automatic fringe tracking (see below). It is clear that possible improvements in this direction will require careful experimental investigation. (2) Automatic Fringe Tracking (AFT). There is a very real possibility that AFT can be used in the near future. Simple tracking for a short baseline has already been proposed by an “astrometric” interferometer (SHAOand STAELIN[1977]), and HARDYand WALLINER[1979] have examined the problem from the point of view of the active control system. It has been pointed out (STEEL[1978]) that if a multiple spectral band detector is used a particularly simple AFT can be based on the fact that the spectrum is “channeled” with fringes, the period of which is inversely proportional to the path error. With AFT considerably longer baselines should be possible. The performance of an AFT system will be limited by the available signal and the integration time of the servo system. As the variation in path is quite slow, very small bandwidths can be used. The signal will depend both on the brightness of the source and the fringe visibility. For very long baselines, of course, many objects will be partially resolved and AFT may be less effective, but it seems likely that if it can be used the chief limitation to resolution will be the availability of suitably large sites for the interferometer. Appendix A: The Photon Counting Statistics The probability that exactly N photons will be counted during a period t is given by the Poisson distribution:
IV, APP. A1
THE PHOTON COUNTING STATISTICS
27 1
where p is the mean counting rate and t the sample time. For convenience we summarize the first few moments of this distribution:
( N )= (CLt),
(A.2a)
( N 2 )= (CLfI2 + (CLf),
(A.2b)
( N 3 )= (CLfI3 + 3(CLtI2+ (CLt),
(A.2c)
(N4)= ( / ~ t ) ~6 +( ~ t+)7~( ~ f ) (~p+f ) .
(A.2d)
We use sharp brackets to indicate the statistical expectation of the number of counts. The mean counting rate at the detector Di during the (k,r) subinterval, k = 1,2, , . . ,M and r = 0, 1, will be proportional to the incident optical power: (A.3) Ijkr=jk[l+(-)i l q k l ' Irlcos(@k+rr/4)], where fk, 1 q k ) and @k are assumed to be constant during each subinterval. The primary data, i l ,ii2 and 4, are defined by eqs. (3.14a, b). The sum til + A2, has the expected value: (el
+ii,)=M-'1(2ajkT)=(27)N";
(A.4)
k.r
here a is the quantum efficiency and No is the expected counting rate in both channels together. Also, since the value of Jk, defined by eq. (4.19), will have an average value (J) very close to rd2/4, one can write
No = 2(aEord2/4).
(A.5)
To find the expected value of 4 we use the fact that for broadband thermal radiation the counting statistics of the two detectors are for all practical purposes independent and uncorrelated. Then we have the following result:
(4) = M-'
1(aT)2(11kr-1Zkr)') k.r
= M-'
Z[((aT)(Zlkr -12kr))2+((aT)(11kr +12kr))l k.r
= ( N o T ) * W ' lY12+NoT,
where the factor
is defined by
(A.6)
212
MICHELSON STELLAR INTERFEROMETRY
[IV, APP.A
The apparent coherence is defined by c2:
c2= i j -(A, + AJ
(A.8a)
and the normalized apparent degree of coherence c$ is
c$ = 4c2/(fi, + f i ~ ) ~ .
(A.8b)
It follows that the expected value of the coherence will be
(a= (77(2
*
(A.9)
IY12.
The variance of E 2 , Var{E2} or u f ~can , be determined from (A.lO): Var { E 2 }
+ A2)}-
= Var {ij}+Var {(A,
2 Cov {ij, (iil + ii2)}.
(A.lO)
The covariance occurs because 4, A l and ii2 are not statistically independent. The evaluation of eq. (A.lO) requires the use of the fourth order moments of the Poisson distribution, and one finds the following: Var {?}= M - ’ [ 4 ( a ~ ) ~+12)(11 ( 1 ~ -12)2+2(a~)(II +12)2], (A.ll) where
1Ilkrckr.
= M-I
(A.12)
k.r
If we define the photon fluctuation A2No, M
b2No =
1 ( N o- 2aTk)’
(A. 13)
k=l
and the factor M
h3 = ( T d 2 / 4 )
1
Jk
(HkI2,
(A.14)
k=l
it follows that Var {c?’}= ( ~ T / T ) ( N ( )+(No7)h3 T ) ~ [ ~ ( y 1 2 + A 2 N o / N ~ ] , (A.15) where T = (27)M is the total integration time. The variance of the normalized degree of coherence will be Var {C$}
= Var {I?’}/(N~)T)~.
from which the signal to noise ratio SIN can be determined:
(A. 16)
IV, App. Bl
THE ANGLE OF ARRIVAL SPECTRUM
273
Appendix B: The Angle of Arrival Spectrum We find the temporal frequency spectrum W,,(f) of the relative fluctuations in the angle of arrival when there are two apertures of finite diameter d separated by a large distance D. In this case the power spectrum will be just twice the spectrum for the fluctuations at a single aperture, since correlations between the two can be ignored. This latter spectrum can be found by a modification of Tatarski’s method for estimating the total power in the fluctuations. If (x, y) = r are Cartesian coordinates in the receiving plane, transverse to the direction of propagation of the light through the atmosphere, and if @(r, t ) is the phase of the wavefront at a given point, then the angle-ofarrival vector is simply the gradient a ( r , t ) = -k-’ V@(r,t ) .
(B.1)
The mean phase tilt averaged over a finite aperture of area 2 is thus
(cf. TATARSKI [1971] eq. 55.14). The space-time correlation function of the vector a ( r , t ) will be, in general, a tensor quantity, but under the assumption that @(r,t ) is an isotopic, homogeneous process one can demonstrate that the cross terms vanish, and one may therefore use the scalar correlation function, which when averaged over the aperture takes the form B a ( 7 ) = ( k x ) - ’ ~ ~ l ~V’(@(r, V * t)@(r’, t+7))d2rd2r’,
(B.3)
where the operator V V’=a’/ax ax‘+a2/ay ay’. The bracketed quantity in the integrand of eq. (B.3) will be recognized as the correlation function B,(r-r’, 7) of the phase @. The time dependence of B, will be assumed to obey Taylor’s hypothesis of frozen turbulence, which asserts that the temporal variation is dominated by the effect of the turbulent eddies being swept by the receiver at some characteristic transverse velocity uL. Under this assumption B@(r-r’, 7)=B,(r-r’+u17, O ) , and eq. (B.3) becomes
At this point it is convenient to introduce the spatial Fourier spectrum of
274
[IV,APP.B
MICHELSON STELLAR INTERFEROMETRY
Ba which we shall call F@(K):
With the help of eq. (B.4) one can express B, as an integral over the spatial wavenum ber vector:
-m
Here the function V(K)is defined as V(K)= 2-l I l d ’ r exp {iK r } ; P
for a circular aperture of radius R one has
I V(K)I’ = A N K I R ) .
(B.6)
The V-function represents the filtering action of the finite aperture: spatial components of the phase which have wavenumbers K >> R-’ will be greatly attenuated and will make no significant contribution to the angle of arrival fluctuations. It follows from the well-known properties of power spectra that the power spectrum for the angle of arrival fluctuations is
w,(f)=
dt exp(-2rrift}Ba(t) -m
(B.7) -m
Now it is reasonable to assume that the phase fluctuations will be isotropic, in which case the spectrum Fa will depend only on the magnitude of K. If we introduce the polar coordinates ( ~ , 4 )and , note that K ’ U,=
K V , COS
(4 - +),
where $ is the angle made by ul to the original x-axis, one obtains
w,(f)=2?rk-’PdKF,(K)A:(KR)K3
d4~
4 7 - 2 ~ f ) . (B.8)
( K COS u ~
IV, APP. BI
THE ANGLE OF ARRIVAL SPECTRUM
275
The angular integration over the delta-function is readily done, and one finds after a suitable change of variables that
where K~ = 2~f/v,. The power spectrum for the angle-of-arrival difference will be WD, (f) = 2 W, (f). All that remains is to find a suitable expression for the spectrum of the This of course must come from a physical model phase fluctuations F@(K). for the propagation of light in the turbulent medium. Assuming the usual Komolgorov model, one finds for a turbulent layer at a distance L that (TATARSKI [1971] eq. 46.37a)
nk2 sin ( ~ L / k ) j--I? F @ ( K=)0.132 -C:L 1+ K exp{-K2/K:}, 4 (KL/k)
{
3
(B.lO)
where C: is the refractive index structure parameter and K , is a high frequency cut-off. In the “inertial subrange” it is assumed that K , >> K >> L-’, so that F @ ( K )K-?. As this region supplies the greatest contribution to the integral in eq. (B.9), we are justified in approximating F@ this way so that W,,(f)=0.132
2n2
1
-C ~ L K -df(l+ ~ f 2 ) - 2 A ; ( K o R m ) . (B.11) u,
The total power in the angle-of-arrival difference spectrum is found by integrating WD, (f): (B.12) This integral is readily evaluated, and if Fried’s parameter is introduced we have a;, = 0.680(d/ro)s(A/d)2.
(B.13)
Eq. (B.ll) can then be written in the alternative form
WD,(f) df=O.l84p-’&,
6
dt(1+t2)-’A:(pG2) dp
(B.11’)
where p = f/fo and fo = v,/nd. Apart from an obvious factor of four, the total power given by (B.13) agrees closely with the value calculated by GREENWOOD and FRIED[19761.
276
MICHELSON STELLAR INTERFEROMETRY
[IV
One may also confirm that for f < f o the power spectrum exhibits an f-' behavior, while for f >> fo it falls off at the expected rate of f-';.
Acknowledgement
One of the authors (W. J. T.) would like to thank the Australian Research Grants Committee for their generous support.
References BEDARD,G., 1967, Proc. Phys. SOC.(London) 90, 131. BLUM,E.-J., 1959, Ann. Astrophys. 22, 140. BORN,M. and E. WOLF, 1975, Principles of Optics (5th ed., Pergamon). S. M. POLLAINE, C. D. ORTHand R. A. MULLER,1978, BUFFINGTON, A., F. S. CRAWFORD, Science 200, 489. CURRIE,D. G . , S. L. KNAPPand K. M. LIEWER,1974, Astrophys. J. 187, 131. DAINTY, J. C. and R. J. SCADDAN, 1974, Mon. Not. R. astr. SOC. 167, 69. DAINTY,J. C. and R. J. SCADDAN, 1975, Mon. Not. R. astr. SOC.170, 519. DAVIS,J., 1979, in: High Angular Resolution Stellar Interferometry (I.A.U. Colloquium No. SO), eds. J. Davis and W. J. Tango (Astronomy Department, School of Physics, University of Sydney). DAVIS,J . and W. J. TANGO,1979, eds., High Angular Resolution Stellar Interferometry (I.A.U. Colloquium No. 50) (Astronomy Dept., School of Physics, University of Sydney). DE WOLF,D. A., 1973, J. Opt. SOC.Am. 63, 657. EDLEN,B., 1953, J . Opt. SOC.Am. 43,339. ELSMORE,B. and M. RYLE,1976, Mon. Not. R. astr. SOC.174, 411. FARRELL,E. J., 1966, J. Opt. SOC.Am. 56, 578. FEINUP,J. R., 1978, Optics Lett. 3, 27. FINSEN,W. S., 1951, Mon. Not. R. astr. SOC.111, 387. FINSEN,W. S., 1954, Circ. Repub. Obs. Johannesburg 114, 240. FINSEN,W. S., 1971, Astrophys. Space Sci. 11, 13. FIZEAU,H., 1868, C.R. Acad. Sci. Paris 66, 934. FRIED,D. L., 1965, J. Opt. SOC.Am. 55, 1427. FRIED,D. L., 1966, J. Opt. SOC.Am. 56, 1372. FRIED,D. L. and G. E. MEVERS,1974, Applied Optics 13, 2620; also Applied Optics 14, 2576. GETHNER, J. S. and G. W. FLYNN,1975. Rev. Sci. Instr. 46,586. 1972, Astrophy. J. 173, L1. GEZARI.D., A. LABEYRIE and R. V. STACHNIK, GREENAWAY, A. H. and J. C. DAINTY,1978, Optica Acta 25, 181. GREENWOOD, D. P. and D. L. FRIED,1976, J. Opt. SOC.Am. 66, 193. HALL,J. L., 1978, Science 202, 147. HANBURY BROWN,R., 1974, The Intensity Interferometer (Taylor and Francis, London). HANBURY BROWN, R., J. DAVISand L. R. ALLEN,1974, Mon. Not. R. astr. SOC. 167,121. HANBURY BROWN,R. and R. Q. TWISS,1956a. Nature 177, 27. HANBURY BROWN,R. and R. Q. Twrss, 1956b, Nature 178, 1046.
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HARDY,J. W., 1978, Proc. I.E.E.E. 66, 651. HARDY,J. W. and E. P. WALLNER,1979, in: High Angular Resolution Stellar Interferometry (I.A.U. Colloquium No. SO), eds. J. Davis and W. J. Tango (Astronomy Department, School of Physics, University of Sydney). HOGGE,C. B. and R. R. B u n s , 1976, IEEE Trans. Antennas Propagat. AP-24, 144. LABEYRIE, A., 1970, Astron. and Ap. 6, 85. LABEYRIE, A., 1976, in: Progress in Optics, Vol. XIV, ed. E. Wolf (North-Holland). MANDEL,L., 1959, Proc. Phys. SOC.74, 233. MICHELSON, A. A., 1891, Nature 45, 160. MICHELSON, A. A. and F. G. PEASE,1921, Astrophys. J. 53, 249. National Academy of Sciences/National Research Council, 1967, Synthetic Aperture Optics (Defense Documentation Center). NOLL,R. J., 1976, J. Opt. SOC.Am. 66, 207. OKE,J. B. and R. E. SCHILD,1970, Astrophys. J. 161, 1015. RODDIER,C., 1976, J. Opt. SOC.Am. 66,478. RYLE,M., 1952, Proc. R. SOC.London 211, 351. SALEH,B., 1978. Photoelectron Statistics (Springer-Verlag) Ch. 5. SCOIT, F. P., 1963, in: Basic Astronomical Data, ed. K. Aa. Strand (University of Chicago Press). SHAO,M. and D. H. STAELIN,1977, J. Opt. SOC.Am. 67, 81. STEEL,W. H.. 1978, private communication. TANGO,W. J., 1979a, Optica Acta 26, 109. TANGO.W. J.. 1979b. in: High Angular Resolution Stellar Interferometry (I.A.U. Colloquium No. SO), eds. J. Davis and W. J. Tango (Astronomy Department, School of Physics, University of Sydney). TANGO,W. J. and R. Q. Twiss, 1974, Applied Optics 13, 1814. TATARSKI, V. I., 1971, The Effects of the Turbulent Atmosphere on Wave Propagation (Israel Program for Scientific Translations, Jerusalem). Twrss, R. Q., 1969, Optica Acta 16, 423. TWKS,R. Q. and W. J. TANGO,1977, Rev. Mex. Astr. Astrofis. 3, 35. Twrss, R. Q. and W. T. WELFORD,1973, Optics Commun. 7, 103. WALLNER, E. P., 1977, J. Opt. SOC.Am. 67, 407. WALLNER,E. P., 1978, private communication. WESLEY,M. L., 1976, J. Applied Meteorology 15, 43. WICKES, W. C. and R. H. DICKE. 1973, Astron. J. 78, 757. WICKES,W. C. and R. H. DICKE,1974, Astron. J. 79, 1433. YOUNG,A. T., 1974, Astrophys. J. 189, 587.
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E. WOLF, PROGRESS IN OPTICS XVII @ NORTH-HOLLAND 1980
V
SELF-FOCUSJNG MEDIA WITH VARIABLE INDEX OF REFRACTION BY
A. L. MIKAELIAN Quantum Electronics Section, Pop00 Society, Moscow, USSR
Prof. of Moscow Physico-Technical Institute with preface by academician
A. M. PROKHOROV Nobel Laureate
To the memory of Rem. V. KHOKHLOV
CONTENTS PAGE
$ 1 . INTRODUCTION . . . . . . .
. . . . . . . . . . . .
283
$ 2. FOCUSING INHOMOGENEOUS MEDIA WITH CEN-
. . . . . . . . . . . . . . . .
288
3. FOCUSING LAMINATED INHOMOGENEOUS CYLINDRICAL MEDIUM . . . . . . . . . . . . . . . . . .
297
9 4. FLAT LAMINATED INHOMOGENEOUS FOCUSING MEDIA. . . . . . . . . . . . . . . . . . . . . . . .
311
9 5. EXPERIMENTS . . .
. . . . . . . . . . . . . . . .
321
$ 6 . METHODS FOR THE CALCULATION OF INHOMOGENEOUS FOCUSING MEDIA . . . . . . . . .
324
TRAL SYMMETRY . . (i
.
$ 7 . QUASI-REGULAR CYLINDRICAL INHOMOGENEOUS
MEDIA.. . .
. . . . . . . . . . . . . . . . . . . .
332
9 8. CONCLUSION. . . . . . . . . . . . . . . . . . . . . 342 REFERENCES. . . .
. . . . . . . . . . . . . . .. . . .
343
Preface In recent years there have been many successful investigations of the optical elements, formed by inhomogeneous media. Among them are lenses and fibers; because of its wide usage for optical communication systems, the SELFOC must be specially mentioned. Optical elements with variable refractive index have a number of peculiar characteristics that are of great interest. They can be realized in a two-dimensional form, using thin film technology. Such elements are very important in the domain of integrated optics, being widely used in systems devoted to communication, reception and processing. The development of integrated optics led to a demand for two-dimensional elements, for example modulators, deflectors, connectors, switches, filters, etc. The present article is devoted to the problem of wave propagation in nonuniform focusing media. Besides the classical SELFOC, first described by the author in 1951, other new types of SELFOCS are considered here. The methods used for the solution of the inverse problem of geometrical optics are also discussed. Such methods may be successfully applied for calculations of different optical elements. It is my hope that this article will be useful not only for specialists who are interested in questions of the utilization of transparent media with given distributions of the refractive index for the realization of different types of optical elements, but also for those who are engaged in investigations of phenomena arising from the nonuniformity of the laser media in processes of generation and amplification of coherent radiation. A. M. PROKHOROV Lebedev Institute, Moscow, USSR April, 1978
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8 1. Introduction 1.1. STATE OF THE FIELD
In the last 15-20 years articles on the analysis of inhomogeneous media and their applications in optics have been appearing more frequently than ever, with various proposals concerned with new optical elements formed by inhomogeneous dielectrics and descriptions of the technology for the manufacture of transparent media whose refractive index varies according to a given law. A number of specific applications of inhomogeneous media have been developed. Among them the self-focusing optical waveguides are of special interest because they are convenient for transmission of information in wide-band systems. The problem of production and application of lenses with variable refractive index has also been widely discussed. Thus a new field of optics originated, concerned with the application of inhomogeneous media to different optical elements and to systems manufacturing. This new field includes not only focusing systems, lenses and waveguides with variable refractive index, but also a number of new systems that are of interest for processing and transmitting information. The interest in the use of inhomogeneous media has arisen not casually but is connected to a considerable degree with the development of lasers and with other advances in coherent optics. A number of possible laser applications have become of immediate importance, particularly in radiooptics, in laser engineering, in holography, in integrated optics and in various related areas.
1.2. HISTORICAL REVIEW
Maxwell was among the first to consider inhomogeneous media in optics, when, in 1854, he described a lens called “Fish-eye” (MAXWELL [1854]). This lens is a dielectric sphere whose refractive index decreases
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Fig. 1.1. Maxwell lens. The rays radiated from any point placed on the sphere meet at the symmetrically opposite point (MAXWELL [18543.
from the center to the periphery according to the law
n ( R )=
n (0) 1+ ( R / R J 2'
where Ro is the radius of the sphere, and n ( 0 ) is the refractive index in the center. This lens ensures that rays from any point located on the surface of the sphere converge to a diametrically opposite point, as shown in Fig. 1.1. Nearly a hundred years later LUNEBURG [1944] analysed a more common type of spherical inhomogeneous medium with central symmetry where the conjugate point (focus) is situated outside the lens, as shown in Fig. 1.2. Naturally, in this case the variation of the refractive index is different and depends upon the focal lengths F1 and F2. The Luneburg lens reduces to the Maxwell lens when F , = F2 = R,. In the best known Luneberg lens one of the foci is located on the sphere (F,= Ro),and the other is located at infinity (F2= 03). In this case n ( R )= n(0)d2-(R/Ro)*,
I
I Fig. 1.2. Generalization of the Maxwell lens suggested by Luneburg; the foci are placed [ 19441). outside the sphere (LUNEBURG
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and by moving the source onto the lens surface, one can scan in a wide angle range without any distortion. The properties of the Luneburg lens make it particularly attractive in connection with the microwave range scanning problem, widely discussed in the 1950’s (FELDand BENENSON [1959]). Several kinds of Luneburg lenses were realized in the microwave range. The inhomogeneous medium was obtained in different ways, particularly by means of “artificial dielectric” (KOCK[19651). In 1951 the author described a self-focusing cylindrical medium with axial symmetry. This medium represents a dielectric waveguide where the refractive index decreases from the center to the periphery as the inverse hyperbolic cosine (MIKAELIAN [19511):
Here r is the radius of the cylinder and n(0) is the refractive index along the cylinder axis. In this case the multiple focusing of rays in propagation takes place, as shown in Fig. 1.3. This self-focusing waveguide, called SELFOC, is now widely applied as an optical fiber for wideband signal transmission, as well as for high resolution image transmission. The cylindrical media with axial symmetry, as well as with central symmetry, serve as a variable refractive index lens. It is easily seen that a section of a self-focusing waveguide in Fig. 1.3 represents a focusing lens. This lens was studied in detail and is known as the “Mikaelian lens’’ in Soviet literature (FELD and BENENSON [1959], ZELKIN and PETROVA [1974], ZHOK and MOLOTSCHKOV [1973]). The focusing properties of the medium, described by the expression (1.3), were first investigated experimentally in the microwave range (MIKAELIAN [19511). The laminated inhomogeneous media, described above, are the simplest ones. In 1952 the author studied more complicated cylindrical media with the refractive index depending on two coordinates (longitudinal and
Fig. 1.3. Self-focusing cylindrical waveguide (SELFOC). If the refractive index decreases along the radius as an inverse hyperbolic cosine function, the rays radiated from the axial [1951]). point source are periodically focused on the axis (MIKAELIAN
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transversal), and he showed that there exists among them an infinite number of self-focusing waveguides (MIKAELIAN [1952bl). We will now consider some of the investigations relating to the practical realization of waveguides made of inhomogeneous dielectric. The first attempts to produce inhomogeneous optical media was made in 1964. BERREMAN [1964a,b], MARCUSE and MILLER[1964] tried to create a self-focusing waveguide as a hollow pipe, filled with inhomogeneously heated gas. In the simplest case a cold gas is driven through a heated pipe, with the gas temperature smoothly decreasing from the periphery to the center. This causes gas density variation, and, therefore, refractive index variation. Under certain conditions Marcuse and Miller succeeded in the realization of refractive index variation according to the quadratic law along the radius, a variation that is equivalent to that associated with the first and second terms of the expansion of the exact law. Because of their complexity, such “gas self-focusing waveguides” did not find practical applications. Attempts at creating a self-focusing waveguide of fiber-glass turned out to be more successful. Such a waveguide was made by UCHIDA, FURUKAWA, KITANO,KOIZUMI and MATSUMURA [1970]. The creation of a fiberglass with variable refractive index was the decisive factor for the establishment and further development of the branch of optics that is connected with applications of homogeneous media. In recent years different types of self-focusing waveguides of practical interest have been realized. In particular they are beginning to be used as “optical cables” in communication systems. This application is of current importance, since further development of common cable communication lines is restrained by severe economic difficulties. It is important to mention that the manufacturing process of the glassfiber with variable refractive index makes it possible to produce not only the simplest laminated self-focusing waveguides, but also waveguides with longitudinal variation of the refractive index, and some lenses of the types that we have mentioned. It is likely that inhomogeneous media will find many useful applications in the relatively near future. 1.3. SUMMARY
This present review can be divided into two parts. In the first part laminated self-focusing media are analysed. These are the simplest
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inhomogeneous media, where the refractive index depends only upon a single coordinate. First we will consider lenses with central symmetry (0 21, the Maxwell lens and the Luneburg lens being the most interesting among them. In § 3 cylindrical media with circular symmetry are considered. Among them is the optical waveguide with refractive index varying as the inverse hyperbolic cosine. In P 4 two-dimensional focusing media will be discussed. The second part of the review is devoted to the more complicated focusing media, that are characterized by the refractive index variation. Methods for the calculation of these media are stated in 0 6. In § 7 some examples of the application are given for the calculation of cylindrical waveguides with the refractive index, that varies in the radial direction, as well as in the longitudinal direction. It is necessary to mention that these methods, although developed a long time ago, were not used until recent times. They are of particular interest in connection with the application of inhomogeneous media in optics, and may be useful for the design and calculation of new types of light-guides, lenses and other elements with variable refractive index. The subject under review is of interest not only for optics, but also for a number of other spheres of science and engineering, e.g., for ocean physics and geophysics, for atmospheric physics and radiophysics, etc. There is an enormous literature devoted to various phenomena involving inhomogeneous media. Many of these investigations were not concerned with optics; nevertheless, they contain useful information for our problem, especially in connection with theory, as is shown in books of ALPERT, GUINZBURG and FEINBERG [1953], TOLSTOY and CLAY [1966] and BREKHOVSKIKH [1957]. Naturally it is impossible to analyse all these studies in a single review, and a simple enumeration of them seems to be inappropriate. For this reason we will only consider in the present review some of the basic investigations that are directly concerned with applications of inhomogeneous media in optics. In the last years I discussed certain aspects of this problem with Rem. V. Khokhlov on several occasions. Rem. V. Khokhlov was especially interested in cylindrical inhomogeneous media, whose refractive index depends on both coordinates. H e studied such cases in order to gain some understanding of a number of phenomena connected with wave propagation in a laser medium, particularly in the presence of appreciable nonlinearities. Naturally, these discussions were very useful and they
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aided in making the present review more purposeful. I wish to mention that the idea of writing this review is to a large degree due to Khokhlov who read some of the sections (3.2, 3.3, 6.1, 6.2, 0 7). On behalf of myself and his numerous friends in the field of quantum electronics this review is dedicated warmly to his memory.
0 2. Focusing Inhomogeneous Media with Central Symmetry 2.1. MAXWELL LENS
As we already mentioned, the Maxwell lens was among the first known lenses with variable refractive index. It has been studied thoroughly and its characteristics have been analysed in a textbook on optics (BORNand WOLF[1964]) as well as in the textbooks on the antenna theory (FELDand BENENSON [1959], ZELKIN and PETROVA [1974]). For this reason we will only state the main characteristics of this lens. The Maxwell lens may be realized either as a sphere whose refractive index depends on the radius according to the law (1.1) or as a flat disk, which is the cross-section of a sphere by a plane that passes through its center. The refractive index of the disk in the polar coordinates is expressed by the same formula as the refractive index for the sphere, i.e., as
where ro is the disk radius (Fig. 2.1). If the refractive index on the outside of the sphere is assumed to be equal to unity, the refractive index in the center of the lens is equal to two. The ray paths are determined by the equation x z + y2+2yroctg y-r;',= 0,
(2.2)
where x = r cos y, y = r sin y (see Fig. 2.1). The expression (2.2) is the equation of the circles, symmetric about the y-axis. The extreme beams, radiated out of focus at the angle *$T,are the semi-circles with the center in the initial point (Fig. 2.1). The properties of the Maxwell lens can be conveniently studied in bipolar coordinates 6, q. The equation of ray paths is then simply q = const. (MIKAELIAN [1952b]).
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Fig. 2.1. A flat Maxwell lens.
Half of a Maxwell lens may serve as a focusing lens, since it transforms the homocentrical bundle of rays into a parallel one (Fig. 2.2). The field intensity distribution in the aperture, i.e., for x =0, can be calculated with the help of the equation (2.2) of the rays. The distances between the lens center and any point in the aperture is given by
We may use the relation
P ( Y )d r = I ( Pdp, )
(2.4)
where P(y)represents the polar diagram of the source radiation, located
l Fig. 2.2. Focusing Maxwell lens.
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACHON
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in the focus of the lens and Z(p) is the field energy distribution to be determined at the lens outlet. From (2.3)
and substituting from (2.5) into (2.4), we find that
2 1 2 I ( p ) = P ( y ) - cos23 = - P( y ) ro 2 To 1+(p/rJ2
*
We must take into account the fact that the refractive index at the lens output varies from point to point according to the formula (2.1). A certain part of the wave energy will be reflected from the output plane of the lens, the refractive index varying from the lens center to its edges. For this reason the field distribution varies in the aperture; it can be taken into account by means of the factor
which is the transmission ratio. It follows that the intensity distribution at the Maxwell lens output is determined by a product of three functions,
P ( Y ) * I ( P ) * Tb),
(2.8)
normalized to unity. The function P ( y ( p ) )depends only on the source, and the function T ( p )depends not only on the lens, but on the degree of coordination between the lens and the environment. For instance, if n(0) = 2, the refractive index with r = ro is, according to eq. (2.1), equal to unity and no special procedure is needed in order to decrease the reflection from the lens output plane; the function T ( p )in the center will have the value 8/9, and at the edge it will reach unity. Therefore it will decrease somewhat the irregularity of the field distribution that is determined by the function I ( p ) . This function characterizes the lens and is determined by the equation of ray paths. For the Maxwell lens:
The Maxwell lens was until recent times not realized in microwave technology nor in optics. Nowadays a technology for the manufacture of
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29 1
Fig. 2.3. Flat lenses with central symmetry may be manufactured from the glass cylinder with the variable refractive index.
cylindrical glass fibers with variable refractive index can be utilized for manufacturing plane Maxwell lenses (cf. Fig. 2.3). Finally we mention an interesting peculiarity of the Maxwell lens: If a parallel bundle of rays impinges on the lens, it is diffracted to a considerable degree, as shown in Fig. 2.4. It is evident that in some cases in laser optics, it is desirable to have the most widely spread ray possible instead of a narrow one.
2.2. LUNEBURG LENS
Unlike the Maxwell lens the Luneburg lens was manufactured in different versions to operate in centimeter and decimeter ranges. In this connection it has been thoroughly studied both experimentally and theoretically. We will state here the results that are of interest for the optical bandwidth. The Luneburg lens is shown in Fig. 1.2. It may be regarded as a generalization of the Maxwell lens to the case when the two conjugate
I \
Fig. 2.4. Scattering of the plane wave penetrating through the spherical Maxwell lens.
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
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foci are situated outside the lens on a straight line, which passes through the center of the lens. In a general case this lens transforms a spherical bundle of rays radiated out of the focus into a spherical bundle of rays, converging to the other focus, as shown in Fig. 1.2. LUNEBURG determined the refractive index of this lens with central symmetry [1944] in a rather complicated parametric form. The equation describing the function n( r ) in a case of a flat lens with the radius, equal to unity ( r O = 1) and the refractive index at the edge also equal to unity, is given by with where
In a particular case with F2 = m we have = eq(N.
F,)
= Ne-q(N. F , )
(2.13)
For practical purposes this is a particularly interesting case.* Fig. 2.5 shows that by moving the source along the arc of radius F, it is possible to scan the radiation pattern formed by the lens. It is important to note that in this case the diagram is not distorted at all.
Fig. 2.5. Luneburg lens for the scanning.
* A table of the values of q ( N , F ) for 1 S F S 2 and for O S N S 1 is given in the book of ZELKINand PETROVA[1974].
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Fig. 2.6. An example of Luneburg flat lens as a scanner; the end of a fiber moves along the arc.
The flat modification of the Luneburg lens, as well as the Maxwell lenses, may be easily realized in optics by means of modern technology (cf. Fig. 2.3). It is possible to design different optical devices (including the devices for integrated optics), that can scan the narrow laser beam in a single plane. Fig. 2.6 shows the simplest example of the design, where the source of radiation is the glass fiber, excited by a laser. The faintly directed radiation out of the glass fiber is transformed by the lens into a narrow beam, which may be deflected, moving the fiber end along the arc of the radius F,. If F, = 1, i.e., if the focus is located on the surface of the lens, the dependence n ( r ) on r is rather simple. In fact the expression (2.12) gives, in this case,
and from this
n ( r )= J i - 7 .
(2.16)
In this modification of the Luneburg lens the scanning of the ray is implemented by moving the source along the lens surface (Fig. 2.7). The Luneburg lens is of great importance for application in optics, not only in connection with the problem of scanning. For instance, it may be easily used for the creation of specific radiation patterns.
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACI'ION
1'
I"'
X
6 I
I
I
I
Fig. 2.7. Scanning Luneburg lens with a focus placed on the surface.
In Fig. 2.7 it is shown how the wave, striking the Luneburg lens, is automatically transformed into a flat wave. The intensity of such waves may be fixed by a receiver, which moves on the lens surface. The Luneburg lens has also been found to be applicable as an analyzer of field structures and for field expansion in a spatial spectrum (ANDERSON,DAVIS,BOYDand AUGUST[1977]). As already mentioned the Maxwell lens is a particular case of a generalized Luneburg lens, with F, = F2= 1. It follows from (2.10) and (2.15) that (2.17) from which it follows 2 n(r) = 1+r2'
(2.18)
This result is in agreement with eq. (2.1) with r,= 1, n(O)=2. The Luneburg solution (2.10) corresponds to a case in which the refractive index at the edge of the lens is equal to unity with the focus, located either on the lens surface or outside the lens, and n(r) being a continuous function, monotonically decreasing from the center of the lens to its edge. MORGAN [1958, 19591 and TORALDO DI FRANCIA [1961] found another solution, free of the above restrictions. It differs from (2.10) by a supplementary factor, that facilitates the analysis of the Luneburg lens and is of great interest as well.
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Fig. 2.8. The variation of the refractive index in Luneberg lenses with the foci F, = 1.1, F2 = m. 1, ordinary lens; 2, lens with the uniform external layer; 3, lens with the nonuniform external layer (ZELFTN and PETROVA[1974]).
As an example consider the curves n(r) for different lenses with the same foci (F,= 1.1, F 2 = m ) , as shown in Fig. 2.8. The curve 1 corresponds to the ordinary Luneburg lens, and the curves 2 and 3 were obtained for the lenses with the upper surface layer. The introduction of a homogeneous layer from r = 0.87 up to r = 1, that has the refractive index n = 1.15 causes a decrease of the range of variation n(r); however, the refractive index in the center increases to 1.4. With the introduction of an inhomogeneous surface layer from r = 0.79 up to r = 1, the refractive index in the center is not maximum; its value is less than the refractive index at the edge. The solutions obtained by Morgan and Toraldo di Francia, include also the Luneburg lens with a focus situated inside the lens. This lens, called the modified Luneburg lens, was analysed for the first time by BRAUN [1955] and GUTMAN [1954]. It has been shown that the refractive index of the lens is determined by the following expression: (2.19)
Fig. 2.9 shows the dependence n ( r ) for the modified lens with F, = 0.5, F2 = m, n(0) = 2.24. The dotted line corresponds to the lens with homogeneous surface layer, with n = 2, in the range of r from 0.5 to 1; n(0) at the center of this lens is equal to 2.35. The equation of the ray paths of the Luneburg lens with the radius r, and with the focus, situated at the surface, is x 2 - 2xy ctg y
+ y’( 1+ 2 ctg’ y ) = 4.
(2.20)
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Fig. 2.9. Refractive index in a modified Luneburg lens, F , = 0.5, F2 = 00; 1, without external layer; 2, with uniform external layer (GUTMAN [1954], BRAUN[1955],ZELKIN and PETROVA [ 19741).
This equation describes the family of ellipses with the angle y at which the ray emerges from the focus as a parameter. It is interesting to note that for the extreme rays, that correspond to y = *;T,the above equation transforms to x 2 + y 2 = r & i.e.;to the equation of a circle. The intensity distribution of the output of the Luneburg lens may be calculated by the same method as for the Maxwell lens and is determined by eq. (2.8). For an ordinary Luneburg lens with the radius equal to unity and with the focus situated on the surface (Fig. 2.7) yo = sin y,
dy, = cos y dy.
(2.21)
Consequently, (2.22) i.e., the intensity increases from the center of the lens to its edges. It should be noted that unlike the Maxwell lens, the Luneburg lens may be easily matched with the environment, if the refractive index on the lens surface is equal to unity. As it was noted before, the flat Maxwell and Luneburg lenses may be realized in the optical bandwidth. The application of these lenses is important for integrated optics (KOGELNIK[1975]). Some steps in this direction have already been made (ZERNIKE [19741, ANDERSON, DAVIS, BOYDand AUGUST [1977]).
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9 3. Focusing Laminated Inhomogeneous Cylindrical Medium 3.1. INTRODUCTION
We have already seen that inhomogeneous media with central symmetry, both volume and flat, have two foci and may be used as lenses with the variable refractive index of different types. We now consider the focusing properties of the regular laminated inhomogeneous cylindrical medium. We imply by this term that the properties of the medium are the same at every point on the z-axis. As we did in the case of inhomogeneous media with central symmetry, we will analyse the simpler case of ray focusing in a cylindrical medium with circular symmetry, where the refractive index depends only upon the radius. For this purpose we introduce cylindrical Coordinates r, 8, z, and analyse the laminated inhomogeneous medium with the distribution of the refractive index n(r, 0) = n ( r ) . It is evident that if we find the solution n ( r ) when the spherical wave of the source point, situated on the z-axis, gradually transforms to a flat wave in the course of propagation, we effectively determine the cylindrical laminated inhomogeneous medium, with multiple ray focusing. Indeed, in such a medium there are an infinite number of foci, and, in the course of propagation, the rays will be repeatedly focused, as it is shown in Fig. 1.3. The laminated inhomogeneous medium with the above mentioned characteristics is practically a cylindrical waveguide with variable refractive index, in which all the rays have actually the same optical length. For this reason such a self-focusing waveguide can be successfully used for both image transmission in a wide bandwidth, and image transmission with high resolution. A section of such a waveguide is a lens with variable refractive index, which has been thoroughly studied both theoretically and practically. Thus, laminated inhomogeneous cylindrical media, unlike media with central symmetry, have much wider usefulness in practical application.
3.2. SELF-FOCUSING CYLINDRICAL WAVEGUIDE
The first self-focusing waveguide, or, as it is now frequently called, “selfoc”, was described by the author in 1951. It was made as a dielectric
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
z =l?
Ea0
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L =4
Fig. 3.1. Laminated inhomogeneous cylindrical SELFOC. Refractive index depends only on the radius, i.e., n ( r ) = n(0)lch fn-r; rays are periodically focused at the planes z = 0,2,4, . . . (MIKAELIAN [19513).
cylinder, in which the refractive index depended upon the radius, according to the expression (1.3). In such a waveguide the rays, radiated at different angles by a point source placed on the waveguide axis, are focused repeatedly in the points z = 0 , 2 , 4 , . . . , shown in Fig. 3.1. We might mention that the established law (1.3) of the inverse hyperbolic cosine is the only possible one for the realization of a self-focusing cylindrical waveguide with circular symmetry, i.e., a laminated selffocusing waveguide. The expression (1.3) can be obtained by solving Euler equations in cylindrical coordinates: ds
(n
fi) - n r g r =$,
ds
where
and the equation of ray paths is defined by r ( z )and O(z). The boundary conditions can easily be determined by examination of Fig. 3.1. In the initial point, where the source of the spherical waves are situated, all the rays are radiated from a point at different angles to the
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z-axis. Hence for z
299
= 0,
r = 0,
drldz = tg y,
(3.3)
where y is the angle that the ray at the source makes with the axis of the cylinder. With z = 1 the rays become parallel, and consequently
(3.4) It is easy to see that since all the rays are radiated from the same point and the waveguide has cylindrical symmetry, the path of every ray will be placed at the same diagonal plane, i.e.,
The equation of every ray path, expressed with the help of an independent variable z , will be expressed in the parametric form r =r(z),
8 = 8 ( z )= const.
(3.6)
Taking into account (3.5) we find that the system (3.1) reduces to
an ds
ar '
(3.7) (3.8)
Using (3.2) the second equation can be written as: drldz =J(n(r)lC,)2-1.
(3.9)
Next, the boundary condition (3.4) will be used. One then finds
c1=n(r)lz=l= n(5),
(3.10)
where 6 is a value of r in the plane z = 1. Substituting the value of C in the expression (3.9), we obtain, after integration (MIKAELIAN [195 13):
(3.11) It is easy to check that the law of variation of the refractive index, expressed by (1.3), is the solution of this integral equation.
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rv, § 3
Using (3.1 1) we may readily determine the equation of ray paths. It has the form (MIKAELIAN [ 195 13): ?r
(3.12)
where y is a parameter that fixes a ray of the family. In our case the value of y defines the angle between the ray direction, radiated from the initial point, and the positive 2-direction. We can easily check that the equation (3.12) describes, in fact, the family of ray paths, shown in Fig. 3.1. We have shown that if the refractive index of a dielectric waveguide decreases with increasing distance from the cylinder axis according to the inverse hyperbolic cosine law, the wave, while propagating, is periodically focused in the points spaced at equal intervals. It is important to emphasize that there does not exist any other cylindrical waveguide with central symmetry, that possesses ideal focusing characteristics.
3.3. MIKAELIAN LENS
It is easy to see from Fig. 3.1 that a section of a self-focusing waveguide is a lens with variable refractive index (Fig. 3.2). The refractive index of the lens decreases in radical direction according to the inverse hyperbolic cosine law, in conformity with the expression (1.3). This behavior is shown graphically in Fig. 3.3. The equation of ray paths is determined from eq. (3.12). According to the adopted scale the focal length of the lens, as well as its thickness, is equal to unity.
Fig.
3.2. Mikaelian lens (FELDand BENENSON [1959], ZELKIN and PETROVA [1974]).
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-40
-4.6
-42 4 8
-03
0
0,4
ll8
1.2
301
t6. 40'
Fig. 3.3. Refractive index distribution in a cylindrical SELFOC (MIKAELIAN [195 11).
The intensity distribution at the lens output is determined by the expression (2.8). From the expression of ray paths (3.12) we find that tg y = sh i r p ,
(3.13)
and hence (3.14)
As is easily seen, the field intensity decreases from the lens center to the periphery according to the same law as the variation of the refractive index. The scanning characteristics of this lens were studied by ZELKIN and ANDREEVA [1968]. Y. A. Zaitsev has carried out calculations for the case, when the focus was moved off the lens surface. Such a lens was called a generalized Mikaelian lens (FELDand BENENSON [1959]). An ordinary lens is a particular case of this more general lens. An expression for the refractive index of the generalized lens, calculated by Zaitsev, is as follows (FELDand BENENSON [1959]): n ( x )=
n(O)d,- D,(sec y - 1) do ch ( ~ / 2 d o ) ( -xX I ) '
(3.15)
where x, is the coord.inate of the crossover of the inner lens surface and a ray that has a coordinate x at the lens output (see Fig. 3.4). The values x
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Fig. 3.4. Generalization of Mikaelian lens suggested by Zaitsev and Kelleher, Goatley [1959]). (FELDand BENENSON
and x1 are connected by the following relation:
(3.16) For the given values of x1 we calculate the value x from (3.16), and then find n ( x ) according to the expression (3.15). It is easily seen that x,=O with Do=O, and we obtain the expression (1.3), that corresponds to the ordinary lens. Both of the lenses described above have been thoroughly studied by different authors. The most detailed description is given in the book of ZELKIN and PETROVA [1974].
3.4. WAVE PROPAGATION IN A SELF-FOCUSING MEDIUM
As we already mentioned, in a self-focusing waveguide the ray paths have the same optical length. Owing to this fact, distortions caused by the blurring of the impulse in a wideband signal, become minimum. Therefore the self-focusing waveguide is of great interest for the creation of optical communication lines in a wide bandwidth. It is evident that the multiple-ray focusing in the course of wave propagation becomes more apparent if the conditions for applicability of geometrical optics are fulfilled more strictly. Even in the first experiments these conditions were fulfilled quite well. In particular, the waveguide diameter was several hundreds wavelengths, and the pattern of ray paths,
FOCUSING LAMINATED INHOMOGENEOUS CYLINDRICAL MEDIUM
303
shown in Fig. 3.1, was seen quite clearly. If we use a point source while transmitting a signal, or if we excite the waveguide by a plane wave, which impinges on a butt-end along the waveguide axis, the distortions, caused by the waveguide dispersion, in the ideal case, should be absent. However, in the course of propagation, an excitation of new types of oscillations will always be present, and there will be energy exchange among them, because any real optical line inevitably has different heterogeneities (bendings, joints, heterogeneousness of the medium, etc.). In consequence, the information transmission in a self-focusing waveguide will be realized in different modes, and the analysis of the dispersion characteristics of a waveguide naturally becomes very important. The multimode waveguide is equivalent, in the final analysis, to a waveguide excited by many point sources, situated off the waveguide axis. One can consider this multimode waveguide also as a waveguide excited by many plane waves, which impinge on the butt-end at different angles. The analysis of the dispersion of a self-focusing waveguide requires a strict solution of the problem of wave propagation in a cylindrical medium with the refractive index, varying according to the hyperbolic cosine law. In other words, the problem is to find the spectrum of initial oscillations of a self-focusing waveguide. For this case we have the wave equation (3.17)
where y is a constant of propagation, and m is an integer, we obtain: (3.19) Next we set
f(r) = $(r)
*
rf
(3.20)
and obtain from (3.19)the following equation for f(r): (3.21)
304
[V, 8 3
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
After substituting for n(r), we obtain the equation (3.22) Unfortunately this equation has not been analysed, and hence the exact expressions for the modes of self-focusing waveguides are not known. Approximate solutions may be obtained by expanding n*(r): n2(r)=
[
n'(0) = n2(0) 1(ch (27rrlL))'
r)2 +
f
(z - ] ry*
..
(3.23)
Taking into account the first two terms in (3.23), we obtain a parabolic waveguide, which has been studied by a number of authors ARCUSE USE and MILLER [1964], MARCATILI [1967], STREIFER and KURTZ[1967]). In this case
This equation reduces to the equation of Whittaker. The solution, with r = 0, is expressed in terms of hypergeometrical functions. With
27r y 2 = c ) n 2 ( 0 ) - - K()n(0)(4~+2rn +2),
L
(3.25)
where s = 0,1,2, . . . ; rn = 0 , 1 , 2 , . . . ; these functions approach zero as r + 00, i.e., they correspond to propagating waves. In this case they are reduced to the polynomials of Laguerre LJm), and the solutions of the equation (3.24) are given in terms of Laguerre functions 27T k m ( r ) = r m- e x p {-%,,n(0)-r2}. L
L:"'(K,,n(0)~r2).
(3.26)
Consequently, in accordance with the expression (3.18), the propagating modes are of the form ,,\r). us.m= +/,m (r) . elmeel(wt--Yv
(3.27)
The structure of a mode field with circular symmetry (rn = 0) has an amplitude maximum in the waveguide center, which considerably decreases with the moving away along the radius. For a lowest mode (s = 0) this decrease is monotonic. For s = 1 , 2 , 3 * * the amplitude decrease is of oscillating character, the number of zeros coinciding with the value of the index s. The amplitude of the modes with rn = 1 vanishes at the center.
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FOCUSING LAMINATED lNHOMOGENEOUS CYLINDRICAL MEDIUM
305
It is easy to see that ideal focusing cannot be achieved with the parabolic distribution of the refractive index, because different modes move with different velocities along the waveguide. In a case of modes with low indexes (which in terms of geometrical optics correspond to paraxial rays), one may assume that:
(2s + rn - 1)<
(3.28)
The distribution constant is given by
r
2T L
yS.,, = K o n ( 0 ) 1--(4s+2m+2)~K,,n(O)--(2s+rn+l).
(3.29) Consequently the velocity of propagation is constant for a lower mode group, satisfying the condition (3.24), i.e., ao
Vsvm = -- v(0).
(3.30)
aYs.m
As was mentioned above, the exact solution of the wave equation has not yet been found for the case when the distribution of the refractive index varies according to the inverse hyperbolic cosine law. We can approximate to this distribution after analysing the wave equation for the case represented by (3.23),where the first three terms in the expansion of the inverse hyperbolic cosine law in powers of r/L are taken into account. Considering the third item as the perturbation of the parabolic waveguide, we obtain (KAWAKAMI and NISHIZAWA [1968]): y:,
= K Z ( 0 -2K(O) )
+
(Fr-
2T L
-(2s + rn + 1)
[6s2+ 6 s ( m + 1) + (rn + l)(m + 2)],
(3.31)
or
(3.32) where
K ( 0 )= Kon(0)= 2 ~ / A ( 0 ) .
(3.33)
Ideal focusing occurs when the group velocity of all the modes is the same. In this case one may disregard in (3.32) the last term provided that: f(l-rn2)~~[L/h(0)-(2s+rn+1)]2.
(3.34)
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
rv, 5 3
It is evident that the expression (3.34) holds only for a very small group of modes, which are defined by the following condition:
2s + WI + 1 - L/h(0).
(3.35)
Consequently the focusing becomes better in comparison with the case of the “parabolic waveguide”.
3.5. IMAGE TRANSMISSION IN A SELF-FOCUSING MEDIUM
The possibility of employing a self-focusing waveguide for image transmission is very attractive. We have already seen that a point object, situated at the waveguide axis images perfectly at the planes, where the rays are focused. If a point object is not situated at the center, aberrations appear and will grow with displacement of the object away from the waveguide axis. This means that in the process of imaging, the resolution at the imaging plane becomes lower from the center of the waveguide to its edges. In order to make a quantitative analysis of the distortions, which appear in a laminated self-focusing waveguide during image transmission, it is necessary to determine the ray paths, radiated out of a point r = To, at the plane z =0, then find a trace, formed by the intersections of these rays with the imaging plane, and estimate the blurring of the image of the point. In order to find the ray paths it is necessary to solve the Euler equations (3.1) for a cylindrical medium, which has refractive index falling off in the radial direction according to the law (1.3). Assume that the point source is situated at the plane z = 0 with r = ro and 8 = O0, i.e., xo = ro cos O0, yo = ro sin O0, and let cos ao, cos Po, cos yo be the direction cosine of the angles, at which the rays are radiated out of a point source. Taking into account that in our case
n ( r ) d d d s = n(ro) cos yo,
(3.36)
we may rewrite the first two Euler equations as follows: (3.37)
(3.38)
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307
Substituting for dd/dz in the eq. (3.37), we readily find that
nzy
z + ( x o ~ ~ ~ ~ o - y o c o ._--.s 1a o ) 2 1 = 0. cos2 y o r2 cosz yo n’(ro)
(3.39)
Let Co denote the integration constant which we chose to be equal to the value of the square brackets at the boundary z = 0. If we use the fact that xo cos a0
+ yo cos p o
ro cos 70
(3.40)
9
it can be easily seen that the integration constant Co= -1. Hence instead of (3.39) we can write: 2
1
n2(r)
1-
(xo cos po- yo cos a0)’
rZ cos’ yo
.
(3.41)
Integrating once more, we obtain: lo‘dz =
1
dr/d---
0
1 nz(r) cos’ yo nz(ro)
- (xo cos po - yo cos aOl2 , r2 cos’ yo
(3.42)
where, for our case,
n(r)/n(ro)= ch $m0/ch$m.
(3.43)
So, we have integrated the system of Euler equations (3.37) and (3.38) and we can now easily calculate the path of every ray radiated out of the For this purpose, it is necessary to point r,, do at the angles ao, Po, yoyo. calculate the integral (3.42), i.e., to find r ( z ) . After substituting for r(z) into the eq. (3.38) and integrating
we can find d ( z ) . Knowing the path equations of all the rays r ( z ) , d ( z ) , we can determine the points of intersection of these rays and the image plane; we can find the dimensions of the spot, and estimate the distortions of image transmission. A system of eqs. (3.42) and (3.44) may be solved by a method of successive approximations. This method is rather accurate in the case of small objects. In this case we can assume an equation of ray paths (3.12), corresponding to a point object location on the waveguide axis as a
308
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
rv, 5 3
zero-order approximation. After substituting this approximate solution in (3.42) and integrating, we find the first-order approximation e ( z ) . It is interesting to note that in the meridional planes of a cylindrical waveguide, when x o cos po - yo cos (Yo = 0,
(3.45)
the system of eqs. (3.42), (3.44) reduces to a single equation:
(3.46) Having integrated this equation, we obtain the equation of the family of ray paths r(z, r,), which differs from (3.12) only by the initial point. This means that meridional rays, i.e., the rays belonging to a plane passing through the waveguide axis, are situated in this plane and are not distorted in the course of propagation. For instance, if the rays are radiated out of the point r,, do, they will gather again in the point r, = ro, Oh = O0. It is an interesting characteristic of a laminated self-focusing waveguide. In particular, it means that in a two-dimensional case, i.e., in the case of the flat laminated waveguide (interesting for applications in integrated optics), the image transmission of a linear object is free of aberrations. This fact will be discussed in more detail below. We note another property of the approximate solution of the Euler equation. We may again apply the method of successive approximations and we can estimate the distortions of the ray paths, radiated at a small angle to the meridional plane. A quantitative analysis of aberrations of the image in a self-focusing waveguide was made by RAWSON, HERRIO-IT and MCKENNA[1970]. The system of eqs. (3.37), (3.38) was solved. In this case at the initial plane z = O the ray parameters were given, and the point of this ray with the image plane z i m = 4 was calculated. The aberrations of the image were defined by the values (Fig. 3.5): - eo (3.47) zim where r, 8, are the ray coordinates at the image plane, 6, are the radial aberrations, and 6, are the tangent aberrations. In this way two groups of rays were analysed. The first group includes the meridional rays, which are radiated out of a point r,, e0 at different angles, but belonging to a meridional plane, crossing the waveguide axis and an object point (Fig. 3.6a).
r,
6, =-
- r,
2im
and
a,=-, eh
v, 0 31
FOCUSING LAMINATED INHOMOGENEOUS CYLINDRICAL MEDIUM
IXO
x
309
1111
Fig. 3.5. The rays, radiated at different angles from the point r,, Oo in the plane z = 0, have no common point in the image plane, but form the blurred focus.
The second group consists of rays that are radiated at different angles at the plane normal to the meridional plane (Fig. 3.6b). As was mentioned above, the meridional rays remain in the plane 8 = &, and they converge to the same point in the image plane. The aberrations of the image are caused by the second group of rays, which are displaced both in the radial direction and in the tangent direction. Accordingly there exist radial and tangent aberrations. We will present some quantitative results. If the waveguide radius is equal to O.lz,,, and rob is equal to 0.952, i.e., the object point is situated near to the periphery, the average radius of a spot is 0.012,. Consequently, the linear image resolution is near 10, i.e., very small. Evidently
Fig. 3.6. (a) The rays, radiated from the point r,, Oo, are situated in the meridional plane and have no aberrations; (b) the rays, radiated at different angles in a plane, normal to a meridional one, have the maximum aberrations.
310
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
rv, 5 3
the distortions will decrease with decreasing distance from the waveguide axis, remaining, however, quite considerable. For instance, if an image is situated near the axis, i.e., rob is equal to 0.095 of the waveguide radius, the image resolution will be equal to 30, i.e., it only increases three times. The calculation shows that a notable improvement of image quality is obtained with decrease of the waveguide radius. If a waveguide radius becomes ten times greater, i.e., equal to O.Olz,,, and rob remains equal to 0.95 of the waveguide radius, the aberrations become thirty times as low; hence the number of resolution elements in the image increases to 1000. In the final analysis the authors of the above mentioned paper conclude that the resolution with the image transmission depends to a considerable extent upon the ratio of the waveguide radius to its length and increases as this ratio decreases. Indeed, this unexpected conclusion follows from the calculations. Note should be taken, however, that all the calculations were made in the approximation of geometrical optics. Taking into account diffraction phenomena, it is necessary to correct considerably the conclusions mentioned above. In the case where the waveguide radius constitutes only 0.01 from its length, the dimensions of a spot at the image plane are not determined by aberrations, but rather by diffraction. For instance, for a waveguide with a diameter equal to l m m , i.e., to 1000m, the dimensions of a spot at the image plane will be about
AOSz, 2r
-=--
Az, 4 r
at the wavelength of 1m, i.e., in a waveguide of 5 cm length it constitutes 25 m, and in a waveguide of 50 cm length it constitutes 250 m. The first case corresponds to 40 spots, and the second case corresponds only to 4 spots. It is the very case, where the waveguide radius constitutes 0.01 from its length, and as a result of aberrations, the number of the resolution spots in the image equals 1000. As it is easily seen, the nature of the image is determined only by diffraction, and the calculation of aberrations is not necessary. Hence in analysing the problem of image transmission in a flexible self-focusing waveguide of 1 meter length or longer and when the ratio of the waveguide radius to its length does not exceed 1 per cent, it is necessary to take diffraction into consideration. If this ratio exceeds ten per cent, in short waveguides or in the case of unflexible waveguides (when they are rather thick), the image resolution depends on the aberrations, and diffraction may be ignored. Evidently in this case one is dealing with image transfer with the help of a self-focusing waveguide, or
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311
even with the help of a laminated inhomogeneous lens, rather than with transmission.
8 4. Flat Laminated Inhomogeneous Focusing Media 4.1. FLAT SELF-FOCUSING WAVEGUIDE
A flat model of a self-focusing waveguide is of great interest for practical applications, especially in view of the progress in integer optics. Such a waveguide was analysed for the fist time by MIKAELIAN [1951], the problem being formulated as follows: it was necessary to find a law of variation of the refractive index of a flat laminated inhomogeneous medium. According to this law the rays were periodically focused in the course of propagation, as it is shown in Fig. 4.1. The problem was considered in the approximation of geometrical optics and was reduced to the solution of the following integral equation:
with the following boundary conditions:
z = 0 , 2 , 4 * - * : X = O , dx/dz=tgy ~ = 1 , 3 , 5 * * * :X = t ,
(4.2)
dx/dz=O.
[1951]), that the refractive index As a result it was stated (MIKAELIAN of a flat self-focusing waveguide varies according to the inverse hyperbolic cosine law n ( x ) = n(0)lch f m ,
(4.3)
X
Fig.
4.1.
Multiple focusing of the rays in the laminated inhomogeneous medium with n = n(x) (MIKAELIAN [1951]).
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V, § 4
and the equation of the family of ray paths is defined from (4.1) and may be written as
2 x(z,[)=-Arsh
“
(4.4)
where [ determines a particular ray and is numerically equal to the maximum ray deviation from the z-axis. It is easy to prove that sh &r[ = tg y. Consequently eq. (4.4) coincides with the equation of rays of a cylindrical self-focusing waveguide in a meridional plane. A section of a self-focusing waveguide is a focusing lens (Fig. 4.2). As it is easily seen, the refractive index of a medium varies according to the hyperbolic cosine law* for a flat waveguide, as in the previous case. The theory of electromagnetic wave propagation in a medium with the above law of refractive index variation, i.e., the wave theory of a flat self-focusing waveguide, was developed comparatively recently (KORNHAUSER and YAGHISAN [1967], KAWAKAMI and NISHIZAWA [1968]). In essence this theory does not lead to any new results, but it helps to clarify the unique focusing properties of a laminated inhomogeneous medium with the refractive index that varies according to the hyperbolic cosine law. X n=
n(z)
’, I I I I
0
I
IZO I I
I I
I
n = const
--
-
t
-
-
I Fig. 4.2. A flat focusing lens with the variable refractive index (MIKAELIAN [1951]). *.The possibility of partial focusing of elastic waves in such a medium was mentioned in a paper by SLICHTER [1932], concerned with the analysis of seismic phenomena in the ground. In this paper a flat vertical cut of ground is analysed, and a boundary between the earth and the atmosphere is assumed to be a straight line, It was shown that if a source of elastic waves was situated on the boundary of the section (x = 0), and the phase velocity in a half-space of the ground (x > 0) varied with the depth according to the hyperbolic cosine law, the part of “rays” which penetrated into the earth, gathered on the earth’s surface in a common point.
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FLAT LAMINATED INHOMOGENEOUS FOCUSING MEDIA
2=2
Z=4
2=6
313
2=8
Fig. 4.3. A point source, placed in any point of the laminated medium, images without any distortion (ZELKIN and ANDREEV [1969]).
Before we state some results of the wave theory, we consider an interesting property of a flat self-focusing waveguide. It turns out that if a point source is shifted in a transverse direction, the rays will still be periodically focused in the planes z = 2 , 4 , 6 . . . , in the course of propagation, as it is shown in Fig. 4.3. Hence such a medium possesses ideal imaging characteristics. For example, if a (one-dimensional) object is situated in the plane z = 0, perfect image is to be observed in the planes z = 4 , 8 , 1 2 . . . . In the planes z = 2,6,10 * the image is also perfect, but is inverted. Evidently if a flat self-focusing waveguide is not excited by a point source, and is excited by a plane wave impinging at any angle (Fig. 4.4), the wave will not be focused in the course of propagation.* However the phase distribution, which exists in the initial plane z = 0, will be repeated in the planes z = 4 , 8 , 1 2 . .. . In the intermediate planes z = 2 , 6 , 1 0 * the wavefront will be plane, but inverted at the angle (360”-2a).
-
* The fact that there is no ray focusing with the waveguide transmitting an inclined plane wave, is of practical interest in quantum electronics. For instance, a laser medium becomes considerably inhomogeneous as a result of intense pumping and nonlinear effects in the process of oscillation and amplification of high powers and destruction of the laser medium, which would begin in the areas of ray focusing, is possible. These phenomena can be considerably reduced by application of “waveguide resonators” with the excitation of a laser medium by an inclined wave instead of conventional Fabry-Perot resonators. Such waveguide resonators were analysed in a series of papers (MIKAELIAN and DIACHENKO [ 1 9 7 2 , 19741).
314
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
I
I
I
I
I
I
I
I
rv, 8 4
I
E
I
I
I
I
I z -2
z =4
z -0
I
Fig. 4.4. A front of the plane wave propagating in a two-dimensional self-focusing waveguide reproduces periodically.
The characteristics of the laminated medium, mentioned above, were studied in detail by ZELKINand ANDREEV [1968], ZELKIN and PETROVA [1974]. If we carefully analyse Fig. 4.3 and Fig. 4.4, we can easily see that a time interval, which is necessary for a ray to pass from the plane z = 0 to any parallel plane (for instance, z = 2), is independent of the output point xo and the output angle yo. In other words, in a flat self-focusing waveguide all the rays have the same optical length. In terms of the wave theory this means that all the waveguide modes have the same velocity of propagation. Let us try to obtain this result from the solution of the waveguide equation for an inhomogeneous medium E ( x ) :
AE + K2(x)E= -grad
(E- griE) -grad (Ex ; ;), =
-
(4.5)
where the time dependence is assumed to be e-j''". For the simplest case, when the electric field has no component along the x-axis, the right-hand side of the equation is equal to zero, and the equation reduces to:
a2Ey a2E, -+7+ K2(x)EY= 0, ax2 az
(4.6)
v, I41
315
FLAT LAMINATED INHOMOGENEOUS FOCUSING MEDIA
where
K 2 ( x )= %n2(x) = K',
n2(0> ch2 (2lrxlL)'
(4.7)
Here L is the path period along the waveguide axis (see Fig. 4.1). The solution is of the form
E,(x, z ) = $(x)ejY',
(4.8)
where y is a constant of propagation. On substituting from (4.8) into (4.6) we obtain the following equation for $(x):
&+ dx2
( K 2 ( x )- y2)$ = 0.
(4.9)
Let us introduce a new variable
5 = th (2lrxlL).
(4.10)
We then obtain from (4.9) the equation d2$ - 2*-d$ + (1 - E2) 2 dt
a
where U(U+
1) =
[
U(U
(&Y,
m2 + 1) -1-62
(4.11)
L y. 21r
(4.12)
rn =-
h(0) is a wavelength in a medium with the refractive index n(O), i.e.:
K,n(O) = K ( 0 )= 2n/A(O).
(4.13)
Equation (4.11) can be transformed to the Gauss equation, the solution of which may be expressed in terms of hypergeometrical functions. Since we are interested only in propagating waves, the field in the infinity must be assumed to vanish, i.e.: $(x) + 0 with
x + fm,
(4.14)
For this to be the case we must have: u - m = p,
p = 0,1,2 -
*
-.
(4.15)
It follows that the constant of propagation of different modes will be given by the following relation:
316
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
w,14
Under the condition of (4.15) the hypergeometrical series, defining the solution of eq. (4.11), terminate and can be expressed in terms of Jacobi polynomials. Consequently the field structure of the propagating modes will be characterized by the following function: dP Jl,(x) = CP((1-t2)-mr,/2 -(1 - 6 2 ) ’ j dtP
(4.17) 6 = th ( 2 m x l L )
Usually L >> A (0). Consequently v is a very large number and may be considered to be an integer. In this case the Jacobi polynomial degenerates into the associated Legendre polynomial and the solution will take the form (KAWAKAMI and NISHIZAWA [1968]):
E(x, z, t) = cL,(x) exp
-
= P L r n ( t h F x ) exp (-j(ot = P;&(th$x)
*
- m p Tz
exp (-j(wt-
and for pth mode
(4.19) It is easily seen that p determines the number of wave periods along the x-axis. The solution (4.18) is practically the same as (4.17), but it is more transparent. It is readily seen, for instance, that the mode with the index p=O is the lowest mode. Further it can be noted that the greater the number p, the greater will be the number of zeros in the interval (-w, +w) along the x-axis. In view of the properties of the associated Legendre polynomials with an odd p, the field in the center is equal to zero and is given by an odd function of “x”. With an even “p” the field is symmetrical with respect to the waveguide axis. With any m # 0 the field at infinity is zero because the argument of the polynomial tends to unity. The curves of the field distribution for the first three types of oscillations are shown in Fig. 4.5. The constant of propagation of the pth mode is equal to:
27F
27F
YP=r%=A(O)-F
27F
27F p=K(O)--p. L
(4.20)
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317
f
475
45
425
Fig. 4.5. Field distribution for the first three modes; L = 1 m, Y = 1, n(0) = 2.
It follows from this expression that the group velocity, i.e., the velocity of the propagation along the waveguide axis for all the modes, is the same and is equal to: am
(4.21)
u, =- = u(O), aY
where u ( 0 ) is a velocity at the waveguide axis. Irrespective of the number of modes, the field structure is periodically focused with the interval, equal to L. For instance, with z =0,
(4.22) where C, is an amplitude of the pth mode. With z = L E(x,L)=~C,+,(x)exp
=eiZT”E(x,O). (4.23)
P
As we can see, the field has a constant phase factor (the same for all the modes). It easily follows that a focusing takes place with z =$L. We see that the wave equation leads to the same result as predicted by geometrical optics.
318
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V, 8 4
It is important to note that according to (4.16), there exists a finite discrete spectrum of the free oscillations of the waveguide, because
Im,I< v.
(4.24)
This means that even if the waveguide is infinite in transverse direction, the dimensions of the spot in a focus will be finite. Naturally in the course of propagation the focusing will be repeated at intervals equal to L, and the dimensions of the spot will remain constant. It is interesting to consider Gaussian field distribution of the wave, impinging on the boundary z = 0 between the inhomogeneous and homogeneous media. The analysis of the cases of a final aperture when the very slow blurring of a focus occurs in the course of wave propagation, is of practical interest as well.
4.2. PARABOLIC WAVEGUIDE
A medium in which the refractive index varies according to the quadratic law is not a self-focusing medium, and from this point of view it is of no practical interest. Nevertheless one must pay attention to a plane parabolic waveguide, because, first, it is easily realized, and second, it differs slightly from a self-focusing waveguide in the case of paraxial rays. The propagation of the electromagnetic waves in a flat medium with the parabolic law of the refractive index variation, was apparently analysed for the first time by KORNHAUSER and HELLER [1963]. In particular, they noted focusing of paraxial rays, and have shown that this case corresponded to the addition of lower modes in phase, which could be obtained from a wave solution of the problem. Analysis of a parabolic [1964], GORDON [1966] etc. waveguide was also carried out by MARCATILI STREIFER and KURTZ[ 19671 established the relation between the solutions for flat and cylindrical waveguides with the help of asymptotic methods. They used conditions under which the associated Laguerre polynomials are transversed to Ermit polynomials, which described the free oscillation of a flat parabolic waveguide. Let us consider the solution for a “parabolic waveguide” in the approximation of geometrical optics (MIKAELIAN [19771). Suppose that
n ( x ) = n(0”
27r [1 -51 ( 7 x) ]
- Ax2] = n(O>
(4.25)
V, § 41
319
FLAT LAMINATED INHOMOGENEOUS FOCUSING MEDIA
and evaluating the integral (4.1), we find the ray paths in the following form: x(z,
6) = 6 cos (m=) z
=
2T L
6 cos - z
1 1 -4(2715/L)2
(4.26) *
We assume that the rays impinge normally on the butt-end of the parabolic waveguide, i.e., with z=o:
x=c,
X’=O.
(4.27)
As it is clear from eq. (4.26) the rays propagate along the cosine curves, the period of which depends upon the ray input coordinate. Consequently, they gather not in a point, but in the section
(4.28) Taking into account that the focal length F=$L, we determine the “relative blurring” in the first focus, as
(4.29) and it follows that
(4.30) There is more blurring at the next focus, which occupies a section, three times as long (Fig. 4.6). In the course of propagation A F reaches its
Fig. 4.6. Blurring of a focus in a “parabolic” waveguide.
320
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
rv, 5 4
maximum and then begins to reduce, with further periodic alteration near an average value. Naturally for paraxial rays, when ,$<< 1, defocusing is not so obvious. We give two numerical examples. Let, for instance, An/n(O) = 0.1. Then = 5 mm, with 6, =0.5 mm, F = 1.5 mm, AF=0.15 mm; and with trnX F = 15 mm, A F = 1.5 mm. If the inhomogeneousness of the medium is we obtain for these cases with very faint, for instance, An/n(O) = trn,= 0.5 mm, F = 50 mm, A F = 5 p m ; with = 5 mm, F = 50 cm, A F = 5 0 p m . The very first of these examples resembles the case of a real optical waveguide. It shows that in practice, for instance, for constructing optical communication lines, it is necessary to use dielectric waveguides with refractive index which varies according to the hyperbolic cosine law. With a parabolic waveguide we obtain considerable distortions. The second example corresponds to the case of the solid-state laser, where the inhomogeneity of the refractive index of the laser medium is connected with light pumping, and also with nonlinearities, appearing at large amplitudes of the electric field. The inhomogeneity of the refractive index and the focusing connected with it, are the causes of a series of phenomena, observed during oscillation and amplification of laser radiation. The effects of destroying the inner layers of laser crystals may be noted here, and also the observation of some foci with very strong pumping, the effects of selffocusing, etc. One can have an idea about these phenomena from the review article of AKHMANOV, SUKHORUKOV and KHOKHLOV [1967]. GLOGE and MARCATILI [19731 have analysed a cylindrical waveguide, surrounded by homogeneous medium. The distribution of the refractive index was characterized by a term which had the power of different integers (1,2,4,10) rather than being a quadratic term. The authors analysed the dispersion of such a waveguide and have stated the necessary accuracy needed for realization of the refractive index, in which the impulse blurring was small. In conclusion it is necessary to note that KORNHAUSER and YAGHJIAN [19671 have analysed wave propagation in a flat inhomogeneous medium with the refractive index
em,
(4.31) This distribution is of the same character as the distribution, considered above; with /3 = 0 it corresponds to a self-focusing waveguide.
v, 8 51
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321
8 5. Experiments Self-focusing properties of a medium with refractive index, that varies according to the inverse hyperbolic cosine have been first studied experimentally by MIKAELIAN [1951]. The experiment was made in the centimeter waveband, and the system, which was used as an inhomogeneous medium, consisted of two conducting surfaces, with a horizontally polarized wave propagating between them (Fig. 5.1). The required law of the refractive index variation, realized by changing the distance between these two conducting surfaces, was n(x)=J1-(h0/2b)’,
(5.1)
where b is a distance between the surfaces, and ho is a wavelength of the oscillator. It follows that
The distance between the surfaces was varied from 27 mm in the center to 18mm at the edge. The inverse hyperbolic cosine law was realized exactly. In all these cases the refractive index decreased 1.75 times at the edge. A section of this “selfoc” is shown in Fig. 5.1. The focal length was 700 mm. The waveguide was excited by a small horn placed at the input, and the amplitude and phase distributions at different cross-sections were measured. The agreement between the experimental and theoretical
Fig. 5.1. Realization in the microwave range of the self-focusing waveguide with the [195 13. refractive index, varying as an inverse hyperbolic-cosine function (MIKAELIAN
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V, 5 5
results was rather good; it was only necessary to suppress reflection of the rays from the sides of the waveguide, because the rays used in the experiment were radiated at sufficiently. large angles with respect to the waveguide longitudinal axis. A later experiment was made also in the centimeter waveband. KELLEHER and GOATLEY [1955] studied the properties of the generalized Mikaelian lens, made of the artificial (full of “holes”) dielectric. The lens was made from leaves of plexiglass, 6.4 mm thick, in the shape of dishes with cylindrical holes distributed in such a way that in the radial direction the required law of the refractive index variation was realized. The diameters of the holes varied from 6.4 mm to 9.6 mm, and they corresponded to the variations of the refractive index from 1.61 at the center to 1.24 at the edge of the lens. The dishes were laid closely to one another and formed a 7-cm-length section of a circular cylinder. The lens diameter was equal to 25.6 cm. The source was placed at a 25.6 cm distance from the output plane of the lens. With this the isocandle diagrams, characterizing the focusing properties of the lens, were analysed (Fig. 5.2). It was stated that the characteristics of the radiation remained sufficiently good in a rather wide bandwidth. The width of the main petal corresponded to the formula B,,5 = S S A / D , where D is the lens diameter (FELDand BENENSON [19593). At 7000MHz it produced 9O, and at 15000 MHz it produced 4.4”. For both of these cases the level of the side petals did not exceed 20 dB and 19 dB, respectively. The lens also has rather good characteristics with the source moved out of the focus. It was stated that during the scanning the best results were received in the case where the source was moved along the arc with the summit in the lens center.
I
I
L6
70
VI
o(
-
Fig. 5.2. Realization of the generalized Mikaelian lens in the microwaves (FELD and BENENSON [1959], KELLEHERand GOATLEY[19553).
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323
UCHIDA, FURUKAWA, KITANO,KOIZUMI and MATSUMURA [1970] were the first to realize a self-focusing waveguide, made of glassfiber with variable refractive index in the optical range. Even in the first experiments the properties of such a waveguide were tested in detail; also the possibility of its use for wideband signal transmission and image transmission was confirmed, as well as for producing short-focused lenses and other applications. The tested waveguide was a glassfiber with diameter of some hundreds of micrometers and with length of about a meter. The phenomenon of multiple wave focusing was observed clearly, the period length reaching 15 mm with the excitement of the waveguide by a unimode red-wave He-Ne laser. This experiment was made with the waveguide diameter equal to 0.7 mm, and the refractive index decreased in the radial direction according to the parabolic law and yielded 1.6 on the waveguide axis. It is important to point out that, as a rule, the diameter of a self-focusing waveguide is several wavelengths; for the given example it exceeds the wavelength by more than 1000 times. Therefore, during propagation, the focal blurring, caused by the difh-action phenomena, was negligible. In the above experiments the rays were paraxial, because a highly coherent laser was used. Therefore, the defocusing during the propagation caused by the refractive index deviation from the inverse hyperbolic cosine law, was not large and was not observed with the length within some meters. The distribution of the self-focusing waveguide field structure was also tested, under different conditions. In particular, noticeable distortions of the field structure were absent, when the waveguide bending had the radius of curvature equal to 175 mm. Different variants of lenses representing the waveguide sections were also investigated. One of them had a 0.25 mm diameter and a focal length of 0.44 mm; another had a 1 mm diameter and a focal length of 1.9 mm. The experiments, described above, mark the beginning of new technologies, for the manufacture of optical waveguides with variable refractive index. By now many papers have been published, in which the results of the researches of different types of waveguides with variable refractive index are given. We will consider some of them. One of the main problems connected with glassfiber improvement relates to the analysis of the mechanism of losses and the design of the systems with small losses. These problems were analysed in detail in a
324
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paper of DIANOV [1977], which also included an extensive bibliography. A lightguide with extremely small losses was realized by BELOV,GURIANOV, DEVIATYKH, DIANOV, NEUSTRUEV, NIKOLAICHIK, PROKHOROV, HOPINand YUSHIN[1977]. In particular the losses on 1.6 m were found to be about 0.7 dB/km. In recent years many papers were published, concerning specific applications of optical fibers. For instance, BELOVOLOV, BUBNOV, GURIANOV, DEVIATYKH, DIANOV, PELIPENKO, PROKHOROV and SISAKIAN [19771 analysed a communication system of computer blocks with fiber length up to 750 meters. GULIAEV, GRIGORIANTS, IVANOV, POTAPOV, SOSNIN, TREGUB, CHAMOROVSKY, ELENKRIG and KORENEVA[ 19771 analysed the dispersion of a light impulse in a communication line. ALIABIEV, BASOV,ZARETSKY, KARTSEV, KLIMOV, KURNOSOV, MATSVEIKO, MOROZOV, POPOV,S A ~ A R O V , and YASHUMOV SAFIULINA, SEMYONOV, SERGEEV, STELMAKH, SHIDLOVSKY [19771 have described an 8-channel communication line between the computer blocks. The review by BRONFIN, ILYIN, KARAPETIAN,LIFSHIZ, MAXIMOVand S A ~ A R O[1973] V concerns some of the new optical elements made of inhomogeneous dielectric. Investigations of a similar kind have been carried out by different groups in many countries.
8 6. Methods for the Calculation of Inhomogeneous Focusing Media 6.1. INTRODUCTION
The models of focusing media described so far are classical and widely known. They are of great interest and are important for practical application. In the case of focusing media with spherical symmetry as well as in the case of laminated inhomogeneous cylindrical medium, the problem was stated as a variational one and was solved under consideration that the refractive index varied only in one direction. It is evident that such a method may be used in a more general case where the refractive index depends upon two or even three coordinates. However, in this case the calculation becomes very complicated. On the other hand, it is obvious that there exist many different self-focusing media which are useful in practice. That is why the proposal of any method for finding and investigating new types of focusing media is
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325
of interest. It turns out that one can easily achieve success in representing the problem as the inversion problem of geometrical optics, i.e., the problem of constructing inhomogeneous media with given ray paths. This problem was first treated by the author in such a form in connection with the investigation of lenses with variable refractive index (MIKAELIAN [1951, 1952al). Quite efficient methods have been found (MIKAELIAN [1952b, c]), which may be applied not only for the calculation of focusing inhomogeneous media, but also for the calculation of elements made of nonuniform dielectrics. The statement of the problem, mentioned above, is adequate to situations encountered in practice. Actually the calculation of lenses, waveguides, telescopes or other devices made of inhomogeneous media reduces to the investigation of the distribution of the refractive index on the given wavefront trnasformation. Usually in this case two surfaces placed in a nonuniform medium are considered. The first of these surfaces is the “source”, which is characterized by a known amplitude and phase distribution of the field. For this case of focusing media, this surface reduces to a point. For the second surface the field distribution is given, which must be realized. The problem is to find the appropriate refractive index. It is clear that this problem reduces to that of constructing inhomogeneous media for the given rays, because the ray paths determine amplitude and phase distribution for any given surface.
6.2. METHOD BASED ON INTEGRATION OF EULERS EQUATIONS
Consider the case of a two-dimensional medium, with the distribution of the refractive index n ( z , x). The Euler equation, which determines ray paths, is
where n ( z , x) = In n ( z , x); x(z, 5 ) is the equation of ray paths, and 5 is a parameter, which labels a ray of the family. In the case of focusing media, where all the rays have a common point (i.e., with x(0,e) = 0), the tangent of the outlet angle may be chosen as a parameter. It is equal to
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
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In order to find a medium in which the ray paths coincide with the given family of curves, it is necessary, as was pointed out above, to solve the Euler equation and to determine the function n(z, x). It must be noted that the formulation of this problem is unacceptable for the single ray, because in this case we can find the refractive index only along one ray. In order to find n(z, x ) in any point of the plane z, x, it is necessary that eq. (6.1) be valid along all the curves of the given family. In this case it is necessary to use the Euler equation in the form
where
(6.4) and to find nl(z, x) = n,[z, x(z, 5)] in the form of n,[z, 5(z, x)] where 5 is the equation of the family curves solved relative to 6. Then one obtains:
(6.5)
Taking these expressions into account, we may rewrite the Euler equation in the form:
where
(6.7) The function $(z, 0, as well as f and cp, are known. To find the solution of eq. (6.6) let us consider the system of the ordinary differential equations that are equivalent to the equation (6.6). It has the following form:
It is easy to see that two independent solutions of this system are equal to
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327
z = F , ( [ ) + C , ; n, = F 2 ( z ,[ ) + C 2 ;thus the general solution is w ( C , , C , ) = 0. Solving it relative to n , and substituting 5 = 5(z, x) from the expression of the family of rays, one can obtain the general solution in the following form (MIKAELIAN [1952a]): n , ( z , x) = F2[z, 5(z, x)l+
@b-F1[5(2, x)l>,
(6.9)
where @ is an arbitrary function. Thus, the eq. (6.9) gives the solution of the inverse problem of geometrical optics, and it allows us to find a medium where the given family of curves is the family of ray paths. It was noted that the given family of curves which represents the ray paths, determines the distribution of intensity and phase on any surface (in a two-dimensional case on any curve), encountered by the wave. Actually the phase distribution is determined by the difference of optical lengths of rays, and the intensity distribution is specified by the density of rays crossing the surface. This fact means that the media, determined by the solution (6.9), can realize the given transformation of the wavefront. To illustrate this method let us try to find all of the inhomogeneous media in which the ray paths coincide with the straight lines going from the origin of the coordinate system. Let us write the equation of the rays in the form: x
=&.
(6.10)
We can then determine the functions
Substituting (6.11) into the system (6.8) we obtain the equation (6.12) Two integrals of this system are equal to
and lead to the well known solution for n ( z , x): n ( z , x) =
@(a). (6.14)
It is seen that the refractive index can vary along the rays in an arbitrary manner.
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SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
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As another example let us take the family of curves to be represented by the expression
-
sh x =sh 5 sin z ,
(6.10')
which characterizes the ray paths in laminated cylindrical SELFOCS. One then has:
and substituting in (6.8), we obtain (6.12')
The integrals corresponding to this system are
-
tg z ch 5 = C1,
n ( z , x) = e"l = C,/cos z .
(6.13')
The general solution therefore is:
and one can obtain the following formula for the refractive index (MIKAELIAN [1951, 19521): 1 n(Z,X)=-@(tgz cos z
*ch()=cos z
@(J
sin2 z +sh2 x), cosz
(6.141)
where @ is an arbitrary function. Hence the wave, propagating in the media, described by (6.14') focuses periodically at the points z = 0, T,2 7 ~* . As we see, there are many cases of inhomogeneous media where the ray paths are the same. It is easy to see that the simplest case n(x) = n(O)/ch x, which does not depend on longitudinal direction and characterizes a laminated SELFOC, is a particular case of the general solution (6.14'). The method, used above, has the following feature: The Euler equations are solved by taking n ( z , x) and its derivatives along the ray paths. However, it is possible to choose a slightly different way and to take the value of x' and x" at the points where n ( z , x) is calculated. In this case
--
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METHODS FOR CALCULATION OF INHOMOGENEOUS FOCUSING MEDIA
one obtains instead of (6.3), the following equation:
an, an, az ax
f(Z, x)+cp(z, x)----0.
(6.15)
Here
E ( Z , X ) is the equation of the ray paths solved with respect to 6. The system, corresponding to (6.15), is as follows: (6.17) The solution of (6.17) is obtained in the same way as (6.8). The choice of which system to use in a practical application depends on the form of the given function that describes the family of curves. It should be noted that for the wide class of ray paths determined by
-
L ( z , x) = L X b ) L , ( z ) ,
(6.18)
it becomes possible to calculate an integral of the system (6.17) or (6.18). This allows us to obtain a general solution for the refractive index. For instance if we solve the system (6.17), we find from (6.18) that
(6.19) Hence we can separate the variables in calculating the integrals, i.e.,
(6.20) In this way the first solution of the system may be written as: S , ( z , x) = c,.
(6.21)
From eq. (6.21) one can determine z in the form z = F(x, C,) and one then obtains the second integral of the system (6.17):
(6.22) Hence the general solution may be written in the form (MIKAELIAN
330
[1952bl)
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
1
n l ( z , x) = If[F(x,Cl), XIdx
1
+ @(S1(z, x)),
[V,§ 6
(6.23)
C,=S,(z. x )
where @ is an arbitrary function. It is easy to see that the problem reduces to the calculation of a single integral. Let us consider this method in connection with the example described above. Writing the equation of ray paths in the form L ( z , x ) = sh xlsin z,
(6.24)
and using the formula (6.20) one finds that S ~ ( Zx ,) = C1 =ch X/COS Z.
(6.25)
Substituting the function f(z, x) = -th x in the equation (6.23) and calculating the integral, we obtain the general solution: 1 chx
ch x @(-)cosz
n ( z ,x ) =-
.
(6.26)
It is seen that this solution agrees with (6.14‘).
6.3. METHOD BASED ON THE PRINCIPLE OF INHOMOGENEOUS MEDIA SIMILARITY
The following “principle of inhomogeneous media similarity” was formulated by MIKAELIAN [1952b]: If the inhomogeneous medium where the wavefront of the propagating wave is determined by sl(z, x) = const. has the refractive index n , ( z , x), then in a medium whose refractive index is n ( z , x) = n,(z, x)@(sl(z,x)), the wavefront remains the same. Hence if some particular solution of the inversion problem of geometrical optics is known, i.e., if the distribution of the refractive index for certain family rays has been determined, this principle makes it possible to determine the general solution of the problem. This solution will determine all the inhomogeneous media with the same family of rays. The principle of similarity for inhomogeneous media follows from the eikonal equation. It becomes especially clear if the solution of the problem of calculation of the inhomogeneous media by the given ray
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33 1
paths is found, using the method of coordinate systems. In accordance with this method the orthogonal system of coordinates U1, U , is used. The coordinate curves coincide with the ray paths. In this case the curves U1= const. represent the wavefront. Under these considerations the Euler equation can be easily integrated and one can obtain the general solution of the problem (MIKAELIAN [1952]) in the form
(6.27) where @ is a random function, and the metrical coefficient is equal to:
hl(ul,u,) = J(a~/au,)z+(ax/aul)z.
(6.28)
The structure of this solution illustrates the principle of similarity quite clearly. It becomes obvious that the argument of the arbitrary function in the solutions given above, is the eikonal function. The application of the similarity principle simplifies the construction of inhomogeneous media with given ray paths. It becomes useful especially in cases where any particular solution is known or may be easily found. For example consider the particular solution of laminated SELFOC: n ( z , x) = n l ( x ) = n(0)lch x.
(6.29)
The equation of the ray paths is defined by the expression (6.24). In order to find an orthogonal family of curves, that is the wavefront, it is necessary to solve the equation: dz -- dx --
aL/az aL/ax
(6.30) a
Using (6.24), we obtain:
Sl(z, x) = ch xlcos z.
(6.31)
Hence according to the stated principle the general solution may be written in the form n(0) chx (6.32) n ( z , x) =chx cosz
@(-)
and agrees with (6.26). In the general case, when only an equation of the ray paths is given, the particular solution may be found from the eikonal equation. But it is necessary to substitute the formula of the wavefront in the eikonal equation. As we have seen before, the wavefront is determined by the
332
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first integral of the system (6.8) or (6.17), which is equivalent to the equation (6.30). There is another method of solution, based on the principle of similarity. It can be used in the case when one wishes to find an inhomogeneous focusing medium irrespective of the ray paths. Let us require that the source, placed in the point -zo, to be imaged ideally in the point +z,, and let us try to find the simplest particular solution for n ( z , x). It is clear that any odd function depending only on z (for instance: z , sin z , sh z etc.) may be used as a particular solution. In other words the function must satisfy the condition nl(z) = -nl(-z). It is evident that such particular solutions are of no practical interest. But they can be used for finding the corresponding family of curves, because the Euler equation is easily integrated in this case. The result is
(6.33) It is then possible to find the equation of the wavefront, because n , ( z ) edz = const.,
(6.34)
and, in accordance with the principle of similarity, one can then construct the general solution for n(z, x). Finally it is possible to obtain from this solution the most interesting ones. In conclusion, we note that the methods of constructing the inhomogeneous media for the given ray paths were described for twodimensional media. The generalization of these methods to the threedimensional case is not difficult. It must be pointed out that the Euler equation was written in the form, which allows for the use of any other system of coordinates U , , U,, having the metric coefficient h,( U1, UJ, h2(U1,U2). It is clear that in this case only the functions f and rp will change, the formula remaining the same.
# 7. Quasi-Regular Cylindrical Inhomogeneous Media 7.1. INTRODUCTION
The methods of determining inhomogeneous media with given ray paths allow us to discuss a more complicated set of optical waveguides
QUASI-REGULAR CYLINDRICAL. INHOMOGENEOUS MEDIA
333
with variable refractive index. This set of waveguides is characterized by the dependence of the refractive index not only upon the transverse coordinate, but also upon the longitudinal one. It was found (MIKAELIAN [1952a]) that there is an infinite number of SELFOCS among such waveguides, with different laws of n(r, 2). As we already mentioned the problem of finding new types of SELFOCS can be formulated as an inverse problem of geometrical optics, that is, the problem of constructing inhomogeneous media with the given ray paths. In the present section this method is used for the investigation of some special types of waveguides, with the variable refractive index (MIKAELIAN [1952c, 19781). Such waveguides are of interest not only for use as transmission lines, but for the design of new optical elements of inhomogeneous dielectrics (lenses, telescopes, focons etc.). These elements, made of waveguide sections, have an additional degree of freedom due to the refractive index variation. This degree of freedom allows us to realize the necessary wavefront transformations with lesser aberrations. This problem is an important one for the development of glass fiber systems for optical communication and for data processing systems, for integrated optics, etc. 7.2. SELF-FOCUSING WAVEGUIDES (SELFOCS)
Consider first the self-focusing waveguides where ray paths have a constant period as shown in Fig. 1.3. Since we have assumed n(r, cp, z ) = n ( r , z), the pattern of ray paths will be the same at all diagonal planes, because of circular symmetry. It is therefore sufficient to investigate a two-dimensional case with ray paths being plane curves as shown in Fig. 7.1.
0
2
Z=f z =3 2-5 Fig. 7.1. Ray paths in a diagonal plane of a cylindrical self-focusing waveguide with the axial symmetry. The refractive index is determined by the function n(r, 2 ) .
334
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V, I 7
According to the given scale the ray focusing occurs on the z-axis in the points z = 0 , 2 , 4 , 6 * * . For the planes z = 1 , 3 , 5 * the deviation of each ray reaches the maximum value 5 (for the corresponding beam). Naturally the paths must be symmetrical relative to the z-axis. It is evident that the equation of the ray paths satisfying the above requirement may be given in different forms. This leads to a great number of cylindrical waveguides (with various n(r, z)) where the rays are periodically focused in the course of propagation. Consider some examples. To begin with we represent a family of curves of Fig. 7.1 in the form of an equation, that determines the ray paths for the simplest case of a laminated medium, where the refractive index does not depend on z. This example was investigated in 0 6 . In this case the refractive index is given by (MIKAELIAN [1952~1):
Hence for a cylindrical waveguide we have
1 n(r, z ) =-ch ilrr
o(-)
(7.2)
Here and below O is an arbitrary function. The solutions (7.1) and (7.2) show that there are many self-focusing waveguides (cylindrical and plane, i.e., two-dimensional), where the rays propagate along the same paths. It is evident that both of the expressions (7.1) and (7.2) contain the inverse hyperbolic cosine law, corresponding to the laminated waveguide. The refractive indexes of all SELFOCS of this group vary periodically in the direction of propagation, but their period can be different. This can be easily seen on comparing the following types of self-focusing waveguides: cos &rz n(r, z) =(Cr-) ch $ m chgm ’
(7.3)
1 n(r, z ) = ch $ r r
(7.4)
n(r, z)=%th(-)
chfm
chim
,
COS$TZ
where C1, C,, C3 are constants (greater than unity).
(7.5)
QUASI-REGULAR CYLINDRICAL INHOMOGENEOUS MEDIA
335
In the next example we will write the equation of the ray paths in the form x(z) = 6 sin $rz.
(7.6)
This equation also corresponds to Fig. 7.1; i.e., the focal points are located at z = 0 , 2 , 4 * * * . The tangent angle of the ray output is given by (7.7)
tgy=dx/dzl,,,=$.rr[.
Using the method described in 0 6, it is easily shown that for this case the self-focusing waveguides are specified by the following refractive index function:
Varying the function @, one can find rather interesting variants of the refractive index laws. All of them naturally depend upon both coordinates, the dependence on the longitudinal coordinate being periodic. Consider now the self-focusing waveguides, in which the ray paths are parallel to the z-axis in focal points z = 0 , 2 , 4 * * * . This means that the output angle for all of the rays is equal to zero (see Fig. 7.2). As an example let us write the equation of the ray paths as x(z) = S(sin t
~z)~,
(7.9)
where “p” is a parameter, different from unity. (For an odd integer “p” the foci of all the rays coincide with the point of inflection and for even integer “p” they are the points of minima.)
0
-
Z
Fig. 7.2. Self-focusing waveguide with the parallel rays in a focal point.
336
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V, § 7
At each focus the tangent of the outlet angle, tgy=-
(7.10)
is also equal to zero. This is the reason why the excitation of such self-focusing waveguides seems to be realized only by a plane wave (more precisely, by the light source with a plane equiphase surface) in the cross-sections z = 1,3,5. As can be seen, if we move the source slightly out of the initial point (to the value of A=),the output angle will be given by (7.1 1) For example, 7r2
tg y = t 5 , A Z L
tg y = &?A:
for p = 2 ,
(7.12)
for p = 3.
(7.13)
In the case of incoherent sources (that is, without phase center) such waveguides may be more advantageous from the point of view of excitation efficiency than the convenient ones, considered above. The laws of the refractive index distribution for this group of SELFOCS may be found with the help of the method used in the previous case. The result is
It was pointed out that corresponding sections of the self-focusing waveguides represent lenses with variable refractive index. The laminated lens is a special case among the infinite variety of such lenses. For the laminated lens, as well as for the lens with the variable refractive index, where the ray paths are the same, the intensity distribution is defined by the expression (3.14). For lenses, made of self-focusing waveguides, in which the ray paths are determined by the expression (7.6), the field distribution is approximately the same: I(~)=+7rcosZy=;7r1+I1 47r
2 .
s
(7.15)
v, Q 71
QUASI-REGULAR CYLINDRICAL INHOMOGENEOUS MEDIA
337
Fig. 7.3. Lens made of the inhomogeneous medium with the parallel rays in the focal point. The field distribution in the aperture can be regulated by displacement of the source.
For the third group of lenses, formed by sections of self-focusing waveguides, in which ray paths are parallel in the focal points (Fig. 7.3), the field distribution appears to be more homogeneous, (7.16) and can be controlled by appropriate displacement of the source from the focus.
7.3. TELESCOPIC WAVEGUIDES
Consider now waveguides with variable refractive index, where the optical length of the rays is practically the same, although the rays are not focused during propagation. An example of such a family of loci is shown in Fig. 7.4 (at a diagonal plane, as was the case earlier). One has now
Fig. 7.4. Telescopic waveguide with the variable refractive index. The rays have the same [1978J). optical length but they have no focal points (MIKAELIAN
338
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V,8 7
periodically repeated synphasic wave front. Such waveguides can be used not only as optical transmission lines, but also as telescopes with variable refractive index, which are the waveguide sections from z = 1 to z = 3+4k, where k is an integer. For this reason we describe them by the term telescopic. For exampre, let us represent the equation of the rays as sh ~
Q X = sh $m$(B+sin ~ T Z ) ,
(7.17)
where B is a constant. It differs from the equation (6.10’), which describes the paths of the self-focusing waveguides, by an additional term which represents a shift of each ray, radiated out of the initial point, along the x-axis; the greater the parallax, the larger the output angle. The equation of the ray family x(z, 6) can be more conveniently written as (7.18)
where, as before, each ray of the family ray is specified by the ordinate x =Ai with z = 1. The “magnification factor” of the simplest telescope is given by the following ratio:
V =XI,=?= A,/Ar sh x Iz =3
(7.19)
With B = 1 the aperture is zero at the plane z = 3, and the waveguide becomes a self-focusing one with the rays parallel at the focus. With B < 1 the waveguide remains a telescopic one, but its ray paths have a somewhat different form (see Fig. 7.5). For this case one must take the modulus of the ratio (7.19). The laws of the refractive index variation of cylindric waveguides with the “telescopic” mode of ray paths (eq. (7.17)) can easily be found by applying the method, presented in section 6.3. We then find that
(7.20)
It is easily seen that with B = O this expression reduces to (7.2), and a telescopic waveguide becomes a self-focusing one.
v, 5 71
QUASI-REGULAR CYLINDRICAL INHOMOGENEOUS MEDIA
0
339
2
Fig. 7.5. Telescopic waveguide with the ray crossing.
For the next model of telescopic waveguides we write the ray-paths equation in the following form (in accordance with Fig. 7.4): x -5B = 5 sin &rz,
(7.21)
x =- Ai (B +sin&rz).
(7.22)
or
B+l
In this case the simplest telescope, formed by the waveguide section from z = 1 to z = 3, has the magnification factor (7.23) One can show that inhomogeneous cylindrical media, corresponding to the telescopic waveguides of this group, are determined by the refractive indexes n(y, z ) =
JW cos 4 T Z
4
(cos i T Z ) B + ' exp (1 +sin &rrz)"
(7.24) (-$a:)].
With B = 0 this expression reduces to (7.8), and the waveguide becomes self-focusing. With B = 1 the telescopic waveguide also becomes selffocusing, but the rays become parallel at the focus. It is easy to see that the expression (7.24) with B = 1 agrees with the expression (7.14) with p=l. Note should be taken that the telescopes with variable refractive index have some remarkable properties. In particular, as it can be seen from
340
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
rv.57
(7.19) and (7.23), they may be made much more “short-focused’’ than the conventional ones.
7.4. IRREGULAR WAVEGUIDES WITH VARIABLE REFRACTIVE INDEX
Consider now waveguides with variable refractive index, whose properties are varied longitudinally. We will call them irregular. Strictly speaking, the waveguides described above are not regular either, with the exception of the special case of a laminated waveguide, where the refractive index varies only in the radial direction. But in these waveguides the longitudinal refractive index variation is of the periodic character. Therefore they can be considered regular on the average (during the period). Irregular self-focusing and telescopic waveguides may be tested by the methods applied above. Here it is interesting to note two cases. In the first case the ray paths are characterized by the amplitude, varying in the course of propagation (Fig. 7.6), and for the second case the ray paths are characterized by the variable frequency (Fig. 7.7).* Let us discuss a representative example: an irregular self-focusing waveguide, where the ray paths correspond to the curves plotted in Fig.
0
Fig. 7.6. Irregular waveguides with the variable amplitude of the rays (MIKAELIAN [1978]); (a) self-focusing waveguide; (b) telescopic waveguide.
* Naturally other types of irregular waveguides are possible with the ray paths not quasi periodic, but represented by more complicated curves. The common methods of calculation are given in 86.3.
V, 8 71
QUASI-REGULAR CYLINDRICAL INHOMOGENEOUS MEDIA
341
Fig. 7.7. Irregular waveguides with the crossing of the rays; (a) self-focusing waveguide; (b)telescopic waveguide.
7.7a, which may be represented by the equation: x (z) = tePZ* sin fm,
(7.25)
where p is a parameter, characterizing the form of the ray paths. With p = 0 this equation reduces to (7.6) and corresponds to the case discussed in 07.2. It follows from eq. (7.25) that all of the rays, radiated from the initial point at different angles y, cross in the points z = 0 , 2 , 4 , 6 * * * . In this case tg y = dz dx
I
z
= tepz( p =
sin f n z +;ITcos f 7 ~ z ) I ,==f&r, ~~
(7.26)
~
and the equation of the ray paths can be written as: 2
-
x(z, y) = - tg y epz . sin frz. 7T
(7.27)
Equiphase planes, where all the rays are parallel to the axis of the waveguide, correspond to the values of z, which are the solutions of the equation p sin f v z + f m cos f 7 ~ = z 0.
(7.28)
For the simplest particular case p = &r, the equiphase planes correspond to the values of z =+,$, * * . The sections of the waveguide under consideration between the equiphase planes are telescopes. The simplest telescope “magnification factor” is equal to en = 23. Calculations show
342
SELF-FOCUSING MEDIA WITH VARIABLE INDEX OF REFRACTION
[V,S 8
that this group of irregular waveguides corresponds to a dielectric cylinder where the refractive index varies according to the law:
n(r, z ) =
J;[
(sin
z
2
+ cos
z)I2 + sin2 7T z 2
From the above expressions and from Fig. 7.6 one can see that the waveguide sections from z = 0 to z = 1.5, from z = 1.5 to z = 2, from z = 2 to z = 3.5, etc., are focusing lenses, with properties different from those of self-focusing waveguides with a constant period. With p # 4 7 ~the waveguide calculation is much more complicated. The self -focusing and telescopic waveguides with variable period have remarkable advantages (Fig. 7.7). They are likely to be used as “delay systems”. SODHA, GHATAK and MALIK[19713 investigated the geometrical-optics approximation in the case where E ( r , 2) = E ~ - E ~ ( Z ) T ’ . They obtained solutions for several simple functions ~ ~ ( 2 ) . It is interesting to note that similar distributions of the dielectric permeability takes place in laser media, when self-focusing of the propagating pulse occurs (AKHMANOV, SUKHORUKOV and KHOKHLOV[1967]).
8 8. Conclusion The theory and the methods of calculation were considered for inhomogeneous media in connection with the problem of lenses and waveguides with a variable refractive index. We have investigated some characteristic types of waveguides, discussed their properties and peculiarities; it was shown that different waveguide sections can be used as various optical elements. All these results may be regarded as consequences of the method of constructing inhomogeneous media for given ray paths. Naturally this method can be used in many other fields and, in particular, it solves the problem of finding new types of lenses (and other optical elements) with variable refractive index that ensure the appropriate transformations of wayefronts. It is evident that this method is applicable to cylindrical media with circular symmetry, as well as to any three-dimensional media.
VI
REFERENCES
343
In the final analysis, the success of the development of this new and important trend seems to depend on the progress of the technology of manufacturing inhomogeneous media with prescribed refractive index variation. The investigations in this domain, which began about ten years ago with the manufacture of the simplest optical elements of inhomogeneous dielectric, are, therefore, of special significance.
References AKHMANOV, S. A,, A. P. SUKHOROKOV and R. V. KHOKHLOV,1967, Selffocusing and Diffraction of Light in the Nonlinear Media, Uspekhi Fiz. Nauk 93, September. M. A. KARTSEV, I. I. KLIMOV, V. D. ALIABIEV, B. V., N. G. BASOV,A. A. ZARETSKY, KURNOSOV,A. A. MATSVEIKO, V. N. MOROZOV, Y. M. POPOV,D. K. SATTAROV, S. S. SAFIULINA, A. S. SEMYONOV, A. B. SERGEEV, M. F. STELMAKH, R. P. SHIDLOVSKY and I. V. YASHUMOV, 1977, Eight-Channel Optical-Fiber for Communication Line between Computer Units, Quantum Electronics 4, N7, 1610. and E. L. FEINBERG, 1953, Propagation of Radio Waves ALPERT,Y. L., V. L. GUINZBURG (Izdatelstvo GJITL, Moscow). ANDERSON, D. B., K. L. DAVIS,J. T. BOYDand R. R. AUGUST,1977, Comparison of Optical-Waveguide Lens Technologies, IEEE Quant. Electronics QE-13, N4, 275. BELOV,A. V., A. N. GURIANOV, G. G. DEVIATYKH, E. M. DIANOV, V. B. NEUSTRUEV,A. V. NIKOWCHIK, A. M. PROKHOROV, V. F. HOPINand A. S. YUSHIN,1977, Glass Fiber Optical Waveguide with Losses Less than 1dB/km, Quant. Electronics 4, N9,2051. G. G. DEVIATYKH, E. M. DIANOV, V. BELOVOLOV, M. I., M. M. BUBNOV, A. N. GURIANOV, I. PELIPENKO, A. M. PROKHOROV and I. N. SISAKIAN, 1977, A Study of Fiber-optic Systems for Communication Between Computer Units, Quant. Electronics 4, N11, 2456. BERREMAN, D. W., 1964a, A Lens of Light-Guide Using Connectively Distorted Thermal Gradients in Gases, Bell Sys. Tech. J. 43, 1469, July. BERREMAN, D. W., 1964b, A Gas Lens Using Unlike Counter Flowing Gases, Bell Syst. Tech. J. 43, 1476, July. BORN,M. and E. WOLF,1964, Principles of Optics (2nd ed., Pergamon Press, London and New York). BRAUN,E. N., 1955, Radiation Characteristics of Spherical Luneburg Lens, Trans. IRE AP-4, N2. L.M., 1957, Waves in Layered Media (Izdatelstvo Academii Nauk SSSR). BREKHOVSKIKH, BRONFM, F. B., V. G. ILYIN,G. 0. W E T I A N , V. Y.LIFSHIZ,V. M. MAXIMOVand D. K. SATTAROV, 1973, Focusing Optical Elements with Regular Distribution of the Index of Refraction, Zh. Pricladnoy Spektroskopii 18, 523. DIANOV, E. M., 1977, A Study on Creation of Athermalized Laser Glasses and Glass Fiber Lightguides with Small Losses (FIAN SSR im. Lebedeva, Moscow). 1959, Antenna and Fiber Design, vol. 2 (Izdatelstvo FELD,Y. N. and L. S. BENENSON, W I A imeni Zhukovskogo). 1973, Multimode Theory of Graded-Core Fibers, Bell GLOGE,D. and E. A. J. MARCATILI, Sys. Tech. J. 52, N9, 1563. GORDON,J. P., 1966, Optics of General Guiding Media, Bell Syst. Tech. J. 45, February, 321.
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GULIAEV, Y. V., V. V. GRIGORIANTS, G. A. IVANOV, V. T. POTAPOV, V. P. SOSNIN,D. P. TREGUB,Y. G. CHAMAROVSKY, B. B. ELENKFUG and N. A. KORENEVA, 1977, Experimental Observation of the Light Pulse Dispersion in a Gradient Fiber Glass Having M-Shaped Refractive Index Distribution, Quant. Electronics 4, N l l , 2464. GUTMAN, A. S., 1954, Modified Luneburg Lens, J. Appl. Phys. 25, N7. S. and J. NISHIZAWA, 1968, An Optical Waveguide with the Optimum DistribuKAWAKAMI, tion of the Refractive Index with Reference to Waveform Distortion, IEEE Trans. MTT 16, 814. KELLEHER, K. S. and C. GOATLEY,1955, Dielectric Lens for Microwaves, Electronics, N8, August, 142. KOCK,W. E., 1965, Sound Waves and Light Waves (Doubleday, New York). KOGELNIK, H., 1975, An Introduction to Integrated Optics,'IEEE Trans. M'TT 23, N1, 1. KORNHAUSER,E. T. and F. S. HELLER,1963, A Soluble Problem in Duct Propagation, in: Proc. Symp. on Electromagn. Theory and Antennas, Copenhagen, 1962, ed. Jordon (Pergamon Press, London and New York) p. 891. KORNHAUSER, E. T. and A. D. YAGHJIAN, 1967, Modal Solution of a Point Source in a Strongly Focusing Medium. Radio Sci. 2, 209, March. LUNEBURG, R. K., 1944, Mathematical Theory of Optics (mimeographed, Brown Univ., Providence, R.I.; reprinted by University of California Press, Berkeley and Los Angeles, 1964). MARCATILI, E. A. J., 1964, Modes in a Sequence of Thick Astigmatic Lens-Like Focusers, Bell Syst. Tech. J. 43, November, 2887. MARCATILI, E. A. J., 1967, Off-Axis Wave-Optics Transmission in a Lens-Like Medium with Aberration, Bell Syst. Tech. J. 46,January, 149. MARCUSE, D. and S. E. MILLER,1964, Analysis of a Tubular Gas Lens, Bell Syst. Tech. J. 43, June, 1759. MAXWELL, J. C., 1854, Cambridge and Dublin Math. J. 8, February, 198; The Scient Papers 1, Paris, 1927. MIKAELIAN, A. L., 1951, Using of Layered Media for Focusing Waves, Dokladi Academii Nauk SSSR 81,N4, 569; Lens Antennas with Variable Index of Refraction (Izd. MEIS). MIKAELIAN, A. L., 1952a, General Method of Inhomogeneous Media Calculation by the Given Ray Traces, Dokladi Acad. Nauk 83, N2, 219. MIKAELIAN, A. L., 1952b, Method of Calculation of Inversion Problem of Geometrical Optics, Dokl. Acad. Nauk 86, N5, 963. MIKAELIAN, A. L., 1952~.Using of Coordinate System for Calculation of Media Characteristics for the Given Ray Traces, Dokl. Acad. Nauk 86, N6, 1101. MIKAELIAN, A. L. and V. V. DIATSCHENKO, 1972, Stabilisation of Waveform in High Distorted Solidstate Media, Pisma GETF 16, N1, 25, 5.07. MLKAELIAN, A. L. and V. V. DIATSCHENKO, 1974, Lasers, Using Waveguide Resonators, Kvantovaia Electronica 1,N4, 937. MIKAELIAN, A. L., 1977, On Selfoc Dielectric Waveguides, Kvantovaia Electronica 4, N2, 467. MIKAELIAN, A. L., 1978, Optical Waveguides with Variable Index of Refraction, Optica i Spectroskopia 44, N2, 310. S. P., 1958, General Solution of the Luneburg Lens Problem, J. Appl. Phys. 29, MORGAN, N9, 1358. S. P., 1959, Generalisation of Spherically Symmetric Lenses, Trans. IRE AP-7, MORGAN, N4, p. 342. RAWSON,E. G., D. R. HERRIOTTand J. MCKENNA, 1970, Analysis of Refractive Index Distributions in Cylindrical, Graded Index Glass Rods (GRIN Rods) used as Image Relays, Applied Optics 9, N3, 753.
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SLICHTER,L. B., 1932, The Theory of the Interpretation of Seismic Travel-Time Curves in Horizontal Structures, Physics 3, N6, December, 273. SODHA,M. S., A. K. GHATAKand D. P. S. MALIK,1971, Electromagnetic Wave Propagation in Radially and Axially Nonuniform Media: Geometrical Optics Approximation, J. Opt. SOC.Am. 61, N11, 1492. STREIFER, W. and C. N. KURTL,1967, Scalar Analysis of Radially Inhomogeneous Guiding Media, J. Opt. SOC.Am. 57, June, 779. TOLSTOY,I. and C. S. CLAY,1966, Ocean Acoustics, Theory and Experiment in Underwater Sound (McGraw-Hill Book Company, New York). TORALDO DI FRANCIA,G., 1961, Spherical Lenses for Infrared and Microwaves, J. Appl. Phys. 32, N10. UCHIDA,T., M. FURUKAWA, I. KITANO,K. KOIZUMIand H. MATSUMURA, 1970, Optical Characteristics of a Light-Focusing Fiber Guide and Its Applications, IEEE J. QE-6, N10, 606. ZELKIN,E. G. and V. A. ANDREEV,1968, Ray Scanning in Mikaelian Lens, Sbornic “Antennas” 31, N4 (Izdatelstvo “Sviyz”).. 1974, Lens Antennas (Izdatelstvo “Sov. Radio”). ZELKIN,E. G. and R. A. PETROVA, ZERNIKE,F., 1974, Luneburg Lens for Optical Waveguide Use, Optics Communications 12, N4, 379. ZHUK,M. S. and U. B. MOLOTSCHKOV, 1973, The Design of Scanning and Widebandwidth Lens Antennas and Fibers (Izdatelstvo “Energia”).
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AUTHOR INDEX A ABELLA, I. D., 87, 136, 158, 160, 190,232 G. S., 190, 195, 196, 197, 207, AGARWAL, 232,233 AGOSTINI, P., 87, 158 G. P., 220, 233 AGRAWAL, AKHAMANOV, S. A., 228, 229, 231, 233, 320, 342, 343 ALEKSOFF, C. C., 66, 82 ALIABIEV, B. V., 324, 343 ALLEN,L., 190, 194, 195, 233 ALLEN,L. R., 242, 276 ALPERT,Y.L., 287, 343 K., 232, 233 ALTMANN, AMINOF, C. G., 122, 160 ANDERSON, D. B., 294, 296, 343 ANDREEV, V. A., 301, 313, 314, 345 AOKI,T., 189, 233 APANASEVICH, P. A., 190, 191, 233 A n , J., 120, 160 ARECCHI, F. T., 179, 181, 188, 189, 233 ARMSTRONG, J . A., 188, 233 ARMSTRONG, L., 199, 233 ARSENAULT, H., 7, 9, 35, 82, 83 ARSLANBEKOV, T. U., 200, 201, 233 AUGUST,R. R., 294, 296, 343 AVAN,P., 109, 158
B BABIN,A. A., 229, 233 BACH,A., 196, 209, 237 P., 190, 233 BACHMANN, E. V., 142, 158 BAKLANOV, BALLARD, G. S., 6, 18, 46, 82 BANDILLA, A., 209, 233 BARGER,R. L., 121, 158 BARJOT,G., 87, 158 BARK,A,, 181, 233 BARNES, C. W., 6, 83 BASAK,S., 215, 238
BASOV,N. G., 324, 343 F., 154, 158, 159 BASSANI, M., 110, 117, 119, 139, 141, 142, BASSINI, 148, 150, 155, 158 J., 128, 159 BAUCHE, BAUMAN, Z., 203, 233 BEDARD,G., 253, 276 BEUAEV,Yu.N., 229, 233 BEWAEVA, N. N., 229, 233 BELKE,S., 189, 233 BELOV,A. V., 324, 343 BELOVOLOV, M. I., 324, 343 C., 187, 233 BENDJABALLAH, BENENSON, L. S., 285, 288, 300, 301, 302, 322, 343 BEN-URI,J., 19, 83 BERGQUIST, J. C . , 135, 159 BEROFF,K., 126, 127, 128, 134, 158 BERREMAN, D. W., 286, 343 M., 166, 233 BERTOLOTI~, Z., 195, 233 BIALYNICKA-BIRULA, BIEDERMANN, K., 76, 83 BIJL,D., 74, 82 F., 87, 91, 95, 101, 102, 104, 106, BIRABEN, 107, 108, 110, 115, 117, 119, 120, 121, 123, 124, 125, 126, 127, 128, 130, 132, 133, 134, 138, 139, 141, 142, 148, 150, 151, 154, 155, 158, 159 S., 230, 233 BIRAUD-LAVAL, BISCHEL,W. K., 128, 159 J. E. 100, 137, 138, 159, 160 BJORKHOLM, N., 120, 127, 159, 160 BLOEMBERGEN, BLUM,E.-J., 246, 276 BOLTON, P. R., 134, 159 BONCH-BRUEVICH, A. M., 87, 159 BORDE,C., 117, 159 BORN,M., 169, 233, 247, 252, 276, 288, 343 BOYD,J. T., 294, 296, 343 D. J., 189, 203, 233 BRADLEY,
347
348 BRASLAU, N., 215, 237 BRAUN,E. N., 295, 296, 343 BRAUNEROVA, Z., 224, 236 BREKHOVSKIKH, L. M., 287, 343 BREWER, R. G., 136, 138, 159 BRONFIN, F. B., 324, 343 BROSSEL, J., 87, 159 W., 209, 236 BRUNNER, BRYNGDAHL, O., 18, 82 BUBNOV, M. M., 324, 343 A., 269, 276 BUFFINGTON, BULLOUGH, R. K., 190, 234 F. V., 153, 154, 159 BUNKIN, BURCKHARDT, Ch. B., 35, 61, 82 B U ' I T E J. ~ ,N., 61, 82 BUTIX,R. R., 265, 277
AUTHOR INDFX
CURRIE,D. G., 242, 276 D
DAGENAIS, M., 184, 192, 193, 235 J. C., 250, 263, 276 DAINTY, R., 6, 8, 11, 18, 19, 20, 23, 24, DANDLIKER, 25, 28, 29, 31, 33, 35, 36, 37, 40, 43, 44, 46, 47, 48, 49, 50, 53, 54, 56, 58, 59, 60, 65, 68, 69, 70, 74, 76, 77, 78, 79, 80, 82, 83, 203, 238 DARICEK, T., 203, 234 DAVIDSON, F., 184, 234 DAVIS,J., 242, 243, 276 DAVIS,K. L., 294, 296, 343 DE, M., 18, 83 V., 179, 181, 183, 188, 233, DEGIORGIO, 234 C N. B., 200, 201, 233 DELONE, CAGNAC, B., 87, 91, 101, 102, 106, 107, DENTINO,M. J., 6, 83 G. G., 324, 343 108, 110, 115, 117, 119, 120, 121, 123, DEVIATYKH, D. A., 265, 276 124, 125, 126, 127, 130, 133, 134, 138, DE WOLF, E. M., 324, 343 139, 141, 142, 148, 150, 155, 158, 159, DIANOV, V. V., 313, 344 DIATSCHENKO, 161 DICKE,R. H., 194, 234, 241, 277 P. J., 190, 234 CANDREY, DINEV,S., 201, 214, 235 CANTRELL, C. D., 181, 234 DI PORTO,P., 222, 224, 234 CARMICHAEL, H. J., 193, 193, 234, 238 D i m , S. N., 199, 200, 234 S., 184, 234 CARUSOTTO, DIXON,R. W., 45, 83 CHAMAROVSKY, Y. G., 324, 344 K. N., 231, 233 DRABOVICH, CHAMPAGNE, E. B., 24, 82 DRUMMOND, P., 193, 210, 234 N., 209, 234 CHANDRA, DUBAS,M., 69, 83 CHARTIER, G., 230, 233 DUBESTKY, B., 142, 158 CHATURVEDI. S., 210, 234 V. P. (CHEBOTAJEV, V. L.), DUCAS,T. W., 120, 160 CHEBOTAYEV, M., 136, 159 87, 89, 100, 114, 142, 143, 146, 158, DUCLOY, DUONG,H. T., 126, 159 159, 161, 190, 236 DuPONT-ROC, J., 107, 109, 158, 159 CHEN,K. M., 128, 159 DURST,F., 65, 83 A. S . , 228, 229, 231, 233 CHIRKIN, S. H., 140, 161 DWORETSKY, CHMELA,P., 222, 234 J., 195, 198, 225, 234, 235 CHROSTOWSKI, E S., 201, 235 CHUDZYNSKI, CLAY,C. S., 287, 345 EASLEY, G. L., 123, 160 COHEN-TANNOUDJI, C., 92, 107, 109, 143, EBERLY, J. H., 190, 194, 195,233,234,236 146, 158, 159, 160 ECKSTEIN, J. N., 122, 159 COLLIER, R. J., 35, 61, 82 N. P., 137, 138, 160 ECONOMOU, COLLINS, c. B., 129, 160 EDLBN,B., 253, 276 CORTI, M., 183, 188, 234 J. C., 190, 234 EILBECK, COURTENS, E., 140, 160 EK, L., 76, 83 CRANE,R., 6, 45, 82 ELENKRIG, B. B., 324, 344 CRAWFORD, F. S . , 269, 276 ELIASSON, B., 69, 70, 74, 16, 77, 78, 79, 82 CROSIGNANI, B., 222, 224, 234 ELSMORE,B., 251, 276
AUTHOR INDEX
EMEUANOV, V. I., 196, 234 ENDO,Y., 189, 233 T. C., 121, 158 ENGLISH, ENLOE,L. H., 6, 83 E w , R. K., 61, 83 EVERY,I. M., 209, 234 EZEKIEL,S., 121, 159, 191, 238
F FABRE,C., 109, 158 FAJN,V. M., 194, 234 A. G., 200, 201, 233 FAJNSHTEJN, FANO,U., 107, 159 E. J., 266, 276 FARRELL, FEINBERG, E. L., 287, 343 FEINUP, J. R., 246, 276 FELD,M. S., 100, 136, 159 FELD,Y. N., 285, 288, 300, 301, 302, 322, 343 A. I., 122, 159 FERGUSON, FIELD,J. R., 181, 234 FILSETH, S. V., 128, 159 FINSEN, W. S., 241, 276 FIZEAU,H., 241, 276 FLUSBERG, A., 101, 128, 130, 159 FLY", G. W., 253, 276 J. J., 154, 158, 159 FORNEY, FORTSON,E. N., 122, 128, 160 FOSSATI BELLANI, V., 76, 83 FOURNAY, M. E., 18, 83 FREED,K. F., 193, 237 FREEDHOFF, H. S., 229, 234 FREEMAN, R. R., 137, 138, 160 FRUDMAN, G. I., 229, 233 FRIED,D. L., 242,256. 258,261, 275, 276 R., 195, 234 FRIEDBERG, D., 122, 160 FROLICH, FUGGER,B., 122, 160 FURUKAWA, M., 286, 323, 345 G GALE,G. M., 123, 159 GANIEL,U., 222, 224, 234 R. L., 215, 234 GARWIN, J. A., 128, 159 GELBWACHS, J. S., 253, 276 GETHNER, GEZARI,D., 242, 276 A. K., 342, 345 GHATAK, GIACOBINO, E., 98, 101, 102, 104, 106, 107, 121, 123, 124, 127, 128, 130, 132, 133, 134, 138, 154, 158, 159
349
GIBBEN, J. D., 190, 234 GIBBS,H. M., 191, 234 GIORDMAINE, J. A., 202, 234 GLAUBER, R. J., 166, 168, 171, 179, 234 GLENN,W. H., 189, 234 GLODZ,M., 201, 235 GLOGE,D., 320, 343 C., 322, 344 GOATLEY, GOEPPERT-MAYER, M., 87, 159, 196, 234 GOUAEV,Yu.D., 228, 229, 233 GOODIER, J. N., 57, 84 J. W., 7, 9, 83 GOODMAN, J. P., 137, 138, 160, 318, 343 GORDON, GRABNER, L., 87, 160 A. H., 250, 276 GREENAWAY, GREENWOOD, D. P., 275, 276 GRIGORIANTS, V. V., 324, 344 GROVE,R. E., 121, 159, 191, 238 GRYNBERG, G., 87, 91, 101, 102, 104, 106, 107, 108, 110, 115, 116, 120, 121, 123, 124, 125, 126, 127, 128, 130, 132, 133, 134, 138, 139, 141, 142, 148, 150, 154, 158, 159, 161 V. L., 287, 343 GUINZBURG, GULIAEV, Y. V., 324, 344 GUPTA,R., 126, 161 GUPTA,S., 181, 234 A. N., 324, 343 GURIANOV, GUTMAN, A. S., 295, 296, 344
H HAHN,E. L., 136, 159 HAKEN,H., 185, 188, 219, 234 HALL,J. L., 135, 159, 244, 276 HAMAL,K., 203, 234 HANBURY BROWN,R., 242, 276 HANSCH,T. W., 100, 120, 122, 123, 128, 135, 159, 160, 161 HARDY,J. W., 269, 270, 277 S., 190, 235 HAROCHE, HARPER,C. D., 127, 128, 160 S. R., 101, 128, 130, 136, 159, HARTMANN, 160, 195, 234 HARVEY, K. C., 120, 123, 127, 129, 130, 134, 159, 160 HASSAN, S. S., 195, 235 R. T., 127, 160 HAWKINS, HEIN,H., 201, 235 HEITLER,W., 92, 160 HELLER,F. S.,318, 344
350
AUTHOR INDEX
HERCHER, M., 190, 237 D. R.,308, 344 HERRIOTT, HERRMANN, J., 187, 189, 190, 191, 231, 235 HEUMANN, E., 214, 235, 238 HIROTA,O., 217, 235 HOGGE,C. B., 265, 277 HOPIN,V. F., 324, 343 HORAK,R., 180, 236 V., 87, 160 HUGHES,
I IKEHARA, S., 217, 235 Yu. A,, 228, 235 ILINSKY, ILYIN,V. G., 324, 343 INDEBETOUW, G., 67, 83 INEICHEN, B., 6, 18, 24, 33, 35, 36, 37, 46, 47, 48, 49, 50, 53, 54, 56, 58, 59, 60, 68, 69, 74. 77, 78, 79, 80, 82, 83 ITEN, Y.,18, 84 ITOH,P. D., 65. 83 IVANOV, G. A., 324, 344 1 JAKEMAN, E., 179, 193, 235 JAKES,W. C., 6, 83 JAVAN, A,, 100, 159 JITSCHIN, W., 128, 160 JOHNSON, B. W., 129, 160 JONES,P. F., 128, 159 JONES,R.,74, 82 JORTNER, J., 140, 160
K KARAPETIAN, G. O., 324, 343 KARCZEWSKI, B., 195, 198, 215, 234, 235 M. A., 324, 343 KARTSEV, A., 87, 159 KASTLER, KATO,Y., 127, 128, 160 KATZ,J., 40, 83 KAWAKAMI, S., 305, 312, 316, 344 KELDYSH, L. V., 200. 235 KELLEHER, K. S., 322, 344 KELLY,P. J., 128, 159 KERSCH, L. A., 18, 83 V. A,, 87, 159 KHODOVOI, KHOKHLOV, R. V., 320, 342, 343 KHOO,I. C., 128, 159 S., 220, 232, 235, 237 KIELICH, A., 184, 235 KIKKAWA, C., 196, 236 KIKUCHI,
KILIN,S. Ja., 190, 191, 233 KIMBLE, H. J., 184, 190, 191, 192, 193, 235 KITANO,I., 286, 323, 345 KLAUDER, J. R.,166, 168, 235 KLEINSCHMIDT, J., 214, 215, 235 Yu.L., 188, 196, 234, 235 KLIMONTOVICH, KLIMOV, I. I., 324, 343 KLYSHKO, D. N., 221, 228, 235 KNAPP,S. L., 242, 276 KNESEL, L., 228, 236 KNIGHT,P. L., 199, 235 KOBE,D. H., 154, 160 KOCK,W. E., 285, 344 KOGELNIK, H., 117, 160, 296, 344 KOGELSCHATZ, U., 24, 83 KOIZUMI, K., 286, 323, 345 KORENEVA, N. A., 324, 344 KORNHAUSER, E. T., 312, 318, 320, 344 A. S., 188, 235 KOVALEV, M., 220, 225, 235 KOZIEROWSKI, KRASINSKI, J., 201, 214, 235 KRJUKOV, P. G., 189, 235 KRYSZEWSKI, S., 225, 235 KUBECEK, V., 203, 234 KUMAR, S., 196, 235 KURNIT, N. A,, 136, 160 V. D., 324, 343 KURNOSOV, KURTZ,C. N., 304, 318, 345 KUSCH,P., 126, 160 T. J., 203, 236 KUSNETSOVA, L LABEYRIE, A., 242, 276, 277 LALOE,F., 122, 160 P., 196, 199, 200, 233, LAMBROPOULOS, 234, 236 P. S., 188, 235 LANDA, LANZL, F., 61, 83 C., 201, 202, 236 LECOMITE, G., 97, 160 LECOMPTE, LEE, S. A,, 128, 135, 159, 160 LEITE,J . R., 136, 159 LETOKHOV, V. S., 189, 190, 235, 236 M. D., 120, 123, 127, 128, 159, LEVENSON, 160 LI, T., 117, 160 LIAO,K. H., 126, 161 LIAO,P. F., 100, 137, 138, 159, 160 LIBERMAN, S., 126, 159 LIEWER,K. M., 242, 276
AUTHOR INDEX
LIFSHIZ,V. Y.,324, 343 LIN,L. H., 35, 61, 82 LOHMANN, A. W., 18, 83 R., 166, 196, 205, 208, 209, 231, LOUDON, 236, 237 W. H., 166, 236 LOUISELL, S., 7, 9, 83 LOWENTHAL, LOY,M. M., 137, 138, 160 LUNEBURG, R. K., 284, 292, 344
M MAINFRAY, F., 87, 158 MAINFRAY, G., 97, 160, 201, 202, 236 MAIEWSKI, W., 201, 235 MALIK,D. P. S., 342, 345 MANDEL, L., 166, 170, 179, 184, 188, 190, 191, 192, 193, 234, 235, 236, 237, 253, 277 C., 87,97, 158, 160,201,202,236 MANUS, E. A. J., ‘304, 318, 320, 343, MARCATILI, 344 D., 286, 304, 344 MARCUSE, MAROM,E., 11, 19, 20, 23, 25, 28, 29, 31, 40, 83 MARR,G. V., 129, 160 A. V., 200, 201, 233 MASALOV, MASEK,V., 52, 53, 83 MASSIE,N. A., 52, 83 J., 52, 53, 54, 56, 58, 59, 60, 82, MASTNER, 83 MATE,K. V., 18, 83 H., 286, 323, 345 MATSUMURA, A. A., 324, 343 MATSVEIKO, MAXIMOV, V. M., 324, 343 J. C., 283, 284, 344 MAXWELL, J., 308, 344 MCKENNA, MCNEIL,K. J., 196, 217, 236 MEHTA,C. L., 168, 181, 196, 220, 233, 234, 235, 236 MEISEL,G., 120, 123, 127, 128, 159, 160 A., 65, 83 MELLING, MEVERS,G. E., 242, 258, 276 MEYSTRE, P., 193, 234 A. A., 241, 277 MICHELSON, A. L., 285, 286, 288, 298, 299, MIKAELIAN, 300,301,311, 312, 313, 318, 321, 325, 327, 328, 329, 330, 331, 333, 334, 337, 340, 344 MILLER,S. E., 286, 304, 344 MIRZAEV, Ag. T., 181, 236
35 1
MISTA,L., 223, 224, 236 B. K., 225, 236 MOHANTY, MOHR,U., 209, 236 B. R., 190, 196, 197, 236 MOLLOW, MOLOTSCHKOV, U. B., 285, 345 MORAW,R., 82, 83 MORGAN, S. P., 294, 344 MOROZOV, V. N., 324, 343 MOSSBERG, T., 101, 128, 130, 159 MOSTOWSKI, J., 199, 236 MO-ITIER,F. M., 6, 8, 11, 18, 19, 20, 23, 25, 28, 29, 31, 46, 47, 48, 49, 68, 69, 74, 77, 78, 79, 82, 83 S., 140, 160 MUKAMEL, MULLER,R. A., 269, 276
N NAYAK,N., 225, 236 NELSON,D., 52, 83 NEMES,G., 213, 236 V. B., 324, 343 NEUSTRUEV, NEW,G . H., 189, 236 NIEMAX, K., 129, 160 A. V., 324, 343 NIKOLAICHIK, NISHIZAWA, J., 305, 312, 316, 344 NOBIS,D., 77, 83 NOLL,R. J., 260, 262, 277 A., 203, 234 NOVOTNY, 0
OKE,J. B., 255, 277 OMONT,A., 107, 160 ORTH,C. D., 269, 276 OSBORN, R. K., 196, 236
P PAISNER, J. A., 190, 235 PASCALE, J., 134, 160 PASCU, M. L., 129, 160 PAUL,H., 209, 236 PEASE,F. G., 241, 277 V. I., 324, 343 PELIPENKO, A. M., 171, 236 PERELOMOV, PERINA,J., 166, 168, 180, 193, 220, 223, 224, 226, 228, 231, 232, 236 PERINOVA, V., 168, 180, 220, 223, 224, 226, 228, 236 PERRIN,J. C., 67, 83 V. M., 228, 235 PETNIKOVA, R. A., 285 288, 292, 295, 296, PETROVA, 300, 302, 314, 345
352
PIKE,E. R., 193, 235 PINARD, J., 126, 159 PINARD, M., 122, 160 PITLAK,R. T., 82, 83 POLITCH, J., 19, 83 POLLAINE, S. M., 269, 276 POLUEKTOV, I. A., 190, 236 PONATH, H.-E., 190, 219, 232, 236 POPESCU, D., 129, 160 POPESCU, I., 129, 160 POPOV,Yu. M., 190, 236, 324, 343 POTAPOV, V. T., 324, 344 POULSEN, O., 135, 159 PRAKASH, H., 209, 234 PRITCHARD, D., 120, 160 PROKHOROV, A. M., 324, 343 PRYPUTNIEWICZ, R., 69, 83 PUSEY,P. N., 193, 235
Q QUATTROPANI, A., 154, 158, 159
R RACAH,G., 107, 159 RAHMAN, N. K., 199, 233 RAJAPOV, L., 181, 236 RAMSEY, N. F., 143, 160 RAWSON, E. G., 308, 344 REHLER, N. E., 195, 234, 236 REINISCH, R., 230, 233 RENTSCH, S., 214, 215, 235 RENTZEPIS, P. M., 202, 234 RESSAYRE, E., 196, 236 RHODES, C. K., 128, 159 RICCA,A. M., 189, 233 RISKEN, H., 188, 236 RITZE,H. H., 209, 233 ROBERTS, D. E., 122, 128, 160 RODDIER, C., 263, 277 ROJTBERG, V. S., 190, 236 ROKNI,M., 215, 238 ROUSSEAU, D. L., 140, 161 RUBINSTEIN, C. B., 6, 83 RUDD,M. J., 65, 84 RYLE,M., 246, 251, 276, 277
S SAFIULINA, S. S., 324, 343 SAKURAI, K., 189, 233 SALEH, B., 254, 277
AUTHOR INDEX
SALOUR,M. M., 127, 143, 146, 147, 159, 160
SANCHEZ, F., 97, 160, 201, 202, 236, 237 SARGENT, M., 188, 237 S A ~ A R O D. V , K., 324, 343 SAUER,K., 190, 233 SCADDAN, R. J., 263, 276 SCHAWLOW, A. L., 120, 123, 127, 159, 160 SCHIFF, L. J., 184, 237 SCHILD, R. E., 255, 277 SCHLUTER, M., 61, 83 SCHRODER, H. W., 122, 160 SCHUBERT, D., 189, 214, 233, 235 SCHUBERT, M., 172, 173, 190, 197, 199, 207, 208, 211, 212, 213, 214, 215, 216, 218,219,221,229,230,232,236,237 SCHUDA, F., 190, 237 SCHUMANN, W., 69, 83 SCHUTTE,F. J., 222, 232, 237 SCOTT, F. P., 251, 277 SCULLY, M. O., 188, 237 SEMYONOV, A. S., 324, 343 SENITZKY, I. R., 196, 237 SERGEEV, A. B., 324, 343 S ~ V I G NL., Y , 18, 83 SHAMIR, J., 19, 83 SHAO,M., 270, 277 SHAPIRO, S. L., 202, 234 SHEDNOVA, A. K., 231, 233 SHEN,Y. R., 196, 197, 210, 218, 220, 237 SHIDLOVSKY, R. P., 324, 343 SHIOTAKE, N., 18, 84 SHISHAYEV, A. V., 87, 114, 143, 146, 159, 161 SHOEMAKER, R. L., 136, 138, 159 SIMAAN, H. D., 196, 205, 208, 20Y. 231, 237 SISAKIAN, 1. N., 324, 343 SI'ITIG,E. K., 45, 83 SLICHTER, L. B., 312, 345 SLUSHER, R. E., 190, 237 SMIRNOV, D. F., 190, 237 SMIRNOVA, T. N., 201, 202, 237 SMITH,A. W., 188, 233 SMITH,S. R., 181, 233 SOBELMAN, I., 96, 160 SODHA,M. S., 342, 345 SOLIMENO, S., 222, 224, 234 SOLLID, J. E., 12, 83 SOMMARGREN, G. E., 61, 83
AUTHOR INDEX
SONA,A., 76, 83 SOROKIN, P. P., 215, 237 SOSNIN,V. P., 324, 344 M. D., 176, 237 SRINIVAS, R. V., 242, 276 STACHNIK, STAELIN, D. H., 270, 277 STEEL.,W. H., 270, 277 L. E., 128, 159 STEENHOEK, STEIN,L., 122, 160 STEINER, M., 121, 160 STELMAKH, M. F.,324, 343 K. A., 69, 70, 74, 83, 84 STETSON, STEVENSON, W. H., 45, 47, 84 STOICHEIF, B. P., 127, 128, 129, 130, 134, 159, 160 STOLER,D., 222, 237 STREIFER, W., 304, 318, 345 STREY,G., 232, 233, 237 STROUD,C. R., 190, 237 SUDARSHAN, E. C. G., 166, 168, 179, 235, 236 SUKHORUKOV, A. P., 231, 233, 320, 342, 343 SURGET,J., 19, 84 SUSSE,K. E., 190, 191, 234 SUZUKI, N., 184, 235 P., 232, 237 SZLACHETKA, SZOKE,A., 140, 160
353
D. P., 324, 344 TREGUB, TREHIN,F., 123, 161 TRIEBEL, W.. 214, 235, 238 TRIVEDI, S., 196, 233 TRCTSHIN, A. S., 190, 237 TRUNG,T. V., 222, 232, 237 TSEKERIS, P., 126, 161 TSURUTA, T., 18, 84 TUNKIN, V. G., 228, 229, 233 Twiss, R. Q., 242, 244, 251, 276, 277
U UCHIDA,T., 286, 323, 345 J. C., 40, 84 URBACH,
V VANDEPLANQUE, J., 134, 160 VARNER,J. R., 67, 84 VASILENKO, L. S., 87, 114, 143, 146, 159, 161 VAUGHAM, J. M., 193, 235 VELZEL,C. H. F., 12, 84 T. N. C., 191, 234 VENKATESAN, VEST,C. M., 61, 77, 83, 84 VIALLE,J. L., 126, 159 VILLAEYS, A. A,, 193. 237 VOGEL,W., 172, 173, 216, 230, 237 VOGLER,K., 189, 233
W T TALLET,A., 196, 236 TANAS,R., 220, 225, 235 TANGO,W. J., 242, 243, 244, 250, 268, 276, 277 TANZLER,W., 232, 237 TATARSKI, V. I., 258, 259, 273, 275, 277 M. M., 188, 237 TEHRANI, TEICH,M. C., 196, 237 J., 87, 158 THEBAUT, THOMAS,A., 67, 83 E. A., 201, 202, 237 TIKHONOV, TIMOSHENKO, S. P., 57, 84 TINDLE,C. T., 221, 238 TOBIRASHKU, S. S., 200, 201, 233 TOLSTOY,I., 287, 345 TORALDO DI FRANCIA, G., 294, 345 N., 196, 209, 237 TORNAU, H. C., 136, 139, 160 TORREY, TOSCHEK, P., 89, 98, 100, 160 TOTIZEBEN,W., 215, 235
WAGGONER, A. P., 18, 83 R., 122, 128, 135, 136, 159, WALLENSTEIN, 160, 161 WALLIS,G., 190, 233 WALLNER, E. P., 266, 268, 270, 277 WALLS,D. F., 190, 193, 195, 196, 210, 217, 221, 231, 234, 235, 236, 237, 238 H., 121, 160, 191, 238 WALTHER, WATRASIEWICZ, B. M., 65, 84 WEBER,H.P., 202,203,211,212,213,238 WEBER,K. H., 129, 160 WECHT,K. W., 202, 234 WEIDLICH, W., 219, 234 WEIDNER,M., 189, 235 WELFORD,W. T., 251, 277 WELLING, H., 122, 160 WELSCH,D., 190, 191, 235 WESLEY,M. L., 256, 277 WESSEL,J. S., 128, 159 WEST,J. B., 121, 158 WHEATLEY,S. E., 127, 160
354
AUTHOR INDEX
WHERRETT,S. R., 129, 160 WHITHELAW, J. H., 65, 83 WICKES,W. C., 241, 277 WIEDERHOLD, G., 215, 237 WIEMAN,C., 128, 135, 160, 161 WIENECKE, J., 187, 235 WIGNER,E. P., 176, 238 WILHELMI,B., 187, 189, 197, 199, 207, 208, 211, 212, 213,214, 215, 218, 229, 233, 235, 237, 238 WILLIAMS, P. F., 140, 161 WINTER,J. M., 87, 161 WOERDMAN, J. P., 128, 161 WOLF, E., 166, 169, 176, 179, 233, 236, 231, 238, 241, 252, 276, 288, 393 WOLGA,G. J., 196, 237 WORLOCK, J. M., 87, 161 Wu, F. Y., 121, 159, 191, 238 Y YAGHJIAN, A. D., 312, 320, 344
YANG,K. H., 154, 161 YARIV, A., 222, 224, 234 YASHUMOV,I. V., 324, 343 YATSIV,S., 215, 238 YEUNG,E. S., 128, 159 YOSHIMURO, T., 184, 235 YOUNG, A. T., 249, 277 YUEN,H. P., 172, 215, 216, 238 YURSHIN,B. Ya., 143, 146, 159 YUSHIN,A. S., 324, 343
Z ZACHARIAS, H., 128, 159 ZADEH,L. A., 213, 238 ZARETSKY,A. A., 324, 343 ZELKIN,E. G., 285, 288, 292, 295, 296, 300, 301, 302, 313, 314, 345 ZERNIKE,F., 296, 345 ZHUK,M. S., 285, 345 ZYGAN,K., 121, 160
SUBJECX INDEX A 171, 178, 183, 191, 203,205, 214, 221, 228, 263, 267, 273 aberration, 24, 309 cross-spectral density, 169 absorption - lineshape, 113 D -, multiphoton, 89, 110, 196, 199, 201, density matrix, 91, 98, 106, 107, 117, 131, 203, 209, 211,213, 231 175, 185, 216 -, two-photon, 87, 95-97, 105, 113, 148, Dirac distribution, 114, 116 202,207, 209, 214, 215 Doppler broadening, 87-89, 93, 100, 110, acoustic wave, 52 138, 148, 151 Airy function, 13 dephasing, 137 analytic signal, 7, 166 - effect, 113, 115, 116, 150 antenna theory, 288 antibunching, 165, 171, 192, 193, 209, 217, - shift, 65, 88, 93, 140 - velocimetry, 65 220, 222, 223 width, 99, 115, 137 aperture function, 260 autocorrelation function, 9, 10, 12-14, 41
E
B
eikonal equation, 331 elasticity, modulus of, 70 electric-dipole approximation, 92, 152, 184 emission, two-photon, 215 Euler equation, 306, 307, 325, 326, 328, 331, 332
Bessel function, 13 Bose-Einstein distribution, 208 Bourdon tube, 78 Bragg cell, 62 - condition, 51 bunching, 165, 171, 220 C
Clebsch-Gordan coefficients, 94, 105 coherence, 165, 167, 175 -, degree of, 37, 39, 168, 249, 253, 262 - function, 168, 169 optical, 139 -, spatial, 87, 229 temporal, 87 - time, 180, 199 - volume, 169 coherent state, 171-174, 209, 222 --, atomic, 189 -_ , global, 177 --, two-photon, 137, 172, 173, 216, 217 correlation function, 8, 96, 166, 167, 169-
-. -.
F Fabry-Perot etalon, 122, 125, 149 resonator, 123 fiber optics, 45 fluorescence, 129, 131, 133, 191, 202, 203 FM-demodulator, 65 --receiver, 65 Fokker-Planck equation, 177, 220. 226 Fourier transform, 8, 9, 29, 95, 122, l!$O, 176. 250 frequency up-conversion, 217, 218, 221, 224, 225 Fresnel approximation, 28, 33 fringe contrast, 37, 43, 61 - effect, 29 -, interference, 32, 48
355
356
SUBJECT INDEX
- localization, 12 - pattern, 28 - tracking, 270 - visibility, 246, 253, 270 G
50-54, 59, 61-64, 75-77, 82 -, Michelson stellar, 239, 242, 243, 247,
252, 255, 261, 262, 268-270
-, radio, 246, 251 -, speckle, 242, 250 ionization, multiphoton, 97 irradiance, 258, 259
Gauss equation, 315 Gaussian mode, 117 generating function, 207 geometrical optics, 305, 310
J Jacobi polynomial, 3 16
H
K
Hamiltonian, 91, 92, 153, 155, 184, 185, 194, 204, 215, 216 Heisenberg equation, 194 - picture, 167, 177, 221 Hilbert space, 167 hologram, 3, 11, 18, 19, 21-25, 28, 32, 34, 35, 38, 40-42, 44, 46, 47, 52, 55, 57, 64. 67, 76 -, computer-generated, 3 - efficiency, 55 -, high speed, 32, 40 -, two-reference-beam, 22, 51, 56, 61. 6 3 holography, 283 --, double exposure, 26, 41, 44, 62, 65 -, pulse, 65 -, heterodyne, 66 -, real time, 27 -, two-reference-beam, 26, 66 Hooke’s law, 70 hyperfine structure, 105, 126, 127
Kerr-cell, 62
I impulse response function, 29, 42 inhomogeneous media, 283, 287, 288, 311, 324, 330, 332, 343 intensity fluctuation, 187 -, mutual, 34, 36, 37, 39 integrated optics, 283 interferometer, amplitude, 242 intensity, 242 -, phase-switched, 246 -, Rayleigh, 252 -, stellar, 249 interferometry, heterodyne, 1, 3-6, 11, 13, 18, 30, 33, 36, 38, 40, 43, 46, 50-54, 59-61, 63, 64, 75-77. 82 -, holographic, 1, 3-6, 10-13, 16, 18, 20, 24, 25, 30, 32, 33, 36, 38, 40, 43-45,
-.
L Laguerre function, 304, 318 Langevin force, 189 laser, continuous wave, 40 -,dye, 62, 87, 120, 121, 123, 129, 135 -, He-Ne, 42 -, pulsed, 40, 62 ruby, 42 -, two-photon, 215, 216 Legendre polynomial, 316 Lorentzian function, 114-1 16 - shape, 94, 113, 120, 132 lineshape, 95 Luneburg lens, 284,285,287, 291-296
-.
M master equation, 99 Maxwell equations, 167, 168 - lens, 284, 287-291, 293, 294, 296 Maxwellian distribution, 114, 116 Michelson interferometer, 122 - visibility, 229 Mikaelian lens, 285, 300-302, 322
N nonlinear optics, 163, 165, 196 number state, 209 0 optical echo, 190 - free induction decay, 190 - nutation, 190 oscillator strength, 94, 109, 111
P parametric amplification, 217, 218, 221, 222
SUBJECT INDEX
- generation, 229 paraxial approximation, 24 - rays, 305, 318, 320 perturbation theory, time dependent, 92 photodetector, 45, 47 photoefficiency, 180 photoelectric effect, 171 photon distribution, 207 idler, 218, 224 -, pump, 218 signal. 2 18. 222, 224 - statistics, 199, 270 Pockels-cell, 62 Poisson distribution, 172, 187, 208 - ratio, 70 - statistics, 253, 254, 264 positronium, 116 power spectrum, 256, 257, 264 P-represention, 168, 175-178, 184, 192,
-.
-.
220 pupil function, 29
Q quadrupolar transition, 103 quasi-Landau level, 128
R Rabi frequency, 111, 139 - nutation, 139 - oscillation, 200 Racah coupling, 104 radiant intensity, 169 Raman excitation, 140 - scattering, 165, 229-232 Ramsey fringes, 136, 146, 147 Rao-Cramer theorem, 266 Rayleigh length, 117 reflection coefficient, 7 resonance fluorescence, 190, 192, 195 rough surface, 7, 9, 11-13, 34, 61 Rydberg state, 109, 120, 127, 128 S
saturable absorber, 186, 189 Schrodinger state, 155, 172, 184 second-harmonic generation, 122, 218, 225, 226 selection rules, 102 self focusing, 285, 286, 302 -- waveguide, 285, 297, 298, 303, 304,
357
306,308, 311,313,318,323,333-338, 340, 342 --induced transparency, 190 shot noise, 266, 267 shutter, acousto-optical, 62 electro-optical, 62 signal-to-noise ratio, 46, 54, 55, 255, 263, 270 speckle, 9-11, 13-15, 18, 30, 32, 33, 40, 43, 242 - field, 16, 17 - noise, 17 - pattern, 41, 42, 57 spectroscopy, 116 -, high resolution, X7 multiphoton, 85,X7 -, saturated absorption, 135 two-photon. 121, 151. 152 spin echo, 136 spontaneous emission, 186, 194 Stark shift, 107, 129 Stokes mode, 231 sum-frequency generation, 228 superradiance, 194
-.
-. -.
T transfer function, atmospheric, 260 transition probability, 94, 97 turbulence, atmospheric, 241,247, 253,255
U uncertainty principle, 90, 172
V van Cittert-Zernike theorem, 249 van der Waals interaction, 134 Voigt profile, 118
W waveguide. parabolic, 318, 319 - self focusing (see self focusing waveguide) -, telescopic, 337, 339, 340, 342 Wigner-Eckart theorem, 102 - 65 coefficient, 105, 132
.
Y Young’s experiment, 169 2
Zeeman effect, 127 - shift, 128
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CUMULATIVE INDEX - VOLUMES I-XWI ABEL~S, F., Methods for Determining Optical Parameters of Thin Film 11, 249 ABELLA,I. D., Echoes at Optical Frequencies VII, 139 ABITBOL,C. I., see J. J. Clair XVI, 71 AGARWAL, G. S., Master Equation Methods in Quantum Optics XI, 1 AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion IX, 235 ALLEN,L., D. G. C. JONES,Mode Locking in Gas Lasers IX, 179 AMMANN, E. O., Synthesis of Optical Birefringent Networks IX, 123 ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, 21 1 XI, 245 ARNAUD, J. A., Hamiltonian Theory of Beam Mode Propagation BALTES,H. P., On the Validity of Kirchhoff’s Law of Heat Radiation for a Body in a Nonequilibrium Environment XII, 1 BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationI, 67 Free Diffraction Images S., Beam-Foil Spectroscopy BASHKIN, XII, 287 BECKMANN, P., Scattering of Light by Rough Surfaces VI, 53 BEVERLY111, R. E., Light Emission from High-Current Surface-Spark Discharges XVI, 357 BLOOM,A. L., Gas Lasers and their Application to Precise Length Measurements IX, 1 BOUSQUET,P., see P. Rouard IV, 145 BRUNNER, W., H. PAUL, Theory of Optical Parametric Amplification and Oscillation xv, 1 O., Applications of Shearing Interferometry BRYNGDAHL, IV, 37 O., Evanescent Waves in Optical Imaging BRYNGDAHL, XI, 167 BURCH,J. M., The Metrological Applications of Diffraction Gratings 11, 73 CAGNAC,B., see E. Giacobino XVII, 85 CASASENT,D., D. PSALTIS,Deformation Invariant, Space-Variant Optical Pattern Recognition XVI, 289 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 CLAIR,J. J., C. I. ABITBOL,Recent Advances in Phase Profiles Generation XVI, 71 CLARRICOATS, P. J. B., Optical Fibre Waveguides-A Review XIV, 327 COHEN-TANNOUDJI, C., A. KASTLER,Optical Pumping v, 1 COLE,T. W., Quasi-Optical Techniques of Radio Astronomy XV, 187 CREW, A. V., Production of Electron Probes Using a Field Emission Source XI, 221 CUMMINS, H. Z., H. L. SWINNEY, Light Beating Spectroscopy VIII, 133 XIV, 1 DAINTY,J. C., The Statistics of Speckle Patterns DANDLIKER, R., Heterodyne Holographic Interferometry XVII, 1 DECKERJr., J. A,, see M. Harwit XII, 101 DELANO,E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 359
3 60
CUMULATIVE INDEX
DEMARIA,A. J., Picosecond Laser Pulses IX, 31 DEXTER,D. L., see D. Y.Smith X, 165 DREXHAGE, K. H., Interaction of Light with Monomolecular Dye Layers XII, 163 DUGUAY,M. A., The Ultrafast Optical Kerr Shutter XIV, 161 EBERLY,J. H., Interaction of Very Intense Light with Free Electrons VII, 359 ENNOS,A. E., Speckle Interferometry XVI, 233 FIORENTINI, A., Dynamic Characteristics of Visual Processes I, 253 FOCKE,J., Higher Order Aberration Theory IV, 1 FRANCON,M., S. MALLICK,Measurement of the Second Order Degree of Coherence V1, 71 FRIEDEN,B. R., Evolution, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX,311 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I, 109 GAMO,H., Matrix Treatment of Partial Coherence 111, 187 GHATAK, A. K., see M. S. Sodha XIII, 169 GIACOBINO, E., B. CAGNAC,Doppler-Free Multiphoton Spectroscopy XVII, 85 V. L., see V. M. Agranovich GINZBURG, IX, 235 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 11, 109 GNIADEK, K., J. PETYKIEWICZ, Applications of Optical Methods in the DiffracIX, 28 1 tion Theory of Elastic Waves VIII, 1 GOODMAN, J. W., Synthetic-Aperture Optics GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 HARWIT, M., J. A. DECKERJr., Modulation Techniques in Spectrometry XII, 101 HELSTROM, C. W., Quantum Detection Theory X, 289 HERRIOTT, D. R., Some Applications of Lasers to Interferometry VI, 171 HUANG,T. S., Bandwidth Compression of Optical Images x, 1 JACOBSSON, R., Light Reflection from Films of Continuously Varying RefracV, 247 tive Index JACOUINOT, P., B. ROIZEN-DOSSIER, Apodisation 111, 29 JONES,D. G. C., see L. Allen IX, 179 KASTLER, A,, see C. Cohen-Tannoudji v, 1 IV, 85 KINOSITA, K., Surface Deterioration of Optical Glasses KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, 1 F., The Elements of Radiative Transfer 111, 1 KOTTLER, IV, 28 1 KOITLER,F., Diffraction at a Black Screen, Part I: Kirchhoff’s Theory VI, 33 1 KOITLER,F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory 1,211 KUBOTA,H., Interference Color XIV, 47 LABEYRIE, A., High Resolution Tehcniques in Optical Astronomy XI, 123 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XVI, 119 LEE, W-H., Computer-Generated Holograms: Techniques and Applications VI, 1 LEITH,E. N., J. UPATNIEKS, Recent Advances in Holography XVI, 1 LETOKHOV, V. S., Laser Selective Photophysics and Photochemistry VIII, 343 LEVI,L., Vision in Communication LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch V, 287 of Physical Optics VI, 71 S., see M. Francon MALLICK,
CUMULATIVE INDEX
36 1
11, 181 MANDEL,L., Fluctuations of Light Beams XIII, 27 MANDEL,L., The Case for and against Semiclassical Radiation Theory XI, 303 MARCHAND, E. W., Gradient Index Lenses xv, 77 MEESSEN,A., see P. Rouard VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Quasi-Classical Theory of Laser VII, 231 Radiation XVII, 279 MIKAELIAN, A. L., Self-Focusing Media with Variable Index of Refraction I, 31 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design V, 199 MURATA,K., Instruments for the Measuring of Optical Transfer Functions VIII, 201 MUSSET,A., ‘A. THELEN,Multilayer Antireflection Coatings XV, 139 OKOSHI,T., Projection-Type Holography VII, 299 OOUE,S., The Photographic Image xv, 1 PAUL,H., see W. Brunner PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 VII, 67 PEGIS,R. J., see E. Delano V, 83 P. S., Non-Linear Optics PERSHAN, IX, 28 1 PETYKIEWICZ, J., see K. Gniadek V, 351 PICHT,J., The Wave of a Moving Classical Electron XVI, 289 PSALTIS,D., see D. Casasent RISEBERG,L. A., M. J. WEBER, Relaxation Phenomena in Rare-Earth XIV, 89 Luminescence VIII, 239 RISKEN,Statistical Properties of Laser Light 111. 29 ROIZEN-DOSSIER, B., see P. Jacquinot ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical XIII, 69 Aberration Measurements of the Human Eye IV, 145 ROUARD,P., P. BOUSQUET, Optical Constants of Thin Films x v , 77 ROUARD,P., A. MEESSEN,Optical Properties of Thin Metal Films IV, 199 RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave XIV, 195 RUDOLPH,D., see G. Schmahl VI, 259 SAKAI,H., see G. A. Vanasse XIV, 195 SCHMAHL. G., D. RUDOLPH,Holographic Diffraction Gratings SCHUBERT, M., B. WILHELMI, The Mutual Dependence between Coherence XVII, 163 Properties of Light and Nonlinear Optical Processes XIII, 93 SCHULZ,G., J. SCHWIDER, Interferometric Testing of Smooth Surfaces XIII, 93 SCHWIDER, J., see G. Schulz X, 89 SCULLY,M. 0.. K. G. WHITNEY, Tools of Theoretical Quantum Optics I. R., Semiclassical Radiation Theory within a Quantum-Mechanical SENITZKY. XVI, 4 13 Framework XV, 245 SIPE,J. E., see J. Van Kranendonk X, 229 SITTIC, E. K., Elastooptic Light Modulation and Deflection XII, 53 SLUSHER, R. E., Self Induced Transparency VI, 21 1 SMITH,A. W., see J. A. Armstrong SMITH,D. Y., D. L. DEXTER,Optical Absorption Strength of Defects in X, 165 Insulators x. 45 SMITH,R. W., The Use of Image Tubes as Shutters V. K. TRIPATHI, Self Focusing of Laser Beams in SODHA,M. S., A. K. GHATAK, XIII, 169 Plasmas and Semiconductors V, 145 STEEL,W. H., Two-Beam Interferometry
362
CUMULATNE INDEX
STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere IX, 73 STROKE, G. W., Ruling, Testing and Use of Optical Gratings for HighResolution Spectroscopy 11. 1 SVELTO,O., Self-Focusing, Self-Trapping, and Self-phase Modulation of Laser XII, 1 Beams VIII, 133 SWINNEY, H. H., see H. 2. Cummins XVII,239 TANGO, W. J., R. Q. TWISS,Michelson Stellar Interferometry V, 287 TAYLOR, C. A., see H. Lipson VII, 23 1 TER-MIKAELIAN, M. L., see A. L. Mikaelian VIII, 201 THELEN,A,, see A. Musset VII, 169 THOMPSON, B. J., Image Formation with Partially Coherent Light XIII, 169 TRIPATHI, V. K., see M. S. Sodha J., Correction of Optical Images by Compensation of Aberrations TSUIIUCHI, 11, 131 and by Spatial Frequency Filtering XVII,239 Twiss, R. Q., see W. J. Tango VI, 1 UPATNIEKS, J., see E. N. Leith VI, 259 VANASSE,G. A., H. SAKAI,Fourier Spectroscopy I, 289 VANHEEL, A. C. S., Modern Alignment Devices VAN KRANENDONK, J., J. E. SIPE, Foundations of the Macroscopic ElecXV, 245 tromagnetic Theory of Dielectric Media XIV, 245 VERNIER,P., Photoemission XIV, 89 WEBER,M. J., see L. A. Riseberg IV, 241 WELFORD,W. T., Aberration Theory of Gratings and Grating Mountings XIII, 267 WELFORD,W. T., Aplanatism and Isoplanatism XVII, 163 WILHELMI,B., see M. Schubert X, 89 WITNEY,K. G., see M. 0. Scully WOLTER,H., On Basic Analogies and Principal Differences between Optical I, 155 and Electronic Information X, 137 WYNNE,C. G., Field Correctors for Astronomical Telescopes V1.105 YAMAJI,K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in InterferVIII, 295 ence Miscoscopy XI, 77 YOSHINAGA, H., Recent Developments in Far Infrared
