PROGRESS IN OPTICS V O L U M E XIV
EDITORIAL ADVISORY BOARD
L. ALLEN,
Brighton, England
M. FRANCON,
Paris, France
E. INGELSTAM,
Stockholm, Sweden
K. KINOSITA,
Tokyo, Japan
A.
Erlangen, Germany
LOHMANN,
M. MOVSESSIAN,
Armenia, U.S.S.R.
G. SCHULZ,
Berlin, D.D.R.
W. H. STEEL,
Chippendale, N.S. W., Australia
W. T. WELFORD,
London, England
PROGRESS IN OPTICS VOLUME XIV
EDITED BY
E. WOLF University of Rochester, N.Y., U S A .
Contributors J. C. D A I N T Y , A. L A B E Y R I E
L. A. RISEBERG, M. J. WEBER
M. A. D U G U A Y , G. S C H M A H L , D. R U D O L P H P. J. V E R N I E R , P. J. B. C L A R R I C O A T S
1976 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK . OXFORD
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C O N T E N T S O F V O L U ME 1(1961) THE MODERNDEVELOPMENT OF HAMILTONIAN OPTICS.R . J . PEGIS. . . . I. 1-29 I1. WAVE OPTICS AND GEOMETRICAL OPTICS IN OPTICALDESIGN.K . MIYAMOTO 31-66 I11. THEINTENSITY DISTRIBUTION AND TOTAL ILLUMINATION OF ABERRATION-FREE . . . . . . . . . . . . . . . . . . 67-108 DIFFRACTION IMAGES.R . BARAKAT D . GABOR . . . . . . . . . . . . . . . . . 109-153 IV . LIGHTAND INFORMATION. DIFFERENCES BETWEEN OPTICAL AND V . ON BASICANALOGIESAND PRINCIPAL ELECTRONIC INFORMATION. H . WOLTER. . . . . . . . . . . . . . . . 155-2 10 COLOR.H. KUBOTA. . . . . . . . . . . . . . . . . . 211-251 VI . INTERFERENCE VII. DYNAMIC CHARACTERISTICS OF VISUAL PROCESSES. A . FIORENTINI . . . . . 253-288 VIII . MODERNALIGNMENT DEVICES.A . C. S. VAN HEEL . . . . . . . . . . . 289-329
C O N T E N T S O F V O L U M E I1 (1963) I. I1. 111. I v.
v.
VI .
RULING.TESTINGAND USE OF OPTICAL GRATINGSFORHIGH-RESOLUTION 1-72 SPECTROSCOPY. G . W . STROKE. . . . . . . . . . . . . . . . . . . . THEMETROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS. J . M . BURCH 73-108 . . . . . . 109-129 DIFFUSIONTHROUGH NON-UNIFORM MEDIA.R . G. GIOVANELLI CORRECTION OF OPTICAL IMAGES BY cOMpENSATION OF h E R R A T I O N S AND BY SPATIALFREQUENCY FILTERING. J . TSUJIUCHI . . . . . . . . . . . . . 131-180 FLUCTUATIONS OF LIGHTBEAMS.L . MANDEL . . . . . . . . . . . . . . 181-248 METHODS FOR DETERMINING OPTICAL PARAMETERS OF THINFILMS. F.A B E L ~249-288
C O N T E N T S O F V O L U M E I11 (1964) THEELEMENTS OF RADIATIVE TRANSFER. F. KOTTLER. . . . . . . . . . 1-28 AFODISATION. P . JACQUINOT AND B. ROIZEN-DOSSIER . . . . . . . . . . 29-186 111. MATRIXTREATMENT OF PARTIAL COHERENCE. H . GAMO. . . . . . . . . 187-332 I. I1.
C O N T E N T S O F V O L U M E IV (1965) I. 11. 111. IV . V. VI . VII .
HIGHERORDER ABERRATION THEORY.J . FOCKE. . . . . . . . . . . . 1-36 APPLICATIONS OF SHEARING INTERFEROMETRY. 0. BRYNGDAHL . . . . . . 37-83 SURFACE DETERIORATION OF OPTICAL GLASSES. K . KINOSITA. . . . . . . 85-143 OPTICALCONSTANTS OF THIN FILMS. P . ROUARD AND P BOUSQUET . . . . 145-197 THEMNAMOTO-WOLF DIFFRACTION WAVE.A . RUBINOWICZ. . . . . . . 199-240 ABERRATIONTHEORY OF GRATINGS AND GRATING MOUNTINGS. w.T . WELFORD 241-280 AT A BLACK SCREEN. PART I : KIRCHHOFF’S THEORY. F. KOTTLER281-314 DIFFRACTION
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C O N T E N T S O F V O L U M E V (1966) I. OPTICALPUMPING. C. COHEN-TANNOUDJI AND A . KASTLER. . . . . . . 1-81 I1. NON-LINEAR OPTICS.P . S . PERSHAN. . . . . . . . . . . . . . . . . 83-144 INTERFEROMETRY. W . H . STEEL. . . . . . . . . . . . . . . 145-197 I11. TWO-BEAM FUNCTIONS. K. IV . INSTRUMENTSFOR THE MEASURINGOF OPTICALTRANSFER MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199-245 V. LIGHTREFLECTION FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE INDEX. R . JACOBSSON . . . . . . . . . . . . . . . . . . . . . . . . . . . 247-286 DETERMINATION AS A BRANCHOF PHYSICAL VI . X-RAY CRYSTAL-STRUCTURE OPTICS. H . LIPSONAND C . A . TAYLOR . . . . . . . . . . . . . . . . 287-350 VII . THE WAVEOF A MOVINGCLASSICAL ELECTRON. J . PICHT . . . . . . . . 351-370
C O N T E N T S O F V O L U M E VI (1967) 1-52 RECENT ADVANCES IN HOLOGRAPHY. E. N . LEITHAND J . UPATNIEKS. . . I. . . . . . . . . 5369 I1. SCATTERING OF LIGHTBY ROUGHSURFACES. P. BECKMANN OF THE SECOND ORDER DEGREE OF COHERENCE. M . FRANCON I11. MWLIREMENT AND S . MALLICK .......................... 71-104 Iv. DESIGNOF ZOOM LENSES.K . YAM^ . . . . . . . . . . . . . . . . . 105-170 OF LASERS TO INTERFEROMETRY. D . R . HERRIOTT . . . . 171-209 V . SOMEAPPLICATIONS STUDIES OF kT"TarITY FLUCTUATIONS IN LASERS.J . A . ARMVI . EXPERIMENTAL STRONG AND A . W . SMITH . . . . . . . . . . . . . . . . . . . . . . 21 1-257 SPECTROSCOPY. G . A . VANASSEAND H . SAKAI . . . . . . . . . 259-330 VII . FOURIER THEORY. F. VIII . DIFFRACTIONAT A BLACKSCREEN.PART11: ELECTROMAGNETIC KOTTLER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331-377
C O N T E N T S O F V O L U M E VII (1969) I.
MULTIPLE-BEAM INTERFERENCE AND NATURAL MODESIN OPENRESONATORS. G. KOPPELMAN. . . . . . . . . . . . . . . . . . . . . . . . . . 1-66 I1. METHODS OF SYNTHESIS FOR DIELECTRIC MULTILAYER FILTERS. E . DELANO AND R . J . PEGIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67-137 I. D. ABELLA. . . . . . . . . . . . 139-168 111. ECHOESAT OPTICALFREQUENCIES. IV . IMAGEFORMATION WITH PARTULLYCOHERENT LIGHT.B. J . THOMPSON. . 169-230 V. QUASI-CLASSICAL A . L. MIKAELIAN AND M . L . THEORYOF LASERRADIATION. TER-MIKAELIAN .......................... 231-297 IMAGE.S . Oom . . . . . . . . . . . . . . . . . 299-358 VI . THEPHOTOGRAPHIC OF VERYINTENSE LIGHTWITH FREEELECTRONS. J . H . EBERLY 359415 VII . IN~RACTION
C O N T E N T S O F V O L U M E VIII (1970) SYNTHETIC-APERTURE O ~ n c s J. . W . GOODMAN . . . . . . . . . . . . . THEOPTICAL PERFORMANCE OF THE HUMAN EYE.G . A . FRY. . . . . . . SPECTROSCOPY. H . Z . C u m m s AND H. L . SWINNEY. . . 111. LIGHTBEATING COATINGS. A . MUSSET AND A . THELEN. . . IV. MULTILAYER ANTIREFLECTION PROPERTIES OF LASERLIGHT.H . RISKEN. . . . . . . . . . . V. STATISTICAL VI . COHERENCE THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY. T. YAMAMOTO ..................... L . LEVI. . . . . . . . . . . . . . . . . . VII . VISIONIN COMMUNICATION. COUNTING. C. L. MEHTA . . . . . . . . . VIII THEORYOF PHOTOELECTRON I.
I1.
1-50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
C O N T E N T S O F V O L U M E I X (1971) I. I1.
111.
IV . V. VI . VII .
GAS LASERSAND THEIR APPLICATION TO PRECISE LENGTHMEASUREMENTS. A . L. BLOOM . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-30 PICOSECOND LASERPULSES. A . J . DEMARIA . . . . . . . . . . . . . 31-71 OPTICALPROPAGATION THROUGH rn TURBULENTATMOSPHERE. J. W. STROBMN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-122 NETWORKS. E . 0. AMMANN . . . . . 123-177 SYNTHESIS OF OPTICAL BIREFRINGENT IN GASLASERS. L.ALLENAND D. G .C . JONES . . . . . . . 179-234 MODELOCKING CRYSTAL OPTICS WITH SPATIAL DISPERSION. V. M . AGRANOVICH AND V. L . GINZBURG. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235-280 APPLICATIONS OF OPTICAL METHODS IN THE DIFFRACTION m 0 R Y OF ELASTIC AND J . PETYKIEWICZ . . . . . . . . . . . . . . . 281-310 WAVES K . GNIADEK
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WII. EVALUATION. DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL SIGNALS. ON USEOF THE PROLATE FUNCTIONS. B. R . FRIEDEN BASED . . . . . . . . 31 1407
CONTENTS O F VOLUME X (1972) I.
BANDWIDTH COMPRFSSION OF OPTICAL IMAGES T . S . HUANG . . . . . . . 1-44 THEUSEOF IMAGE TUBESAS SHUTTERS. R . W . SMITH. . . . . . . . . . . 41-87 QUANTUM OPTICS.M .0. SCULLY AND K .G . WHITNEY 89-1 35 TOOLSOF THEORETICAL Iv. FIELD CORRECTORS FOR ASTRONOMICALTELESCOPES.C .G . WYNNE. . . . . 137-164 V. OPTICAL ABSORPTION STRENGTH OF DEFECTSIN INSULATORS. D .Y. SMITH AND 165-228 D . L. DEXTER. . . . . . . . . . . . . . . . . . . . . . . . . . . VI. ELASTOOPTIC AND DEFLECTION. E . K . SITTIG. . . . . 229-288 LIGHTMODULATION DETECTION VII . QUANTUM THEORY. C . W. HELSTROM . . . . . . . . . . . . 289-369 I1. I11.
CONTENTS O F VOLUME XI(1973) I. I1. I11.
Iv. V.
VI . VII.
MASTER E@UATION METHODS IN QUANTUM OPTICS. G. S. AGARWAL . . . . RECENT DEVELOPMENTS IN FAR INFRARED SPECTROSCOPIC TFCHNIQUES. H. YOSHINAGA ........................... INTERACTION OF LIGHT AND ACOUSTIC SURFACE WAVES. E. G. LEAN. . . . 0. BRYNGDAHL . . . . . . . . EVANESCENT WAVESIN OPTICALIMAGING. PRODUCTION OF ELECTRON PROBESUSINGA FIELDEMISSION SOURCE.A . V . CREWE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OF BEAM MODEPROPAGATION. J . A . ARNAUD. . . . HAMILTONIAN THEORY GRADIENT INDEX LENSES. E. W. MARCHAND . . . . . . . . . . . . . .
1-76 77-122 123-166 167-221 223-246 247-304 305-337
CONTENTS O F VOLUME XI1 (1974) I.
SELF.FOCUSING. SELF.TRAPPING. AND SELF-PHASE MODULATION OF LASER 1-51 BEAMS.0. SVELTO. . . . . . . . . . . . . . . . . . . . . . . . . I1. SELF-INDUCED TRANSPARENCY. R . E . SLUSHER. . . . . . . . . . . . . 53-100 111. MODULATION TECHNIQUES IN SPECTROMETRY. M . HARWIT.J. A . DECKER JR . 101-162 OF LIGHTWITH MONOMOLECULAR DYELAYERS. K . H .DREXHAGE163-232 Iv. INTERACTION V. THEPHASE TRANSITION CONCEPTAND COHERENCE IN ATOMICEMISSION. R. 233-286 GRAHAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. BEAM-FOIL SPECTROSCOPY. S . BASHKIN . . . . . . . . . . . . . . . . 287-344
C O N T E N T S OF VOLUME XI11 (1976) I.
ON THE VALIDITY OF KIRCHHOFF’S LAWOF HEATRADIATION FOR A BODYIN A NoNEQuILmRIuM ENVIRONMENT. H . P. BALTSS . . . . . . . . . . . . . 1-25 I1. THECASEFORAND AGAINST SEMICLASSICAL RADUTIONTHEORY. L . MANDEL 27-68 OF THE I11. OBJECTIVEAND SUBJECTIVE SPHERICAL ABERRATION MEASUREMENTS J . L . CHRISTENSEN . . . . . . . . . . . HUMAN EYE.W. M . ROSENBLUM, 69-91 IV . INTERFEROMETRIC TESTINGOF SMOOTH SURFACES. G . SCHULZ.J . SCHWIDER 93-167 V. SELFFOCUSING OF LASER BEAMSIN PLASMASAND SEMICONDUCTORS. M . s. SODHA . A. K . GHATAK . . . . . . . . . . . . . . . . 169-265 ..V. K . TRIPATHI VI . APLANATISM AND ISOPLANATISM. W. T. WELFORD . . . . . . . . . . . . 267-292
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PREFACE The present volume of PROGRESSIN OPTICS,just like its thirteen predecessors, contains review articles covering recent researches in optics and related subjects. The first article, by J. C . Dainty, deals with the statistics of speckle patterns, i.e., with the statistics of the random variation of intensity that is produced when a highly coherent light beam is reflected or transmitted by an optically rough surface. In view of the increasing utilization of laser light it has become imperative to gain a good understanding of speckle phenomena. The article presents the most important aspects of the underlying theory. The second article, by A. Labeyrie, provides a review of modern high resolution techniques employed in optical astronomy. It gives an account of the methods that are being gradually developed for the purpose of overcoming the limitations that the earth’s atmosphere imposes on the resolving power for astronomical observations in the optical region of the electromagnetic spectrum. An introductory section, dealing briefly with the history of the subject, is followed by an account of the main features of atmospheric turbulance and of its effects on the degradation of optical images. Various aspects of direct stellar interferometry are then discussed. Classic stellar interferometry, originating in the pioneering investigations of Fizeau, Michelson, Anderson and Pease is reviewed and its modern refinements and modifications are described. Other interesting new developments, including Labeyrie’s own important contributions in the area of speckle interferometry are then presented. The article also includes accounts of the synthetic aperture technique, of intensity interferometry and of heterodyne interferometry. Rare-earth-activated materials are being more and more frequently utilized in quantum electronic devices such as lasers, quantum counters and infrared-to-visible upconvertors. The third article in this volume, contributed by L. A. Riseberg and M. J. Weber, deals with relaxation phenomena in rare-earth luminescence. It provides a survey of the energy levels
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and the excitation and decay modes of the rare earths and also covers such topics as relaxation by radioactive decay, multiphonon processes and ion-ion interactions. Examples of some applications are also given. The fourth article, written by M. A. Duguay discusses a useful recent application of picosecond laser pulses, namely the development of ultrafast shutters based on the optical Kerr effect. This device utilizes optically induced birefringence to obtain gating times of the order of a few picoseconds. After a discussion of gating in different substances and the factors that limit the resolution, some applications of the ultra-fast shutters are described. Among them are their use in ultra-high speed photography, which has made it possible, for example, to photograph a light pulse in flight and the development of a technique for the sampling of ultra-short optical signals by means of which molecular fluorescence signals can be displayed on the picosecond time scale. A relatively new technique for making diffraction gratings is described in the fifth article, contributed by G. Schmahl and D. Rudolph. The grating profile is provided by the intensity distribution of holographically produced interference fringes that are “stored” on a glass blank coated with a thin film of photoresist. After a presentation of the basic principles of such holographic diffraction gratings, their production is described. The properties of such grating are then discussed and their performance is compared with that of gratings of more conventional type. The sixth article, by P. J. Vernier, is concerned with a basic question concerning the photoelectric effect, namely the origin of photoelectrons. It has long been known that generally only a very thin layer of an irradiated solid gives rise to photoelectrons. A more accurate knowledge is, however, required in connection with efforts to improve photocathodes for use in photometry and for a precise interpretation of the results of electron spectroscopy. In this article the theoretical foundations of this subject are first discussed. A detailed review is then presented of researches on the escape depth of photoelectrons, and experimental and theoretical results are compared. Investigations on surface photoexcitations are also reviewed. The last article in this volume, contributed by P. J. B. Clarricoats, presents an account of theoretical researches on optical properties of fibre waveguides. Since about the early 19503, when glass fibres appear to have been first seriously considered as optical elements, much research has been conducted in this field. Today optical fibres promise to play an important role in the field of telecommunications. In this article the basic properties of optical fibre waveguides of various types are discussed and their relative
PREFACE
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merits are brought out. The modal as well as the ray methods of treatment are employed in analysing their properties. This volume attests once again to the vigor and the breadth of current research in optics. Department of Physics and Astronomy University of Rochester Rochester, N . Y . 14627 July 1976
EMILWOLF
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CONTENTS I . THE STATISTICS OF SPECKLE PATTERNS by J . C. DAINTY (London) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. NORMAL SPECKLE PATTERNS . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 First order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Second order statistics . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Statistics of the measured intensity. . . . . . . . . . . . . . . . . . . . 3. PARTIALLY COHERENT ILLUMINATION ..................... 3.1 Spatial coherence - Fraunhofer plane . . . . . . . . . . . . . . . . . 3.2 Spatial coherence - image plane . . . . . . . . . . . . . . . . . . . . . 3.3 Temporal coherence . . . . . . . . . . . . . . . . . . . . . . . . . . 4. SURFACE-DEPENDENT FEATURES OF SPECKLE PATTERNS . . . . . . . . . . . . . 4.1 Depolarising surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 A small number of scatterers . . . . . . . . . . . . . . . . . . . . . . 4.3 Slightly rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 5. CONCLUDING REMARKS ........................... ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 5 5 8 13 18 18 21 25 33 33 35 41 44 44 44
I1. HIGH RESOLUTION TECHNIQUES IN OPTICAL ASTRONOMY by A . LABEYRIE (Meudon) 0 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . 0.1 Introduction . . . . . . . . . . . . . . . . . 0.2 History. . . . . . . . . . . . . . . . . . . . 1. ATMOSPHERIC OPTICS . . . . . . . . . . . . . . . 1.1 The atmospheric heterogeneity . . . . . . . . . 1.2 Wave deformations and shadow patterns . . . . . 1.3 The speckrled structure of images. . . . . . . . . 1.4 The MTF for short and long exposures . . . . . 2. DIRECTINTERFEROMETRY . . . . . . . . . . . . . . 2.1 Basic principles . . . . . . . . . . . . . . . . 2.2 Visibility modulus determination. . . . . . . . . 2.3 Quantum noise . . . . . . . . . . . . . . . . 3. INTERFEROMETER DESIGNS AND RESULTS . . . . . . . . 3.1 The Fizeau and Michelson interferometers . . . . 3.2 Photoelectric Fizeau interferometers . . . . . . . 3.3 The speckle interferometer . . . . . . . . . . . 3.4 Interferometry with two telescopes. . . . . . . .
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4 . THEIMAGERECONSTRUCTION PROBLEM . . . . . . . . . . . . . . . . . . 4.1 The visibility phase problem with direct interferometry . . . . . . . . . . . 4.2 The triple interferometer . . . . . . . . . . . . . . . . . . . . 4.3 The seeing compensation approach . . . . . . . . . . . . . . . . . . . . 5. CONSTRUCTION OF A SYNTHETIC-APERTURE ARRAYOF OPTICALTELESCOPES . . 6. INTENSITY INTERFEROMETRY . . . . . . . . . . . . . . . . . . . . . . . 7. HETERODYNE INTERFEROMETRY . . . . . . . . . . . . . . . . . . . . . . 8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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111. RELAXATION PHENOMENA IN RARE-EARTH LUMINESCENCE by L . A . RISEBERG (Waltham. Massachusetts) and M . J . WEBER(Livermore California)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. INTRODUCTION 2. HISTORICAL DEVELOPMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Spectra . . . . . . . . . . . . . . . . . . . . . . . 2.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. RARE-EARTH ENERGY LEVELS . . . . . . . . . . . . . . . . . . . . . . . 4. EXCITATION AND DECAYIN RARE-EARTH SYSTEMS. . . . . . . . . . . . . . . 5. RADIATIVE DECAY .................... ......... 5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Electric-dipole transitions: Judd-Ofelt theory . . . . . . . . . . . . . 5.1.2 Magnetic-dipole and electric-quadrupole transitions . . . . . . . . . . 5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 6. MULTIPHONON RELAXATION . . . . . . . . . . . . . . . .. . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Temperature dependence . . . . . . . . . . . . . . . . . . . . . 6.3.2 Energy gap dependence . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Host dependence . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 5d + 4f relaxation . . . . . . . . . . . . . . . . . . . . . . . . 7. COOPERATIVE RELAXATION. . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Ion-ion energy transfer . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Energy migration . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 8. SELECTED APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Upconversion phosphors . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. CONCLUDING REMARKS . . . . . . . . . . . . . . . . . .. . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91 93 93 95 98 102 106 106 106 109 111 116 116 117 124 124 127 129 132 133 133 133 135 141 142 150 151 154 155 156
IV . THE ULTRAFAST OPTICAL KERR SHUTTER by M . A.DUGUAY (Murray Hill. New Jersey)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. INTRODUCTION 2. THEULTRAFAST OPTICALKERR SHUTTER . . . . . . . . . . . . . . . . . . 2.1 Gating in CS, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Gating in nitrobenzene . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gating with subpicosecond pulses . . . . . . . . . . . . . . . . . . . .
.
163 165 165 169 170
CONTENTS
2.4 Gating in glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Time response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Collinear gating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Self-focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Transverse gating . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. ULTRAHIGH SPEEDPHOTOGRAPHY ...................... 3.1 Light photographed in flight . . . . . . . . . . . . . . . . . . . . . . 3.2 Gated picture ranging . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Ultrahigh speed framing photography . . . . . . . . . . . . . . . . . . 4. SAMPLING OPTICALSIGNALS .......................... 4.1 F.1uorescence lifetime measurements . . . . . . . . . . . . . . . . . . . 4.2 The echelon technique . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The optical sampling oscilloscope (OSO) . . . . . . . . . . . . . . . . . 4.4 Multichannel sampling with detector arrays . . . . . . . . . . . . . . . . 5. CONCLUDING REMARKS ........................... ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REFERENCES.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
173 174 174 175 176 177 177 180 182 183 183 185 186 188 191 192 192
V . HOLOGRAPHIC DIFFRACTION GRATINGS by G . SCHMAHL and D . RUDOLPH(Gottingen) 197 t . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. THEORETICAL CHARACTERISTIC^ OF SPECTROSCOPIC DIFFRACTIONGRATINGS . . . . 198 3. BASICPRINCIPLES OF HOLOGRAPHIC DIFFRACTION GRATINGS . . . . . . . . . . . 200 201 3.1 Interference fringe system . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Accuracy of the interference fringe system . . . . . . . . . . . . . . 201 207 3.1.2 Interference arrangements . . . . . . . . . . . . . . . . . . . . . 3.1.3 Improvement of the ruling accuracy by superposition of identical recon207 structed wavefronts . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Frequency and wavelength stability of the laser light . . . . . . . . . 213 214 3.2 Photoresist layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 PRODUCTION OF HOLOGRAPHIC GRATINGS . . . . . . . . . . . . . . . . . . . 216 4.1 Gratings with symmetrical groove profiles . . . . . . . . . . . . . . . . 217 4.2 Gratings with asymmetrical groove profiles . . . . . . . . . . . . . . . . 219 223 5. PROPERTIES OF HOLOGRAPHIC GRATINGS AND COMPARISON WITH CLASSICAL GRATINGS 5.1 Wavefront interferogram, resolution and instrumental profile of plane gratings 224 226 5.2 Scattered light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.3 Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 5.4 X-ray gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 5.5 Gratings with imaging properties . . . . . . . . . . . . . . . . . . . . 6. FURTHER IMPROVEMENTS OF HOLOGRAPHIC GRATINGS . . . . . . . . . . . . . . 242 242 ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . PHOTOEMISSION by P. J . VERNIER (DijoN) 247 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 1.1 The 3-step model . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 2. THEORETICAL BASISOF THE PE . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Calculation of the density of absorbed photons (DAP) from the bulk dielectric 250 constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
XVI
2.2 Dielectric constant and microscopic processes in a solid . . . . . . . . . . . 2.2.1 Collective motion of the electrons . . . . . . . . . . . . . . . . . . 2.2.2 Collective motion of the ions . . . . . . . . . . . . . . . . . . . . 2.2.3 One-electron excitations (direct transitions) . . . . . . . . . . . . . 2.2.4 One-electron excitations (non-direct transitions) . . . . . . . . . . . 2.3 Photoexcitation coefficient . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fresnel equations and the DAP . . . . . . . . . . . . . . . . . . . . . 2.4.1 Validity of the Fresnel equations and spatial dispersion . . . . . . . . 2.4.2 Validity of the Fresnel equations, surface roughness and plasma oscillations 2.5 Surface photoexcitation . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Surface photoexcitation from bulk states (SPBS) . . . . . . . . . . . 2.5.2 Photoexcitation from surface states . . . . . . . . . . . . . . . . . 2.5.3 Surface absorption, Fresnel equations and DAP . . . . . . . . . . . 2.6 The electron escape probability . . . . . . . . . . . . . . . . . . . . . 2.6.1 Coulomb repulsion between electrons . . . . . . . . . . . . . . . . 2.6.2 Phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Electron-hole recombination . . . . . . . . . . . . . . . . . . . . 2.6.4 Transmission by the surface . . . . . . . . . . . . . . . . . . . . 2.7 Theoretical determination of the escape probability . . . . . . . . . . . . 2.7.1 Electron-electron interaction and the ballistic approximation . . . . . . 2.7.2 Diffusion equation and electron-hole recombination in negative electron affinitv. (NEA) Dhotocathodes . . . . . . . . . . . . . . . . . . . . 2.7.3 De-excitation of photoelectrons by phonon scattering only . . . . . . . 2.7.4 De-excitation of Dhotoelectrons bv both Dhonon and electron-electron interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.5 The escape probability and analysis of the experimental data . . . . . . 2.8 Photoemission and many-body effects . . . . . . . . . . . . . . . . . . 2.9 One-step theories of photoemission . . . . . . . . . . . . . . . . . . . 3. EXPERIMENTAL DETERMINATION OF THE ESCAPE DEPTHOF THE PHOTOELECTRONS . . 3.1 Estimation of the escape depth from one photoyield . . . . . . . . . . . 3.2 Estimation of the escape depth from the variation of the ratio of front Y + to back Y - yield versus the thickness zo of thin films . . . . . . . . . . . . . . 3.3 Estimation of the escape depth from the variation of the photoyield of thin films with thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Estimation of the escape depth from the back and front photoyields of one thin film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Estimation of the escape depth from the variation of the photoyield with the angle of incidence. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Estimation of the escape depth from the PE of substrate through a coating layer 3.7 Non-photoelectric methods . . . . . . . . . . . . . . . . . . . . . . 3.8 Escape depth in negative electron affinity (NEA) photocathodes . . . . . . 3.9 Estimation of the escape depth from band bending considerations . . . . . 3.10 Estimation of the elastic escape depth for high energy electrons . . . . . . . 4 . SURFACE PHOTOEXCITATION . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Detection of a surface effect from the thickness dependence of the photoyield of thin films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Detection of a surface effect from the polarization dependence of the photoyield 4.3 Evidence for a surface photoemission from the spectral yield distribution . 4.4 Evidence for the photoemission from surface states obtained from photoelectron energy distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
-
252 253 253 253 255 255 256 258 260 261 261 262 263 263 264 267 268 268 270 270 272 273 214 275 275 277 279 280 281 284 286 288 290 292 295 299 302 305 306 307 312 314 319 321 32 1
CONTENTS
XVII
VII . OPTICAL FIBRE WAVEGUIDES . A REVIEW by P. J . B . CLARRICOATS (London) 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . FIBRES W I CORE ~ A N D CLADDING POSSESSING UNIFORM REFRACTIVE INDEX. . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Characteristic equation . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Approximate solutions of the characteristic equation . . . . . . . . . . . 2.4 Ray interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Validity of core and cladding mode approximations . . . . . . . . . . . . 2.6 Group delay and pulse dispersion in the absence of mode coupling . . . . . 2.7 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Powerflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Attenuation due to a lossy layer . . . . . . . . . . . . . . . . . . . . 2.10 Leaky modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 1 Attenuation due to bends . . . . . . . . . . . . . . . . . . . . . . . 2.12 Mode coupling due to perturbations . . . . . . . . . . . . . . . . . . 2.13 Coupling due to bends . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Mode coupling and pulse dispersion. . . . . . . . . . . . . . . . . . . 2.15 Reduction of pulse dispersion by intentional inhomogeneities . . . . . . . . 2.16 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.1 Incoherent . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.2 Coherent . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. FIBRES WITH NON-UNIFORM REFRACTIVE INDEX. . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The W-fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Graded index fibres . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fibres with parabolic index variation . . . . . . . . . . . . . . . . 3.3.3 Impulse response of graded index fibres . . . . . . . . . . . . . . 3.4 Fibres with ring-shaped refractive index profiles . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDG~MENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
329 331 331 333 334 338 339 342 350 352 354 357 363 366 366 369 374 375 315 376 381 381 382 383 383 384 389 396 399 400 400
AUTHOR INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
403
SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . .
412
CUMULATIVE INDEX
VOLUMES I-XIV . . . . . . . . . . . . . . . . . 420
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E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
I THE STATISTICS OF SPECKLE PATTERNS BY
J. C . DAINTY Queen Elizabeth College, Campden Hill Road, Lonaon, W8 7AH U.K.
CONTENTS PAGE
0 1. INTRODUCTION . . . . . . . . . . . . . . . . . . .
3
0 2 . NORMAL SPECKLE PATTERNS
5
. . . . . . . . . . .
5 3 . PARTIALLY COHERENT ILLUMINATION
. . . . . .
18
0 4. SURFACE-DEPENDENT FEATURES OF SPECKLE PATTERNS . . . . . . . . . . . . . . . . . . . . . .
33
9: 5 . CONCLUDING REMARKS . . . . . . . . . . . . . .
44
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . .
44
REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
44
9
1. Introduction
Light with a fair degree of spatial and temporal coherence incident on an optically rough surface produces a reflected or transmitted beam that has a random spatial variation of intensity. This intensity distribution is called a speckle pattern. Fig. 1 shows the speckle pattern obtained at a distance of 1 m from a 1 mm diameter area of ground glass illuminated by a He-Ne laser. In this case the speckle pattern is of high contrast, and has a characteristic scale, or speckle “size”, approximately equal to the diameter of the Airy disc that would be produced in the absence of the ground glass (i.e. approximately 1 mm).
Fig. 1. Speckle pattern produced in the Fraunhofer plane of an optically rough diffuser illuminated by a He--Ne laser.
The statistical properties of a speckle pattern depend, in general, on both the coherence of the incident light and the statistics of the scattering surface or medium. In the laboratory, speckle patterns are usually produced by 3
4
THE STATISTICS OF SPECKLE PATTERNS
[I, §
1
highly coherent light incident on relatively large areas of optically very rough surfaces and this case is in fact an exception to the general rule; the statistics of such speckle patterns do not depend on the detailed surface properties and we shall refer to these as normal speckle patterns. The statistics of such spatial patterns are very closely related to those of the temporal fluctuations of thermal (Gaussian) light sources (JAKEMAN [19741). The general dependence of speckle statistics on the coherence of the incident light and the nature of the scatterer has led to several applications in the measurement of coherence and scattering parameters. One of the first recorded observations of a speckle pattern was by EXNER [1877,1880] who sketched the form of the pattern produced by candlelight incident on a glass plate on which he had breathed. The non-monochromaticity of this light source caused the pattern to have a radially fibrous structure and this feature was extensively discussed in the early literature (VON LAUE[1914,1916,1917], DE HAAS[1918a, b], BUCHWALD [1919], RAMAN [1919], RAMACHANDRAN [1954]). The mathematical basis for much of the analysis of the statistics of speckle patterns was established by Lord RAYLEIGH [1880,1918,1919]. The detailed first and second order statistics of normal speckle patterns formed in the Fraunhofer plane were fully evaluated by VON LAUE[1914, 19161; in particular he calculated general expressions for the second order probability density function of the intensity and the autocorrelation function of the intensity. Early work on speckle patterns is reviewed by HARIHARAN [1972]. Following the invention of the laser the phenomenon of speckle was re-discovered and a large number of short papers describing various simple properties were published. These are included in a bibliography on speckle compiled by SINGH[1972]. In this article we shall concentrate our attention on the first and second order statistics of speckle patterns. 0 2 is concerned with normal speckle patterns formed in perfectly coherent light. In this section, patterns formed in the image plane and in the Fraunhofer plane of a diffuser are considered separately, although of course a general theory could be used to cover both cases; with the appropriate assumptions in each case, the statistics are in fact identical. The statistics in the near-field are not evaluated as they are essentially the same as those in the far-field except for regions very close to the diffuser (ELIASSON and MOTTIER[1971]). The effect of partially coherent illumination on speckle statistics is discussed in § 3, and surface dependent aspects are considered in 5 4. Random intensity patterns produced by volume scatterers such as the atmosphere are not explicitly included in this article. The question of
1,
NORMAL SPECKLE PATTERNS
§ 21
5
propagation through scattering media is very much more complex than the relatively simple cases described here and has only partially been solved (CHERNOV [1960], TATARSKI [1961], STROHBEHN [1971]). Finally, many of the details of the scattering process and the nature of the scattering surface are excluded from the discussion below and are described by BECKMANN and SPIZZICHINO [1963] and BECKMANN [1967].
8 2. Normal Speckle Patterns 2.1. FIRST ORDER STATISTICS
We shall consider first a speckle pattern formed in coherent light in the Fraunhofer plane, as shown in Fig. 2. The effective complex amplitude of 4
Fig. 2. Formation of a speckle pattern in the Fraunhofer plane.
the scattered light in the scattering plane may be written as
where N is the number of independent scatterers, aj is the modulus of the scattered wave due to thejth scatterer, pj is the phase of the scattered wave, and S(<)S(q) is the two-dimensional Dirac delta function. The complex amplitude A(<,q) is a random process with the following [1963, 1975cj): assumptions made (GOODMAN
6
THE STATISTICS OF SPECKLE P A T T E R N S
CL § 2
i) the scatterers are randomly distributed over the area of the diffuser with uniform probability, ii) the aj are statistically independent random variables, iii) the /Ij are statistically independent random variables, are uniformly distributed in the interval - n to n (rough surface approximation) and are independent of the aj, iv) the polarisation of the incident wave is unaltered. It should be noted that these assumptions apply to the effect of the scattering medium on the incident field thus circumventing a detailed consideration of the interaction of the field and scatterer. In the so-called Beckmann model (BECKMANN and SPIZZICHINO [19631) the statistical properties of the scattering surface are specified and the field immediately after the scatterer is deduced making appropriate assumptions about the interaction. This more general approach however can only be pursued in detail for scatterers with a Gaussian distribution of surface heights and does not indicate the conditions under which we might expect surfaceindependent (normal) speckle patterns. It can be shown that for a Gaussian distribution of surface heights with an autocorrelation function whose scale width is very much less than the overall dimensions of the scatterer the Beckmann model gwes the same result as the Goodman model for the statistics of the speckle intensity fluctuation; in addition it also predicts the shape of the envelope of the average intensity in the Fraunhofer plane. Making the usual far field assumptions, the complex amplitude in the observing plane A(x,y) may be written as,
where the unimportant phase factor has been ignored. Substituting for A ( < ,q ) and evaluating the Fourier transform we obtain,
It is clear from this expression that the complex amplitude in the observing plane is given by the sum of a large number of random phase and amplitude vectors. As a result of the central limit theorem (see for example CHANDRASEKHAR [1943], MIDDLETON [1960]) the random process A(x,y) tends to a complex Gaussian process. The real and imaginary parts of the field are identically distributed with zero mean and variance s2/2 and at any single point they are statistically independent. The joint probability density
1 9 0
21
NORMAL SPECKLE PATTERNS
I
function for the real and imaginary parts of the field, denoted by A R and A , respectively, is given by
Provided that the phase of the scattered field at the scatterer is uniform in the interval -n to and that the number N of scatterers is very large, the complex amplitude of the speckle pattern will be a complex Gaussian process regardless of whether the scatterers have a uniform or random modulus. The Convergence of the process to a Gaussian form will in general depend on the statistics of the aj and this is discussed further in 84.2. In many practical situations the number of scatterers is so large that problems of convergence are not important. If the real and imaginary parts of the field have a joint Gaussian distribution, then it follows using a probability transformation that the modulrs has a Rayleigh distribution, the intensity has a negative exponential distribution and the phase is uniformly distributed in the interval - n to n :
=o =o
A
(4)
I
where Z is the intensity, 4 is the phase, and (I) = s2 is the ensemble average (mean) intensity. The probability density function for intensity (eq. (5)) is usually the most relevant distribution in practice; subject to the constraints on intensity of a finite mean and positivity, the negative exponential distribution has maximum entropy and indicates that a normal speckle pattern is totally random. Experimentally the probability density function for intensity p(Z) is found to be negative exponential to a high degree of accuracy (MCKECHNIE [1974b]), as shown in Fig. 3. The first order statistics of speckle patterns formed in the image plane of a rough diffuser illuminated by coherent light are the same as those in the
8
-
-
THE STATISTICS OF SPECKLE PATTERNS
I
statistical
I
fluctuation I
I
h\
\
5
6
vertical INTENSITY Ye change
7 8 scale xl00
Fig. 3. A measured histogram based on some 23000 intensity measurements taken from a speckle pattern. Because the sampling aperture was small in relation to the speckle size, the histogram should have the same form as the negative exponential reference curve which is shown (MCKECHNIE [1974b]).
Fraunhofer plane, provided that a large number of scatterers lie with the area of the point spread function of the imaging system in object space. This condition is likely to be satisfied in many practical situations, but will not hold for high resolution optical systems. The area of the point spread function of an aberration-free lens is -approximately L2/NA2where NA is the numerical aperture, whereas the minimum possible phase decorrelation area of any diffuser is approximately L2 ;the maximum number of scatterers contributing to a particular image point is therefore approximately equal to 1/NA2.This maximum number may be further reduced depending upon the actual phase decorrelation area of the diffuser used in practice. We cannot expect the statistics of the intensity to follow the negative exponential form for optical systems with numerical apertures greater than approximately 0.1. The resulting statistics for small numbers of scatterers and other surface dependent features are discussed in 0 4. 2.2. SECOND ORDER STATISTICS
We again consider separately speckle patterns formed in the Fraunhofer plane and in the image plane. The second order statistics of the scattered field describe, in general terms, the spatial structure of the field. For speckle patterns formed in the Fraunhofer plane, as in Fig. 2, it is intuitively obvious
1,
§ 21
NORMAL SPECKLE PATTERNS
9
that the dimension of the finest structure in the scattered field is inversely related to the effective diameter of the scatterer (just as the dimension of the Airy disc is related to the diameter of the diffracting aperture). The most general second order statistic of the intensity is the second order probability function, p(Z1,I,) and was first derived by VON LAUE [1916]. If 1, and Z2 are the intensities at two points (xl ,yl) and ( x 2 , y 2 )in the Fraunhofer plane, then
where 4ois the modified Bessel function of zero order and CI2 is the modulus of the autocorrelation function of the complex amplitude and is given by
-m
where S(<,q) is the intensity distribution at the scattering plane. In the derivation of (7) and (8), we assume that a large number of randomly phased scatterers lie within the scattering aperture. It can be seen that p(Z, ,Z2) is completely defined in terms of the autocorrelation function C , ; it is a property of Gaussian processes that all probability density functions are completely specified by the autocorrelation function. Thus in practice the only second order statistic we need to evaluate is the autocorrelation function or its Fourier transform, the Wiener spectrum. The autocorrelation function of the intensity in the Fraunhofer plane of a rough diffuser has been derived by many authors (VON LAUE[1916], GOODMAN [19631, GOLDFISCHER [19651, SUZUKIand HIOKI[19661, Ross [1970] and YAMAGUCHI [1972, 19733); here we follow Goodman’s approach. The autocorrelation function of the complex amplitude C,(xl ,yl) is defined as CA(X1,Yl) = (A(x+x,,y+y,)A*(x,y)), (9) where ( ) denotes the ensemble average. (For a statistically stationary speckle pattern the ensemble average may be replaced by a space average.) The complex amplitude in the Fraunhofer plane of a rough scatterer is given by eq. (2), and combination of equations (2) and (9) leads to
where the various cross-products reduce to zero upon taking the average.
10
THE STATISTICS OF SPECKLE P A T T E R N S
[I,
§ 2
If the scatterers are considered to be packed sufficiently closely on the scattering area, then the scattering intensity Ja,j12 can be represented by a continuous function S(5, q ) and we may write m
-00
The intensity function S(t,q) is proportional to the incident intensity distribution over the scattering area and zero elsewhere, so that once this distribution is known the autocorrelation function of the complex amplitude of the speckle pattern is also known apart from a multiplicative constant. The autocorrelation function of the intensity fluctuation is defined as C,(X,l
Y1) =
< m + x , , Y f Y 1 ) m Y)>-<1>2.
(11)
The first term on the right-hand side of eq. (1 1) can be expressed in terms of C,(x, ,yl) using a moment theorem for complex Gaussian processes due to REED[I9621 :
=
+ (z:z,>.
(12)
The result is CI(X19 Y,)
= IC,(X,l
Y1)l2
(13)
and therefore the autocorrelation function of the intensity fluctuation in a normal speckle pattern is given by m
-m
where it is assumed that S(5, q) is suitably normalised. The Wiener spectrum (or power spectrum) W(u,v) of the intensity fluctuation in the Fraunhofer plane, which in general terms describes the spatial frequency content of the scattered intensity, is equal to the Fourier transform of the autocorrelation function : m
-m
1,
§ 21
N O R M A L SPECKLE P A T T E R N S
11
since u =
ir
W(U,v)dudu = C(0,O) = o2 = ( I ) ’ .
-02
Since both the autocorrelation function and Wiener spectrum depend only on the intensity distribution across the scattering aperture, the above analysis applies also to planes other than the Fraunhofer plane, provided that a large number of scatterers contribute to the intensity at any point in the plane. For a uniformly illuminated circular scattering area of radius r, the autocorrelation function and Wiener spectrum of the intensity fluctuation of the speckle pattern are given by
and
where y l ( x ) is the first order Bessel function and w, is the cut-off spatial frequency given by w, = 2rfAD. Several thousand values of the speckle pattern intensity are required to accurately verify the above results and the experimental measurements of HOHN[1968] and DAINTY [1970] only approximately verified the theore[1974a1 tical results. However recent experimental results by MCKECHNIE shown in Fig. 4 indicate that the expressions given above are verified to a high degree of accuracy. The results of Fig. 4 also show that the roughness of the scatterer does not influence the Wiener spectrum (or autocorrelation function), provided of course that the illumination is perfectly coherent, that the phase fluctuation at the scatterer is uniform in the interval --n: to 71 and that a large number of scatterers contribute to the intensity at any point in the Fraunhofer plane. The autocorrelation function and Wiener spectrum of speckle patterns produced in the image plane of rough diffusers may be found either by using an extension of the Fraunhofer case (ENLOE[1967]) or by straightforward application of linear filter and square law detection theory (BURCK-
12
THE STATISTICS OF SPECKLE P A T T E R N S
T
I
I ERROR
I DUE
~
I
TO S T A T I S T I C A L
0
X
x
I
-~
' 7 .
FLUCTUATION
CL § 2
ROUGH DIFFUSER 0 = 10 MICRONS FINE '8 u0.g 0,
-
z .L. 2
0 SPATIAL FREQUENCY ( t )
U N I T S OF
-
Fig. 4. Measured Wiener spectra of the intensity of speckle patterns observed in the Fraunhofer plane of two optically rough diffusers. The solid line is the form predicted using eq. (17) (MCKECHNIE
[1974a]).
HARDT [19701, DAINTY [19701, LOWENTHAL and
ARSENHAULT [19703). The
latter approach is described here. Let the optical system be represented by a linear filter whose isoplanatic amplitude point spread function is P(x, y ) and transfer function is T(u,v ) ; the transfer function is equal to the pupil function H(5, q) suitably scaled,
T(u,4 = f w f u , u-4,
(18)
where f is the focal length. If the complex amplitude distribution of an object is A ( x , ,yl), then the complex amplitude distribution in the image A'(x,y) is given by A'(X>Y)
=
~ ~ ~ ~ ~ , . Y l ~ P ~ x - x l ~ Y - Y ~ ~ d x l d Y l . -m
Suppose that the object distribution A @ , ,yl) is a stationary statistical process with a Wiener spectrum WA(u,v).The spectrum of the output amplitude of the linear filter is given by a well-known result (MIDDLETON c19601), WA(U, 4
=
IT(%U)I2W(~, v).
(19)
The Wiener spectrum of the intensity in the speckle pattern W(u,v) is found from the Wiener spectrum of the complex amplitude WL(u,v)by
1 3 5
21
13
NORMAL SPECKLE P A T T E R N S
applying a result derived by RICE[19541 in his analysis of the square law detection problem :
WL(ul, vl)WL(ul +u, v1
W(u,v ) =
+
0) du,
du, .
(20)
-w
If the diffuser structure is very much finer than the diameter of the point spread function (this condition is also necessary for Gaussian statistics for the image amplitude), the Wiener spectrum of the object amplitude WA(u,u) will be constant for the values of (u, u) for which IT(u, u)I2 is significantly non-zero, and combination of equations (18) to (20) gives, for this special case of a “white” noise object, r
m
r
-m
where 4 is a normalisation constant. Comparison of eq. (15) for the Fraunhofer plane and eq. (21) for the image plane shows that the Wiener spectra have the same form in each case. For speckle patterns produced in the Fraunhofer plane the intensity distribution of the illumination is the quantity that determines the spectrum, whereas in the image plane it is the squared modulus of the pupil function. For an unshaded circular imaging pupil of radius r, the autocorrelation function and Wiener spectrum of the intensity fluctuation are given by equations (16) and (17) respectively, with the distance D replaced by the focal length f. The autocorrelation function has the same form as the intensity distribution of an Airy disc, and this fact has given rise to the general rule-of-thumb that the speckle “size” is of the same order of magnitude as the size of the Airy disc; note however that the autocorrelation function and hence the speckle “size” is independent of any aberration of the imaging system. When imaging a general diffuse object, the object amplitude is not statistically stationary, and the statistics of the image are described by non-stationary functions ; this is discussed by LOWENTHAL and ARSENHAULT [1970] as an extension of the above arguments for the stationary case. 2.3. STATISTICS OF THE MEASURED INTENSITY
The statistical properties we have evaluated so far all relate to the complex amplitude or intensity at one or more points in a speckle pattern. In prac-
14
T H E S T A T I S T I C S OF S P E C K L E P A T T E R N S
CL
D
2
tice, speckle patterns are more likely to be averaged over some non-zero area by, for example, a scanning aperture and we shall call this averaged or integrated intensity the measured intensity. The measured intensity Z’(x, y ) is related to the intensity of the speckle pattern Z(xl, yl) by a convolution formula, Z’(x7y) = ~ ~ ~ ( x l . ~ ~ ~ ~ ~ ~ - ~ l , ~ - ~ l(22) ) d X , d y -a
where B ( x , y ) is the intensity point spread function of the averaging device normalised such that the volume of B ( x , y ) is unity. The Wiener spectrum of the measured intensity fluctuation W’(u,v ) is simply related to that of the speckle pattern intensity :
W’b, v) = W(u,u)lb(u, 412,
(23)
where b(u, v ) is the Fourier transform of B ( x , y ) . The variance of the measured intensity $t is given by CT; =
{J --m
W‘(u,u)dudv
W(u,u)lb(u, v)I2dudu.
=
(24)
--m
The Wiener spectra of the intensity fluctuation in normal speckle patterns formed in the Fraunhofer and image planes are given by equations (15) and (21) respectively. GERRITSON, HANNAN and RAMBERG[1968] calculated the ratio (Z)/aa (a “signal-to-noise” ratio) for square pupils and scanning apertures. For a circular imaging pupil of radius r and a circular scanning aperture of radius a, eq. (24) becomes,
where
k=
2a diameter of scanning aperture Rayleigh resolution limit of lens 0.61Aflr.
The quantity k is a measure of the aperture size relative to the speckle “size”, since we showed in Q 2.2 that the speckle “size” is of the same order of magnitude as the Airy disc. In Fig. 5, ob/(Z)is plotted as a function of the relative aperture diameter for both top-hat and Gaussian scanning apertures.
15
NORMAL SPECKLE PATTERNS
L O 21
01 0
I
2
6
4
8
10
1
12
K
Fig. 5 The standard deviation of the intensity fluctuations relative to the mean intensity for top-hat and Gaussian scanning apertures for a speckle pattern formed by an optical system with a top-hat pupil function. For the top-hat aperture, -, k = (diameter of scanning k = (diameter aperture)/(Rayleigh resolution limit of lens). For the Gaussian aperture, of aperture at l/e points)/(Rayleigh resolution limit of lens).
If the scanning aperture is very large compared to the speckle ‘‘size’’ then the variance is given approximately by c$
N
rJ
W(0,O)
(b(u,v)lz dudv.
-W
For unshaded pupils and scanning apertures, this expression reduces to
where d b is the area of the scanning or averaging aperture, and d pis the area of the pupil in spatial frequency units (i.e. d,= nvz/ln2f2). Finding the first order probability density function for the measured intensity, p(I’), is not straightforward. The problem is analogous to that of square-law detection and low-pass filtration in the one-dimensional time domain, which was analysed by KACand SIEGERT [19471 and SLEPIAN [1958]. The extension* to the case of the measured intensity in a speckle
* All three authors in fact made errors in extending the results of K A c and SIEGERT[1947] ; these arose as a result of failing to include all aspects of the fact that the amplitude is a complex Gaussian process.
16
T H E S T A T I S T I C S OF S P E C K L E P A T T E R N S
[I7 §
2
pattern was made by CONDIE [19661, DAINTY [19711 and BARAKAT [19731. Equa.tion (22) for the measured intensity may be rewritten in terms of the complex amplitude in the speckle pattern A(x, ,yl) as, I’(x, Y ) =
I.j
14x1, Y1)I2B(x-x1, Y-Y1)dX, dY,.
-02
The problem in findingp(1’) is that I.’is a weighted sum of correlated random variables. By expanding A(xl ,yl) in terms of orthogonal functions (the Karhunen-Lokve expansion), the measured intensity can be expressed as a weighted sum of independentrandom variables and following this approach the probability density function for the measured intensity is given by
where E, are the eigenvalues of the homogenous Fredholm equation Y) =
.r9r
C A ( ~ - ~ I , Y . - Y ~ )Yl)$n(Xl, B ( ~ , , y,)dx1dy1,
(28)
-w
with c0 2 c1 2 E~ . . . and C,(x,y) is the autocorrelation function of the complex amplitude in the speckle pattern. It is in general difficult to calculate the eigenvalues from eq. (28); however, if the eigenvalues E, are plotted as a function of n it is often found that for a wide range of C,(x,,y,) and B(x,,y,), the eigenvalues are approximately equal to E~ up to a value of n N no, and approximately equal to zero for n > no. With this approximation, the probability density function for the measured intensity becomes (CONDIE[19661, SCRIBOT C19741),
‘s
p(T) = 2n
-w
exp (-izZ’) dz, (1-i.z~~)”
which, upon integration, gives the gamma distribution,
It can also be shown that, in this approximate case, (1) =
Eon0
1,
Q 21
17
NORMAL SPECKLE PATTERNS
and 0;
= &;no.
Equation (29) can be therefore rewritten as
where no is interpreted as the number of independent correlation cells (speckles) within the scanning aperture. It is interesting to note that this approximate expression for the probability density function of the measured intensity has also been derived using a heuristic argument (GOODMAN [1965]) that is again an extension ofwork in the one-dimensional time domain (RICE[1954], MANDEL [1959]). The scanning aperture is regarded as consisting of no independent correlation cells, the intensity being taken as constant within any one cell and statistically independent of the intensity in all other cells. The intensity of each correlation cell is assumed to have a negative exponential distribution and therefore the total measured intensity is approximately a gamma variate, as in eq. (30), where no is chosen such that the variance of the approximate and exact distributions are equal :
no
0
(31)
= (Z)2/oi.
1.0
2.0
Fig. 6 . The approximate probability density function of a normal speckle pattern scanned by an aperture. This distribution has two parameters, the mean intensity (assumed to be unity) and the standard deviation ubof the intensity fluctuation (relative to the mean intensity).
18
THE STATISTICS OF SPECKLE P A T T E R N S
CL § 3
In Fig. 6 the approximate probability density function is drawn for a mean measured intensity of unity and various values of a b / ( z ) ; for a b / ( z ) --t 1, the distribution tends to the negative exponential form and for a b / ( z ) --t 0 it tends to a Gaussian form. An important difference between the approximate distribution given by eq. (30) and the exact distribution given by eqs. (27) and (28) is that the approximate distribution depends on the Wiener spectrum of the speckle only through a weighted integral (eq. (24)) whereas the exact distribution depends on the form of the Wiener spectrum. It is found however that the agreement between the approximate and exact formulae is close regardless of the form of the speckle Wiener spectrum (MANDEL[1959], BEDARD, CHANGand MANDEL[1967], SCRIBOT [1974]).
0 3. Partially Coherent Illumination 3.1. SPATIAL COHERENCE
-
FRAUNHOFER PLANE
Speckle patterns were first observed in situations where the light incident on the scatterer was both spatially and temporally partially coherent, and the properties of such patterns are somewhat less straightforward than those for perfectly coherent illumination. For each case of spatial and temporal partial coherence we again make the subdivision into speckle patterns formed in the region of the Fraunhofer plane and those formed in the image plane, although a perfectly general theory may of course be used to apply to both cases (PARRY[1975b]). In Fig. 7 we show an optical system that might be used to form a speckle pattern in the Fraunhofer plane of a diffuser illuminated by spatially partially coherent light. The diffuser is uniformly illuminated by light which has a mutual coherence function r((,,tZ,q l ,q 2 ) given by the
so urce
L1
L2
Fig. 7. Formation of a speckle pattern produced in the Fraunhofer plane ofa diffuser illuminated by spatially partially coherent light.
1,
o 31
PARTIALLY COHERENT ILLUMINATION
19
inverse Fourier transform of the source intensity s(x,y ) suitably scaled snd normalised. The area of the diffuser that contributes to the speckle pattern is limited by the extent of the lens L2 whose intensity transmittance is equal to the squared modulus of the pupil function, Ill(<,v ] ) I 2 . To find the statistics of the speckle pattern intensity Z(x,y) formed in this case, we must first obtain a relationship between Z(x,y ) , s(x, y ) , H ( 5 , v ] ) and Z(c, v ] ) , where this last function is the complex amplitude at the pupil due to the diffuser. There are two conceptually different approaches that have been used to obtain this relationship. The first (LABEYRIE [1970], DAINTY[1973]) is based on the fact that Z ( x , y ) is an image, albeit a very poor one, of the source intensity and the usual equations for imaging of self-luminous (incoherent) objects apply ; the effective pupil function is equal to the product Z ( < , q ) H(<,v]).This approach is useful when the diffuser is weak, as in the case of the atmosphere; the speckle pattern only exists in a small region of the x-plane (see 94.2). The second, more conventional, approach is to follow the mutual coherence function as it propagates through the system (Ross [1969], ASAKURA, FUJIIand MURATA [1972], FUJIIand ASAKURA [1973,1974a]) and this is outlined below. Assuming that the illumination is quasimonochromatic and that the mutual coherence function is stationary, then (BORNand WOLF[1970]), I(x, Y ) =
ij.
m1-52,
?l-YIZ)H(Sl, ?)Z(tl,? ) H * ( 5 2 , v ] 2 ) 2 * ( 5 2 3 r z )
-m
Making the substitutions u = - 51-4;2 , v=- 91 - Y 2
Af
Af
we obtain
x exp { - 2 4 u x
+ uy)} du dv
20
THE STATISTICS OF SPECKLE P A T T E R N S
[I,
0 3
which further simplifies to 1(x7Y ) =
S(X,
Y)
e
1
rJ
/2
H(Afu, Afu)Z(Afu,A f u ) exp { - 2ni(ux + uy))dudu ,
-m
where 8 denotes convolution. However the second term in the convolution is simply the normal speckle pattern that would be obtained in perfectly coherent light (provided of course that the complex amplitude transmittance of the diffuser Z ( x , y ) satisfies the constraints given in 9 2). Thus, I(x7 Y ) = 4x9 Y ) 63 I&, Y),
(33)
where Ic(x,y)is the speckle pattern intensity that would be obtained with perfectly coherent illumination. Equation (33) relates the intensities of speckle patterns produced in the Fraunhofer plane in spatially partially coherent and fully coherent illumination, and the first and second order statistics of the partially coherent case follow quite simply. The effect of the non-zero source extent s(x,y) is to blur out and to reduce the contrast of the fully coherent speckle pattern. Comparison of eq. (33) with eq. (22), shows that the source distribution s(x,y) in the partially coherent case plays the same role as the scanning aperture B(x,y) in the case of the measured intensity in coherent light. Thus the exact first order probability density function is given by eqs. (27) and (28) with s(x, y ) substituted for B(x, y), or approximately by
where no is the number of (coherent) speckles that lie with the source function s(x, y). The Wiener spectrum of the intensity fluctuation in a speckle pattern formed by spatially partially coherent illumination of a diffuser is given by
wu,
0) =
K(U,W
W u , nfo)12
(34)
where r(5,q) is the mutual coherence function and Wc(u,u) is the Wiener spectrum of a coherent speckle pattern. Equation (34) is the basis of a method of measuring the mutual coherence function ;providing the diffuser satisfies the conditions laid down in 0 2, the function Wc(u,u) is of a known form (eq. (21) in the present notation) and by measuring W(u,v ) the squared modulus of r(<, q) can be found. An example of an experimental result is given in Fig. 8; the same basic method is also used in astronomy for the determination of stellar diameters and binary star separations.
I,
5
21
PARTIALLY COHERENT ILLUMINATION
31
separation
2
mm
Fig. 8. An experimental measurement of the spatial coherence of a light source using speckle [1973]). patterns (FUJIIand ASAKURA 3.2. SPATIAL COHERENCE
-
IMAGE PLANE
The statistics of the intensity in the image of a spatially partially coherently illuminated diffuser are slightly more difficult to find than those in the Fraunhofer case. Referring to Fig. 9, the image intensity is given by
C%Y)
=~~~~(~l-xz,Y~-Yz)h(x-x~,Y-Yl~ -m
source
diffuser (0bjed)
lens
image
Fig. 9. Formation of a speckle pattern produced in the image plane of a diffuser illuminated by spatially partially coherent light.
22
THE STATISTICS OF SPECKLE PATTERNS
[I,
0
3
where h(x,y ) is the amplitude point spread function of the imaging system and Z(x,y) is the complex amplitude transmittance of the diffuser. In terms of the effective source J(u, v) = s(a/AD,b/AD), m
n
J(u, 4
=
n
j j T(x, y ) exp {
- 2zi(ux
+ vy)) dx dy,
-m
this may be written as I(x, Y ) =
?;s
J(u,v)lQ(x,Y; u, 41, du dv,
(36)
-m
where Q(x,y , u, v) is the amplitude in a speckle pattern produced by an oblique coherent wave incident on the diffuser at angle (nu, nu), Q ( x , Y ; ~0 ), =
S:F
h(x-x,,y-yi)Z(xi,y,)ex~ f - 2 z i ( u x i + v ~ i ) ) d x i d ~ i .
-00
Equation (36) states that the speckle pattern produced in the image plane in spatially partially coherent illumination is just the (weighted) sum of the intensities of speckle patterns produced by different angles of coherent illumination. The exact first order probability density function is therefore given by an expression similar to eq. (27), where the eigenvalues E, are given by an expression similar to (28). The approximate probability density function is given by equation (30), where the value of no is approximately equal to the ratio of the diameter ofthe point spread function to the diameter of the coherence patch at the diffuser. In an optical system which has an illuminating condenser of numerical aperture NA, and an imaging objective numerical aperture of NA, then no N Y = NA,/NAo (for no B 1). The Wiener spectrum of the intensity fluctuation in a speckle pattern produced in a spatially partially coherent optical system has been derived for optically very rough surfaces by DAINTY [19701and YAMAGUCHI [19731, and for surfaces of an arbitary magnitude of roughness but with a Gaussian phase profile by FUJIIand ASAKURA [1974c]; this latter case is discussed in 8 4.3, and only the result for a very rough surface is found here. Equation (35) for the image intensity may be rewritten in the form
I(X) =
rJ
V ( x , , ~~)Z(x-x,)Z"(x-x,)dx,dx,,
-m
PARTIALLY COHERENT ILLUMINATION
23
where
One dimensional notation is used here to simplify the derivation. The autocorrelation of the intensity fluctuation Z’(x) = Z(x) - ( I ) is given by C(x,)
=
+
(Z’(x)Z’(x x,)
= ({v(xl
3
xZ)v(x3, x4)
- ,xi
x ((Z(x-x,)Z*(x-x,)Z(x+x, - (Z(X-
xJZ*(x- xz))(Z(x
- x , ) Z * ( x + x , -x,))
+xo - X3)Z*(X+xo- x,))}
x dx, dx, dx, dx,.
(37)
It is now assumed that the object amplitude Z ( x ) is a complex Gaussian process so that applying the theorem of REED[1962] to the fourth order moment we obtain, m
-00
x (Z*(X-X,)Z(X+X,-X~))} dx1 dxZdx3dx4,
r{
or in terms of the autocorrelation of the diffuser complex amplitude Cz(x),
c(xO)
=
v(xl, x 2 ) v ( x 3 ,
x~)(c:(x1-x4+xO)
-00
x C,(X, - ~3
+x,)}
dx, dx, dx3 dx4.
(38)
The Wiener spectrum of the intensity fluctuations is the Fourier transform of eq. (38), and reduces to
K(u1)W u + u , ) m - u1, -(u + o
n u + u1, u1) du, , (39)
where Wz(u)is the Wiener spectrum of the object amplitude, and 9-(ul, u,) is the transmission cross-coefficient (BORNand WOLF[1970]) and is given by
24
THE STATISTICS OF SPECKLE PATTERNS
[I?
03
+
J(u)H(u u,)H*(u - u2)dul du2.
Since F ( u l , u2) = Y * ( - u z , - u , ) , eq. (39) reduces further to
w.4=
s-40
~ , ( u , ) ~ , ( u + u , ) l ~ ( u + u , %)12d% ,
(40)
and for a diffuser whose lateral structure is small compared with the point spread function of the imaging system (a “white-noise” diffuser), IS(U+ ul,ul)l2 du, .
(41)
It should be noted that the result of eq. (41) for a white noise diffuser is not restricted to the case of Gaussian statistics for the diffuser complex amplitude transmittance ; if the function Z(x, y ) has an autocorrelation function that is a delta-function (i.e. white noise), then it can be shown that eq. (41) must follow regardless of the statistics of Z(x,y). In Fig. 10, the scale value W(0)of the Wiener spectrum is plotted as a function of the coherence parameter Y Y =
radius of effective source radius of entrance pupil
NA, --
NA, .
Fig. 10. The zero spatial frequency value of the Wiener spectrum of the intensity fluctuation in a speckle pattern formed by an aberration-free partially coherent optical system with a top-hat effective source distribution, plotted as a function of the coherence parameter 9’ (DAINTY [1970]).
1,
o 31
PARTIALLY COHERENT ILLUMINATION
25
The quantity W(0) gives an indication of the variance 0’ of the speckle pattern, and it is clear from the figure that the speckle contrast remains high for Y < 1. This result is also confirmed by FUJIIand ASAKURA [1974c], and is discussed further in $4.3. 3.3. TEMPORAL COHERENCE
Much of the early discussion on the nature of speckle patterns was concerned with the effect of the non-monochromaticity of the light source (DE HAAS[1918], RAMACHANDRAN [1943]). Speckle patterns formed in the Fraunhofer plane of diffusers illuminated by all lines of an argon laser are shown in Fig. 11 for three surfaces of different roughness (a colour plate is given by MARTIENSSEN and SPILLER[1965]). Speckle patterns formed by non-monochromatic sources have two notable features that distinguish them from normal (monochromatic) speckle patterns. Firstly there is a radial structure which depends-both on the temporal coherence of the source and on the roughness of the diffuser; secondly, the contrast of the pattern is a function of position in the Fraunhofer plane and also depends on the temporal coherence and surface roughness. For a surface whose r.m.s. height variation q,is greater than one wavelength but less than the coherence length Lc of the incident radiation (see Fig. 12 (a)), each wavelength forms a normal speckle pattern that is fairly well correlated with the speckle patterns produced by neighbouring wavelengths, at least near the centre of the pattern. The speckle size is of course related directly to the wavelength and the overall effect is to produce a radial structure in the total pattern. The “length” of a speckle in the radial structure depends upon the magnitude of the relative phase changes from point to point on the diffuser as a function of angle and this depends on the r.m.s. height variation oh; for a large surface roughness the relative phase change is large for a small change in angle and the radial structure is less apparent (see Fig. 11). The contrast of the pattern is also less for surfaces whose r.m.s. roughness is greater than the coherence length of the incident light (see Fig. 12 (b)) because at any point in the Fraunhofer plane fewer scatterers give coherent contributions and more give incoherent contributions. For normal incidence and observation at the centre of the Fraunhofer plane, GOODMAN [1963] showed that high contrast is only achieved if oh < Lc. More complete treatments of the statistics of speckle patterns formed in the Fraunhofer plane in temporally partially coherent (“white”) light have been given by GOODMAN [1963], FUJIIand ASAKURA [1974a], PARRY
26
THE S T A T I S T I C S O F S P E C K L E P A T T E R N S
Fig. 1 1 . Speckle patterns produced in the Fraunhofer planes of diffusers with three surface roughnesses illuminated by an argon laser (wavelengths present 514, 496, 488, 476 nm); (a)uh u lpm,(b)u, z 3pm,(c)uh u PARRY [1974b]).
21
PARTIALLY COHERENT ILLUMINATION
(a1
U,
<
Lc
(b)
up,
>
Lc
Fig. 12. Surfaces whose r.m.s. height variation uh are less and greater than the coherence length L,.
[1974a, b, 1975a, b] and PEDERSEN [1975a, b]. The results given so far in this article have applied to all surfaces whose r.m.s. roughness is greater than one wavelength, regardless of the detailed statistical properties of the surface. However we can no longer accept this naive picture of the scattering surface and it is necessary to introduce surface dependent parameters using the Beckmann model (BECKMANN and SPIZZICHINO [19631). [1975b]). We consider the scattering geometry shown in Fig. 13 (PEDERSEN The surface lies in the (x,y)~ x p l a n eand has a profile h(x) whose r.m.s. fluctuation is CT,, > A. The surface is illuminated by a unit amplitude plane wave with wave vector k , , and the scattered wave k is observed in the Fraunhofer plane, where (kl = Ik,J = 27c/A. The scattered amplitude is a
\
incident wave
Z I
surface profile h(L0
Fig. 13. Diffraction geometry for the formation of speckle patterns (PEDERSEN C1975b-j).
28
T H E STATISTICS OF SPECKLE P A T T E R N S
11, 5
3
function of k , and k only through the difference q = k - k , as multiple reflections are ignored. The problem of finding the statistics of the scattered field for surfaces of arbitrary profile h(x) is difficult. If the surface roughness is greater than one wavelength and we illuminate the surface with monochromatic light then as we have seen in 52 it is possible to obtain a useful statistical description of the scattered field. However, it should be noted that the description given in Q 2 does not explain the roughness dependent angular correlation of speckle patterns (ARCHBOLD and ENNOS[19721). In this present case we make two important assumptions about the surface profile. Firstly we assume that h(x) is a stationary Gaussian random variable with zero mean and autocorrelation function c h ( x 1 - x2); the first order characteristic function is therefore given by al(t)= (exp (ith)) = exp ( - p1h 2t 2). (42) Secondly it is assumed that the total area of the illuminated surface is large compared to the area of correlated surface structure (i.e. large compared to the extent of c h ( x 1 - x2)) so that there are many correlation areas within the illuminated area. Provided that these two assumptions are made, it can be shown that (regardless of the form of c h ( x 1 - x2)) the normalised autocorrelation function of the scattered intensity is locally stationary and given by C(Aq)
=
@wz)C(Aqx)?
(43)
where Aq = q1 - q 2 , Aqz is the projection of Aq on the z axis, and Aqx is the projection of Aq on the x plane. C(Aq,) is the normalised autocorrelation function of the intensity in a “normal” speckle pattern which in the present notation is given by S(x) exp (- iAqx . x) d2x
I
where S(x) is the intensity distribution of the incident light. It is useful at this point to let q
=
k-k,
=
k(n-no)
=
km
where k is the wavenumber (k = 27cn/A) and m = n - n 0 is the change in propagation direction between the incident and scattered wave.
1,
D
31
29
PARTIALLY COHERENT ILLUMINATION
If a polychromatic speckle pattern is formed using light whose spectral intensity distribution is S ( k ) , the intensity of the observed speckle pattern Z(m) is simply the sum of the monochromatic patterns I(km) suitably weighted by S ( k ): I(m) = lLm2(k)I(km)dk. It follows that the mean intensity ( I ( m ) ) is given by
and provided that this mean is locally stationary over regions of speckle correlation, the normalised angular autocorrelation function Cw(m ,m z ) of the white light speckle intensity fluctuation is given by
,
CJm,
9
m2)
=
jSS(k1)S(k2~C(k2m2 - k, mhdk, dk,,
(46)
-m
where S ( k ) is normalised so that Sfm S(k)dk = 1. To gain further insight into the white light speckle angular autocorrelation function it is useful to consider a particular example in which the spatial intensity distribution S(x) of the illuminating light is Gaussian with r.m.s. radius r/2. It follows from eqs. (42) to (44) that (F'EDERSEN [1975b]), C(Aq) = exp { - r21AqXl2 - oiAq,Z}. We further assume that the normalised spectral density S ( k ) is also Gaussian with an r.m.s. spectral bandwidth W around a mean wavenumber ko , 1 S(k) = -. exp { -(k-l~,)~/2W~}. WJZ We now introduce a path vector s = rm,+o,m,e,, where
is the r.m.s. path deviation of the light contributing at the point m in the speckle pattern and e, is a unit vector in the z direction. Equation (46) for the angular autocorrelation of the intensity can now be evaluated to give
30
THE STATISTICS OF SPECKLE PATTERNS
[I.
P3
Let the mean path vector be s = (sl +s,)/2 and a difference path vector be d = s 2 - s l . In the limit d = 0 eq. (48) reduces to C,(O; s) = a2/(I>2 =
l/JiXmy,
(49)
where s is in general given by eq. (47); for normally incident illumination and viewing at the centre of the Fraunhofer plane s = 2a, and we obtain a2/(I>2 = l/Jiqmq.
(50)
This simple expression clearly shows how the speckle contrast decreases as either the surface roughness a),or the illumination bandwidth Wincreases. If we now restrict our problem still further by assuming that we are using relatively narrow band sources such that W
In order to show how the speckles are elongated and in what way this elongation depends on the surface and illumination properties, it is convenient to assume that the illumination is normal to the scattering surface and consider paraxial diffraction angles. We then have (Fig. 14), s = i s x , s,,
SJ
={re, 0, 2
4
9
z
Fig. 14. Diffraction geometry for the calculation of polychromatic speckle correlation with normal incidence (F’EDERSEN [1975b]).
I,
D
31
31
P A R T I A L L Y COHERENT I L L U M l N A T I O N
where the x-axis lies along s, and 8 is the polar diffraction angle. Similarly,
d = {rAOX,rAO,, 0}, where A8, and A8, are the components of the change in diffraction angles in the radial x-direction and the azimuthal y-direction respectively. Equation (51) becomes
Cw(A@;8)= Cw(0;8)exp It can be seen from eq. (52) that the degree of angular correlation is locally stationary with an elliptical Gaussian correlation function. In the azimuthal direction, the width of the correlation function (i.e. the speckle width) is determined only by the normal diffraction value,
In the radial x-direction, the correlation function is lengthened by the process of angular dispersion and we have
This is illustrated in Fig. 15. The elongation of the speckles in the radial direction due to angular dispersion is given by (ge/%sp
=
2(wc/k,)',
(53)
,"'y diffr
I
/ /
> I
I
I
!
I
I
f
0,
I
*I
Fig. 15. Illustration of the correlation regions in polychromatic speckle patterns. On axis the correlation region is diffraction-limited and circular. Off axis the angular dispersion causes [197Sb]). the region to be radially elongated (PEDERSEN
32
THE STATISTICS OF SPECKLE PATTERNS
CL § 3
where w, is a correlation bandwidth given by
w, =
w / d T Z .
(54) It is clear from eqs. (53) and (54) that the speckle length in the radial direction depends on both the surface roughness a, and the illumination bandwidth W. The probability density function of intensity in a polychromatic speckle pattern can be found by considering the pattern to be the sum of a number of partially correlated monochromatic speckle patterns, the degree of correlation depending upon position and surface roughness. Using an analysis similar to that given in $2.3 for the first order statistics of the measured intensity in a normal speckle pattern, it can be shown that the probability density function of intensity in a polychromatic pattern is approximately given by a gamma variate (eq. (30)) or by exact expressions similar to eqs. (27) and (28) (PARRY[1975a]). The above results indicate that the observation of white light speckle patterns formed in the Fraunhofer plane may produce useful measures of surface roughness and this has been suggested by SPRAGUE[1972] and TRIBILLON [19741, both of whom present preliminary experimental results. However, it must be borne in mind that the above analysis contained a large number of assumptions. In particular it was assumed that single scattering occurred (first Born approximation) and that the surface could be characterised either by a Gaussian distribution of height with non-zero correlation length, or by a “white-noise” distribution; it is not clear whether these assumptions are valid in many practical scattering problems. Speckle patterns formed in the image plane of a diffuser illuminated by spatially coherent, polychromatic light have been studied by ELBAUM, GREENBAUM and KING [1972], GEORGE and JAIN[1972, 1973, 19741, and MCKFXHNIE[1975]. The main application here is the reduction of speckle in the images of diffuse objects. The analysis of the statistics is very similar to the above case for the Fraunhofer plane. The main results are (i) the speckle pattern is statistically stationary for a uniform diffuse object provided that the aberrations of the imaging system are not field-dependent, (ii) the speckle contrast decreases as the bandwidth W of the illumination increases and as the surface roughness a, increases, (iii) the speckle pattern depends on the aberrations of the imaging system. GEORGE and JAIN[1974] and MCKECHNIE [1975] have given detailed analyses of the statistics using Goodman’s model of the scattering suface.
S U R F A C E - D E P E N D E N T FEATURES
9
33
4. Surfacedependent Features of Speckle Patterns
In earlier sections we have stressed that speckle patterns resulting from certain surfaces being illuminated by monochromatic spatially coherent light are essentially independent of detailed surface properties. The main requirements were (i) the surface does not alter the polarisation of the incident light, (ii) a large number of scattering centres contribute to the amplitude at any point in the observation plane, and (iii) the phase of the scattered wave is random in the interval --n: to (or oh > A). In monochromatic light it was shown that the value of the surface roughness oh did not influence the speckle statistics provided that condition (iii) was followed, although in polychromatic light g,, does influence the speckle statistics. In this section we shall consider the statistics of speckle patterns formed when the above requirements are not fulfilled and we shall see that in general the properties of speckle patterns depend in a very complicated way on surface properties.
4.1. DEPOLARISING SURFACES
When plane-polarised monochromatic light is incident on a scattering medium, the transmitted or reflected amplitude in general consists of components of the field that are parallel and perpendicular to the incident field; these components have in general unequal intensities and an arbitary correlation factor. A discussion of mechanism of depolarisation and its relation to the scattering medium is outside the scope of this article (BECKMANN and SPIZZICHINO [1963]). As far as the effect of depolarisation on the statistics of the resultant speckle pattern is concerned, we may reduce the problem to that of an addition of two partially correlated speckle patterns of unequal intensity (BARAKAT[1973], GOODMAN [1975a, c]). Let A , , II and A , , I , be the amplitude and intensity of the speckle patterns produced by the parallel and perpendicular components respectively. The correlation coefficient for the amplitude is given by
Experimentally we can measure a correlation matrix of intensities (CHAKRA-
34 BORTY
THE STATISTICS O F S P E C K L E P A T T E R N S
CL D 4
[1973], GEORGE, JAIN and MELVILLE [1975])
However, for a large number of scattering centres the complex amplitude in each speckle pattern is a complex Gaussian process and using the results of REED[1962] cmn = Ipmn12,
or pmn =
Jcntn . ~ X (i4mnL P
where 4mnis a phase factor. It turns out that the phase factor is of no consequence in this case, but has an important influence when a similar analysis is applied to the addition of three or more correlated speckle patterns GOODMAN [1975a, c]). The hermitian coherency matrix whose elements are (AmA,*)can therefore be written
In problems where we have the sum of correlated random processes, the standard method of solution involves expanding the random process in an orthogonal series such that the sum is of independent random processes; standard theorems of probability theory can then be applied to yield p ( l ) , the probability density function. Applying this to our example yields (GOODMAN [1975a, c]), for distinct eigenvalues,
=
0
otherwise,
or for identical eigenvalues, c1 = c2
=
0
= c0
otherwise,
where E, are the eigenvalues of the coherency matrix. An example is given in Fig. 16 for the cases of c12 = 0 ( E ~= E~ = O S ) , c12 = 0.6 ( E ~= 0.887, E* = 0.113) and c12= 1 (el = 1, E~ = 0) with the assumption that the two mean intensities are equal.
35
SURFACE-DEPENDENT FEATURES
I
I
I
I
1.0
2.0
I
I 30
I Fig. 16. The probability density functions for the intensity of the sum of two partially correlated normal speckle patterns with equal mean intensity and c,* = 0, 0.6 and 1.0 (GOODMAN [1975a, c]).
4.2. A SMALL NUMBER OF SCATTERERS
If only a small number of scatterers contribute to the amplitude at a point in the observation plane then the central limit theorem cannot be applied and the complex amplitude will not have a complex Gaussian distribution. In this case the statistics of the scattered field will depend on the statistics of the scattering medium. This dependence is discussed below. A small number of scatterers also implies that the autocorrelation function of the amplitude of the scattered wave immediately behind the scatterer is not small compared to the dimensions of the illuminated area. In all cases considered so far in this article we have assumed that this correlation function is small in extent relative to the illuminated area (i.e. diffusers relatively of fine structure) and so we first consider the effect of allowing this autocorrelation function to increase in area, but at the same time assuming that the number of correlation areas is sufficiently large to apply the central limit theorem. Consider the optical system in Fig. 17. A plane wave is incident on a random medium which imposes a complex amplitude distribution A ( t ) at the entrance pupil of a lens whose pupil function is H(5). If the lens is aberration-free then the amplitude in the observation (Fraunhofer) plane
36
THE STATISTICS OF SPECKLE P A TTER N S
random medium
A([)
[I. § 4
5
H(r)
I(x)
Fig. 17. Formation of a speckle pattern in the Fraunhofer plane for a random complex amplitude with a correlation function of non-zero extent.
is thefinite Fourier transform of A(<),but in general the amplitude in the observation plane is the Fourier transform of the product A ( ( ) H ( ( ) . We have included this dependence on the pupil function as this problem first arose in an application in astronomy where the atmosphere is the random medium and the effect of telescope aberrations is of practical interest (DAINTY [1973,1974]). The autocorrelation function of the amplitude A(5) is defined as
and is independent of if the random wavefront A(<)is statistically stationary. Because the extent of C,(() is not considered to be small compared with the limiting aperture of the pupil function, the intensity distribution in the observation plane (the Fraunhofer plane of A ( ( ) and also the h a g t plane of the point source) is concentrated in a region near to the optic axis; within this region a speckle-like pattern is seen. The Wiener spectrum of the intensity distribution is equal to (li(u)I2), where i(u) is the Fourier transform of the intensity distribution I(x). Thus the Wiener spectrum is given by W(4 = rSH*(;,)rr(C)H(i,+;)H’li2+<~ --m
where u = (/A$ It is clear from this equation that the Wiener spectrum of the speckle depends on the fourth order moment of the complex amplitude of the scattered field. If this complex amplitude is a “white” noise process, the fourth order moment can be expressed in terms of delta functions, and the expression for the Wiener spectrum reduces to that given by either
1,
o 41
SURFACE-DEPENDENT FEATURES
37
eq. (15) or (21) in 92. To consider the effect of the autocorrelation of the complex amplitude on the Wiener spectrum of the speckle intensity, we consider a particular example in which it is assumed that A(5) is a complex Gaussian process (DAINTY [1973]). The fourth order moment can then be written in terms of the autocorrelation function and eq. (55) becomes, after some rearrangement,
-m
This expression is essentially the same as one given by ENLOE[1967] for an analogous case where a partially resolved diffuser is imaged by an optical system and also as eq. (59) below. The first term on the right-hand side of eq. (56) is governed by the average intensity in the observation plane and the second term describes the intensity fluctuation. For a “white” noise field C,(t) = S(t), and the intensity fluctuation term becomes identical to that given in eqs. (15) and (21) in $ 2 . In Fig. 18 we give an example of the effect of C,(t) on the Wiener spectrum of the intensity fluctuation (second term in eq. (56)). The autocorrelation function is assumed (for computational convenience) to have a top-hat form with a diameter that is some fraction R of the diameter of the optical system (DAINTY [1974]). In Fig. 18(a) and (b) the effect of defocus is shown for R = 0.2 and R = 0.1 respectively. Clearly the speckle pattern is independent of lens aberration only in the limit R + 0 (i.e. a “white” noise field A ( ( ) ) . The problem of finding the first order statistics when N is finite was first examined by Lord RAYLEIGH [1919] and more recently his results were applied by BURCH [1969]. More general analyses with application to the scattering by liquid crystals have been given by JAKEMANand PUSEY [1973a, b, 19751 and a summary of their results is presented below. It is assumed that the scatterer is a deep phase screen such that the phase of the scattered light immediately after the screen has a Gaussian distribution with a variance 082 >> 1. The complex amplitude in the Fraunhofer plane is given by
38
THE STATISTICS OF SPECKLE PATTERNS
1.0
0
I 0‘ 0
W
2.0
4\ I
I
1*o
W
2-0
Fig. 18. Wiener spectra of the intensity fluctuation for speckle patterns observed in the Fraunhofer plane of a defocussed lens for defocus values of 0, 1 in,2/n and 4/nwavelengths. TJpper (a), R = 0.2; lower (b), R = 0.1 The parameter R is equal to the diameter of the correlation area relative to that of the optical system (DAINTY [1974]).
where N is$nite, aj(x, y ) is a scattering factor, and pj is the random phase from thejth scatterer and is independent of the phases of all other scatterers. Equation (57) describes a finite random walk with variable step length studied by Lord RAYLEIGH [1919] and others. Assuming that all the aj can be described by the same distribution function, the following expressions for the first two moments of the intensity can be found:
(0= N(.2>, (1’) = N ( a 4 ) + 2 N ( N - 1)(a2)’.
Higher moments can also be found. Evaluating the moments of a j yields an expression for the ratio of the variance of the speckle intensity to the
39
S U R F A C E - D E P E N D E N T FEATURES
square of the mean intensity (JAKEMAN and PUSEY[1975]), o2
~- - 1-
(02
2 N
~
082 +exp 4N
where k = 27-c/A, r8 is the phase correlation length, and 8 is the angle observation (for normal incidence). Clearly as N -,co , 02/(Z)2 tends to the Gaussian value of unity. However for finite N this ratio (the speckle “contrast”) depends on the angle of observation, the variance 0; and correlation scale length rg of the phase immediately after the scatterer and of course on N itself. Some experimental results (PUSEYand JAKEMAN [1975]) for the variation of speckle contrast with angle of observation 8 and the number of scatterers for a liquid crystal scatterer are given in Figs. 19 and 20 respectively; these broadly confirm eq. (58).
L
0
I
0.2
I
I
0.4
I
I
I
06
sin2e Fig. 19. Angular dependence of the speckle contrast for a small number of scatterers (PUSEYand JAKEMAN [1975]).
The normalised angular autocorrelation function of the intensity,
can also be found using eq. (57) and evaluating the appropriate moments
40
THE S T A T I S T I C S OF S P E C K L E P A T T E R N S
[I,
0
4
T
3
2
1
E Area-’
x103
( rn-2)
Fig. 20. Dependence of speckle contrast on illuminated area of diffuser for a fixed size of and JAKEMAN[1975]). inhomogeneity (PUSEY
of a j yields (JAKEMANand PUSEY [1975]) C,(A8)+
4 exp { - k2riA82/16}
-
4N
where small 8 and 8‘ have been assumed for simplicity, A 8 = O-Q‘, and where C,(A8) is the autocorrelation function of the normal speckle pattern produced when N + co as given by eq. (14). Clearly eq. (59) gives the correct result of C(8,6’) -+ C,(A8) as N + 00. However for finite N two characteristic scale lengths appear in the autocorrelation function. The first is the normal speckle “size” governed by the total extent of the scattering volume, whilst the second depends on the variance and scale length of the phase immediately behind the scatterer. This result is essentially the same as that given in eq. (56) but derived and used in a different context (DAINTY [1973]). Some experimental measurements of C(6,e’) in the scattering by liquid crystals are shown in Fig. 21 and broadly confirm the above theoretical results. The statistics of speckle patterns produced when only afinite number of scatterers contribute to the pattern at any point clearly depend
41
S URF ACE - DE P E NDE NT FEATURES m
'r
t
01
10
2 0 volts
1
I 20
I
I 30
I
O2
I
40 degrees
I
I
50
J
Fig. 21, Angular autocorrelation functions in the Fraunhofer plane for speckle patterns produced by many scatterers (100 v) and a few scatterers (20 v) (PUSEY and JAKEMAN [19751).
on the properties of the scattering surface and may be of use in determining these properties. 4.3. SLIGHTLY ROUGH SURFACES
The statistics of speckle patterns considered throughout this article have been derived on the assumption that the surface roughness is greater than one wavelength. If oh< ;1then the speckle pattern will contain information on the surface properties; several authors have suggested that useful surface parameter information is contained in speckle patterns produced by slightly rough surfaces (ALLENand JONES[19631, CRANE { 19703, GOODMAN [1973,1975b, c], PEDERsEN [1974], FuJIIand AsAKURA [1974b, c], WELFORD [19751). It is convenient to first examine the speckle patterns formed in coherent light in the image plane of a slightly rough diffuser. Such a speckle pattern can be considered to consist of a uniform beam arising from a specularly reflected component added coherently to a diffusely reflected beam which forms a normal speckle pattern. The relative intensities of the two components will depend on the properties of the scatterer. The probability density function of the intensity of a speckle pattern produced in the image plane of a slightly rough diffuser will be similar to that obtained when a
42
T H E S T A T I S T I C S OF S P E C K L E P A T T E R N S
[I,
94
uniform beam is added coherently to a speckle pattern. This problem has already been investigated by GOODMAN [1967] and DAINTY[1972]. The real and imaginary parts of the complex amplitude in the speckle pattern have a Gaussian distribution with non-zero mean and the distribution of intensity and phase can be found using the appropriate probability transformation (DAINTY[1972]). In Fig. 22(a) we show some experimental measurements of the probability distribution of intensity p(1) for four surfaces, and in Fig. 22(b) theoretical curves for p(1) are shown for a range of specular/diffuse beam ratios. The agreement between the overall forms of the two sets of curves is gcod and indicates that the simple approach outlined above may be
I r =lo
IRRADIANCE
I
Fig. 22. (a) Experimental probability distributions of speckle intensity in the images of slightly rough diffusers (FUJIIand ASAKURA [1974b]). (b) Probability density functions for a range of specular/diffuse beam ratios (DAINTY [1972]).
1,
o 41
43
S U R F A C E - D E P E N D E N T FEATURES
adequate for practical purposes. The main difficulty lies in establishing a link between the surface parameters and the ratio of specular/diffuse transmittance (or reflectance). In order to establish this link it is again necessary to use a surface model that gives a Gaussian distribution of phase in the scattered wave immediately after the diffuser ; the standard deviation of the phase is cr+ and the scale length of the phase correlation function is rb. Using this model FUJIIand ASAKURA [1974c] evaluated o / ( l ) as a function of both the relevant surface parameters and the degree of spatial coherence. Their theoretical and experimental results are summarised in Fig. 23. The surface parameter of importance is (for reflected light)
The final term on the right hand side of eq. (60), (oh/r,&is closely related to the r.m.s. gradient of the surface height distribution and this equation indicates that a simple measure of speckle contrast in the image cannot uniquely determine the surface roughness. Recently OHTSUBO and ASAKURA [1975] have observed that the speckle contrast is somewhat smaller in the image plane than in out-of-focus planes. GOODMAN [1975b, c] has shown that this is because the scattered field does not have circular Gaussian statistics (as assumed above) and that
0
0.5
1.0
1.5
Fig. 23. Speckle contrast as a fmction of roughness characteristic a:/rb for four conditions of partial spatial coherence (a) - (d) (FUJI[and ASAKURA[1974c]).
44
T H E STATISTICS O F S P E C K L E P A T T E R N S
CI
the degree of non-circularity depends on the surface roughness. The dip in speckle contrast through focus is due to change from circular to noncircular statistics and back again. It may also be useful to study the intensity at the centre of the Fraunhofer plane within the first bright ring of the Airy disc for a slightly rough surface. A similar statistical variation to that shown in Fig. 22 would be obtained, but in this case one would not expect on simple physical grounds any strong dependence on the r.m.s. gradient of the surface height. Further advances are awaited on this aspect of scattering from rough surfaces.
8
5. Concluding Remarks
We began this review article by stressing that in a large number of practical situations normal speckle patterns are produced ; the statistics of these speckle patterns are well-established both theoretically and experimentally and in particular they do not depend on the detailed scattering properties of the scattering medium. When examining the effects of polychromatic illumination on speckle statistics we found the value of the r.m.s. height variation of a scattering surface strongly influenced the statistical properties of the scattered intensity. Finally in 54 it was seen that the general relationship between the scattered intensity and the scattering surface may be very complicated, even for surfaces with a Gaussian distribution of surface heights. However, it must be stressed that through the whole review we have glossed over the details of the interaction of an electromagnetic wave with a scattering medium and have used relatively naive models of scattering surfaces. Hopefully this unsatisfactory state of affairs in the subject as a ‘whole will be rectified by thL: advances of future workers. Acknowledgements
I wish to thank Professor W. T. Welford of Imperial College, London for his encouragement, advice and criticism of my work on speckle patterns over a number of years. I also wish to thank Dr. G. Parry for his advice on the original manuscript. References ALLEN,L. and D. G. C. JONES, 1963, Phys. Lett. 7,321. ARCHBOLD, E. and A. E. ENNOS,1972, Opt. Acta 19, 253. ASAKURA, T., H. FUJIIand K. MURATA,1972, Opt, Acta 19, 273. BARAKAT, R., 1973a, Opt. Acta 20,729.
11
REFERENCES
45
BARAKAT, R., 1973b, Opt. Commun. 8, 14. BECKMANN, P., 1967, in: Progress in Optics, Vol. VI, ed. E. Wolf (North-Holland). BECKMANN, P. and A. SPIZZICHINO, 1963, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon/MacMillan, London/New York). BEDARD, G., J. C. CHANGand L. MANDEL,1967, Phys. Rev. 160, 1496. BORN,M. and E. WOLF,1970,Principles ofoptics (fourth ed., Pergamon Press, London, N.Y.). BUCHWALD, E., 1919, Ber. Deut. Phys. Ges. 21, 492. BURCH,J. M., 1969, in: Optical Instruments and Techniques, ed. J. Home-Dickson (Oriel Press, Newcastle-upon-Tyne). BURCKHARDT, C. B., 1970, Bell Syst. Tech. J. 49, 309. CHAKRABORTY, A. K., 1973, Opt. Commun. 8, 366. CHANDRASEKHAR, S., 1943, Rev. Mod. Phys. 15, 1. CHERNOV, L. A,, 1960, Wave Propagation in a Random Medium (Dover Press, New York). CONDIE,M. A., 1966, Thesic Stanford University. CRANE,R. B.. 1970, J. Opt. SOC.Am. 60,1658. DAINTY, J. C.. 1970, Opt. Acta 17, 761. DAINTY, J. C., 1971, Opt. Acta 18, 327. DAINTY, J. C., 1972, J. Opt. SOC. Am. 62, 595. DAINTY,J. C., 1973, Opt. Commun. 7, 129. DAINTY,J. C., 1974, Mon. Not. R. Astr. SOC.169, 631. ELBAUM, M., M. GREENBAUM and M. KING,1972, Opt. Commun. 5, 171. ELIASSON, B. and F. M. MOTTIER,1971, J. Opt. SOC.Am. 61, 559. ENLOE,L. H., 1967, Bell Syst. Tech. J. 46,1479. EXNER,K., 1877, Sitzungsber. Kaiserl. Akad. Wiss. (wein) 76, 522. EXNER,K., 1880, Wiedemanns. Ann. Physik 9, 239. FUJII,H. and T. ASAKURA, 1973, Optik 39, 99. FUJII,H. and T. ASAKURA, 1974a, Optik 39, 284. FUJII,H. and T. ASAKURA, 1974b, Opt. Commun. 11, 35. FUJII,H. and T. ASAKURA, 1974c, Opt. Commun. 12, 32. GEORGE, N. and A. JAIN,1972, Opt. Commun. 6, 253. GEORGE, N. and A. JAIN,1973, Appl. Opt. 12, 1202. GEORGE, N. and A. JAN, 1974, Appl. Phys. 4, 201. 1975, Appl. Phys. 6, 65. GEORGE, N., A. JAINand R. D. S. MELVILLE, GERRITSEN, H. J., W. J. HANNAN and E. G. RAMBERG, 1968, Appl. Opt. 7, 2301. GOLDFISCHER, L. I., 1965, J. Opt. SOC.Am. 55, 247. GOODMAN, J. W., 1963, Stanford Electronics Lab. TR2303-1 (SEL-63-140). GOODMAN, J. W.. 1965, Proc. I.E.E.E. 53, 1688. GOODMAN, J. W., 1967, J. Opt. SOC.Am. 57, 493. GOODMAN, J. W., 1973, in: Remote Techniques for Capillary Wave Measurement, eds. K. S. Krishnan and N. A. Peppers, Stanford Research Institute Report, 2nd April. GOODMAN, J. W., 1975a, Opt. Commun. 13, 244. GOODMAN, J. W., 1975b, Opt. Commun. 14, 324. GOODMAN, J. W., 1975c, in: Laser Speckle and Related Phenomena, ed. J, C. Dainty (Springer-Verlag, Heidelberg). HAAS,DE W. J., 1918a, Koninklijke Acad. van Wetenschappen (Amsterdam) 20, 1278. HAAS,DE,W. J., 1918b, Ann. Phys. IV 57, 568. HARIHARAN, P.; 1972, Opt. Acta 19, 791. HCIHN,D. H., 1968, Optik 27, 353. JAKEMAN, E., 1974, in: Photon Correlation and Light Beating Spectroscopy, eds. H. Z. Cummins and E. R. Pike (Plenum Press, New York). JAKEMAN, E. and P. N. PUSEY,1973a, Phys. Lett. 44A, 456. JAKEMAN, E. and P. N. PUSEY,1973b, J. Phys. A 6, L88. JAKEMAN, E. and P. N. PUSEY, 1975, J. Phys. A 8, 369.
46
THE S T A T I S T I C S O F S P E C K L E P A T T E R N S
LI
U C , M. and A. J. F. SIEGERT, 1947, J. Appl. Phys. 18, 383. LABEYRIE, A,, 1970, Astron. and Astrophys. 6, 85. LAUE,M. VON,1914, Sitzungs. Akad. Wiss. (Berlin) 44, 1144. LAUE,M. VON,1916, Mitt. Physik. Ges. (Zurich) 18, 90. LAUE,M. VON,1917, Verhandl. Deut. Phys. Ges. 19, 19. S. and H. ARSENHAULT, 1970, J. Opt. SOC.Am. 60, 1478. LOWENTHAL, T. S., 1974a, Optik 39, 258. MCKECHNIE, MCKECHNIE, T. S., 1974b, Thesis, University of London. MCKECHNIE, T. S., 1975, in: Laser Speckle and Related Phenomena, ed. J. C. Dainty (SpringerVerlag, Heidelberg). MANDEL, L., 1959, Proc. Phys. SOC.74, 233. MARTIENSSEN, W. and E. SPILLER,1965, Naturwiss. 52, 53. MIDDLETON, D., 1960, An Introduction to Statistical Communication Theory (McGrawHill, New York). OHTSUBO, J. and T. ASAKURA, 1975, Opt. Commun. 12, 30. PARRY, G., 1974a, Opt. Acta 21, 763. PARRY, G., 1974b, Opt. Comniun. 12, 75. PARRY. G., 1975a, Opt. Quant. Elect. 7, 3 18. PARRY,G., 1975b, in: Laser Speckle and Related Phenomena, ed. J. C. Dainty (SpringerVerlag, Heidelberg). PEDERSEN, H. M., 1974, Opt. Commun. 12, 156. PEDERSEN, H. M., 1975a, Opt. Acta 22, 15. PEDERSEN, H. M., 1975b, Opt. Acta 22, 523. PUSEY,P. N. and E. JAKEMAN, 1975, J. Phys. A 8, 392. RAMACHANDRAN, G. N., 1943, Proc. Ind. Acad. SOC.(A) 18, 190. RAMAN, C. V., 1919, Phil. Mag. 38, 568. RAYLEIGH, Lord, 1880, Phil. Mag. 10, 73. RAYLEIGH, Lord, 1918, Phil. Mag. 36, 429. RAYLEIGH, Lord, 1919, Phil. Mag. 37, 321. REED,I. S., 1962, I. R. E. Trans. Info. Th. IT-8, 194. RICE,S. O., 1954, in: Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover Press, New York). Ross, G., 1969, Opt. Acta 16, 611. Ross, G., 1970, Phil. Tran,. Roy. SOC.268A, 177. SCRIBOT, A. A,, 1974, Opt. Commun. 11, 238. SINGH,K., 1972, Pubblicazioni dell 'Instituto Nazionale de Ottica 27, 197. SLEPIAN, D., 1958, Bell Syst. Tech. J. 37, 163. SPRAGUE, R. A., 1972, Appl. Opt. 11, 2811. STROHBEHN, J. W., 1971, in: Progress in Optics IX, ed. E. Wolf (North-Holland). SUZUKI, T. and R. HIOKI,1966, Jap. J. Appl. Phys. 5, 807. TATARSKI, V. I., 1961, Wave Propagation in a Turbulent Medium (Dover Press, New York). TRIBILLON, G., 1974, Opt. Commun. 11, 172. WELFORD, W. T., 1975, Opt. Quant. Elect. 7, 413. YAMAGUCHI, I., 1972, Optik 35, 591 and 36, 173. YAMAGUCHI, I., 1973, Optik 37, 141.
E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
I1
HIGH-RESOLUTION TECHNIQUES IN OPTICAL ASTRONOMY BY
A. LABEYRIE Observatoire de Paris, 92190 Meudon, France
CONTENTS PAGE
Q 0. INTRODUCTION . . . . . . . . . . . . . . . . . . .
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Q 1. ATMOSPHERIC OPTICS .
. . . . . . . . . . . . . . .
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Q 2 . DIRECT INTERFEROMETRY . . . . . . . . . . . . .
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Q 3 . INTERFEROMETER DESIGNS AND RESULTS . . . . .
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Q 4 . THE IMAGE RECONSTRUCTION PROBLEM . . . . . .
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5 5. CONSTRUCTION OF A SYNTHETIC-APERTURE ARRAY O F OPTICAL TELESCOPES. . . . . . . . . .
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Q 6. INTENSITY INTERFEROMETRY . . . . . . . . . . .
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Q 7 . HETERODYNE INTERFEROMETRY . . . . . . . . . .
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Q 8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . .
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REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
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0
0. Introduction
0.1. INTRODUCTION
Continued progress in the art of building optical instruments has resulted at about the time of Newton, Herschel and Foucault, in telescopes having better optical quality than the atmosphere. From then on, the subsequent evolution which led to today’s giant telescopes failed to further improve the resolving power of optical observations through the atmosphere. A few obstinate physicists however succeeded in showing that the atmospheric degradation of images can be avoided to some degree by using special observing techniques. Used to observe a few favorably bright stars, these techniques have indeed demonstrated principles which can yield high resolution information beyond the normal atmospheric cut-off. The techniques utilize approaches known as stellar interferometry, intensity interferometry and lunar occultation. The state of the art in these fields has been reviewed by Hanbury Brown in 1968. The present review is more specifically concerned with the more recent developments of stellar interferometry, and the corresponding instruments, generally referred to as coherent synthetic-aperture systems. Pioneering work by A. A. Michelson established the feasibility of large resolution gains, but lack of a mature technology long prevented followers from even repeating Michelson’s observations. Only very recently did the progress of sensors and electronics allow improvements to the original Michelson interferometer. In the last few years, the progress of coherent optics allowed a better understanding of the speckle phenomenon in stellar images, which in turn triggered spectacular developments in both post- and pre-detection image processing methods. The operation of two independent telescopes as a Michelson interferometer, achieved a few months before this review was concluded, indicates that the technology is now ready for operating large arrays of telescopes as optical synthetic apertures. As already experienced at radio wavelengths, telescope arrays are likely to improve enormously the resolution and the luminosity of observations. Their operation from ground-based sites will help designing similar systems for space use. 49
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0.2. HISTORY
The history of attempts to penetrate the turbulent atmosphere dates back to 1868, when Fizeau proposed to observe Young’s fringes in stellar images. Following initial attempts by Stephan, this suggestion was brillantly developed by Michelson using aperture masks on the Yerkes refractor and the 100-inch Hooke reflector. Later,Michelson, Pease and Hale built successively two models of a synthetic aperture system having respectively 20 and 50 feet span, and in which the observer’s eye served as the sensor. Each system involved a pair of small apertures supported by a single steel structure, designed to be as rigid as possible while being mobile around the polar axis of an equatorial mount. After the equality of optical paths had been carefully adjusted, fringes could be observed in stellar images. As we shall see, the phase shifts introduced by the atmosphere induced fast displacements of the fringes, which decreased their apparent contrast. Nevertheless, fringe contrast measurements were possible and these gave apparent stellar diameters for several bright red stars. The observations were difficult and time-consuming, as can be realized from the fact that none has ever succeeded in repeating these experiments. Pease’s measurements of stellar diameters were however confirmed recently by different methods benefiting from the large aperture of the 5-meter Hale telescope, and thus easier to implement. The level of activity in this field dropped near zero between 1930 and 1958. In 1958, Hanbury Brown and Twiss proposed the intensity interferometer method as an alternate approach intended to avoid the problems arising with direct interferometry. In spite of the method’s inherently low sensitivity, the patient work of Hanbury Brown and Davis at Narrabri (Australia) resulted in a major resolution breakthrough. The angular diameters of the 32 brightest southern stars were measured, with resolution of the order of a millisecond of arc. In the meanwhile, interferometric observing techniques underwent considerable development at radio wavelengths. The synthetic-aperture arrays of radio telescopes built in this period by Ryle and his collaborators began to surpass the traditional one-second limit of resolution with optical telescopes, and quickly achieved arc-second, as different groups began to use heterodyning techniques with antennas spaced thousands of kilometers apart. Interest in the direct synthetic-aperture approach at optical wavelengths revived in the year 1965, presumably in relation with the general progress of coherent optics and laboratory interferometry which the invention of lasers has triggered. Summer schools held at Wood’s Hole in 1966 and
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1967, followed by a symposium at Tucson in 1970, stimulated interest in long-baseline interferometers such as proposed by MILLER[19661. A belief emerged that modern technology could solve the problems encountered by Michelson and Pease, and that coherent arrays of telescopes could be operated at optical as well as at radio wavelengths. Several groups, particularly in the USA, USSR and France, began to tackle these problems, investigating in particular the use of artificial sensors to replace the human eye. In 1970, speckle interferometry established a generalized form of Fizeau interferometry, which utilizes the full aperture of large telescopes. Used at the 200-inch Hale telescope by GEZARI, LABEYRIE and STACHNIK [1972], the method provided a confirmation of Pease’s measurements as well as additional hints on color-dependance and limb-darkening effects relating to the two largest apparent stellar disks, Betelgeuse and Mira Ceti. The method also provided diffraction-limited measurements on a dozen binary stars, unresolvable by conventional techniques. Following these initial results, progress in the technology of sensors, and particularly the development of photon-counting television by BOKSENBERG [19721, resulted in considerable gains in both the sensitivity and the accuracy of data reduction. At this point a number of articles discussed further possibilities for improvements, particularly in the direction of image reconstruction. Computer simulations as well as laboratory experiments produced particularly encouraging results in relation with the approach known as “rubber telescope imaging”. In this approach, atmospheric disturbances are actively corrected by servo devices requiring a bright star in the observed field. In the meanwhile, work continued in the direction of long-baseline systems. Preliminary designs were made by MILLER [1966] for a 100-meter interferometer, while the present author attempted to produce interference with two telescopes. The latter project reached its goal in the recent months. It is now undertaken to build a coherent array of telescopes.
0 1. Atmospheric Optics 1 . 1 . THE ATMOSPHERIC HETEROGENEITY
The turbulent behaviour of the atmosphere has been extensively studied in the recent years. Chernov, Tatarsky, LEEand HARP[1969], LAWRENCE and STROHBEHN [19701 and others developped the theory of wave propagation in turbulent air, while HUFNAGEL and STANLEY [1964], FRIED[1966],
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KORFT, DRYDEN and MILLER[1972] studied the corresponding effects on images. Experimental measurements of scintillation phenomena were made by YOUNG[1969], MIKESELL [1955] and PROTHEROE [1961]. Phase effects were observed by RODDIER and RODDIER [1973] and VERNINand RODDIER [19731. A variety of mechanisms occurring in clear air account for the disturbances which affect optical waves propagating horizontally or vertically. This review is mainly concerned with vertical propagation, and the relevant body of knowledge may be briefly summarized as follows : Air at a uniform pressure and temperature is in principle an excellent optical medium. Apart from the microscopic fluctuations responsible for Rayleigh scattering, the Brownian motion alone introduces negligible wide-scale phase fluctuations on optical waves. However, temperature is not uniform in the atmosphere, and its fluctuations produce optical phase fluctuations on propagating waves, in response to the local variations of the temperature-sensitive refractive index. This temperature effect is predominant at optical wavelengths, but other causes such as humidity or partial pressure fluctuations may contribute at other wavelengths, such as in the infra-red. The temperature fluctuations are caused by a variety of turbulence mechanisms, depending on wheather conditions and terrain topography. Among these are thermal convection, interface turbulence between layers at different temperatures, wind shear turbulence, wake turbulence (downwind from certain”mountain peaks), etc.. As reviewed by LUMLEY and PANOVSKY [1964], the theory of turbulence evolved by fluid dynamics since the 1950’s ascribes a power law to the spatial power spectrum of density fluctuations. The exponent value generally accepted is that obtained theoretically by Kolmogoroff, namely - 11/3. The scale size for turbules ranges between an “inner scale” amounting to a millimeter or so in air at sea level, and an “outer scale” on the order of 10 to 100 meters at sea level. Reasonable agreement exists between theoretical results and experimental measurements, although certain points remain to be clarified. The turbulence parameters which are most significant with respect to optical effects are : 1. the size and the size distribution of turbulence cells; 2. the RMS density fluctuation, integrated along the line of sight; 3. the altitude of turbulent layers; 4. the apparent lifetime of turbules along a given line of sight. In the presence of winds, this lifetime differs from the intrinsic lifetime, since turbules are carried across the line-of-sight by the wind. Concerning the spatial distribution of turbulence cells, these are usually
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53
organized in random isotropic fashion. However, they sometimes take the form of waves resembling those on water surfaces, and similarly generated by gravity oscillations at the interface between atmospheric layers having different densities (GAVIOLA [19491). The temporal frequency spectrum of atmospheric turbulence is of particular concern to interferometric observers. This is largely governed by wind speeds at ground and altitude levels. Lifetimes vary between 1 and second, depending on wheather conditions. 1.2. WAVE DEFORMATIONS A N D SHADOW PATTERNS
The optical wave received at ground level from a point source located above the atmosphere has both phase and amplitude perturbations. The amplitude fluctuation pattern, also referred to as the shadow pattern, is responsible for the well known twinkling of stars. Because absorption cannot account for the effect, it is generally interpreted as resulting from the action of high-altitude turbulent layers according to elementary stioscopic or Scblieren effects. Both geometric ray deflection and interference may contribute in this effect. A simple method for viewing the high-altitude turbulent layers responsible for the shadow pattern was used extensively by BOYER[unpublished] for wind monitoring purposes. It consists in observing the lunar limb with a small, amateur-type telescope. By defocusing slightly the eyepiece outward, it is possible to focus on the turbulent layers and to see the flowing stream of turbules, their invisible phase pattern being translated into an amplitude pattern by a Schlieren effect. The altitude, velocity and direction of the turbulent flow may thus be determined. More elaborate methods involving large mirrors and shadow pattern correlations have recently been developped by Roddier and his collaborators (MARTIN,BORGNINO and RODDIER [1975]). Phase corrugations on the wave are generated by turbulence at all altitudes, but with increased efficiency at decreasing altitudes. The low-altitude phase cells are observed easily on the aperture of large telescopes when conducting knife-edge tests. In addition to the steady cell flow outside of the dome, a stationary turbulence component occurring inside the dome is also generally observed. RMS values for the fast phase fluctuation have been estimated to vary in the range from 1 to 20 radians. Available data do not cover the long-wave components of phase corrugations. 1.3. THE SPECKLED STRUCTURE OF IMAGES
On good observing nights, small telescopes in the 2 to 10 cm range of
54
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HIGH-RESOLUTION TECHNIQUES
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1
aperture sizes are smaller than seeing cells. Thus the phase is nearly uniform over their aperture. Consequently, these telescopes are nearly diffraction limited, and star images have the appearance of the classical Airy disk. Considering now the other extreme case of very large instruments, their aperture may cover several thousand seeing cells, having random phases. The Airy-disk pattern is completely destroyed under such conditions ; and classical diffraction experiments, in accordance with elementary Fourier transform analysis, show that the angular spread of the image is of the order of Ajd, d being the characteristic size of seeing cells. 12-cm cells thus correspond to one arc-second as the approximate width of the image projected onto the sky plane; and this figure is rather typical of average observing conditions with large instruments in the best observatory sites. Only very exceptionally are seeing cells larger than 24 cm; and this explains why larger instruments do not benefit from their theoretical resolution performance. Some astronomers, especially binary star observers, have long remarked on the existence of a fast-moving fine structure inside the typical one-second image (Fig. 1). Recently (LABEYRIE [1970]), this structure has been interpreted as a speckle phenomenon, same as the well known effect in diffused laser beams, and this identification led to the interferometric method called “speckle interferometry” (sect. 3.3). As is apparent in the recent review of DAINTY [1976], the speckle phenomenon has been extensively studied in the recent years. In the astronomical case, it results from the fact that any point inside the star image indeed receives coherent contribu-
L
(a)
(b)
Fig. 1. The corrugated optical wave and corresponding image; (a) the full aperture of a large telescope produces a speckled spread function; (b) a Fizeau mask with two small apertures produces Young’s fringes in the image.
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tions from many of the phase cells in the aperture plane. The resulting amplitude at the image point considered is thus a sum of many vibrations with random phases. Summing vibrations with random phases has been a classical problem since Rayleigh: the squared modulus of the sum may have any value between 0 and IZaI2,a being the amplitude of the individual vibrations. In the image plane, the particular value found in the summing process depends on the image point considered since individual vibration phases vary when the point is displaced in the image plane by more than l/a,a being the angular aperture. The scale size for intensity variations in the image plane is thus l/u.As discussed by GOODMAN [1965], in the case and MILLER[1972], in the of laser speckles, and by KORFF,DRYDEN astronomical case, the addition of coherent, but randomly phased, vibrations corresponds to a two-dimensional random walk in the complex plane if component vibrations are represented by vectors in this plane. The sum vector is distributed according to a Gaussian law, and the ampli-
Fig. 2. Speckled images of a non-resolved star (Vega) obtained simultaneously in different colors at the 200-inch telescope. Colors (and spectral bandwidths) are, going clockwise from top left : red (500 A), yellow (250 A), green (250 A) and blue (250 A). Residual atmospheric dispersion is apparent in the blue image. Exposure time is 0.01 second.
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tude A has a Rayleigh distribution of the form ,,/%exp(-A/z2). The distribution of intensities I is exponential e-'/z2. This result is independant of the model assumed for seeing cells, provided random phase variations are present. Indeed, the appearance and the 2"d order statistical properties of laser speckle do not depend on the type of scatterer used. Different scatterers, or qualities of seeing in the stellar case, change only the size of the image envelope, without affecting the statistics of the fine speckle pattern within it. This important property simplifies appreciably simulation experiments carried out in the laboratory or on computers, since real atmospheric seeing does not have to be exactly reproduced. The analysis applies also regardless of aperture shape, and in particular to segmented or multiple apertures such as may be encountered in syntheticaperture systems (§ 2 and Q 4). The shape of speckles however depends on the aperture geometry: the aperture stop may be represented by a multi-
Fig. 3. Synthetic aperture systems and corresponding spread functions : top row - apertures consisting of 1, 2 and 6 telescopes, as well as giant monolithic aperture; middle row - corresponding spread functions, in the absence of atmosphere and optical aberrations; bottom row short-exposure, monochromatic, spread-functions in the presence of the atmosphere. This is a laboratory simulation result obtained by photographing a monochromatic point source (a laser-illuminated pinhole) through a mild diffuser and aperture mask. The conditions simulated correspond to 1.5 'meter telescopes and typical 1 sec. seeing.
n, §
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57
plicative term applied to the infinite incident wave, and the speckles in the image plane are thus convolved, in complex amplitudes, with the diffractionlimited spread function. Speckled images recorded at the Palomar 200-inch telescope are reproduced in Fig. 2. The laboratory equivalent of syntheticaperture systems involving several large telescopes produces the speckled images in Fig. 3. These images were obtained as photographs of a monochromatic point source, using a multi-aperture diaphragm and a mild diffuser in front of the camera lens. Compared to the speckles from the monolithic aperture, these contain an additional finer interference structure which takes the form of fringes with two apertures, of a honeycomb pattern with 3 apertures, etc.. The number of speckles contained within the image envelope increases with aperture size D as (D/dj2.When dealing with low-altitude seeing, no ensemble translation of speckles occurs in the image even when seeing cells are carried as a rigid pattern by the wind flow. Instead, speckles appear and disappear locally much like vapor bubbles at the surface of boiling water. It is apparent from the video images recorded at Palomar that the lifetime of speckles increases at increasing wavelengths, from near ultra-violet to near infra-red. In small telescopes, few seeing cells are present across the aperture at any instant, which results in few speckles in the images. Consistently with simple statistics, experience shows that such images are subject to rapid wander and size fluctuations. In such cases, image selection or exposure triggering techniques such as used by RATT [1957], may improve markedly the image sharpness. Telescopes up to a meter in aperture diameter may thus produce nearly diffraction-limited images durilig a few milliseconds every hour or so. Using short-exposure electronographic photographs of binary stars, R ~ S C HWLERICK , and BOUSSUGI~ [1961] have shown that the instantaneous speckle patterns are identical for closely spaced pairs but different in the case of widely spaced pairs. The angular extent over which speckle is invariant, called the isoplanatic patch, is on the order of 3 to 10 arc-seconds. Patch size .is mainly dependant upon the presence of turbulent layers at high altitudes, since these layers are crossed in different regions by the light beams coming from different stars into the telescope. Temporal and spatial coherence have both been assumed in the above analysis. The size d of speckles being proportional to wavelength, some degree of monochromaticity is indeed required for purity of the pattern. If atmospheric and telescopic aberrations are smaller than the wavelength 2, the requirement may be written I/d2 > D/d. In white light, at the 200-inch telescope, the image of a star close to zenith shows contrasted speckles
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1
at the center of the typical one-second envelope. The few central speckles are surrounded by radially-oriented coloured streaks generated by the chromatic spreading of speckles at higher interference orders. This is well observable visually in conditions of moderate wind speed, using a strong eyepiece. It takes a filter with 200 fingstrom or narrower bandpass to observe pure speckles all the way to the edge of a one-second image. When pointing at a star lower toward the horizon, dispersion tends to elongate the speckles and it takes a much narrower filter to remove this effect unless some form of prismatic compensation is used. The experimentally-observed proportionality of speckle size to wavelength supports the diffractioninterference interpretation of the speckle phenomenon, as opposed to the ray-deflection interpretation which had sometimes been proposed before the advent of lasers and speckle theory. The optical components of large astronomical telescopes are rarely made to the accuracy meeting the Rayleigh criterion. Residual coma, spherical aberration and astigmatism amounting to 0.5 arc-second are usually tolerated since the effect of the atmosphere is even worse. Such transverse aberrations, as long as they remain inferior to seeing effects, have no influence on the speckle patterns : somewhat paradoxically, bright speckles retain their similarity with Airy peaks in the presence of aberrations which would destroy the Airy peak if they acted alone. Realistic simulations of astronomical speckle phenomena may be carried out in the laboratory using a bright artificial star, a sheet of polyethylene or other diffusing material representing the atmosphere, a lens aperture to represent the telescope, and filters. In addition to laboratory simulations, a number of authors have used computers to derive image spread functions by Fourier-transforming random distributions of phase cells generated across some circular aperture. In all cases, the speckle patterns obtained resemble closely those obtained with large telescopes. 1.4. THE MTF FOR SHORT AND LONG EXPOSURES
The speckled or fringed structure in short-exposure images disappears when dealing with long exposures, due to the averaging of energy distributions in time-dependant speckle patterns. The short-exposure MTF, defined as the Fourier transform of the intensity distribution in the instantaneous spread function, has been studied experimentally (GEZARI, LABEYRIE and STACHNIK [19721) and theoretically (KORFF,DRYDEN,MILLER[1972], DAINTY [1973]). As shown in Fig. 4,it features a central peak and surrounding feet extending all the way to the diffraction-limited cut-off frequency.
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(MT
F
Fig. 4.Telescope-atmosphere MTF for alarge aperture : A, diffraction-limited MTF; B, single short exposure; C, long exposure; D, quadratic average from many short exposures.
It has been shown that the RMS profile, obtained by averaging the squared modulus of successive Fourier transforms, is identical to that for the diffraction-limited MTF except in the central region where a central peak is added. The long-exposure MTF, obtained as the simple time-average of the short exposure MTF, or equivalently as the Fourier transform of the long-exposure image, consists of the central peak only, the feet being cancelled in the averaging process. Rather than being Fourier transformed, the star image may be autocorrelated. The central peak appearing in the short-exposure case may be shown to be identical to what would be obtained under diffraction-limited conditions. It follows that the speckle pattern has certain similarities with a random array of diffraction-limited images, i.e. Airy peaks in the case of a circular aperture. Under certain conditions of oceanic storms where organized swell is replaced by random waves, sailors have learned to fear the sudden appearance of “monster waves”, much higher than average (ADLARD COLES[19671). Similarly, and because speckled electromagnetic fields are governed by the same Rayleigh statistics as gravity-induced oscillations at liquid surfaces, there is a rare occurrence of exceptionally bright speckles in astronomical images. These are not dangerous, fortunately, and may in fact be exploited for diffraction-limited imaging purposes.
Q 2. Direct Interferometry 2.1. BASIC PRINCIPLES
As discussed in section 1.4, the stellar images produced by large telescopes
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generally feature an envelope inside which a finer interference structure is present under adequate conditions of temporal coherence and of exposure. The angular size of the envelope is on the order of one to several seconds of arc depending on atmospheric conditions, while the scale size i for the smallest interference details is inversely related to the aperture size d according to the relation i = i / d . With existing large telescopes such as the 200-inch telescope, the interference structure is thus 50 times finer than the one-second image. Ordinary observing procedures ignore the interference structure and thereby do not take any advantage of its presence. Their angular resolution is therefore limited to about one or two seconds, although certain exposures have sometimes succeeded in showing 0.3 sec detail on bright objects such as the sun or planets (it has sometimes been attempted to improve the resolution of long-exposure images with deconvolution procedures, but the more or less Gaussian spread function in this case does not lend itself to significant resolution gains with the noise levels encountered in typical astronomical images). Interferometric observing, on the other hand, concentrates more on the fine interference structure than on the envelope. Indeed, this fine structure contains information on object structures having a comparatively fine scale, and which escape detection if one observes only the envelope. This results, according to widely accepted results of the theory of coherence, from the fact that the interference structure is a spread function which is convolved, in intensities, with the function representing the brightness distribution on the object. Indeed, different object points (within the isoplanatic patch) contribute ideptical patterns in the image. These patterns are translated in the image plane relative to each other in accordance with the source geometry. For ordinary sources (i.e. spatially incoherent sources, which have a coherence time shorter than the exposure time) these contributions add in energy, thereby reducing the contrast or visibility of the interference features according to a convolution operation. Unlike the Gaussian spread functions mentioned earlier, the more complicated interference structures lend themselves to successful deconvolutions down to the size of the finest interference detail, i.e. down to the diffraction limit. The spread function is, however, not completely known, being timedependant, and it is thus generally impossible to directly apply deconvolution procedures. Instead, it is possible to use power spectrum or autocorrelation analysis, but these techniques extract only part of the high resolution information. When dealing in particular with the simple and historically important case of a Fizeau-type apertured telescope, on which a mask reduces the
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aperture to a pair of small holes, the interference features consist of Young’s fringes (Fig. 1b). The fringes oscillate in response to atmospheric phase shifts, but remain detectable visually under most circumstances of moderate wind. Their contrast is, however, decreased in case of celestial objects larger than the fringe spacing projected onto the sky. The RMS fringe contrast, measured at different aperture separations, can be used to obtain a “visibility curve” which relates to the object’s structure through a Fourier transform. This is the type of analysis which was used successfully by MICHELSON and PEASE [1921] to obtain the first measurements of stellar diameters with baselines up to 15 meters. The response time of the human eye, on the order of 0.1 second, is sufficiently short under favorable circumstances to effectively freeze the fringe motion for efficient visual work. Optical paths in both beams must, however, be equalized to within a few wavelengths when observing in white light. This proved to be a difficult requirement in systems involving segmented rather than monolithic optics : not only are monolithic telescope mirrors very stiff and accurately figured to provide optical path equality, but they are efficiently supported in flotation cells designed to maintain the exact figure within a few wavelengths at all observing angles. In comparison, segmented optics systems such as used by MICHELSON and PEASE[1921] suffer from considerable flexibility requiring frequent readjustments of path equality. 2.2. VISIBILITY MODULUS DETERMINATION
It is of interest to establish in better detail the exact relationship between fringe contrast and object structure. In this section, light will for most purposes be assumed to be effectively monochromatic. More precisely this means that its coherence time is longer than the differential light propagation delays occurring in the interferometer, but shorter than the exposures. Spectral filtering with a bandwidth of the order of an Angstrom produces effectual monochromatic light in most practical circumstances where optical path differences are less than a millimeter and exposures are longer than a microsecond. Within the framework of Zernike’s coherence theory, the concept of degree of mutual coherence has classically been used to relate fringe visibility measurements and object structure (BORN and WOLF [1970]). As an alternate treatment we will use here an equivalent derivation more directly adapted to multiple and large apertures. If S(x,y,A, t ) is the intensity spread function in the focal plane of the instrument, then the intensity in the image Z(x,y , A, t ) of an incoherent object characterized by the apparent
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intensity distribution O(x,y ) is obtained through the convolution I = S 00. The spread function consists of Young’s fringes in the case of a FIZEAU [1868] or MICHELSON [1920] interferometer, of speckles in the case of a single large aperture, of fringed speckles in the case of two large apertures (Fig. 3), of “honeycombed” speckles with 3 large apertures located according to a triangular geometry, etc.. It has been mentioned in section 2.1 that direct image summing, i.e. of the form J t I ( x , y I, , t)dt, results in a loss of the high-resolution detail. Instead, the fringe or speckle information may be preserved and extracted by summing either the power spectra or the autocorrelation functions of images. This becomes apparent by writing the relations simultaneously in the image space and in the Fourier space:
The summed atmospheric term in eq. ( 3 ) tends towards the well defined limit mentioned in sect. 1.4. In the Fourier space, this limit consists of the diffraction limited MTF with an additional central peak. This term being known, frow experiment or theory, a division gives the modulus of the object function, which corresponds to the visibility curve obtained by Michelson and Pease. This general procedure can be applied regardless of aperture geometry and gives spatial information on the object with diffraction-limited MTF characteristics. It can be used not only with a Fizeau interferometer or the full aperture of a large telescope, but also, in principle, with a coherent array of large telescopes (Fig. X). It is interesting to note that early observers seem to have attributed magic virtues to the Fizeau screen, without realizing that information is actually gained when this aperture plate is removed from the top of the telescope. However, some justification of Fizeauaperturing practice lies in the fact that the Fizeau screen simplifies the image structure so that it can be processed by the eyes and brain of the visual observer. Instead the considerable information content in speckled images from a large aperture exceeds the data processing power of the human eye-brain system.
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A
B
C
Fig. X (additional): Principle of synthetic-aperture telescope : the giant (for example 100meter) parabolic mirror in A may be apertured with a multi-aperture mask as shown in B while retaining the same limiting resolution. B is optically equivalent to the array of mirrors shown in C if component mirrors are accurately adjusted in tilt and axial position to reproduce the B geometry with 4 4 accuracy. This difficult tolerance can be relaxed to a few microns when using filtered light. C is also equivalent to the coudi. arrangements in Figs. 12 and 13.
One major limitation to interferometric observing has to do with extended objects, or more generally with objects consisting of more than a few pixels* the size of the diffraction limit. For these objects, the convolution represented by eq. (1) results in a very faintly contrasted image requiring exceptional signal-to-noise ratio for a valid reduction. 2.3. QUANTUM NOISE
The above theory does not take into account the noise component appearing when few photon-events are recorded in each image. However, this happens to be a serious practical limitation to the accuracy of measurements in most practical cases, since the requirements for short exposures, narrow spectral bands and slow focal ratios (highf-numbers) do result in photon-starved images. The problem is especially relevant when observing faint objects with the new generation of image sensors working in the photon-counting mode (BOKSENBERG [19721). Because amplification is virtually noise-free in these receivers, the discrete nature of photon-events becomes the dominant source of noise. The low level image is seen in the form of few bright scintillations occurring on a dark background, and distributed in time and space according to a compounded Poisson law. When observing fringes at decreasing illumination levels, it eventually becomes impossible to decide whether fringes are present or not. If the
*
“Pixel” is a generally adopted word which means “picture element”
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0
3
fringe pattern were fixed, prolongated integration would solve the problem. The fringes are however moving in our case, and higher order statistical averages must be performed to extract the fringe signal. The analysis procedure described in section 2.2 may still be used, but the validity of eq. (3) under such conditions may be questioned. Classical results pertaining to compounded Poisson distributions show that the equations are still valid at low level if care is taken to remove the only distorsion arising in the summed autocorrelation function, a narrow central peak appearing at low level due to the correlation of each photon-event with itself. Assuming low levels, pure photon noise, and many images, the signal-to-noise ratio in the summed autocorrelation function is found to be
N,N~N~
(4)
where N , is the number of pho.ton-events per speckle, N , is the number of speckles in the image, and Ni is the total number of images used to build up the summed autocorrelation (LABEYRIE [19741). Applied to speckle interferometry observations with a 200-inch aperture, this expression leads one to expect a limiting magnitude in excess of 20. A more detailed noise analysis published by DAINTY[19741 yields similar conclusions. Not surprisingly, the limiting magnitude derived is much fainter than that found by GUSKOVA and KOROLKOV [19731for the case of photoelectric interferometers involving two small apertures. Because size determinations for the faintest cosmological objects are of crucial importance to modern cosmology,appreciable effort is currently made to reach the above sensibility limits of interferometric observing. Applied now to multi-aperture systems such as described in 8 2.4 and 8 4, expression (4) predicts limiting magnitudes in excess of 15, assuming the 11 to 10 A spectral bandwidth imposed for the fainter objects by the more severe temporal coherence requirements for segmented systems.
0 3. Interferometer Designs and Results 3.1. T H E FIZEAU A N D MICHELSON INTERFEROMETERS
The original Fizeau interferometer, used for the first time by Stephan at Marseilles in 1873, involved only the aperture screen as special equipment on the telescope. A strong eyepiece was used to observe the fringes. Telescope optics are usually sufficiently good to meet the one-micron or so tolerance on path equality which allows fringe observations in white light.
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Problems are created by the prismatic dispersion of the atmosphere when observing stars at low elevation. Michelson, Pease and Anderson first used a Fizeau interferometer at the Mt Wilson 100-inch reflector (1920-1933). In order to avoid difficulties with a 100-inch mask, they installed a small mask some distance above the focal plane, in the converging beam. It could be rotated in position angle, and atmospheric dispersion was corrected with a glass plate that could be tilted. This system failed to resolve any stellar disk but was successful in resolving the conspicuous spectroscopic binary Capella, spaced by approximately 0.05”. The fringes were found to disappear for certain baseline orientations, allowing precise measurement of the separation and position angle between the component stars. This was the first instance, and is still one of the very few cases, where stellar masses could be determined directly. Several variants of this interferometer have continued to be used by double s t q observers (FINSEN [1964]), but their contribution to double star observing in general has been of somewhat secondary importance in comparison with conventional visual work. In order to increase the possible baseline span beyond the maximum size of available telescopes, Michelson equiped the 100-inch reflector with a 20-foot beam. The beam carried four flat mirrors designed to reflect the light beams in perioscopic fashion. These could be positioned, with about 50 microns accuracy, for optical path equality. Further adjustment for
Fig. 5. Diagram of the 50-foot Michelson interferometer built by Hale and Pease at MtWilson.
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fringe acquisition in white light was achieved by observing the star image through a direct-view prism. Reference fringes of adjustable contrast were provided by an auxiliary Fizeau interferometer for visual contrast measurements, and corrections had to be made for a systematic effect which decreased the apparent fringe contrast at long baseline settings. In spite of appreciable operating difficulties, Michelson and Pease succeeded in resolving and measuring nine stars. These angular measurements confirmed the enormous linear dimensions of objects such as Betelgeuse in comparison to our sun, and gave the foundations for a scale of stellar temperatures. A second interferometer having a 50-feet beam was later constructed by Hale and Pease at Mt Wilson (Fig. 5). Similar in its principle to the 20-foot system, the larger interferometer failed to produce many additional results for reasons which are not completely clear but seem related to operational difficulties greater than were expected. During a recent visit to Mt Wilson, I found the interferometer well preserved in its building. In spite of some very excellent design features, it appears that the 50-foot cantilever beam may have suffered from poor stability about its symmetry axis. The instrument could be revived at moderate cost, and modern electronics for guiding and fringe sensing could certainly make its operation much easier than in Pease’s days. It is not clear, however, whether the modernized interferometer could compete with recent instruments involving two independently mounted telescopes. 3.2. PHOTOELECTRIC FIZEAU INTERFEROMETERS
Photomultiplier tubes are hardly more sensitive than the human eye for short exposure work at medium illumination levels, but they have faster response and better photometric accuracy. Before the newer generation of television sensors appeared, these advantages led several groups to develop photoelectric fringe sensors replacing the human eye in Fizeau-type systems. ELLIOTT and GLASS[1970] have used a picket-fence mask to generate a photoelectric signal from Young’s fringes, but most workers in the field have preferred beam-splitter arrangements to obtain a flat interference field (Fig. 6), following Michelson who had already considered using beam-splitters for variants of the basic Fizeau arrangement. He apparently found no advantage in doing so for visual work, but the single-pixel nature of photomultiplier tubes obviously makes it easier to work on flat interference fwlds than on Young’s fringes. This advantage of beam-splitter arrangements no longer holds with the new generation of multi-pixel sensors. A wide variety of configurations are possible with beam-splitter arrange-
INTERFEROMETER D E S I G N S A N D RESULTS
61
-14-1-1-
(C)
Fig. 6 . Types of single-pixel photoelectric fringe sensors: a, Kosters prism (K) arrangement used by Currie with two polarizers (P), the waves made to interfere have opposite orientations in the figure plane; b, system used by Cagnet, in which waves are identically oriented and polarizers unnecessary; c, “standing wave phototube” used by the author, in which the thin S11 film probes a standing-wave pattern produced by the two beams; d, picket-fence mask used by Elliott and Glass in a Young’s fringe pattern.
ments. These fall in several classes depending on their symmetry properties : 1. lateral shear; 2. radial shear; 3. axisymmetricflip; 4.centrosymmetric flip. Lateral shear applied to the aperture of a large telescope produces an interference field in which the interferometric baseline is everywhere the same. A single-pixel sensor could suffice to probe the entire wave in the absence of seeing. In the axisymmetric case no interference occurs in natural light since interference maxima for one of the polarization vectors correspond to minima for the other polarization. This situation arises in all single-pass
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3
beam splitter arrangements where the two incident beams are subjected to equal numbers of reflexions before meeting the beam-splitter. The phenomenon has been responsible for appreciable frustration in some attempts at operating stellar interferometers. Once the effect is understood however, fringes may be retrieved easily by using an additional mirror or polarizing element. In the centrosymmetric case, a two-dimensional display of the object’s visibility function may be obtained directly if the waves are tilted so as to produce narrow fringes : the fringe contrast varies locally in proportion to the local visibility value. The shadow pattern on the wave however destroys somewhat this display. Devices belonging to the first class have been used on telescopes by KULAGIN[1970], CAGNET [1973] and C. RODDIER[1971]. The fringe signal in the Cagnet system is obtained as the difference between the outputs of two photomultipliers located on each side of the beam splitter (Fig. 6b). The device was operated in the Fizeau mode at the Haute Provence 193 cm reflector. C. RODDIER [19711 worked with a different system which in fact uses the full aperture of the telescope ; interference thus tends to vanish when using large apertures with a single-pixel sensor. CURRIE,KNAPPand LIEWER[1974] have developped an axisymetry interferometer (Fig. 6a) which they mounted on the 100-inch and 200-inch telescopes of the Hale observatories. They used a differential detection scheme similar to that of Cagnet, but working in the photon-counting mode with an on-line digital processor. With a pair of 2-cm apertures and 10 Angstroms spectral bandwidth, they were able to measure visibility curves on Betelgeuse and alpha Hercules. In this system, polarizers are used to avoid the above mentioned polarization incoherence problem associated with axisymmetry devices. The system appears to work at rather low counting rates, and this leads to expect considerable sensitivity for future large-aperture, multi-pixel, devices working in the photon-counting mode. Centrosymmetry systems have been operated in the laboratory by different authors. BEAVERS [1963] has experimented with several forms of photoelectric fringe sensing on the reduced-scale version of the Michelson 20-feet interferometer which he has built. Yet another type of single-pixel sensor was used by the author concurrently with a Cagnet-type device in initial attempts with the two-telescope interferometer at Meudon. As shown in Fig. 6c, the device is based upon a special photomultiplier tube (built to the author’s specifications by the collaborators of Prof. Lallemand at Observatoire de Paris). The S11 photocathode may be illuminated from both sides with plane waves. If these are coherent, they interfere in the form of a standing-wave pattern
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which is probed by the semi-transparent and very thin S11 film. For flat interference, the photocathode plane has to be parallel to the nodal planes of the standing waves much as is the case with the beam-splitter films used by CAGNET [1973] and CURRIE,KNAPPand LIEWER[1974]. If the phase is made to oscillate with 180” amplitude by an auxiliary mirror drive, the phototube signal may be submitted to lock-in detection, thus providing with a single phototube the equivalence of the two-photomultiplier systems mentioned above. Used concurrently with the Cagnet-type sensor, the standing-wave sensor was generally preferred. However, both sensors were discarded after a photon-counting television sensor was built and its superiority was recognized (sect. 3.3). 3.3. THE SPECKLE INTERFEROMETER
The principle of speckle interferometry follows immediately from the discussion of image speckles (sects. 1-3 and 2-2). As proposed by Labeyrie in 1970, the method consists in recording speckled images at the focus of a large telescope. The images are then analyzed statistically according to eq. (3), to obtain diffraction-limited information in the form of a twodimensional “visibility function” formally similar to Michelson’s visibility curve. Although the very existence of speckles appears to have long remained ignored by many experienced astronomers, including perhaps Michelson himself, the principle had already been used on a some,qhat intuitive basis by double-star observers working visually on large refractors. With adequate training, and because of the moderate aperture size, the brain of these observers could perform the second order statistical analysis required, and detect stellar companions spaced by 0.1’’ under 0.5” seeing conditions. They could notice the double character of the granules or “condensations” which moved inside the image of close binary stars. It has however been impossible to record the phenomenon until receivers more sensitive than photographic plates appeared. Using a Lallemand-Duchesne electronographic tube, ROSCH,WLERICK and BOUSSUG~~ [1961] were first to succeed, and started a double-star program with the one-meter telescope at Pic du Midi. The speckle interferometer which I use at the prime focus of the 200-inch [1974]), telescope is represented in Fig. 7. As described previously (LABEYRIE it includes essentially a magnifying lens and a field-grating arrangement serving as a tunable filter, and also to correct atmospheric dispersion. Different types of sensors have been used successively: 1. photographic film with an image intensifier; 2. a standard television camera equipped
70
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I
\
Irn
F300
&
Fig. 7. Speckle interferometer used at the prime focus of the 200-inch telescope M, fieldfinder mirror, with 0.1 mm hole; L, magnifying lens; G, concave field grating (Jobin-Yvon holographic type) in plane of magnified image;U, spherical mirror; S , spectral mask; TV, television camera ; R, auxiliary spectrum-viewing lens ;C, geometry-calibration grid. The optics provides magnification and tunable filtering (color and bandwidth are separately adjustable). Atmospheric dispersion is corrected by translating TV axially and rotating the complete system about the telescope axis.
with a SIT tube; 3. a photon-counting television camera. Because standard television image frequencies are close to the optimum values, determined by atmospheric frequencies, television-type sensors are particularly efficient for this application. Initially, optical analog techniques were used to reduce speckle interferometry data. This was required on account of the high information content in the numerous two-dimensional images which had to be processed. The images recorded on film, or transferred from video tape to film, were Fourier transformed in a laser processor, and their power spectra were summed by multiple-exposing a photographic plate. 200 to 6000 exposures were typically used in the summation for residual noise on the order of a few per cent. Fringes in the summed pattern were interpreted as evidence of stellar duplicity, while attenuation of the outer edges indicated a resolved stellar disk (Fig. 8). The power spectrum of the object could, in principle, be obtained by dividing the summed power spectra obtained respectively for the object and an unresolved reference star. In practice, the difficulty of insuring the required sensitometric gamma resulted in unavoidable photometric distorsions in the processed data. Nevertheless, the sensitivity and two-dimensional character of the method permitted a number of findings: 200 objects were observed as of December 1974, 90 of which in the course of only two nights. Twelve stars were found to be binaries (LABEYRIE, BONNEAU, STACHNIK and GEZARI [1974]), allowing stellar mass determinations in five cases. Most of the supergiant stars resolved by Michelson and Pease were also resolved in spite of slightly inferior resolution with the 200-inch telescope. Two of these (Betelgeuse and Mira Ceti) were found to feature a markedly limbdarkened profile, the width of which increases from red to blue wavelengths
Fig. 8. Stellar structure evidenced from time-averaged power spectra. Speckled images on top with corresponding power spectra below. From left to right : Betelgeuse (resolved disk), Capella (resolved binary), unresolved reference star. The power spectra presented here are relatively noisy due 2 to the small number of frames used in the average.
N -4
Fig. 9. Typical integrated power spectra of 200-inch images. obtained optically, showing resolution of six sellar disks and two binaries. Object-reference pairs are indicated by a bar. The alteration in the case of p Canis Majoris is believed to result from aberrations resulting from flexure of the 200-inch mirror for certain orientations. A mirror mask suppresses the bright central peak.
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(BONNEAUand LABEYRIE [1973]). All the resolved disks were found to be revolution symmetrical, with the possible exception of Mira (Fig. 9). It is currently attempted to improve the image analysis procedures. High-speed digital correlators have been used in real time, but they provided only one dimension. A two-dimensional technique applicable to digital video signals at extremely low pulse rates has been worked out for work on faint objects or attenuated bright ones. As discussed in sect. 2.3, it appears indeed that interferometric information may be recovered at extremely low counting rates, down to 2 photon-events per image, or even less. Part of the information in high-level analog images must be sacrificed due to rate limitations in digital processing techniques. It appears that beam attenuation, for the brighter objects, is the optimum way of sacrificing information if it allows digital reduction in the photon-counting mode. The expression (4) shows that minutes of observation suffice to provide adequate signal-to-noise ratio at counting rates on the order of 100 per image, which are compatible with on-line digital processing. A software and a hardware autocorrelation system using a special algorithm are currently developped along these lines at Meudon. R. Lynds and his collaborators (1973) at Kitt Peak National Observatory have developped a digital two-dimensional correlator which they use to reduce their analog speckle interferometry data. Steps have been taken in a different direction by STACHNIK and NISENSON [1973] : they use electro-optic crystals as transducers for optical reduction of speckle data in real time. 3.4. INTERFEROMETRY WITH TWO TELESCOPES
Because self-supporting structures such as the 50-foot interferometer at Mt Wilson cannot be extrapolated for baselines on the order of 100 meters, it has been suggested by MILLER [I9661 and others to use separately mounted collectors for long-baseline work. Miller has studied configurations involving a pair of heliostats and a central station equipped with optical delay lines. Following this general philosophy, I have constructed an interferometer utilizing two telexcopes. Installed at Nice, the system recently produced fringes on Vega (LABEYRIE [1975]). The instrument was intended to test design concepts suitable for future extrapolation toward long baselines, large component apertures, and progressive growth into an array including perhaps 40 telescopes. It consists of two 25-cm telescopes located on each side of a laboratory building, along a 12 meter baseline oriented in the
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03
North-South direction. The telescopes have a Cassegrain-Coudt configuration, both caude beams being received in the laboratory building at focal ratio f: 3000, and recombined on an optical table as shown in Figs. 10, 11 and 12. The table is mobile on tracks to compensate the optical path variations occurring as the star follows its diurnal motion. It carries a twin .autoguider system and a roof-mirror arrangement for recombining the two images in order to produce Young’s fringes similar to those observed with his instrument by Michelson. The telescope mounts are of a special altitude-altitude design providing adequate stiffness with a minimum number of coude flats. Considerable care has been taken to avoid possible mount vibrations. The synthetic image, or a fringed spectrum, are observed either visually or with the photon-counting television camera. The camera is interfaced to a PDP8 minicomputer through a preprocessor. The fringes observed repeatedly on Vega were found to be of very good contrast, suggesting that the subjacent limestone soil at Nice provides adequate stability. Also, it appears that the narrow coudk beams are essentially insensitive to turbulence
Fig. 10. The two-telescope interferometer at Nice. Narrow coudC beams from both telescopes are received in the central building, where they recombine to produce Young’s fringes in the synthetic image. The telescopes have special alt-alt mounts built from heavy-gage materials for dimensional stability. Tracks are currently being designed for a variable baseline.
INTERFEROMETER D E S I G N S A N D RESULTS
Fig. 11. One of the two tclescopes operated as a Michelson interferometer. The coudii beam exits through the hearing visible in front. Also visible are the massive secondary spider, yoke and concrete support.
in the horizontal path, implying that no piping should be required with this configuration at long baseline settings. In contrast with some of the beam-splitter arrangements used to recombine beams, the simple optical configuration can be adapted for work with N telescopes. This requires replacing the roof-mirror by a pyramid mirror. Optical delay lines such as proposed by Miller may prove necessary
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[n. 0 4
,=I Fig. 12. Optical layout of the Meudon/Nice two-telescope interferometer: Tn, TS- Northland South telescope; M- primary mirror cf= 850 mm); m- Cassegrain secondary cf= 7.5 mm); F- coude flat; L-field lens; rm-roof mirror in pupil plane; D-dichroic mirror; TV1-guiding camera; bl-bilens serving to separate the North and South guiding fields; S and P- slit and direct view prism used for fringe acquisition; TV2- photon counting camera (tunable filter or disperser not represented); Tr- tracks on which table moves (programming mechanism not represented).
in this case. Making accurate contrast measurements on inherently variable fringes has been a standing problem since Michelson and Pease. Systematic errors may be caused by incomplete temporal coherence, atmospheric dispersion, polarization effects, differential field rotation, vibrations, insufficiently short exposures, and guiding errors. The problem of avoiding all these effects has not yet been solved at Nice, but it appears that the use of larger component apertures with digital reduction methods in the photon-counting mode should attain a level of accuracy sufficient for many astrophysical problems. Steps are currently being taken to replace the small telescopes by 60-inch ones. Also, railway tracks are being installed for variable and progressively longer baselines.
0 4. The Image Reconstruction Problem 4.1. THE VISIBILITY PHASE PROBLEM WITH DIRECT INTERFEROMETRY
The equations in sect. 2.2 show how the autocorrelation function of objects, or equivalently their power spectrum, may be obtained. This is generally insufficient for reconstructing images. The information missing is the relative phase of the varied spatial frequency components, i.e., the exact location of the component fringe patterns on the sky plane. In the
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71
Fizeau interferometer for example, the phase cannot be determined since the fringes observed for any baseline value oscillate in such a way that no average position exists. Due to turbulence and tracking instabilities, it is not practicable to pin-down some average fringe position on the sky plane. Thus, fringe positions on the sky plane at different baseline settings cannot be compared. A similar problem is classically encountered in the interpretation of X-ray diffraction data for mapping crystal structures. The X-ray spot diagrams do not contain phase information, and considerable effort has been invested in trying to obtain indirectly this information in order to deduce the electron density distribution in crystals. The absence of a well defined average fringe position may be explained by the random walk character of the operation which consists in adding fringed images. This addition has the form c i [ l + sin(2nx/s + q,)] which is equal to N+xisin(2nx/s+cp,). The latter term may be represented by a sum of randomly oriented vectors in the complex plane, and the resulting phase keeps varying wildly as N increases. MCGLAMERY [19671 and others have proposed to average directly the phases cpi . This might be feasible in the absence of amplitude variations. However, these variations cut the fringe contrast repeatedly during the integration period, thus creating 360" ambiguities which are likely to affect the average result. Although the visibility phase information is generally necessary to reconstruct images, there are a number of special object geometry cases where the phase is not needed if a certain a priori knowledge of the object exists. Such cases include : 1. centrosymmetrical objects, for which the Fourier spectrum is purely real; 2. objects with a reference star in the immediate vicinity. The application of speckle interferometry in the latter case has been discussed and explored through laboratory experiments by BATES, GOUGH and NAPIER [1973]. When a reference star is present at a suitable distance, the autocorrelation function indeed contains a pair of sideterms consisting of cross-correlations between the object and the reference star. The reference star being a delta function, these side terms turn-out to be reconstructed images of the object. Chosing which side term corresponds to the actual object orientation is normally impossible, unless one uses the image envelope, and this introduces a 180" ambiguity in the orientation of the object. LIUand LOHMANN [19731 have discussed the case of speckle interferometry on objects which contain a dark background interrupted by relatively small islands, one of these at least being a point star. They show that a high-resolution image may be reconstructed by utilizing image envelopes to eliminate unwanted cross-correlation terms. Stellar configurations
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04
suitable for imagery by the last two methods are not too unfrequent due to the high occurrence of multiple star systems. With the new generation of very sensitive receivers, many sources should become amenable to highresolution imaging by these methods. We shall see in section 3.3 that these observations are potentially more sensitive than those with pre-detection compensation systems. One more possible method, which has not been very much investigated yet, involves the exceptionally bright speckles which are likely to appear sometimes in the image according to Rayleigh statistics. These superspeckles may be considered as diffraction-limited images of the source. 4.2. THE TRIPLE INTERFEROMETER
Rogstadt proposed to use for visibility phase measurements in the optical region the 3-antenna method worked out by JENNISON [1967] at radio wavelengths. The concept was further developped by GOODMAN and RHODES[1973] with the help of laboratory and computer experiments. The principle of the triple interferometer may be explained in the context of a Fizeau interferometer with 3 apertures instead of the conventional two. Holes are obturated in sequence according to a circular permutation scheme in such a way that only two holes are used simultaneously. The permutation time is shorter than atmospheric lifetimes. Three possible fringe systems are recorded in sequence, with a cycle time shorter than the atmospheric lifetime. With a point source, whenever atmospheric phase shifts are so arranged that fringe systems 1 and 2 coincide, then every second maximum of 3 should also coincide with the maximas of 1 and 2. Lack of coincidence implies that the source is not a point source, and gives the relative phases of its Fourier components at spatial frequencies f and 2f. Due to lack of experience in real astronomical situations, it is unclear yet how this method will compare with the seeing compensation and superspeckle selection methods. 4.3. THE SEEING COMPENSATION APPROACH
One of the most attractive approaches to diffraction-limited imaging through the atmosphere is that suggested by BABCOCK [1953]. The idea is to remove atmospheric phase fluctuations by means of active phasing devices. The servo loop originally proposed by Babcock involved a rotating knife-edge to map wave defects, with a television camera and an Eidophor optical transducer for applying phase corrections on the wave. The state
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CONSTRUCTION OF SYNTHETIC-APERTURE ARRAY
79
of the art has long been incompatible with such attempts, but the idea has recently been revived by different groups at Berkeley (Lawrence Radiation Laboratory), Itek Corp. and Hughes Labs.. MULLERand BUFFINGTON 119741 at Berkeley, in collaboration with DYSON[1974] used computer simulations to study the performance of arrays having a few dozen aperture elements with individually controllable phases. A parameter suitable for deriving an error signal was found to be the intensity at one preselected image pixel: the first aperture is phased to maximize intensity, than the second, etc., and a few cycles of adjustments performed within the seeing lifetime suffice to increase dramatically the image sharpness, even though no attempt is made to suppress the shadow pattern on the aperture. A different system, resembling more the original Babcock device has been demonstrated in the laboratory by HARDY,FEINLrEB and WYANT [1974] at Itek. In this seeing compensator, a wavefront-shearing interferometer serves to measure the phase distribution on the wave. A simple analog computer derives correction signals, and these are supplied to a deformable piezoelectric mirror. The spectacular image improvement already obtained with this other simple system in its present stage of development suggests a bright future for the seeing compensation approach. A rather modest limiting magnitude, of the order of 10 to 13, has been predicted for devices of that kind. However, for bright enough objects, not only imaging cameras but also spectrographic equipment should benefit from reconcentrating the energy which is scattered by the atmosphere.
0 5. Construction of a Synthetic-Aperture Array of Optical Telescopes In the past few years, interest has arised in building optical telescopes much larger than those in existence. As discussed by CODE119731, it has generally been felt that the conventional monolithic approach is not suitable for building such giant instruments. The construction costs with the conventional approach indeed appear to increase faster than collecting area. Instead, it has been seriously envisaged to build arrays of optical telescopes resembling those used in radio astronomy. ODGERS and RICHARDSON 119723, for example, have proposed an incoherent array consisting of fourty 1.5 meter telescopes. In their project, a synthetic image is produced at a common focus where the coudC beams meet. The usable field is small but the array is primarily intended for feeding light into a spectrograph. The advantages and limitations of telescope arrays have been discussed by CODE119731, primarily from the incoherent synthesis point of view. Code pointed out a unique advantage of array over monolithic telescopes : they
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can grow, i.e., it is possible to start with a relatively modest system and progressively expand it once satisfactory operation is demonstrated. However, the greatest potential interest of the array approach lies perhaps in the coherent synthesis applications. Interferometric baselines on the order of 100 meters can indeed be envisaged with an array since system cost depends mostly on the total collecting area and not on the telescope spacing. C6mpOnent telescopes for an array do not need to involve unduely novel techniques, except perhaps for the coudC beam arrangements, which should require as few reflexions as possible. Telescopes should preferably be movable for a flexible baseline geometry, unless their individual size is excessive. Optical path equality can be achieved by means of tunable delay lines, as discussed by MILLER[1971]. Fringe detection in the image may be achieved by speckle interferometry or using any of the numerous possible beam-splitter arrangements. Figure 3 shows, under conditions of laboratory simulation, the possible appearance of images produced by a two-telescope arrays and a 6-telescope array, component apertures being on the order of 1.5 meter. More complicated interference structures would be observed within the speckles in the case of more than 6 telescopes, but the image analysis procedures used with a single telescope remain valid. Fig. 13 shows the general layout of an array proposed by the author. The array will have variable baseline geometries. Altitude-altitude or spherical mounts are envisaged for the telescopes. These are 1.5 meter Cassegrainians with a small interchangeable secondary and single-flat
Fig. 13. Proposed synthetic-aperture array of telescopes. Narrow coudB beams propagate from each telescope into the central station, where they recombine. The telescope mounts represented consist of ferro-cement spheres tracked on fluid pads. They are expected to provide better dimensional stability than conventional coudi. mounts. Sphere surfaces are precision ground. Conventional 1.5 meter (60-inch) telescope optics are mounted inside the spheres.
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coudt focus. Because the mounts are not equatorial, photoelectric or computer-controlled guiding is needed. Also, the coudt flat must be rotated along the alt-2 axis during observing. This is automatically achieved by a gear mechanism, and satisfactory operation has been achieved with the prototype telescopes installed at Nice. The array design is such that telescopes may be utilized : 1. individually, for conventional observing with each telescope separately ; 2. collectively in the incoherent mode, as proposed by ODGERSand RICHARDSON [1972] for their array; 3. collectively in the coherent mode. The Meudon/Nice interferometer is a prototype intended for evaluating the concept. Its operation has been demonstrated and found satisfactory using photoelectric guiding to maintain the superposition of the two star images. The mechanical design of telescope mounts must be unusually stiff in order to
Fig. 14. Cross-section of concrete spherical mount studied for the array of 1.5 telescopes. Reinforced concrete has good vibration damping characteristics, short-term stability and low cost. The outer sphere surface is ground smooth and supported on rollers, water bearings, or piezoelectric pads.
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[IL 0 6
avoid vibrations at the 0.1 micron scale. This implies particularly thick spider arms and a very rigid yoke or sphere. In the prototype system, it was found necessary to sacrifice partially the possibility of operating at low elevation angles in order to meet these requirements. This is no great loss for coherent work, since atmospheric turbulence and dispersion increase sharply at large zenith angles. An attractive alternate design involves spherical mounts. For low-cost production and excellent vibration-damping characteristics, these mounts could consist of a shell made of reinforced concrete or ferro-cement and supported on three rollers or fluid pads, as shown in Fig. 14. This design appears to have attractive advantages, and the unusual tracking problems appear to be quite solvable with the help of modern minicomputers. Among the advantages are : 1. structural simplicity; 2. no dome is needed since the sphere includes self-enclosed laboratory space; 3.’ the spherical mount may be tracked either in the alt-alt, equatorial, or even alt-az modes. A project along these lines is currently being worked out at Meudon. A 3.5 meter ferro-cement sphere has been constructed and will shortly be equipped with a 1.5 meter mirror to develop the spherical tracking technology. A second telescope will then be built, and institutions from different countries will later be invited to contribute additional telescopes for progressive array growth. The system’s luminosity will surpass that of Mt Palomar’s 200-inch instrument if the array grows to include eleven 1.5 meter telescopes or six 2-meter telescopes. Because it is still difficult to predict the maximum baseline dimensions that will be usable, some flat expanse of terrain at least one square kilometer should be selected, in a region having low nebulosity and jet-stream activity in addition to other desirable astronomical characteristics.
0
6. Intensity Interferometry
In order to overcome the problems which stellar interferometry had to face in the years 1950, Hanbury Brown and Twiss proposed the novel method which they called “intensity interferometry”. The principle may be presented in the following elementary fashion, readers being referred to Hanbury Brown’s articles for more details. Neglecting atmospheric effects, which have no influence on this method, the illumination produced on the ground by a stellar source is not uniform if mapped during a period shorter than the coherence time z = l/f of the beam ( f being the frequency bandwidth). The non-uniformity results from interference of light emitted by different parts of the source. Indeed, the
11,
§ 61
INTENSITY INTERFEROMETRY
83
electric field behaves as if the source were spatially coherent during such a short interval. The phase is however not uniform on the source, so that the emitter may be compared to a piece of diffusing glass illuminated by a laser beam. If the diffusing glass is further assumed to fluctuate randomly with lifetime z, an accurate model of the beam’s spatio-temporal structure is obtained. The theory of speckle phenomena mentioned in section 1.3 applies to this case and shows that a distant screen, the terrestrial ground in the present case, is illuminated with a speckle pattern fluctuating with the time-constant z. The size of the speckles d is related to the angular size M of the source by the usual relation d = A/M. This is equivalent to saying that the whole spatio-temporal structure of an incoherent beam is speckled: temporal as well as spatial speckles are present. The beam structure may be described as a flow of random cells or speckles (also known as field modes or coherence cells) propagating in space with velocity c while deforming themselves. Fast detectors located on a transverse surface (the terrestrial ground) see simultaneous intensity fluctuations if they are spaced by less then the transverse dimension of speckles. They see uncorrelated fluctuations in the opposite case. The cell size may thus be determined by comparing the signals from two detectors having a variable spacing. This is the technique used by HANBURY BROWN,DAVIS and ALLEN[1974] at Narrabri observatory. A pair of 6.5-meter light collectors mobile on a 300-meters diameter circular track are each equipped with a fast photomultiplier and narrow-band filter. The two photoelectric signals are multiplied, and the result is integrated for several hours or days until adequate signal-to-noise ratio is obtained. Because of the limited electrical bandwidth in the photomultipliers and amplifier circuits, the optical frequency bandwidth effectively utilized is extremely narrow, on the order of Angstroms. This implies a comfortable tolerance on the dimensional stability of mechanical structures, but also a very inefficient use of the incident energy. For this last reason, the method has been applicable only to the very brightest blue stars up to magnitude 2.5. It nevertheless permitted remarkably accurate measurements on 32 stars with the unrivaled resolution of l o p 3 arc-second. Following the recent results with two telescopes operated as a Michelson interferometer (sect. 3.4), J. Davis and myself have discussed the potential applicability of both methods for work at very long baselines, on the order of 2 kilometers. Whereas baselines as long as one kilometer may be usable with direct interferometry, it seems that propagation of an electrical signal is easier over long distances than the undisturbed coherent propagation of
84
HIGH-RESOLUTION TECHNIQUES
[II,
§ 8
an optical beam. For 2 kilometers baseline, the optical beam produced by a 1.5 meter (60 inch) telescope may be propagated in 10 cm piping (no periscope-type relay lenses being required). Vacuum is also unnecessary, especially if the pipe structure includes thermal insulation. In spite of their remarkably low loss characteristics, state-of-the-art fiber-optics light guides cannot be used since they distroy the temporal coherence. Single-mode fibers may however become available before interferometric baselines grow enough to require their use. It is thus difficult to predict the result of the competition between direct and intensity interferometry in the coming years, although short baselines appear to favor direct interferometry at this time.
0 7. Heterodyne Interferometry Heterodyning techniques have been successfully employed at radio wavelengths for the very-long-baseline observations involving antennas located several thousand kilometers apart. The method consists in beating light from the star with that from a local oscillator, on a suitable sensor. Simultaneous work at two stations using the same oscillator frequency, provides interference information. Heterodyne interferometers for work at 10.6 microns are currently developped by TOWNES and his collaborators [1974] at Berkeley, as well as by GAYand JOURNET [1973] at Observatoire de Paris. At the University of Utrecht, VANDE STADT[1973] and Nieuwenhuyzen build a system intended for work at 3.4 microns. Like intensity interferometry, and for the same reason, the heterodyne approach restricts considerably the spectral band used. Its potential usefulness is generally considered as marginal in the ultra-violet, visible and near infra-red regions where photoemissive sensors are available. In the 2 to 10 micron infra-red range, heterodyning becomes more efficient, but even there it is not yet clear how it will compete with direct interferometry. Work on direct interferometry at 10 microns is carried out at Berkeley by D. Cudaback and J. Franck.
0
8. Conclusions
Present trends in the fields concerned with high-resolution observation at optical wavelengths indicate the likelihood of major improvements in the coming decade. The directions which appear to hold most promise are: 1. synthetic-aperture arrays of telescopes. It is unclear yet how far it will be possible to push this technique which is still in its infancy, but orders
111
REFERENCES
85
of magnitudes will probably be gained in the resolution and luminosity of interferometric work; 2. imaging techniques related to the “rubber telescope” concept may succeed in providing improved images of the bright stars and fainter sources in their vicinity. The corresponding devices may become useful accessories to spectrographic equipment and also to telescopes participating in array work. Satellite-based telescopes such as NASA’s proposed Large Space Telescope also provide remarkable new possibilities. Concerning synthetic aperture arrays, however, it is likely that ground-based systems in the 100-metersrange of size will be operational before equivalent space systems. The experience gathered with them should help designing space systems for completely diffraction-limited performance. Such systems will open a new era in optical astronomy.
References Symposia
“Restoration of atmospherically degraded images”, 1966, Woods Hole Summer Study, Nat. Acad. of Sciences, Nat. Res. Council. “Synthetic aperture Optics”, 1967, Woods Hole Summer Study, Nat. Acad. of Sciences, Nat. Res. Council. “Synthetic Aperture Optics”, 1970, ed. M. W. Stockton, Optical Sciences Center Report 58, Tucson. ADLARDCOLES,K., 1967, in: Heavy weather sailing. ANDERSON, J. A,, 1920, Ap. J. 51, 263-275. BABCOCK, H. W., 1953, Pub. Astr. SOC.Pac. 65, 229. BATES,R. H. T., P. T. GOUGHand P. J. NAPIER, 1973, Astr. and Ap. 22, 319. BEAVERS, W., 1963, Astrol. J. 68, 273. BLUM,E. and M. CAGNET,1961, Comptes Rendus, p. 253-265. BOKSENBERG, A,, 1972, in : Auxiliary Instrumentation for Large Telescopes, ESOjCERN conference, Geneva, 1972. BONNEAU, D. and A. LABEYRIE, 1973, Ap. J. Letters, 181, 1. BORN,M. and E. WOLF,1970, PFinciples of Optics (Pergamon press, 4th ed.). BOWEN,I. S., 1964, Astron. J. 69, 816. W., private communication. BOYER, J. B. and J. W. HARVEY, Kitt Peak Nat. Obs. report. BRECKINRIDGE, CAGNET. M., 1973, Optics Comm. 8, 4. CODE,A. D., 1973, in: Ann. Rev. Astron. Ap., p. 239. D. G., S. L. KNAPPand K. M. LIEWER,1974, Ap. J. 187, 131. CURRIE, DAINTY, J. C., 1973, Optics Comm. 7, 129. DAINTY, J. C., 1974, Mon. Not. R. Astr. SOC.169, 63434. DAINTY,J. C., 1976, in: Laser Speckle and Related Phenomena, ed. J. C. Dainty (Springer Verlag, Berlin) in print. DYSON,F. J., 1974, in preparation. ELLIOT,J. L. and I. S. GLASS,1970, Astron. J. 75, 10. FINSEN,W., 1964, Astron. J. 69, 319-324. FIZEAU, H., 1868, Comptes Rendus, p. 66-934.
86
HIGH-RESOLUTION TECHNIQUES
FRIED,D. L., 1966, J.O.S.A. 56,1372. GAVIOLA, E., 1949, Astron. J. 1178, 155. GAY,J. and A. JOURNET, 1973, Nature 241, 32. GEZARI,D., A. LABEYRIE and R. V. STACHNIK, 1972, Ap. J. Letters 173, L1-L5. GOODMAN, J. W., 1965, Proc. IEEE 53, 1688. GOODMAN. J. W. and RHODES, 1973, private communication. 0. and D. V. KOROLKOV, 1973, Izd-vo “Naouka”, 5,112-118, GUSKOVA, HANBURY BROWN,R. and R. Q. Twlss, 1958, Proc. Roy. SOC.A 248, 199-221. HANBURY BROWN,R., 1968, in: Ann. Rev. Astron. and Ap. 6, 13. HANBURY BROWN,R., J. DAVISand L. R. ALLEN,1974, M.N.R.A.S. 167, 121-136. HARDY, J. W., J. FEINLIEB and J. C. WYANT, 1974, in: Optical propagation through turbulence, OSA meeting, July 1974, Boulder. 1964, J.O.S.A. 54, 52. HUFNAGEL, R. E. and N. R. STANLEY, , JENNISON, R. C., 1967, Introduction to Radioastronomy, Philosophical Library, New York, p. 125-129. KNOX,K. T. and B. J. THOMSON, Ap. J. Letters 182, L133. KOLMOGOROFF, A. N., 1941, D.A.N. SSSR 30 (4), 229. KORFF,D., G. DRYDEN and M. G. MILLER,1972, Optics Comm. 5, 187. KOZLOV, Yu. G., 1968, Opt. and Spectroscopy 25, 424425. KULAGIN, E. S., 1970, Soviet Phys.-Astronomy 13, 6. LABEYRIE, A,, 1970, Astron. and Ap. 6, 85-87. LABEYRIE, A,, 1974, Nouv. Rev. d’Optique 5, 3. LABEYRIE, A,, D. BONNEAU, R. V. STACHNIK and D. G. GEZARI,1974, Ap. J. 194, L147-LI51. LABEYRIE, A., 1975, Japan J. Appl. Phys., to appear. LABEYRIE, A,, 1975, Ap. J. 196, L71-L75. LAWRENCE, R. S. and J. W. STROHBEHN, 1970, Proc. IEEE 58, 1523. LEE,R. W. and J. C. HARP,1969, Proc. IEEE 57, 375. LEIGHTON, R. B., 1966, Scientific American 194, 157. LIU, Y. C. and A. W. LOHMANN, 1973, Optics Comm. 8 , 4 . LUMLEY, J. L. and H. A. PANOVSKY, 1964, The Structure of Atmospheric turbulence (Interscience New York). LYNDS,R., 1973, private communication. MARTIN,F., J. BORGNINO and F. RODDIER, 1975, Nouv. Rev. d’Optique 6, 1. MCGLAMERY, B. J., 1967, J.O.S.A. 57, 293. MICHELSON, A. A,, 1920, Ap. J. 51, 257-262. MICHELSON, A. A. and F. G. PEASE,1921, Ap. J. 53,249-259. MIKESELL, A. M., 1955, U.S. Naval Observatory, 2nd series, 17, p. 141, part I1 MILLER,R. H., 1966, Science 153, 3766. MILLER,R. H., 1971, AURA technical report no. 40,Tucson. MULLER, R. A. and A. BUFFINGTON, 1974, in: Optical propagation through turbulence, OSA meeting, July 1974, Boulder. ODGERS, G. J. and E. H. RICHARDSON, 1972, J.R.A.S. Canada 66,2. PEASE,F. G., 1930, Armour Engineer 16, 125-128. PEASE,F. G., 1931, Ergebn. Exakten Naturwiss. 10, 8 6 9 6 . PLATT,J. R., 1957, Ap. J. 125, 601. PROTHEROE, W. M., 1961, Pub. Univ. Pensylvania (Flower and Cook Observatory) 90,27. ROCCA,A,, F. RODDIER and J. VERNIN,1974, JOSA 64, 1000. RODDIER, C., 1971, private communication. RODDIER, C . and F. RODDIER,1973, J.O.S.A. 63 (6), 661. ROGSTADT, 1967, in: Synthetic Aperture Optics, Woods Hele Summer Study, Nat. Acad. of Sciences, Nat. Res. Council. R ~ S C HJ.,, G. WLERICK and BOUSSUG$1961, Travaux de l’Obs. Pic du Midi, 4. STACHNIK, R. and P. NISENSON, 1973, private communication.
111
REFERENCES
STEPHAN, H., 1873, Comptes Rendus 76. TOWNES, C., 1974, private communication. VANDE STADT,1973, submitted to Astr. and Ap. VERNIN, J. and F. RODDIER, 1973, J.O.S.A. 63 (3), 270. YOUNG,A. T., 1969, App. Optics8, 869. F., 1938, Physica 5,785. ZERNIKE,
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E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
I11
RELAXATION PHENOMENA IN RARE-EARTH LUMINESCENCE BY
L. A. RISEBERG GTE Laboratories Incorporated, Waltham, Massachusetts, 02154, U.S.A.
and
M. J. WEBER* Lawrence Livermore Laboratory, Livermore, California, 94550, U.S.A.
*
Work performed under the auspices of the U.S. Atomic Energy Commission.
CONTENTS PAGE
Q 1. INTRODUCTION
. . . . . . . . . . . . . . . . . . . Q 2 . HISTORICAL DEVELOPMENTS . . . . . . . . . . . .
91
Q 3. RARE-EARTH ENERGY LEVELS . . . . . . . . . . .
98
93
Q 4. EXCITATION AND DECAY IN RARE-EARTH SYSTEMS 102 Q 5. RADIATIVE DECAY
. . . . . . . . . . . . . . . . . 106 Q 6. MULTIPHONON RELAXATION . . . . . . . . . . . . 116 Q 7 . COOPERATIVE RELAXATION . . . . . . . . . . . . .
133
Q 8. SELECTED APPLICATIONS . . . . . . . . . . . . . . 150
9 9. CONCLUDING REMARKS. . . . . . . . . . . . . . .
155
REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
156
0
1. Introduction
The processes active for the relaxation of rare-earth ions in excited electronic states include (1) radiative decay, (2) nonradiative decay wherein the excitation energy is converted into vibrational quanta of the surroundings, and (3) nonradiative transfer of energy between like and unlike ions with possible degradation of the excitation. These relaxation phenomena for rare-earth ions in solids are treated in this article. In the past decade considerable progress has been made in the experimental and theoretical investigation of rare-earth relaxation and in extending our understanding of these processes. Our objective in this review is to provide a coherent summary of the totality of advances made in this area in recent years. The upsurge of interest and activity devoted to relaxation phenomena has been prompted by a combination of circumstances. First, as a result of extensive studies of rare-earth spectra, many of the electronic properties of the rare-earths such as the lower energy level schemes and line intensities are now well established. A good survey of the theoretical understanding of these properties in the mid-1960’s was given by WYBOURNE [1965]. It was natural, therefore, that attention should turn to other questions such as ion-lattice interactions and cooperative phenomena, both of which are essential to a complete understanding of luminescence and relaxation effects. Concurrently, there has been a growth in the utilization of rare-earthactivated materials. These include not only phosphors for photoluminescence, cathodoluminescence, and radioluminescence applications but more recently quantum electronic devices such as lasers, quantum counters, and infrared-to-visible upconverters. Because of these device implications, particular emphasis will be devoted to presenting the current knowledge of rare-earth relaxation phenomena in a form which will be of utility to workers in applied fields. We seek to provide sufficient depth to ensure a complete presentation but not an exhaustive one. Esoteric discussions which would be of interest only to 91
92
RELAXATION PHENOMENA
Cm §
1
specialists are avoided. The approach to the subject is generally phenomenological. This seems appropriate at present since satisfactory ab initio calculations of the rates of most relaxation processes are still beyond our capabilities. The phenomenological approach, on the other hand, has established parameters and guidelines which are useful in predicting and analyzing the behavior of rare-earth systems. Given an ion in an excited electronic state, we shall be interested in the rates and relative importance of the various radiative and nonradiative processes active for relaxation. The excited states of the rare-earth ion to be considered include those of the ground 4fN electronic configuration or of higher-lying configurations, such as 4f'- '5d. Excitation via charge transfer states, while active for some rare-earth phosphors, is more specialized and is not treated explicitly. The excitation may be produced by optical radiation (photoluminescence), electron beam (cathodoluminescence), or other ionizing radiation (radioluminescence) such as X-rays or gamma rays. We shall not be concerned with the specific excitation process involved or its rate, since in general the relaxation phenomena to be treated are common for all forms of initial excitation. The rare earths comprise two series: the lanthanides - lanthanum (57) through lutetium (71), and the actinides - actinium (87) through lawrencium (103). Because of the limited number of investigations and demonstrated uses of actinide ions, the lanthanide ions will be treated almost exclusively. The extension of the concepts to the actinides should be evident. Throughout, the concern will be with the relaxation of excited rare earths present in crystals, glasses, and liquids rather than in the free ion or gaseous state. The behavior of rare earths in crystals has received particular attention in the study of ion-lattice interactions since the phonon properties and eigenstates are better defined. A brief historic review of the evolution of the spectroscopy of rare earths which set the stage for relaxation studies is presented in section 2. This is followed in sections 3 and 4 by a survey of the energy levels and the excitation and decay modes of the rare earths. The next three sections then treat, respectively, relaxation by radiative decay (5), multiphonon processes (6), and ion-ion interactions (7). Finally, in section 8, some typical examples are given of the use of this information in the discovery and development of rare-earth activated luminescent materials for specific applications.
111,
§ 21
HISTORICAL DEVELOPMENTS
8
93
2. Historical Developments
2.1. SPECTRA
In 1907 BECQUEREL[I9071 first made spectroscopic observations of crystalline rare-earth salts at low temperatures. He was intrigued by the observation that at low temperatures the broad absorption bands resolved into a multitude of sharp lines. This feature is indicative of the unique characteristics of these materials which over a period of more than a half century have formed a basis for the rich history of rare-earth spectroscopy. A firm scientific basis for the interpretation of rare-earth spectra was made possible by the theoretical work of BETHE [1930] and KRAMERS [1930] in the late nineteen twenties. An upsurge of interest in experimental studies followed, most notably in work of Spedding and Freed and their collaborators on the absorption spectra, in work on the fluorescence spectra in Germany, and in work in the Soviet Union by Zaidel and his collaborators. This early work was directed toward the location and identification of the large number of energy levels that contributed to the observed complex, sharp-line spectrum. The theory continued to advance, with a notable contribution by VANVLECK[I9371 who explained the origin of the observed optical transitions. The greatest advances in understanding the details of rare-earth spectra came in the decade immediately following World War I1 with the availability, for the first time, of large quantities of rare earths in high purities. The availability of liquid-helium temperatures also permitted experimental solution of previously intractable spectroscopic problems. In the years immediately after the war the work of Hellwege and his collaborators, as well as that of the Dutch investigators, pioneered the interest in this activity. Corresponding theoretical work was carried out, particularly by Jorgensen and by the group at Oxford. The latter work was stimulated by experimental activity in the paramagnetic resonance of rare-earth ions (BLEANEY and STEVENS C19531). With the introduction of the powerful techniques of tensor calculus and crystal-field theory, the stage was set for a major thrust during the late nineteen fifties in the detailed investigation of rare-earth spectra, especially in the precise assignment of energy levels and in the generation of rare-earth wavefunctions in crystals. By the early sixties the Johns Hopkins group, under the direction of DIEKE [1961], had generated a complete set of energy level assignments for all of the trivalent rare-earth ions in the anhydrous trichlorides. These energy level diagrams are summarized in Fig. 2.1.
94
R E L A X A T I O N PHENOMENA
38-
36.
3432. 30. 28.
26. 24.
2220. 18. 16. 14. 12.
10-
8.
, 6. 4.
t
2 -5 02
F5(z
-.
-,*
-3
H,
Ce Pr
4
1%
I-%
.....
r-i 8
I,
-*
-
-r
-2-
6H5/z
-9
-I
IFo
Nd Pm Sm Eu
-
- 2 - 1 - 2 -
'S
7
Gd
Tb
F6
6
Hi5+
Dy
5
I,
Ho
,
Iiq
Er
3
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R
2
FT,~
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Fig. 2.1. Observed energy levels of the trivalent rare earths. The width of the levels indicates the approximate magnitude of the Stark splitting in LaCl, . The semicircles denote those levels from which fluorescence has been observed in LaC1, (after DIEKE [1968]).
114 §
21
HISTORICAL DEVELOPMENTS
95
2.2. RELAXATION
At this time, with the lower energy-level structure of the rare earths established, activity became directed toward some of the more subtle phenomena, among which were the relaxation processes accompanying the luminescence.* A particular stimulus for these studies from the early sixties onward was the development and increasing importance of lasers involving rare-earth ions in crystals, glasses, and liquids. Over the years, the field of relaxation phenomena has received review treatment in various forms. The monumental treatise of EL’YASHEVICH [1953] contained a summary and detailed description of work up to that time. Review papers in the early 1960’s by DIEKE [1961,1963] were in the nature of updated reports. More recently, reviews by Moos [1970], GRANT [1971] and KUSHIDA [1973a, b, c] contained discussions of one or more aspects of this field, and a broad coverage was given in the,book by DIBARTOLO [19681. Although only recently studied in detail, relaxation effects were manifested very early in the study of rare-earth luminescence. The question of the source of the radiative transitions was, as noted above, elucidated by Van Vleck. This model was confirmed experimentally by BROER,GORTER and H ~ ~ G S C H A G[1946]. E N The symmetry properties of the crystal field were invoked in early spectroscopic studies to account for the selection rules and polarization observed in these radiative transitions. With the availability of good energy level assignments and wavefunctions, calculations of oscillator strengths for magnetiedipole transitions could be made. Theoretical calculations of electric-dipole strengths also became possible with the techniques and phenomenological approach introduced by JUDD [1962] and OFELT[1962]. Experimental verification of spectral intensity calculations is now well established. As for nonradiative relaxation phenomena, very early qualitative observations indicated their importance in affecting the luminescence properties. It was first noted in the hydrated salts that only the ions in the center of the rare-earth series exhibited fluorescence. A variety of parameters appeared to be related to their stability, but the importance of the H 2 0 vibrations in luminescence quenching was soon recognized. The [1942] in terms stability of these central ions was explained by HELLWEGE
* In a paper titled “The Puzzle of Rare-Earth Spectra in Solids”, VANVLECK[1937] noted that “Decision between the various alternatives will probably be possible only when a detailed spectroscopic classification and Zeeman data become available for various energy levels in the solid spectrum”. This comment is also apropos for the study of relaxation phenomena.
96
RELAXATION PHENOMENA
CIIh §
2
of the electronic energy that had to be taken up by vibrational energy. The larger the energy separation between the excited state and the nextlower state, and hence the greater the number of vibrational quanta required to conserve energy, the smaller would be the probability for nonradiative decay. Similar observations were made in anhydrous crystalline systems. For example, it was known that in the anhydrous rare-earth trichlorides only those levels separated by at least 1000 cm-’ from the next-lower level were sufficiently stable for radiative decay to be observed. (Fluorescing levels are indicated by half circles in Fig. 2.1 .) In the late 19503, extensive measurements of the lifetimes were begun which led to the quantification of these phenomena. BARASCH and DIEKE[1965] and WEBER[1966] observed that lifetimes were generally shorter, the smaller the energy gap from the fluorescing level to the level below. Subsequently, by combining lifetime data with measurements of quantum efficiencies and calculations of the radiative transition, the actual decay rates for multiphonon relaxation were extracted. Such measurements indicated a systematic behavior of the rates with energy gap, with temperature, and with the vibrational frequency spectrum of the host. Related theoretical activity was directed toward the treatment of phonon-induced relaxation. The earliest considerations of the effects of the dynamic crystalline field were concerned with spin-lattice relaxation between Zeeman levels of the paramagnetic ground state. Waller, van Vleck and Kronig were pioneers in the early theories of the 1930’s and 40’s. ORBACH’S [19613 parametrization of the orbit-lattice interaction led to a detailed and definitive stage of theoretical and experimental work in this area in the early nineteen sixties. The same one and two-phonon relaxation processes operating between crystalline Stark levels accounted for the temperature-dependent linewidths observed in optical spectra (YATSIV [1962], YEN,SCOTTand SCHAWLOW [1964]). Multiphonon relaxation of excited rare earths involves the simultaneous emission of several phonons. The treatment of these processes is a more difficult theoretical problem. KIEL’S[1964] treatment using the first-order dynamic crystal-field interaction in higher-order perturbation theory indicated the limitations of precise calculations. The phenomenological theory introduced by RISEBERG and Moos [19681 accounted satisfactorily for the observed experimental behavior. Further refinements in the treatment of multiphonon relaxation were developed by MIYAKAWA and DEXTER [1970], and FONG,NABERHUIS and MILLER[1972] using an adiabatic theory. These have also yielded satisfactory agreement with experiment,
111,
§ 23
H I S T O R I C A L DEVELOPMENTS
97
again in a similar phenomenological way. Although detailed calculations are prohibitively difficult, these semi-empirical approaches have provided adequate understanding for general considerations governing the rates of multiphonon relaxation. A similar evolutionary development took place in the field of relaxation by ion-ion energy transfer. Again it was observed in the spectroscopic study of rare earths that (1) concentrated materials appeared to show fluorescence quenching relative to diluted samples and (2) energy transfer between different ions in the same lattice could take place prior to fluorescence, that is, energy absorbed by one ionic species appeared as luminescence from a second group of ions. Much of this work was pioneered by the group at Philips Laboratories in Holland and was reviewed in an article by BOTDEN[1952]. A milestone in the understanding of these phenomena was the theoretical work of DEXTER[1953] who adapted the earlier treatment of F~RSTER [1948,1949] for the case of organic luminophors to explain the sensitization of luminescence in solids. Later, DEXTERand SCHULMAN [1954] considered the specialized case of migration of resonant excitation among like atoms to isolated impurities which act as quenching centers. The energy transfer mechanism invoked was the electrostatic interaction between the charge clouds of the two ions; this was expressed as a sum of multipolar products. Observation of radiative processes involving pairs of ions was explained by DEXTER [I9621 through a straightforward extension of his earlier theory. Much of the work on ion-ion interactions in the succeeding ten years was devoted to the measurement of the concentration quenching (VAN UITERT[19661) and the assignment of particular multipolar terms governing the transfer. Exchange interactions and the details of the dynamics of the energy transfer were treated by INOKUTIand HIRAYAMA [1965]. It was subsequently shown that the effects of higher-order terms of the multipolar expansion and exchange must be considered carefully and that the results of concentration dependence studies were limited in their conclusiveness. During this period, particularly in the mid 1960’s, great emphasis was placed on the utilization of these ion-pair energy transfer processes for sensitization of rare-earth lasers (VANUITERT[19661). A further stimulus occurred in 1969-1 971 with the investigation of infrared upconversion phosphors. These studies were predominantly device oriented and tended to test and verify rather than extend the basic understanding of energy transfer processes (AUZEL[1973]).
98
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[III,
Q 3
More recently a number of inroads have been made in the detailed experimental investigation of such processes as resonant energy migration between like ions, diffusion-limited relaxation, and non-resonant transfer between unlike ions. Detailed studies of energy diffusion processes led to some of the first agreements of experiments with theory for electric dipoledipole interactions. Additional theoretical work in this area dealt with some of the subtleties of these phenomena, such as the dependence of non-resonant transfer rates on the energy mismatch; this was later confirmed experimentally. An additional activity during the past few years has been the observation and study of cooperative luminescence processes between similar and dissimilar ions.
9 3. Rare-Earth Energy Levels Rare earths of the lanthanide series are most commonly incorporated into materials in the trivalent state. These ions are characterized by a xenon-like electron structure (ls22s22p63s23p63d'04s24p64d'05sZ5p6) with a partially filled 4f shell. The ground electronic configuration is 4fN; the first excited configuration is 4fN-' 5d. The relative location and energy extent of the 4f and 5d configurations for the tripositive rare earths are shown in Fig. 3.1. Most of the rare earths can also be stabilized in the divalent state in appropriate hosts by the use of special growth and post-growth treatment. EuZ+ and Y b z f , which have half-filled and filled 4f shells, respectively, have the smallest reduction potentials followed by Sm2+and TmZ+.These are the most commonly occurring dipositive ions. The two lowest configurations for the divalent rare earths are shown in Fig. 3.2. Comparing Figs. 3.1 and 3.2, the 4f and 5d states of divalent ions are located at lower energies than for the isoelectronic trivalent ions. Optical line spectra of rare earths arise from transitions between levels of the 4fN configuration. The positions of these levels arise from a combination of the Coulomb interaction among the electrons, the spin-orbit coupling and the crystalline electric field. The resultant splittings of the 4fN configuration are shown schematically in Fig. 3.3. The electrostatic interaction yields terms 2 s + 1 Lwith separations of the order of lo4 cm-'. The spin-orbit interaction then splits these terms into J states with typical splittings of lo3 cm-'. Finally, the J degeneracy of the free ion states is partially or fully removed by the crystalline Stark field, yielding a Stark manifold usually extending over several hundred cm- I .
99
RARE-EARTH ENERGY LEVELS
-
II
x 600
-
700
-
800
-
900
I
- I000
- 1500 - 2000
- 3000 - 4000 - 5000
-
-10000
5
C e Pr
Nd
Pm
Srn
Eu
Gd
Tb
Dy
Ho
Er
Tm
5 _
i
Y b Lu
Fig. 3.1. Approximate extent of the two lowest configurations of the trivalent rare earths. [1968]). White - 4fN;black - 4fN-'5d (after DIEKE
Fig. 2.1 provides a good guide to the location of the J states of the trivalent rare earths since the centers of gravity of the J manifolds exhibit only small variations with host. The order and separation of the levels within a J manifold, on the other hand, vary considerably from host to host. The overall extent of the crystalline Stark splittings is small on the energy scale of Fig. 2.1.
100
[IK 5 3
RELAXATION PHENOMENA
x 700
800 900 1000
I- 1
i500
2000
3000 4000
5000 0000
Nd
0 X LZO
k! Ce
Pm
-
Sm
Eu
Gd
Tb
HO
Er
Tm
Yb
Fig. 3.2. Approximate extent of the two lowest configurations of the divalent rare earths. [ 19681). White-4fN;black-4fN-' 5d (after DIEKE
COULONS
SPIN-ORBIT
CRYSTAL.FIELD
Fig. 3.3. Schematic diagram of the splitting of rare-earth energy levels due to the electrostatic, spin-orbit, and crystal-field interactions.
111, §
31
RARE-EARTH ENERGY LEVELS
101
The complex energy level structures of the rare earths were successfully unraveled with the advent of the powerful tensor operator techniques and crystal-field theory. This work has been thoroughly discussed, for example, by WYBOURNE [1965] and by DIEKE [1968]. The free ion states, obtained by diagonalizing the combined electrostatic and spin-orbit energy matrices, are linear combinations of Russell-Saunders states of the form IfN[ySL]J> =
c C(ySr)lfNySLJ>.
(3.1)
YSL
In this intermediate coupling scheme the total angular momentum J is a good quantum number but the spin and orbital angular momentum numbers S and L are not (this is denoted by the brackets in eq. (3.1)). y includes whatever other quantum numbers are required to specify the states. In the labeling of rare-earth states and levels as in Fig. 2.1, the SLJ designation is. usually taken to be the dominant Russell-Saunders state component in eq. (3.1) or the one in the limit as the spin-orbit coupling goes to zero. The crystal field reduces the (25+1)-fold degeneracy of the above free-ion states and causes a small admixing of J states. Because of the shielding effects of the outer 5s and 5p shell electrons, the crystal-field interaction with the inner 4f electrons is weak. The crystal field can thus be treated as a perturbation on the free-ion states. The crystal-field potential is expanded in a series of spherical harmonic terms of the form
where the factors l$ are parameters describing the strength of the crystalfield components, 0:) are tensor operator components which transform as corresponding spherical harmonics, and the summation is over the i electronsofthe ion. The e a r e related to the products A:(rk)of the commonly used parameters A: and radial integrals ($). The number and types of terms appearing in the expansion in eq. (3.2) are derivable using group theory and the point symmetry at the rare earth site. For the f shell, k is limited to values 5 6. In the above approach the strength of the crystal field is given by a small number of l$ parameters. Attempts to calculate these parameters using lattice sums and including covalency and overlap effects have achieved only very limited success. Therefore the values have been determined experimentally. To do this the energy matrix including the crystal-field interaction is diagonalized using an estimated set of starting parameters.
102
[IK 5 4
RELAXATION PHENOMENA
The resulting predicted energy levels are compared with the observed levels and, by an iterative fitting procedure, the parameters are adjusted to obtain a best overall fit to experiment. When the positions of many levels have been measured, and the site symmetry is high so that only a few terms appear in the expansion in eq. (3.2), root-mean-square deviations of observed and calculated energies as small as 10 cm- have been obtained. The crystal-field parameters have also been interpreted using a superposition model pioneered by NEWMAN[1971]. The field is assumed to arise from a sum of independent contributions from the other ions in the crystal. All processes contributing to the crystal field are included in this parameterization. Once the crystal-field parameters for a given ion-host system have been determined, a complete set of energy levels and eigenstates can be computed. These states are labeled by a crystal quantum number ,u and are of the form
'
If"ySL]Jp)
=
1 C(ySLJJ,)lfNySLJJ,).
(3.3)
YSLJZ
These states can be used to calculate matrix elements for radiative and nonradiative transitions between any rare-earth fN energy levels of interest. Some of the levels from higher-lying electronic configurations (including 5d and 6s) have been studied for rare earths in crystals (see, for example, LOH [1968]). In general, however, the locations of higher-lying configurations are not well established because most of the levels are at energies beyond the readily accessible optical region and above the fundamental absorption edge of many materials. For the 5d states there is a strong interaction between the outer 5d states and the static and dynamic crystal fields. As a consequence, the locations of the 5d states vary by many thousands of cm- in different materials and the optical transitions have large linewidths. The location of states having parity opposite to that of the 4fNconfiguration are of interest since, as discussed in section 5, electric-dipole transitions between 4f states become allowed via admixing with these states. Hence the locations of states such as 4fN-'5d and 4fN-'5g can affect the rate of radiative decay from 4f states.
0 4. Excitation and Decay in Rare-Earth Systems As we have seen in the preceding section, rare-earth ions have a complex energy level structure. Given such a multilevel system, which levels may
5
111%
41
E X C I T A T I O N A N D D E C A Y I N R A R E - E A R T H SYSTEMS
103
be expected to fluoresce, under what conditions, and with what efficiency? Before examining in detail the processes which govern the answers to the above questions, we review the decay schemes and terminology used in describing rare-earth relaxation phenomena. At this point only radiative decay and nonradiative decay by multiphonon emission will be considered since they are the most basic processes active and are always present to some degree. Decay schemes arising from ion-ion interactions are reviewed in the section devoted to these processes. Transitions in the optical region of the spectrum occur between J levels of different terms. The crystalline Stark splitting results in fine structure and groups of closely spaced lines. Since all levels are radiatively coupled to other levels with varying transition probabilities, emission or absorption experiments involving rare earths in crystals may exhibit hundreds of sharp lines. In glasses, because of local variations in the symmetry and range of the surrounding charges, there is usually a distribution of Stark splittings and the structure may not be resolved in optical spectra because of inhomogeneous broadening. Similarly in liquids the averaged field produces broad spectral lines. Early spectroscopic studies were directed toward the rationalization of the locations of spectral lines and their assignment to specific czergy levels. Relaxation studies are concerned with the appearance or absence of emission lines and their intensities. The decay of rare-earth luminescence is principally a result of transitions between J manifolds. Relaxation between levels within a given J manifold is rapid because the level separations are generally within the range of phonon energies and hence one- and two-phonon processes are very probable. These nonradiative processes cause lifetime broadening of rareearth spectra and are active for spin-lattice relaxation as studied in paramagnetic resonance experiments. At 300 OK, the rate of intra-manifold relaxation may bey 10l2sec-'. Because of the fast thermal equilibration among the Stark levels, the J manifold can, in most cases, be treated as a whole when considering the slower radiative and nonradiative decay rates to other Jmanifolds. The radiative and nonradiative transition probabilities from individual Stark levels, however, are not equal and thus, as we shall see, it is sometimes necessary to consider the Boltzmann population and transition probabilities from individual Stark levels. The simplified energy level diagram for a rare earth in Fig. 4.1 illustrates possible decay modes involving transitions between J manifolds. Consider a photoluminescence experiment beginning with the absorption of optical radiation causing excitation from the ground state 0 to one or more of the ensemble of upper levels 3, 4, 5 . When the upper levels are closely spaced,
104
Car, 0 4
RELAXATION PHENOMENA
3
FI uorescence
Absorption
V
2
V
1
Fig. 4.1. Schematic energy level diagram showing the radiative (straight line) and nonradiative (wavy line) decay modes of a rare-earth ion following optical excitation.
relaxation occurs predominantly by a level-by-level nonradiative cascade to level 3 denoted by the wavy line transitions. If the energy gap between levels 3 to 2 is large, nonradiative decay by multiphonon emission is less probable and radiative transitions, from level 3 to terminal levels 2, 1, 0, denoted by straight lines, become important. Levels 2 and 1 subsequently relax to the ground state by multiphonon emission in this example. Fig. 4.1 is a fair representation of many of the relaxation schemes active for the trivalent rare-earth energy levels in Fig. 2.1. The rate of relaxation of an excited J state is governed by a combination of probabilities for radiative and nonradiative processes. The lifetime z, of an excited state a is given by
‘a
b
b
where the summations are for transitions terminating on all final states b. The radiative probability WR includes both purely electronic and phononassisted transitions ; the nonradiative probability WNRincludes relaxation
111,
9 41
E X C I T A T I O N A N D D E C A Y 1N R A R E - E A R T H SYSTEMS
105
by multiphonon emission and effective energy transfer rates arising from ion-ion interactions. The radiative quantum efficiency ua is defined by
It is also equal to the ratio of the number of photons emitted from level a to the number of ions excited by photons into level a. Depending upon the specific rare-earth energy levels and host involved, the relative probabilities for radiative or nonradiative decay between given levels may range from comparable values to the two extremes where W,"b-sz W a y or W$ > W2R. The radiative quantum efficiency of a level may therefore approach zero or unity. Determination of the relative and absolute magnitudes of WR and W N Rconstitutes the basic thrust in the study of relaxation phenomena. Determination of any tweof the quantities z, q, W R ,C W N Rin eqs. (4.1X4.2) is sufficient to derive the other two quantities. Several approaches, theoretical and experimental and combinations thereof, have been applied to accomplish this. As will become evident later, meaningful ab initio calculations of either radiative or nonradiative decay rates are still beyond our present capabilities. Therefore all approaches involve experiment and phenomenological treatment. Experimentally, with the present existence of fast, intense pulsed light sources and selective excitation techniques, measurements of excited-state lifetimes present n6 serious problems. For a second quantity, measurements can be made of the quantum efficiency q, but care is required to avoid the introduction of systematic errors. As an alternative, measurements of the line strengths in absorption have been combined with the Einstein ,443 relations to determine the corresponding emission probability W: . Knowledge of the rare-earth concentration is required, however, to obtain the absolute intensities of absorption spectra. When transitions to several terminal J states are present, relative fluoresWR. This cence intensities must be measured to determine the total 1 becomes tedious if the wavelength and intensity ranges involved become large. Different detectors may be required for different spectral ranges and accurate calibration of the spectral sensitivity of the detection system is required. Success in calculating W$ using the Judd-Ofelt approach and phenomenological intensity parameters has prompted several studies wherein calculations of W,"bare combined with measurements of z, to determine
1
1
106
[m,0 5
RELAXATION P H E N O M E N A
the other quantities in eqs. (4.1) and (4.2). This approach can also be tedious since it generally requires measurements of absorption spectra to determine the best set of intensity parameters and computation of transition matrix elements. In addition, whereas overall the r.m.s. errors between calculated and measured line strength may be small ( 7 lo%), the strengths of individual transitions may exhibit larger variations. As discussed in the following section, the Judd-Ofelt approach works best for the lower-lying states of the 4fNconfigurations.
9
5. Radiative Decay
5.1. THEORY
Optical transitions between electronic states of rare-earth ions are predominantly of electric-dipole nature. Magnetic-dipole and electric-quadrupole transitions are allowed but their contributions to radiative decay are generally small or negligible. Given appropriate eigenstates for the rare earth, the probabilities for magnetic-dipole (MD) and electric-quadrupole (EQ) transitions can be calculated ; similar calculations for electricdipole (ED) transitions are not possible, however, The reasons for this are reviewed below together with the Judd-Ofelt treatment which has been applied to interpret the optical intensities of rare earths. The latter approach, combined with empirical intensity parameters, can be employed to calculate radiative decay probabilities. 5.1 .l. Electric-dipole transitions :Judd-Ofelt theory
The selection rules for electric-dipole transitions are A1 = k 1, A S = 0, IALI, ]AJI 5 21, where for lanthanide series ions I = 3. Transitions between f Nlevels involve no change in parity and hence ED transitions are forbidden by Laporte’s rule. Such transitions become allowed if odd harmonics in the static or dynamic crystal field admix states of opposite parity into 4fN. This can occur statically if the rare-earth ion resides in lattice sites lacking inversion symmetry. Noncentrosymmetric sites occur for the following point group symmetries: c, CI c 2 , CZV, c 3 C ~ c V4 c&, c 6 c6v9 D2d, D3, D3h, Dq, Dg, C3h, s q , T, T d , 0. Consider the admixing caused by the odd-parity terms ( k odd) in the crystal-field expansion in eq. (3.2). To first order, the eigenstate for the ath level of the f Nconfiguration is 7
3
111, §
51
R A D I A T I V E DECAY
107
where and the summation is over states fl of configurations having opposite parity. Possible excited configurations include 4f"-'n'Z', where n'Z' = 5d or 5g, and core excitations 3d94fN+l . Matrix elements of the electric-dipole operator P between the states in eq. (5.1) are given by ($ulPl$b)
=
, [(Eb-E,).-1(6nlP16B)(6,1
v,.,dd16b>
+(~,-~,)-1(6ul~~~d16~)(6,1P16b)1,(5.2) where P,
=
-e
C~~(c;))~. i
Any attempt at ab initio calculations of the matrix elements in eq. (5.2) are plagued by numerous unknown quantities and difficulties. These include : (1) the values for B,"(k odd) in V,, which, as discussed earlier in connection with the even k terms in the static crystal field case, have not been calculated satisfactorily ; (2) the energy denominators, since the energy levels E, of the excited configurations are generally not known; (3) the interconfigurational radial integrals jR(4f)R(n'l')t dr, which require good radial wavefunctions R for their evaluation, and finally ; (4) lengthy computations involving consideration of many states. A decade ago JUDD[1962] and OFELT[1962] independently showed that calculations of ED probabilities could be made tractable by treating the 4fNand the excited, opposite-parity configurations as degenerate with a single average energy separation and by replacing Cil~s)(q5slC~') by the tensor operator component U f i q of even order t. By so doing one can invoke closure over the summation in eq. (5.2). The electric-dipole matrix element for the pth component of polarization can then be expressed simply as ($ulPpl$b)
y(t,4, p)(fNySLJJzIUt!kqlfNy'S'ZJ'J:),
=
(5.3)
q , t even
where the energy denominator (En,r.-E4r)av, the crystal-field parameters, and the radial integrals are all incorporated into the phenomenological parameters Y ( t , q , p ) . The number and type of components A; which enter into the Y ( t , g , p ) can be determined from group theory. Since at most f and g states are involved, k 6 7. Application of this approach to the
108
CIIL §
RELAXATION PHENOMENA
5
intensities of transitions between crystalline Stark levels is dependent upon the availability of reliable crystal-field eigenstates. When ions reside in sites of low symmetry, these are difficult to obtain. In an intermediate coupling scheme, the line strength S = ~ ( $ ~ ~ ~ ~ for ED transitions reduces to a simple expression containing three intensity parameters given by
$ b ) ~ 2
t=2,4,6
The eigenstates are of the form in eq. (3.1) and the matrix elements of U @ ) can be derived (WYBOURNE [1965]) using tabulated doubly reduced matrix elements of NIELSON and KOSTER[1964] and 3-j and 6-j symbols. The expression for the electric-dipole line strength arising from one-phonon vibronic transitions is identical in form to that given in eq. (5.4) (JUDD [19621);hence this contribution is included in the intensity parameters 0,. The oscillator strength f and the absorption cross section 0 for a transition of frequency v are related to S by f ( a J ; bJ’) = [87c2mv/3(2J+ l)he’]S(aJ; bJ’)
and
s
o(v)dv =
7ce2(n2+ 2)’ 9mcn
(5.5)
f
and thus can also be determined once the intensity parameters are known. In eq. (5.6), n is the index of refraction of the host. Due to the ED forbiddeness of f-f transitions, the oscillator strengths for transitions between J states are small, of the order of For relaxation one is interested in the spontaneous emission probability given by A ( d ; bJ’) =
64x4v3x S(aJ; bJ’). 3(25 + i)hc3
(5.7)
x is the local field correction and for ED transitions is approximated by n(n’ + 2)’/9. The radiative lifetime and the fluorescence branching ratios from a level a are defined by b
and
n1,
a 51
RADIATIVE D E C A Y
109
respectively, where the summations are over all terminal levels b. Matrix elements of the tensor operator U ( k ) for intermediate coupling have been tabulated for many ions and transitions. Those for absorption from the ground state have been given for all lanthanide ions by CARNALL, FIELDS and RAJNAK [1968]. Only a limited number of matrix elements for emission have been given, partly because the states from which fluorescence is observed vary with the host. In Table 5.1, references are given which provide values of U ( t ) matrix elements for fluorescence transitions of several rare earths. Although somewhat different sets of intermediate coupled eigenstates may have been used in obtaining the above matrix elements, the resulting values exhibit only small differences, and hence values derived for one material have generally been applied to other materials. This point has been discussed by GASHUROV and SOVERS [1969] and several sets of matrix elements for Er3+ obtained by different workers have been tabulated for comparison by REISFELD, BOEHM,LIEBLICH and [19731. In cases where spin-forbidden transitions become allowed BARNETT via a small admixing of spin states, the matrix elements are sensitive to the eigenstates used. Examples of this occur for Eu3+'andTb3+.Severe truncation of eigenstates can also lead to significantly different values. An attractive feature of the Judd-Ofelt approach is that once the set of intensity parameters has been obtained for a given rare earth-host combination, they can be used to calculate absorption and emission probabilities between any f Nlevels of the system. This includes transitions such as excited-state absorptions which are difficult to measure experimentally. The application and limitations of the Judd-Ofelt approach are discussed later in section 5.2. Electric-dipole transitions between states of 4f and 5d configurations are parity allowed. The oscillator strengths for f 4 transitions are therefore much larger than for f-f transitions with magnitudes of 10-'-10-2 (LOH [1966]). Emission from 5d states, while not common, has been observed for several rare earths (see, for example, WEBER[1973a]); Ce3+ and EuZf are notable examples. Calculations of the probability for radiative decay from 5d states by ED transitions requires eigenstates and interconfigurational radial integrals. Examples of such calculations for Ce3+ have been given by HOSHINA and KUBONIWA [1971] and MANTHEY [1973]. 5.1.2. Magnetic-dipole and electric-quadrupole transitions Magnetic-dipole transitions are parity allowed between states of f Nand subject to selection rules A1 = AS = AL = 0 and IAJ( 5 1 (Otft 0) in the Russell-Saunders limit. The most significant contributions usually involve
110
RELAXATION PHENOMENA
CIII,
85
TABLE5.1 References to intermediate coupling matrix elements and Judd-Ofelt studies for rare-earth ions in crystals References Rare-earth ion Matrix elements
Judd-Ofelt parameters
Pr3 Nd3+ Pm3+ Sm3+ Eu3+ Gd3+ Tb3+ Dy3+ H O+~ Er3+ Tm3+ Yb3+ +
A. WEBER[1968b]. B. KRUPKE[1971]. C. KRUPKE[1972]. D. AXE[1963]. E. WEBER[1967b]. and HOSHINA [1972]. F. KUBONIWA G . WEBER,MATSINGER, DONLAN and SURRATT[1972]. H. WEBER[1967a]. and ORTON[1968]. EVERITT I. CHAMBERLAIN, J. KRUPKE[1966]. and PELLETIER-ALLARD [1971]. K. DELSART and MATSINGER [1973]. L. WEBER,VARITIMOS M. WEBER[1968a1. N. DETRIO[1971]. 0. BECKER[1971]. P. KRUPKEand GRUBER[1965].
intramultiplet transitions. Other transitions become possible because spinorbit coupling admixes S and L states, but these transitions are rarely of importance. The line strength for MD transitions is given by S,,(aJ; bJ') = B2l(f"ySL]JIIL+2S(Jf"[y'S'~]J')12,
(5.10)
where /? = eh/2mc. The matrix elements of the magnetic-dipole operator L + 2 S between SLJ states are easily calculated from formuli given elsewhere (WYBOURNE [19651) and have been tabulated for absorptive transiFIELDSand RAJNAK[1968]. The tions for lanthanide ions by CARNALL,
111,
8 51
RADIATIVE DECAY
Ill
spontaneous emission probability for magnetic-dipole transitions is calculated using eq. (5.10) in eq. (5.7) with xmd= n3. Electric-quadrupole transitions are also parity allowed between states of fN with the selection rules A S = 0; IAL.1, IAJI S 2. Expressions for EQ line strengths have been evaluated for several rare-earth transitions ; however, the resulting probabilities have been found to be several orders of magnitude smaller than those for dipolar processes. Possible enhancement mechanisms for quadrupole transitions have been considered (JGRGENSEN and JUDD [1964]), but thus far no cases have been discovered where electric-quadrupole transitions are significant. Hence they are generally considered to be unimportant for radiative decay. 5.2. EXPERIMENTAL RESULTS
The radiative decay rates of rare-earth ions have generally been derived either from the Einstein A-B relationship between the probabilities for absorption and spontaneous emission or from the Judd-Ofelt approach. In the first case the decay rate A,, for the transition from an excited state a to the ground state 0 is determined from measurements of the associated integrated absorption cross section 0 and the relation (5.1 1)
If radiative decay occurs to other terminal levels, as is frequently the case for rare earths, additional measurements of fluorescence branching ratios Baj are needed. Since A,, : A,, : A,, . . * =
pa, : pal : pa, . . .)
(5.12)
the other A , and the radiative lifetime T , in ~ eq. (5.8) can be found. This [19471 and RINCK[1948] approach was used many years ago by HELLWEGE and more recently by CHAMBERLAIN, PAXMAN and PAGE[1966] and others. Accurate knowledge of the rare-earth concentration is required to determine oOaand careful correction for the spectral response of the apparatus is required to obtain the relative fluorescence intensities and B’s. When these requirements are fulfilled, this method is a convenient one for determining radiative decay rates. Implicit in the above relationship is the principle of detailed balance between the probabilities for absorption and emission. This is considered to be a valid assumption for transitions between fN levels of rare earths,
112
RELAXATION PHENOMENA
“11,
§ 5
although no definitive test has been reported. FOWLER and DEXTER C1962, 19651 have formulated a more general relationship given by
where E,,, is the effective field at the emitting center, g b and go are the degeneracies of the lower and upper electronic levels, and (rya) is the electric-dipole matrix element between component states y and 6 of levels a and b. Possible differences in the matrix elements for absorption and emission should be considered when treating f ++ d transitions of the rare earths. The optical intensities of rare-earth spectra from liquids, glasses, and crystals have been successfully rationalized using the Judd-Ofelt approach. The intensity parameters for a given ion-host combination are derived from a least-squares fit of calculated and observed intensities using as many experimental values as available. Average deviations ranging from 5 to 20 % have been reported. Discrepancies for the intensities of transitions TABLE 5.2 Measured and calculated oscillator strengths for the absorption spectrum of Er3+ in Y,O, (UUPKE [1965]) Oscillator strength f(10-6)
[S’LIJ’
Energy (cm-
I)
measured
calculated
6 520
1.25
10250 12500 15210 18200 19180 20 360
0.34 0.31 1.73 0.34 11.03 1.13
21050
0.40
0.48
24 500 26 350 31 210 33920 34600 36350
0.61 20.55 0.03 0.16 0.57
0.67 21.55 0.07 0.40 0.07 1.22
38750
8.89
10.31
0.09
0.52 MD 0.73 ED 0.27 0.37 1.44 0.24 9.95 1.34
111,
§ 51
RADIATIVE DECAY
113
between individual pairs of J states can, however, be much larger. An example of the fit obtainable between measured and calculated oscillator strengths is shown for Er3+in Y , 0 3in Table 5.2. When levels from different Jmanifolds are not resolved, intensities are attributed to the sum of individual transitions. Application of the Judd-Ofelt theory to trivalent rare earths in liquids has been well illustrated in a series of papers by CARNALL et al. C1965, 19681. Although Eu3+ and Tb3+ were not tested thoroughly, satisfactory agreement was obtained for all ions except Pr3+. PEACOCK [1971,1973] has also made extensive studies of the spectral intensities of 4f" series ions in solution in the context of the Judd-Ofelt theory, again with satisfactory results. Spectral intensities of rare earths in glasses have been investigated by Reisfeld and co-workers, and many results are summarized in a review by REISFELD[1973] and references therein. Recently KRUPKE[19741 has carefully examined the intensities of Nd3 in several glasses used for lasers and has obtained extremely good fits with rms deviations of only 5 % . The resulting Judd-Ofelt parameters were used to calculate fluorescence lifetimes and branching ratios and stimulated emission cross sections. The literature of the Judd-Ofelt treatment applied to rare earths in crystals is more dispersed. A summary of references to intensity studies in crystals is given in Table 5.1. Unlike liquids and glasses, where the Stark splittings are usually not resolved, transitions between individual Stark levels can be observed in crystal spectra. Although most studies have considered only the total transition probabilities between Jmanifolds, in several instances intensities of spectra between Stark levels have been analyzed (see, for example, AXE [1963], KRUPKEand GRUBER[1965], BECKER [19711, DELSART and PELLETIER-ALLARD [19713). A prerequisite for such studies, as noted earlier, is the existence of satisfactory crystal-field eigenstates. Judd-Ofelt parameters are expected to vary throughout the lanthanide series because of differences in the average energy separations of the opposite parity configurations, the interconfigurational radial integrals, and the relative contributions of vibronic intensities. In several studies ( K R U P [1966], ~ CARNALL, FIELDSand RAJNAK[1968], BUKIETYNSKA and CHOPPIN [19703, PEACOCK [19733, WEBER, VARITIMOS and MATSINGER [1973]) intensity parameters have been determined for a number of rare earths in the same liquid or crystalline environment. Some systematic variations in the R, values have been noted and used to predict parameters for adjacent ions in the 4fNseries. Overall, however, the behavior based +
114
RELAXATION PHENOMENA
[Ill,
5
5
upon the results reported thus far are still somewhat unclear and additional systematic studies are needed. Radiative decay probabilities are calculated using Judd-Ofelt parameters and eqs. (5.7) and (5.8). In Table 5.3, calculated radiative lifetimes of the 4F, state of Nd3+ in several different hosts are compared with observed lifetimes. The latter values are those measured at low concentrations and temperatures where a quantum efficiency of unity is expected. In view of the 10% uncertainty in the Judd-Ofelt parameters, the agreement is quite satisfactory. In general, agreement is good when, as in the case of Nd3+, radiative transitions occur to several J manifolds, thus providing a further averaging effect. Since the spontaneous emission probability, eq. (5.7), varies as v3, radiative decay becomes more probable for higher lying states. For 4f levels of rare earths, radiative decay rates range from approximately 10 to lo6 sec-'.
-
TABLE 5.3 Comparison of the radiative lifetime calculated using reported Judd-Ofelt parameters and the observed lifetime for the 4F, state of Nd3+ in different hosts Lifetime (ps) Host calculated LaF, y 2 0 3
Y,AI,O,, YAIO, ED-2 glass
635 270 259 157 372
observed
-
670 260 250 175 350420
The intensities of many transitions involving JAJII 2 have been found to be very sensitive to the small changes in the local environment. This is observed in solids, liquids, and gases. Several possible sources of this hypersensitivity have been considered by J@RGENSEN and JUDD[19641. JUDD[1966] has noted that the B,' component in the expansion of the crystal-field potential affects only the Q2 parameter and thus may account for the environmental sensitivity. This component is allowed for the following site symmetries: C,, C 2 , Czv,C3, C3v,C4, C4v,c6, C6". Vibronic mechanisms have also been proposed, but the behavior appears to be different for crystals, molecular complexes, and gaseous phases (see PEACOCK [1972a] and references therein). Examples of transitions having large U(') matrix elements and exhibiting hypersensitivity are.Nd3+: 41, -, 4G,, Eu3+: 7F, + 'D,, Er3+: 'I.y -, 2 w
111,
0 51
RADIATIVE DECAY
115
and 4G,. In the last case, PEACOCK[1972b] found better results by ascribing separate fi2 values for each of the two hypersensitive transitions. Because the energies of the final states differ significantly, a breakdown of the average f d separation used in the Judd-Ofelt theory was suggested. The occurrence of hypersensitivetransitions should therefore be noted when employing the Judd-Ofelt method to calculate radiative decay probabilities. For some ion-host combinations, charge transfer states occur at energies which are lower than 5d states. This is observed, for example, for Eu3+ in the oxysulfides (STRUCKand FONGER C1971-j). These levels could be responsible for the introduction of opposite parity states. In glasses, REISFELD,BOEHM, LIEBLICH and BARNETT [1973] have found that increasing covalency is correlated with lower charge transfer bands and larger Q2 parameters. Thus far no systematic study and treatment of possible effects of charge transfer states on radiative decay of rare earths has been made. The Judd-Ofelt theory is also applicable to the spectral intensities and radiative decay of actinide series ions. Few intensity and decay rate measurements have been reported for 5fNions. From absorption spectra, it is known that the oscillator strengths of f-f transitions for these ions are generally larger than for 4fNions, and therefore admixing of states from opposite parity configurations may be greater. In summary, the Judd-Ofelt method provides a useful description of rare-earth intensities and can be used to predict radiative decay probabilities. There are several limitations to its accuracy, however, due to assumptions inherent in the theory and to its application in practice. As examples of the latter, when considering transitions between J manifolds, equal ion populations in the Stark levels of the initial manifold are usually assumed and J-state mixing is usually neglected. It should also be noted that whereas the quality of Judd-Ofelt parameter fits are dependent only upon relative absorption coefficients or fluorescence branching ratios, spontaneous emission probabilities and absorption/emission cross sections require accurate knowledge of the rare-earth concentration to be meaningful. The Judd-Ofelt treatment is expected to be most satisfactory for transitions between low-lying levels of fNwherethe approximation of an average energy denominator in eq. (5.2) is most valid. Studies of Pr3+ and Sm3+, for example, have indicated that a single set of parameters is sometimes not sufficient to account for all intensities and a second set of parameters should be used for the higher-lying levels. This is indicative of a possible breakdown of the closure approximation involved in arriving at eq. (5.3).
116
RELAXATION PHENOMENA
5
6. Multiphonon Relaxation
6.1. INTRODUCTION
Nonradiative relaxation between J states can occur by the simultaneous emission of several phonons sufficient to conserve the energy of the transition. These multiphonon processes arise from the interaction of the rareearth ion with the fluctuating crystalline electric field. The crystal field at the ion site is not static but undergoes oscillatory behavior due to the vibrations of the lattice or molecular groups. The lattice vibrations are quantized as phonons having symmetry properties determined by the symmetry of the crystal and excitation energies determined by the masses of the constituent ions and the binding forces. The effects of the oscillating electric field are manifest in the spectroscopy of rare-earth ions in ways other than multiphonon relaxation. These include paramagnetic spin-lattice relaxation and the appearance of line broadening and vibrational sidebands in optical spectra. The earliest treatment of the ion-lattice interaction, both experimentally and theoretically, was carried out for the relaxation of Zeeman levels within the paramagnetic ground state. At low temperatures relaxation occurs by the absorption and/or emission of low-energy acoustic phonons either via direct processes involving a single phonon or via Raman processes involving the inelastic scattering of two phonons. Relaxation of Zeeman levels can also occur via an Orbach process involving consecutive phonon absorption to and emission from a higher-lying Stark level. Transitions between Stark levels can be induced by both acoustic and optic phonons. Since the Stark level separations are usually much larger than Zeeman splittings, the more numerous, higher energy acoustic and optic phonons contribute to direct processes and very rapid relaxation rates are possible. The associated lifetime broadening frequently accounts for the observed linewidths of optical transitions (YATSIV[19621). When the lattice and ion are treated as a coupled system, optical transitions are considered to occur between vibrational-electronic or vibronic states. These states are written in terms of Born-Oppenheimer product wavefunctions
where is the eigenfunction of the electronic state and the Ini) are eigenfunctions of the phonon number operator and the ni are phonon mode occupation numbers. Transitions involving no change in vibrational quantum number n are purely electronic or zero-phonon lines. Transitions
In,
8 61
MULTIPHONON RELAXATION
117
to adjacent levels involving the creation or annihilation of one or more phonons appear as vibronic sidebands to the zero-phonon line. Since the ion-lattice coupling for rare earths in levels of 4fNis weak, only one- or two-phonon sidebands are observed. The vibronic sidebands are of special interest because they reflect the extent of the phonon spectrum and the strength of the ion-phonon coupling for the different vibrational modes. Although multiphonon relaxation processes were postulated and understood qualitatively many years ago, it has only been within the last decade that detailed experimental studies have been carried out and a rational interpretation provided. From early spectroscopic studies of rare earths in LaC1, ,it had become apparent that a level had to be separated from the next lower level by an energy in excess of 1000 cm-' for it to be sufliciently stable to exhibit fluorescence. Since it was known that the phonon spectrum of LaCl, extended to a maximum energy of about 250 cm-', it was evident that a competing nonradiative relaxation process must involve the simultaneous emission of many phonons. Similar behavior regarding the occurrence of metastable fluorescing levels was observed in other crystal systems (DIEKE[1963]). Measurements of fluorescence lifetimes (BARASCH and DIEKE [19653, WEBER[19661) provided semi-quantitative confirmation of the decreasing importance of multiphonon decay with increasing number of phonons -required for energy conservation. This was followed in the mid-1960's by a concerted effort to measure multiphonon relaxation rates for various rare earths in several different host crystals. Based upon these measurements, theories have evolved which describe the essential features of mu1tiphonon relaxation rather well. All theories, however, are necessarily approximate in nature. Multiphonon processes involve a combination of crystal-field theory and lattice dynamics and thus presents the solid-state theorist with a formidable problem defaying definitive treatment. For pedagogical reasons, we proceed in reverse chronological order below, that is, the theory of multiphonon relaxation is treated first followed by a review of the experimental work. 6.2. THEORY
As mentioned above, the treatment of multiphonon relaxation processes has its basis in the dynamic crystal-field Hamiltonian introduced for oneand two-phonon spin-lattice relaxation processes. The extension to multiphonon processes necessarily begins with this basic Hamiltonian. Several
118
Cw §
RELAXATION PHENOMENA
6
theoretical techniques have been employed to calculate the rates of multiphonon processes. The assumption of weak ion-lattice coupling, treatment of the lattice in the harmonic approximation, and neglect of the detailed phonon and electronic properties are common to the approaches used to calculate multiphonon transition rates between 4fNlevels of the rare earths. KIEL[19641 treated the problem first by applying conventional timedependent perturbation theory in higher orders. The intractability of ab initio calculations was immediately evident. RISEBERG and Moos [I19681 extended this approach by developing a phenomenological treatment which was very successful in describing a wide body of experimental data on multiphonon transition rates collected by them, by WEBER[1968a1, and subsequently by other authors. Later theoretical treatments by MIYAKAWA and DEXTER [1970] and FONG,NABERHUIS and MILLER[1972] employed more elegant mathematical techniques and yielded somewhat more elaborate dependences of the rate of multiphonon emission on energy gap and temperature. By using adjustable parameters, these treatments were similarly successful in fitting the existing experimental data. Divalent and trivalent rare earths can also be excited via absorption into states of higher-lying configurations, such as 4fN- 5d, with subsequent relaxation to 4f” states. Because the ion-lattice interaction for the outer d electrons is greater than for the inner f electrons, the weak coupling approximation is not valid and the nonradiative decay rates are faster. d -+ f relaxation of rare earths has been the subject of recent experimental and theoretical investigations (WEBER[1973a] ; LAUERand FONG[19741). Since the phenomenological model provides an intuitively satisfying description of the experimental results and permits a straightforward and generally useful account of multiphonon relaxation rates in a wide variety of different systems, it will be reviewed here. Included, too, are a few cases which appear to be exceptions and where some extension is in order. We begin by expanding the crystal-field Hamiltonian H C F in a Taylor series about the equilibrium ion positions, that is, HCF
= Vc,+
C i
Qi
. Vi V c F + . . . = V&+
1 FQi++ C y ,
jQiQj+
i
. . (6.2)
i. j
where Qirepresents the ith normal mode coordinate. The equilibrium crystal field VCFis identical to the static field given by eq. (3.2) and discussed in section 3 ;the remaining terms in eq. (6.2) constitute the dynamic crystalfield interaction. The Vi . , , are partial derivatives of the crystal-field and are expressible as tensor operator sums in the form given in eq. (3.2) and acting on the electronic states. The Qi are expressible in terms of phonon
111,
§ 61
MULTIPHONON RELAXATION
119
creation and annihilation operators. Thus the dynamic crystal-field interaction involves products of operators acting upon product states of the ion-lattice system given in eq. (6.1). In time-dependent perturbation theory, a process involving the transition from electronic state I$.) to electronic state ($b) with the emission of p phonons can arise in different ways. First, the first-order term in eq. (6.2) can simply be used in pth order perturbation theory or, second, the pth order term in eq. (6.2) can be used in first-order perturbation theory. In the most general formulation, the transition probability also includes contributions from intermediate terms. Taking only the contributions from the two extreme terms, the probability for multiphonon relaxation between states a and b involving p phonons is given by
where . . . l$mn-l) are virtual intermediate states and g(wi) . . . g(wj) are the densities of phonon states. The electronic levels are assumed to be infinitively narrow, and energy conservation is guaranteed implicitly by the &function. The summation is over all possible intermediate states and phonon modes. Although evaluation of eq. (6.3) is possible in principle, in practice only order-of-magnitude estimates are feasible. This is also true for the more elaborate theoretical treatments of multiphonon relaxation. The greatest difficulty arises from lack of detailed information regarding the frequency, polarization, and propagation properties of the vibrations and the associated strength of the ion-phonon coupling coefficients Vi, Vj, . . .. It is possible, as exemplified by GERMAN and KIEL[1973], to discuss relative rates of multiphonon emission in similar systems using estimates based upon eq. (6.3). The phenomenological model derived from eq. (6.3), as shown below, also has significant predictive values. The rate of multiphonon relaxation is temperature dependent. This
120
[UI,0 6
RELAXATION PHENOMENA
arises from two possible sources. The first is the orbit-lattice interaction. For rare-earth ions, however, we assume that the interaction remains harmonic and that the Vi , Vi. , . . . are essentially independent of temperature, at least up to temperatures of several hundred "C.The second and dominant source arises from stimulated emission of phonons as the modes become thermally populated. As we shall see later, a simple single-frequency phonon model frequently yields good agreement with experimental observation. Consider multiphonon relaxation across an energy gap AE to the nextlowest level. The number of phonons p i of equal energy h o i , required to conserve energy, and hence the order of the process, is determined by the condition pihwi = AE.
(6.4)
The temperature dependence from eq. (6.3) is then W ( T )= W,(n,+ 1)P[,
(6.5)
where n, is the occupation number of the ith phonon mode and W , is the spontaneous transition rate, that is, W(T)= W , at T = 0. Replacing n, by its Bose-Einstein average
n, = [exp (ho,/kT)- 1]-',
(6.6)
we have for the temperature-dependent multiphonon transition rate for a single-frequency p-phonon process, WPi(T) =
w,
71
exp (hwi/kT) exp (ho,/kT)- 1
The critical feature in the temperature dependence is the order of the process. This is illustrated in a striking manner in Fig. 6.1 where eq. (6.7) is plotted for processes involving four, five, and six phonons of equal energy and satisfying eq. (6.4). The difference in the curves is substantial. Therefore, although there is actually a spread of phonon energies, it is the order of the process rather than the precise phonon energy distribution which governs the temperature dependence. Since the highest energy and/or most strongly coupled phonons can conserve the energy AE in the lowest order process, they are expected to make the largest contribution to the relaxation rate and its temperature dependence. As pointed out by KISLIUKand MOORE[1967], the initial rate of relaxation W , by lower energy phonons oi may be smaller, but due to the larger orderp,, the temperature dependence in eq. (6.7) is stronger
111,
5 61
MULTIPHONON RELAXATION
121
Fig. 6. I . Theoretical temperature dependences of multiphonon relaxation for the single frequency model; AE = 1500 K and pi = 4, 5 and 6 (after RISEBERG[1968]).
and hence may become dominant at elevated temperatures. (See also, LAUERand FONG[1974].) In a more general treatment, one must sum [1973]. contributions from all phonon modes, as has been done by STURGE Eq. (6.7) is consistent with Sturge’s results in the limit of weak coupling characteristic of rare earths in crystals. The model leading to eq. (6.7) assumes a simple process involving decay between two single discrete levels. In reality, as noted earlier, decay occurs between groups of Stark levels of two J multiplets. If levels of the upper multiplet are designated by the subscript a and levels of the lower multiplet by the subscript b, then the total decay rate from one of the upper levels is x b Wab. Since the levels within the multiplet are in thermal equilibrium, the combined rate is a Boltzmann average of the rates from the separate levels given by
where W, is an individual decay rate from upper multiplet level a to lower multiplet level b, g, is the degeneracy and A , is the energy separation of the ath level from the bottom level of the upper multiplet. The intrinsic temperature dependence of the multiphonon rate w,b is expressed by eq. (6.7).
122
RELAXATION PHENOMENA
[m,§ 6
Eq. (6.8) represents a precise description of the characteristics of the temperature dependence. A similar expression applies to the temperature dependence of the total radiative rate (where w a b is replaced by w a b A&), since the probabilities A,, for radiative transitions between individual Stark levels are also not equal. In practice the individual rates wabcannot be determined with sufficient accuracy to exploit the precision implicit in eq. (6.8). Several authors (PARTLOW and Moos [1967], RISEBERG, GANDRUD and Moos [19673) have successfully fitted experimental results by approximating eq. (6.8) through the introduction of a limited number of effective levels. The most significant value of the temperature dependence, however, is in establishing the order of the multiphonon decay process and the energies of the dominant phonons involved. This can be accomplished adequately using an approximate treatment as shown by the experimental work to be discussed later. In the phenomenological model the energy gap dependence of the multiphonon emission rate arises as a consequence of the convergence of the perturbation expansion. The ratio of the pth order transition rate to the (p- 1)th order rate in a given host, again considering a single-frequency phonon model is simply
+
wpfwp-'
=
E
< 1,
(6.9)
where E is a coupling constant whose magnitude determines the rapidity of the convergence of terms in the perturbation expansion in eq. (6.3). The rate for a pth order process can then be written phenomenologically as
W p= A E ~ ,
(6.10)
where A is a constant and the order p is determined from eq. (6.4). The validity of this approach depends upon the extent to which the individual features of the phonon modes and electronic states are statistically averaged out in higher-order multiphonon processes. In the case of many crystals, the rare-earth vibronic spectrum, which reflects the strength and extent of the ion-lattice interaction, is rather broad and diffuse. When this effective phonon density of states is convolved p times (p > 2) as in eq. (6.3), little structure remains and the exact properties of the phonons become unimportant. The remaining critical parameter is the number of phonons required to conserve energy and, therefore, the energy gap. Similarly, because of the large number of equivalent processes involving many different intermediate states, the symmetry properties of the electronic wavefunction are generally also unimportant. Exceptions can occur, however. Terms in the expansion of the ion-lattice interaction containing
111,
0 61
MULTIPHONON RELAXATION
123
tensor operators with k odd cannot couple levels of the same configuration because of parity. In addition, only those terms will contribute which contain even-order operators satisfying the triangle rule
IJ,-J,I
5k
s IJ,+J,I
(6.11)
for initial and final states a and b. Thus, for example, there are no firstorder matrix elements of the orbit-lattice interaction connecting states with J = 0 and J = 1 and hence relaxation can occur only by higher-order processes. In other cases, eq. (6.11) may restrict the number of terms in eq. (6.2) which are effective. Overall, however, these selection rules are not too restrictive and the properties of the electronic states as well as the phonons are averaged out in multiphonon relaxation processes. The order of the process, under the single-frequency model, is determined by the energy gap and eq. (6.4). The phonon frequency wi ,generally speaking, should be close to the maximum of the phonon spectrum, considering the rapidity of the convergence as expressed by the smallness of E . This is usually true and experimental studies of the temperature dependenceconfirm that theorder ofthe process is often the lowest order consistent with energy conservation and the cut-off frequency of the phonon spectrum. If lower frequency modes are more numerous and/ormore strongly coupled, the dominant multiphonon process may occur in a higher order. Thus, an assignment of the proper value to assume for wi should be made in conjunction with knowledge of the intensity of the vibronic spectrum. A notable omission above is the class of materials where high-energy vibrations arise from the internal modes of well-defined molecular groups. In such crystals and in glasses, the rare earth may be relatively weakly coupled to the molecular vibration. Furthermore, the molecular vibrational frequencies are large (up to 1000 cm-l) and may have very welldefined energies due to the absence of dispersion. The decay by emission of a few such vibrational quanta may have rates comparable with radiative rates in the optical region. There is an additional contribution from the more strongly coupled lattice modes which are lower in energy. As might be anticipated, the above model might not be applicable in such cases because the degree of averaging is smaller. In summary, in the limit of the validity of eqs. (6.4) and (6.10), we can combine the expressions and obtain a phenomenological expression
-
where C and
o!
are positive-definite constants characteristic of a given
124
R E L A X A T I O N PHENOMENA
Cm § 6
material at low temperatures*. The experimental thrust over the past several years has been the determination of w i , E , C and CI for a wide variety of rare-earth doped materials. These results are discussed and summarized below. 6.3. EXPERIMENTAL RESULTS
Methods for experimentally determining pure multiphonon relaxation rates follow from the discussion in section 4. They generally involve measurements of excited-statelifetimes and one or more other quantities. Three main approaches have been successfully employed. One combines absorption data and the relationship between the Einstein A and B coefficients to find the probability for radiative decay which is then subtracted from the fluorescence decay rate to determine the nonradiative or multiphonon rate. (See, e.g., RINCK[1948], and CHAMBERLAIN, PAXMAN and PAGE[1966].) Alternatively, the radiative contribution can be calculated by the JuddOfelt method, as shown by WEBER[1967a, 1968a, b, 1973bl and DELSART and PELLETIER-ALLARD [19731. The third technique involves measurements of relative quantum efficiencies to determine the multiphonon branching ratios and hence the transition rates. This method was introduced by PARTLOW and Moos [1967] and used by RISEBERG and Moos C1967, 1968) to measure rates for an extensive set of rare-earth levels in several different crystals. Variations of these approaches have subsequently been applied by other workers to measure multiphonon rates in additional crystalline materials and, more recently, in glasses. 6.3.1. Temperature dependence The temperature dependence of multiphonon relaxation rates was studied by early workers and provided strong support for the phenomenological approach and the use of a single-frequency model. As an example, Fig. 6.2 shows the temperature dependence of the E(6F*) to D(6F8)multiphonon decay process in LaBr, :Dy3+.The theoretical fit is given by eq. (6.7) with the following parameters : p i = 5,
hw, = 155 cm-’,
Wo = 8 . 8 lo3 ~ sec-’.
* Treatment of multiphonon relaxation by the adiabatic approximation (FREEDand JORTNER [1970], FONG, NABERHUIS and MILLER[1972], STURGE[1973]) predicts a more complicated energy gap law given by Wcc e-hE’h”n/vAE, where the dimensionless Q is defined by (log (AE/(hv)- 1 ) and 5 is a dimensionless displacement.The departure of this law from the simple exponential dependence in eq. (6.12) has thus far not been established experimentally.
m 9
8 61
125
MULTIPHONON R E L A X A T I O N
50
a
c 0
.-
0 c
TPK)
Fig. 6.2. Temperature dependence of the E(6Ft) to D(6Ft) multiphonon decay process in and Moos [1967]). LaC13:Dy3+(after RISEBERC
The energy gap is taken to be that to the uppermost Stark level of the terminal J multiplet. In this case it is not necessary to apply the detailed model of eq. (6.8) because the emitting state consists of two Stark levels separated by only a few cm-'. The theoretical fit clearly indicates that the phonons correspond to the high-energy optical region, and that the process is the lowest order consistent with energy conservation and the high frequency cut-off of the phonon spectrum at 175 cm-'. A more complex situation is provided by the F('F3) to E(5F4,5Sz) relaxation of Ho3+ in LaF,. The theoretical fit to the experimental data in Fig. 6.3 is given by eq. (6.7) with the following parameters:
P, = 6
A, =0
hoi = 3oOcm-' g1 = 0
W,= 2.2 x lo4sec-'
A,
=
150~m-' A , = 5 o O ~ m - I
92 = 1 W, = 0
93 = 5
w3= 0.
The effect of the Stark levels above the decaying level is accounted for by a single non-decaying level at 150 cm-'. This reflects the reduced decay rates for these levels because of the larger gaps. Similarly, the effect of the
126
RELAXATION PHENOMENA
Fig. 6.3. Temperature dependence of the F(’F3) to E(’F,, ’ S 2 ) multiphonon decay process in LaF, :Ho3+ (after RISEBERGand Moos [1968]).
next highest multiplets (G(5F2)and H(3K8))is accounted for by the assumption of five non-decaying levels at 500 cm-’. Although the parameters used in fitting the temperature dependences are only a rough approximation to reality, the model does successfully describe the effects of thermal depopulation of the lowest Stark component of the decaying level. FONG,NABERHUIS and MILLER [19721have performed further fits to temperature dependences taking into account the full Stark structure with rates for individual levels obtained from the gap dependence. Similarly good fits to experiment were achieved. The phonons used in the above fits represent the lowest order process consistent with energy conservation and the cut-off frequency of the phonon spectrum. This result has generally been obtained for a number of crystals where the vibronic spectrum shows strong peaks near the phonon cut-off frequency. There are other crystals, however, for which the situation is more ambiguous. If the most intense peaks in the spectrum occur at lower energies than the extreme upper limit, it is no longer clear which value to assume for the effective phonon energy in the single fre-
111,
0 61
M U L T I P H O N O N RELAXATION
I27
quency model. This is borne out by temperature dependence studies in Y,Al,012 (ZVEREV, KOLODNYI and ONISHCHENKO [1971]) and in YA103 (WEBER[1973b]). In these crystals the temperature dependence cannot be fitted by the lowest possible order. Instead, better fits are obtained using a larger number of lower energy phonons corresponding to the more prominent peaks in the high-energy region of the phonon spectra. In both cases the highest energy phonons appear as weak peaks in the vibronic spectra. Therefore, in obtaining a value for the effective phonon energy, both the cut-off and the shape of the vibronic spectrum should be considered. In most instances, however, it may simply be taken to be the highest value. 6.3.2. Energy gap dependence Measurements of spontaneous multiphonon emission rates for rare earths in a number of different crystalline materials have confirmed the validity of the exponential dependence on energy gap given by eq. (6.12). As an example, data on the multiphonon emission rates in YAlO, at 77 OK are plotted versus energy gap to the next-lower level in Fig. 6.4. The rates for five different ions and fifteen different excited states can be fitted approximately (within a factor of 2) by a simple exponential dependence on the energy gap. (The two exceptions, the 'D1 and 'D2 states of Eu3+, are subject to selection rules and the restrictions of eq. (6.1 l).) From knowledge of the phonon spectrum of YAlO, and for the range of energy gaps studied, a minimum of three and as many as seven or more phonons are active in the relaxation. The close obedience to the exponential energy gap law over a range of approximately six decades provides convincing testimony to the phenomenological approach and the averaging out of the detailed features of the electronic states and phonon modes in these highorder processes. The exponential dependence expressed by eq. (6.12) and illustrated in Fig. 6.4, while applicable to processes involving many phonons, must eventually break down for small energy gaps where relaxation by one- or two-phonon processes is possible. In this regime the statistical averaging that occurs for higher-order processes is no longer prominent; relaxation rates are strongly dependent on the phonon density of states and on the ion-phonon coupling. This is evident from the optical linewidths observed for transitions to individual Stark levels. These widths, when determined by lifetime broadening due to rapid one- or two-phonon processes, correspond to relaxation rates which are much larger than would be obtained by extrapolation of the energy gap dependence (WEBER[1973b1).
-
128
R E L A X A T l O N PHENOMENA
,
I
I
I
Neodymium
I
1000
2000
I
I
3000
0
Europium
0
Holmium
I
I
4000
I
I 5009
Energy gap to next-lower level (cm-1)
Fig. 6.4. Dependence of the rate of multiphonon emission on energy gap to the next lower level [1973b]). for excited states of rare-earth ions in YAlO, at 77 K (after WEBER
For large energy gaps the multiphonon decay rates become smaller and approach the regime where Z A i j > W. If in this limit W is obtained by subtracting A from C1, then uncertainties in A can greatly affect the value and accuracy of W. The multiphonon rates for YA103 were obtained using A's found from Judd-Ofelt calculations. The uncertainty in their values produces the large uncertainties in the resulting W values plotted in Fig. 6.4. Use of relative quantum efficiency measurements is a more desirable experimental approach in this case. The selection rule in eq. (6.11) has generally not been found to be an important factor for multiphonon relaxation when only one or two terms of the ion-phonon interaction are forbidden. This, however, may be because the effect is not expected to be significantly larger than the normally observed variations from the exponential dependence. In cases where
111,
8 61
MULTIPHONON RELAXATION
129
multiphonon relaxation is formally forbidden, as in the case of J = 1 to J = 0 transitions, the rates deviate significantly from the exponential law (see the 'D1datum for Eu3+ in Fig. 6.4). This has been discussed recently by GERMAN and KIEL[I9731 for the ,P1 to 3P0decay in LaCl, : Pr3+.The model usually employed for such processes assumes a virtual intermediate state different from either the initial or final 'state and thus the reduction in the transition rate occurs by virtue of the increased energy denominator. German and Kiel also treated the 'I6 to 3P,decay and proposed that this rate is much faster than that anticipated from an extrapolation of the dependence on energy gap. 6.3.3. Host dependence Multiphonon relaxation rates have now been measured for rare earths in several varieties of crystals. In many instances, data sufficient to illustrate the energy gap dependence has been obtained. This is shown for a selection of crystals in Fig. 6.5. Included in parentheses are the phonon energies, haerf, which, based upon the temperature dependence of multiphonon rates and/or vibronic spectra, appear to be most important for relaxation. Since the rare earth 4f levels exhibit only small changes with host (2100 cm-'), the different rates evident in Fig. 6.5 for a given energy gap are clearly a property of the host via the phonon frequency spectrum and the strength of the ion-phonon coupling. If the rates are plotted as a function of a normalized energy gap AE/ho,,, , which is approximately equal to the number of phonons active, the resulting slopes of the curves will reflect the relative strength of the ion-phonon coupling in the different hosts (RISEBERG and Moos [1968]). The parameters C and a in the expression for the multiphonon emission rate in eq. (6.12) can be determined directly from the exponential energy gap dependences. These parameters for crystals studied to date are listed in Table 6.1. There is a wide variation in C which ranges from 4.5 x lo7 to 1.5 x 10'O sec-'. As noted earlier, the rate does not correspond to the rate for single-phonon processes which are known to be 5 10' ' sec- '. The parameter E in eq. (6.9) can also be determined from the data and the value of haeff derived as noted above. The results are included in Table 6.1. E reflects the strength of the ion-phonon coupling and correlates with the crystalline fields in the various crystals studied. This was strikingly confirmed by results obtained for LaCl, and LaBr, where identical values and Moos [1968]). This is expected since for E were obtained (RISEBERG the structures and crystal fields of these two materials are identical. The difference in multiphonon rates for these crystals in Fig. 6.5 arises only
130
RELAXATION PHENOMENA
I
I
I
f
I
I
I
Energy gap ( cm-')
Fig. 6.5. Spontaneous multiphonon emission rates from excited states of trivalent rare earths as a function of energy gap to the next lower level (after WEBER[1973c]).
from differences in the phonon spectra (via the halogen masses) and thus the order required to conserve energy. Once sufficient data has been obtained for a given host to define the exponential energy gap dependence and parameters in eq. (6.12), the results can be used to predict the rate of spontaneous multiphonon decay from any 4f rare-earth level of interest. The results are generally good to within a factor of 2-3. The only cautionary considerations are the possible selection rule restrictions and the inapplicability of extrapolations to small energy gaps. Multiphonon relaxation in crystals containing high-frequency molecular group vibrations in addition to the lattice phonon continuum requires special consideration. An early treatment of CaWO, (RISEBERG [19683) indicated, at least qualitatively, a convergence of the multiphonon rate with energy gap. REEDand Moos [1973a, b], from a more detailed study
-
111,
8 61
131
MULTIPHONON R E L A X A T I O N
TABLE 6.1 Phenomenological parameters for multiphonon relaxation of rare-earth ions in crystals Crystal
LaCI,
hw,,,(cm-') C(sec-') 260
1.5 x 10"
E
Reference
1.3 x
0.037
RISEBERG, and Moos [1967, GANDRUD RISEBERG and Moos [I9681 ~~~ER~l967al;R~~E~ERGand Moos [I9681 WEBER C1968al; RISEBERG and Moos [19681 RISEBERG and Moos [1968] ZVEREV, KOLODNYI and ONIsHCHENKo c19711
a(cm)
LaBr,
175
1 . 2 10" ~
1.9 x lo-'
LaF,
350
6.6x 108
5.6x 1 0 - 3
0.037 0.14
550
2.7 108
3.8 10-3
0.12
2 '
3'
SrF,
360
3.1 x 10'
4.5 x
0.20
Y,AI,O,,
700
9.7 x 107
3.1 x 10-3
0.045
YAIO, LiYF,
600 400
5,Ox lo9
4.6~ 3.8 x
0.063 0.22
BaY F,
-
3.5 x lo7
4 . 5 ~ 1 0 ~ 4.1 x10-3
-
WEBER[1973b] JENSSEN [1971] JOHNSONand GUGGENHE~M [I9731
of relaxation in YVO,, YAsO,, and YP04, have shown that the rates can be critically dependent on the degree of resonance between the gap energy and the sums of peaks in the vibrational spectrum. Experimental results to date have been limited to crystalline hosts, however, similar dependences of multiphonon emission rates on temperature, energy gap, and host may be expected for rare earths in liquids and glasses. For liquids, in virtually all cases, there are high-frequency molecular vibrations to consider. Relaxation in liquids has been treated by the study of fluorescence lifetimes and efficiencies for a given transition in different solutions (differing in some cases by a single substitution). The vibrational spectrum is thus varied. Results have confirmed, in an approximate sense, the convergence of the multiphonon rates with the order of the process (KROPP and WINDSOR[1965]). Although multiphonon rates have not been extracted explicitly, it is probable that behavior similar to that observed in molecular crystals will be observed. Glasses also contain vibrational groups with frequencies higher than most phonon frequencies in crystals. Although the degree of coupling of these vibrations to rare-earth impurities is not well established, multiphonon decay bridging large energy gaps should be possible. This is reflected, qualitatively, by the smaller number of fluorescing levels observed for rare earths in glasses compared to crystals. Studies of radiative and nonradiative relaxation of rare-earth doped glasses by REISFELD[19731 and co-workers have already shown some evidence of a systematic behavior of multiphonon decay rates similar to that found in crystals.
132
RELAXATION PHENOMENA
CIII,
06
Overall, the phenomenon of multiphonon relaxation is reasonably well understood with a workable theory having predictive value and well supported by extensive experimental work.
6.3.4. 5d + 4f relaxation When excitation occurs via 5d states, 5d 3 4f relaxation is of interest. As shown in Fig. 3.1, the 5d states of trivalent rare earths are located at energies 5 50000 cm-'. Optical transitions to these states are therefore frequently masked by absorption of the host material. The 5d states of divalent rare earths (Fig. 3.2), in comparison, are located at lower energies. For most divalent and trivalent lanthanide ions, levels of the 4fN-' 5d configuration overlap those of the 4fNconfiguration. In such cases ions excited into 5d levels rapidly decay nonradiatively to nearby 4f levels. This is evident from the absence of 5d emission and the relative intensity and quantum efficiency of 5d bands in the excitation spectra of 4f fluorescence. Relaxation of 5d states has been investigated for trivalent rare earths in Y3AI5Ol2(WEBER[1973a]). In this host the crystal-field splitting of the 5d states is large, resulting in low-lying 5d levels. For ions such as Nd3+ and Tb3+,the broad 5d bands overlap 4f levels; no 5d 4f emission was observed and rapid nonradiative relaxation was inferred. For Ce3+ and Pr3+ , however, there are large energy separations from the lowest 5d level to levels of 4f, approximately 16500 and 10000 cm- ',respectively. 5d + 4f fluorescence was observed from these ions and the lifetime and intensity were measured as a function of temperature. The results are shown in Fig. 6.6. At elevated temperatures, the Ce3+ and Pr3+ 5d lifetimes exhibit a rapid decrease. Since the d -+ f energy gap for Ce3 is larger, more phonons are needed to conserve energy and a higher temperature is required before stimulated phonon processes compete with radiative decay. The temperature dependences in Fig. 6.6 are more rapid than for f + f relaxation and cannot be fitted using a single-frequency phonon model. LAUERand FONG [1974] obtained good agreement with the Pr3+ temperature dependence by using an intermediate strength coupling model, thus confirming that the weak coupling approximation is not valid for 5d relaxation. Nonradiative decay between the 5d levels of Ce3+ was also studied and found to be fast, although only a limiting rate of 7 5 x lo8 sec- ' at 77 "K was obtained. For comparison, the extrapolated multiphonon emission rate at 77°K between similarly spaced 4f levels in Y3A15012 is secWhereas intra4f and intra-5d nonradiative transitions occur between initial and final electronic states having the same parity, interconfiguration 5d-4f nonradiative transitions require that at least one odd--f
+
-
COOPERATIVE RELAXATION
133
Fig. 6.6. Temperature dependence of the 5d fluorescence lifetimes and intensities for Ce3+ [1973a]). and Pr3+ in Y,AISO,, (after WEBER
parity vibrational mode be active in the ion-lattice interaction. The A S = 0 spin selection rule should not be important since the spin-orbit interaction causes significant spin state admixing in both configurations. If emission can be observed from several high 5d levels, an energy gap dependence for nonradiative decay similar to those for 4fN levels might be obtained. Generally, however, the 5d levels are suficiently close that, based upon the above rates, a rapid nonradiative cascade to the lowest 5d would be expected to dominate.
0 7. Cooperative Relaxation 7. I . ION-ION ENERGY TRANSFER
7. I.I. Introduction
Paramagnetic ions in solids can be treated as isolated ions only when they are well separated. As the concentration is increased or if non-random distribution occurs, the ion spacing may become sufficiently small that the ions interact. The coupling of adjacent paramagnetic ions can arise via
134
R E L A X A T I O N PHENOMENA
[In, §
7
exchange interactions if their wavefunctions overlap, via super-exchange interactions involving intervening ions, or via various electric and magnetic multipolar interactions. Exchange effects are manifested in deviations from single-ion paramagnetism or in ferromagnetic or antiferromagnetic ordering. Bulk magnetic properties of crystals have been the subject of active research beginning in the early years of this century. With the advent of electron paramagnetic resonance (EPR) techniques after World War 11, it was possible to measure the magnitude of the fundamental interactions between rare-earth ion pairs in solids. The resulting knowledge of the nature and strength of the interactions obtained from dilute crystals could then be applied to explain the magnetic properties of concentrated materials such as magnetic transition temperatures, ordering characteristics in the magnetic phase, and collective excitations (magnons). High-resolution optical spectroscopy of magnetic materials have been directed toward gaining a similar understanding of ion-ion interactions (LEASK[1968]). Until the mid-l960’s, the field of energy transfer in rare earths progressed almost independently of the studies directed toward magnetic properties, despite the fact that some of the basic interactions are the same. The first systematic investigation of ion-ion energy transfer began in the 1940’s. These included observation of (1) sensitized luminescence, wherein excitation into the absorption bands of one class of ions (sensitizers or donors) resulted in emission from a second class of ions (activators or acceptors), (2) fluorescence quenching by relaxation to another ion acting as a nonradiative energy sink, and (3) self-quenching wherein the relative intensity of fluorescence was observed to decrease as the concentration of the activator was increased. These phenomena were observed even in relatively dilute systems, thus suggesting that long-range interactions were responsible. F~RSTER [19483 had introduced a formalism for treating energy transfer in molecules involving the electric dipole-dipole interaction. DEXTER [1953], in a classic paper, considered the role of electric dipole-dipole, electric dipole-quadrupole, and exchange interactions and derived expressions for the transition probabilities and for the luminescence yield as a function of concentration. The Forster-Dexter theory was the starting point for modern research on ion-ion processes for rare earths. Most work up until the late 1960’s was devoted to studying the concentration dependence of the fluorescence quenching and attempting to determine the dominant term in the multipolar expansion of the interaction for a given system (VANUITERT [1966]). INOKUTIand HIRAYAMA [1965] extended Forster and Dexter’s work by treating exchange coupling and including the time dependence of the
n ~0,
71
COOPERATIVE R E L A X A T I O N
135
fluorescence decay when ion-pair interactions are present. This latter feature yields an additional means of identifying the nature of the coupling. Energy transfer from the point of view of the magnitudes of the ion-ion interactions in the ground state, as measured by EPR, has been discussed by BIRGENEAU [19681. Recently KUSHIDA[1973a,b, c] has analyzed ionpair relaxation rates utilizing theoretical estimates of the interaction strengths. Resonant energy transfer between like ions gives rise to spatial migration of excitation. Relaxation by energy migration to quenching centers was proposed by BOTDEN[1952] to account for concentration quenching of fluorescence; the theory for this process was developed by DEXTER and SCHULMAN [1954].Within the past few years, the details and rates of energy migration processes for rare earths have been elucidated further, principally through studies of the time dependent behavior (WEBER[1971]). In addition to these developments, a phenomenological approach has evolved, one involving rate equations but not the explicit form of the interaction. This approach has seen wide application to the understanding of materials whose properties are dependent upon ion-ion coupling, such as infrared-to-visible upconversion phosphors (AUZEL[I 9731). 7.I .2. Theory There are several interactions involving two or more ions which provide the means for energy transfer and cooperative relaxation. Consider first the electric multipolar coupling arising from the Coulomb interaction between the electron charge clouds of two ions. Let rAi and rBj be the coordinate vectors of electrons i a n d j belonging to ions A and B, respectively. The electrostatic interaction is
where R is the internuclear separation and K is the dielectric constant. The various multipolar terms appear from a power series expansion of the denominator. This expansion is most succinctly expressed in terms of tensor operators (KUSHIDA[1973a]) : "2
c;:;
where is a numerical factor dependent on the orientation of the coordinate hxes and Or) is a multipole operator
136
RELAXATION PHENOMENA
Cni, 0 7
(7.3) The leading terms in the expansion in eq. (7.2) are the electric dipole-dipole (EDD), dipole-quadrupole (EDQ), and quadrupole-quadrupole (EQQ) interactions. These have radial dependences of R - 3 , R - 4 , and R-’, respectively. Higher-order terms are generally of negligible importance. The matrix elements of HEsare subject to selection rules AS = 0 and IALI, lAJl 5 k. As was discussed earlier for radiative decay, these rules are substantially relaxed for transitions between states of the 4fN configuration because of the large spin-orbit admixing of SL states and possible J state mixing. Whereas EQ transitions are parity-allowed, ED transitions require an admixing of opposite-parity states into 4fNand are correspondingly weak for rare earths. Because of this forbiddenness, EQQ transitions may be more probable than EDD transitions (AXEand WELLER[1964]). However, due to the more abrupt radial dependence, R-’ versus R - 3 , EDD interactions should dominate at large separations. The intensity of EDD transitions can be treated as in the Judd-Ofelt theory for radiative transitions. A second ion-ion coupling mechanism is the magnetic dipole-dipole interaction
where p i = li + 2si and ( l i ,si), ( I j , s j ) are the orbital and spin operators for the ith andjth electrons of ions A and B, respectively. The selection rules AS, AL, AJ = 0, -t 1 for transitions between 4fNstates are again relaxed by SLJ state admixing. The MDD interaction has the same long-range R T 3 radial dependence as the EDD interaction. Finally we consider the exchange interaction given by
J i j in eq. (7.5) represents the isotropic or Heisenberg component of the exchange integral. More generally Jij is a tensor with additional terms for the anisotropic and asymmetric contributions of the exchange interaction ( E R D ~[1966]). S For rare earths in solids, the large residual orbital angular momentum results in high-rank orbital contributions to Jij expressible in the form
(7.6)
111,s
73
137
COOPERATIVE R E L A X A T I O N
The high-rank terms can make significant contributions. Additionally, such terms relax the selection rules for exchange interactions within 4fNto IASI 5 1, IAL( S 6, IAJI 5 7. Again, because of state admixing, exchange is relatively free of selection rule restrictions.
I
A
B
-
(1)
(1 I
B
A
(b)
Fig. 7.1. Schematic diagrams of cooperative relaxation processes: (a) transfer involving excited state to ground state de-excitation of one ion and excitation of a neighboring ion to the same excited state; (b) transfer involving dissimilar pairs of levels of like or unlike ions (dashed lines for like ions).
138
RELAXATION PHENOMENA
Cm§7
The manner in which the interactions between two like or two unlike ions lead to energy transfer and relaxation is illustrated schematically in Fig. 7.1. The simplest process is shown in Fig. 7.la where an ion A in an excited state (2) decays to its ground state (1) with corresponding excitation of a neighboring ion B from its ground state (1) to (2). If A and B are identical ions, this process involves resonant transfer of energy from ion to ion. While this does not lead to net relaxation, it does give rise to spatial energy migration. In concentrated rare-earth materials resonant transfer can be fast. Discussion of energy migration is postponed to section 7.1.3; below we consider only a single ion pair. Fig. 7.lb depicts a process whereby an excited ion A decays from state (2) to (2’) while ion B, initially in its ground state, is excited to (1’). Here ions A and B may or may not be identical, as indicated by the dashed lines for possible matching levels. In the absence of any additional interactions, energy conservation imposes the constraint of resonance ; that is, (E, - E,,) = (El,-El). In liquids and solids, however, due to the presence of ion-lattice coupling, any energy mismatch may be taken up by the emission or absorption of one or more phonons. The interactions in eqs. (7.2), (7.4), and (7.5) can all be treated to include ion-lattice coupling (ORBACH [19671). The transition probability for energy transfer from ion A to B is given by
where HAB is the ion-pair coupling Hamiltonian and FA(,!?)and FB(E) are normalized line-shape functions for the transitions of ions A and B. Detailed expressions for these probabilities for the cases of EDD, EDQ, and direct exchange coupling were derived by DEXTER [1953]. The overlap integral in eq. (7.7) denotes a resonance model with the transition rate dependent on the degree of overlap of the two line-shape functions. These ion-ion interactions differ in their dependence on ion separation. The radial dependence of the ion-pair transfer rate is derived from the square of HABin eq. (7.7) and is therefore R - 6 , R-’ , and R-’O for EDD, EDQ, and EQQ coupling, respectively, and R - 6 for MDD. Direct exchange involves an exponential decrease of the wave-functions contained in the calculation of Jij and can be treated in terms of a radial dependenceexp (- R/L), where L is an effective Bohr radius of the ground and excited states under consideration (DEXTER[1953]). Thus W A B has an exp ( - 2 R / L ) radial dependence. Dexter correctly concluded that direct exchange, with its exponential radial dependence, is probably too short-range for effective
111,
P
71
COOPERATIVE RELAXATION
139
energy transfer in dilute materials and little attention was paid to the role of exchange until recently. [19531also treated the concentration dependence of the luminesDEXTER cence yield. For ion A, this is defined by
where WAB is given by eq. (7.7) and 7: is the radiative lifetime of state A(2). As the ion concentration is changed, the ion-ion separation and the probability for relaxation both change correspondingly. For a given interaction and functional dependence on R , the average yield yIA for a single-pair model is obtained by taking an integral over all space of qA times the probability that the nearest ion B is at a distance R . This yields OD
ijt)= l - y L
e-Yt
-dt, 1 + tnI3
(7.9)
with y = ynpB,the reduced density, and t = 4zR3/3yn.p B is the density of B ions and yn is defined by 7:WiB = 3yn/47cR3for n = 6 (EDD), n = 8(EDQ), and n = 10 (EQQ). A great body of experimental work over the past decade was devoted to applying the Dexter model to determine which multipolar term of HEs was dominant for energy transfer and luminescence quenching in a particular dopant-crystal system. Van Uitert and co-workers, among others, carried out an extensive series of studies in a wide variety of materials (see GRANT C1971) for a bibliography of this work). Although Dexter's treatment of luminescence yield describes the observed concentration dependences in a reasonable way, the assignments of the radial dependences and dominant interactions in a given case are often ambiguous. In some instances, with the available data, there was not sufficient differences in the theoretical dependences of yI(n) with n. There are several features of the overall approach which made analysis difficult. First, at concentrations large enough to show substantial ion-pair decay, resonant transfer (as in Fig. 7.la) can be exceedingly fast among ions A, particularly since the degree of resonance for ion-pair decay (as in Fig. 7.1b) can be expected to be generally smaller. Therefore, those A ions surrounded by a greater number of B ions than the average will dominate the decay, and the short-range (but stronger) interactions will be enhanced (i.e. EDQ and EQQ). At low concentrations, where the average separation is larger, the longer-range interactions, such as EDD, will be dominant. Thus, in a given material, different interactions dominate for different concentration
140
RELAXATION PHENOMENA
[Ill,
9: I
ranges. In the case where A and B represent different species, for sufficiently high concentrations of A (greater than a few %) resonant transfer within the A system is sufficiently fast that the excitation may be considered equally shared among all of the ions A. A rate equation approach for A 4B transfer is then justified (GRANT [1971]). An assumption of the Dexter approach was that the rate was dominated by transfer to the nearest B ions. An extension to the entire environment including the dynamics of the transfer was formulated by INOKUTIand HIRAYAMA [1965] (hereafter referred to as IH). The difficulty in treating the dynamics of these processes is that for an excited ion A decaying to a random distribution of quenching ions B, the environment of the A ions varies and the ion-pair relaxation rate depends on the particular environment of any excited ion A. Thus the decay of the system of excited ions A is nonexponential. In the IH approach, an ion A is considered to be surrounded by a set of quenching ions Bk at distances Rk. The energy transfer rate from an A ion to the kth B ion is W A B r (Rk). The time dependence of the A excited-state population is then (7.10)
where N is the total number of ions Bk in a finite volume surrounding A, and zA is the intrinsic lifetime of an isolated A ion. Experimentally the observed quantities are proportional to the statistical average 4(t) of p(t) over an infinitely large number of sensitizers. These volume integrals and &t) were calculated for a variety of interaction models by IH. For the inverse power laws characteristic of multipolar interactions, 4(t)is given by +(t) =
[
4, exp -
-r
(1-
-)-(-r’”] 3 C t 0‘
(7.11)
‘A
where C is the acceptor (B) concentration and n = 6,8,10 for EDD, EDQ, and EQQ, respectively. C , is a critical concentration defined by
c, =
m’
(7.12)
where R , is the separation at which the energy transfer rate for an isolated A-B pair is equal to 2, ’. The IH theory also treats the direct exchange interaction where @(t)has the form (7.13)
111,
0 71
COOPERATIVE RELAXATION
141
and y = 2Ro/L. g(z) in eq. (7.13) is a function defined by (-z)"I c m!(m+ 1)4 rn
=
62
(7.14) *
Thus energy transfer can be studied via the observation of the time dependence of thedecay at a particular concentration. #(t) may also be used in the analysis of other quantities. For example, the quantum yield ijA is simply given by (7.15)
and the mean lifetime by
Numerical integration of eqs. (7.15) and (7.16) can be performed as described by IH to determine these quantities as functions of C.
7.1.3. Energy migration Thus far only one-step processes involving resonant energy transfer between donors and acceptors have been considered. Relaxation by energy migration is a multistep process involving resonant energy transfer from one ion to another of the same species in a random walk manner and finally to an acceptor which acts as a quenching center or energy sink. The basic step in the migration is illustrated in Fig. 7.la. Migration becomes increasingly important as the rare-earth content is increased. In concentrated materials or where the rare earth is a constituent of the host, the probability for resonant energy transfer between donor ions may be large. Rapid energy diffusion can lead to a spatial equilibrium of excitation within the donor system. The rate limiting step for the donor relaxation then becomes either the donor-acceptor transfer rate or the acceptor relaxation rate. In the limit of fast diffusion, a simple rate equation model for the donor system relaxation can be used which predicts a simple exponential decay. When the rate of energy diffusion within the donor system is slow but still comparable to the intrinsic decay rate, the donor decay is composed of competing processes. Those excited donors near acceptors relax predominantly by direct ion-pair energy transfer ; those more distant donors, however, must first diffuse into the vicinity of an acceptor before relaxation
142
RELAXATION PHENOMENA
occurs. The time evolution of
[In.
07
4(t)is governed by a diffusion equation
where D is the diffusion constant and v(r - r,) is the probability for energy transfer from an excited donor to the nth acceptor at r,. YOKOTAand TANIMOTO [1967] obtained a general solution for the donor fluorescence decay function including both diffusion within the donor system and donor-acceptor energy transfer via EDD coupling. Their expression at earlier times in the decay reduces to eq. (7.1 1) for n = 6 . The decay at long times after excitation reduces to a simple exponential decay. The asymptotic solution at long times is described by a characteristic lifetime _1 --- 1 +4nN,D, (7.18) T
70
where N , is the density of acceptors and p is a length defined by (DEGENNES C19581) p = 0.68(C/D)* (7.19) which characterizes the relative effectiveness of direct transfer versus migration to acceptors. This solution applies when d e p e a, where d is the donor-donor separation and a is the distance between acceptors. The changing time-dependent decay behavior from an initial nonexponential decay given by eq. (7.11) to an exponential decay with a lifetime given by eq. (7.18) is a distinguishing feature of diffusion-limited relaxation (WEBER [1971]). 7.2. EXPERIMENTAL RESULTS
Since its appearance, the Inokuti-Hirayama theory has been employed extensively to interpret ion-ion energy transfer studies. Shionoya and co-workers have applied the IH model to several rare-earth systems (NAKAZAWA and SHIONOYA [19671; YAMADA, SHIONOYA and KUSHIDA [1972]). Fig.7.2 shows the results for TbtoNd transfer in Ca(FQ,), glass. The quantum yield ijA (referred to as Z/Zo) and a conveniently defined lifetime zh are plotted as a function of concentration. The theoretical fit for n = 8 suggests the dominance of the EDQ interaction. A variety of other combinations of ions in Ca(PO,), glass were examined and all showed more or less comparable agreement with EDQ coupling. The various theoretical dependences of 4(t)and ijA, however, are actually
111,
0 71
143
COOPERATIVE RELAXATION
1.0
Tb - Nd I= 8 0 0 exptl Calc
-
0.8
. 0
CI
I
2
0
-c 0
I 0.6
t
I
' c
y1
3
F
>
a
0.4
20 0.2
0.01
0.1
10
1
I
I
I
0.001
0.01
0.1
cO , I .
C
Fig. 7.2. Theoretically calculated concentration dependences (IH theory) of the emission intensity (or quantum efficiency) Zir, and the lifetime q,/rhoaf donor luminescence for the EDQ interaction. Circles show experimental data for Tb-to-Nd transfer in Ca(PO,), glass (after NAKAZAWA and SHIONOYA [1967]).
very close, and it is often difficult to make an unambiguous determination of the operative interaction. For example, Fig. 7.3 shows the Eu3+ 5Do luminescence decay for the system Y203: 1 % Eu, 4 % Yb. Although the EDQ interaction provides the best fit, other fits are only slightly beyond the experimental error bars. In contrast, Fig. 7.4 shows the luminescence decay of the 'F, level of Yb3+ in YF, : 0.3% Yb, 6% Ho where the fit for the EDD interaction is quite closely followed. Although the IH theory appears to describe such cooperative relaxation reasonably well, it suffers from limitations similar to those which plague the Dexter theory. Because of the requirement that resonant energy migration does not occur, such experiments are valid only at low concentrations. At higher concentrations, different multipolar terms may be expected to begin to dominate because of their different range dependences. KUSHIDA [1973a,b, c] has shown, from a detailed theoretical analysis and comparison of various experimental results, that at lower concentrations the EDD
144
RELAXATION PHENOMENA
I
I
('0.95
-
E'O.Ol
Yb0.04'23'
I
,
I
I
S=6 0
.
I
Eu 5D0: T,, = 880 ps
LNT
s=10
Fig. 7.3. Luminescence decay of the 'Do level of Eu3+ in Y , O , : l % Eu3+, 4 % Yb3+ at liquid nitrogen temperature. The theoretical curves show the IH calculations for EDD, EDQ and KUSHIDA[1972]). and EQQ interactions (after YAMADA, SHIONOYA
term should dominate because of its greater range. At high concentrations (small average separations) the forbiddenness of EDD within 4f"becomes important compared with higher-order terms and the relation EQQ > EDQ > EDD gradually becomes operable. Since the contributions from each of these processes are somewhat comparable, definitive identification of multipolar processes from an IH type analysis is difficult. GRANT[1971], from a theoretical examination of the energy transfer process, has concluded that concentration dependences do not reflect multipolar terms, but rather the number of particles (ions) participating . in the relaxation process. Exchange has never been definitively associated with a cooperative relaxation process for rare earths from fits using the IH model. BIRGENEAU [19681 has discussed the possible contribution of exchange for nearest neighbor (nn) and sixth nearest-neighbor (6nn) coupling. In the nn case, direct exchange was found to be comparable with EQQ, which is expected to be the most important multipolar term for small separations. Depending on the particular system under consideration, either of the two interactions could dominate. For the 6nn case, it was found that superexchange via the intervening ion ligands could be comparable with the long-range multipolar interaction terms (MDD as well as electric). Here the situation
111,
5 71
145
COOPERATIVE RELAXATION
>
t v)
z z W
c
Y0
10'
z
Lu
P W
z
s
\
4
'\
\
\
\
\
\ \
10-2
0
I
1
0.2
0.4
\
I 0.6
0.8
1.0
TIME (mi)
Fig. 7.4. Luminescence decay of the Yb3+ 'F, level in YF,: 1 % Yb3+, 10% H o 3 + .The solid curve is experimental. Theoretical curves for exchange (dashed curve) and EDD (short and long dashed curve) interactions using the IH theory are included (after WATTSand RICHTER [19723).
146
RELAXATION PHENOMENA
[In,
6
7
is even more unclear, since the expected concentration dependence of superexchange would be difficult to analyze. While a dependable analysis of the interactions contributing to cooperative relaxation is still lacking, much work has been done by treating the relaxation phenomena in a more phenomenological fashion. For example, the resonance imposed by the overlap integral of the Dexter model (eq. (7.7)) has led to a variety of experiments to determine the dependence on this parameter. NAKAZAWA and SHIONOYA [19673 compared the transfer rates derived from the IH model for a variety of relaxation processes with the expected dependence on the overlap integral. Using measured linewidths in their calculations, they found very good agreement. One of the implications of the resonance condition is the different rates expected for different Stark components of a given J state. Thus, in addition to line-broadening and line-shift effects, a substantial temperature dependence is introduced as a result of the change in the thermal distributions of ions among the various Stark levels. The temperature dependence can be studied phenomenologically via a straightforward consideration of Boltzmann statistics (ASAWA and ROBINSON [19661). Many systems which exhibit ion-pair decay involve nonresonant processes and large energy mismatches requiring the participation of many phonons to conserve energy. In light of the discussion in section 6 of single-ion multiphonon relaxation, an exponential dependence of the rate on the energy mismatch might again be expected to be operative. Because the additional ion-ion coupling term in multiphonon-assisted pair relaxation may well depend in a substantial way on atomic parameters, the gap dependence might not hold quite so well as in the single-ion case. The statistical averaging over the orbit-lattice interaction should still be relevant. MIYAKAWA and DEXTER [1970] in their theoretical analysis of multiphonon processes derived a cooperative relaxation analog of the multiphonon gap dependence of eq. (6.12). This is given by W
=
Ce-PAE
(7.20)
where p is related to the single-ion multiphonon exponent a by f l = a -y and (7.21) In eq. (7.21) g B and g A are electron-lattice coupling parameters for ions A and B, respectively. YAMADA, SHIONOYA and KUSHIDA [1972] carried out an experimental study of the above energy gap dependence for a variety of rare earth ion-pair systems in Y,O,. Assuming the interaction
n ~ 0, 71
147
COOPERATIVE R E L A X A T I O N
to be EDQ, they determined the cooperative relaxation rate for an interionic separation R A B = lOA (corresponding to a concentration of B = 4 mol. %). The results are shown in Fig. 7.5 ;an exponential gap dependence is obeyed reasonably well with a B value of 2.5 x l o p 3 cm. Taking g B = gA and a value of a of a 3.8 x 10-3 cm from Table 6.1 yields a 8 of 2.2 x cm, which is in excellent agreement with the results of Fig. 7.5.
1o5
I
I
9
Hd5S2) +Sm
Hd5S2) +Tm
a 2
1o4
Y203 LNT
Ed4S312)+Yb
II
a 7
-8 v)
2
103
a
Sm(4G512) +ELI Tm(’G4) +Vb
w
U
v)
z
2I-
> Q
E 2
102
w
10
I
I
1
I
\
ENERGY GAP. (cm-1)
Fig. 7.5. Energy gap dependence of multiphonon assisted energy transfer in Y,O, (after YAMADA, SHIONOYA and KUSHXDA [1972]).
148
RELAXATION P H E N O M E N A
(1) A
t :: : 1: : 82
51
- I (2)
3 . (1)
1 (2’
(1)
(1)
B
A2
(b)
1)I-
hu
A
B lc)
Fig. 7.6. Schematic diagrams of various cooperative relaxation phenomena : (a) multi-ion relaxation, (b) cooperative excitation, (c) Raman luminescence.
111, §
71
149
COOPERATIVE R E L A X A T I O N
Several additional related processes of cooperative excited-state relaxation have been observed experimentally. One is decay involving more than two ions which is indicated schematically in Fig. 7.6a. Here an ion A decays, exciting two neighboring ions B1 and B2 to excited levels, the sum of whose energies equals the transition energy of ion A. PORTER[1968] has identified such a process in LaCI, : Ho3+. The inverse of this process, cooperative excitation, is shown in Fig. 7.6b. Here two ions, A and A’, decay with the energy going into excitation of a third ion. This process and OVSYANKIN [1967]. A third example has been observed by FEOFILOV is cooperative luminescence in which two excited ions decay simultaneously emitting a single photon equal to the sum of the transition energies of the two ions. This process was observed in YbP04 by NAKAZAWA and SHIONOYA [19703. Finally, the process shown in Fig. 7.6c, known as Raman luminescence, was observed in Yb203: Gd3+ by FEOFTLOV and TROTIMOV [1969]. In this case an ion A decays, simultaneously exciting ion B and emitting a photon at an energy equal to the energy difference between ions A and B. Similar processes have been observed involving one rare earth and one iron group ion (VANDER ZIEL[1970]) and have been described as rareearth-terminated chromium fluorescence. The orbit-lattice interaction is operative for the above processes, and there is a rich variety of relaxation phenomena involving the ion-ion interaction in various combinations with radiative and phonon-induced transitions. Theoretical treatments of such processes have been presented by DEXTER [19623. Another area where phenomenological treatment has produced useful descriptions of experimental information arises when the concentrations of donor ions A is large and fast migration averages the environment of acceptor ions B. GANDRUD and Moos [I9681 have studied fast energy diffusion between rare earths in concentrated crystals. Because of the condition of an averaged environment, the use of a rate equation model is appropriate. Transition rates can be simply determined from the measurement of the exponential decay of the excited ions. These approaches have seen wide utilization in studies related to infrared upconversion phosphors (KINGSLEY [1970]; OSTERMAYER, VAN DER ZIEL, MARCOS, VAN UITERT and GEUSIC[19711; WATTS[19701; KUSHIDA [1973bl). These phosphors are discussed in section 8. Several investigations in the past few years have demonstrated the existence of diffusion-limited relaxation. The first study by WEBER[1971] used a chromium-doped europium metaphosphate glass, where excited Eu3 ions +
150
RELAXATION PHENOMENA
CIIL
08
formed the donor system and Cr3+ impurities acted as energy acceptors. When Cr3+ ions were present, the initial Eu3+ fluorescence decay was nonexponential, as given by eq. (7.11); the final decay, however, was exponential with a rate dependent on Cr3+ concentration as given in eq. (7.18). Since the rate of energy diffusion within the Eu3+ system could be varied by changing the temperature, and thereby the number of resonant transitions, it was possible (1) to verify the Da law predicted by combining eqs. (7.18) and (7.19); and (2) to cover the range ofrelaxation from diffusion limited to fast diffusion. VAN DER ZIEL,KOPFand VAN UITERT [1972] conducted similar studies of Tb3+ relaxation in (Y, -xTb,)3Al,CJ,, crystals and again verified the Da dependence for dipolar processes. Because rapid resonant migration is possible in concentrated rare-earth materials, it was concluded that fluorescence quenching could frequently be due to energy -migration to unavoidable contaminants which serve as nonradiative sinks. Other studies of energy migration include those of WATTSand RICHTER[1972] (Yb3+ -, Ho3+) and KRASUTSKY and Moos [1973] (Pr3+ -,Nd3+). In these experiments the concentration of both the donor and acceptor ions were varied, thus changing the rate and relative importance of energy migration. Recently the quantum efficiency of diffusion-limited energy transfer between Ce3+ and Tb3+ was studied in Lal-,-,Ce,Tb,PO, (BOURCET and FONG[19741). Here the diffusion of donor energy involves d-f transitions. A recently developed technique for the observation of energy migration in rare-earth systems involves excitation with a narrow-band laser in a system with significant inhomogeneous broadening (RISEBERG [1973]).In such a system excitation is selective, involving only a narrow energy band within the inhomogeneously broadened system. A narrowed fluorescence signal is observed, corresponding to emission by that class of ions that has been excited. Migration from these ions to other ions can then be studied by observation of the line-narrowed signal. Recently, a time-resolved study of such laser-induced line-narrowing effects was carried out by MOTEGIand SHIONOYA [I9731 for Eu3+-doped Ca(PO,), glass. They observed migration between Eu3+ ions located at sites having different crystal fields, and were able to assign to the transfer process a phononassisted EDD interaction.
0
8. Selected Applications
A significant impetus for the study of rare-earth relaxation during the
111,
§ 81
151
SELECTED A P P L I C A T I O N S
last decade has been the application of rare-earth activated materials as phosphors and in a variety of quantum electronic devices, particularly lasers. For example, the luminescence of Eu3+ in the 600-650 nm region has become the standard red phosphor emission in color television. Other significant uses of rare-earth phosphors include fluorescent lamps and X-ray image intensifying screens. In the case of lasers, a total of eleven different rare earths (Pr3+, Nd3+, Sm2+, Eu3+, Tb3+ D 2 + , 3 + H 3 + Er3+, T m 2 + * 3 +Yb3+, , U3+s4+)have been lased in crystals, glasses and liquids. The single class of lasers enjoying the widest practical utilization today is based on the 4F, + 'Iy emission of the Nd3+ ion at 1.06 pm. In applications involving the luminescence of rare earths, the relaxation processes discussed in this chapter are implicitly involved in a critical way. After excitation, whether by optical, X-ray, cathode-ray, or other means, it is the rates and relative importance of various relaxation processes that determine what the emission spectrum will be, its intensity and efficiency, and the lifetimes of the initial and final states. Below we discuss two representative examples in which relaxation phenomena have been considered explicitly in the design of luminescent devices. 9
Y
9
0
,
8.1. UPCONVERSION PHOSPHORS
AUZEL[1966] was the first to demonstrate the use of sequential excitation whereby infrared light absorbed by Yb3+ ions is converted to visible light emitted by Ho3+, Er3+, or Tm3+ ions. This infrared-to-visible upconversion process for the Yb3+-Er3+ system is indicated schematically in Fig. 8.1. Here an Yb3+ ion excited into the 2F+level relaxes by interacting with an Er3+ ion and raising it to the 41y level. Subsequently, a second excited Yb3+ ion relaxes and raises the previously excited Er3+ to a higher 4F, level. Phonon-induced relaxation across the small energy gaps lying above the 4S, follows. Fluorescence in the green at 540 nm occurs from the 'S, to the 'Iy ground state, thereby completing the upconversion process. The Yb3 concentrations in IR-to-visible upconversion materials are relatively high (usually 10% or greater). Fast resonant transfer of the excitation among the Yb3+ ions is therefore possible and variations in the Er3+ surroundings are effectively averaged out. Because the Yb3+ system sees an averaged environment, the excited-state dynamics can be treated in terms of rate equations, HELVES and SARVER[1969] introduced the use of rate equations in the investigation of these processes. Other workers (KINGSLEY [19701; WATTS[19701; JOHNSON, GUGGENHEIM, RICH and +
152
68
[I..
R E L A X A T I O N PH E NO M E N A
4 2 / :2
A3
4s3/2
s2
- -2FY2
s2
-
.97um
Sl
2
F?/2
Yb3+
Er3+
Yb3+
Fig. 8.1. Schematic diagram of infrared upconversion in the Yb3+-Er3+ system. Sequential relaxation of two Yb ions results in excitation ofvisible luminescence from the Er3+4S, level.
[1972] ; OSTERMAYER, V A N DER ZIEL, MARCOS,V A N UITERT and GEUSIC [1971]; MITA [1972]) have applied this form of analysis to other systems with good success. A general set of rate equations describing the upconversion process (after AUZEL[1973]) are given below. For the population Ai of the ith level of the activator (having n levels), OSTERMAYER
dA. dt
n
i- 1
2= S, I= 1
C
Ajcrji-S2
Aiaij+
Aiwji-
Ajflji-SI
n
C Aiflij-
j= 1
j=i+l
C Aiwij
j= 1
i- 1
n
C
C
j=i+ 1
j=i+l
+Sl
i- 1
n
i- 1
1
AiAjyji-
AiAjyij
j=i+l
(8.1)
j= 1
and for the population S , of the excited level of the sensitizer dS2 dt
__ =
n
SIR-S,
n-1
C C aijAi
w,+
j=i+l
+S,
i=l
n
n-1
j=,
i=l
1
fljiAj.
(8.2)
Since the total populations A and S are constant, one has the additional n condition CAi i=1
=
A;
S,+S,
=
s.
(8.3)
111,
0 81
153
SELECTED A P P L I C A T I O N S
The transition probabilities aij, P i j , y i j , and wij are independent of sensitizer and activator concentrations and are defined as aij = transfer rate from Yb3+ to Er3+ Pij = back transfer rate from Er3+ to Yb3+ y V. . = Er3+ to Er3+ transition rate wij = single-ion decay rate (radiative plus nonradiative) R is the rate of excitation of the Yb3+ ions. Whereas eqs. (8.1) and (8.2) have been written in a more general form than necessary to describe the situation of Fig. 8.1, the equations can readily be specialized to any of the IR upconversion systems. For a complex ion-ion system with many levels, the above set ofequations contains a large number of rate constants. Some of the rates can be determined from measurements of fluorescence lifetimes and quantum efficiencies. Others, such as multiphonon relaxation rates, can be estimated from the guidelines and phenomenological models discussed earlier. In any given system, the large number of unknown parameters in eqs. (8.1) and (8.2) may necessitate the use of other simplifying assumptions. Thus, for example, HEMS and SARVER[1969] reduced the number of levels to be considered and treated only three levels of Er3+ ("Iy,"Ly and "S,). In other cases, back transfer has either been neglected (HEWS and SARVER [1969]) or been considered to be as probable as direct transfer and much faster than other de-excitation processes for either of the levels (KINGSLEY [19701). The justification of such assumptions must ultimately depend upon the relative rates of the associated relaxation processes. Studies of the kinetics are useful in assessing the performance of these systems. One can predict, for example, the dependence of efficiency on 10.1
0.1
1
10
lo2
103
X(Sei1)
Fig. 8.2. Power efficiency E (normalized to radiative quantum efficiency q ) for the Ho3+ green emission in YF, :Yb, Ho as a function of Yb excitation rate X (after WATTS[19703).
154
RELAXATION PHENOMENA
[III,
5
8
input power. By a combination of experimental measurements and phenomenological analysis, WATTS[19701 was able to solve the kinetics and extract the power efficiency of green emission from Ho3+ in YF, : Yb, Ho. The efficiency E is shown plotted against the excitation rate X in Fig. 8.2 normalized to the radiative quantum efficiency q of the 5S2, 5F4state. The significant point of the analysis is the saturation behavior predicted at high values of E . 8.2. LASERS
All of the relaxation phenomena we have discussed (radiative, multiphonon, cooperative) are implicitly relevant to the development and evaluation of the performance of rare-earth lasers. A well-known example is the Nd3+ laser. Referring to the energy level scheme in Fig. 2.1, optical pumping is achieved by absorption into the levels lying above 4F,. For high efficiency the medium should exhibit rapid nonradiative decay from these levels directly to the 'F, level via a cascade of multiphonon emission processes. Multiphonon relaxation from the 4F,, in contrast, should be slow compared to its radiative decay. The strength of the radiative transition from 4F, to 41y determines the stimulated emission cross-section and thereby the threshold, gain, and energy storage properties of the medium. To maintain the characteristics of a four-level system, multiphonon decay from the 'I? to the 41, must be fast compared to decay from 'F, . Finally, if the Nd3' content becomes too high (above 1-5 % , in most cases), selfquenching of the fluorescence sets in, involving 4F, -+ '% decay of one ion with excitation of a neighboring Nd3+ ion to either 'L!+ or 'IT. Various materials do not satisfy the above requirements. For example, soft crystals suffer from bottlenecking in the cascade to the 'F, because optical phonon energies are low and multiphonon rates are slow. In certain glasses, such as the borates, the presence of high-energy vibrational quanta result in partial nonradiative quenching of the 4F+ level. Fortunately, the desirable properties are observed for most hard crystals (high-energy optical phonons) at Nd3+ concentrations of less than a few percent. It is interesting to note that the Nd3+ laser was invented before such refined analysis of the basic relaxation phenomena was made. Utilization of energy transfer to increase the optical pumping efficiency was an early feature of the investigatory stages of rare-earth solid-state lasers. One of the most successful applications of energy transfer is the Ho3+ laser. Increased absoprtion and pumping efficiency were obtained by multiple-doping with Er3+,Tm3+,and Yb3+ to sensitize the Ho3+ 51,
111,
8 91
CONCLUDING REMARKS
155
fluorescence. The approach was first developed for Y3A1501 (JOHNSON, GEUSICand VANUITERT[1966]), and has subsequently been extended to other hosts. Co-doping with high concentrations of Er3+ and Yb3+ yields intense absorption bands which provide large spectral overlap with broadband pump sources. A complex sequence of ion pair processes results in most of the excitation going into the 'I, level of Ho3+.This is the lowest excited J level and remains metastable. The Tm3+,although it also contributes to the absorption, serves mainly to provide additional levels in the fast energy transfer sequence. This sensitized laser is the most efficient optically pumped solid-state laser ever operated. Another more recent example of the use of relaxation phenomena in laser design is LiYF,: Tb3+ (JENSSEN,CASTLEBERRY, GABBEand LINZ [1973]). Optical pumping of Tb3+ is via the levels above 'D, . The 5D3-to5D, gap, however, is too large to be bridged by multiphonon decay. To circumvent this, the concentration of Tb3+was raised to 25 % to increase the probability for 'D3 to 'D, decay via ion-pair relaxation involving resonant or near-resonant transitions of the type 'D3 + 'D4 : 7F6+ 7F0, or 5D3+ 7Fo, : 7F6+ 'D,. Lasing then occurs from 'D4 to 7F5.Since the 7F, to 7F6gap is small, the 7F5terminal laser level is rapidly depleted by multiphonon decay, thus completing a four-level laser scheme. There are many other examples of the importance of relaxation phenomena in applications of luminescent rare-earth materials. Those' discussed above indicate, in a general way, how such processes have been considered in the past. Such considerations no doubt will play a similar role in the development of future materials and devices.
3
9. Concluding Remarks
In t h s chapter, we have attempted to survey the relaxation phenomena affecting the luminescent properties of rare-earth ions in solids. Due to the rapid growth of activity in this field in recent years, the literature is voluminous and therefore no attempt was made to cover it completely. Instead, examples were chosen for their appropriateness in illustrating the discussion. The coverage has therefore been selective in detail, yet broad in scope. The intention has been to acquaint the non-specialist with the subject and to provide the applications-oriented individual with a familiarity with the basic phenomena having implications on his project. In the case of radiative relaxation, the Judd-Ofelt theory provides a useful rationale for parametrizing the phenomena. Through a combination of experimental measurements and theoretical calculations, the radiative
156
RELAXATION PHENOMENA
[I11
transitions of the lower 4fNlevels of rare earths can be satisfactorily accounted for by the worker willing to devote the effort required to carry out such an analysis. In the case of multiphonon relaxation, the phenomenological model provides a powerful description. It is relatively straightforward to apply in most cases, with an accuracy which, while not precise, is adequate for semi-quantitative evaluation and prediction of performance. For cooperative relaxation, the situation is not as satisfactory. The question of which basic interaction dominates cannot be easily answered for a given situation. Nor is it clear what approach can shed light on a more general description of ion-pair interactions. Although qualitative remarks can be made, quantitative predictions of an ion-pair decay rate for a given rare-earth ion concentration in a given crystal are subject to large error. As in most fields that have matured to any degree, it appears that the remaining problems are the most difficult.
References ASAWA,C. K. and M. ROBINSON, 1966, Phys. Rev. 141, 251. AUZEL,F., 1966, C. R. Acad. Sci. (Paris) 262, 1016. AUZEL,F., 1973, Proc. IEEE 61, 758. AXE,J. D., 1963, J. Chem. Phys. 39, 1154. AXE,J. D. and P. F. WELLER,1964, J. Chem. Phys. 40,3066. BARASCH, G . E. and G. H. DIEKE,1965, J. Chem. Phys. 43, 988. BECKER, P. J., 1971, Phys. Status Solidi B 43, 988. BECQUEREL, J., 1907, Radium 4, 328. BETHE,H., 1930, Z. Physik 60,218. BIRGENEAU, R. J., 1968, Appl. Phys. Letters 13, 193. BLEANEY, B. and K. W. H. STEVENS, 1953, Rept. Prog. Phys. 16, 108. BOTDEN, T. P. J., 1952, Philips Res. Repts. 7, 197. BROER,L. J. F., C. J. GORTER and J. HOOGSCHAGEN, 1945, Physica 11, 231. BOURCET, J. C. and F. K. FONG,1974, J. Chem. Phys. 60, 34. BUKIETYNSKA, K. and G. R. CHOPPIN,1970, J. Chem. Phys. 52 (6), 2875. CARNALL, W. T., P. R. FIELDS and K. RAINAK,1968, J. Chem. Phys. 49, 4412, 4424, 4443, 4447, 4450.
CARNALL, W. T., P. R. FIELDS and B. G. WYBOURNE, 1965, J. Chem. Phys. 42,3797. CHAMBERLAIN, J. R., A. C. EVERITT and J. W. ORTON,1968, J. Phys. C 1, 157. CHAMBERLAIN, J. R., D. H. PAXMAN and J. L. PAGE,1966, Proc. Phys. SOC.(London) 89,143. DE GENNES,P. G., 1958, J. Phys. Chem. Solids 7, 345. DELSART, C. and N. PELLETIER-ALLARD, 1971, J. Phys. (Pans) 32, 507. DELSART, C. and N. PELLETIER-ALLARD, 1973, J. Phys. C 6, 1277. DETRIO,J. A., 1971, Phys. Rev. B 4, 1422. DEXTER,D. L., 1953, J. Chem. Phys. 21, 836. DEXTER,D. L., 1962, Phys. Rev. 126, 1962. DEXTER,D. L. and J. H. SCHULMAN, 1954, J. Chem. Phys. 22, 1063. DIBARTOLO, B., 1968, Optical Interactions in Solids (Wiley, New York). DIEKE, G. H., 1961, Spectroscopic Observations on Maser Materials, in: Advances in Quantum Electronics, ed. J. R. Singer (Columbia University Press, New York) pp. 164186.
HI1
REFERENCES
157
DIEKE, G. H., 1963, Energy Levels of and Energy Transfer in Rare Earth Salts, in: Paramagnetic Resonance, ed. W. Low (Academic Press, New York) pp. 237-252. DIEKE,G. H., 1968, Spectra and Energy Levels of Rare Earth Ions in Crystals (John Wiley and Sons, Inc., New York). EL’YASHEVICH, M. A., 1953, Spectra of the Rare Earths (State Publishing House of TechnicalTheoretical Literature, Moscow). ~RDCSS,P., 1966, J. Phys. Chem. Solids 27, 1705. 1967, Appl. Opt. 6, 1828. FEOFILOV, P. P. and V. V. OVSYANKIN, FEOFILOV, P. P. and A. K. TROTIMOV, 1969, Opt. and Spect. 27, 291. FONG, F. K., S. L. NABERHUIS and M. M. MILLER,1972, J. Chem. Phys. 56,4020. FCIRSTER,T., 1948, Ann. Physik 2, 55. F~RSTER,T., 1949, Z. Naturforsch. 49, 321. FOWLER, W. B. and D. L. DEXTER,1962, Phys. Rev. 128, 2154. FOWLER, W. B. and D. L. DEXTER,1965, J. Chem. Phys. 43, 1768. 1970, J. Chem. Phys. 52, 6272. FREED,K. F. and J. JORTNER, GANDRUD, W. B. and H. W. Moos, 1968, J. Chem. Phys. 49, 2170. G. and 0. J. SOVERS,1969, J. Chem. Phys. 50, 429. GASHUROV, GERMAN, K. R. and A. KIEL, 1973, Phys. Rev. 88, 1846. GRANT,W. J. C., 1971, Phys. Rev. B5,648. HELLWEGE, K. H., 1942, Ann. d. Phys. 40,529. HELLWEGE,K. H., 1947, Naturwiss. 34,212. 1969, Phys. Rev. 182, 427. HEWS, R. A. and J. F. SARVER, 1971, J. Phys. SOC.Japan 31, 828. HOSHINA,T. and S. KUBONIWA, M. and F. HIRAYAMA, 1965, J. Chem. Phys. 43, 1978. INOKUTI, H. P., 1971, Phonon Assisted Laser Transitions and Energy Transfer in Rare Earth JENSSEN, Crystals, Technical Report No. 16 (M.I.T. Crystal Physics Laboratory, Cambridge). JENSSEN, H. P., D. CASTLEBERRY,.D. GABBE and A. LINZ,1973, IEEE J. Quant. Elec. QE9,665. 1966, Appl. Phys. Letters 8,200. JOHNSON, L. F., J. E. GEUSICand L. G. VAN UITERT, JOHNSON, L. F., H. J. GUGGENHEIM, T. C. RICHand F. W. OSTERMAYER, 1972, J. Appl. Phys. 43, 1125. 1973, Appl. Phys. Letters 23, 96. JOHNSON, L. F. and H. J. GUGGENHEIM, JI$RGENSEN, C. K. and B. R. JUDD,1964, Mol. Phys. 8, 281. JUDD,B. R., 1962, Phys. Rev. 127, 750. JUDD,B. R., 1966, J. Chem. Phys. 44, 839. KIEL, A., 1964, Multi-Phonon Spontaneous Emission in Paramagnetic Crystals, in : Quantum Electronics, eds. P. Grivet and N. Bloembergen (Columbia University Press, New York) pp. 765-777. KINGSLEY, J. D., 1970, J. Appl. Phys. 41, 175. KISLIUK,P. and C. A. MOORE,1967, Phys. Rev. 160, 307. KRAMERS, H. A,, 1930, Proc. Acad. Sci. Amsterdam 32, 1176. KRASUTSKY, N. and H. W. Moos, 1973, Phys. Rev. BS, 1010. KROPP,J. L. and M. W. WINDSOR, 1965, J. Chem. Phys. 42, 1599. KRUPKE,W. F., 1966, Phys. Rev. 145, 325. KRUPKE,W. F., 1971, IEEE J. Quant. Elec. QE-7, 153. KRUPKE,W. F., 1972, IEEE J. Quant. Elec. QE-8, 725. KRUPKE,W. F., 1974, IEEE J. Quant. Elec. QE-10, 450. KRUPKE,W. F. and J. B. GRUBER,1965, Phys. Rev. 139, 2008. S. and T. HOGHINA, 1972, J. Phys. SOC. Japan 32, 1059. KUBONIWA, KUSHIDA, T., 1973a, J. Phys. SOC.Japan 34, 1318. T., 1973b, J. Phys. SOC.Japan 34, 1327. KUSHIDA, KUSHIDA, T., 1973c, J. Phys. SOC.Japan 34, 1334. LAUER,H. V. and F. K. FONG,1974, J. Chem. Phys. 60,274. LEASK,M. J. M., 1968, J. Appl. Phys. 39, 908.
158
RELAXATION PHENOMENA
[111
LOH,E., 1966, Phys. Rev. 147, 332. LOH, E., 1968, Phys. Rev. 175, 533. MANTHEY, W. J., 1973, Phys. Rev. B8, 4086. MITA,Y., 1972, J. Appl. Phys. 43, 1772. 1970, Phys. Rev. Bl, 2961. MIYAKAWA, T. and D. L. DEXTER, Moos, H. W., 1970, J. Luminescence 1, 106. 1973, J. Luminescence 8, 1. MOTECI,N. and S. SHIONOYA, E. and S. SHIONOYA, 1967, J. Chem. Phys. 47, 322. NAKAZAWA, 1970, Phys. Rev. Letters 25, 1710. NAKAZAWA, E. and S. SHIONOYA, NEWMAN, D. J., 1971, Adv. Phys. 20, 197. NIELSON, C. W. and G. F. KOSTER,1964, Spectroscopic Coefficients for the p", d" and t" Configurations (The M.I.T. Press, Cambridge, Massachusetts). OFELT,G. S., 1962, J. Chem. Phys. 37, 511. ORBACH,R., 1961, Proc. Roy. SOC.A264,458. ORBACH,R., 1967, Phonon Sidebands and Energy Transfer, in: Optical Properties of Ions in Crystals, eds. H. M. Crosswhite and H. W. Moos (Wiley Interscience, New York) pp. 445-455. Jr., F. W., J. P. VAN DER ZIEL,H. M. MARCOS,L. G. VAN UITERTand J. E. OSTERMAYER GEUSIC,1971, Phys. Rev. B3, 2698. PARTLOW, W. D. and H. W. Moos, 1967, Phys. Rev. 157, 252. PEACOCK, R. D., 1971, J. Chem. SOC.A 2028. PEACOCK, R. D., 1972a, J. C. S. Faraday II,68, 169. PEACOCK, R. D., 1972b, Chem. Phys. Letters 16, 590. PEACOCK, R. D., 1973, Mol. Phys. 25, 817. PORTERJr., J. F., 1968, Bull. Am. Phys. SOC.13, 102. REEDJr., E. D. and H. W. Moos, 1973a, Phys. Rev. B8, 980. REEDJr., E. D. and H. W. Moos, 1973b, Phys. Rev. B8,988. REISFELD, R., 1973, Spectra and Energy Transfer of Rare Earths in Inorganic Glasses, in: Structure and Bonding, Vol. 13, eds. J. D. Dunitz, P. Hemmerich, J. Ibers, C. J$rgensen, J. Neilands, R. Nybalm, D. Reinen and R. Williams (Springer-Verlag, Berlin, Heidelberg, New York) pp. 53-98. 1973, Proc. Tenth Rare Earth Res. Conf. REISFELD, R., L. BOEHM, N. LIEBLICH and B. BARNETT, 2, 1142. RINCK,B., 1948, Z. Naturforsch. 3. 406. RISEBERG, L. A,, 1968, Multiphonon Orbit-Lattice Relaxation of Excited States of Rare Earth Ions in Crystals, Ph.D. Thesis (The John Hopkins University, Baltimore). RISEBERG, L. A., 1973, Phys. Rev. A7, 671. and H. W. Moos, 1967, Phys. Rev. 159,262. RISEBERG, L. A., W. B. GANDRUD RISEBERG, L. A. and H. W. Moos, 1967, Phys. Rev. Letters 19, 1423. RISEBERG, L. A. and H. W. MOOS,1968, Phys. Rev. 174, 429. STRUCK,C. W. and W. H. FONGER,1971, Phys. Rev. B4, 22. STURGE,M. D., 1973, Phys. Rev. B8, 6. VAN DER ZIEL,J. P., 1970, J. Luminescence 1, 807. VANDER ZIEL,J. P., L. KOPFand L. G. VAN UITERT,1972, Phys. Rev. B6,615. VAN UITERT,L. G., 1966, Luminescence of Insulating Solids for Optical Masers, in: Luminescence of Inorganic Solids, ed. P. Goldberg (Academic Press, New York) pp. 465539. VANVLECK,J. A,, 1937, J. Phys. Chem. 41, 67. WAITS, R. K., 1970, J. Chem. Phys. 53, 3552. WATTS,R. K. and H. J. RICHTER,1972, Phys. Rev. B6, 1584. WEBER,M. J., 1966, Radiative and Nonradiative Transitions of Rare-Earth Ions: Er3+ in LaF,, in : Physics of Quantum Electronics, eds. P. L. Kelley, B. Lax and P. E. Tannewald (McGraw-Hill Book Co., New York) pp. 35CL360.
IIl]
REFERENCES
159
WEBER,M. J., 1967a. Phys. Rev. 157, 262. WEBER,M. J., 1967b, Relaxation Processes for Excited States of Eu3+ in LaF,, in: Optical Properties of Ions in Crystals, eds. H. M. Crosswhite and H. W. Moos (Wiley Interscience, New York) pp. 467484. WEBER,M. J., 1968a, Phys. Rev. 171, 283. WEBER,M. J., 1968b, J. Chem. Phys. 48, 4771. WEBER,M. J., 1971, Phys. Rev. B4,2932. WEBER,M. J., 1973a, Solid State Commun. 12, 741. WEBER,M. J., 1973b, Phys. Rev. B8, 54. WEBER,M. J., 1973c, Proc. Tenth Rare Earth Res. Conf. 2, 932. WEBER,M. J., B. MATSINGER, V. L. DONLAN and G. T. SURRAIT,1972, J. Chem. Phys. 57,562. WEBER,M. J., T. E. VARITIMOS and B. H. MATSINGER, 1973, Phys. Rev. B8, 47. WYBOURNE, B. G., 1965, Spectroscopic Properties of Rare Earths (Wiley Interscience, New York). YAMADA, N. S., S . SHIONOYA and T. KUSHIDA,1972, J. Phys. SOC.Japan 32, 1577. YATSIV,S., 1962, Physica 28, 521. YEN,W. M., W. C. SCOTT^^^ A. L. SCHAWLOW, 1964, Phys. Rev. 136, A271. YOKOTA,M. and 0. TANIMOTO, 1967, J. Phys. SOC.Japan 22. 779. ZVEREV, G., G. Y.KOLODNYI and A. M. ONISHCHENKO, 1971, Soviet Physics JETP 33,497.
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E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
THE ULTRAFAST OPTICAL KERR SHUTTER BY
MICHEL A. DUGUAY* Bell Laboratories, Murray Hill,New Jersey 07974, U S A .
*
Present address: Sandia Laboratories, Albuquerque, New Mexico 871 15, U.S.A.
CONTENTS PAGE
0 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . 5 ?,. THE ULTRAFAST OPTICAL KERR SHUTTER . . . . . 5 3. ULTRAHIGH SPEED PHOTOGRAPHY . . . . . . . . . 0 4. SAMPLING OPTICAL SIGNALS . . . . . . . . . . . . 0 5 . CONCLUDING REMARKS. . . . . . . . . . . . . . . ACKNOWLEDGEMENTS. . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . .
163 165 177 183 191 192 192
8
1. Introduction
One of the new frontiers to be opened by the laser has been the area of picosecond pulses. This began in 1966 when DEMARIA,HEYNAUand STETSER [19663 reported the emission of ultrashort pulses by a neodymium : glass laser Q-switched by a saturable absorber. Soon thereafter ARMSTRONG [1967] measured these pulses and found them to have a duration on the order of 6 picoseconds, a record at the time in terms of ultrashort optical or electrical pulses. The discovery of picosecond laser pulses stimulated the invention of new techniques for measuring ultrashort optical pulses, notably the two-photon fluorescence technique of GIORDMAINE, RENTZEPIS, SHAPIRO and WECHT [1967]. Picosecond pulses were also applied to scores of experiments ranging from measurements of molecular relaxation times to laser fusion studies. One particularly useful application of picosecond laser pulses, and one that will be the object of this chapter, has been in the development of an ultrafast shutter (or gate) based on the optical Kerr effect (also called AC Kerr effect) by DUGUAY and HANSEN [1969a]. This effect, which had been [1956] well before the laser, and which was predicted by BUCKINGHAM ~ refers to the very short-lived first observed by MAYERand G I R E[1964], birefringence induced in transparent mediums by high-power laser pulses propagating therein. The ultrafast optical Kerr shutter uses this optically induced birefringence to gate light on and off on the picosecond time scale, much in the way in which the conventional Kerr cell uses an electrically induced birefringence to gate light on and off on the nanosecond time scale. To date, gating times as short as 5 ps have been achieved, when using ultrashort pulses from neodymium glass lasers. By comparison, the fastest electrically driven Kerr cells have an open time of 0.5 ns. Moreover, the recent development of subpicosecond pulses in CW modelockcd dye lasers due to SHANKand IPPEN [1974] has enabled these workers to achieve [1974]). gating times as short as 2 ps in a CS2Kerr shutter (IPPENand SHANK The ultrafast optical Kerr shutter has found a number of applications,
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THE ULTRAFAST OPTICAL KERR SHUTTER
[IV,
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1
some of which will be described in this chapter. When used in conjunction with an ordinary camera, the optical Kerr shutter has made possible the stop-motion photography of light pulses in flight. Also, gated picture ranging on the centimeter scale has been demonstrated, opening the possibility of eventually “seeing through” biological tissue, such as the human skin. The Kerr shutter has been applied to the measurement of some interesting molecular relaxation times. For example, by filling the shutter with CS2, and then with nitrobenzene, DUGUAY and HANSEN[1969a] found that it takes nitrobenzene molecules about 32 ps to relax from a state of partial alignment to one of random orientation. In one of the shortest direct time measurements to date, IPPENand SHANK [19751 have recently measured the analogous relaxation time in CS2, and found it to be 2 ps (at 25 “C). Of more general interest has been the use of the ultrafast shutter in displaying the time profile of a variety of optical signals. This has been done by employing one of two methods. The first method, which is patterned after the electronic sampling oscilloscope, consists in using the gate to cut out a short time sample from the signal at a time progressively advancing with each laser shot. By using a photomultiplier behind the gate, a very high sensitivity has been achieved in sampling ultrashort optical signals. With this method the fluorescence decay time of l,l’-diethyl-2,2’-carbocyanine iodide in methanol has been found to be 14k 3 ps, DUGUAY and HANSEN [1969b1. ALFANO and SHAPIRO [19721 have used the same method to measure ultrafast relaxation times in erythrosin and in a tetracene crystal (ALFANO, SHAPIRO and POPE[1973]). The second method features multichannel sampling of a single event optical signal. For this, one takes advantage of the wide angular acceptance (10-20 degrees) of the ultrafast Kerr shutter to cut out and record a multiplicity of samples from the one optical signal. The samples are distributed uniformly from the leading to the lagging edge of the pulse, and their envelope represents a display of the signal pulse shape. This approach has been followed by TOPP,RENTZEPIS and JONES[1971], who have developed the echelon technique for this purpose, and by DUGUAY and SAVAGE [19731 and by VOGEL,SAVAGE and DUGUAY [19741, who have used an “0rgan”fiber array with the shutter in order to build an optical sampling oscilloscope (OSO).With the OSO optical picosecond signals with peak powers on the watt level can be displayed instantly and accurately on an oscilloscope screen. Recently, MOUROUand MALLEY [1974] have used the shutter in a configuration where 512 samples are taken in one laser shot, and they have
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thus obtained very accurate displays of molecular fluorescence signals on the picosecond time scale. Another recent application of the ultrafast gate has been in the measurement of ultrashort X-ray pulses emitted by laser produced plasmas (DUGUAY and OLSEN[1975]). For this the X-rays were converted to light in a plastic scintillator, and the light was measurcd on the picosecond time scale with the optical sampling oscilloscope technique.
5 2.
The Ultrafast Optical Kerr Shutter
2.1. GATING IN CS,
The simplest, and what has been so far the most useful configuration for the ultrafast Kerr gate (or shutter), is shown in Fig. 1. A short cell containing carbon disulfide is placed between two crossed polarizers. In the spectral range 5000-7500 A sheet polarizers (like Polaioid HN22) can be used. Elsewhere in the spectrum good extinction and sizeable angular acceptance ( 15") require the ust of calcite (or mercurous chloride) crystal polarizers. The CS2 cell is made of low birafringence glass and is held gently to avoid strain birefringence. The gating laser pulse is directed into the CS2 cell at a small angle a from the axis AX, along which the signal light to be gated is traveling. When working with Nd: glass picosecond pulse lasers, the gating pulse will typically be 10 ps in duration at a wavelength of 1.06 p. In the absence of the gating pulse the crossed polarizers shut out the signal light; the gate transmission in the OFF state is typically 0.001 %.
-
-
SIGNAL
F
GATING PULSE
Fig. 1. The ultrafast optical Kerr shutter in its simplest form. Polarizers PI and P, are crossed and have their polarization axes at *45" to the plane of polarization of the gating laser pulse. Filter F greatly attenuates the infrared gating pulse to prevent possible damage to P, . In some cases F can be dispensed with (see Momou and MALLEY [1974]).
166
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
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Q
2
The gating pulse (infrared pulse in Fig. 1) is plane polarized in the horizontal plane, as is usually the case with practical lasers which have vertical Brewster faced components. Polarizers P, and P, have their axes at +45" to thehorizontal. Asthegating pulse propagatesthrough the CS, cell, it creates a region of birefringence that travels along with it. That ultrashort portion of the signal light that happens to enter the CS2 cell at the same time as the gating pulse experiences the effect of this moving birefringent region, and as a result undergoes a change in polarization that allows it to be partially transmitted through polarizer P, . The birefringence induced in CS, by a laser pulse whose plane polarized electric field is E(t) in the medium is given by : anll - 6n, = n2.j: mEi2(t')exp [- (t - t')]dt'/c
(2.1)
Here Gn,, and dn, refer to changes in the refractive index in directions parallel and perpendicular, respectively, to the applied field E(t), n 2 B is the AC Kerr effect coefficient,z is the relaxation time of the AC Kerr effect in the gate medium, and the bar over E 2 signifies a local time average over one optical period. In CS, the bulk of the induced birefringence arises when molecules are rotated so that their axis of easy polarizability moves closer to the plane of polarization of the applied field E(t). The relaxation time associated with this motion is about 1.8 ps in CS, at 25°C (BROIDAand SHAPIRO [1967], IPPENand SHANK[1975]). This means that it takes 1.8 ps after the passage of a 6-function gating pulse in order for the molkcules to randomize again, thereby returning the ultrafast gate to its OFF state. This time defines the minimum open time of a CS, shutter. For pulses much longer than z = 1.8 ps, eq. (2.1) can be approximated by:
6nll-6n,
N
nZBF(t)
(2.2)
where the bar indicates a timc average over about 2 ps. The optical signal S(t) is polarized at 45" to the horizontal by P I . After travelling with the gating field E(t) over a distance L in the Kerr medium, the horizontal component Sll(r) of the signal undergoes a phase lag cp relative to the vertical component S,(t) given by : cp = 271(6n1,-6n,)L/A (2.3) .where A is the signal wavelength in vacuum. As in the conventional Kerr cell, the transmission of the device is given by:
T
=
$T, T2 sin2(cp/2)
iv, 9 21
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
167
where T l , 2 is the transmission of polarizer P,,2 for light parallel to its polarization axis. In writing down the factor of 4 we have assumed an unpolarized signal. For a signal polarized along the polarization axis of P I , 4 is to be replaced by unity. Combining eqs. (2.2), (2.3) and (2.4) we get: T(t) =
tT,T2sin2 (nn,,,??(t)L/,?
).
(2.5)
The term F(t)is related to thz instantaneous power density P (in watts/m2) carried by the pulse though a medium of refractive index n by the relation (we use SI units in the formulas throughout this chapter, except where otherwise noted)
EI(t)
=
P(t)/cne,
(2.6)
where 8, = 8.85 x 10-l2 Fd/m and c = 3 x 10’ m/s. Substituting in eq. (2.5) we get: (2.7)
As a specific example let us consider the experiment of DUGUAY and HANSEN [1969a], where infrared ultrashort pulses from a Nd: glass laser are used to gate ultrashort green pulses derived from them by second harmonic generation. The various numbers in eqs. (2.5) and (2.6) are: 3T1T2= 0.14 (HN22 type sheet polarizers from Polaroid Corp.), L = m, 1 = 0.53 x m, n = 1.60 and n2B = 2.2 x m2/V’ as measured for CS2 by MAYERand GI= [I9641 and by PAILLETTE [1969]. Using these numbers in eq. (2.7) we get T(t) = 0.14 sin’ (3.15 x l O - I 3 P )
(2.8a)
or, more conveniently, expressing P in MW/cm2, we have T ( t) = 0.16 sin2 C3.15 x 10-3P(MW/cm2)]
(2.8b)
which is plotted in Fig. 2. As can be seen from Fig. 2 and eq. (2.7) when the gating power density is less than 200 MW/cm2, the transmission Tit) is approximately quadratic in P ( t ) T ( t )N + T ~~,[nn,,LP(t)/(ilcne,)]~. (2.8~) Under these conditions the gate open time is somewhat shorter than the gating pulse. For example, if P(t) is a Gaussian with a full width at half maximum (FWHM) of 14ps, T(t)will also be Gaussian, but with an FWHM equal to 14/$ = 10 ps.
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T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
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-
[IV.
3
2
CS2 SHUTTER
HN22 POLARIZERS T (%I
0
250
SO0
7 50
1000
P(MW/Cm2)
Fig. 2. The transmission T of a CS, ultrafast Kerr shutter at a signal wavelength of 0.53~ is plotted versus the power density in the gating beam. For a shorter signal wavelength the maximum would move towards smaller gating power densities. The CS, shutter has a length L = 1 cm and the polarizers are of type HN22 (made by Polaroid). With calcite crystal polarizers, the peak transmission could reach 50 “4for unpolarized signal light.
In the experiment of DUGUAY and HANSEN [1969a], second harmonic green pulses are used to probe the gate transmission versus time by changing the delay T’ by which the green pulse lags the infrared gating pulse at the gate. The second harmonic pulse has a time profile which is closely approximated by P 2 ( f ) .The green signal Gt4)(7’)which emerges from the gate is therefore given by the convolution
s s
CY4)(t’) = T(t)P2(t-T’) dt.
(2.9)
At gating power densities below 200 MW/cm2, T ( t ) cc P 2 ( t )and we have G(4)(2’)cc P2(t)P2(t- 7‘)dt.
(2.10)
An experimental plot of Gt4’ (7’)is shown in Fig. 3. The shape of the time profile of P ( t ) cannot be derived from G(4)(7’).However, the approximate overall duration of P ( f )can be deduced from that of G(4)(T’). Assuming a Gaussian-shaped P(t), for example, G(4)(7’) is also Gaussian and has the same width as P(t). From this it follows that in the CS2gating experiment of Fig. 3, the gate open time at half maximum was 17/$ = 12 ps.
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
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t 1
0 -50
0
* 50
4 T'(pS)
Fig. 3. The solid line represents the transmitted green light signal G'4'(~')as a function of the delay T' between the powerful gating 1 . 0 6 ~pulse and the weak probing second harmonic (0.53~)pulse. The width of G(4)(~') is 15 ps and is approximately equal to the duration (at half height) of the gating pulse. With gating pulses about 150 MW/cm2 in peak power density, the shutter transmitted 4 % of the green pulses at T' = 0. When the shutter is filled with nitrobenzene, the dotted curve is obtained. The exponentially decaying part of the wrve gives the relaxation time (32+4 ps) of the orientational Kerr effect in nitrobenzene (at 23°C).
-
2.2. GATING IN NITROBENZENE
Also shown in Fig. 3 is a plot ,of G(z) when nitrobenzene is used as the optical gate medium. Nitrobenzene molecules are bigger than CS2 molecules, and once aligned along a certain direction by the AC Kerr effect, it takes them longer to relax to a random orientation state after the passage of the laser pulse. This is evident in Fig. 3 where the gate transmission is seen to decay exponentially with a time constant of 16f2 ps after the gating pulse has left the gate. Since the gate transmission is proportional to the square of the induced birefringence, the relaxation timc associated with the latter is 32+4 ps in nitrobenzene at 25°C. This relatively long relaxation time supports the view that the mechanism responsible for the AC Kerr effect in nitrobenzene is indeed orientational in nature, in agree-
170
T H E U L T R A F A S T OPTICAL KERR SHUTTER
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2
ment with the light scattering studies of STARUNOV, TIGANOV and FABELINSKII [19661 (see also FABELINSKII [19683). 2.3. GATING WITH SUBPICOSECOND PULSES
Recently SHANKand IPPEN [I 9741 have developed a continuously pumped mode-locked dye laser that produces pulses as short as 0.7 ps with a peak power of the order of one kilowatt. By focussing these pulses down to a waist of about 20 p they have been able to drive a CS2 Kerr shutter just about as fast as it can go. Their experimental arrangement is shown in Fig. 4. The opening of the shutter is probed by weaker ( 5 % of the incident PULSE TRAIN INPUT X.0.60p POLARIZER
POLARIZER P, ROTATOR
CHOPPER
LOCK-IN AMPLIFIER
I X-Y RECORDER VARIABLE DELAY
Fig. 4.Gating with subpicosecond pulses has been achieved by IPPEN and SHANK[1974] with this set-up. A continuously pumped dye laser produces a day-long train of pulses at I = 0 . 0 6 ~ . The major part (95 %) of each pulse gates on the K e n shutter, while the minor part (5 %) probes the shutter transmission. The availability of day-long trains of pulses allows the use of highly sensitive lock-in detection techniques.
power) pulses split off from the input beam (wavelength: 0.60 p). This time the pulses are so short that the more exact expression (2.1) for 6nll-an, must be used in eqs. (2.3) and (2.4). The result is that the transmission T(t) is given by: T(t)= +TIT, sin2 [(p(t)/2]
(2.11a)
~ ( t=) ( r ~ ~ ~ n L /m A F) j( ;t f )exp [ - (t- t')/z] dt'lz.
(2.11b)
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171
As one varies the delay z’ by which the probe pulse lags the gating pulse, the transmitted signal G(z’) varies like: G($) cc
jeffiffi
T ( t ) p ( t- z f )dt.
(2.12)
In the limit where the subpicosecond pulse p(t)becomes a &function centered at time I = 0, and for small enough cp’s such that sin cp N cp, these equations simplify to : d t ) a (zLI4 exp (-
(2.13a)
t/z)
~ ( tK) +T, T2(nL/A)’exp (- 2t/z)
(2.13b)
G(z‘) cc T(z’)cc exp (- 2z’/z).
(2.13c)
In this limit the transmitted signal therefore reproduces a decaying exponential characteristic of the rotational relaxation time z of CS, . The curve obtained by IPPENand SHANK[1975] is shown in Fig. 5 . An asymmetry with a decaying tail towards large positive z’ is clearly detectable. L
-
OPTICAL KERR SHUTTER PULSE CORRELATION IN CS,
-3-2-1 0 1 2 3 TIME DELAY (psec)
4
5
Fig. 5. Signal G(T’)transmitted by the CS, optical Kerr shutter when driven and probed by 1.2 ps pulses from a mode-locked CW dye laser (IPPEN and SHANK [1975]). The ordinate is linear and in arbitrary units. The shutter transmission at T’ = 0 is of the order of 0.1 ”/,.
-
The peak power density at the waist is lo8 W/cm’ and the beam waist extends over about 1.0 mm. The product PL therefore falls somewhat short of what is required for efficient gating (see Fig. 2), explaining the low peak transmission (- 0.1 %) observed. In order to get more signal and to bring out the effect of the relaxation time z more clearly, IPPENand SHANK[1975] have also measured the transmission T ( t )when the shutter is biassed half-way to the first maximum. This is achieved by introducing a quarter-wave plate between the crossed
172
T H E U L T R A F A S T O P T I C A L KERR s H U T T E R
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polarizers. For this case eq. (2.1la) becomes
T(t) = $Tl T2 sin2 [44+ (p(t)/2]
(2.14)
where q ( t ) designates as before the laser induced phase lag. For small.& eq. (2.14) approximates to
W )= fiT1 T,(1 +do),
d2.
dt)
(2.15)
The expression for the transmitted signal G(z’) becomes
G(z’) oc
Lm+
(1 q(t))F(t - 7’) dt.
(2.16)
For pulse durations well under one picosecond, G(z’) simplifies in this case to
+
(2.17)
G(z’) a const. exp (- z’/z).
The curve measured by IPPENand SHANK [1975], shown in Fig. 6, portrays a steep rise and an exponential decay indicating a time constant z = 2.1 f0.3 ps, one of the shortest lifetimes to be measured directly by optical techniques. Forthcoming increases in the peak power of subpicosecond pulses promise to render possible the study of light induced refractive index changes in a great variety of liquid and solid systems and on a very fine timescale. These studies should provide a most valuable complement to light scattering techniques (FABELINSKII [19681) in the study of ultrafast molecular dynamics. W
z
o_ UJ
OPTICAL KERR SHUTTER PULSE CORRELATION IN CS, (HALF-WAY BIAS)
v)
-3-2-1 0 1 2 3 4 TIME DELAY (psec)
5
6
Fig. 6. Same as Fig. 5, except that this time the shutter is biassed half-way by inserting a quarter-wave plate between the two crossed polarizers.
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THE ULTRAFAST OPTICAL KERR SHUTTER
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2.4. GATING IN GLASS
In order to achieve ultimately the shortest possible gating times, it will be necessary to use the AC Kerr effect of the type found in glasses, where the relaxation time is expected to be exceedingly short. Recent work by OWYOUNG, HELLWARTH and GEORGE [19723 has shown that the optically induced birefringence in glasses results from a distortion of the atomic electronic clouds, and therefore probably has a response time on the order of one atomic orbital period, i.e. s. With subpicosecond laser pulses now becoming available, SHANKand IPPEN[1974], optical gates based on glasses promise to achieve subpicosecond open times. Gating experiments were carried out by DUGUAY and HANSEN[1970] in fused quartz and in two types of Schott glass: BK-7 and LaSF-7. The curve for the gated green signal G(4)(z’) is shown in Fig. 7 for a sample of BK-7 10 cm in length.
-
Fig. 7. This curve plots the transmitted signal for a glass ultrafast shutter driven by 1 . 0 6 ~ pulses from a mode-locked Nd:glass laser. The probe pulses (d = 0.53~)are harmonically derived from the infrarcd ones. The width of the curve for G‘4)(~’)at half-height is approximately equal to the duration of the infrared pulses. Although harder to gate on, glass Kerr shutters would ultimately allow gating on the femtosecond (10-l5s) time scale.
The curve for G(4)(z’) obtained with LaSF-7 is consistent with a very fast response. Any relaxation time, if present, is less than 5 ps. One disadvantage of glass optical gates is the high power density required to get the same transmission as with a CS2 gate. With LaSF-7 the first transmission maximum occurs at P = 15 GW/cm2 for a one-centimeter thick gate. These experiments have served to measure the nonlinear index of
174
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
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2
birefringence n2, of glasses. By using the value n,,(CS,) = 2.2 x m2/V2, as a reference, the values n2,(LasF-7) = (6f2) x m2/V2, mZ/V2 and n,,(fused silica) = (1.5f0.5) x ~,,(BK-T)= (2+ 1) x lo-’, m2/V2 were obtained (DUGUAY and HANSEN [1970]). 2.5. TIME RESPONSE
In the geometry shown in Fig. 1 two other factors limit the resolution for pulses less than 5 ps in duration. One is the group velocity mismatch between the gating and the gated beams. In CS2 a green light pulse will fall behind an infrared (2 = 1.06 p) pulse by 2.0ps after 1cm of travel. In LaSF-7 glass that number would be 1.8 ps. A tenfold reduction in the gate thickness down to one millimeter would reduce this differential delay to 0.18 ps in LaSF-7. However, the power density required to reach the first transmission maximum would then be increased to 150 GW/cm2, a relatively large value, but one that is well within the state of the art. 2.6. COLLINEAR GATING
Another factor lengthening the time response of the optical gate is the spread in arrival times at the gates for different parts of the infrared gating beam, due to the finite angle a between gated and gating beams. As can be seen in Fig. 8, this spread amounts to (d tana)/c, where d is the gating
Fig. 8. Illustration of the time smearing factor (d tan a)/c introduced by the small angle tl between gating and gated beams. In the figure the gating and gated pulses are assumed to be nearly delta-functions and are represented by the solid and crossed areas. When the bottom parts of the two beams enter the gate in perfect synchronism, the upper parts are (d tana)/c apart. In the experiment of DUGUAY and HANSEN[1969a], this factor amounted to 2.0 ps.
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T H E U L T R A F A S T O P T I C A L KERR SHUTTER
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beam diameter. In the experiment of DUGUAY and HANSEN[1969b], the numbers were d = 0.5 cm, LY = 0.08, so that (d tan LY)/C was 1.4 ps. This geometrical time-smearing effect can be completely eliminated by going over to a collinear geometry for the gated and gating beams. In the arrangement of M o u ~ o uand MALLEY[1974], shown in Fig. 9(a), this was achieved by letting the gating beams go through the first polarizer P, of type Polaroid HN22, which is almost completely transparent to 1.06 p radiation.
--
-
SIGNAL LIGHT c
Fig. 9. One way to avoid the time smearing factor of Fig. 8 is to employ a geometry where the gating and gated beams are collinear. (a) In the geometry of MOUROUand MALLEY[1974] the gating beam at 1 . 0 6 ~is sent through polarizer P, (type Polaroid HN22) where it suffers negligible absorption, and therefore undergoes very little change in polarization (i.e., it stays plane polarized at 45" from the axis of P,). The signal beam is coupled in by means of a dichroic dielectric mirror. (b) In the set-up used by Yu and ALFANO [I9741 a dichroic mirror is used to couple the gating beam into the shutter.
Another geometry, used by Yu and ALFANO [1974], shown in Fig. 9(b), makes use of dichroic mirror M to reflect the gating pulse into the gate, and let the signal through. In this case, for best extinction of the crossed polarizers, angle @ should be kept to a minimum, and the polarization axis of P,should be either in the plane of Fig. 9 or normal to it. Correspondingly, the infrared pulse should be plane polarized at 45" to the horizontal plane. 2.7. SELF-FOCUSING
Self-focusing does not normally occur in a properly designed ultrafast
176
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
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Kerr gate. Nevertheless, it is worth looking at the conditions under which it might arise. Self-focusing, which was predicted by KELLEY [1965], followGARMIRE and TOWNES [1964], refers ing the self-trapping work of CHIAO, to the phenomenon whereby a powerful laser beam induces in a medium a refractive index change that tends to focus parts of the beam to a number of points. The change in refractive index seen by a plane polarized laser field E(t) is given by
6nll = n , P ( t )
(2.18)
where n2 is the nonlinear index of the medium (for CS, n , = 3 x nZB= 1.5 x lo-” mZ/VZat 1.06 p). In the case of the spatially inhomogeneous and high power laser beams used in practice for gating, the self-focussing theory of SUYDAM [1973] (see also CAMPILLO, SHAPIRO and SUYDAM[1973, 19741) is applicable. Both theory and experiment find that parts of a laser beam will self-focus after a minimum distance zfgiven by (2.19)
where 6 describes the radial inhomogeneity of the beam and is equal to ~ 6 E o ~ / ~Eo E obeing ~ , the peak value of the optical field. With Nd: glass lasers 6 values in the range 0.03-0.3 can be expected. Assuming a gating power density of 500 MW/cm2 (for which n , F = 1.7 x lop5), and substituting in eq. (2.19), we find zf ranges from 1.5 to 3.0 cm for gating at 1.06p. In the one-centimeter length of a typical ultrafast shutter, there is little danger therefore of seeing any part of the gating beam self-focus. 2.8. TRANSVERSE GATING
In their study of self-focused filaments in CS,, SHIMIZU and STOICHEFF [1969] used the geometry shown in Fig. 10, which was later recognized by MALLEY and RENTZEPIS [1970] as a useful one for the ultrafast gating of optical signals. In +he experiment of SHIMIZU and STOICHEFF [1969], the birefringence induced by the 1.06 p ultrashort pulse was probed by photographing the green light (second harmonic) transmitted through the crossed polarizers. This way both the temporal and the transverse spatial profile of the 1.06 p beam in CS, was recorded. MALLEY and RENTZEPIS [19701 used the same geometry to display the time profile of a 0.63 p pulse generated by a rhodamine 6G laser pumped by the second harmonic of the 1.06 p pulse (Nd: glass laser). As the 1.06 p gating pulse propagates
U L T R A H I G H SPEED P H O T O G R A P H Y
177
and STOICHEFF [I9691 to study self-focused filaments in CS,, Fig. 10. Set-up used by SHIMIZU and as a method of displaying the 1 . 0 6 ~pulse. When used as a Kerr gate, the transverse geometry provides an open slit moving from A to B at the speed of light in CS, . In photographing the transmitted signal light, axis AB becomes a time axis and one can use this set-up to display the signal (MALLEY and RENTZEPIS [1970]). Self-focusing of the gating beam can be a problem in a transverse geometry Kerr gate.
from A to B in Fig. 10, the Kerr gate first opens at A and then later on at B. This way the time profile of the pulse incident from the top appears as a spatial intensity profile on the film. The transverse geometry has also been used by FISCHER and ROSSMANITH [19731 in studies of synchrotron radiation, and by RICHARDSON and SALA [I9731 in their ultrafast framing photography of laser produced plasmas, which we will return to later. TOPP,RJZNTZEPIS and JONES [1971] have combined a transverse Kerr shutter with a spectrograph in order to time resolve the spectrum of an optically pumped rhodamine 6G laser. Self-focusing is a problem that is more severe in a Kerr gate of transverse geometry. Because the minimum opening time at a given z is limited by the transit time nd/c, it is necessary to keep d as small as possible. For 5 ps resolution, for example, one must have d = 1.0 mm. In order to get the same phase retardation 640 as one had in Fig. 1 over L = 1 cm, it is necessary to increase the gating power density tenfold to 5 GW/cm2. As a result of both the smaller diameter and the increased power density, the gating beam can undergo self-focusing within the long CS2cell dimension (- 1 cm) and break up in filaments as in the experiment of SHIMIZU and STorcHEw [19691. This leads to severe inhomogeneities in the gate transmission.
0 3. Ultrahigh Speed Photography 3.1. LIGHT PHOTOGRAPHED IN FLIGHT
A unique application of the ultrafast shutter has been in the photography of light in flight (see DUGUAY and HANSEN [19701, DUCUAY and MATTICK
178
THE U L T R A F A S T OPTICAL X E R R SHUTTER
r1v, 4 3
Fig. 11. Experimental set-up used for the ultrahigh-speed photography of light pulses in flight. Ultrashort infrared (1.06~)pulses from a Nd: glass laser open the ultrafast Kerr shutter for about 10 ps. The green pulses are harmonically derived from the infrared pulses. As they pass through a cell of milky water, light scattering makes them brightly visible from the side, so that they can be stop-motion photographed in flight (see Fig. 12). Filter F is made of infrared absorbing glass that is transparent to the visible; it attenuates the 1 . 0 6 ~pulse by a factor of 1000.
-
[1971] and DUGUAY[1971]). The set-up that was used to accomplish this is shown in Fig. 11. An ultrashort pulse of green light is directed into a cell containing a colloidal suspension of milk particles in water. The milk particles greatly increase the instantaneous light scattering that occurs in pure water, thereby making the green pulse brightly visible from the side. (A laser pulse propagating in vacuum would not be visible from the side.) A 35 mm camera, whose mechanical shutter has been manually opened, is placed behind a conventional ultrafast CS2 shutter. The latter is driven by an infrared pulse (A = 1.06 p) produced by a mode-locked Nd: glass laser. The green pulse is harmonically derived from the infrared pulse. The lengths of the paths followed by the two pulses are adjusted in such a way that when the shutter opens for about 10 ps it captures a picture of
IV, §
31
ULTRAHIGH SPEED PHOTOGRAPHY
179
Fig. 12. An ultrashort pulse of green laser light is photographed “in flight” as it propagates from right to left through a cell of milky water. The scale is in millimeters. The shutter open time was about 10 ps. The red spot on the left side of the picture is the impression made on the high speed Ektachrome film by the infrared laser pulse used to activate the ultrafast Kerr shutter and incompletely attenuated by filter F in Fig. 11.
the green pulse in midflight through the milky water cell, as shown in Fig. 12. Thus the green “light bullet” is stop-motion photographed in flight. The red round spot to the left of the cell results from the direct head-on impact onto the film made by the infrared pulse incompletely attenuated by filter F in Fig. 11. Thus Fig. 12 also provides a pictorial representation of second harmonic generation from infrared (in red) to green. In principle the ultrafast photography of light in flight constitutes one of the most direct ways of displaying ultrashort laser pulses. Pulses obtained from a mode-locked Nd: glass laser stand out bright and well isolated,
180
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
“v,
03
the contrast between the brightness of the pulse and that of the background being better than 200 to 1. Even though the ultrashort pulse display shown in Fig. 12 is one of the best obtained so far, it still is far from perfection because the shutter remains open for about the same time (- 10 ps) as the green pulse duration. In 10 ps light moves 2.2 mm in water, and as a consequence, the picture of the green pulse is blurred out. One cannot recover the precise shape of the green pulse. What would be needed in this case would be a 0.3 ps gating pulse (perhaps from a mode-locked dye laser) and a glass Kerr shutter because the green pulse is known to have a subpicosecond substructure (SHAPIRO and DUGUAY [19693). Nevertheless, this technique has proved useful in the study of weaker satellite pulses accompanying the powerful ultrashort pulses generated by model-locked Nd : glass lasers (see DUGUAY and MATTICK [197I]). 3.2. GATED PICTURE RANGING
The technique of gated picture ranging has been used on the nanosecond time scale to improve the visibility of targets, such as airplanes and ships, obscured by fog or other obstacles. A powerful nanosecond light pulse is sent out from the observation point towards the target. At a preselected later time an electronic image converter tube is gated on for a few nanoseconds and only the echo image scattered back by the target is recorded. Earlier (or later) echoes from the fog corresponding to closer (or farther) ranges are not recorded because the picture tube is gated off at those times. With picosecond and now even subpicosecond laser pulses, the same technique can be applied on the centimeter and even millimeter scales by using the ultrafast Kerr shutter to gate the echoes. In a feasibility experiment using the set-up shown in Fig. 13(a), a second-harmonic green pulse was sent through a piece of thin paper tissue (“facial” tissue) towards a target carrying the stylized drawing of a bell (shown unobscured by the tissue in the upper left corner of Fig. 13(b)). When photographed under room light illumination (see Fig. 13(b), upper right corner), the, target is completely obscured by the tissue. When the green pulse is sent through the tissue and the ultrafast shutter is turned on at the right time by the infrared pulse, only the echo from the target is recorded by the camera. Two results are shown in the lower part of Fig. 13(b). In certain parts of the human body the skin is partially transmitting to light, and veins, for example, can be seen. With the recent achievement of 0.5 ps laser pulses and 2 ps Kerr gating times, a spatial resolution of better
U L T R A H I G H SPEED P H O T O G R A P H Y f
181
TARGET
1 cm
Fig. 13. (a) Schematic description of set-up used for gated picture ranging through a piece of paper (or "facial") tissue. An ultrashort green laser pulse first illuminates the tissue and then 33 ps (1 cmjc) later the target. The target echo, which carries the image information, lags the tissue echo by 66 ps. (b) The top left picture shows the target under room lighting when the tissue is removed. With the tissue back in place the target is completely invisible under room lighting. The bottom two pictures show two examples of gated picture ranging through the tissue. The target is visible, but the passage of the image carrying echo through the tissue has degraded the quality of the picture (the ultrafast shutter itself does not degrade the picture, see DUCUAY and MATTICK[1971]).
than 1 mm should be possible in gated picture ranging, and greatly improved vision though the skin appears feasible in those parts of the body where the skin is 1-2 mm thick. A clearer picture of veins and arteries under the skin might help the diagnosis of certain diseases and injuries. The outcome of such experiments remains to be seen.
182
THE ULTRAFAST OPTICAL K E R R SHUTTER
[rv,
03
3.3. ULTRAHIGH SPEED FRAMING PHOTOGRAPHY
RICHARDSON and SALA[19731 have combined an ultrafast Kerr shutter with an electronic streak camera in order to multiframe photograph a laser produced plasma (or laser “spark”). In their experiment a train of about 30 pulses, spaced by 6.7 ns and produced by a mode-locked Nd: glass laser, is focused in air, causing it to break down (to “spark”). Part of the beam is used to drive an ultrafast Kerr shutter of transverse geometry. Thus the shutter opens for about 10 ps every 6.7 ns. Light emitted by the growing laser spark is focused into and through the shutter onto an electronic streak camera. In this camera the image is electronically swept downward at a speed of up to 2.5 mm per nanosecond. Consequently, the series of discrete images that emerge from the shutter at 6.7 ns intervals are recorded at progressively lower positions on the film. As can be seen in Fig. 14, the sequential images are well separated from one another and one can follow the growth of the laser spark.
Fig. 14. Picosecond framing photography of a laser produced “spark” in air achieved by combining an electronic streak camera with an ultrafast optical Kerr shutter. The latter is opened for 10 ps every 6.7 ns. In the 6.7 ns interval during which the Kern shutter is opened, the position of the image is electronically swept down in the streak camera tube, so that the next 10 ps frame transmitted by the Kerr shutter is recorded well below the preceding image. (a) Five frames in the initial stage of optical breakdown in air. (b) Five frames about 200 ns after breakdown. RICHARDSON and SALA[19731.
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SAMPLING OPTICAL SIGNALS
183
In electronic high speed framing cameras, the framing time is limited to about 300 ps. Since the Kerr shutter is capable of gating times down to 2 ps, the technique of RICHARDSON and SALA[1973] clearly represents a major improvement in ultrafast framing .photography.
8
4. Sampling Optical Signals
4.1. FLUORESCENCE LIFETIME MEASUREMENTS
The sampling technique used in the electronic sampling oscilloscope was developed by JANSSEN [19501. The beauty of this technique resides in the fact that both the highest sensitivity and the fastest time response can be achieved simultaneously in displaying weak electronic signals. The key elements needed in electronic sampling are an ultrafast gate (the shortest gating times are 20 ps at present) and an amplifier to amplify the sample cut out at a gven time from the signal. An ultrafast optical Kerr shutter (or gate) and a photomultiplier constitute the analogous key elements in applying the sampling technique to optical signals. Since a Kerr shutter transmission of 50% can in principle be achieved when using crystal (e.g., calcite) polarizers, and since photomultipliers have quantum efficiencies as high as 30%, this optical sampler is close to the theoretical limit of sensitivity. The optical sampling arrangement used by DUGUAY and HANSEN [1969b] is shown in Fig. 15. The CS2 gate is driven, as before, by 1.06 p pulses about 10 ps in duration generated by a Nd: glass laser. The 0.53 p second harmonic pulses are sent into a cell containing a cyanine dye dissolved in
-
I
I
Fig. 15. Picosecond fluorescence decay times were first measured by using an ultrafast shutter together with a photomultiplier in order to do point by point sampling of optical signals (DUGUAY and HANSEN[1969]). The fluorescent dye is excited by green pulses about 10 ps in duration and emits a fluorescent signal at 1 = 0 . 7 5 ~ On . a given laser shot, a 10 ps sample is sliced from the incoherent fluorescence signal and is detected by the photomultiplier. As the delay is vaned from shot to shot, the entire signal can be sampled as a function of time.
184
[IV,
THE ULTRAFAST OPTICAL KERR SHUTTER
§ 4
methanol or acetone; the dye is 1, l’-diethyl-2,2’-carbocyanine iodide (commonly known as DDI). Some of the spontaneous fluorescence light emitted by the DDI is collected by a lens and directed into the ultrafast Kerr shutter. At each firing of the laser, a sample (- 7 ps wide) is cut out from the fluorescence signal at a given point on the waveform. With each laser firing the relative delay between the gating and the excitation pulses is changed, so that after many shots the whole signal has been sampled and displayed. An example of a fluorescence waveform display obtained by this method is shown as the full line in Fig. 16.
-cn
-DDI IN METHANOL
I-
1
PROMPT RESPONSE(SHG)
4
I3
n I-
0
I
I
a
I -40
\ I
-20
I
I
I
0
20
40
I 60
I
80
I 100
T I M E t ( pSeC 1
Fig. 16. The solid line shows the fluorescence signal emitted by the dye DDI sampled as a function of time. Time is measured relative to the arrival time of the green pulse which excites the fluorescence. The dotted line represents the “prompt” response of the measuring system. It is obtained by removing the dye and by sampling the green pulse itself (it is the same as G(4)(2’) in the text). A deconvolution of the prompt curve from the fluorescence curve gives a decay time of 14+3 ps for DDI dissolved in methanol or acetone.
By using a similar arrangement, ALFANO,SHAPIROand POPE [I9731 have measured a 145 ps fluorescence decay time in a tetracene crystal at room temperature. In a more recent experiment using a collinear gating arrangement, Yu, Ho, ALFANO and SEIBERT [1975] have measured a 60 ps lifetime in photosystem 1 of spinach. One problem that has plagued the users of this shot by shot sampling technique is the time consumed in displaying one optical signal. Nd: glass lasers typically fire once a minute, and because of pulse height and duration fluctuations, several shots must be taken for each delay value in order to
IV,
(i 41
SAMPLING OPTICAL SIGNALS
185
obtain a statistically significant display. For this reason workers in the field have sought to develop multichannel sampling schemes, whereby all samples are taken in one laser shot. This will be the topic of the following three sections. 4.2. THE ECHELON TECHNIQUE
In the echelon technique developed by TOPP, RENTZEPISand JONES [I9711 (see Fig. 17), the signal beam is reflected from an echelon reflector. SIGNAL OR PROBE PULSE
X=l.OSp
Fig. 17. The echelon technique developed by TOPP,RENTZEPIS,JONFS [ 19713 makes use of an echelon reflector to divide the signal beam into a number of segments spaced apart in time by typically 3 ps. One ultrashort sample is cut out by the ultrafast shutter from each segment. The samples are recorded on photographic film (as shown above) or by a linear array of photodiodes.
The echelon step is chosen so that the various segments of the reflected signal are progressively delayed at intervals of 4 ps, for example, as they enter the shutter. When the shutter opens for a time dictated by the infrared gating pulse, one ultrashort sample is cut out from each segment of the signal beam. These samples are recorded photographically and by photodensitometry, a plot of segment height vs. number (Fig. 18) gives the time profile of the signal. Recently this method has been improved by NETZEL, RENTZEPIS and LEIGH[1973] by using a linear array of photodiodes instead of film to record the signal segments. TOPP,RENTZEPISand JONES [1971b] have also combined the echelonshutter technique with a spectrograph to time resolve the spectrum of the stimulated emission from a rhodamine 6G laser. Interesting data on the photobleaching of rhodopsin have been receniy obtained by using these techniques (see NETZEL, RENTZEPIS and LEIGH[19733).
186
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
CIV,
§ 4
Fig. 18. Experimental result obtained with the echelon technique. In this example the signal beam is the second harmonic of the 1 . 0 6 ~gating beam. The various segments of the green pulse probe the opening of the ultrafast shutter.
4.3. THE OPTICAL SAMPLING OSCILLOSCOPE (OSO)
Oscilloscope displays of picosecond optical signals have been obtained in experiments done by DUGUAY and SAVAGE[I9731 and by VOGEL, SAVAGE and DUGUAY [19741. In the experimental set-up of DUGUAY and SAVAGE[1973] (see Fig. 19) the green pulse ( A = 0.53 p) to be displayed is sent into a cell containing highly dilute milk. Light pulses scattered from a set of 10 discrete points (only 4 are shown in Fig. 21) are coupled through an ultrafast gate into a set of 10 optical fibers. The 10 scattered pulses replicate the shape of the incident green pulse. The spacing between successive points A, B, C, . . . is such that the scattered light pulses arrive at the gate with delays progressively longer by 3.9 ps increments. When the shutter is opened by a single ultrashort 1.06 p pulse, samples are cut out from the 10 scattered pulses. These samples are centered at 3.9 ps intervals from the leading to the lagging edge of the green pulse shape. Another and more efficient way of obtaining 10 pulses replicating the shape of the incident pulse, is by using Vogel’s multibeamsplitter (see VOGEL,SAVAGE and DUGUAY [19741) shown in Fig. 20. The spacing between pulses is determined by the thickness of the glass etalons used. With 3.3 mm thick glass etalons, the pulses are 4.1 ps apart. Vogel’s multibeamsplitter
o
IV, 41
187
SAMPLING OPTICAL SIGNALS
t
yi
FIBER ARRAY
i5ns
Fig. 19. Set-up used to optically sample ultrashort laser pulses and display them on a real-time oscilloscope. The pulse of green ( A = 0.53~) light to be displayed comes down from the top left and enters a cell containing highly diluted milk (or a Ludox silica suspension type LS). The ultrafast shutter is driven by the infrared pulse shown. The shutter opens only once and cuts out a slice (or sample) from each scattered pulse. The samples are centered at 3.9 psec intervals from the leading to the lagging edge of the incident green pulse. The fiber array transforms the spatially distinct samples at points A‘, B‘, C‘, . . . . into temporally distinct pulses on the oscilloscope screen. The fiber ends at A‘, B , C‘, . . . . are butted against a glycerin-wetted glass window (not shown) for good optical coupling.
has the advantage over the echelon (see TOPP,RENTZEPIS and JONES [1971]) in this application, of replicating the signal pulse not only temporally but also spatially. When the signal beam has spatial inhomogeneities, this insures that all replica pulses entering the shutter are identical in shape. The 10 samples cutout by the ultrafast gate are sent into 10 optical fibers, cut to progressively longer lengths and giving delays of 5, 15,25, 35, . . ., 95 ns. This set of optical fibers if referred to as an “organ” array, by analogy with the progressively longer pipes of the musical organ. The out-
188
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
[IV,
0
4
VOGEL'S MULTIBE
n
~-
/ n /n /n
J
OPTICAL SIGNAL
Fig. 20. Multibeamsplitter developed by VOGEL,SAVAGE and DUCUAY [I9741 to subdivide an incident optical signal into 10 replica pulses (only four shown above, for clarity), identical except as to height to the incident pulse, and spaced apart by 4.1 ps in this case. In the absence of the antireflection-coated etalons E, ,E,, . . ., El,, the replica pulses would be synchronous at the shutter. The ten identical glass etalons introduce a uniform interpulse delay of 4.1 ps. The ten replica pulses are directed towards the ultrafast Kerr shutter for multichannel sampling.
put ends of the fibers are placed near the face of a single photomultiplier of 3 ns response time, the output of which is displayed on a fast (0.8 ns risetime) oscilloscope. This way the 10 samples, which were taken within the same 5-10 ps interval, are spaced out at 10 ns intervals for detection and display. An example of a display obtained with Vogel's multibeamsplitter is shown in Fig. 21. The second harmonic pulse at A = 0.53 p is being sampled here. If the shutter had opened for, say, 2 ps in this case, the envelope of the samples would represent the green pulse shape. For that laser shot, however, the duration of the 1.06 p gating pulse was more like 16 ps, leading to a gate open time of about 10 ps, a duration about equal to the green pulse width. The envelope of the sample pulses in Fig. 21 represents in fact the function G(4)(7)given in eq. (2.10). The sensitivity of the OSO with Vogel's beamsplitter was such that pulses with peak powers down to a few watts could be displayed. With various improvements (see VOGEL,SAVAGE and DUGUAY [19741) sensitivity down to the level of ten milliwatts in optical power seen possible in practice. 4.4. MULTICHANNEL SAMPLING WITH DETECTOR ARRAYS
Recently MOUROUand MALLEY[I9741 have implemented an elegant way of measuring picosecond fluorescence decay times. Their set-up is shown in Fig. 22. Second harmonic pulses (A = 0.53 p) about 4 ps in dura-
IV,P 41
SAMPLING OPTICAL SIGNALS
1
189
4 . 1 ~ 8SAMPLING STEP
(a) PULSE DISPLAY
(b) GATED CALIBRATION Fig. 21. (a) Sampled display of an ultrashort laser pulse obtained with a Vogel multibeamsplitter, an ultrafast shutter and an organ array as shown in Fig. ,20. The envelopqof the samples constituks the displayed shape. This shape is the convolution G‘4’(5’) described in the text involving the green pulse shape ( - 12 ps wide) and the shutter transmission function T(t) (also 12 ps at half maximum in this sample). The width of the display at half maximum is about equal to the duration of the infrared pulse that drove the shutter in this example (17 ps). The ordinate scale points up the excellent sensitivity available with this technique. (b) Picture obtained when the ultrafast shutter is manually opened by removing P, . Unequal heights reflect residual ineqiialities in the coupling and transmission of the various channels.
,.,
190
T H E U L T R A F A S T O P T I C A L KERR S H U T T E R
“v,
P4
QATING PULSE F
.LINEAR ARRAY OF DE
w
Fig. 22. Set-up used by MOUROUand MALLEY[1974] to measure ultrafast relaxation times. A second harmonic green pulse 4ps in duration propagates through the dye solution (erythrosin in water) leaving an exponentially decaying tail of fluorescent light in its wake. This fluorescence tail is imaged through the ultrafast shutter onto an array of 512 photodiodes. When the shutter is opened for 4 ps by an infrared pulse, a record of the tail is captured by the linear diode array. N
2 4 -
z a
K a K
s 3 -
K
a
J
a
At = 4 psec
E 2 v)
0
x K
0 0 W
530nm
I -
K
-
0
t(psec)-
0
100
Fig. 23. Result obtained by MOUROUand MALLEY119741 using the arrangement of Fig. 22. The risetime of the fluorescence signal from erythrosin B in water (3 x 10- M) appears limited by the widths of the excitation and gating pulses .,( 4 ps). When the dye is replaced by milky water, a stop-motion image of the second harmonic (530 nm) pulse is projected onto the linear photodiode array: the sharp curve obtained gives the prompt response of the system, as in Fig. 16.
IV,
§ 51
CONCLUDING R E M A R K S
191
tion derived from a Nd: glass laser are passed through a cell containing a molar solution of erythrosin in water. The green beam has been collinated down to a diameter of 1 mm. The spontaneous fluorescence light emitted along this narrow track is imaged through collinear Rerr gate onto a linear array of 512 photodetectors (only 5 are shown in Fig. 19 for clarity). Each photodetector collects light from a small volume element situated along the fluorescence track. The light pulses emitted by each volume element reach the ultrafast shutter after a delay proportional to the distance along AZ. When the shutter is opened by the 1.06 p pulse (-6 ps in duration) it cuts out one sample from each pulse. Just as in the optical sampling oscilloscope, the samples are spaced uniformly from the leading to the lagging edge of the pulse shape. The result obtained by Momou and MALLEY [1974] for erythrosin is shown in Fig. 23. The number of samples taken is large enough to make the recorded trace appear continuous. Mourou and Malley found a fluorescence signal risetime of 4f 1 ps, that is essentially equal to the time resolution of the apparatus. The implication is that the true fluorescence risetime (i-e., the risetime under deltafunction pulse excitation) is less than 4 ps. Another way of looking at the Mourou-Malley experiment is from the point of view of ultrahigh speed photography (see DUGUAY and HANSEN [1970]). The green pulse (or light “bullet”) leaves in its wave a tail of fluorescence light that is stop-motion photographed in flight by the Kerr shutter and detector array. The latter replaces the film and records one only horizontal line of the picture, but that is all that matters here. Momou and MALLEY [1974] have also used the ultrafast shutter and their detector array in conjunction with a spectrograph to do time resolved spectroscopy of the spontaneous emission of rhodamine 6G on the picosecond time scale.
0 5. Concluding Remarks The ultrafast Kerr shutter has established itself as a useful instrument in fields of studies involving picosecond laser pulses. The shutter has been driven by pulses derived from a variety of lasers, including COz laser pulses (see DUGUAY and SAVAGE [1973], OWEN,COLEMAN and BURGESS[1973]). The potential of this instrument in studies of ultrafast molecular dynamics has been left largely unexploited so far, probably for a number of reasons, one of which certainly being the difficulty and expense of producing stable powerful laser pulses. If the day comes when laser pulses achieve the reliability and flexibility of electronic pulses, use of the ultrafast Kerr
192
THE U L T R A F A S T O P T I C A L K E R R S H U T T E R
CIV
shutter will become an easy task, a task that will not involve the difficulties and dangers of high voltage pulses used in conventional Kerr cells. The use of subpicosecond pulses in driving the shutter (IPPENand SHANK [19751) has opened a most intriguing new frontier where the electronic Kerr effect will certainly be called into play. Thus, one century after the discovery of the DC Kerr effect (KERR[1875]), a closely related effect, the AC Kerr effect, is playing an active role at the pinpoint of technology. Acknowledgements
I would like to acknowledge the help of Mrs. Jeri Romaine and Mr. A. Savage in preparing the manuscript for publication. References ALFANO,R. R. and S. L. SHAPIRO, 1972, Optics Commun. 6, 98. and M. POPE,1973, Optics Comm. 9, 388. ALFANO,R. R., S. L. SHAPIRO 1967, Phys. Rev. 154, 129. BROIDA,H. P. and S. L. SHAPIRO, BUCKINGHAM, A. D., 1956, Proc. Phys. SOC.B69, 344. CAMPILLO, A. J., S. L. SHAPIRO and B. R. SUYDAM, 1973, Appl. Phys. Lett. 23,628. and B. R. SUYDAM, 1974, Appl. Phys. Lett. 24, 178. CAMPILLO, A. J., S. L. SHAPIRO and C. H. TOWNES, 1964, Phys. Rev. Lett. 13,479. CHIAO,R. Y ., E. GARMIRE and H. HEYNAU, 1966, Appl. Phys. Lett. 8, 174. DEMARIA, A. J., D. A. STETSER DUGUAY, M. A., 1971, American Scientist 59, 550. DUGUAY, M. A. and J. W. HANSEN,1969a, Appl. Phys. Lett. 15, 192. M. A. and J. W. HANSEN,1969b, Optics Commun. I , 254. DUGUAY, DUGUAY, M. A. and J. W. HANSEN, 1970, NBS Special Publication No. 341, pp. 45-49 (Gov. Printing Office, Washington, D.C.). DUGUAY, M. A. and A. T. MATTICK,1971, Appl. Optics 10, 2162. DUGUAY, M. A. and A. SAVAGE, 1973, Optics Commun. 9.212. DUGUAY, M. A. and J. N. OLSEN,1975, Picosecond X-ray Pulses, IEEE J. Quant. Elect. QE11, 170. FABELINSKII, 1. L., 1968, Molecular Scattering of Light (Plenum Press, New York). FISCHER, R. and R. ROSSMANITH, 1973, IEEE Trans. Nucl. Sci. NS-20, 549. J. A., P. M. RENTZEPIS, S. L. SHAPIRO and K. W. WECHT,1967, Appl. Phys. Lett. GIORDMAINE, 11. 216. JANSSEN, J. M. L., 1950, Philips Tech. Rev. 12, 52. IPPEN,E. P. and C. V. SHANK,1975, Appl. Phys. Lett. 26. 92. KELLEY, P. L., 1965, Phys. Rev. Lett. 15, 1005. KERR,J.. 1875, Phil. Mag. 50, 337. MALLEY. M. M. and P. M. R~NTZEPIS. 1970. Chem. Phys. Lett. 7, 57. MAYER,G. and F. GIRES.1964. C. R. Hebd. Seanc. Acad. Sc. Paris 258, 2039. MOUROU. G. and M. M. MALLEY,1974, Optics Commun. 11, 282. NETZEL, T.. P. M. RENTZEPIS and J. LEIGH, 1973, Science 182, 238. and T. J. BURGESS, 1973, Appl. Phys. Lett. 6,272. OWEN,T. C., L. W. COLEMAN OWYOUNG, A,, R. W. HELLWARTH and N. GEORGE, 1972, Phys. Rev. B5,628. PAILLETTE, M., 1969, Annales de Physique 4, 671. RICHARDSON, M. C. and K. SALA,1913. Appl. Phys. Lett. 23. 420.
IVf
REFERENCES
193
SHANK,C. V. and E. P. IPPEN, 1974, Appl. Phys. Lett. 24. 313. S. L. and M. A. DUGUAY, 1969, Phys. Lett. 28A, 698. SHAPIRO, SHIMIZU,F. and B. P. STOICHEFF, 1969, IEEE J. Quant. Electr. QE-5, 544. and I. L. FABELINSKII, 1966, Zh. Eksp. i Teor. Fiz. Pisma STARUNOY, V. S., E. V. TIGANOV Redaktsiyu 4, 262 [Engl. Trans.: Soviet Phys. JETP Letters 4, 1761. SUYDAM, B. R., 1973, NBS Special Publication No. 387 (Gov. Printing Offce, Washington, D.C.) pp. 42-48. and R. P. JONES, 1971a, J. Appl. Phys. 42, 3415. TOPP, M. R., P. M. RENTZEPIS TOPP,M. R., P. M. RENTZEPISand R. P. JONES,1971b, Chem. Phys. Lett. 9. 1. VOGEL,G. C., A. SAVAGE and M. A. DUGUAY, 1974, IEEE J. Quant. Electr. QE-l0,642. Yu, W. and R. R. ALFANO.1974, Opt. Electronics 6, 243. Yu, W., P. P. Ho, R. R. A L F A N OM.~SEIBERT, ~~ 1975, Biochimica et Biophysica Acta 387,159.
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E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
V
HOLOGRAPHIC DIFFRACTION GRATINGS BY
GUNTER SCHMAHL and DIETBERT RUDOLPH Uniwrsitiifs- Sternwmte, Giittingen,
Germany
CONTENTS PAGE
9 1 . INTRODUCTION. . . . . . . . . . . . . . . . . . . 5 2. THEORETICAL CHARACTERISTICS OF SPECTROSCOPIC DIFFRACTION GRATINGS. . .
0
.. .. . . .
197
198
3. BASIC PRINCIPLES OF HOLOGRAPHIC DIFFRACTION G R A T I N G S . . . . . . . . . . . . . . . . . . . . . . 200
5 4. PRODUCTION OF HOLOGRAPHIC GRATINGS . . . . 216 0 5. PROPERTIES OF HOLOGRAPHIC GRATINGS AND COMPARISON WITH CLASSICAL GRATINGS. . . . . 223
5 6. FURTHER IMPROVEMENTS OF HOLOGRAPHIC GRATINGS..
. . . . . .
. . . . . . . . .
.
. . . 242
. . . .
.
. . . 242
. . . . . . . . . . . . . . .
.
.
,
ACKNOWLEDGEMENTS. . . . . . . . . . REFERENCES. . . .
. .
242
0 1. Introduction The first experiments with grating like structures were probably made by the American astronomer David RITTENHOUSE [17861 in Philadelphia. He used parallel hairs laid in a fine screw and observed diffraction effects of light. The first ruled optical gratings were produced by Joseph VON FRAUNHOFER [1821/22], who also discovered the fundamental properties of optical diffraction gratings. Comprehensive reviews of the history, theory and manufacture of gratings have been given by KAYSER [1900], STROKE [1963, 19671 and HARRISON [19733. Because of the severe mechanical problems of ruling gratings many alternative methods of production have been considered. MICHELSON [19271 suggested producing gratings by photographing stationary waves using Lippmann plates. At the National Physical Laboratory in England BURCH and PALMER [1961] made gratings by photographing fine interference fringes and measured the shift caused by processing of photographic emulsions. LABEYRIE [1 9661 suggested using layers of bichromated gelatine, Daguerre layers, photoconductive layers in connection with sputtering techniques or thermoplastic layers as recording materials. None of these methods gave gratings suitable for practical spectroscopic use. Stimulated by the fact that in high resolution stellar spectroscopy with large telescopes it is neccessary to use large diffraction gratings of good quality, the authors of the present article proposed making diffraction gratings holographically by using photoresist layers. The first results, obtained in 1967, already demonstrated the good optical quality of such holographically made diffraction gratings (RUDOLPH and SCHMAHL [1967a, b, c, 19681). In the meantime several groups have started working in this field and holographic diffraction gratings compete with classically ruled gratings from the visible to the ultraviolet and soft X-ray regions. The scope of this article is to describe the basic method of making gratings holographically, to describe the results obtained and to compare these results with those obtained from gratings produced by traditional means. 197
198
HOLOGRAPHIC DIFFRACTION GRATINGS
[v. 8 2
0 2. Theoretical Characteristics of Spectroscopic Diffraction Gratings The notation for the basic grating equations is introduced in this section and a brief summary of the theory of spectral image formation by optical gratings is given (STROKE [1963]). The relation between the wavelength and the angle of diffraction is given by the grating equation ma1 sin i+sin i' = -, a
where i and 'i are the angles formed by the wave propagation vectors of the incident and the diffracted waves with the normal to the grating surface. The grating spacing is a, the wavelength of the diffracted radiation 1and the spectral order m. The angle of diffraction 'i is positive if the incident and diffracted wavefronts are on the same side of the normal and negative if they are on opposite sides of the normal. The angular dispersion, 6, is given by
m 1 di' g=-=____-(sini sin it). dA cos ilea I cos i'
+
In autocollimation i = 'i (2.2a) The linear dispersion in the focal plane of a focussing system of focal length f is di' L = - . f. dI Assume that a grating is used in a spectrograph with a very narrow entrance slit; there will then be no difference between coherent and incoherent illumination. According to the Rayleigh criterion two spectral elements with wavelengths I and 1 A 1 can be separated if the maximum of the diffraction pattern of the first spectral element coincides with the first minimum of the second spectral element. For a monochromatic plane wave with a width A = W . cos'i (see Fig. 1) the first minimum of the diffraction pattern is located at
+
A Po =
A
2 = w .cos -
(2.4)
With A = W .cos 'i and di' = Po in equation (2.2) and using equation (2.4)
”3
§ 21
SPECTROSCOPIC DIFFRACTION G R A T I N G S
199
Fig. I .
the resolving power, R,is seen to be expressible in the form
1 A1
w 1
R = - = -(sin i + s h i’) = N . m = A * 6, where N is the total number of grooves. In the focal plane of the spectrograph the complex amplitude of the diffraction pattern of a monochromatic plane wave is given by
where the angular coordinates of the diffraction pattern in the focal plane is given by fi = n x / ( A * f / A and ) q = u/(A/2),the normalised width of the diffracted plane wave. The complex amplitude of the diffracted monochromatic plane wave g(q) is given by
Here d(q) is the phase of the diffracted wavefront which can be derived from wavefront interferograms. d(q) is constant for a plane wave. We also assume that the amplitude is constant, which means that the efficiency and polarisation properties are uniform over the whole area of the grating. With %(q) = 1
The intensity distribution of the diffraction pattern is given by
I@)
1
=-
4n
%*(/9) = -
?I
CB>” ___
200
HOLOGRAPHIC DIFFRACTION G R A TIN GS
cv. 4 3
where ‘$I@) is normalised so that the total intensity is 1. Wave front aberrations A(?) are introduced by periodic, random and progressive errors in the rulings and also by variations in the flatness of the blank or in the layer in which the grooves are made. Aberrations also occur if ‘$I(?) is not constant. All aberrations give rise to deviations from the theoretical diffracted intensity distribution. High quality diffraction gratings should therefore have the following properties : a) A high ruling accuracy; i.e. periodic, random and progressive ruling errors have to be as small as possible. Periodic errors result in ghosts in the spectrum, random errors give broad wings to the diffraction pattern i.e. scattered light over a broad wavelength region. Progressive ruling errors (also known as error of run) over parts of the grating give rise to deviations from the theoretical line profile near the centre of the line. b) Good optical quality of the grating blank and the layer in which the individual grooves are made. c) A well defined profile for the individual grooves to obtain high efficiency and small amount of light scattering. d) The profile of the grooves should be uniform over the entire grating to obtain constant intensity over the diffracted wavefronts when using the grating in a spectrograph or a spectrometer.
0 3.
Basic Principles of Holographic Diffraction Gratings
A hologram made by superposition of two monochromatic waves and subsequent storage of the resultant intensity distribution is an optical element with dispersive and imaging properties. The most simple holograms can be made by using two plane waves, a plane wave and a spherical wave or two spherical waves. In principle the process for making gratings holographically is very simple. Glass blanks polished to optical tolerances are coated with a thin film of photoresist which is capable of forming well defined profiles of the individual grooves of a grating and which can also store an interference fringe system with accuracy and durability. The process of constructing gratings with laser light and photoresist layers could as well be called “interferographic”. However, the term holographic grating is more useful than the term interference grating especially for gratings with imaging properties and for gratings with improved accuracy made with identical reconstructed wavefronts.
V, I 31
BASIC PRINCIPLES
20I
3. I . INTERFERENCE FRINGE SYSTEM
3.1.1. Accuracy of the interference fringe system
The accuracy of the interference fringe system necessary for making gratings holographically will be discussed in this article in the case of an interference pattern for plane gratings with equidistant spacings. If we assume that the blank and the photoresist layer are perfectly plane the accuracy of the diffracted wavefronts will only depend on the ruling accuracy. For holographic gratings the ruling accuracy is determined by the accuracy of the interference system used to make the gratings.
Fig. 2. Formation of interference fringes in the xz-plane.
In Fig. 2 two monochromatic beams, with direction of propagation parallel to the x, y plane are superposed
and
(3.1) S,(x, y ) = %(x, y) . exp
If these beams are parallel beams we can write 271 $1
x sin i,
=
2n: $2
=
(34
x sin i'.
From Fig. 2 i = a+/?, 'i = or-8. 'i is negative if i and 'i are on opposite sides of the normal in the xy-plane. Since we are only interested in the spatial frequency and not in the position of the interference maxima and
202
cv, § 3
HOLOGRAPHIC DIFFRACTION GRATINGS
minima we can choose p l = p2= 8. The superposition yields an intensity distribution I(x) = F e x p ( i ( b , +
.b*
exp (-i
7
t ) ) +Bexp(i(9,+
Tt))j
(41+T t ) ) +B*exp (-i (42+
7
t))]
= A Z + B 2 + 2 A . B cos (41-42).
(3.3)
Equation (3.3) corresponds to equally spaced fringes. If A = B the maximum intensity variation from zero intensity in the minima to Z(x) = 4A2 in the maxima is obtained. From equations (3.2) and (3.3) the spacing of the fringes can be obtained as
-
a=
sin (cl+fi)+sin (a-fi)
A
2 sin clcos j? '
(3.4)
Using a symmetrical system (fi = 0) and I = 0.4 pm one obtains fringes with the following spacings (Table 1). For a perfect plane grating the grating spacing is constant over the whole area of the grating. Variations in the spacing yield aberrations, A(?), in the diffracted wavefronts and consequentlydeviations from the ideal intensity distribution in the spectrum. TABLE1 Grating spacing as function of the intersection angle a for I = 0.4 pm
uc"1
a b l
grooves/mm
1.1 11.5 53.1
10 1
100 1000 4000
0.25
The production of interference fringes with exactly equal spacings would require perfectly plane wavefronts. However, these cannot exist for two reasons: Firstly, in practice optical elements must be used to produce the wavefronts and these are never perfect. Secondly, because finite beams must be used divergence due to diffraction at the apertures is introduced. Wavefronts diffracted by high quality spectroscopic diffraction gratings should have no aberrations of high spatial frequency. Aberrations of low spatial frequency should not exceed 1/20, if 2 is the minimum wavelength used. In holographic production of gratings one normally uses high quality
v, 0 31
203
BASIC PRINCIPLES
optical elements, which do not show aberrations of high spatial frequency. Common aberrations of such elements are of very low spatial frequency. A holographic grating made with wavefronts having such aberrations has, for example, a ruling error Ax over the length I of the grating. This means that the position of one groove at the end of a ruled interval of the length 1 differs by Ax from the ideal position. For a grating used in autocollimation such a ruling error yields a wavefront aberration A(?) = 2 * Ax sin i'.
(3.5)
TABLE 2 Values of Ax for various values of i' and A($. The quantities Ax, i' and d(q) denote the ruling error, the angle of diffraction, and the diffracted wavefront aberration (for 1= 0.4 pm), respectively \
15"
30"
45"
60"
0.19 0.08
0.10
0.04
0.07 0.03
0.06
1/10 1.120
0.04
0.02
0.01
0.01
44
0.02
Table 2 shows values of Ax for various values of i' and A(?) for A = 0.4 pm. The larger i', the better Ax has to be. In addition, the value of Ax has to be smaller as the wavelength at which the grating is used decreases. When constructing the grating holographically according to Fig. 2, the wavefronts of the two beams S1and S2 will have aberrations due to the optical elements used. We assume here that S, has an ideal wavefront and S2 has an aberration A(?). Then a grating produced with these two wavefronts - and on an ideal blank and photoresist layer - reconstructs the beam S2 - without considering amplitude factors - when the grating is illuminated with the wavefront S1.We can call the reconstructed beam the + 1 order. This example demonstrates that for the construction of a holographic grating which will be used in the first (or low) order the optical elements with which the beams S1 and S2 are produced do not neccessarily have to be much better than the collimator and the camera of the spectrograph in which the grating is mounted. To make gratings holographically which are used in higher orders (echelle gratings) the accuracy of good optical elements is not sufficient. The reason is, that the wavefront aberrations for a grating (in autocollimation) are given by
204
HOLOGRAPHIC DIFFRACTION GRATINGS
[v. § 3
Because Ax cc a - with a given wavefront aberration the ruling error Ax increases linearly with increasing grating spacing - it follows that d(q) cc m. Assuming an echelle grating with 100 grooves/mm used for 0.5 pm in the 17th order with i = 'i = 60" the aberrations are seventeen times larger than for a high line density grating to be used in the first order for 1 = 0.5 pm made with the same beams. Nevertheless it is possible to obtain the ruling accuracy necessary for holographic echelle gratings. This point will be discussed in section 3.1.3. Independent of the source of aberrations of ideal plane wavefronts - caused by zone errors, diffraction or misalignment of the optical elementsone can consider two extreme possibilities. The first case is the superposition of two slightly divergent beams, the second case is the superposition of one slightly divergent beam with one slightly convergent beam (RUDOLPH and SCHMAHL [1970)). For a grating with a progressive ruling error one can write
where x is the width of the grating up to the nth groove, measured from the middle of the grating. Ax is the deviation of the nth groove from its ideal position. For the change of the grating spacing as a function of n it follows from eq. (3.7) that
-=I d2x
_ 4 -_ 1 nv-2
dn2
(v-2)!
v=2
(3.8)
The term d1 corresponds to a progressive linear ruling error and yields an astigmatism of the grating, and higher terms of 6, influence the spectral resolution. The superposition of two divergent (or two convergent) beams yields an interference field which can be described as a section of a family of two-sheet hyperboloids of revolution (hyperbolic case). In Fig. 3 two divergent beams originate from the two points F1 and F2 which are the focal points of th$ hyperboloids of revolution that are the geometrical loci of the interference maxima. The hyperboloids are given by x2/a: -y2/ai - z2/a; = 1 with 2al = Ir, -r21 = nl, n = 0, 1, 2 ... and a: = a; = cz -$n2A2. In a plane parallel to the xy-plane z = h the geometrical loci of the interference maxima are hyperbolas
-tn2A2)).
xz = $n2A2(1 + ( y 2 +h2)/(c2
(3.9) The spacings a of the interference maxima, which are given by the angle
V,
o 31
205
BASIC PRINCIPLES
Fig. 3.
+
2a = F1PF2 ,are considered as a function of x in the vicinity of the plane x = 0, i.e. the divergences of both beams are nearly equal. In all practical
cases c is in the order of several kilometers, that means c > nl. With y = s and s2+h2 = b2 we obtain from eq. (3.9)
+ ..
.))
(3.10)
and hence on taking the square root we obtain the following expansion for x: x =
4 ( I + F') b2 *. n +
g(l+ $)-i*
n3+.
2
. ..
(3.11)
We note that eq. (3.11) contains n in odd powers only. For the change of the grating spacing as a function of n it follows from eq. (3.11) da -= dn
d2x dn2
6A3b2
-w
$)-i
*n+
....
(3.12)
A comparison of eq. (3.12) with eq. (3.8) shows that 6 , = 0, that means, that in the hyperbolic case the grating is -to the first approximation - free of aberrations. The superpositionofonedivergent beam withoneconvergentbeam yields an interference field which can be described as a section of a family of ellip soids of revolution (elliptic case). In Fig, 4 one divergent beam originates from the point F, and m e beam converges to the point F,. F, and F, are the focal points of the ellipsoids of revolution that are the geometrical loci of the interference maxima. The ellipsoids are given by x2/a: +y2/ai +
206
HOLOGRAPHIC DIFFRACTION GRATINGS
t
Fig. 4.
z 2 / u : = 1 w i t h 2 a 3 = ( r , + r 2 ) = 2 c + t * A a n d a ~ = a ~ = c - t - A + $ t 2 AI 2n .a plane z = h parallel to the xy-plane the geometrical loci of the interference maxima are circles
x2 = - y2
+( t k+y12)(1- h2(C ++A))-
2).
(3.13)
The interference pattern is therefore a section of a zone plate pattern (SCHMAHL and RUDOLPH [1969]). A grating made with one divergent and one convergent beam is, therefore, a section of a zone plate at very large values of t. We may, therefore, restrict the discussion to determine the spacing a as a function of x for y g: x and z -jc x. In Fig. 2 for z = y = 0, /3 = 0 it follows from eq. (3.13) that
( i l r+ ...- ).
x =(tAc)3 1 +
-
-
(3.14)
Because of the assumption that the grating is a section with large zone numbers t can be replaced by t = to+n, with n = running number of the grating; n = 0 represents the middle of the grating. With t = to +n and expansion of the square root eq. (3.14) gives
In the expansions (3.14) and eq. (3.15) higher powers of A/toc are neglected. The change in grating spacing as a function of n follows from eq. (3.15) within an accuracy better than 10% for all practical cases: da d2x = dn dn2
--&A,)*
*
to%+
. . ..
(3.16)
A comparison of eq. (3.16) with eq. (3.8) shows, that 6, # 0; that means in the elliptic case that the grating has a linear progressive ruling error. In Table 3 numerical values for Ax (cf. eq. (3.7)) are given for gratings with
V, Q
31
201
BASIC PRINCIPLES
TABLE3 Values of Ax calculated for gratings in the hyperbolic and elliptic cases.
1
100
500 1000 2000
J
0.3 0.1 0.05
a width W = 400 mm and grating spacings in the range 10 pm to 0.5 pm. In the two cases the beams were assumed to have a divergence or angle of aperture 2y = 0.2 seconds of arc. The result is that in the hyperbolic case the accuracy is much higher than in the elliptic case. The physical reason for this effect is that in the first case the intersection angle 2c1of the two wavefronts is nearly constant over the whole beam diameter. It follows that it is not absolutely neccessary to use nearly plane wavefronts to make gratings of high quality. The wavefronts may have certain aberrations, but they must be identical in the two beams. With usual optical elements it is impossible to obtain identical wavefronts. This, however, is possible by holographic reconstruction and will be discussed in section 3.1.3. 3.1.2. Interference arrangements Fig. 5 shows some optical arrangements suitable for the construction of normal holographic gratings. In Fig. 5a a lens 0, a spatial filter S and a spherical or parabolic mirror P are used to produce a plane wave, parts of which are reflected by plane mirrors PI to generate the interference fringe system which is recorded in the plane E. Fig. 5b shows an arrangement with two off-axis mirror systems. In Fig. 5c an arrangement is shown suitable for making large gratings (with a central aperture), e.g. for coud6 spectrographs of large telescopes.
3.1.3. Improvement of the ruling accuracy by superposition of identical reconstructed wavefronts As discussed in section 3.1.1 the accuracy of an interference field, made with commercially available good optical elements, is sufficient for holographic gratings for use in the visible region in a low order. This is no longer true for spectroscopic gratings for use in higher orders, for spectroscopic gratings constructed with visible light and used at much shorter wave-
208
[v.
HOLOGRAPHIC DIFFRACTION GRATINGS
5
3
PI
P
2a
3Yv D
Fig. 5. Optical arrangements.
lengths and for gratings wiih a low line density for use as scales. To enhance the ruling accuracy of such gratings one can use the following procedure (SCHMAHLand RUDOLPH[1970]): Each wavefront affected with aberrations can be assumed to consist of parts of wavefronts with different
v, 8 31
BASIC PRINCIPLES
209
divergences. If one succeeds in superposing wavefronts, the parts of which have in each case the same divergence, i.e. identical wavefronts, the accuracy of the above discussed hyperbolic case can be obtained over the whole surface of the grating. Identical wavefronts can be made by reconstruction using two holograms. The principle is illustrated in Fig. 6: One hologram H,, is made using the two beams R, and R 2 , and a second.hologram H32
recording process
reconstruction
Fig. 6. Principle of the reconstruction of identical wavefronts.
using the beams R3 and R2. Illuminating the hologram H12with R1 and the hologram H3, with R; , the beam R2 is reconstructed twice, except for amplitude factors. The superposition of the two beams R2 yields the grating H22. In a model experiment holograms H12 and H3, were made with the beams R1, R2 and R 3 . The wavefronts of all three beams had aberrations of several wavelengths. These aberrations were introduced by using bad optical elements. IgFig. 7a and Fig. 7b Moirb patterns from the holograms H,2 and H32 are shown. The Moire patterns were made by illuminating the holograms with an interference fringe system made with two plane wavefronts having aberrations 5 A/lO. The Moire patterns demonstrate that these holograms- acccrding to the fact that they are made with distorted wavefronts - have large ruling errors. From these two holograms two identical wavefronts R2 were reconstructed according to Fig. 6. Superposition of these two wavefronts yielded the hologram H22. Fig. 7c shows the Moire pattern of this hologram which demonstrates the high ruling accuracy
210
HOLOGRAPHIC DIFFRACTION GRATINGS
,
Fig. 7a. Moire pattern of the hologram H I .
Fig. 7b. Moirt5 pattern of the hologram H,,
V,
Q
31
BASIC PRINCIPLES
21 1
Fig. 7c. Moire pattern of the hologram H,,
of this hologram. Figs. 7a-c show - in agreement with theoretical considerations - that the gain in accuracy obtained by this method is about two orders of magnitude, if the aberrations are not greater than a few wavelengths. In Fig. 6 as well as in the model experiment discussed, the aberrations introduced by the blanks and the resist layer of the holograms H I , and H,, have been neglected. This effect can be taken into account by providing the hologram H I with the aberrations of the blank of the hologram H,, and vice versa. In Fig. 8 an example of an optical arrangement suitable for such an experiment is shown. In Fig. 8a the laser beam L is divided by a beamsplitter 1. A quasi-plane wavefront is generated by the mirror 4 via the mirror 2 and a lens and a spatial filter 3. After reflection on the blank 5, provided with a resisi layer and coated with a metal layer, and the mirror 6 this wavefront is combined with a spherical wave produced with the help of mirror 8 and’ a lens and a spatial filter 9. With this interference pattern blank 7, coated with a photoresist layer, is exposed. The undeveloped resist layer on blank 7 is then made reflective and the reflective coating on the resist layer of blank 5 is stripped off. In the next step the blank 5 is replaced by blank 7 and exposed by an interference figure according to Fig. 8b. After removing the reflective coating
212
[V>§
HOLOGRAPHIC DIFFRACTION GRATINGS
3
from blank 7 both the exposed resist layers of blanks 5 and 7 are developed and recoated. As in Fig. 8c the holograms on blanks 5 and 7 reconstruct two beams with identical wavefronts which are superposed. The resulting interference pattern with highly improved ruling accuracy is recorded in a resist layer on the blank 12.
Fig. 8a.
a2
. ,L
\
213
BASIC PRINCIPLES
Fig. 8a-c. Optical arrangements for producing holographic gratings with improved accuracy by the use of identical reconstructed wavefronts.
3.1.4. Frequency and wavelength stability of the laser light Laser beams have a finite spectral width which affects the contrast K in the interference figure, given by K = (I,,,,,-Zmin)/(Zmax Zmin).Zmax, Zminare the intensities of the interference maxima and minima respectively. When the path length difference A between the two interfering beams is zero, the contrast is unity under optimal conditions. The contrast of the interference fringes as a function of the path length difference depends on the width and the profile of the laser line used. When the path length difference is small compared to the coherence length, the contrast is nearly independent of the profile of the spectral line. The coherence length can, therefore be expressed by the formule L z 71. c/Av where Av is the width of the spectral line. To make a good grating we assume that the contrast of the interference fringes does not decrease by more than 5 % from the centre to the edge of the interference pattern. In this case the necessary coherence length is L x 10 A. If A is zero at the centre of the interference pattern, i.e. at x = 0 in Fig. 2, the path length difference at x = I = 0.5 W with j3 = 8 is given by A = 0.5 W sin CI.For a grating with W = 400 mm and with 2000 grooves/mm made with the laser line A = 457.9 nm it follows that A = 92 mm and L = 10 A = 920 mm or Av = lo9 Hz. In addition to a small width the laser line must have a high frequency and wavelength stability. According to eq. (2.1) the relative variation of the spacing is
+
214
HOLOGRAPHIC DIFFRACTION GRATINGS
cv, § 3
given by Aala = A]./). = Av/v. For a fixed number of interference maxima 2 N, i.e. with W = 2 N u = 2 1, the change of the width of the grating is given by A l / l = Aa/a. Let vo be the frequency of the centre of the line at the time t o and vo+Av the frequency of the centre of the line at to+At, At being the exposure time. To obtain sufficient ruling accuracy the Nth interference maximum should not shift by more than 0.05 a during the time At. It follows that AI.= (Aa/a)l= (Av/v)lS 0.05 a. In the above mentioned example, with W = 21 = 400 mm, a = 0.5 pm the required frequency stability is given by Av/v = 1.25 x When using the laser line A = 457.9 nm and A = 350.7 nm, Av = 80 MHz and Av = 100 MHz respectively. Such a high frequency stability in connection with a narrow line width can be obtained by use of an oven stabilised Ctalon in the laser cavity (DOWLEY [1971]) which filters one axial mode. The frequency interval between two axial modes for a laser running inTEM,, is given by Av = c/2RL, where R, is the length of the laser cavity. The width of an axial mode is a small fraction of this interval. With high power CW Ar' and Kr' lasers for which R, = 145 cm, the interval between two axial modes is lo8 Hz. Using an oven stabilised &talonwith such lasers and only one axial mode - with a corresponding long coherence length of several meters - the frequency stability was found to be better than 75 MHz. The frequency stability was measured by using a confocal scanning interferometer. Even with high frequency stability the wavelength of the laser light can be shifted by temperature and pressure variations in the air surrounding the laser, because of the variation of the refractive index. For example a variation in the pressure of amount Ap = 0.1 mm Hg yields a wavelength variation of AAjL = 4 x lo-*, a variation of temperature A T = 0.1 K yields Ai/A = Under normal weather conditions pressure variations are usually much smaller than Ap = 0.1 mm Hg during the exposure times necessary to make gratings holographically and hence, pressure stabilisation is not necessary. It is, however, useful to stabilise the temperature. 3.2. PHOTORESIST LAYERS
To fulfill the four requirements mentioned in 92 it is necessary to use a low noise recording material with which-it is possible to obtain well defined profiles of the individual grooves and to preserve the high accuracy of an interference fringe system. Such recording materials can be found in the family of photoresists (cf. e.g. CLARK[1973]). Normal photographic emulsions of silver halides in gelatine and other light sensitive layers based on gelatine are useless for this purpose because it is not possible to obtain
V,
o 31
215
BASIC PRINCIPLES
sufficient dimensional stability in a gelatine layer. Photoresist layers are widely used in microcircuit technology. There are two types of photoresists, namely positive- and negative-workingresists. In positive resists the exposed parts show an enhanced solubility in appropriate developing agents relative to unexposed parts. The contrary is true for negative resists. Photoresist layers can be deposited onto optical substrates in optical quality of nearly any desired thickness. For thin layers, i.e. layers with a thickness of less than about one micron, well known spinning techniques can be employed. An important point is that for positive-working photoresists, for example, the changes which occur when the resist is exposed take place in the molecular structure so that no grain effects are observed and the resolution is extremely high. Normally photoresists have their peak sensitivity in the ultraviolet. Consequently, to make gratings with photoresist and laser light a high energy density in the ultraviolet or blue wavelength region is necessary. For large gratings it follows that lasers with rather high powers are necessary. In Fig. 9 a characteristic curve for the positive working photoresist Shipley.AZ 1350 is shown. In this figure the depth of removed resist
I
0
50
100
150
200
250
E
300
350
[rnJ/ccm2]
Fig. 9. Characteristic curve for the positive working photoresist Shipley A 2 1350.
216
HOLOGRAPHIC DIFFRACTION GRATINGS
CV-0
4
is plotted against the energy density E of the exposure with A = 457.9 nm for a development time of 15 seconds, using concentrated AZ 1350 developer. This curve has been obtained with structures of low spatial frequencies. In this case the depth d can approximately be written as d = ~ , . ( Z * t ) ~ + c ~ with E = Z.t, where E is the energy density and t is the exposure time. c1 and c2 depend on the manner that the resist was treated before the exposure, e.g. drying and aging, and development conditions i.e. concentration of the developer and development time. Fig. 9 shows, that to obtain a depth of 0.2 pm, for example, one needs an energy density of 2 x lo6 erg/cm2. In comparison to that the fine grain holographic emulsions Agfa Scientia 8 E 56 and 8 E 75 only need about 2 x 10' erg/cm2 for a good hologram. Deeper modulation with the same energy density than shown in Fig. 9 can be obtained with longer development times or stronger developers. For example d can be doubled by increasing the development time by a factor of four. By use of the developer AZ 303 one can even obtain a gain in sensitivity by a factor of about five. Making holographic gratings with high line densities, however, normally rather thin layers are used, the adhesion of which can be critical when using too strong a development process.
0
4. Production of Holographic Gratings
The following lines of an Ar+ laser are suitable for the production of holographic gratings: 351.1 nm, 363.8 nm, 457.9 nm, 488.0 nm and 514.5 nm (preferably in combination with second harmonic generation). The lines 350.7 nm and 356.4 nm of a Kr' laser and the lines 325.0 nm and 441.6 nm of helium-cadmium-lasers are also suitable. Widely used in combination with positive working photoresist such as AZ 1350 are the UV lines of Ar' and Kr+ lasers and the line 457.9 nm. With commercially available CW-lasers one can now obtain powers of 0.1 up to 1 watt in these lines. The very strong line 488.0 nm, which can be obtained with a power of several watts, is especially suitable for use in combination with photopolymers sensitised for the blue wavelength region, e.g. Kodak Ortho resist. With the above mentioned powers typical exposure times for gratings of a width of about W = 200 mm made in AZ 1350 are of the order of a few minutes. Consequently the stability requirements concerning temperature and vibration are higher than for normal holographic experiments but less drastic than for ruling engines. As an example the conditions in the Optical Laboratory of the Gottingen Observatory are briefly discussed :The optical bench is set up on a combination of springs and damping
V, Q
41
P R O D U C T I O N OF H O L O G R A P H I C G R A T I N G S
217
elements and is located in a well isolated chamber. It was verified by optical methods that the amplitudes of the residual vibrations of the interference system were less than 20 nm. The temperature stabilisation of the interference chamber is performed by heat exchangers, using temperature controlled brine, resulting in a temperature stability 0.01 < AT < 0.1 K. To avoid thermical and mechanical disturbances in the interference chamber it was necessary to set up the laser outside the chamber and to stabilise the environment to about f1 K so as to obtain the necessary frequency stability of the Ar" and Kr' lasers. 4.1. GRATINGS WITH SYMMETRICAL GROOVE PROFILES
The normal process to make holographic gratings with optical arrangements according to Fig. 5 yields transmission gratings in photoresist on glass blanks with symmetrical groove profiles. By coating them with thin metal layers, e.g. aluminium, these gratings can be converted into reflection gratings. With an appropriate thickness of the resist layers and a sinusoidal intensity distribution of the interference fringes one normally obtains, after development, a sinusoidal groove profile, as shown in Fig. 10. One possible explanation for the formation of nearly ideal sinusoidal groove
Fig. 10. Scanning electron micrograph of a grating with sinusoidal groove profiles.
d
s
M U
a
4
a
X
0
cl
0 0
s
m
N
C
C .-
4 L
M
iz
W
k
.-N
W
3 'E
.-
k
U
4 k
M ii;
V,
P
41
219
PRODUCTION O F H O L O G R A P H I C G R A T I N G S
profiles, despite the non-linear characteristic curve shown in Fig. 9, may be a non-isotropic solution process of the exposed photoresist for structures of high spatial frequency. Using resist layers with a thickness small compared with the grating spacing one can obtain square wave (laminar) profiles for the production of phase gratings. Coating with a metal layer, e.g. chromium, and subsequent stripping yields gratings with square wave profiles in metal on glass. Other methods, e.g. etching processes, can also be used. Fig. 11 shows a microphotograph of such a grating in metal on glass. After coating with a metal layer with a high atomic number, e.g. gold, such gratings with groove densities of three hundred to several thousands per millimeter are well suited as gratings for grazing incidence for the soft X-ray region. Fig. 12 shows a raster scan photograph of the groove profiles of such a grating with a grating spacing a = 1.67 pm and a groove height of about 10 nm. Advantages of such gratings without any organic material are their resistance against thermal and mechanical stresses. This is especially important when using gratings with high power radiation. Gratings in metal on glass with low grating spacings can be used as scales, especially if made with a high ruling accuracy, using the method described in section 3.1.3. The good uniformity of holographically made scales (a) in comparison to classically ruled scales (b) is demonstrated in Fig. 13. The transmission was measured using a microphotometer. Tmsmission
'1
50
p/,]
a=8pm
a'
/
50
c 0
1
1
>
I
2
y
Imml
Fig. 13. Comparison of the uniformity o f a holographically and a classically made scale.
4.2. GRATINGS WITH 4SYMMETRICAL GROOVE PROFILES
In the spectral range from the infrared to the ultraviolet holographic
220
[V? § 4
HOLOGRAPHIC DIFFRACTION GRATINGS
gratings with sinusoidal groove profiles have high efficiency values only if the grating spacing is comparable to the wavelengths used, as will be shown in section 5.3. For gratings with spacings large compared with the wavelength, high efficienccy values can only be obtained by using sawtooth groove profiles. There are several methods of producing holographic gratings with such profiles. The first method is to produce gratings with a triangular profile by inclining the grating blank to the direction of the
7 \
Photoresist
hr
fringes
Blank
\
-
v
‘
development
Fig. 14. Method to make holographic gratings with sawtooth groove profiles. Refraction in the resist layer and the blank has not been taken into account.
interference fringes (SHERIDON [1968)), as shown in Fig. 14. The grating spacing is given by a
=
1112 sin a cos p = 11‘12 sin a‘ cos p’.
(4.1)
The blaze wavelength A, is given by the distance between the interference maxima and minima:
1, = 2h = I’/sin a’
=
11/n sin a.
(4.2)
The primed symbols are used for the beams in a medium with the refractive index n, i.e. glass and photoresist. The grating spacing and the blaze wavelength can be changed easily by changing the angle of incidence p, the laser wavelength k and the intersection angle of the interfering beams S, and S 2 . If the refractive index of the media through which the interfering beams pass were constant, change in the inclination angle p only - without changing J. and CI - would change the grating spacing and not the blaze wavelength 11, . Because of the different refractive indices of air, glass and photoresist it is necessary, in practice, to change at least two parameters if one wishes to change the grating spacing without changing the blaze wavelength or vice versa. The method that we just discussed has been successfully applied (HUTLEY
v, Q 41
P R O D U C T I O N OF HOLOGRAPHIC G R A T I N G S
22 1
Fig. 15. Scanning electron micrograph of a grating with sawtooth groove profiles. made by the method illustrated in Fig. 14. (Reproduced from HUTLEY[1974a] Fig. 2.)
[1974], NAGATA and KISHI[1974]). Fig. 15 shows a asymmetric groove profile of a grating with 800 grooves/mm, measured with a scanning electron microscope. The method has two disadvantages. Firstly it is necessary that one of the interfering beams passes through the blank from behind. This requires blanks with a good optical homogeneity to prevent aberrations of the wavefront. Secondly it is impossible to make concave gratings with a large f-ratio with this method. Another method for making asymmetric groove profiles is to use a Fourier synthesis method (SCHMAHL and RUDOLPH[1974], SCHMAHL [1975]). Sawtooth intensity profiles can be made by superposing n fringe systems of the form
The first two terms with an appropriate phase relation already yield a good approximation to an ideal sawtooth profile. In practise it is, however, very difficult to superpose two fringe systems with the required accuracy when using different optical arrangements for the different fringe systems. For a 100 mm grating with, e.g., 600 grooves/mm the accuracy of the adjustment of the mirrors has to be better than 0.1 second of arc. To overcome this difficulty one can use the following method. With a symmetric arrange-
222
HOLOGRAPHIC DIFFRACTION GRATINGS
2
1
I’ 2’
P
s i n d Z =2 sin Ul
Fig. 16. Arrangement for making holographic gratings with sawtooth groove profiles.
ment, i.e. B = 0 in Fig. 2, the number of grooves per mm I/ais proportional to sin c1. To double the line density one has, therefore, to double sin a. This can be done exactly according to Fig. 16. A grating g with a grating spacing 2a is illuminated with a monochromatic parallel wavefront at normal incidence. The + 1 and - 1 orders diffracted respectively by the parts 1
Fig. 17. Scanning electron micrograph of a grating with sawtooth groove profiles made with the arrangement illustrated in Fig. 16, (I = 1.67pm.
V.
P
51
C O M P A R I S O N WITH C L A S S I C A L G R A T I N G S
223
and 1’ of the grating g form a fringe system at b, with the spacing a, whereas the 2 and - 2 orders diffracted respectively by the parts 2 and 2‘ form a fringe system with the spacing a/2. The proper phase relation between the two fringe systems can be obtained, for example, by using a plane-parallel plate p in a part of the parallel wavefront or by evaporating a step onto the section 2’ of the grating g. The method also works by replacing the grating g of Fig. 16 by two identical gratings g, and g, made either on one blank polished to optical tolerances or on two separate blanks. The latter arrangement has the advantage, that the gratings g, and g, have to be only about twice as large as the grating b. The asymmetric profiles shown in Fig. 17 were made by successive exposure of a photoresist layer arranged at b with the two fringe systems according to Fig. 16. For the shown profiles the measured ratios of the intensity in the first positive to the first negative order was 17 at the blaze wavelength. One advantage of this method is, that it is possible to produce sawtooth profiles not only on flat but also on curved blanks, e.g. it is possible to produce concave gratings with sawtooth profiles and uniform blaze over the whole area. Finally it should be mentioned that it is possible to make asymmetric groove profiles by copying holographic gratings using visible light, UV or X-rays and by using ion etching processes.
+
0
5. Properties of Holographic Gratings and Comparison with Classical Gratings
Since the first holographic gratings for serious spectroscopic use were made in 1967 in the Optical Laboratory of the Observatory of the University of Gottingen such gratings have been competitive with classically ruled gratings from the visible to the ultraviolet and soft X-ray regions. Today it is possible to realise large gratings with a width of more than 600 mm, to reach line densities of more than 10000 lines per millimeter, to make gratings with a very low amount of straylight and completely free of ghosts. It is possible to obtain good efficiency values for gratings with symmetrical groove profiles and high line densities. In addition it is possible to attain high efficiency values independent of the line density by making asymmetric groove profiles holographically. Holographic gratings can be formed independently of substrate curvature and, in principle with any desired surface variation of grating frequency. Furthermore replicas of holographic gratings can also be made.
224
H O L O G RAP H I C DIFFRACTION G R A T I N G S
cv, 8 5
5.1. WAVE-FRONT INTERFEROGRAM, RESOLUTION AND INSTRUMENTAL PROFILE OF PLANE GRATINGS
One method of obtaining information about the optical quality of gratings is to examine wave-front interferograms. The spacing between two maxima or two minima in an interferogram is called one fringe and corresponds to a wa:le-front aberration d(q) of one wavelength of the light used to make the interferogram. According to eq. (3.5) a wavefront aberration d(q) of p fringes of a grating measured in autocollimation corresponds to a ruling error Ax = p ,442 sin .)'i Normally wavefront interferograms are made with large Michelson interferometers.
-
P
Fig. 18. Grating interferometer for producing wavefront interferograms
To avoid large beam splitters - necessary in a Michelson interferometer we used a grating interferometer as shown in Fig. 18 (SCHMAHL [1973]). A parallel beam S1 of laser light is produced by using the lens 0 and the parabolic mirror P. The grating G1is arranged so that the first order S2 is diffracted in the opposite direction of propagation of S1 (first arm of the interferometer). The zero order of the grating G1,the beam S 3 (second arm of the interferometer) illuminates the grating G2, which can be tested in the zero, first and higher orders in autocollimation. The interfering beams S2 and S4 collimated by the mirror P and reflected by the beam splitter TP produce the interferogram, which can be recorded in the plane E. The light used has to have a coherence length of more than twice the distance between the gratings G, and G2 and, if possible, less than four times this distance to avoid multiple-beam interferences. The optical quality of the grating interferometer can be tested by use of a plane mirror of good optical quality instead of grating G 2 .Fig. 14 shows, as an example, a wavefront interferogram from a holographic reflection grating made in our laboratory with 1500 grooves/mm and a width of 100 mm, examined
COMPARISON WITH CLASSICAL GRATINGS
"
225
T
Fig. 19. Wavefront interferogram of a grating with 1500 grooves/mm, made with the arrangement illustrated in Fig. 18.
in the first order with 1 = 457.9 nm. The interferogram shows that the wavefront aberrations are smaller than 1/10. From the wavefront interferograms one can deduce that the instrumental profiles are symmetrical and that the spectral resolution reaches the theoretical values. Fig. 20a and Fig. 20b show photoelectric recordings of hyperfine structure of the mercury lines 435.8 nm and 546.1 nm in the second order in autocollimation, made with a grating having ruled width 180 mm and 1465 grooves/mm, made in the Gottingen Laboratory. The measurements were made with an 8-meter spectrograph in the Solar Tower of the Gottingen Observatory. The theoretical resolving power of the grating is 527000 in the second order. The measurements were made with 94% of the grating surface - because of vignetting by the autocollimation lens of the spectrometer and with an entrance slit width of WE,= 20 pm, an exit slit width of W,, = 10 pm and an uncooled gas discharge lamp, corresponding to W , x 0.4 nm. With W& = W&,,+ W&+ W&+ W t one obtains for the observed width of a single isotope component at half maximum Webs = 1.2 & 0.1 nm for the wavelength 435.8 nm. This value is in full accordance with the measured widths of the single
226
[v, 8 5
HOLOGRAPHIC DIFFRACTION GRATINGS
1
Hg4358 m = 2 180
x
1465
130mm2. llrnm
Hg 5461 m = 2 180
x
130md
1465 llmm
h Fig. 20a. Hyperfine structure of the mercury Fig. 20b. Hyperfine structure of the mercury line 546.1 nm. line 435.8 nm.
components of Fig. 20a and means that the grating has full theoretical resolving power. 5.2. SCATTERED LIGHT
In a grating spectrograph or spectrometer the light which is observed in between the orders and which is in excess of that due to Fraunhofer diffraction of an ideal grating can occur as ghosts, grass and diffuse scattered light. Ghosts and grass are typical attributes of classically ruled gratings and are caused respectively by periodic and random errors in the positions or depths of the grooves. Much work has been done to describe quantitatively ghosts and grass. The better way, however, is to avoid such impurities completely which is indeed the case for holographic gratings. The only impurity that occurs with holographic gratings is diffuse scattered light, caused by the roughness of the grating surface. In practical cases it is necessary to know to what extent spectral measurements are influenced by scattered light. For this purpose one should know the intensity of scattered light as a fraction of the intensity of an emission line in a certain order. Unfortunately the scattered light arising from a grating and measured in a grating spectrograph or spectrometer does not only depend on the quality of the grating but also on geometric parameters such as widths and heights of entrance and exit slits, area of the grating and linear dispersion.
V, B
51
221
COMPARISON WITH CLASSICAL GRATINGS
The best way to measure the scattered light is to test a grating in the same arrangement as used for spectral measurements (HUTLEY[19731). Nevertheless, to give an idea of the amount of scattered light from a high quality holographic grating we compared the scattered light from a grating with that scattered from a mirror by measuring both under the same conditions in a spectrometer. Fig. 21 shows the results of our measurements and demonstrate that the amount of light scattered from our gratings is comparable to that scattered from good mirrors and is not caused by ruling errors. These results are in good agreement with measurements of other authors (HUTLEY [1974], HUNTER[1975j, PIEUCHARD and FLAMAND [1975]). The low level of scattered light from holographic gratings is especially important in the cases of Raman spectroscopy and high resolution stellar absorption
loo
'I
lo-' 10-2..
lo*
"
1cP-
*.
Fig. 21. Comparison of scattered light of a holographic grating and of a mirror, both coated with aluminium.
5890
5896 --A
Fig. 22. Solar spectrum near the sodium resonance lines.
[A1
228
HOLOGRAPHIC DIFFRACTION GRATINGS
cv, 5 5
spectroscopy (SCHMAHL and RUDOLPH[1972]). Fig. 22 shows a photoelectric record. of the solar spectrum near the sodium resonance lines, taken in the middle of the solar disc. The NaD, line shows a residual intensity of only 6 % without any rectification. This test was made with a spectral range of about 200 nm entering the spectrometer. Up to now comparable low residual intensities of these lines could be obtained with conventional gratings only with a strongly reduced spectral range and/or double pass systems. Fig. 23 shows a comparison of a conventional and a holographic concave grating in grazing incidence. Both had a radius of curvature of about 2 meters. The conventional grating had 1200grooves/mm and the holographic grating had 1800 grooves/mm. Though one has to bear in mind that an exact comparison would require equal grating spacings for both gratings, the results indicate that the signal-to-noise ratio is much better in the case of the holographic grating (HUNTER[1975]). INTENSITY
INTENSITY
loo{
loo{ 9080-
WAVELENGTH X
181.987 Kr
__c
HOLOGRAPHICALLY RULED GRATING
MECHANICALLY RULED GRATING
Fig. 23. Comparison of a conventional and a holographic concave grating. (Reproduced from HUNTER[I9751 Fig. 16.)
5.3. EFFICIENCY
One of the most important properties of diffraction gratings is the efficiency. The absolute efficiency is defined as the ratio of the diffracted flux in a given order to the incident monochromatic flux. Often the term “relative efficiency” or “groove efficiency” is used, that is absolute efficiency divided by the reflectivity of the surface layer, e.g. aluminium. , It will be shown in this section that holographic gratings have efficiency values comparable to those of classically ruled gratings. In combination
V,
8
51
COMPARISON W I T H CLASSICAL G R A T I N G S
229
with the good straylight properties that we just discussed, holographic gratings show, therefore, signal to noise ratios which are normally much better than those of classical gratings. The properties of gratings which determine the efficiency in a given spectral region are groove profile and line density. If one wants to calculate the efficiency of gratings one has to solve Maxwell equations for the diffracted electromagnetic fields with the grating surface as boundary. The solution of this problem is rather difficult and many authors have worked in this field. In recent years good agreement between theoretical and experimental results have been obtained (PETIT[19663, LOEWEN, MAYSTRE, MCPHEDRAN and WILSON[19751). A detailed discussion of this work relating to holographic gratings is beyond the scope of the present article. As discussed in section 4.1 two beam interference arrangements normally yield gratings with symmetrical groove profiles of sinusoidal shape. With such profiles high efficiency values can be obtained if the ratio of the wavelength 1 to the grating spacing a is in the region 0.7 5 1/a S 1.5. Especially in this region, where gratings normally are used in the first diffraction order - e.g. in the visible with groove densities between 1200 grooves/mm and 2400 grooves/mm - experimental and theoretical results show that the efficiency of holographic gratings with symmetrical groove profiles is comparable to the efficiency of classically ruled gratings with sawtooth profiles. Fig. 24'shows a comparison between gratings with 1800 grooves/mm, measured in the first order. The efficiency in this region depends strongly on the polarisation properties of the incident beam.
90 80 -
/
/-\\
< '.
7060-
50 LO-
\
30-
I
/
20- I
\
\
\
\
\
\
Fig. 24a, b. Comparison of the efficiency values of a classical (a) and a holographic (b) grating with sinusoidal groove profiles. Both gratings have 1800 grooves/mm. (Reproduced from FLAMAND [1975] Fig. 3.)
230
v,§ 5
HOLOGRAPHIC DIFFRACTION GRATINGS Absolute efficiency
la00 llmm
1N
90
80 70
-
-
60 50
20 -
40 30-
A $
0
10
''
300
LOO
5;)O
6W 7M)
800
&m]
Fig. 24c. Efficiency of a holographic grating with sinusoidal groove profiles and 1800 grooves/mm. (Reproduced from HUTLEY[1974b] Fig. 6.)
Relative Fnicimcy
Relative efficiency
"/-I 1260 llmn
-.
/
b],,' 70
E,
r--.
,/-\, ,
-
50 40-
I
b
500
600
700
800
900
'\
' '-'
10-
0,
400
3M)
500
\
'..Ell
600
700
A [nml
r%1
800
A
903
[d
(b)
(a) Relative efficiency
1580 I/mm
\
I
01 300 LOO
..
,
//
Relative efficiency
c
t
80 -
70
['/;I
-
3600 llmm
60.
i: 20 10.
fi
*
Fig. 25. Efficiency values of holographic gratings with sinusoidal groove profiles and with 1260, 1580,2090 and 3600 grooves/mm.
v, § 51
C O M P A R I S O N WITH C L A S S I C A L G R A T I N G S
231
Therefore one has to avoid regions which show strong anomalies, i.e., rapid variations of efficiency over a comparatively short range of wavelength. Fig. 25a to Fig. 25c show the efficiency in the visible region of three holographic gratings with 1260, 1580 and 2090 grooves/mm and symmetrical groove profiles, measured in an arrangement near autocollimation in the first order (MIKELSKIS C19731). Fig. 25d shows the efficiency of a grating with symmetrical'groove profiles and 3600 grooves/mm, measured in the ultraviolet in the first order. The curves demonstrate that the peak efficiency values occur in the range 0.8 5 ,?/a ,< 1, which means that the peak efficiency shifts to shorter wavelengths with increasing line densities. Measurements of the efficiency as function of the angle 6 between incident and diffracted beam show that the peak efficiency shifts to longer wavelengths with increasing 6 relative to measurements in autocollimation -(6 = 0). For example, the peak efficiency is shifted by about 80 nm for 6 = 34" for a grating with 1560 grooves/mm with respect to the position for peak efficiency measured for 6 = 0. These results are in contradiction to calculations of the peak efficiency with increasing S made under the assumption that the blaze of classical gratings with sawtooth profiles results from the fact that the facets of the grooves act as small mirrors which reflect the light in the same direction as the grating is sending it by diffraction. Fig. 25 shows that when J./a < 0.8 strong anomalies occur, in this case for gratings used in the first order in autocollimation. Such anomalies are characteristics of all gratings, irrespective of whether they are made mechanically or holographically. The anomalies were first mentioned by Wood. A general theoretical treatment shows that the Wood anomalies are actually of two distinct types, a resonance type and a form first mentioned by Rayleigh, which appears at wavelengths due to the emergence or reentry and OLINER [1965]). of another spectral order at the grating surface (HESSEL In special cases these two types are merged together. Anomalies are connected with plasma waves in the electron gas in the metal coating of the gratings. Such plasma oscillations are known as surface plasmons (TENGand STERN[1967], HUTLEY [1973]). Anomalies are well suited to test electromagnetic grating theories, which should reproduce the observed anomalies with regard to strength and location in the spectrum. When A/a < 0.8 one can obtain high efficiency values only with asymmetric sawtooth groove profiles. As shown in section 4.2 such profiles can be made holographically. Efficiency values of various gratings with sawtooth profiles made by HUTLEY [1974] according to the method of SHERIDON are given in Fig. 26 and Fig. 27. Fig. 26 shows the efficiency of different
232
cv, 8 5
HOLOGRAPHIC DIFFRACTION GRATINGS
Mirror
I Kx)
I
200
I
300
I
100 Wavelength
200
D
300
trim)
Fig. 26. Efficiency values of holographic gratings with sawtooth groove profiles. A) 1200 grooves/mm, B) 1570 grooves/mm, C) 600 grooves/mm, D) 800 grooves/mm. (Reproduced from HUTLEY[1974a] Fig. 5.)
gratings with 600 grooves/mm to 1570 grooves/mm, measured in the first order in the vacuum ultraviolet. Fig. 27 shows the distribution of light among the various diffracted orders of grating A in Fig. 26. The results given, demonstrate that holographic gratings can fully compete with classically ruled gratings concerning the efficiency. This is not only true for gratings with symmetrical sinusoidal groove profiles in the range 0.8 < A/a < 1.5, but also for holographic gratings with asymmetric sawtooth profiles. 5.4. X-RAY GRATINGS
As discussed in section 4.1 gratings for the soft X-ray region can be made holographically in metal on glass form without any remaining organic material. It has been found possible in this way to control groove profile accurately and so produce square wave (laminar) grooves with heights of
C O M P A R I S O N WITH C L A S S I C A L G R A T I N G S
233
Spectral order
Fig. 27. Light distribution among the various diffracted orders of grating.A) of Fig. 26. Relative efficiency (full line) and absolute efficiency. (Reproduced from HUTLEY[1974a1 Fig. 7.)
the grooves, showing phase cancellation in the zero order (RUDOLPH, SCHMAHL, JOHNSON and SPEER[1973]). Fig. 28 shows the efficiency of such a grating with 294 grooves/mm and with groove heights of 22 nm, measured at 4.5 nm. The efficiency is greater than that of most mechanically ruled gratings used in the soft X-ray region (SPEERand RUDOLPH [1974]). The holographic process gives the possibility of overcoming the limitations of classically ruled gratings used at grazing incidence. The limitations arise from the fact that such gratings are almost universally ruled on spherical blanks, for use in Rowland circle geometries, with grooves equally spaced along the chord. Spherical aberration imposes a severe upper limit on the allowed focsl ratio, typicallyf/80 tof/100 for maximum resolution in this region. Furthermore, correction of astigmatism is not usually possible. New design possibilities of holographic gratings derive from the fact that such gratings can be formed independently of substrate curvature, and, in principle, with any desired surface variation of grating frequency.
234
HOLOGRAPHIC DIFFRACTION GRATINGS
CV>§
5
Fig. 28. Absolute efficiency of a 2 meter radius concave grating with 294grooves/mm at 4.5 mm in the _+ 1 orders (right hand scale). The Lero order (left hand scale) shows modification due to phase cancellation. (Reproduced from JOHNSON [1975].)
An analysis has shown (HABER[1950]) that primary astigmatism in Rowland circle mountings can be corrected by a suitable choice of toroidal grating blank radii. This solution has been realised by making a reflection grating for grazing incidence on a toroidal blank. Fig. 29 shows such a
Fig. 29. Toroidal grating for grazing incidence.
COMPARISON WITH CLASSICAL GRATINGS
235
toroidal grating with 600 grooves/mm and a ruled area of 8 x 45 mm2. The minor radius is 5.65 mm (SPEER,TURNER, JOHNSON, RUDOLPHand SCHMAHL [1974]). The result is, that the threshold sensitivity for photographic recording has been improved by a factor of about 35 in comparison to the classical case. Correction of spherical aberration can be achieved by suitable surface variation of groove frequency. Using such solutions it is possible to realise concave gratings with an improved focal ratio. 5.5. GRATINGS WITH IMAGING PROPERTIES
It is well known that it is possible to combine in one optical element the diffractive properties of a grating with the focussing properties of a spherical mirror (ROWLAND [18831). Classical concave gratings are ruled mechanically with equidistant grooves along the chord and have been discussed in detail (BEUTLER [1945], NAMIOKA [19611). To reduce the aberrations, especially the astigmatism, it has been proposed to rule gratings with nonuniform groove distribution (ROWLAND[19023, SAKAYANAGI [19671). Successful attempts to produce mechanically concave gratings of this type have been made in recent years (GERASIMOV, YAKOVLEV, PEISAKHSON and KOSHELEV [1970], HARADA,MORIYAMA and KITA[1975]). The basic element of a hologram is a zone plate, i.e. a grating with variable spacing and curved lines which has dispersive and imaging properties, even if recorded on a plane surface. The imaging properties can be improved by making an aplanatic system ie., one satisfying the Abbe sine condition, for certain wavelengths. This can be done by use of a zone plate recorded on a spherical blank (MURTY[1960], MURTYand DAS p9711). In principle it is possible to minimise the aberrations of gratings having imaging properties in a given wavelength region by an optimal choice of the following parameters : Curvature of the blank and surface variation of the spacing frequency. This can be done by a suitable choice of the recording wavelength, the geometry during the recording process, and by use of suitable wavefronts, e.g. plane waves, spherical waves or appropriate aspherical waves. A general treatment of concave gratings with variable spacings and curved lines has been given by NODA,NAMIOKA and SEYA [1974a]. It is evident that it is possible to make holographic concave gratings on spherical blanks with equidistant spacings along the chord and with straight grooves by the use of two recording plane waves as indicated in Fig. 2, in a symmetrical arrangement (B = 0). These gratings have imaging properties and aberrations identical to those of classical
236
HOLOGRAPHIC DIFFRACTION GRATINGS
CV?
B
5
concave gratings. The size of gratings produced holographically is only limited by the size of the optics used and not by any mechanical limitations. With sinusoidal groove profiles they have the same high efficiency values in the appropriate A/a region as holographic plane gratings. With sawtooth profiles made by Fourier synthesis high efficiency values can also be obtained in other regions. Unlike most classically ruled concave gratings, which are often ruled as tri-partite gratings especially for large f-ratios, the efficiency of holographic concave gratings is quite uniform over the entire grating surface. Furthermore, holographic concave gratings have the same good properties concerning scattered light and are completely free of ghosts, that is, they have a good signal to noise ratio. In Fig. 30 a holographic grating of ruled area 8.6 x 26.8 cm2 and a classical concave
Fig. 30. Photograph of a conventional grating of ruled area 6.5 x 4 em2 and a holographic grating of ruled area 8.6x26.8 cm'. (Reproduced from HUNTERf19751 Fig. 10.)
grating of ruled area 6.5 x 4 cm2 are shown. Both have 2400 grooves/mm and a radius of curvature of 85 cm.In Fig. 31 and Fig. 32 a comparison with the gratings shown in Fig. 30 is made (HUNTER[1975]). This comparison demonstrates that the efficiency values are comparable, but also that the efficiency of the holographic grating is much more uniform in spite of the larger aperture. Fig. 33 shows another comparison of a holographic and a classical grating in the XUV (HUNTER[1975]) and demonstrates that especially in classically ruled tri-partite gratings large variations of the efficiency over the grating surface occur whereas holographic gratings of the samef-ratios are quite uniform. In addition to these classical concave gratings special types of holo-
237
COMPARISON W I T H CLASSICAL G R A T I N G S
2000
1500 WAVELENGTH
(1)
MEASURED EFFICIENCY OF A HOLOGRAPHIC GRATING AT 15' ANGLE
OF INCIDENCE. POS. FIRST ORDER ORDER
A.
0,
ZERO ORDER
NEG. FIRST
POS. F I R S T OROER GROOVE EFFICIENCY --...--...-.....
I500 WAVELENGTH t i )
1100
2000
MEASURED EFFICIENCY OF A CONVENTIONAL GRATING AT 15. ANGLE OF INCIDENCE.
NEG. FIRST ORDER
0,
ZERO ORDER
X,
POS. FIRST ORDER
A.
NEG. FIRST ORDER GROOVE EFFICIENCY
Fig. 31. Comparisons of the gratings of Fig. 30. a) Efficiency of the holographic grating. b) Efficiency of the classical grating. Zero ( x ), positive (A), and negative ( 0 )first orders at 15" angle of incidence. The dotted line represents the negative first order groove efficiency. [1975] Fig. 13 and Fig. 12.) (Reproduced from HUNTER
238
CV, 8 5
HOLOGRAPHIC DIFFRACTION GRATINGS
40-
0
+I
-I
-
7-I- BEAM WIDTH
0
J
60 -
50 -
-
+I
s40 t 0
z
E30-
LL LL
u20-
'0
+BEAM WIDTH
O
-
A
Fig. 32. Comparison of the gratings of Fig. 30. a) Holographic grating. b) Conventional grating. Efficiency maps of the zero and the positive and negative first orders at 144 nm. The angle of incidence is 15". The two large peaks on either side of the zero order of the conventional grating are caused by specular reflections from the unruled edges. (Reproduced from HUNTER[1975] Fig. 14 and Fig. 11.)
graphic concave gratings with reduced aberrations have been made. The first proposals and results in this field of special concave gratings were given by CORDELLE, FLAMAND, PIEUCHARD and LABEYRIE [19691. Further investigations and special designs of gratings with imaging properties for the visible, UV and soft X-ray regions have been made by several groups (e.g., POUEY[1975], SPEER,TURNER, JOHNSON, RUDOLPH and SCHMAHL [1974], NODA,NAMIOKA and SEYA[1947b], NIEMANN, RUDOLPH and SCHMAHL [19741).
COMPARISON W I T H CLASSICAL G R A T I N G S
239
HOLOGRAPHIC GRATING
I -
POSITIVE ORDERS
1
NEGATIVE ORDERS
0
0
5 GRATING SURFACE
10 crn
Fig. 33a. Efficiency maps of a holographic grating at 121.6 nm in the zero order and positive and negative first and second orders. 1200 grooves/mm, gold coating, and 1 m radius of curvature. (Reproduced from HUNTER[I9751 Fig. 8.) B8L x 573 A I O
10. -
We will now discuss some special types of gratings and their imaging properties. 1. Gratings on a spherical blank recorded with two spherical wavefronts : By choice of suitable construction parameters it is possible to construct concave gratings for Rowland circle geometry, with highly reduced aberrations. Especially it is possible to completely avoid astigmatism and some types of coma and spherical aberration for one particular wavelength. As a rule of thumb one can say that the astigmatism can be reduced over a large
240
cv. 5 5
HOLOGRAPHIC DIFFRACTION GRATINGS
wavelength region by about one order of magnitude compared to classical concave gratings in Rowland mounting. Special solutions are discussed by FLAMAND [1975], NODA, NAMIOKA and SEYA[1974b] for example. Special properties of concave gratings made on a spherical blank can be obtained in the case when one of the spherical waves used for construction originates from the center of curvature of the blank. The basic principle for such gratings has been treated by MURTY[1960]. Holographic gratings of this type have been discussed by CORDELLE, FLAMAND, PIEUCHARD and LABEYRIE [1969]. The principle of this type of gratings is shown in Fig. 34. The
H
Fig. 34. Principle of holographic concave gratings with non-equidistant spacings and three stigmatic points.
grating is made with two spherical waves of wavelength L o , originating from 0 and A . After processing a concave reflection grating G is obtained. If one illuminates this grating in the same geometrical arrangement with polychromatic light originating from A , one can easily see from the holographic principle that the wavelength l ois focussed stigmatically at 0. According to the grating equation the wavelength 2A0 is focussed stimatically at A'. Furthermore, MURTY[1960] has shown that a third point A is a stigmatic image of A' if OA = m R with OA' = R/m.A circle with the points 0,A and A with the given conditions is called the circle of Apollonius and divides AA' harmonically both internally and externally. By use of the grating equation one can show that under these conditions the wavelength focussed at A is given by E. = (m+ l)& in the first diffracted order. It is possible to locate the polychromatic source at each of the three points 0, A' and A. Table 4 gives the wavelengths of the resulting stigmatic images. It is not essential to use waves originating from 0 when recording the pattern. Concave gratings with three stigmatic points are also obtained
V,
o 51
24 1
C O M P A R I S O N WITH C L A S S I C A L G R A T I N G S
TABLE 4 Wavelengths and positions of stigmatic images, according to Fig. 34, for the first spectral order, I , = recording wavelength Image location
Source location 0 0
A A
1=0 I = A, I =ml,
A 3, = I , I = 21, I = (m
+ 1)I,
A
I
= mi,
a = (m+i)n, I = 2mI,
using waves originating from A' and A, which yields a similar kind of matrix as shown in Table 4 (FLAMAND [1975]). The line H in Fig. 34 is the horizontal or sagittal focus. The curved line V is the vertical or tangential focus, i.e. - at least for smallf-ratios of the grating - V is the focus of the spectral lines. But it must be stressed that for large apertures especially severe aberrations occur between the three stigmatic points. These aberrations must be calculated for every particular arrangement. 2. The holographic method allows the construction of gratings with imaging properties on extremely curved substrates. One example has been discussed in section 5.4. 3. Up to now examples where the recording waves were either plane or spherical have only been discussed. Another possiblity is - as already mentioned - the use of aspherical waves for the recording process, to correct aberrations of gratings with imaging properties. A special case of diffraction gratings are zone plates, i.e., circular gratings with radially increasing line density. Zone plates with large zone numbers can be realised holographically by superposition of two spherical waves or one spherical wave and one plane wave. Such high power zone plates can be used for imaging and/or spectrometric purposes in the soft X-ray region (X-ray microscopy, X-ray astronomy, X-ray spectroscopy) (RUDOLPHand SCHMAHL [1967], SCHMAHL and RUDOLPH[1969]). If such zone plates are constructed with visible light and are used in the X-ray region large spherical aberration occurs. This aberration has been corrected by use of aspherical recording wavefronts ( R U ~ L P[H 19743, NIEMANN, RUDOLPH and SCHMAHL [1974]). This method can, in principle, be applied to other gratings with imaging properties.
242
HOLOGRAPHIC DIFFRACTION GRATINGS
0
[v
6. Further Improvements of Holographic Gratings
Holographic plane and concave gratings with high groove densities and sinusoidal groove profiles have only been widely used up to now. These gratings have efficiency values comparable to those of classically ruled gratings but have less scattered light and are completely free of ghosts. Such gratings have, therefore, an improved signal to noise ratio. As discussed in section 4.2 it is also possible to make gratings with sawtooth groove profiles holographically. These processes are more complicated than the process of making gratings with sinusoidal groove profiles. Holographic gratings with sawtooth profiles will, therefore, only be used widely in the future when replica gratings in large series are available. Furthermore, it can be expected that considerable progress will take place in the field of gratings with imaging properties. Application of holographic X-ray gratings is just in the beginning and it can be expected that it will be possible to enhance the Ctendue of spectrometers in the grazing incidence region. As shown in section 4.1 the holographic methods allow the production of scales of high accuracy, especially by use of identical wavefronts, as discussed in section 3.1.3. Although so far not used in practice, this could be important in the future development of metrology.
Acknowledgements
The authors wish to thank Mr. W. R. Hunter, U. S. Naval Research Laboratory, Washington, Dr. M. C. Hutley, National Physical Laboratory, England and Dr. J. A. Flamand, SocietC Jobin Yvon, France for sending us new results. We wish to thank Dr. R. L. Johnson, Imperial College of Science and Technology. London, for reading the manuscript. We are also indebted to Mrs. H. Brandt and Mr. R. Spindler for the illustrations and p ho t ogrd ph s. References BEUTLER, H. G., 1945, J. Opt. SOC.Amer. 35, 31 I . BURCH,J. M. and D. A. PALMER, 1961. Optica Acta 8. 73. CLARK,K. G., 1973, Electronic Components June-September, 553. CORDELLE, J., J. FLAMAND, G. PIEUCHARD and A. LABEYRIE, 1969, Aberration-Corrected Concave Gratings Made Holographically, in : Optical Instruments and Techniques, ed. J. Home Dickson (Oriel Press, 1970). DOWLLY, M. W., 1971, Coherent Radiation, Technical Bull. Nr. 106. FLAMAND. J.. 1975. Rev. Physique-Chimie. in press.
vl
REFERENCES
243
FRAUNHOFER, J. v., 1821/22, Denkschrift der kgl. Akademie Miinchen, 8, 1-76. F. M., E. A. YAKOVLEV, I. V. PEISAKHSON and M. V. KOSHELEV, 1970, Opt. SpecGERASIMOV, troc. 28, 423. HARER, H., 1950, J. Opt. SOC.Amer. 40. 153. and T. KITA,1975, Suppl. Jap. Journal of Appl. Phys. 14-1, 175. HARADA, T., S. MORIYAMA HARRISON, G. R., 1973, Appl. Opt. 12, 2039. HESSEL,A. and A. OLINER,1965, Appl. Opt. 4, 1275. HUNTER, W. R., 1975, Journal of the Spectroscopical Society of Japan 24, Suppl. Nr. I . 37. HUTLEY, M. C., 1973a, National Physical Laboratory Report MOM 1, January 1973. M. C., 1973b, Optica Acta 20, 771. HUTLEY, HUTLEY. M. C.. 1974a. Blazed Interference Diffraction Gratings for the Ultraviolet. in : Vacuum Ultraviolet Radiation Physics, eds. E. Koch, R. Haensel, C. Kunz (Pergamon/Vieweg. Hamburg) p. 713. M . C.. 1974h. Sci. Prog. Oxford 61. 301. HLITI.I.Y. JOHNSON. R. L., 1975, Ph. D. thesis (University of London). KAYSER, H., 1900, Handbuch der Spektroskopie, Bd. I (S. Hirzcl, Leipzig). LABEYRIE, A., 1966, Quelques Nouvelles Methodes en Holographie. MCmoire pour obtenir le Diplome #Etude Superieurs de Sciences Physiques (University of Paris/Orsay). LOEWEN, E., D. MAYSTRE. R. MCPHEDRAN and 1. WILSON,1975. Suppl. of the Jap. Journal of Appl. Phys. 14-1, 143. MICHELSON, A. A,, 1927, Studies in Optics (Phoenix Books, The University of Chicago Press) p. 104. MIKELSKIS, H., 1973, Diplomarbeit, Gottingen. MURTY,M. V. R., 1960, J. Opt. SOC.Amer. 50, 923. MURTY.M. V. R. and N. C. DAS,1971. J. Opt. SOC.Amer. 61, 1001. NAGATA.H. and M . KISHI,1974, Production of Blazed Holographic Gratings by a Simple Optical System, in: Suppl. Jap. Journal of Appl. Phys. 14-1, 181. NAMIOKA, T., 1961. Choice of Grating Mountings Suitable for a Monochromator in a Spacc Telescope, in: Space Astrophysics, ed. W. Liller (McGraw Hill, New York, Toronto. London) p. 228 ff. NIEMANN. B., D. RUDOLPHand G. SCHMAHL, 1974. Optics Communications 12, 160. NAMIOKA and M. SEYA,1974a. J. Opt. SOC.Amer. 64,1031. NODA,H., T. NAMIOKA and M. SEYA,1974b, J. Opt. SOC.Amer. 64, 1043. PETIT,R., 1966, Rev. Opt. 6. 249. 1975, Suppl. Jap. Journal of Appl. Physics 14-1, 153. PIEUCHARD, G. and 1. FLAMAND, POUEY,M., 1974, Journal of the Spectroscopical Society of Japan 24, Suppl. Nr. I , 67. RITTENHOUSI-:. D.. 1786. Trans. Amer. Phil. SOC.2. 201. ROWLANI), H. A,, 1883, Amer. J. Sci. (3), 26, 87. ROWLAND, H. A., 1902. Phys. Papers, The Johns Hopkins University Press, Baltimore. RUDOLPH, D. and G. SCHMAHL, 1967a. Umschau in Wissenschaft und Technik 67, 225. RUDOLPH, D. and G. SCHMAHL, 1967b. Mitt. Astron. Ges. 23, 46. 1967~.German Patent Application Nr. 1623803. RUDOLF,D. and G. SCHMAHL, RUDOLPH, D. and G. SCHMAHL, 1970, Optik 30,475. RUDOLPH, D., G. SCHMAHL, R. L. JOHNSON and R. J. SPEER,1973, Appl. Opt. 12. 1731. RUDOLPH, D., 1974, Bundesministerium fur Forschung und Technologie, Forschungsbericht W 74-07. SAKAYANAGI. Y., 1967, Science of Light 16, 129. SCHMAHL, G. and D. RUDOLPH, 1968, Mitt. Astron. Ges. 24, 41. SCHMAHL, G. and D. RUDOLPH,1969, Optik 29, 577. SCHMAHL. G. and D. RUDOLPH, 1970, Optik 30. 606. SCHMAHL, G. and D. RUDOLPH, 1972, Stellar Spectroscopy With Holographic Gratings, in : Proceedings of the ESO/CERN-Conference on Auxiliary Instrumentation for Large Telescopes. eds. S. Laustsen and A. Reiz (Geneva. June 1972).
244
HOLOGRAPHIC DIFFRACTION GRATINGS
cv
SCHMAHL, G., 1973, Bundesministerium fur Forschung und Technologie, Forschungsbericht T 73-1 6. SCHMAHL, G. and D. RUDOLPH,1974, German Patent Application Nr. P 2433800, 9. SCHMAHL, G., 1975, Journal of the Spectroscopical Society of Japan 24, Suppl. Nr. I , 3. SHERIDON, N. K., 1968, Appl. Phys. Lett. 12, 316. SPEER,R. J., D. TURNER, R. L. JOHNSON, D. RUDOLPH and G. SCHMAHL, 1974, Appl. Opt. 13, 1258. SPEER,R. J. and D. RUDOLPH,1974, Design and Performance. of Soft X-ray Reflection Gratings formed Holographically, in : Vacuum Ultraviolet Radiation Physcis, eds. E. Koch, R. Haensel. C. Kunz (Pergamon/Vieweg, Hamburg) p. 709. STROKE,G. W., 1963, Ruling, Testing and Use of Optical Gratings for High Resolution Spectroscopy, in: Progress in Optics, Vol. 11, ed. E. Wolf (North-Holland, Amsterdam). STROKE, G. W., 1967, Diffraction Gratings, in: Encyclopedia of Physics, ed. S. Fliigge (Springer, Berlin). TENG,Y. Y. and E. A. STERN,1967, Phys. Rev. Lett. 19, 511.
E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
VI
PHOTOEMISSION BY
P. J. VERNIER Laboratoire de PhoioelectriciiP, Facult6 des Sciences (M.I.P.C.), Universit6 de Dijon, Dijon, France
CONTENTS PAGE
9 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . 5 2 . THEORETICAL BASIS OF THE PHOTOEMISSION (PE) 6 3. EXPERIMENTAL DETERMINATION OF THE ESCAPE
247 250
DEPTH OF THE PHOTOELECTRONS . . . . . . . . . 279
9 4. SURFACE PHOTOEXCITATION . . . . . . . . . . . . 9 5 . CONCLUSION . . . . . . . . . . . . . . . . . . . . .
319
ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . .
321
REFERENCES .
321
. . . . . . . . . . . . . . . . . . . . . .
305
0
1. Introduction
During the first years after its discovery in 1887 by Hertz, photoemission (PE) was investigated by experiments of Hallwachs, Elster and Geitel and Lenard. The results of these experiments suggested to Einstein the concept of photon and his well-known equation that has been verified later on by Millikan. Early studies that led to high-yield photocathodes still in use (Ago-Cs, SbCs,) are well exposed in classical monographs (HUGHESand DUBRIDGE [1942], SUHRMANN and SIMON [1958], ZWORYKIN and RAMBERG [19493). The investigations have been pursued for nearly a century in order to produce higher-yield photocathodes in a wider spectral range and with a better reliability. SOMMER [1970], who played an important role in the development of trialkaline photocathodes, gave a very complete review of the experimental knowledge and of possible technical applications. Quite recently a new role has been found for PE in electron spectroscopy namely the investigation of electron energy levels and the transitions between them. After the pioneering work of Spicer, nearly all elements and many alloys and compounds have been measured. The spectral range investigated has widened during recent years and now extends to X-rays. Applications to chemical analysis gradually became reliable and are now commercially exploited (ESCA). Several review papers on electron spectroscopy have been published recently (GCJRLICH and SUMI[ 19703, N. V. SMITH [1971]). (See also SHIRLEY [1971].) In this paper we shall only be concerned with the origin of the photoelectrons. We shall deal with different theories, techniques and materials. For a systematic review of important points such as technical applications, electron spectroscopy and its relations with band structure, work function (for this last point see RIVIERE [1969]), etc. we will refer elsewhere. It was established a long time ago that the part of a solid responsible for PE is generally a very thin superficial layer. A precise knowledge of the exact origin of the photoelectrons is fundamental both to improving the 247
248
PHOTOEMISSION
CwS
1
photocathodes used in photometry and to interpreting the results of electron spectroscopy. Strictly speaking, the solid is a whole and any modification of one atom induces a change of all solid state properties, e.g. of the PE induced by a given incident light beam. Nevertheless because of screening effects it is possible, within some approximation: to construct models, where the PE current is analysed as a sum of local contributions. Another interest of this local analysis is to obtain a quantitative evaluation of the screening effects that can be used as intermediate results to determine more fundamental processes. 1.1. THE 3-STEP MODEL
Early theories dealt with metals as free electron gases and with PE as a surface phenomenon (FOWLER [1931], MITCHELL [1934,1935,1936]). The emission of an electron was the result of the simultaneous interaction of an electron of the solid with the surface and with the electromagnetic field. Later theoretical (FAN[1945]) and experimental studies led workers to describe PE as a volume process and to divide it into 3 parts: 1) Absorption of one photon and transfer of its energy to one excited electron. 2) Transport of the excited electron towards the surface. 3) Transport of the excited electron across the surface. The contribution of a small volume 6V to the PE is then the product of three factors : the total number of absorbed photons, the proportion of absorbed photons that lead to the excitation of one electron, the escape probability of an electron excited in 6V. The first two factors describe step 1, the last factor describes steps 2 and 3. Insofar as we can make 6 V tend to zero, the photoyield Y can be written as n
where D(r) is the density of absorbed photons per incident photon (DAP) at r and p(r) is the electron escape probability for an electron excited at r . To investigate the photoelectron energy distribution (PED) it is useful to define the probability for an absorbed photon of energy hv to excite an electron to an energy between E and E dE, which we can write in a normal-
+
%§
11
INTRODUCTlON
249
ized form as (1.2) PO
where
Po =
s
/?(E,hv)dE.
We shall also define the probability for an electron excited at energy E' at r to emerge with an energy between E and E+ dE, as p(E', E, r)dE. The number of electrons emitted with an energy between E and E+ dE is then
s
n(E)dE = dE D(r)/?(E',hv)p(E, E', r) dE' d3r, and
s
Y = n(E)dE.
(1.3)
(1.4)
To obtain a more general validity for the expressions (1.1) and (1.3) we may try to introduce into the DAP D(r) a surface term of the form A(z) where z is the distance from r to the surface. A depends on the polarization and on the angle of incidence of the light. Of course, no transport process through the solid needs to be considered and the escape probability associated with the surface absorption ps is merely the ratio of the number of emitted electrons to the total number of excited electrons. p, might not be the limit of p(r) as z tend to zero and P(E, hv) can take different forms 8, and P, for surface and volume effects. Two objections can be made against the 3-step model as it is described here.
1) Is it possible to separate the 3 steps? This objection appears with a special clarity in the case of a surface effect. The separation of the electron transport through the surface and of the photon absorption may seem difficult because no absorption occurs if the electron is not perturbed by the surface. As a matter of fact the decomposition of the surface PE into several steps is rather artificial. Nevertheless /?, and A&z) are well defined and it is convenient to use a step model to compare surface and volume PE. We shall later discuss one-step formalism but the 3-steps description is still mostly used for the analysis of experimental data. We must note, howevei., that from a physical point of view the 3steps are not necessarily independent. For instance, many-body
250
PHOTOEMISSION
[Vl,
9: 2
processes such as electron-electron interactions do perturb more or less both light absorption and transport phenomena. 2 ) Is it possible to consider light absorption in a volume element that tends to zero, so that the integrals (1 . I ) and ( 1 . 3 ) are valid? We may try to give a significance to the DAP in a volume that tends to zero by considering the expectation values of the density of current and the electric field. But beside the doubtful significance of such concepts, they would be of very little use in the interpretation of experiments. The DAP is a macroscopic quantity, if we calculate it from macroscopic constants such as the index of refraction or the dielectric constant. This implies that we consider the solid as homogeneous. It is then certainly meaningless to use the DAP to calculate the absorption within a volume smaller than one crystalline cell. Furthermore we must note that the determination of the dielectric constant and of the index of refraction either from theoretical considerations or from experimental data is always made in the assumption of a quasi-infinite solid. The application of the concept of DAP to one crystalline cell requires, therefore, much caution, especially if the cell is. near the surface. We shall give a special discussion of the surface absorption that cannot be dealt with, if the solid properties are described by the dielectric constant alone. The 3-step model will be our guide in this study. In a first part, we shall discuss separately the theoretical basis of the density of absorbed photons (DAP) and of the electron escape probability that appears in the expressions (1 . l ) and (1.3) and we will describe how both contribute to determining the origin of the photoelectrons. We shall give special attention in the study of the DAP to surface effect. We shall also discuss briefly the more elegant one-step theories that have been developed recently. Sections 3 and 4 deal with the analysis of the experimental results. Insofar as PE is a pure volume effect, the DAP can be calculated from purely optical data and needs no discussion here. In section 3, we shall discuss the experimental determinations of the escape probability of the photoelectrons in the light of the results of our section 2. In section 4,we shall discuss the possibility of a surface effect.
5 2.
Theoretical Basis of the Photoemission (PE)
2.1. CALCULATION OF THE DENSITY OF ABSORBED PHOTONS (DAP) FROM
THE BULK DIELECTRIC CONSTANT
PE is generally measured for solid slabs bounded by two parallel planes
VL
9
21
THEORETICAL BASIS OF THE P E
251
or for semi-infinite solids i.e. slabs of infinite thickness. The DAP is then a function of the distance z from the surface. We shall consider here the DAP in slabs illuminated by a monochromatic parallel incident beam of frequency v = 0427~that induces in the slab an electric field of complex amplitude E(r) at the point r . We shall disregard the magnetic absorption that is not important in PE. The response of the solid to the electric field is then described by the complex amplitude either of the density of electric current j(r) or of the electric displacement R(r). These quantities are related by j(r) = ioR(r),
(2.1)
if we include in the density of current the displacement current. Within the linear approximation, the most general relation between E(r) and R(r) has the form:
s
R(r) = E(r, r’)8(r’)d3r’. The DAP at r is given by: D(r) =
(2.2)
Re j(r)&*(r) - Im R(r)l*(r) 2Nho 2Nh ’
where N is the number of incident photon per unit time. As defined by (2.3) the DAP may have a microscopic significance and can be used to define the absorption in a volume 6V arbitrarily small. But in this paper we shall not make use of this possibility. In practice screening effect make E(r, r’) tend to zero when lr-r’l tends to infinity. To define the DAP in a volume 6V large compared with the size of the crystalline cell we may use a local dielectric constant that is generally complex, E(r) = E1(r)-iiE2(r), and write
We may also introduce the index of refraction n = v - i K defined by :
(2.4)
252
PHOTOEMISSION
[Vl,
§ 2
where E , is the dielectric constant of vacuum. We then write:
where i, is the angle of incidence of the incident light beam, I the wavelength in vacuo, and &, the complex amplitude of the electric field in the incident beam. The smallest volume where (2.8) can be applied is one crystalline cell, but this is only a very rough approximation if it is not embedded in a set of identical cells. This restriction is still more important if we wish to go beyond the local approximation by introducing the wave-vector-dependent dielectric constant E ( q , o).If the gradient of the componentspf q r ) is large enough, so that 4 r ) is significantly different from &(r’)for values of r - r’, which do not cancel E(r, r’), the expressions (2.5) and (2.8) cannot be substituted for (2.2) and (2.3). This occurs when the screening length in the solid is not much greater than the wavelength of the light. E ( q , o)can then be defined as the ratio of the Fourier transforms of R(r) and &(r)in an infinite solid. This adds new restrictions to the use of E ( q , o)in the calculation of the dielectric (mean) response and the DAP in a finite volume SV, if 6V is near the surface. Even in an isotropic or cubic material, the isotropy of the relations between b(r) and R(r) is destroyed by the definition of the light wave vector q. E ( q , o)then becomes a tensor. In the simple case when q is real and the material is isotropic, we have a cylindrical symmetry and we can define scalar longitudinal and transverse dielectric constants EL(q, o)and +(q, o) that are the components of the dielectric tensor in a particular reference sys[19631). We may note that the cylindrical symmetry is generally tem (STERN broken when q is complex (heterogeneous wave). 2.2. DIELECTRIC CONSTANT A N D MICROSCOPIC PROCESSES IN A SOLID
In principle the dielectric constant can be calculated by considering every independent excitation that can be induced in an infinite solid by an electromagnetic wave. Each excitation affords a contribution to the absorption i.e. to eZ. is deduced from the dispersion curve of cZ by the Kramers-Kronig relations (STERN[19631). Independent excitations are very difficult to define because of the large number of degrees of freedom that exist in a solid, and it would be most difficult to define a complete set of independent excitations that would completely describe the response of the solid. In practice the components of absorption are described as damped excitation i.e. excitations that decay into one another.
v1, § 21
T H E O R E T I C A L BASIS OF T H E PE
253
Because E~ describes an infinite solid, we have only to consider bulk excitations that occurs in an infinite solid. For convenience they can be described in a finite volume if periodic boundary conditions are applied to the parameters that describes the excitation. A complete a priori calculation of e2 has never been attempted for real solids but more limited calculations of the contributions of the different excitations permits us to identify them in the experimental results. 2.2.1. Collective motion of the electrons Every description of the free or quasi-free electron gas can be used to describe this component.It can be approximatedin aciassicalway from three parameters ;the density of electrons, their relaxation time and their effective mass. This leads to the classical Ketteler-Helmholtz formula. Quantic treatment of the interacting electron gas can also be given. Resonant oscillations of the electrons occurs when the incident electromagnetic wave can be coupled with a spontaneous mode (plasma oscillation, quantized in plasmons). This occurs for the frequency (and wave vector, when spatial dispersion is important), that cancels or nearly cancels the dielectric constant. If the plasma oscillations are damped i.e. if the plasmon can spontaneously decay into another type of excitation, the resonant character of the response to the electromagnetic wave is also damped. In addition to the volume plasma resonance observed in an infinite solid for E = 0, a surface plasma resonance can be observed in a semi-infinite solid bounded by an insulator of dielectric constant ci when E+Ei
=
0
(see STEINMANN [19681). 2.2.2. Collective motion of the ions The collective motion of the ions can be described as a set of phonons. Optical phonons can be coupled with the electromagneticwave and contribute to the absorption. This effect is important in the infra-red but not in the spectral range where PE occurs. 2.2.3. One-electron excitations (direct transitions) In the volume theory of PE a photon can lead to the emission of an electron only if its energy brings one electron up to an energy level above vacuum level. This process, especially in semiconductors, is often called electron-hole pair creation because the electron brought up to a state above the Fermi level EFleaves under EFa vacant state or a hole.
254
PHOTOEMISSION
[VL
5
2
Generally, this process is treated in the one-electron approximation. The electronic state of an infinite solid is described by a set of occupied Bloch states, with wave functions i,hk(r) = uk(r)eik‘r.For each band u k ( r ) is a different function which has the crystal periodicity. We shall assume that each wave function is normalized in a unit volume and that a complete set of wave functions is obtained by using cyclic conditions at the boundaries of a unit volume of appropriate form. We assume that the dimensions of the crystal cell are much smaller than the unit length. The density of permitted states in k space is then 4n3/V, ,where V, is the volume of the crystalline cell. At temperature T, the occupation probability for the $k level is a function of its energy E , (2.10) The complex part of the one-electron contribution to the dielectric constant is given by:
s., s., F
f ( E k ) [1-f(Ekt)]Okk,d3kd3k’
E2a
=
(IW12 )
__
(2.11)
where (IS(r)12) is the mean value of the squared modulus of the complex amplitude B ( r ) in the volume 6V for which gZolis defined. When the vector potential of the electromagnetic field has the complex amplitude A(r), the transition probability O k k , from the state $k to the state $ k , is Okk,
The matrix element
=
2n
is =-($k’l
lJ&kk?1’6(Ek,
’
m
- E k - ho).
( A ( r ) .P 3 - P * A(r))I$k)3
(2.12)
(2.13)
where p = (h/i)V, is the momentum operator. The integral (2.11) is extended over the complete first Brillouin zone for k and k’ and a contribution of type (2.1 1) must be added for each combination of a partially or fully occupied band and a partially or fully vacant band. In the long-wavelength approximation, we can consider E ( r ) to be uniform in 6 V , so that (2.14) It follows immediately from the periodicity of uk(r) ..and ukr(r)that
VI,
§ 23
T H E O R E T I C A L B A S I S O F T H E PE
255
# 0 only if k = k (momentum-conservation selection rule). Had we not taken k and k' in the same Brillouin zone, the selection rule would have been
d k k '
k = k'+K,
(2.15)
where K belongs to the reciprocal lattice of the crystal. In the free electron model, uk and uk' are constant, if k and k' are taken in the appropriate Brillouin zones. Therefore d k k , = 0 and no absorption occurs in this approximation.
2.2.4. One-electron excitation (non-direct transitions) Other processes can lead to the conversion of the photon energy into a one-electron excitation (or electron-hole pair creation). The coupling between the occupied state & and the vacant state (f/k. can occur through lattice deformation. The emission or the absorption of one or several phonons insures the momentum conservation. The energy transferred to the electron differs from the photon energy by the energy of the phonons. Coupling between rClk and (f/k, can also occur through other many-body effects e.g. the electron-electron interaction or the readjustment of the conduction band due to the created positive hole (SPICER C19671, DONIACH [19701). Any breakdown of the model which assumes independent Bloch electrons involves the breakdown of the k conservation selection rule. Later on we shall consider surface effects that cannot be included in a description of the solid by the dielectric constant.
2.3. PHOTOEXCITATION COEFFICIENT
Our survey of the possible photon absorption mechanisms in the bulk of a solid is not complete. We did not take into account multiphoton absorption, multielectron excitation and exciton resonances that would not easily fit in a 3-step process. In any case the interactions between the different excitations make our analysis approximate. The one-step theories of PE were developed to answer this objection. In principle one can deduce from our survey an estimate of the photoexcitation coefficient Po and its distribution between the different electron energies p(E, v). p(E,hv) is the most useful intermediate between the observed PED n(E, hv) and the microscopic processes that many authors investigated through PE experiments. b(E, hv) differs from n(E,hv) first because of the elimination of the electrons of energy smaller than the vacuum
256
PHOTOEMISSION
[VL
52
level E, and also because it includes inelastic scattering during the transport to the surface. In the different interactions of light with the solid that can be observed in PE data, we must distinguish 1) those which produce one-electron excitation, 2) those which compete with one-electron excitation, and 3) those which only modify the distribution of the electric field and the DAP in the solid. The collective motion of electrons, insofar as it is damped only by one-electron excitations or light emission, is of the third type. The plasma resonance iq a particular form of this collective motion, and we could apply the expressions (2.6) and (2.7) without any reference to the plasmon concept. We can also consider that a photon excites a plasmon that decays either into a one-electron excitation or a reflected or scattered photon. The plasmon appears there as an additional step in the PE process and the excitation of a plasmon must not be included into the calculation of p(E,hv) as competing with one-electron excitations. If a collective oscillation of the electrons is damped by thermalization of its energy, the process that decreases B(E, hv) is in fact the thermalization. 2.4. FRESNEL EQUATIONS AND THE DAP
The Fresnel equations permit the calculation of the electric field excited by an incident plane wave in a solid. At the interface of two semi-infinite homogeneous media, characterized by indexes of refraction no and n, the reflection coefficient is no cos i , -n, cos i, r, = (2.16) no cos i, + n , cos i, ’ for s polarization, and r =
n1 cos i, -no cos i, n , cos i , +no cos i, ’
(2.17)
for p polarization. Here i, and i, are respectively the angles of incidence of the wave vectors k , and k , in the incident medium (generally vacuum) and in the photoemitting solid. They are related by Snell law
no sin i, = n, sin i,. Absorption always occurs in a photoemitter, hence n , , i , and k , are always complex.
"'9
§ 21
257
T H E O R E T I C A L B A S I S O F T H E PE
In a semi-infinite photocathode the DAP at a distance z from the surface is D(z) =
4 n v ~1(- rr*) exp ( - 4 ~ / 4 , 1cos i,
(2.18)
where 11, =
v-iK,
and p and q are the positive solutions of (2.19), (2.19)
r is given by (2.16) or (2.17) depending on the state of polarization of the light. In thin film of thickness zo we must take into account the reflection of the light on both faces of the film. For s polarization l+r, 1 + r, r: exp ( - 2iq,z,)
+
x {exp (- iq, z ) r: exp (iq,[z
- 2z,]))
where rs and rl are the coefficient of reflection for the electric field, given by (2.16) when the incident light beam encounters respectively the first and the second face of the film and 4, =
2714 cos i,
(2.21)
A
is the z component of the wave vector q in the photoemitter. Because of absorption q, is always complex. q, that appears in (2.18), is the imaginary part of 1qZ/2n.For p polarization, because the angle of refraction i, in the photoemitter is complex, we must add separately the energy densities Dp,(z) and D,,(z) associated with the tangential (x) and normal ( z ) components of the electric field:
+
(1 ~,)(cosi,/cos io) 1 rprb exp ( - 2iq, z,)
+
x cos i, x {exp ( - iq, z)
+ TI,exp (iq,(z - 22,)))
12
,
258
DP&)
1{
PHOTOEMISSION
471~1~ (1+ rp)(cosi,/cos i,) I cos i, 1+ rprb exp (- 2iq, z,)
= ___
/j
x sin i, x { exp ( - iq, z ) - rb exp (iq,(z - 22,))) ;
(2.22)
rp and rl, have for p polarization the same significance as rs and ri for s polarization. The calculations of the photoyield by COQUET and VERNIER[1966] and by many other authors since that time have been based on equations (2.20) and (2.22). 2.4.1. Validity of the Fresnel equations and spatial dispersion The Fresnel equations (2.16) and (2.17) represent the solution of the Maxwell equations when two homogeneous media described by two local indexes of refraction are separated by a plane discontinuity. The well-known continuity of the tangential components of the electric and magnetic fields (CTCF) that is used as boundary condition to derive eqs. (2.16) and (2.17) is a consequence of the Maxwell equations at singular points where the discontinuity of the index of refraction occurs. If spatial dispersion is not negligible we can try to substitute for the local index of refraction the wave vector dependent one, n(q, 0). Well inside the solid a complete set of solutions of the Maxwell equations includes the plane waves with a wave vector q that satisfies the dispersion equation q2
=
w2 -
C2
n2(q,0).
(2.23)
The wave vector of the transmitted wave q associated with a given incident wave must be deduced from (2.23) and the Snell laws (equality of the tangential components of the incident, transmitted and reflected waves). New effect introduced by spatial dispersion arise from the possible multiplicity of solutions for q. HOPFIELD and THOMAS [1963] explained observed anomalies in the reflectivity of CdS for photon energies near the exciton resonance by such an effect. This situation is similar to the phenomenon of double refraction in anisotropic media. To determine the amplitude of both transmitted waves, the incident one is divided into two components, linearly polarized in the simplest case, elliptically in the general case. Each component is coupled with one transmitted wave by the CTCF and a generalization of Fresnel equations can be found for it. Such a decomposition has not been found
VI, §
21
T H E O R E T I C A L B A S I S O F T H E PE
259
in the case of spatial dispersion and the CTCF associated with the dispersion equation (2.23) are insufficient to determine the amplitude of several transmitted waves. HOPFIELD and THOMAS [19631 and many later authors introduced an additional boundary condition by examining the microscopic process that is responsible for the spatial dispersion e.g. Hopfield and Thomas cancel the contribution of the exciton to the electric polarization at the surface. MELNYK and HARRISON [1970] introduced an additional boundary condition, namely the continuity of the normal component of the electric field, to account for the excitation of longitudinal plasma oscillations in metals in addition to the standard transverse waves. For p polarization at high angles of incidence, they predicted oscillatory variations of the transmittance and absorptance of thin films of K with photon energy above the plasmon resonance. ANDEREGG, FEUERBACHER and FITTON [I9711 observed such variations in the spectral distribution of the photoyield. The justification of the additional boundary conditions raised a good deal of controversy. AGARWAL, PATTANAYAK and WOLF[1971a, b, 19741 substituted for the additional boundary condition a coupling between the transmitted waves at the surface. VERNIER [1973] noted that the use of boundary conditions implies that the transition layer, where the bulk index of refraction of the homogeneous medium does not represent the solid properties, has a negligible thickness. We may ask whether the standard CTCF can be applied across this layer, when the local approximation breaks down. The rigorous solution is to solve the coupled equations that relate the motion of the charges and the electric field in the vicinity of the surface. Such a solution has been given for a free electron gas by SAUTER [1967], FORSTMANN [1967], and FUCHS and KLIEWER [1969]. FUCHS and KLIEWER [1969] calculated the impedances 2, and 2, that appears in the standard expressions of the reflectance for p polarization; R , = for s polarization; R, =
cos io-z, cos i, z,
(2.24)
1-2, cos i, 1+z, cos i,
(2.25)
+
These expressions would be equivalent to Fresnel equations (2.16) and (2.17) if Z , and Z, could be deduced from a single dielectric constant E for every angle of incidence by (2.26)
260
PHOTOEMISSION
Z,
=
( - -sin’io
[VI,
,
Ei
2
(2.27)
This requirement is, however, not fulfilled. Moreover Fuchs and Kliewer found a surface absorption term. We note here that such surface absorption implies that the normal component of the Poynting vector is not continuous at the surface. Therefore the CTCF is no longer valid. The process of surface absorption considered by Kliewer and Fuchs does not lead to one-electron excitation and therefore to PE. Later on we shall note other processes of surface absorption that lead to PE and that are also inconsistent with the CTCF and, therefore, with the Fresnel equations.
2.4.2. Validity of the Fresnel equations, surface roughness ana plasma oscillations The Fresnel equations have been established for a perfectly smooth surface. If the surface is slightly rough, the incident light produces scattered waves in addition to the reflected and transmitted ones. The Fresnel equations can still be valid if the total intensity of scattered waves is negligible. This can occur for relatively large roughness if no resonance occurs. Surface plasma oscillations can be dealt with as a special form of scattered wave and the breakdown of Fresnel equations when the incident wave is coupled with plasma oscillations is a very popular mean of proving their existence. JASPERSON and SCHNATTERLY [1969], ENDRIZ[19731and DAUDE, SAVARY and ROBIN[1972] plotted the reflectance of different metals (Ag, Al, Mg) versus the photon energy and observed a dip for plasma resonance that increases with roughness. CALLCOTT and ARAKAWA [ 19741 observed a similar dip in the plot of the reflectance of films of Li versus the angle of incidence. They were able to fit the observed values of the reflectance for all angles of incidence with one value of the complex index of refraction only for hv > 6 eV. In both cases the breakdown of the Fresnel equations was attributed to an effect of roughness that becomes especially large when surface plasmons can be excited. Arakawa and his coworkers gave a special attention to the excitation of surface plasmons in metallic gratings that can explain the Wood anomaly. It is beyond the scope of this paper to give a complete review of the plasmon phenomena (see RITCHIE [1973]). We shall only note that plasmons can be excited on perfectly smooth surface when a metallic film is deposited onto the face of a prism or semi-cylinder on which a beam of light is totally reflected (OTTO[1968,1970]). In that case the surface plasmon represents the evanescent wave that is excited in the film by the incident wave. The
VI, §
21
T H E O R E T I C A L BASIS OF T H E P E
261
calculation of the frustrated total reflection by glass when it is coated with a metallic film can be performed with Fresnel equations. Of course in that case too the Fresnel equations are only valid when the surfaces are smooth enough. 2.5. SURFACE PHOTOEXCITATION
In the one-electron approximation, we can distinguish two effects : 1) Surface photoexcitation from bulk states (SPBS) due to the perturbation of bulk states by the surface. 2) Photoexcitation from surface states. 2.5.1. Surface.photoexcitation,from bulk states (SPBS)
Each bulk Bloch function $(r) that represents the electronic state of the infinite solid must be corrected in a semi-infinite solid by a factor Xk(r), $(r) = Xk(r)uk(r)eik'r>
(2.28)
where Xk(r)= 0 when r is outside the solid Xk(r) = I when r is inside the solid. In a first approximation xk(r) may be assumed to be discontinuous at the surface; in a better approximation Xk(r) varies in a continuous manner from 0 to 1 in a transition layer. The substitution of (2.28) in the expression (2.14) of the matrix element implies that (2.29) is replaced by &kkb
(2.30)
f AAkk'
where
A
-
he A = ~ <Xk(r)uk,(r)eik'.r( ~ ~ - b(r.) (V,Xk(r))luk(r)eik"). mo
(2.31)
We assumed here that the gradient of the electron wave function is much steeper than the gradient of the electric field. This assumption might not be valid if the screening length is very short ; to calculate the matrix element the expression (2.13) should then be used instead of (2.14). SCHAICH and ASCHCROET [19711 noted that the surface contribution
262
PHOTOEMISSION
[VI, § 2
cannot be separated from the volume contribution because the quadratic character of the photoelectric response produces interference effects. The and A A k k ‘ can take any value difference between the arguments of between 0 and 2.n. But, if k = k’, AMkk,is generally much smaller than d k & b so that the surface perturbation does not significantly modify the matrix elements associated with direct transitions. If k # k’, &kkb = 0 and we can write A&kkp
= d’h?kke.
Therefore, in a first approximation, we can justify the separation of surface from volume made by ENDRIZand SPICER [1971b], by writing
An important feature of the SPBS is that in a first approximation Vrzk(r)is a vector normal to the surface. We may then write
@9 . (V, X&))
= &,(r)W,
(2.33)
where €,(r) is the normal component of the complex amplitude of the electric field b ( r ) . If the surface is perfectly plane, only an oblique p-polarized wave can produce surface absorption. For normally incident light or for s waves, only the roughness of the surface or the periodic variation of the normal due to the disposition of the superficial atoms can produce SPBS. This characteristics of SPBS has been well known since the early theory of MITCHELL [1934]. 2.5.2. Photoexcitation from sui-face states In a semi-infinite solid we find surface states in addition to bulk Bloch states. They were first introduced by TAMM[1932] and characterized by wave functions quite similar to (2.28), with a complex wave vector k = k,-ik,.
(2.34)
The imaginary part k, is normal to the surface and directed towards the interior of the solid. Intrinsic surface states are associated with dangling bonds and are expected to disappear with surface contamination. Of course, surface contamination can introduce impurity surface levels. DAVISON and LEVINE[19701 reviewed the theorical and experimental work on intrinsic surface states. Recently, MONCH[1973] reviewed the special case of Si surface. Experimental evidence for surface states is most easily obtained when they lie in the forbidden band of a semi-conductor e.g., through band
VI, 5
21
T H E O R E T I C A L B A S I S O F T H E PE
263
bending effects (ALLEN and GOBELI [1962]). But surface states have been theoretically predicted at other energy levels in the band scheme both in semi-conductors and metals (BORTOLANI, CALANDRA and KELLY[19731, APPELBAUM and HAMANNC19731, GURMAN and PENDRY C19731, HEINE [19721). We are concerned here with PE from surface states. It is generally difficult to discriminate between PE from intrinsic surface states and other levels that occupy the same position in the band scheme. No polarization test is available and the best test is the dependence of the observed phenomena on an exposure to very small amounts of gases. 2.5.3. Surface absorption, Fresnel equations and DAP We have seen that collective motion of the electrons and the quantum transitions between one-electron states can induce surface absorption. We have also seen that surface absorption implies a breakdown of the CTCF and, therefore, of the Fresnel equations (2.16) and (2.17). But this effect is expected to be small and has always been neglected in the calculations of the volume contribution to the DAP. We may hope to give an adequate account of the surface effects by including the surface term A6(z) in the DAP. 2.6. THE ELECTRON ESCAPE PROBABILITY
The absorption of one photon at r produces excited electrons distributed in energy according to the function P(E,hv). To define completely the effect of the absorbed photon we should also give the distribution of the wave vectors of the excited electrons. Different interactions suffered by the excited electron prevent it from being emitted with the energy E. Their effect is characterized by the functions p(E', E, r) and p(r), that have been defined in 91.1. If the photo-excitation can be assumed to be isotropic, p(E', E, r) and p ( r ) are independent of the angle of incidence and of the state of polarization. They depend only on the distribution function B(E, hv). Because of the influence of the distribution of the wave vectors, the escape probability may be quite different for photoexcited electrons and for electrons injected by other means. Most experimenters assume, without discussion, that (2.35) p o is the escape probability of an electron excited just at the surface and L is the escape depth or attenuation length. The fit of the experimental data when (2.35) is substituted into (1.1) may be taken as a sufficient
264
PHOTOEMISSION
[VI,
52
justification of (2.35). Then L appears as a phenomenological parameter that measures the effective depth that contributes to the PE. A more fundamental justification of (2.35) requires an examination of the interactions involving an excited electron during its transport towards the surface. 2.6.1. Coulomb repulsion between electrons 'The Coulomb repulsion between electrons cannot be fully accounted for, if the electronic state of the solid is represented as a set of independent Bloch electrons. Because of this effect we must consider that an electron in an excited state has a finite lifetime z,. If no other process occurs to deexcite a Bloch electron, the mean free path may be written as
(2.36) where ug is the group velocity associated with the electron. Two types of Coulomb scattering have been considered: those due to interaction with one-electron states (electron-electron scattering) and those arising from the excitation of plasmons. a ) Electron-electron scattering or electron-hole pair creation is the interaction of the excited electron with a less energetic electron which is, with a very high probability, under the Fermi level E F .Because of this interaction the photon energy is then shared between two excited electrons or in other terms between two electron-hole pairs. The energy lost by the excited electron of energy E is most often in the range of magnitude ( E - EF)/2.Within a few eV of the threshold one interaction generally brings both electrons under the vacuum level E,. For high energy photons (in the UV or X-ray regions) both electrons may be brought above the vacuum level. This possibility can be described from a phenomenological point of view by p ( v ) and p(E', E, Y), if we consider these functions as mean numbers of emitted electron per absorbed photon. p ( r ) can, in principle, become larger than unity, but many absorbed photons lead to no PE at all.
b ) Plasmon scattering results from the interaction of the excited electron with a set of conduction electrons. A collective motion of the conduction electrons is then induced. We can describe the interaction macroscopically, in terms of the dielectric constant. This type of interaction becomes important if a plasma resonance can be excited and the loss of energy is then quantized into surface or volume plasmons. Plasmons are found in the spectrum of characteristic energy losses when charged particles are sent through thin films (see review by PRADAL, GOUTand FABRE [1965]).
"I,§
21
265
T H E O R E T I C A L B A S I S OF T H E PE
QUINN[1962] calculated the mean free path of an excited electron of energy E and momentum p in a free electron gas with respect to electronelectron scattering (lee)and plasmon scattering (l,,). He found a decrease of lee when E increases in the low energy range. This result has been generalized by BERGLUND and SPICER[19641. They calculated the lifetime z(E) as a function of the density of state p(E). They assumed that the probability for an electron of energy E to transfer an energy AE to another electron of energy Eo is proportional to
P(Eo + AE)p(E - AE),
+ t
I
5
'I)
E
Fig. 1. Mean free path of an electron of energy E above the Fermi level with respect to electron-electron interaction in gold, calculated by KROLIKOWSKI and SPICER[19691 (continuous curve), and by SZE,MOLLand SUGANO[1964] (dotted curve). Experimental values of the escape depth for photons of energy hv = E, obtained by: CROWELL, HOWARM,LABATE and SPITZER[1962], x SZE, MOLLand SUGANO[1964], 0 KATRICH and SARBEI[1961], + VERNIER,COQUETand BIGUEURE [1966], V PONG.SUMIDA and MOORE[1970], A MEAD [1962] (non photoelectric method), * KANTER[1970] (non photoelectric method).
266
CVI, §
PHOTOEMISSION
2
if both final levels E,+AE and E-AE are vacant, i.e., lie above the Fermi energy EF (random-k approximation). Therefore 1
( E - Eo@
6,
ZEF-E
(2.37) dAEp(E,)p(E, + AE)p(E- AE). dEo J E F - E . Results of the calculation of I(E) by KROLIKOWSKI and SPICER [ 19691 using (2.37) are plotted together with experimental data (see Q 3 or Fig. 1 for gold). KANE[1967] developed the random approximation in the case of
z(E) CC
A x-
G-
-5ar
& ,a-
# $6-
4-
2-
I
I
I
I
I
VI, §
21
THEORETICAL BASIS OF T H E PE
261
Si and discussed its validity by comparing the results with Monte-Carlo calculations, based on a more refined model. Quinn found for plasmon scattering in a free electron metal (2.38) where h o p is the plasmon energy, p F is the momentum of an electron at the Fermi energy and a, is the Bohr radius. Fig. 2 represents the variations of the mean free paths l,, and lee with the electron energy E - EF above the Fenni level as calculated by SMITH and FISCHER [1971] from eqs. (2.37) and (2.38). Fig. 2 also represents the variations of the mean free path 1 = lpelee/(lpe -k lee) that results from the combination of plasmon and electron-electron scattering. The strong decrease of Ipe occurs when lee is already very small and is not seen in 1 but the situation might be quite different in other metals. The experimental data from Smith and Fischer will be discussed in section 3.1. In a general discussion of the electrons energy losses during PE MAHAN [19731 has shown that surface excitations, and especially surface plasmons, may be important. FEIBELMAN [1973] investigated the validity of the assumption of a mean free path independent of the distance z to the surface. He found for Auger electrons (of a few hundreds of ev), that near the surface the increase of the mean free path with respect to the bulk plasmons is partly compensated by the decrease of the mean free path with respect to surface plasmons. But the generalization of such a result to low energy electrons might not be possible. 2.6.2. Phonon scattering The Bloch theorem has been proved in the approximation of fixed ions. The interaction of electron with lattice vibrations is represented by phonon assisted transitions of each electron from one Bloch state to another. The energy lost by an electron in one interaction equals that of the created phonon, typically a fraction of kO,, where 0, is the Debye temperature. In principle the annihilation of a phonon that increases the energy of the electron is possible but practically negligible in metals. For most excited electrons kOD is much smaller than the energy of the electron above the Fermi level and many phonon interactions are necessary to reduce the energy of an excited electron below the vacuum level. Since the mean momentum exchanged between the electron and the phonon is large and the energy loss small the phonon interaction may be often approximated as elastic.
268
PHOTOEMISSION
[VI,
82
The mean lifetime zp and the mean free path I, of the electron with respect to phonon creation can be defined as for scattering by electrons. We may write I, = vgzp.
zp is the main parameter that determines the conductivity of metals. From conductivity measurement one finds that lp lies in the range of a few 100 A. But the energy range of the excited electrons are quite different in PE and in conductivity. We may also note that the conductivity depends on exchanges of momentum whereas the PE depends mainly on exchanges of energy. In semiconductors, if the difference between the energy of the excited electron and the bottom of the conduction band is not larger than the band gap, both electron-electron and plasmon scattering are forbidden. In this case phonon-scattering can produce a thermalization of the excited electrons at the bottom of a conduction band. After a sufficient number of phonon interactions, the excited electrons, which are trapped in the conduction band, attain an equilibrium distribution when the number of created phonons equals the number of annihilated phonons. The probability of occupation of an energy level E of the conduction band is then proportional to exp (- E/kT). The total number of electrons in the conduction band is determined by the transitions that bring an electron into and out of the conduction band, (photon absorption and recombination with holes). This process is very important in the negative electron affinity (NEA) photocathodes, where the vacuum level E, lies below the bottom of the conduction band, because every thermalized electron can be emitted. 2.6.3. Electron-hole recombination In semi-conductors an electron can recombine with a hole with, or without, radiation. The lifetime q, of the excited electron with respect to such a process is generally several orders of magnitude greater than z, and z,. But in NEA photocathodes the other interactions are completely forbidden by the gap and the electron-hole recombination is the only process that affects electrons trapped at the bottom of the conduction band.
2.6.4. Transmission by the surface When an electron reaches the surface in a Bloch state, defined by an energy E‘ and a wave vector k’, it has a probability p,,(k’) of escaping. E’ and k’ are generally different from the initial energy and wave vector of the electron after excitation. p&’) can be easily calculated only in the
VI, 5 21
269
T H E O R E T I C A L B A S I S O F T H E PE
case of a free electron gas bounded by an ideal surface potential barrier. If k' is taken in the appropriate Brillouin zone, the electron momentum is hk' and we may write
(2.39) The z components of the wave vector inside (ki) and outside (k;) the solid are related to the work function W, by
The transmission probability (2.39) may be expressed as a function of the angle of incidence 8 of the excited electron at the surface and of the limit angle 8, given by
w,C O S ~eo = w,,
(2.41)
where W, is the kinetic energy of the electron. The transmission probability is given by cos 8' - (cos2 8' - cosz Q,)+ , cos 8' (COSZ e' + cosz eo)+
+
=o
when 8' < Bo, when 8' > 8,.
(2.42)
We may try to extend the expressions (2.41) and (2.42) to the case of Bloch electrons; 8' then represents the angle of incidence of the group velocity and W, the energy of the electron above the bottom of the conduction band : W, = E - E ,
(2.43)
W, = E , - E , .
(2.44)
We may note that 8, is smaller in metals than in semi-conductors. This extension is not valid for NEA photocathodes. At best we could give 8, = 4 2 and pE.(O')= 1. Neverthelesstheresultmaybequalitativelycorrect and contribute to the high yield of NEA photocathodes. Most authors merely define the escape probability p o for an electron excited at the surface by taking z = 0 in p(r).p o becomes then an empirical parameter which can include the effect of an activating surface layer. This is done in expression (2.35). Let us recall here that if surface photo-excitation occurs, the correspond-
270
PHOTOEMISSION
CVI,
52
ing escape probability ps, that has been defined in section 1 . 1 , may be different from po . 2.7. THEORETICAL DETERMINATION OF THE ESCAPE PROBABILITY
The escape probability p ( r ) results from the combination of several scattering processes and of the transmission through the surface. The general calculation is quite complicated and cannot lead to general formulae except in the most simple approximation. 2.7.1. Electron-electron interaction and the ballistic approximation
In many cases the electron-electron interaction is the dominant scattering process (I, zs- I, e.g. in metals). For small photon energy the energy loss of the excited electron in one scattering even brings it under the vacuum level E,. If the expressions (2.41) to (2.44) are valid, the escape probability for an electron that has been excited in a state of energy E with a group velocity making an angle 6 with the surface is PE(e, r,
= PE(@
exp [ - z/l(E) cos
(2.45)
where pE(6) is given by (2.42) and 1(E) is the mean free path for an electron of energy E. For an isotropic excitationpE(r)may be written for an electron of energy E as (2.46) At least in metals the limit angle 6 , defined by eqs. (2.41), (2.43) and (2.44) is small for small photon energies and we may write, (2.47) (2.48) The escape probability increases sharply with E because of 8,. Therefore most of the emitted electrons have been excited from quite near to the Fermi level. If the mean free path I(E) does not vary too sharply with E, we can approximate (2.48) by
Pb) = Po exp [ - z / U , + h41.
(2.49)
The escape depth L that appears in (2.35) is, therefore, approximately the
VI, 5
21
271
T H E O R E T I C A L BASIS OF T H E PE
mean free path for an electron excited from the Fermi level. p o depends only on the photon energy and can be written as (2.50) In the assumption of a slow variation of I with the electron energy CROWELL, SPITZER, HOWARTH and LABATE [1962] substitute for the expression (2.49) L = ICE, + hv, +$hv - hv,)],
(2.51)
where hv, is the photoelectric threshold. KROLIKOWSKI and SPICER[1969] calculated how the observable PED n(E, hv) depends on the scattering processes in the ballistic approximation. In this case the scattering has only substractive effects on the initial distribution of the electrons p(E,hv). These authors approximated the expression (2.42) by pE(6)= 1 when 6 c B0, =0
(2.52)
when 6 > 6,,
and integrated the contribution of the different layers to the photoelectric current. They assumed a normal incidence illumination with an absorption coefficient a. They described the escape probability by the expression (2.45). After integrating over 6 they found that
with
"
f(6,, a, E ) = - 1-cos 8,2
___
al(E)
In
( +al(E) 1
cos 6 ,
)].
(2.53)
Here 6 , depends on the energy Eand is given by (2.41). To apply the expression (2.53) KROLIKOWSKI and SPICER use the expression (2.37). The ballistic approximation breaks down when secondary electrons keep enough energy to get out of the solid. KANE[1967] performed a numerical calculation of the energy distribution of secondary electrons in Si, after [19691 attributed a structure scattering by pair creation. SMITHand SPICER observed in the energy distribution of photoelectrons excited in alkali metals by 10.2 eV photons to electrons that have lost energy by plasmon excitation.
212
[w,4 2
PHOTOEMISSION
2.7.2. Diffusion equation and electron-hole recombination in negative electron affinity ( N E A ) photocathodes At the opposite extreme from the (2.6.1)ballistic approximation the mean free path may be larger for electron scattering than for phonon scattering. Before being brought under vacuum level, the excited electron is then subjected to a large number of phonon scattering events. In each event it loses much of its momentum but very little of its energy. In NEA photocathodes the transport of the excited electrons can be described as elastic scattering of the electrons by phonons. The excited electrons are quickly thermalized into the bottom of the conduction band (see section 2.6.2) and their lifetime q, in that state is limited only by the electron-hole recombination, because the average energy gain by phonon annihilation exactly balances the average loss by phonon creation. The thermalization time can be neglected with respect to zh. The density f(z) of excited electron at the depth z obeys the diffusion equation (JAMESand MOLL[19659,
9 d2f ~
dz2
f = POD(Z). + zh -
(2.54)
Here fi0D(z) is the density of electrons excited by unit time and 9 is the diffusion coefficient. The density of photoelectric current is given by the current at z = 0, df I = -9e-. dz
(2.55)
The surface properties of the cathode define boundary conditions that we can describe by an escape probability p o . The surface treatment necessary to obtain the NEA is the deposition of an activating film in which inelastic scattering occurs and reduces p o . In a semi-infinite solid we can deduce from (2.54) and (2.55) that
[
m
y =
.o
Po exp (-Z/L)PoD(Z)dZ,
(2.56)
where Ld is the diffusion length L,
(2.57)
= (Th9)f.
Eq. (2.56) can also be obtained by substituting from (2.35) into (1.1) and the diffusion length can be identiJied with the escape depth. In a thin film of thickness zo we can deduce from (2.54) that Y
=
s
+
+
[ A exp (- z/Ld) B exp ( z/Ld)]PoD(z) dz.
(2.58)
VL
5 21
T H E O R E T I C A L B A S I S OF T H E P E
213
Here A and Bare constants which can be deduced from boundary conditions at the interfaces of the film. The quantity in brackets can be identified with the escape probability p ( r ) in the expression (1.1). We can consider that the terms A exp (-z/Ld) and Bexp ( + z/Ld) result from successive back and forth diffusions in the film with an escape probability p o at the surface and an absorption probability p A at the substrate interface and we can write
2.7.3. De-excitation of photoelectrons by phonon scattering only In semi-conductors, when the NEA is not achieved and when the pair creation is forbidden by a band gap larger than the electrons affinity, the de-excitation of the photoelectrons results from successive small energy losses in phonon scattering events. The age theory derived by Fermi to calculate the neutron transport in nuclear reactors was applied by HEBB [1951], LYE and DEKKER[1957] to the transport of secondary electrons. They assumed that phonon creation was the only energy loss process for the electrons and that an electron of energy E above the bottom of the conduction band E, loses a fraction ( of E - E, in each phonon creation. They found for the escape probability p(z) = po Erfc 12/22;),
(2.60J
where
>J,
Erfc (y) = -
exp (- x’) dx,
(2.61)
and zF is the age of the electron after excitation,
(2.62) 9 is the diffusion coefficient due to phonon scattering. lp is the mean free path for phonon scattering. Figure 3 represents the functions Erfc(x) and e- 2x. Within the standard accuracy of photoelectric data these functions cannot be distinguished from one another and the expression (2.60) can be approximated by (2.35) with
L =
2;.
(2.63)
214
PHOTOEMISSION
\
Erfc (X\
Fig. 3. Comparison of the functions Erfc (x) -and exp (-2x) -----
2.7.4. De-excitation of photoelectrons by both phonon and electron-electron interactions BARTELINK,MOLL and MEYER[1963] introduced in the Fermi age equations an absorption term that can describe the electron-electron interaction. When the energy loss in phonon scattering becomes negligible, they could approximate the escape probability by the equation (2.35) with the escape depth,
(2.64)
KANE[1966] calculated, in a one-dimensional model, the rate of phonon creation and electron-hole pair creation as a function of the mean free paths lp and I,. He also calculated the phonon energy loss distribution that distorts the observed PED from P(E,hv). DUCKETT [1968] showed on a random-walk model that Kane had well approximated the 3-dimensional case. LANGRETH [19711derived results quite similar to Kane’s in the modem formalism of Green functions. BALLANTYNE [19721 included the energy losses due to phonon scattering and the variation with the photon energy hv of the complex dielectric constant el(hv)- ie,(hv) in the expression of the photoyield near threshold. He could thus fit the experimental results in a wider spectral range than
VI,
§ 21
T H E O R E T I C A L B A S I S O F T H E PE
275
with the expressions of KANE[1962]. Insofar as the mean free path I@) is a slowly varying function of the electrons energy E, the pair production scattering determines the overa!l number of emitted electrons but does not affect the form of the spectral yield nor the PED near the threshold. For a non negligible energy loss Ephin phonon scattering, he found that if the final states at threshold are not at an extremum in the conduction band then
x (hv-hv,)3;
(2.65)
x (hv - hv,)2.
(2.66)
and that if Ephis negligible, then
The expressions are the same for direct and indirect transitions, only the value of the threshold hv, is different. The expression (2.65) gave quite good results for semi-conductors. In the most general case a simple explicit form of the escape probability cannot be given. STUART,WOOTENand SPICER[I9641 and STUARTand WOOTEN[19671 performed numerical calculations for various values of Eph,1, and lp. The results roughly agree with simple models.
2.7.5. The escape probability and analysis of the experimental data. The simple expression (2.35) has been justified in many simple cases aad we may expect that it provides a reasonable basis for the analysis of the experimental data. The expression (2.60) is probably as good, but the integration of (1.1) can only be made by numerical methods if (2.60) is substituted into (1.1). For this reason nearly all authors have used (2.35) with the exception of HOFFMANN and DEUTSCHER[ 1970) and HIRSCHBERG and DEUTSCHER [19683. 2.8. PHOTOEMISSION AND MANY-BODY EFFECTS
In the 3-step model, the break down of the independent electron approximation has been introduced as scattering processes that determine the transport properties. Many-body effects have also been introduced into the determination of the photoexcitation probability B as photon absorption processes that compete with the production of one-electron excitation. We have seen that the decay of the various excitations of the solid into one another may bring about difficulties . in the interpretation
216
PHOTOEMISSION
[w Q
2
of P(E, v). The transition probabilities given by the expressions (2.1 l), (2.12), (2.13) have been derived within the one-electron approximation. The k conservation selection rule in the volume effect is a direct consequence of (2.13). Such a treatment may not be sufficient and we need to introduce also the many-body effects in the calculation of the transition probability 6 j ) k k . . The indirect transitions with phonon creation may be considered as a transition between two Bloch states of different wave vectors that is allowed by a subsequent scattering of the electron in the final state by the lattice. In a more general way, we may associate with each scattering process of the excited electron a perturbation of the transition probability O k k . .The perturbation may be an increase or a decrease only when the transition probability is not zero without scattering e.g., for direct transitions. In most cases the perturbation merely adds a new term to the absorption. Indirect transitions with phonon creation have a quite low probability in comparison to direct transitions and have been observed by optical means only in the spectral range where no direct transition can occur. Because the mean free path of the electrons for the electron interaction is often much shorter than for the phonon interaction we may expect in the first case a much stronger perturbation that, quite strangely, is seldom taken into account. A semi-empirical way of taking into account the dependence of photon absorption on the scattering processes is to introduce damping in the Bloch waves that are used to calculated matrix element (2.13). We may attribute to the initial state $k a quasi-infinite lifetime and a finite lifetime z to the final state. The theory of Weisskopf and Wigner, quoted by DAVYDOFF [1965], when adapted to our notations gives, instead of (2.12), (2.67) The expression (2.67) can be very useful for comparing the results of optical or PE measurements with the information deduced from escape depth measurements. It must be applied with great caution when several scattering events follow one another; z must be taken as some sort of an empirical parameter that represents the time necessary for a complete thermalization of the absorbed photon energy. In the dynamical theory of electron diffraction (DEDERICHS [19721) damping of the electron wave is often introduced by substituting for the classical Bloch wave (2.9) the expression t,bk = U k ( r ) exp
(ik . r) exp (- k , . r).
(2.68)
VI,
ii 21
T H E O R E T I C A L B A S I S OF T H E P E
211
The addition of an imaginary component ik, to the wave vector k is equivalent, in principle, to the introduction of a finite lifetime, if k, has the same direction as the group velocity associated with the-k state and if (2.69) where I is the mean free path of the Bloch electron. If we substitute (2.68) into the expression (2.13) or (2.14), lA,','tlis no longer zero for k # k'. We can divide k and k into two components respectively parallel and perpendicular to kI, k = k,+kN,
k
= ki+kh.
The k conservation rule that could be expressed in the approximation (2.13) by
lAkk,Iz = A(k)G(k-k')
(2.70)
becomes (2.71) Based on the same principle as the Wigner and Weisskopf expression (2.67), the expression (2.71) suffers the same limitations because of multiple scattering. We can nevertheless use it to predict a correlation between the value of the escape depth of the electrons and the importance of non-direct transition that were first introduced by Spicer. We note here that I A k , ' ! l 2 as deduced from (2.71) has a tail of non negligible values for quite high values of IkN-k&l. Another way of breaking the validity of the k conservation rule is to assume an interaction of the excited electron with the hole created below the Fermi level by its excitation (SPICER[1967], DONIACH [1970]). A rigorous treatment of the problems of light absorption requires the determination of independent excitations of infinite lifetime that presently are far from being known in the most general case. 2.9. ONE-STEP THEORIES OF PHOTOEMISSION
In the microscopic interpretation of the PE process the different steps certainly depend on one another and the division into steps appeared to
218
PHOTOEMISSION
[VI,
5
2
several authors as artificial. For that reason, several formalisms have been derived to treat PE as a one-step process. MAKINSON [19491developed a theory of surface PE as a one-step process. In order to include in the expression of the photo-yield a factor representing the transmission of the electrons, Makinson introduced the coupling of the electrons of the solid with the sets of electron waves obtained by associating with each incoming wave the reflected and transmitted ones. In more recent work the set of waves associated with each incoming wave has been completed by the diffracted waves and its coupling with the electrons of the solid has appeared as a basic element of the one-step theories of PE. ADAWI[1964] and MAHAN[1970] have introduced the methods of scattering theory for the surface effect and the volume effect respectively. They treated the PE as inelastic scattering of the electrons of the solid by photons. Mahan used the asymptopic form of the Green functions to calculate the photoelectric current dI emitted within a solid angle dQ. He obtained the result (2.72) The integral is extended over the range of wave vectors ki associated with occupied states 4i in the solid, p is the momentum of the emitted electron, 4’ is a wave function that represents an incoming electron wave with a momentum p directed within dQ, along with the reflected, diffracted and transmitted waves. (4’*\9\&) is the element of the T matrix that couples the initial state with the state +’*. The operator 9 is given by the Lippmann-Schwinger equation. If V is the operator that represents all interactions that have been neglected in describing the electron states by stationnary wave functions c$~ and &’, we may write 9
=
(2.73)
V+VGo9,
where Go represents the Green operator. In practical calculation, 9 must be expanded,
9 = V+VG,V+VG,VG,V+
....
Each term represents a scattering of definite multiplicity.
(2.74)
w, D 31
THE ESCAPE D E P T H OF THE PHOTOELECTRONS
279
Mahan developed his calculation mainly for q5i representing free electrons but the expression (2.71) is still valid when q5i represents Bloch electrons. Let us note here that we may include in the double scattering term of (2.72) double scattering by the electromagnetic field, and thus include two-photon photoemission. The same T-matrix elements appear in the Mahan theory of PE and in the theory of Leed and Auger emission. In the latter cases the calculations are much more advanced than in PE (see for instance TONG,RHODIN and TAIT[1973]) and could be probably used in PE. In recent years, other onestep formalisms have been developed by SUTTON[19701, SCHAICH and ASCHCROFT [1970], THORNBER [1971], HERMEKING [1972,1973], TZOAR and GERSTEN[1973], GERSTEN and TZOAR[1973], CAROLI,LEDERERROZENBLATT, ROULETand SAINT-JAMES [19733, MEESSEN [19731. Based on more rigorous principles than the 3-step model, they have been able to predict new structures in the PE of X-rays (NOZIERES, DEDOMINICIS [19691). But up to now no one-step theory of PE has been able to include in a proper manner the effects of electron-electron scattering. In the papers of MAHAN [19701and SCHAICH and ASCHCROFT [19703electron scattering is introduced just like in the 3-step model. In any case the functions that appear in the 3-step model may be used at least as phenomenological parameters to interpret the experimental data. We may hope that advances in the one-step theory will afford a better link between the functions that can be deduced from experimental data and the microscopic processes.
0 3.
Experimental Determination of the Escape Depth of the Photoelectrons
In this section we shall assume that surface absorption is negligible and analyse the photoelectric data on the basis of expression (1.1). The consistency of our results will provide evidence of the validity of our assumptions. In the in the next section we will discuss the evidence for surface PE. The DAP D(r) can be calculated for thick films or bulk emitters by means of the simple expression (2.18) when the index of refraction is known from preliminary optical data. For thin films the more complicated expressions (2.20) and (2.22), that take into account the interferences between the beams obtained by multiple reflexions on both faces, are required. The thickness of the film must have been deduced from optical or any other physical data. For a review of optical measurements see VERNIER [1973]. If the escape probability is assumed to have the form (2.35), the measurement of one photoelectric yield Y gives one equation between Po, po,L
280
PHOTOEMISSION
[VI,
53
and known quantities, such as the angle of incidence i, , the index of refraction v - iK, and the thickness zo,
Y
= J>z)/?,p,
exp
(<)
dz = Y(v,
IC, zo,
Do, p o , L).
(3.1)
The integration of (3.1) when D(z) is given by (2.20) or (2.22) was performed by COQUETand VERNIER[1966], VERNIER, GOUDONNET, CHABRIER and CQRNAZ [1971], PEPPER [1970]. 3.1. ESTIMATION OF THE ESCAPE DEPTH FROM ONE PHOTOYIELD
An estimate of L is possible with one equation (3.1) if &p0 can be deduced from theoretical considerations. Thus, SMITH and FISCHER [19711 obtained the mean free path of the unscattered photoelectrons produced in Cs and Rb by photons in the range 2-11 eV. They separated by energy analysis the unscattered electrons that are distributed in a quite narrow peak about the mean energy E- E, abovk the level E, of the bottom of the conduction band. They approximated the average transmission of the surface for electrons of energy E by assuming that every electron with a wave vector within the escape cone is transmitted, and found
1-cos 8, Po(J9 =
,
where 8, is given by (2.41) In the spectral range investigated by Smith and Fischer (2 < hv < 1 1.2 eV, 4 < E- E, < 13 eV), the mean free path is very small for Cs ( I 2~ 1.5 A). This value is in good agreement with qualitative estimates obtained by other methods (MAYER [1961], VERNIER, M. PAUTY and F. PAUTY [1969]). The theoretical predictions of QUINN[1962] and THOMAS [1957], based on deexcitation by plasmons and by hole-electron pair creation, give values of the mean free path somewhat larger, but also very small (Fig. 2). For such small values of the mean free path surface phenomena could be very important but the estimate of Smith and Fischer provides a very useful order of magnitude. We may note that the escape depth L of the photoelectrons might be appreciably larger than the mean free path I for the largest photon energy when scattered electrons can escape. KROLIKOWSKI and SPICER[1969,1970] estimated the mean free path of photoelectrons in Cu (I = 22 A for electrons 8.6 eV above Fermi level) and in Au (Fig. 1) by a less direct method that is based on the same principle. We have seen (section 2.6.1) how the variations of the mean free path I(E) of an electron
VI,
9 31
T H E E S C A P E D E P T H OF THE P H O T O E L E C T R O N S
28 1
with its energy E is related to the density of states (ODS). We also have seen (section 2.7.1) that the observed photoelectron energy distribution (PED) depend on both 1(E) and ODS. Krolikowski and'spicer obtained the energy dependence of the mean free path within a constant factor, by fitting it with all PED data. They obtained the constant factor from one absolute yield for hv = 8.6 eV by assuming the escape conditions described in section 2.7.1. GESELL and ARAKACVA [I9711 deduced the attenuation length of unscattered photoelectrons in A1 and Mg by two methods similar to that of Smith et al. and that of Krolikowski et al. respectively. They also used the variation of the yield with the angle of incidence of light (see section 3.5). They found a steep decrease of the attenuation length from several hundreds of Bngstrom for electrons of 5.6 eV above the Fermi level to a few Angstrom for electrons of 10.2 eV. This decrease is much sharper than the one caland ASHLEY [1965]. culated by the theory of RITCHIE Even if the scattered electrons are eliminated by energy analysis, the models used to estimate the transmittance of the surface for the excited electrons are very rough. Moreover, the correct calibration of a source of light especially in the ultra-violet range is quite delicate, Therefore this method of estimation of the escape depth is exposed to large systematic errors and more redundant data give much safer results. To obtain both p o and L we need two independent equations of type (3.1), i.e.,we must illuminate the sample in two ways such that the associated DAP are not proportional. Taking the ratio of the associated photoyield eliminates po and the absolute calibration of the light source is no longer necessary. This has most often been accomplished with several thin films of different thicknesses or with one thin film illuminated through the substrate (back illumination) and directly from vacuum front illumination. With particular materials and in definite spectral ranges the special yield distribution or the yield variation with the angle of incidence may give sufficient data. 3.2. THE ESTIMATION OF THE ESCAPE DEPTH FROM THE VARIATION OF THE RATIO OF FRONT Y+ TO BACK Y - YIELD VERSUS THE THICKNESS zo OF THIN FILMS
An elementary analysis of the DAP gives for large values of zo and for front illumination D+(z) = ( l - R + ) exp (-zz,a);
(3.3)
282
[v1, §
PHOTOEMISSION
3
for back illumination one obtains D-(z)
=
(1 -R-)exp (a(z-zo)).
(3.4)
Here R+ and R- are respectively the reflectances for front and back illumination. a = (4n;q)/L is the absorption coefficient, where q is given by (2.19); q = IC for normal incidence. Assuming R+ = R-,SZE,MOLLand SUGANO [1964] found that
+
Y+ 1 aL exp ( - az,) - exp ( - z,/L) X Y1 - a ~ l-exp(-(a+l/L)zo) '
01
'
///' / ""I
"' / -' '00
I
I
m
I
(3.5)
l -
VI, § 31
If L
-SK
T H E ESCAPE D E P T H OF THE PHOT OE L E CT RONS
283
l/a we may approximate (3.5) by
Plotting the experimental value of In( Y +/ Y -) versus zo gives, without any preliminary optical measurement, the value of tl from the slope of the asymptote. Its intercept on the x-axis gives L (Fig. 4). The validity of the expressions (3.5) and (3.6) may be seriously compromised when R - is large because the beam reflected on the front face of the film produces dear the free face of the film stationary waves, that cannot be neglected in the calculation of Y - even for large values of zo . Then the rigorous expressions (2.20) and (2.22) should be substituted for (3.3) and (3.4). (See the introduction to section 3.) Because of its simplicity, the original method, based upon the expression (3.6), has quickly become very popular. Gold has been often measured and can be taken as a reference because of its chemical inertness. The scatter in the results is high (Fig. 1) but the fit can be considered as good enough if we note that the structure of the film, the surface state and the work function may not be the same in every experiment. KATRICH and SARBEI [1961], SZE, MOLLand SUGANO[1964], VERNIERand COQUET[1965] measured the escape depth for photoelectrons emitted into vacuum by gold films with different work functions. CROWELL, SPITZER,HOWARTH and LABATE[1962], and SZE, MOLLand SUGANO [1964] were able to measure the escape depth for photon energy smaller than 1 eV by measuring the PE into a semi conductor (respectively Si and Gap). It would have been interesting to investigate the variation of the escape depth with the work function to test the deexcitation mechanism (in the ballistic approximation the escape depth should not depend on the work function). But the results are not sufficiently reliable. One result is clearly established in gold. The escape depth is found to decrease when the photon energy increases. This is in good agreement with the calculation of KROLIKOWSKI and SPICER [1969]. The escape depth obtained for other metals lies in the same range of magnitude as for gold, 20-80 A in the photon range 4 5 eV (HALLand MEE [1970] Fe, Mn, Ca) (BISNER, ROBOZand BARNA[1971] In). The method is probably not accurate enough to attribute the observed variations of L to differences between the metals. As for gold ( L = 740 & 60 A) Crowell et al. have found for photoemission from Ag, Pd and Cu into Si in the infra red spectral range (0.7-1.05 eV) values of the escape depth in the range of a few hundreds of Angstriims
284
PHOTOEMISSION
[VI,
5
3
-=
( L = 440k60 A for Ag, L = 170k30 A for Pd, 50 A < L 200 A for CU). The same method can be applied in the far ultra-violet as long as transparent substrates exist i.e. down to the LiF transparency cut off at 11.2 eV. PEISNER, QUEMERAIS, PRIOL and ROBIN[I9731 applied it to amorphous films of Ge (Fig. 4). 3.3. ESTIMATION OF THE ESCAPE DEPTH FROM THE VARIATION OF THE PHOTOYIELD OF THIN FILMS WITH THICKNESS
The pioneering works of MAYER [1961] and of his school, of H. THOMAS [1966] and of REPENBRING [1966] have demonstrated the volume character of PE from the variations of the photoyield of thin films with thickness. They gave a first order of magnitude of L (a few tens of ingstrom in Na, K and Rb, a few ingstrom in Cs). VERNIER, COQUET and BIGUEURE [1966] were able to fit the result of the calculation from (3.1) on the measured variation of the photoyield of gold films by a proper choice of the escape depth L (Fig. 5). The best fit was obtained for the same value of L for back and front yield, sand p polarization. The fit was good up to the thinnest
'h
I
m
I
4 x .
zo(N
5
Fig. 5. Variation of the photoyield of a thin film of gold with its thickness, as calculated by [1966] for 1 = 2804 A, i, = 65" and direct illumination polarization. COQUET and VERNIER The circles represent the experimental data after VERNIER, COQUET and BIGUEURE [1966]. The photoyield of a film of infinite thickness has been taken as unity.
VI,
P
31
T H E E S C A P E D E P T H OF T H E PHOTOELECTRONS
285
films that could be obtained with a continuous structure. The values L confirmed the earlier results of VERNIER and COQUET [1965] by the simple method (3.2). MAYER, BLANARU and STEPHEN [I9701 obtained in a similar way the escape depth in Rb (Fig. 6). We may note in their results the quite high values of the escape depth for small photon energy and the steep decrease of L when hv becomes larger than the plasmon energy.
Fig. 6. Escape depth of the photoelectrons in Rb as a function of photon energy after MAYER, BLANARU and STEPHEN [1970].
The fit between the theoretical and experimental curves that represent Y versus film thickness is not always good. It cannot be good if the structure or the work function of the film depends on its thickness (SHU'LMAN, STRAKOVSKAYA and NEMCHENOK [1969], GAUDART [19733). PONGet al. [1966,1967,1970,1972] extended the method into the far ultra-violet. They measured the photoyield of films of increasing thicknesses with illumination through the LiF substrate. For insulators CuBr (PONG[1966]), KBr (PONG[1967]) he first deposited a transparent film of gold. Pong found L = 180 A in KBr, L = 30 10 A in PbTe and 51 A
286
PHOTOEMISSION
[VL
5
3
for CuBr for 8 eV < hv c 11 eV. We may note that scattering by electronhole pair creation may be more important in CuBr than KBr where the band gap is quite important. PONG,SUMIDAand Moo- [1970] found L = 40+ 10 A for gold and L = 230 A for A1 in the spectral range 5.5-10.2 eV. The value for A1 is in good agreement with the theory of STUART and WOOTEN [19673 if the mean free path for electron interaction is Z, = 500 A and the mean free path for phonon interaction ,Z = 130 A. For gold the value of L seems quite large when compared with the results of VERNIER, COQUET and BIGUEURE [1966] and of PIERCEand SIEGMANN [1974]. 3.4. ESTIMATION OF THE ESCAPE DEPTH FROM THE BACK AND FRONT PHOTOYIELDS OF ONE THIN FILM
In the previous methods L has been calculated from data measured on several films that were assumed identical in every respect (structure, surface state) except in thickness. Such a set of films may be difficult to produce, especially from the high yield compounds used for industrial photocathodes. Especially in that case it may be safest to deduce the index of refraction from data measured on the film itself and not taken from the literature. BURTON[19471 measured, in thin films of SbCs,, the transmittance, the reflectance and the front and back yields for normal incidence. He deduced from these data the absorption coefficient and gave for the escape depth L = 250 A. HIRSCHBERG and DEUTSCHER [1968] measured the same data and in addition the thickness by the Tolansky method. They deduced the complex index of refraction from optical measurements. They analyzed the photoyields for front and back illumination on the basis of expression (1. l), but described the escape probability by (2.60). They found T$ = 150 A in the photon energy range 2-3 eV. This means that in the assumption of an escape depth given by the usual expression (2.35), they would have found L N Ti. HOFMANN and DEUTSCHER {1970] analysed multialkali layers in a quite similar manner and found T$ = 300 A in the spectral range 1.8-2.4 eV. We note the large value of the escape depth found in these semi-conductors with respect to the low value generally found in metals. VERNIER, GOUDONNET, CHABRIER and CORNAZ [1971] measured in thin films of amorphous Ge transmittance, the reflectance front and back yield for several angles of incidence and for both polarizations. They deduced the index of refraction and the thickness of each film from the transmittances and reflectances. For every wavelength, all the optical data
"1,
§
31
287
T H E E S C A P E D E P T H OF T H E P H O T O E L E C T R O N S
were consistent with one value of the thickness for each film. The index of refraction did not depend on the film thickness zo when zo > 80 A. It can therefore be assumed that the films were homogeneous and reproducible. They deduced one value of the escape depth L for each angle of incidence and for each polarization by comparing the observed value of Y + / Y with the result of calculation from the substitution of (2.20) or (2.22) into (3.1). They have found L = 4Of. 10 A for every angle of incidence and polarization in the photon energy range 4-5 eV. GOUDONNET, TRUITARD and VERNIER[1973] applied the same method to silver in the vicinity of the plasma resonance. For clean films the threshold is at hv, N 4 eV and all data in the range 4-4.5 eV are consistent with an escape depth L = 40& 10 A (Fig. 7). When a silver film is actived by a submonolayer of Cs, the escape depth is not changed for hv > 4 eV. The photoelectric
I
3;
I
3.6
, I 3.7 i.0
I
,
39
4
I
!+I 42 43
>,4 45 RKXCN E r€ml a / ]
>
Fig. 7. Estimated values of the escape depth of photoelectrons from the photoyield of a 280 b; thick film of Ag as a function of photon energy, for normal incidence 0 ; for s-polarization and an angle of incidence of 30" x and 60" 0 ;for p-polarizationand an angle of incidence of 30" and 60" A.
+
threshold falls below hvb = 3 eV, but for hv < 4 eV, L seems to depend on the polarization of light. We may try to attribute this anomaly to an anisotropy of the photoelectric excitation that involves a real dependence of L on polarization, but we shall discuss in section 4.2 interpretations based on a surface effect.
288
PHOTOEMISSION
CVI.
§ 3
CHABRIER, GOUDONNET, DOLIZY, ESCHARD and VERNIER[19731 adapted the same method to thin films of SbNa,K(Cs), deposited into vacuum sealed cells of appropriate form, by taking into account the additional reflexions due to the windows. In the spectral range 1.5-3 eV, they found results consistent in order of magnitude with HOFMANN and DEUTSCHER [19701 ( L = 300 A). They did not find any variation of L with polarization but found that L depended considerably on the samples that are quite difficult to obtain in a reproducible manner. 3.5. ESTIMATION OF THE ESCAPE DEF'TH FROM THE VARIATION OF THE PHOTOYIELD WITH THE ANGLE OF INCIDENCE
This could be a satisfactory method for bulk material and in the far ultra-violet for thin films when the most general methods cannot be used. The ratio of the photoyield Y(i,) for the angle of incidence i, and Y(0) may be expressed as a function p,,(L), with the expression (3.1). The relative uncertainty A p l p on the experimental estimate of p is determined by the apparatus. At best we may hope that A p / p is a few percent, but at high angles of incidence A p / p would be much larger. An estimate of L is possible if p( co)- p ( 0 ) is larger than Ap. QUEMERAIS, PEISNER,F ~ O L and ROBIN[1973] calculated p,(L) for bulk materials and for thin films. They noted that pio(L)depends more strongly on the state of polarization of the light than on L. Therefore, any uncertainty in the polarization induces an error Ap larger than p ( c o ) - p(O), that makes the determination of L impossible. GOUDONNET, CHABRIER, VERNIER [1974] calculated ( p ( a)- p(O))/p(O) and found it larger than any obtainable value of A p / p except when both the imaginary and the real parts of the index of refraction are very small (typically smaller than 0.1-0.3). This occurs only in a few solids, and in a very narrow spectral range in the far ultra-violet. GESELLand ~ A K A W A[1971], h A K A W A , BRAUNDMEIER, WILLIAMS, HAMMand BIRKHOFF [1974] measured the variation of the photoelectric yield Y of films with the angle of incidence i, in the far ultra-violet. The state of polarization of the light had been measured for each wavelength. They sought for an estimate of the index of refraction v - ilc, the thickness of the film zo and the escape depth of the electrons L by fitting the experimental Y versus i, curve with the theoretical expression (3.1). Gesell and Arakawa were able to place L between 4A and 12 A for A1 at hv = 21.2 eV. ARAKAWA, BRAUNDMEIER, WILLIAMS, HAMMand BIRKHOFF [19741 obtained a good estimate of v and ic for carbon films in the photon energy range 15-80 eV but they obtained an estimate of L only for hv > 30 eV in agreement with the results of GOUDONNET and CHABRIER [1974].
w § 31
T H E ESCAPE D E P T H O F T H E PHOTOELECTRONS
289
An electromagneticwave can also be induced in a film by frustrated total reflection, when a light beam is sent from the substrate at an angle of incidence i, larger than the limit angle. The reflectance has a very profound dip for a critical angle of incidence i2pand for p-polarization, when surface plasmons in the film can be coupled to the incident light (OTTO[1968,19701). The dip in the reflectance is associated with a peak in the curve that represents the photoyield Y - ( i 2 )versus the angle i , . MACEK,OTTOand STEINMANN [19721 observed this peak for hv = 5 eV and A1 films deposited and ARAKAWA [I9751 observed this peak onto a quartz prism. CALLCOTT for 5 eV < kv c 10 eV with A1 films deposited onto a LiF semicylinder. They calculated the ratio of the photoyields Y-(i2)/Y-(O) from the expression (3.1) in the Pepper's formulation for different values of L (Fig. 8) and observed that the peak value Y - ( i z P ) / Y - ( Owas ) very sensitive to the value of L. The distributions of the electromagnetic energy D(z) that has been calculated by Macek et al. and by Callcott et al. is very different for i, = i2p and for other angles of incidence. Therefore the comparison of the experi-
'2
Fig. 8. Comparison of experimental values of the angular yield ratio Y - ( i 2 ) / Y - ( 0 ) with values calculated from the expression (3.1). No possible choice of film thickness zo and escape and depth L can account for the yield observed at the resonance peak (after CALLCOTT ARAKAWA [1975]).
290
PHOTOEMISSION
[VI,
§ 3
mental result with the calculation gives an estimate of L. Macek et al. found L = 5 0 k 8 A for hv = 5 eV. Callcott and Arakawa also measured the photoyield Y + ( i , ) for light incident from vacuum at the angle of incidence i,. They obtained from the ratio Y-(O)/Y+(O)(section 3.4), the estimate L = 40+ 10 for hv = 5 eV and L = 15 + 5 A for hv = 10 eV. But the measured value of the ratio Y - ( i2J Y - ( O ) was larger than the result of the calculation, whatever be the assumed values of L and zo. Like in the experiments of CHABRIER, CORNAZ, GOUDONNET and VERNIER [1970], all values of the photoyield cannot be explained within the schema of a pure volume effect with one value of the escape depth. Here too an explanation can be found in a contribution of surface effect (cf. section 4.2).
3.6. DETERMINATION OF THE ESCAPE DEPTH FROM THE PE OF A SUBSTRATE THROUGH A COATING LAYER
The principle of such methods is to observe the transparency of a thin film for the electrons excited in its photoemissive substrate. This transparency is proportional to e-zo/L,where zo is the films thickness and L is the escape depth of the film. LEWOWSKI, BASTIE and BIZOUAFCD [1970] deposited thin films of alkali halides onto photocathodes of Al. For an illumination at hv = 4.89 eV the films were transparent and no absorption occurred in them. Plotting In Y against the thickness of the film, they obtained a straight line with slope l / L . The values of L are roughly proportional to the volume of the crystalline cell when one alkali halide is replaced by another, L = 10 A for NaCl, L = 110 A for CsI. These values are much lower than the result obtained by PONG[1967] in KBr for hv = 8 eV ( L = 180 A). Differences in the transport mechanism could explain this result. In Pong experiments the electron-hole pair creation may be forbidden in KBr, because the electron affinity is smaller than the band gap. The escape depth limited by phonon scattering could then be quite high (SPICER[1960]). For Lewowski et al. the electrons excited at lower levels might have to jump from trap to trap. PONG[1972] illuminated a photocathode of CuI through its LiF substrate. He covered it with a thin film of amorphous Se and plotted the logarithm of the emitted current versus the Se-film thickness zo . For small values of zo the contribution of the Se film to photoexcitation was negligible and the curve was a straight line with slope l/L. The measured escape
V
L
31 ~
THE ESCAPE D E P T H O F THE PHOTOELECTRONS
29 1
depth of the electron in Se was L = 40+ 10 A for hv = 7.8 eV; the mean energy of the emitted electrons was then 4.7 eV above the Fermi level. PONGand SMITH [1973] replaced the selenium by copper phtalocyanin and found then L = 11 A for electrons 1.5 eV above the Fermi level (for hv = 7.8 ev). When a film and its substrate both contribute to the photoexcitation, the contributions of the film and the substrate sometimes can be separated [1970], deposited by an energy analysis of the photoelectrons. EASTMAN thin films of yttrium onto a substrate of gold and measured the variation of the integrated intensity Z of the d peak of gold as a function of the Y film thickness. The experimental data could be fitted with the law
with L' = lOA in the photon energy range 5-8 eV. The same method can be applied to thin films of any substance, if its PED has no structure in the d band of gold. By substituting Gd or Ni for Y , Eastman obtained the same value L' = lOA for hv = 7 eV. We must note that such a method gives the elastic escape depth L'; the escape probability without energy loss is proportional to exp (- z/L'). L' may be well approximated by the mean free path 1. But the escape depth L for all emitted electrons, as defined by (1.1), may be much larger, especially for high energy electrons excited by soft X-rays or by electron bombardment. Applications of the Eastman's method to this energy range will be discussed in section 3.10. CAMPAGNA, PIERCE,SATTLER and SIEGMANN [1973] measured the spin polarization P of the photoelectrons emitted by a ferromagnetic material placed in homogeneous magnetic field normal to its surface, (3.7) where nt and nl are the respective numbers of spin up and spin down photoelectrons. Of course if copper is substituted for the magnetic material then P = 0. Many important results about magnetism have been obtained from such measurements of P.We are concerned here with the possibility of separating the electrons originating from a ferromagnetic and a nonmagnetic material. PIERCE and SIEGMANN [1974] deposited thin films of copper of increasing thickness zo on a substrate of Ni and measured the variation of P with z,,. For a uniformly magnetized film and a constant photoelectric current, P was found to be proportional to the number of electrons excited in Ni. Pierce and Siegmann could thus obtain the trans-
292
PHOTOEMISSION
1
1
I
I
1
5
10
15
20
25
Cu THICKNESS (A)Fig. 9. Spin polarization P of the electrons emitted by an Ni sample coated with a film of Cu of thickness z., The rectangular fields represent the statistical uncertainties for both P and zo. Fields with the same cross hatching are for films successively evaporated on the same Ni and substrate. The solid curve is a least square fit by an exponential curve (after FIERCE SIEGMANN C1974-J).
parency exp( -z,/L) and the escape depth L for the Cu film (Fig. 9). Pierce and Siegmann deduced in a similar manner the escape depth in Ni from the variation of P for the electrons emitted by a substrate of Cu covered with a thin film.of Ni. Pierce and Siegmann found L = 1 1 A in Cu for electrons 5.2 eV above the Fermi level. 3.7. NON-PHOTOELECTRIC METHODS
In the experiments O f EASTMAN [1970], LEWOWSKI, BASTIE and BIZOUARD [1970] and PONG[1972], PONGand SMITH[1973] the emitted electrons
VI,
§ 31
THE ESCAPE DE P T H OF THE PHOTOELECTRONS
293
are excited in a material and deexcited in another. To investigate the escape probability any type of excitation can be substituted for photoexcitation. We must note, however, that the energetic and angular distribution of the excited electrons may not be the same as in PE. KANTER[1970] injected electrons into self-supporting films of Ag, Au and Ag with a low energy electron gun (Fig. 10). He measured the current transmitted by the film with a collector and he could eliminate, with a
0
target Film
F""" c o l k ~
Fig. 10. Principle of the apparatus of KANTER [1970]. The beam is collimated by a 0.02 Tesla magnetic induction. The beam diameter is about 1 mm. The beam could be moved, with the help of deflection plates, across the film surface so that the detection of pin holes and other film nonuniformities was greatly facilitated. The normal component ofthe beam energy spread, as measured at the collector without a target inserted, was 0.5 eV between the 10% and 90 % points of the collector current versus retardation voltage curve. Typical bombarding currents were 2 x 10-'A. Current leaving the film was in the 10-"-IO-'4A region and were measured with a vibrating reed electrometer. The noise current was about 10-14A.
retarding potential technique, the electrons that had suffered large energy losses. Kanter plotted the logarithm of the electron current versus the film thickness, and found a straight line of slope l/Lo. Here Lo is the elastic escape depth L', when the scattered electrons are eliminated, and the total escape depth L, when all transmitted electrons are collected. In this energy
294
[VI, § 3
PHOTOEMISSION
range, where L and L .' are nearly equal, we may expect the escape depth measured for electrons injected with a kinetic energy E, to be approximately the same as for photoelectrons excited by photons of energy hv = E, WF, where WF is the work function of the film. In Fig. 1 the results of Kanter are plotted for gold at that energy. Kanter found no significant difference between the values of the escape depth for Al, Au and Ag. He found a decrease from L = 4OA for an electron energy of E = 5 4 e V above the Fermi level down to 15-20A for E = 10 eV. The experiments of Kanter determine the electron interaction processes more precisely than PE, because the angular and energy spreads of the electrons are smaller. The main difficulty is to avoid holes and other defects in the films. The injection of hot electrons into a thin film may be obtained in sandwiches of metal-insulator-metal thin films (Fig. 11). Such sandwiches emit electrons into vacuum when an appropriate voltage V is applied between the metal films. Electrons of the base metal film are transported across the insulator either by thermoinjection into the conduction band or by the tunnel effect. They are then transported across the outer metal film and are emitted into vacuum. We may assume the transport probabilities across the insulator and the outer metal film to be proportional to exp( - zI/L,)
+
outer metal film
base metal film
-+ii substmte
Fig. 11. Scheme of a sandwich electron emitter.
VI,0 31
T H E ESCAPE D E P T H O F T H E P H O T O E L E C T R O N S
295
and exp(-z,/L,), respectively, where z, and z, are the thickness of the insulator and the outer metal films, and L, and L , are the corresponding escape depths. Many investigators deduced L, and L, from the variation of the emitted current with z, and 2,. To compare the results with photoelectric estimates of the escape depth, we could assume that most electrons attain the outer metal film with an energy eV above the Fermi level. However, this assumption is not consistent with the observations by HICKMOIT [1963, 19651, VERDERBER and SIMMONS [19673, NIQUET, VERNIER and HARTMANN (1970) of emitted electrons when V is smaller than the work function of the outer metal. Moreover, the small values generally found for L, (5 to 24 A in Al,O, according to SAVOYE and ANDERSON [19673, NIQUET,VERNIERand HARTMANN [1970], COLLINS and DAVIES[1963], KANTERand FEIBELMAN [1962]) suggest very strong energy losses in the insulator. Only a few authors have found larger values of L, (HICKMOTT [1965]: Ll =200 A in Al,O,, GOULD,HOGARTHand COLLINS [1973]: L, = 400 to lo00 A in SiO,). The average energy of the electrons injected in the metal might be much lower than one eV. Because the escape depth of the electrons is a decreasing function of their energy, we might expect that the attenuation length measured in the outer electrodes of the sandwiches is generally greater than that obtained from photoelectric data (MEAD[1962], COLLINS and GOULD[1971], SAVOYE and ANDERSON [1967], NIQUET,VERNIERand HARTMANN [19701, COLLINS, EDGEand LEGG[19721). 3.8. ESCAPE DEPTH IN NEGATIVE ELECTRON AFFINITY (NEA) PHOTOCATHODES
We have shown in section 2.7.2 that in NEA photocathodes, the escape depth is the diffusion length of thermalized electrons at the bottom of the conduction band. A restriction has been made on the validity of the expression (2.35) in thin films, whose thickness is not much greater than the diffusion length, because electrons can be diffused back by the rear face of the film. In that case, (2.59) should be substituted for (2.35). Several non-photoelectric methods can be used to determine the diffusion length (HACKETT[1972], FRANKand GARBE[1973], ASHLEY, CARR,ROMANO-MORAN [1973]). The published values range from 0.1 p to 10 p. The special features of the transport process and of optical excitation in NEA emitters allow special photoelectric methods to determine the escape depth of the photoelectrons. In GaAs and other 3-5 compounds, the conduction band contains two minimum energy levels at the points r and X of the Brillouin zone (Fig. 12).
296
PHOTOEMISSION
Fig. 12. Band scheme of GaAs, showing the excitation and thermalization of electrons in for I .4 ihv 1.7 eV (a) and for hv > 1.75 eV (b and c) (after JAMES and MOLL[19691).
r and X minima of the conduction band,
-=
For 1.4 eV < hv < 1.7 eV all excited electrons are near the r level and, except for a small path before thermalization, they undergo the same transport process, characterized by the same escape probability
P
= Pr ~ X (P - z / L r ) .
(3.8)
The escape probability at the surface p r and the escape depth Lr are independent of the photon energy. The photoyield Y for 1.4 eV < hv < 1.7 eV depends on hv only because the absorption constant a depends on hv. We may write
The plot of 1/ Y versus 1/ a (Fig. 13), for that spectral range is a straight line that intersects the x axis at the abscissa - L,. Its slope is 1/pr . Except for very strong doping a results from direct interband transitions and does not depend on the sample. a can, therefore, be taken out of the literature. Many determinations of the spectral yield of GaAs and other 3-5 NEA
VI,8
31
THE ESCAPE DEPTH O F THE PHOTOELECTRONS
291
Fig. 13. 1/Y versus l/aplot for Zn doped GaAs crystal (density of Zn atoms p = 2.8 x 10'' [1969a]). ~ r n - ~with ) , different Cs, 0 coverages (after GARBE
Fig. 14. Diffusion length L, in GaAs as a function of the Zn doping concentration p and crystal growth process (after GARBE [1969a]).
298
PHOTOEMISSION
CVh§
3
photocathodes lead to determinations of L, and p r . EDEN,MOLLand SPICER[1967] found for GaAs L, = 0.15 p. GARBE[1969a] found that for Zn-doped GaAs L, depends on the doping and on the fabrication process (Fig. 14). JAMESand MOLL[1969] found, independently, quite similar results. SCHADE, NELSON and KRESSEL [1971] found L, = 2 to 7 p in slightly Ge doped crystals and ASHLEY,CARRand ROMANO-MORAN [1973] found by Hackett’s methods diffusion lengths up to 23 p. FRANK and GARBE[1973] showed that HACKETT’S method and photoelectric methods give consistent results. Many other 3-5 materials have been obtained with NEA, binary InP and Gap, ternary GaInAs and GaAsSb, quaternary InGaAsP (for a review see BELLand SPICER[1970], WILLIAMS and TIETJEN[1971], SOMMER [1973]). The NEA was first obtained in cleaved monocrystals, but it has been also obtained on epitaxic thin films, used as semi-transparent photocathodes (ANTYPAS, JAMESand UEBBING [1970], LIU, MOLLand SPICER[1970], GARBE[1973]). In a quite general way the diffusion length of the electron L, found in these materials is much smaller than in GaAs. It is generally about 0.1 p or less. The reason is very likely not fundamental and improvements in the fabrication process could probably increase L, . In GaAs, when hv > 1.7 eV a fraction F, of the excited electrons are thermalized in the vicinity of the X level. JAMESand MOLL[I19691 and GARBE[1969a] separated in the PED two contributions to the photoyield. The first one Y, is due to emission from r level, the second one Y, from X level. The lifetime of the electrons in the X level is limited by diffusion towards the r level. The transport of the X electrons is characterized by a diffusion length L, and an escape probability P, for the X electrons that reach the surface. From the joint diffusion equations for X and r electrons it has been found that
Y, =
PXF, 1 l/aL,
+
(3.1 1)
Plotting 1/ Y, versus l / a for hv > 1.7 eV gives L, just as L, was obtained for hv < 1.7 eV (Fig. 13). To obtain the other constants James Moll and Garbe fitted the experimental curve to the expression (3.10). They found in both works that L, z 0.03 p is much shorter than L,. Presently Si has been activated to NEA only on (100) crystallographic faces by Cs-0 treatment (MARTINELLI [1970], RICHARD [1973]), or R b O
VI,
§ 31
T H E ESCAPE D E P T H OF THE PHOTOELECTRONS
299
treatment (MARTINELLI [1973]). The emission process is roughly the same as in 3-5 materials. The electrons are excited by photon absorption and rapidly thermalized into the bottom of the conduction band, which is in Si at the X point of the Brillouin zone. An expression of type (3.9) can, therefore, be used to analyze the spectral yield of NEA Si in the same manner as NEA GaAs for hv between 1.4 eV and 1.7 eV, e.g. by plotting 1/Y versus l / a . The diffusion length, depending on the sample and its doping ranges between 2 and 18 pm. SOMMER [I9731 noted that the diffusion length in Si is usually larger than in 3-5 compounds but that the high thermoelectronic emission reduces the practical application of Si NEA photocathodes. We must note here that the very high escape depths and the very high quantum yields that results are made possible by the NEA where the escape depth is identical with the diffusion length of the thermalized electrons. Whenever the NEA is not achieved, the escape depth in the same materials drops to much smaller values e.g. for Si near threshold 10 to 30 A, as we shall see in the next paragraph. 3.9. ESTIMATION OF THE ESCAPE DEPTH FROM BAND BENDING CONSIDERATIONS
Because of surface states and because surface doping -is different from volume doping, the distance 6E from the valence band to the Fermi level depends on the distance z to the surface (Figs. 15 and 16). This effect makes it possible to obtain NEA. When the electron affinity is positive VANLAAR and SCHEER [1962] calculated the spectral yield distribution on the assump-
hS; (4 WKh band
€3
--____
Ferrni level
Lbkcce band
-Cz
0
Fig. 15. Scheme of the band profile of a p-type Si crystal with positive electron affinity.
300
PHOTOEMISSION
Fig. 16. Band scheme just after cleavage of the degenerate n-Si cristal (density of donors 10’’ cm-’) for which WAGNER and SPICER [I9721 measured the PED represented in Fig. 17.
tion that each layer of thickness dz contributes to the emitted current the amount d l a (hv - E , - 6E(z))4exp (- z/L) dz.
(3.12)
They calculated the profile of the band E ( z ) by solving Poisson equation, and integrated dZ. For p-type silicon the result of the calculation depends sufficiently strongly on L to allow an L determination by comparison with the experimental data. Van Laar and Scheer have found L = 15 A in the spectral range 5-6 eV. GOBELIand ALLEN[1962] distinguished in the contribution of each layer a term associated with indirect excitation with a threshold hvi(z) = E0+6E(z) and a term associated with direct excitation with a threshold
VI, 0
31
301
T H E E S C A P E D E P T H OF T H E P H O T O E L E C T R O N S
hvd(z)= hv,(z)+ 0.3 eV. They assumed for each layer a contribution
s
dY = (c,(hv-hv,(z))~+c,(hv-hv,(z)) exp (-z/L)dz
(3.13)
to the photoyield. They calculated hvi(z),integrated over z, and compared the result with the experimental result of Scheer and Van Laar. They found L = 25 i-5 A in the same energy range. WAGNER and SPICER[19721 investigated just cleaved degenerate n-type samples of Si (10" As atoms per cm3, 0.001 SZacm) for hv in the range 6-12 eV. After an exposure to oxygen at very low pressure (lo-'' torr), they observed several changes in the PED N(E) (Fig. 17), viz.
-6
-5
-4 -3 -2 INIT~ALELECTRON ENERGY lev)
-1
0
Fig. 17. PED for n-Si just after cleavage (1) in ultrahigh vacuum and after exposure to residual gases, (2) for hv = 10.2 eV (after WAGNERand SPICER [1972]).
a. The structures A and B disappeared. The peak B was not observed in every cleave. Wagner and Spier attributed these structures to surface states lying at 0.5 eV and 1.1 eV under Fermi level. We shall discuss surface states in the next section. b. The low energy edge of the PED is shifted by 0.6 eV because of a decrease of the work function of 0.6 eV.
302
PHOTOEMISSION
[VI,
§ 3
c. The two structures C and D attributed to direct transitions from the valence band, are shifted by only 0.2 eV. Wagner and Spicer interpreted the results of Fig. 17 by the band bending represented on Fig. 16. On clean n-type Si the free bonds at the surface induce a space charge and a negative band bending. When the cleavage has been exposed to oxygen the bonds are saturated and the band becomes straight; the distance from the vacuum level to the Fermi level is then decreased by 0.6eV. Because of the very short escape depth, the electrons excited in the flat band region cannot attain the surface. Wagner and Spicer reported the observed shift of 0.2 eV for the structures C and D on the band profile calculated by GOBELI and ALLEN[1962] and found L = 12 A for h = 10 eV. This value is consistent with the value of 25 A that had been found by Gobeli and Allen near threshold. VILJOEN, JAZZAR and FISCHER [I19721 modulated the band bending of n and p-type GaSb with infra-red illumination. The photoelectric threshold for clean GaSb samples was about 5 eV. They measured the variation of the infra-red modulation of the photoyield with the UV photon energy and deduced from it an estimate of L in GaSb. They found L = 100 A, larger than the value expected from comparison with similar semi-conductors.
3.10. ESTIMATION OF THE ELASTIC ESCAPE DEPTH FOR HIGH ENERGY ELECTRONS
METZGER[1965] and DUCKETTand METZGER[1965] measured the spectral distribution of the photoyield of the alkalihalides and found a drastic decrease of the photoyield when the photon energy hv becomes larger than twice the gap Eg i.e. when excited electrons can induce pair creations. If hv is further increased (above approximately 3 4 ) the photoyield increases again and may become larger than unity because of the emission of scattered and secondary electrons. The importance of pair scattering and secondary emission is better seen in the PED (BLECHSCHMIDT, SKIBOWSKI and STEINMANN [1970]). For hv > 13 eV the peak due to valence band decreases and for hv > 14 eV a peak due to scattered electrons appears (Fig. IS). When scattered electrons can be emitted, the elastic escape depth L' is more significantly related to microscopic processes than to the total escape depth defined by the expression (1.1). BAERand LAPEYRE [1973] deduced L' from the energy distribution of photoelectrons emitted by IK in the photon range 7-22 eV. They assumed
T H E ESCAPE D E P T H OF THE PHOTOELECTRONS
SCATTEED ELECTFDNS
303
VALENCE
I
PHOTOELECTRoN AND PHcrroN ENERGY Fig. 18. Energy distribution of photoelectrons emitted by IK for various photon energy. Each curve is indexed by the photon energy in eV (after BLECHSCHMIDT, SKIBOWSKI and STEINMANN
[1970]).
that the intensity of the elastic peak Z, from the valence band is given by (3.14)
where c1 is the optical absorption coefficient for a photon of energy hv and L'(hv)is the elastic escape depth for electrons excited at a level hv above the valence band. They assumed that the hv dependence of L' could be and SPICER[1969]). They calculated by the expression (2.37) (KROLIKOWSKI could then fit the observed variation of the intensity of the elastic peak with the result of calculation based on eq. (3.14). They found that L' = 16A for hv = 14 eV. The electron-spectroscopy of photoelectrons excited by soft X-rays has been applied to chemical analysis (ESCA). Characteristic peaks of each element are found in the PED at hv - Eb, where Eb is the binding energy of a core electron. BAER,HEDEN,HEDMAN,KLASSON and NORDLING [19701 measured the variation with gold thickness zo of the chromium 3p peak (Eb= 43 eV) and of the gold 4 f 3 peak (Eb= 83 eV) in energy distribution of the electrons emitted by a chromium substrate covered by a gold film (Fig. 19). When zo increases, the chromium peak is expected to decrease as - zo/L',
(3.15)
304
PHOTOEMISSION
[VI. §
3
and symmetrically the gold peak is expected to increase as 1 -e-Zo/L'.
(3.16)
We may assume that the elastic escape depth L' in gold has the same value for both chromium and gold peaks because the binding energy E,, is in both cases small with respect to the photon energy. The theoretical curve agrees with experimental data for L' = 2 2 k 4 8,and hv = 1253 eV (Fig. 19).
Fig. 19. Variation with gold thickness of the intensity of the gold x and chromium 0 peaks for photoelectrons emitted by a chromium substrate. covered by a gold thin film illuminated by 1252 eV photons. The continuous curves are calculated for L.' = 22A (after BAER.HEDEN. HEDMAN, KLASSONand NORDLING [1970]).
X-rays also induce the emission of Auger electrons with an energy independent of hv and characteristic of the chemical elements because of the HEDMAN, BERNDTSSON, deep holes that they create in the solids. KLASSON, NILSSON, NORDLING and MELNIK[1972] used both Auger electrons and directly photoexcited electron peaks, to determine the escape depth of electrons in gold and alumina. They found in gold L' = 19 8, for E = 0.9 k e V a n d L f = 3 7 8 , f o r E = 1.4keV,inaluminaL'= 1 3 8 , f o r E = 1.4keV and L' = 22 8, for E = 3.9 keV. STEINHARDT, HUDISand F'ERLMAN [19721 investigated L' in a thin film of carbon dtposited onto a gold substrate. They found L' = 10 8, for E = 920 eV and L' = 13 8, for E = 1169 eV. The same method applies when Auger emission is induced by electron bombardment. We may assume a uniform excitation in the ?art of the film and substrate that are responsible of the Auger emission, because the incident electrons have a penetration depth greater than L' both because L' increases with electron energy and because secondary electrons can also
Q
"1.
41
305
S U R F A C E PHOTOEXCITATION
induce Auger emission. SEAH[1972] discussed the validity of eqs. (3.15) and (3.16) and the significance of L' for description of the intensities of Auger peaks induced in a substrate coated with films of different thicknesses. PALMBERG and RHODIN [19681 deposited Ag epitaxial films on a monocrystal of Au. They found for the elastic escape depth in Ag L' = 4 8, for E = 72 eV (Au peak) and L' = 8 8, for E = 362 eV (Ag peak). RIDGWAY and HANEMANN [1971] deposited Fe onto a Si monocrystal and found L' = 7.4 8, for the escape depth of 91 eV electrons in Fe. SEAH[I9721 investigated L' in films of Be, Ag and Cu deposited onto substrates of Be, Ag and Cu. The estimate of L', obtained from the increase of the layer peak and from the decrease of the substrate peak as the film thickness increases, are of the same order of magnitude but the second ones seemed less reliable. They are given in parenthesis in the Table 3.1. In the same table we have given the results obtained by TARNGand WEHNER[1973] with Auger electron emitted by W substrates covered with Mo and Mo substrates covered with W. JACOBIand H ~ L Z[1971] L measured the Auger emission of self supporting carbon films both for transmission and reflection of primary electrons with energy ranging from 100 to 2500 eV. They found for the unscattered escape depth L' = 7.5 & 1.5 8, for electron emitted at 262 eV. This value is in good agreement with the results of STEINHARDT, HUDISand PERLMAN [1972]. We abstracted in Table 3.1 the results obtained for the elastic escape depth of high energy electrons. 'TABLE 3.1 Elastic escape depth of electrons (in d;) Energy(eV)-+ 48 Material
1
Be
60
110
120 262
4.7 (8.6)
935
10.0
12.7
1169 1400 1736 3900
13
'412 0 3
7.5
C
cu Ag Au Mo W
350 355 920
10
22
13
(6.1) 9.4 5.2 (7.2)
6.7 6
0 4. Surface Photoexcitation A surface contribution to photon absorption may arise, within a
306
PHOTOEMISSION
CVL
P4
one-electron theory, from two distinct effects, surface PE from bulk states (SPBS) and PE from surface states. We shall analyse experimental data on the assumption that surface and volume effects can be separated in the intensity of the emitted current. When necessary, we shall use the oneelectron approximation. The surface photoexcitation is more likely detected by photoemission than by any other of its physical consequences (e.g., reflectance or transmittance studies), because the small value of the escape depth reduces the relative importance of bulk phenomena with respect to surface ones. In spite of that, and even for quite small values of the escape depth, most analyses of PE data have been based on the assumption of a pure volume photoexcitation and calculated from the bulk value of the dielectric constant. The consistency of the calculated escape depths with all experimental data is an evidence of the quite small value of the surface effect with respect to the volume one. This result must be especially noted because in early theoretical work on PE from metals, the free-electron model led theorists to consider only the surface effect. 4.1. DETECTION O F A SURFACE EFFECT FROM THE THICKNESS DEPENDENCE O F THE PHOTOY IELD OF THIN FILMS
This first photoyield-versus-thickness curves for thin films of alkali metals obtained by Mayer and his school (see the review by THOMAS [1966]) are of historical impbrtance because they showed that the volume effect represents at least the main part of PE. An additive surface absorption term could be detected in the photoyield versus thickness if the film could be produced in the bulk structure and bounded by two parallel planes for the smallest thicknesses. In practice several monolayers at least are necessary to approach this model. Up to quite thick films, the roughness and work function may depend on the thickness and often on other uncontrolled quantities. An agreement between the thickness dependences of the photoyield, as it is measured and as it is calculated with the assumption of a volume effect by means of the expression (3.1), can be a negative test for surface effect. We must note that it is sensitive only if the thinnest “good films” have a few monolayers and it is difficult to be sure of the quality of such films. VERNIER, COQUET,BIGUEURE [1966) and MAYER,BLANARU and STEPHEN [1970] have found no important contribution of the surface effect in gold an Rb respectively. In Cs theoretical (QUINN[1962]) and experimental (VERNIER, M. PAUTY, F. PAUTY[1969]), SMITHand FISCHER [1971]) considerations have shown
VI, 0
41
SURFACE PHOTOEXCITATION
307
that the escape depth of photoelectrons is very small, at most a few monolayers. This fact would make the surface effect easier to detect if it exists. The steep increase of photoyield at zero thickness, observed by MAYER [1961], is consistent both with volume and surface photoexcitation and we shall need further experimental tests to decide whether the mechanism of the absorption, that can lead to photoemission only in the surface layers, is different from bulk absorption. 4.2. DETECTION OF A SURFACE EFFECT FROM .THE POLARIZATION DEPEN-
DENCE OF THE PHOTOYIELD
As predicted by the earliest theories (section 2.5.1), in a surface effect due to the surface perturbation of bulk states (SPBS) the only active component of the electric field is normal to the surface. Therefore in a pure SPBS the ratio Yp/Ysof the photoyields for p and s polarizations should be infinite for a perfectly smooth surface. For the assumption of a pure volume effect in a semi-infinite solid or in an opaque film, Y p / Y ,should equal the ratio (1 - Rp)/( 1 -R,)of the fluxes that penetrates into the solid, where R, and R, are the reflectances for the p and s polarizations. In a thin film, the volume effect assumption leads to expressions of Yp/Y, that differs from (1 - Rp)/( 1 - R,) because of interferences and can be deduced from eq. (3.1). For very small escape depths it is sufficient to consider the ratio of the density of electromagnetic energy just below the surface for p and s polarizations. The equality of Y p / Y swith the value, that is calculated in the assumption of a pure volume effect ((1 - RJ( 1 - R,)for opaque films), affords a good negative test for surface effect. No surface effect was detected by the polarization effect in experiments of IVESand BRIGG[1936,1938], MAYERand THOMAS [1957], METHFE~SEL [1957] in alkali metals and by VERNIER, PAUTYand BERTHELEMY [1965] in gold, silver and aluminium. Only recently anomalies have been found that could not be explained by the standard theory of the volume effect. The detection of a surface contribution could be expected if one or several of the following conditions is satisfied : (a). If the electron model is a good approximation because it predicts no volume effect, e.g., for Al, Mg, alkali metals, (b). If the volume effect requires the violation of the k conservation selection rule. This can occur near the threshold, (c). .If a special distribution of the electro-magnetic field concentrates the energy in the form of an electric field normal to the surface. This occurs near surface plasma resonance for p-polarization (Fig. 20) (CHABRIER,
308
PHOTOEMISSION
[VI,
D4.
GOUDONNET, TRUITARD and VERNIER[19731, CALLCOTT and ARAKAWA [1975]). Surface roughness may help this concentration (see section 2.4.2). This resonance can be described (ENDRIZ[1973]) as an additional step in the interaction of an incident photon with the solid; the photon excites a surface plasmon that subsequently decays into a one-electron excitation (or a reflected photon). The distinction between surface and volume photoexcitations is based on the same principle at plasma resonance and at other photon energies. We must know whether the electric field, whether or not coupled with plasma oscillations, excites photoelectric transition, because of surface perturbation or because of standard bulk processes. The plasma resonance of silver is especially popular because it is in the nearest ultra-violet and the chemical reactivity is much smaller in silver than in alkali metals. The only drawback of silver is that the work function of the clean metal is larger than the plasmon energy. Therefore PE experiments at plasma frequency need a surface coating with a sub-monolayer of an electropositive metal (Cs, Ba, or Al) or compound (Cs,O, CsF or BaO). For either volume or surface effect the photoyield Y , and the ratio Y,/ Y, have a peak at the plasma frequency just as do the absorption coefficient A , and the ratio of absorption coefficients APIA,. But the amplitude of the peak is not consistent with an isotropic excitation in the volume of silver (HOFMANN and STEXNMANN [1968]). FONTENEAU, PAUTYand VERNIER[I9711 have found Yp/A, = YJA, for a light frequency larger than the plasma resonance but Y,/A, < YJA, in the region of plasma resonance. They also observed this anomaly when the light frequency is smaller than plasma resonance in the thickest films. A contribution of the SPBS to PE would produce a discrepancy of the opposite sign so that the explanation should probably be sought in the direction of PE from surface states localized in the activating layer (aluminium oxidi%d by residual gases). The experiments of CHABRIER, GOUDONNET, TRUITARD and VERNIER [I9731 quoted in section 3.4, which led to a determination of the escape depth, afford a more complete set of data to investigate this anomaly of silver. For light frequency above the plasma resonance the consistency of the values of escape depth for p and s polarizations is equivalent to the result of Fonteneau et al., Y,/A, = Y J A , . The results described by Chabrier et al. as a larger escape depth for p polarization implies the opposite of Fonteneau et al., Yp/Ap > YJA, . Here the sign of the anomaly is consistent with a SPBS. This interpretation is confirmed by the appearance of the anomaly at plasma frequency where the electromagnetic energy is concentrated in the normal component of the electric field (condition (c)) (Fig. 20). The anomaly also appeared at light frequencies smaller than the
309
3500
<
35
3400
I
I
36
3,7
33ao
3220
3 m
3 m
I
38
3,9
4
4)
52
XoO
I
ArA)
I
,
,
4.3 h3leVl
Fig. 20. Variation of the density of electrostatic energy per incident light power, near the surface associated with 3 components of the electric field in a 280 b; thick silver film illuminated at an angle of incidence i, = 60”. ------ .r component parallel t o the surface (p wave), -z component normal to the surface (p wave), - - - - - - - - - - y component parallel to the surface (s wave).
plasma resonance, where the condition (c) might be satisfied. But the sensitivity of the anomaly to the composition of the surface does not allow us to rule out a PE from surface localized impurity levels. ENDRIZand SPICER[1971b] observed in aluminium values of the ratio Y,/ Y, larger than expected in a volume effects below surface plasmon and energy for hu < 10.5 eV. ENDRIZand SPICER[1971a,b], FLODSTR~M ENDRIZ[1973] and ENDRIZ[1973] discussed in detail the expected contribution of the surface effect (SPBS) with the model developed by MITCHELL
310
PHOTOEMISSION
[VI,
9
4
[1934,1935,1936] and SCHIFFand THOMAS [1935], ENDRIZand SPICER [1971] have shown that roughness causes strong decrease of the reflectance and increase of the photoyield near plasma resonance. We have seen (section 3.4, Fig. 8) that CALLCOTT and ARAKAWA [1975] observed a misfit between the expression (3.1) for the volume effect and the observed values of Y-(i2)/Y-(0) about the angle of incidence iz,, at which plasma resonance occurs. They explained this anomaly by a contribution of the surface effect that amounts up to 65 % because, like in silver, the plasma resonance concentrate the light energy in the electric field component that is normal to the surface (CHABRIER, GOUDONNET, TRUITARD and VERNIER [19731). The investigations quoted in this section about the polarization dependence of the photoyield of alkali metals were consistent with a pure volume effect. Discrepancies appeared only recently in the work of MONIN[1973] and MONINand BOUTRY [1974]. Monin measured the index of refraction of opaque alkali films by ellipsometry and the quantum yield Y. He calculated the “true quantum efficiency”, i.e., the ratio Q = Y/(1 -R) (Figs. 21 and 22). Just after the deposition of films at 77 O K (under ultra-high vacuum, of course) Q was roughly independent of the polarization of the light as was expected in the assumption of a volume isotropic excitation (Table 4.1). But if the film was reheated to ordinary temperature Q decreased more for the s than for the p polarization (Table 4.1). TABLE4.1 Value of the true quantum efficiency that gives maximum polarization dependence for alkali opaque filmsdeposited a 77” K before and after reheating, for both polarization Q , and Q,, at wavelengths 1, in lngstroms (after MONINand BOUTRY[1974]) cs
1, Cold films
Reheated films
lo4 P, lo4 Q,
lo4 Q, lo4 Q,
Rb
K
Na
245
3200 880 710
3880 17 9.1
3340 230 64
5000 0.95 0.75
4550
4300
28 21
270
5000
0.8
4360 16
0.5
2
The results of Table 4.1 are consistent with an important contribution of the surface effect (SPBS) in reheated films of Rb and K and a less important one in reheated films of Na. But we cannot explain why no surface effect occurs before reheating. A precise knowIedge of the structural change in the bulk and in the surface could help the interpretation.
SURFACE PHOTOEXCITATION
31 1
Fig. 21. Spectral distribution of the "true quantum yield" of an opaque film of Rb after deposition at 77 K and after reheating at 195" K * for s and p polarization (after MONIN [19733).
+
Qp and Q, for Cs are not very different and this suggests that the relative contribution of the surface effect is very small although the contribution of the volume effect is reduced by the very small' value of the escape depth that is confirmed here by the small value of both Qp and Q, (several orders less smaller for other alkali metals). But the distinction between the volume effect and emission from surface states becomes rather semantic when the escape depth is reduced to a few or even to one atomic layer (section 3.1). In such thin a surface layer the usual approximations of volume effect may be quite rough, especially the description of the photon absorption by the bulk dielectric constant and the assumption of an electric field discontinuous at the surface. It seems quite difficult then to deal with the surface layer by using the model of an homogeneous solid limited by an abrupt surface. Every observed anomaly of the ratio Yp/Y,seems up to now to depend on structural parameters that are not completely controled. If these param-
312
PHOTOEMISSION
[VI, 5
4
Fig. 22. Spectral distribution d the “true quantum yield” Q of an opaque film of Cs after deposition at 77” K + and after reheating at 195” K * for s and p polarization (after MONIN [1973]).
eters concern the roughness the explanation could be found in SPBS. If these parameters concern the surface impurities the explanation would lie rather in photoemission from surface states. 4.3. EVIDENCE FOR SURFACE PHOTOEMISSION FROM THE SPECTRAL YIELD DISTRIBUTION
In metals the good results obtained with the free electron model in many fields, e.g.,electrical conductivity, led to the idea that no volume photoeffect could occur and photoemission has been explained in the early theories as a pure surface effect. When the possibility of a volume effect was recognized (FAN[1945]), the k conservation selection rule led one to expect a larger threshold hv, for the volume effect than hv, for the surface effect, because in the latter case only the tangential component of the electron wave vector k is to be conservedwhen the photon energy is absorbed by the electron. WEISSLER [19561 interpreted the spectral yield distribution
VI,
D
41
SURFACE PHOTOEXCITATION
313
of several materials (W, Mo, Pt) with two thresholds hv, and hv,. The experiments described in the last paragraph did not confirm this interpretation but the surface effect is still expected to afford a more noticeable contribution near the threshold (ENDRIZ [19731, CHABRIER, GOUDONNET, TRUITARD and VERNIER [I19731, CALLCOTT and ARAKAWA [19751). The steep increase of the photoyield, that Weissler had interpreted as a threshold for volume effect, is now attributed to a reflectance decrease that permits a better penetration of the light into the solid when the photon energy becomes larger than the plasmon (GdkLICH [1959]). An alternative explanation of the PE at low photon energy, when no k conservative transition exists, lies in indirect transitions with phonon creation or annihilation or in non-direct transitions where the momentum conservation is insured by other many-body effects (see section 2.8). GARTLAND, BERGEand SLAGSVOLD [19731 compared the spectral distribution of the photoyields Yo for normal incidence and Y700 for an incidence of 70". As expected they have found Yo to be proportional to ( h v ~ h ~ , ) ~ near the threshold, but this was not so for Y700.They attributed the ) ~ to a contribution difference between Y700and the expected (hv - h ~ , law of the surface effect. A calculation of the penetration of light for an angle of incidence of 70" would be necessary to confirm this interpretation. In a semi-conductor high values of the photoyields require transitions from the valence band. A threshold must then be observed when the photon energy equals the difference hvi between the vacuum level and the top of the valence band. Another threshold hv, has been considered by ALLEN and GOBELI [19621, because the corresponding transitions are ,generally not direct, and they assumed much higher transition probability when direct transitions are possible. When photoelectrons come from surface states, a tail can be observed in the spectral yield distribution below the threshold hvi . To give a correct interpretation of this tail we must take into account the band bending induced by the space charge due to surface states and, of course, a possible shift of the spectral yield distribution due to a change of the work function. In a theoretical analysis KANE[19621 has made a survey of 11 different possible surface and volume emission processes and associated with eaw one a contribution to the spectral yield:
q = Ci(hV--hVi)P*.
(4.1)
The threshold hvi and the exponent pi depend on the process. VANLAAR and SCHEER [1965] fitted the measured photoyield of a cleaned slightly
314
PHOTOEMISSION
[VI.
D
4
p-doped Si crystal with the expression
Y
=
C,(hv-E,)~+C,(hv-E,)~
(4.2)
with E , = 4.85 eV, E, = 5.40 eV. In such a crystal they expected no band bending and they attributed the first term to a surface effect and the second to a volume effect. FISCHER [1968] fitted the spectral yield curve of different 3-5 compounds with the expression
Y
=
C,(hv-E,)+C,(hv-E,).
(4.3)
Fischer ascribed the linear term to direct transitions between valence and conduction band but could give the emission from surface states as the only possible interpretation of the second term. We must note here that the Kane expression (4.1) is nothing more than the first term of a development that is certainly not valid in a very wide photon energy range. BALLANTYNE [19723 derived expressions that take into account electron scattering and are expected to be valid in a wider energy range (see section 2.7.4). LAUDB[19731 and SEBENNE, GUICHAR, BOLMONT and BALKANSKI [19731 have seen in the low energy tail of the spectral distribution of the photoyield from cleaved crystals of Si pieces of structure that disappeared when the cleavage (made under ultra-high vacuum), was exposed to the residual gases or to oxygen. We shall return to these investigations in the discussion of the results obtained by PED with the same material.
4.4. EVIDENCE FOR THE PHOTOEMISSION FROM SURFACE STATES OBTAINED FROM PHOTOELECTRON ENERGY DISTRIBUTIONS
The PED has been the most direct way to investigate the energy levels of electrons in a solid. If a piece of structure obseryed in the PED appears or disappears when a very small amount of gas is adsorbed we think at first of a surface localized level. We reported in section 3.9 and Fig. 17 the PED observed by Wagner and Spicer for highly n-doped samples of Si. The attribution of the peaks A and B to surface states is strongly suggested by arguments other,than their disappearance after an exposure to oxygen. Their distance to the Fermi level and their intensity are independent on the photon energy, in contrast to the peaks C and D that have been attributed to direct interband transitions. Wagner and Spicer estimated a density of surface levels of about n, = 8 x lo4 cm-’.
VI, §
41
SURFACE PHOTOEXCITATION
315
EASTMAN and GROBMAN [1972] measured the PED for cleaved surfaces of lightly n-doped crystals of Si ( l l l ) , Ge (111) and GaAs (110). The resistivity of the Si and Ge crystals were respectively 5 0 * cm and 4 Q * cm. They used synchrotron radiation in the spectral range 7-25 eV. Like Wagner and Spicer, Eastman and Grobman found a structure near the Fermi level that disappeared after an exposure of the sample to the residual gases for a few hours (lo-'' torr). Except for the peak B, that has not been found by Eastman and Grobman, the estimated density of surface states in Si is roughly the same in both experiments. RANKEand JACOBI[I9731 have observed oxygen sensitive structure in the PED from polar faces of GaAs crystals ((111) Ga and (1 11) As), that they attributed to surface states. In Si samples that had been exposed to residual gases, Eastman and Grobman observed the appearence of a broad band of filled states centered at 7 eV below tbe vacuum level, that they attributed to extrinsic levels of (Si-0). Eastman and Grobman gave a rough estimation of the escape depth of bulk excited electrons by comparing the number of electrons, that had been excited in bulk and in surface, and assuming the same matrix element for both transitions. They obtained L = 40 A for hv = 8.5 eV, L = 17A for hv = 10eV and L = 6 A for hv = 12eV. The agreement with other estimates (WAGNER and SPICER[1972], GOBELIand ALLEN [ 19621, KANE [1967]) is satisfactory, although the difference between the values of L for hv = 8.5 eV and hv = 12 eV seems quite large. The cleavage of a crystal under an ultra-high vacuum is not a sufficient condition for obtaining a perfect or perfectly reproducible surface. ERDUBACK and FISCHER [I9721 observed the surface of cleaved crystals of Si by LEED and by Auger electron spectroscopy, as they determined the PED for photons of low energy before and after annealing at different temperatures. In agreement with the observations of ALLENand GOBELI[1962], CALLCOTT [19671, FISCHER [19681, and EASTMAN and GROBMAN [19721, before annealing, they observed no structure that could be attributed to PE from surface states, as expected if the levels associated with the peak B of Wagner and Spicer were present. Erduback and Fischer observed such structures only after annealing. The LEED pattern of the Si surface which was of type 2 x 1 just after cleavage became 1 x 1 after annealing at 550 "K and 7 x 7 after annealing at 800 OK. SEBENNE, GUICHAR, BOLMONTand BALKANSKI [1973] deduced from the spectral yield distribution of n-Si and its modification by an exposure to oxygen a density of surface levels in agreement with results described above. But LAUDB[19731 suggested that the structures previously attributed to surface states should be attributed to direct transitions from the r;,
316
PHOTOEMISSION
[VL
04
level of bulk Si. The disappearance of the structure with contamination should be explained by surface inelastic scattering. In metals PE from intrinsic and extrinsic surface states has been invoked to explain structures in the PED that were sensitive to residual gases. FORSTMANN and HEINE[1970] suggested intrinsic surface states for anomalous structures previously observed for Ni and Cu e.g., by CALLCOTT and MAC RAE[1969] who had measured the PED for (1 11) faces of Ni [I9711 believes monocrystals that had been controled by LEED. EASTMAN that these structures should rather be attributed to PE from extrinsic surface states. EASTMANand CASHION {1971] measured the PED for thin films of Ni exposed to oxygen and CO and found characteristic electronic levels at 5.5 eV below the Fermi level for oxygen and at 7.5 and 10.7 for CO. BAKERand EASTMAN [19731 also observed characteristic impurity levels when (1 11) and (100) faces of a W monocrystal were exposed to 0 and CO. They could control the structure of the absorbed films by LEED and Auger spectroscopy, and identify in the PED from CO absorbed layer a peak at -8.9 eV due to CO molecule. In metals most of the effects, that could reveal a gap between the energy bands associated with each direction of the wave vector k, are masked by overlapping, when all directions of k add their contributions. Nonetheless VERNIER, COQUET and BOURSEY [19681predicted a possible increase of the photoelectric threshold for crystallographic faces of a crystal such that the vacuum level lies in the band gap associated with k normal to this face. FEUERBACHER and FITTON[1973] suppose in that case that a new type of surface effect occurs with direct transition of a bulk Bloch state to a free electron state in vacuum. A surface coupling between these states is possible because in a semi-infinite solid each Bloch state in the solid includes an exponentially damped part outside the solid and each free electron wave outside the solid includes a damped part inside the solid. A determination of the PED for one direction of emission can separate the overlapping bands and is especially useful to detect the surface states associated with one direction of k. The energy distribution for the electrons emitted normal to the surface within an angle of 12-15" was measured by WACLAWSKI and PLUMMER [19721 with a retardation potential technique and FEUERBACHER and FITTON [1972,1973] with a 127"cylindrical analyser. FEUERBACHER and FITTON[1972] observed (100) faces of W monocrystal and WACLAWSKI and PLUMMER [19721 observed polycristalline samples, but purified the sample by a heat treatment, that induced a dominant (100) orientation. In both publications 7.7 eV and 10.2 eV photons induced a PED with a peak that has been attributed to electrons excited from a sur-
317
S U R F A C E PHOTOEXCITATION
12 11 x)
9
8
12 11 x)
3
5
I
-> 3L
12h
rlz
4 a
1
3
Fig. 23. Energy distribution of photoelectronsemitted normally to (IOO), (1 10) and ( 1 1 1 ) faces of a monocrystal of W for several photon energies (after FEUERBACHER and FITTON [1973]).
318
PHOTOEMISSION
CVL
5
4
face level at 0.4 eV below the Fermi energy. Feuerbacher and Fitton made the - 0.4 eV - peak disappear by an exposure of W to H, . Waclawski and Plummer observed the same effect with H,, 0,, N, and CO and noticed at the same time appearance of peaks characteristic of the absorbed gas. FEUERBACHER and FITTON [1973] gave the energy distribution of the photoelectrons emitted normal to (loo), (1 10) and (1 11) faces of a monocrystal. They confirmed the results obtained earlier with (100) faces and found very different distributions for the other faces (Fig. 23). The surface peak 1 that had been found at -0.4 eV for (100) face, does not exist for other orientations. This confirms its surface origin. The structure no. 2 could be explained at first sight by non-direct transitions from bulk levels, but Feuerbacher and Fitton attributed it to the photoexcitation of bulk states allowed by surface, because it is observed only for (100) faces. The peaks no. 4 and 6, observed for face (1 lo), can be associated with a strong density of levels at E4 = -0.3 eV and E, = - 1.4 eV but no Bloch state exists at the energy E4+ hv and Es hv on the line T N of the band diagram (Fig. 24). Feuerbacher and Fitton explain these structures by the surface coupling between the Bloch state inside the solid and the free electron state outside the solid that we have previously considered in this paragraph. Other structures 3,8,9,10,11 can be readily interpreted as direct transitions. and CALLCOTT [1975].) (See also TURTLE
+
a!-
(A)
(do)
($0)
r
(000)
~
~~
(1%
Fig. 24. Band structure of tungsten after Christensen. (Quoted by FEUERBACHER and FITTON [1973]).
VI, § 51
CONCLUSION
0
319
5. Conclusion
The surface excitation, that was alone considered in the early works based on the free electron model, appears now as a minor contribution that can be proved only by very refined technique. In a good approximation the PE is localized by electron scattering within a more or less thin superficial layer. The attenuation length of a light beam is generally at least in the range of one hundred Angstrom and is larger than the escape depth of the electrons. The escape depth of the electrons varies in a very large range according to the photon energy. It depends to a quite small extent on the material when the electron-electron interaction is not forbidden by a band gap. As was first recognized by SPICER[1960] in alkali-antimonides, much higher values of the escape depth have been obtained for small photon energy in semi-conductors when the electron affinity is smaller than the band gap. The drop of the elastic escape depth in IK by one order of magnitude when the electron energy becomes large enough to allow scattering by pair production (BAERand LAPEYRE [1973]) is another instance of the dominant efficiency of the electron-electron interaction. Still higher values of the total escape depth are obtained when the electron affinity is negative. These abnormal values, which violate the general rule, have, of course, a special importance in the fabrication of light detectors. We have represented in Fig. 25 the general trend of the variation of the escape depth with energy. The number of published data is so high that we have not been able to indicate individual measurements. For high energy photons quite a large difference appears between the inelastic and elastic escape depths. In principle, the values represented in Fig. 25 are associated with elastic escape depth. But we must be aware that the distinction between elastic and inelastic escape depth is based on a finite energy resolution of the experimental apparatus. Obviously the inelastic escape depth results from the combination of an increasing number of mean free paths and in far ultra-violet and soft X-ray range the inelastic escape length may be several orders of magnitude larger than the elastic one. The escape depth is often reduced to a few monolayers (less than one for Cs). Then the dependence of the light absorption process on the electron scattering is certainly very important for purely optical as well as for photoelectric phenomena. The least perturbation, that we can expect, concerns “direct” transitions. When electron scattering becomes more important, increasing differences between initial an final wave vectors of the electron are allowed. It should be much more satisfactory to treat such
320
PHOTOEMISSION
1
1
10
I
ZT)
ELECTRCN ENERGY ABOVE FERMI LEVEL
[Vl,
I
m
lev)
05
>
Fig. 25. General behavior of the variation of the escape depth with electron energy.
cases within a one-step theory and we may hope that one-step theories will be soon able to take into quantitative account the electron-electron scattering. Of course, the surface is one of the most important elements in photoemission. The work function is determined by surface phenomena and the interaction of an incident electron with the surface can include elastic or inelastic scattering that cannot be fully described by the work function alone. The PE has much to receive from and much to contribute to progress in surface physics. Here too, it would be desirable to gather into a one-step theory the interaction of the electrons with the surface, with light and with other elements of the solid. Especially when the escape depth is very small, it is very crude to describe the absorption of light by the surface layer with the same dielectricconstant as the absorption by the bulk. The interpretation of reflectance data is subject to the same objection as the interpretation of photoelectric data, especially in the case of cesium.
VII
REFERENCES
321
We may hope for great progress in PE and in surface physics because of the advances in ultra high vacuum technology, LEED, Auger electron analysis and PE experimentation. With cross-checking of several techniques we may hope to determine the position and the nature of the surface atoms and their interactions with light and electrons and to describe them with realistic models. Acknowledgement
The author wishes to thank T. A. Callcott for many stimulating suggestions as well as critical reading of the manuscript. References ADAWI,I., 1964, Phys. Rev. M A , 788. AGARWAL, G. S., D. N. PATTANAYAK and E. WOLF,1971a, Optics Commun. 4, 255; 1971b, Phys. Rev. Lett. 27, 1022. G. S., D. N. PATTANAYAK and E. WOLF,1974, Phys. Rev. B10,1447. AGARWAL, ALLEN,F. G. and G. W. GOBELI,1962, Phys. Rev. 127, 150. and B. FITTON, 1971, Phys. Rev. Lett. 27, 1565. ANDEREGG, M., B. FEUERBACHER G. A., L. W. JAMESand J. J. UEBBING,1970, J. Appl. Phys. 41, 2888. ANTYPAS, J. A. and D. R. HAMANN, 1973, Phys. Rev. Lett. 31, l,M. APPELBAUM, ARAKAWA,E. T., A. J. BRAUNDMEIER, M. W. WILLIAMS, R. N. HAMMand R. D. BIRKHOFF, 1974, Communication at the 4th Intern. Conf. on Vacuum-Ultra-Violet Radiation Physics, Hamburg. 1973, Appl. Phys. Lett. 22,23. ASHLEY,K. L., D. L. CARRand R. ROMANO-MORAN, 1973, Phys. Rev. Lett. 31, 304. BAER,A. D. and G. J. LAPEYRE, J. HEDMAN, M. KLASSON and C. NORDLING, 1970, Solid StateCommun. BAER,Y., P. F. HEDEN, 8, 1479. 1973, J. Vac. Sci. & Technol. 10, 223. BAKER,J. M. and D. E. EASTMAN, J. M., 1972, Phys. Rev. B6, 1436. BALLANTYNE, D. J., J. L. MOLLand N. I. MEYER,1963, Phys. Rev. 130, 972. BARTELINK, J. EDGECUMBE and R. L. MOON,1971, Appl. Phys. BELL,R. L., L. W. JAMES, G. A. ANTYPAS, Lett. 19, 513. BELL,R. L. and W. E. SPICER,1970, Roc. IEEE 58, 1788. BELL,R. L. and J. J. UEBBING,1968, Appl. Phys. Lett. 12, 76. BERGLUND, C. N. and W. E. SPICER,1964, Phys. Rev. 136 A, 1030. ~LECHSCHMIDT,D., M. SKIBOWSKI and W. STEINMANN, 1970, Phys. Stat. Sol. 42,61. V., C. CALANDRA and M. J. KELLY,1973, J. Phys. C6, 349. BORTOLANI, J. A,, 1947, Phys. Rev. 72, 531. BURTON, CALLCOTT, T, A., 1967, Phys. Rev. 161, 746. CALLCOTT, T. A. and E. T. ARAKAWA, 1974, J. Opt. SOC.Amer. 64, 839. 1975, Phys. Rev., B11, 2750. CALLCOTT, T. A. and E. T. ARAKAWA, CALLCOTT,T. A. and A. U. MCRAE,1969, Phys. Rev. 178, 966. M.. D T. PIERCE, K. SATTLER and H. C. SIEGMANN, 1973, J. Phys. 34, C 6-95. CAMPAGNA, B. ROULETand D. SAINTJAMES, 1973, Phys. Rev. B8, CAROL!,C., D. LEDERER-ROZENBLAIT, 4552. CHABRIER, G., J. CORNAZ, J. P. G o m o m and P. J. VERNIER,1970, Optic. Commun. 1,391. CHABRIER, G.,J. P. GOUDONNET, J. F. TRUITARD and P. J. VERNIER, 1973, Phys. Stat. Sol. (b) 60.K 23.
322
PHOTOEMISSION
IVI
CHABRIER, G., J. P. GOUWNNET, P. DOLIZY,G. BCHARD and P. J. VERNIER,1973, Acta Electron. 16, 203. COLLINS,R. A. and L. W. DAVIES,1963, Appl. Phys. Lett. 2, 213. COLLINS,R. A. and R. D. COULD,1971, Solid State Electronics 14, 805. COLLINS,R. A.. 1. A. EDGEand K. 0. LEGG,1972, Phys. Stat. Sol. (a) 9, 309. COQUET,E. and P. J. VERNIER,1966, C.R. Acad. Sc. 262, 1141. CROWELL, C. R., W. G. SPITZER, L. E. HOWARTH and E. E. LABATE, 1962, Phys. Rev. 127,2006. CROWELL, C. R. and S. M. SZE, 1967, Physics of Thin Films 4, 325. DAUDE,A., A. SAVARY and S. ROBIN,1972, J. Opt. SOC.Amer. 62, 1. DAVISON, S. G. and J. D. LEVINE,1970, Solid State Physics 25, 1. DAVYDOFF, A. S., 1965, Quantum Mechanics (Pergamon Press) p. 309. DEDERICHS, P. H., 1972, Sol. State Phys. 27, 135. DONIACH, S., 1970, Phys. Rev. B 2, 3898. DUCKETT,S. W., 1968, Phys. Rev. 166, 302. DUCKETT,S. W. and P. M. METZGER,1965, Phys. Rev. 137A, 953. EASTMAN, D. E., 1970, Sol. State Communications 8, 41. EASTMAN, D. E., 1971, Phys. Rev. 3B, 1769. EASTMAN, D. E. and J. K. CASHION, 1971, Phys. Rev. Lett. 27, 1520. EASTMAN, D. E. and W. D. GROBMAN, 1972, Phys. Rev. Lett. 28, 1378. EASTMAN, D. E. and W. F. KROLIKOWSKI, 1968, Phys. Rev. Lett. 21, 623. EDEN,R. C., J. L. MOLLand W. E. SPICER,1967, Phys. Rev. Lett. 18, 597. ENDRIZ,J. G., 1973, Phys. Rev. B 7, 3464. ENDRIZ,J. G. and W. E. SPICER,1971a, Phys. Rev. Lett. 27, 570. ENDRIZ,J. G. and W. E. SPICER,1971b, Phys. Rev. B 4, 4159. ERBUDAK, M. and T. E. FISCHER, 1972, phys. Rev. Lett. 29, 732. FAN,H. Y., 1945, Phys. Rev. 68, 43. FEIBELMAN, P. J., 1973, Surf. Sci. 36,558. FEUERBACHER, B. and B. FITTON, 1972, Phys. Rev. Lett. 29, 786. FEUERBACHER, B. and B. FITTON, 1973, Phys. Rev. Lett. 30,923. J. S. E~CHER and B. F. WILLIAMS, 1972, J. Appl. Phys. 43, FISCHER, D. G., R. E. ENSTROM, 3815. FISCHER, T. E., 1968, Surf. Sci. 10, 399. FISCHER. T. E.. 1969. Surf. Sci. 13. 30. FLODSTR~M, S. A. and J. G. ENDRIZ,1973, Phys. Rev. Lett. 31, 893. FONTENEAU, J. Y., M. PAUTYand P. VERNIER,1971, C.R. Acad. Sci B 272,441. FORSTMANN, F., 1967,Z. Phys. 203,495. FORSTMANN, F. and V. HEINE,1970, Phys. Rev. Lett. 24, 1419. FORSTMANN, F. and J. B. PENDRY,1970, Z. Phys. 235, 75. FOWLER, R. H., 1931, Phys. Rev. 38,45. FRANK, G. and S. GARBE, 1973, Acta electronica 16, 237. FUCHS.R. and K. L. KLIEWER,1969, Phys. Rev. 185, 905. GARBE,S., 1969a, Solid State Electronics 12, 893. GARBE,S., 1969b. Phys. Stat. Sol. 33, K 87. GARBE,S., 1970, Phys. Stat. Sol. (a)2, 497. 1970, Proc. Third Intern. Symp. on Gallium Arsenide and Related GARBE,S. and G. FRANK, Compounds, p. 208. GARTLAND, P. O., S. BERGEand B. J. SLAGSVOLD, 1973, Phys. Rev. Lett. 30,916. GAUDART. L., 1973. J. Phys. 34,C 6 9 7 . J. I. and N. TZOAR,1973, Phys. Rev. 8B. 5671. GERSTEN, GERSTEN, J. 1. and N. TZOAR,1974, Phys. Rev. 9B. 4038. GESELL, T. F. and E. T. ARAKAWA, 1971, Phys. Rev. Lett. 26, 377. 1971, Report TM 2617 of Oak Ridge National Laboratory, GESELL,T. F. and E. T. ARAKAWA, Oak Ridge, Tenn.
VI1
REFERENCES
323
GOB~LI, G. W. and F. G. ALLEN,1962, Phys. Rev. 127, 141. WRLICH, P., 1959, Advances in Electronics and Electron Physics 11, 1. G~RLICH, P. and K. SUMI,1970, Physica Status Solidi 2, 427. GOUDONNET, J. P., G. CHABRIER and P. VERNIER, 1974, Communication at the Intern. Conf. on Vacuum-Ultra-Violet Radiation Physics, Hamburg. GOULD, R. D., C. A. HOGARTH and R. A. COLLINS,1973, J. Non Crystalline Solids 12, 131. GURMAN, S. J. and J. B. PENDRY, 1973, Phys. Rev. Lett. 31, 637. HACKETT, W. H., 1972, J. Appl. Phys. 43, 1649. HALL,C. K. and C. H. B. MEE,1970, Congrts Intern. sur les couches minces. Levide, supple-
. ment no. 147. HEBB,M. H., 1951, Phys. Rev. 81, 707. HEINE,V., 1972, Proc. R. SOC.Lond. A331, 307. HERMEKING, H., 1972, Z. Phys. 253, 379. HERMEKING, H., 1973, J. Phys. 6,2898. HICKMOTT,T. W., 1963,X Appl. Phys. 34, 1569. HICKMOTT, T. W., 1965, J. Appl. Phys. 36, 1885. HIRSCHBERG, K. and K. DEUTSCHER, 1968, Phys. Stat. Sol. 26, 527. HOFMANN, H. H. and K. DEUTSCHER, 1970, Z. Phys. 236, 288. HOFMANN, J. and W. STEINMANN, 1968, Phys. Stat. Sol. 30, K 53. HOPFIELD.J. J. and D. G. THOMAS,1963, Phys. Rev. 132, 563. HUGUES,A. L. and L. A. DUBRIDGE,1942, Photoelectric phenomena (Mac Craw Hill). IvEs, H. E. and H. B. BRIGGS,1936, J. Opt. SOC.Amer. 26, 247. IVES,H. E. and H. B. BRIGGS,1938, J. Opt. SOC.Amer. 28, 330. JACOBI,K. and J. H ~ L Z L1971, , Surf. Sci. 26, 54. JAMES, L. W., G . A. ANTYPAS,J. EDGECUMBE, R. L. MOONand R. L. BELL,1971, J. Appl. Phys. 42,4876. and R. L. BELL,1973, Appl. Phys. JAMES,L. W., G. A. ANTYPAS;R. L. MOON,J. EDGECUMBE Lett. 22, 270. JAMES,L. W. and J. L. MOLL, 1969, Phys. Rev. 183, 740. JASPERSON, S. N. and S. E. SCHNATTERLY, 1969, Phys. Rev. 188, 759. KANE,E. O., 1962, Phys. Rev. 127, 131. KANE,E. O., 1966, Phys. Rev. 147, 335. KANE,E. 0.. 1967, Phys. Rev. 154, 624. KANTER,M. and P. J. FEIBELMAN, 1962, J. Appl. Phys. 33, 3580. KANTER,M., 1970, Phys. Rev. B 1, 522. KATRICH, G. A. and 0. G. SARBEI,1961, Sov. Phys. Solid State 3, 1181: KLASSON,M., J. HEDMAN, A. BERNDTSSON, R. NILSSON,C. NORDLING and P. MELNIK,1972, Phys. Scripta 5, 93. KLEIN,W., 1969, J. Appl. Phys. 40,4384. KROLIKOWSKI, W. F. and W. E. SPICER,1969, Phys. Rev. 185, 882. KROLIKOWSKI, W. F. and W. E. SPICER,1970, Phys. Rev. 51, 478. LANGRETH, D. C., 1971, Phys. Rev. 3B, 3120. LAUDB,L. D., 1973, J. Phys. 34, C 6 3 5 . LEIVOWSKI, T., P. BASTIEand M. BIZOUARD, 1969, C.R. Acad. Sci. B 268, 1 10. LEWOWSKI, T., P. BASTIEand M. BIZOUARD, 1970, Phys. Stat. Sol. (a) 2, 847. LIU, Y.Z., J. L. MOLLand W. E. SPICER,1970, Appl. Phys. Lett. 17, 60. LYE, R. G. and A. J. DEKKER,1957, Phys. Rev. 107, 977. MACEK,C. H., A. OTTOand W. STEINMANN, 1972, Phys. Stat. Sol. (b) 51, K59. MAHAN,G. D., 1970, Phys. Rev. Lett. 24, 1068. MAHAN,G. D., 1970, Phys. Rev. B2, 4334. MAHAN,G. D., 1973, Phys. Status Solidi B 55, 703. MAKINSON, R. E. B., 1949, Phys. Rev. 75, 1908. MARTINELLI, R. U , 1970, Appl. Phys. Lett. 16, 261.
324
PHOTOEMISSION
CVI
MARTINELLI, R. U., 1973, J. Appl. Phys. 44,2566. MAYER.H.. 1961, Symp. of the electric and magnetic properties of thin metallic layers, Leuwen, Belgium. and H. STEFFEN, 1970, Thin Solid films 5, 389. MAYER,H., D. L. BLANARU MAYER,H. and H. THOMAS, 1957, Z. Phys. 147,419. MEAD,C. A,, 1962, Phys. Rev. Lett. 8, 56. MELNYK, A. R. and M. J. HARRISON, 1970, Phys. Rev. B 2, 835. S., 1957,Z. Phys. 147,442. METHFESSEL, METZGER,S. W., 1965, J. Phys. Chem. Solids 26, 1879. MITCHELL, K., 1934, Proc. Roy. SOC.A M , 442. MITCHELL, K., 1935, Proc. Camb. Phil. SOC.31, 416. MITCHELL, K., 1936, Proc. Roy. SOC.A153,513. M ~ N C HW., , 1973, in: Festkdrperprobleme XIII, ed. H. J. Queisser (Pergamon Press) p. 241. MONIN,J., 1973, Acta Electronica 16, 139. MONIN,J. and G. A. BOUTRY. 1974, Phys. Rev. B 9, 1309. 1970, C.R. Acad. Sci. Ser. B 270, 1234. NIQUET,G., P. J. VERNIERand P. HARTMANN, 19,69, Phys. Rev. 178, 1097. N O ~ E R EP. S ,and C. T. DE DOMINICIS, PALMBERG. P. W. and T. N. RHODIN,1968, J. Appl. Phys. 39, 2425. OTTO,A,, 1968, Phys. Stat. Sol. 26, K99. OTTO,A., 1970, Phys. Stat. Sol. 42, K37. M. PRIOLand S. ROBIN,1973, J. Phys. 34, C 6-9. PEISNER,J., A. QUEMERAIS, PEISNER, J., P. ROBOZand P. B. BARNA,1971, Phys. Stat. Sol. (a) 4, K187. PEPPER, S. V., 1970, J. Opt. Soc. Amer. 60,805. F. J., 1966, in: Basic problems in thin films, eds. R. Niedermayer and H. Mayer PIEPENBRING, (Gottingen, Vandenboeck and Ruprecht) p. 325. 1974, Phys. Rev. B 9, 4035. PIERCE,D. T. and H. C. SIEGMANN, E. W. and J. W. GADZUK, 1970, Phys. Rev. Lett. 25, 1493. PLUMMER, PONG,W., 1966, J. Appl. Phys. 37, 3033. PONG,W., 1967, J. Appl. Phys. 38, 4103. PONG,W., 1972, J. Appl. Phys. 43,60. PONG,W. and J. A. SMITH,1973, J. Appl. Phys. 44, 174. and G. Mm-, 1970, J. Appl. Phys. 41, 1869. PONG,W., R. SUMIDA T D. FABRE, 1965, J. Phys. 26, 372. ~ D A LF.,, C. G ~ Uand and S. ROBIN,1973, C.R. Acad. Sci. B 276,753. QUEMERAIS, A., J. PEISNER, M. PRIOL QUINN,J. J., 1962, Phys. Rev. 126, 1453. W. and K. JACOBI,1973, Solid State Commun. 13, 705. RANICE, RICHARD, J. C., 1973, Acta Electron. 16, 245. RIDGWAY, J. W. T. and D. HANEMANN, 1971, Surf. Sci. 24, 451. RITCHIE,R. H., 1973, Surf.Sci. 34, 11. RITCHIE,R. H. and J. C. ASHLEY, 1965. J. Phys. Chem. Solids%, 1689. RIVIERE,J. C., 1969, in: Solid State Surface Science 1, ed. Mino Green (Marcel Dekker, N.Y.) p. 179. SAUTER, F., 1967, Z. Phys. 203,488. SAVOYE, E. D. and D. E. ANDERSON, 1967, J. Appl. Phys. 38, 3245. SCHADE, H., H. NELSONand H. KRESSEL, 1971, Appl. Phys. Lett. 18, 121. W. L. and N. W. ASHCROFT,1970, Solid State Commun. 8, 1959. SCHAICH, W. L. and N. W. ASHCROFT, 1971, Phys. Rev. B 3, 2452. SCHAICH, SCHIFF, L. I. and L. H. THOMAS,1935, Phys. Rev. 47, 860. SEAH,P., 1972. Surf. Sci. 32, 703. SEBENNE, C., G. GUICHAR, D. BOLMONT and M. BALKANSK~, 1973, J. Phys. 34, C 6-35. SHIRLEY, D. A., 1971, Ed., Electron Spectroscopy, Proc. of an Intern. Conf. held at Asilomar, Pacific Grove. California (North-Holland. 1972). A. R., S. E. STRAKOVSKAYA and R. L. NEMCHENOK, 1969, Bull. Acad. SC. USSR, SHUL‘MAN, Phys. Ser., English transl. 33, 463.
VI1
REFERENCES
325
SMITH,N. V., 1971, Critical Rev. Solid State Sci. 2, 45. 1971, Phys. Rev. B 3, 3662. SMITH,N. V. and G. B. FISCHER, , Phys. Rev. 188, 593. SMITH,N. V. and W. E. S P I C ~ R1969, SOMMER, A. H., 1970, Photoemissive Materials (John Wiley). SOMMER, A. H., 1973, J. Phys. 34,C 6 5 1 . SPICER,W. E., 1958, Phys. Rev. 112, 114. SPICER,W. E., 1960, J. Appl. Phys. 31, 1077. SPICER,W. E., 1967, Phys. Rev. 154, 385. STEINHARDT, R. G., J. HUDISand M. L. PERLMAN, 1972, Phys. Rev. B 5, 1016. STEINHARDT, R. G., J. HUDISand M. L. F’ERLMAN, 1971, Proc. Intern. Conf. on electron spectroscopy, ed. D. A. Shirley (North-Holland) p. 557. STEINMANN, W., 1968, Phys. State Sol. 28, 437. STERN,F., 1963, Sol. State Phys. 15, 299. STUART,R. N. and F. WOOTEN,1967, Phys. Rev. 156, 364. STUART,R. N., F. WOOTENand W. E. SPICER,1964, Phys. Rev. 135, A 495. SUHRMAN,R. and H. SIMON,1958, Der lichtelektrische Effekt und seine Anwendungen (Springer Verlag). SUTTON,L., 1970, Phys. Rev. Lett. 24, 386. SZE,S. M., J. L. MOLLand T. SUGANO,1964, Solid State Electronics 7,509. TAMM, I. and S. SCHUBIN,1931, Z. Phys. 68, 97. TAMM, I., 1932, Phys. Z. Sowj. 1, 733. TARNG. M. L. and G. K. WEHNER,1973, J. Appl. Phys. 44, 1534. THOMAS, H., 1957, Z. Phys. 147, 395. THOMAS, H., 1966, in: Basic problems in thin films, eds. R. Niedermayer and H. Mayer (Gottingen, Vandenboeck and Ruprecht) p. 307. THORNBER, K. K., 1971, Phys. Lett. 34A, 205. TONG.S. Y.. T. N. RHODINand R. H. TAIT,1973, Phys. Rev. 8, 421. TURTLE,R. R. and T. A. CALLCOTT, 1975, Phys. Rev. Lett. 34,86. TZOAR,N. and J. 1. GERSTEN,1973, Phys. Rev. 8B, 5684. 5. J. and R. BELL,1968, Proc. IEEE 56, 1625. UEBBING. VANLAAR,J. and J. J. SCHEER,1962, Philips Research Reports 17, 101. VERDERBER, R. R. and J. G. SIMMONS, 1967, Radio Elec. Eng. 33, 347. VERNIER, P. J., 1973, Acta Electronica 16, 181. VERNIER, P. J. and E. COQUET,1965, in: Basic problems in thin film physics, eds. R. Niedermayer and H. Mayer (Gottingen, Vandenboeck and Ruprecht) p. 328. G. CHABRIER and J. CORNAZ, 1971, J. Opt. SOC.Am. 61, 1065. VERNIER, P. J.. J. P. GOUDONNET, P. J., E. COQUETand M. BIGUEURE, 1966, C. R. Acad. Sci. B 262, 1728. VERNIER. P. J., E. COQUET and E. BOURSEY, 1968, J. Phys. 29, Suppl. 2-3,534. VERNIER, P. J., J. P. GOUDONNET, G. CHABRIER and J. CORNAZ, 1971, J. Opt. SOC.Am. 61,1065. VERNIER, VERNIER, P. J., M. PAUTYand J. F. BERTHELEMY, 1965, in: Optical properties and electronic structure of metals and alloys, ed. F. Abeles (North-Holland) p. 323. P. J., M. PAUTYand F. PAUTY,1969, J. Vac. Science and Technology 6,743. VERNIER, VILJOEN, P. E., M. S. JAZZARand T. E. FISCHER,1972, Surf. Sc. 32, 506. 1972, Phys. Rev. Lett. 29, 783. WACLAWSKI, B. J. and E. W. PLUMMER, WAGNER,L. F. and W. E, SPICER,1972, Phys. Rev. Lett. 28, 1381. WEISSLER, G. L., 1956, Photoionization in gases and photoelectric emission from solids, in: Handbuch der Physik, ed. S. Flugge 21, 341. WILLIAMS, B. F. and J. J. TIETJEN,1971, Proc. IEEE 59, 1489. WOOTEN,F., T. HUENand R. N. STUART,1966, Optical properties and electronic structure of metals and alloys, ed. F. Abeles (North-Holland) p. 332. 1949, Photoelectricity and its application (J. Wiley, ZWORYKIN, V. K. and E. G. RAMBERG, New York).
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E. WOLF, PROGRESS IN OPTICS XIV 0 NORTH-HOLLAND 1976
VII OPTICAL FIBRE WAVEGUIDES - A REVIEW BY
P. J. B. CLARRICOATS Department of Electrical ana Electronic Engineering, Queen Mary College, University oflondon, England
CONTENTS PAGE
0 1 . INTRODUCTION'. . . . . . . . . . . . . . . . . . . 329 6 2. FIBRES WITH CORE A N D CLADDING POSSESSING UNIFORM REFRACTIVE INDEX . . . . . . . . . . . 331
9 3 . FIBRES WITH NON-UNIFORM REFRACTIVE INDEX 381 ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . 400 REFERENCES .
.......................
400
9 1. Introduction During the period 1969-75, the use of optical fibres for telecommunication purposes has grown from a position of speculative research into one of commercial reality. This growth has been largely due to the efforts of those engaged both in fibre manufacture and in the development of light sources. Against this background many workers have been engaged in research into the theory of optical waveguides and it is with their contribution that the present review is concerned. Work on optical fibres goes back at least to the early 1950's and a resume of the early history is to be found in the book by KAPANY [1967]. Credit for the proposal to use a cladded optical fibre for telecommunications purposes is attributed to L o and HOCKHAM [1966], whose classic paper appeared in July 1966. Thereafter much of the early work aimed at the development of long-haul systems was undertaken in the United Kingdom at the British Post Office and by Standard Telecommunication Laboratories. By 1969, workers in the U.S.A., especially at the Bell Telephone Laboratories, in Germany and in Japan, were enlarging their efforts. At first, fibre attenuation was prohibitively large and solid-state laser sources inefficient and of very short lifetime. A major breakthrough KECK occurred in 1970 with the announcement in the U.S.A. by KAPRON, and MAURER [I9701 of the Corning company, of a quartz fibre with losses below 20 dB km-'. Thereafter, other laboratories also achieved excep tionally low loss fibres based on quartz. In 1972, another low loss fibre was announced using an organic liquid (tetrachloroethylene) as a core material. This was proposed independently by OGILVIE,ESDAILEand KIDD [1972] in Australia and by KAISER,TYNES, CHERINand PEARSON [1972] in U.S.A. Their announcement was soon followed by one by ~ A Y N E and GAMBLING [I9721 in England, who proposed the use of a related organic liquid with even lower loss. In Japan, effort was concentrated mainly on fibres with graded refractive index since these have the considerable advantage of neutralising modal pulse dispersion, which is the principal disadvantage associated with multimode fibres. Their researches were 3 29
330
OPTICAL FIBRE WAVEGUIDES
[VIL
0
1
concentrated mainly around the material borosilicate whose loss, although larger than that found in quartz, now comes within the range required for longhaul systems. Theoretical work on optical fibres for telecommunication purposes was initiated in the United Kingdon in 1965, by Kao, although interest
A Monomode fibre with cylindrical core
Monomode fibre with triangular core -1 k - b
Doubly clad fibre also called W-fibre
Graded refractive index fibre
Multimode fibre with ring shaped refractive index
Single material fibre
Fig. 1. Optical fibre configurations.
VII,
0 21
FIBRES W I T H CORE A N D C L A D D I N G
33 1
in related propagation problems associated with vision had been undertaken earlier by ENOCH[1961], SNITZER and OSTERBERG [1961] and by BIERNSONand KINSLEY [1965]. A major study by SNYDER [1969], paved the way for the analysis of many fundamental problems in optical waveguides and his team in Australia have continued with their investigations subsequently. At Bell Telephone Laboratories the theoretical contributions* of Arnaud, Gloge, Marcatili, Marcuse, Miller and Personick, were especially relevant, while in Germany, Unger* and his co-workers made many important investigations. In a fast moving field, the task facing the author of a review is especially difficult as a subtle change in technology can rapidly shift the emphasis which should be given to a specific topic. Several excellent review papers have already appeared (see references) and in one of them MILLER,MARCATILI and TINGYE LI [1973] have treated the topics of systems, sources and fibre manufacture. None of these are discussed here, concentration being given instead to those fundamental electromagnetic aspects which underlie propagation in fibres. A number of relevant books are cited in the references and it is known that others are in preparation. For this reason, I have stopped short of developing many of the equations in their entirety and refer the interested reader to other works if more detail is required. It has also proved impossible to treat all of the fibre configurations shown in Fig. 1, instead attention is confined to those of Fig. 1(A), (C), (D), (E) and (F). Readers interested in the configurations of Fig. 1(B) and (G) should consult references by DYOTT,DAYand BRAIN[1973], TYNES [19741, MARCATILI [19741 and STANDLEY and HOLDEN [19741.
92. Fibres with Core and Cladding Possessing Uniform Refractive Index 2.1. INTRODUCTION
Fig. 2(a) shows a cylindrical fibre with uniform refractive index in both core and cladding regions together with a lossy outer layer. With an appropriate choice of refractive indices and core radius, the fibre may support only one mode with field closely confined to the core, see Table 1. Higher order modes, if excited, can be attenuated if a lossy outer layer is present because their fields are significant beyond rz . This is the case with the monomode fibre. In the multimode fibre, more than one mode has its
*
References to the numerous contributions of these workers will be given in later sections.
332
O PT I CAL F I B R E W A V E G U I D E S
C
Fig. 2. Optical fibre configurations:(a) fibre with finite cladding boundary, (b) infinite cladding approximation, (c) cladding mode approximation.
TABLE 1 Typical fibre parameters
I .53 1.53 1.465
1.458
1.5 1.5 1.458 1.415
2 x 2 x ~ O - ~ 7 xi0-3 4.3 x lo-’
0.3 0.3
0.17 0.42
1.o
30 1.75
35
2.21 66 2.21 108.
single mode 2200 single mode 5540
field confined to the core region and in practical fibres designed to support many modes, the condition is usually accomplished by making the core radius much larger than in the monomode case. A first step in evaluating the transmission properties of a fibre involves the calculation of the propagation coefficients of the modes. An exact analysis for the configuration of Fig. 2(a) has been made by CLAWCOATS and CHAN[1973], however, their study serves to validate the widely used infinite cladding assumption which reduces the configuration to that of Fig. 2(b). A further assumption applies to modes below their core-mode cut-off when the fields are virtually unperturbed by the presence of the core, the fibre can then be represented by Fig. 2(c). Studies of the propagation behaviour of a cylindrical dielectric embedded in an infinite medium of different permittivity as in Fig. 2(b) or 2(c) began
vn, 0 21
FIBRES WITH CORE A N D C L A D D I N G
333
with a theoretical treatment by HONDROS and DEBYE [1910]. These were followed much later by theoretical and experimental investigations relevant to the propagation of microwave signals along dielectric rod waveguides and to the preliminary analysis of circular waveguides containing ferrite rods. The possible use of the structure as an optical waveguide is attributed and OSTERBERG [1961] whose investigations stimulated interest to SNITZER in optical fibre waveguides for telecommunication applications. 2.2. CHARACTERISTIC EQUATION
The general form of the characteristic equation for modes with azimuthal dependence* n in the structure of Fig. 2(b) is given below, the derivation of which is presented in numerous texts and papers (see references),
where
For surface waves to be guided without radiation loss by a straight uniform fibre E2 < B < E l , a condition which defines the regime of trapped modes or surface waves. From the above relations we obtain
+
U 2 W 2 = k2r& -El)
=
k2r: (n: - n;) = V 2 = (kr,)2(N.A)2.
(2)
Since in optical fibres, n, x n 2 , we have the relationship V x kr, n,
2An (T) = kr, nl(2A)f w kr, n1 8,
(3)
where A = (n, - n 2 ) / n l , 8, = complement of the critical angle for total internal reflection and N.A. = the fibre numerical aperture. Real values of are obtained, provided that the refractive indices n, and n2 are real, n, > n2 and V exceeds the cut-off value for the mode as given in Table 2. When n = 0, the boundary conditions at r = rl may be
B
*
Note: Throughout this text n is the azimuthal mode number which should not be confused with refractive indices n l , 2 . . ..
334
OPTICAL FIBRE W A V E G U I D E S
TABLE2
U,and U, for first 28 modes (nl x
0 2.405 3.832 5.136 5.520 6.380 7.016 7.588 8.447 8.654 8.772 9.761
n2)
2.405 3.832 5.136, 5.520 6.380 7.016 7.588 8.477, 8.654 8.772 9.761 10.173 9.936 11.065
satisfied with either E, = 0 (TE,, modes) or H, = 0 (TM,, modes) otherwise both longitudinal components of electric and magnetic field are required and the modes are classified as HE,, or EH,,. However, when n , x n 2 , the longitudinal components of field are of order smaller than the transverse components, as first noted by SNYDER[1969a, b]. Under these conditions, the HE,, and EH,- 2 m modes are almost degenerate and the transverse fields of the combination are almost linearly polarised. 2.3. APPROXIMATE SOLUTIONS OF THE CHARACTERISTIC EQUATION
On introducing the condition n , x n2 into equations (1) and (2)
With the use of recurrent relations for Bessel functions, then
UJ,(U) --
n l WKn(W) JnTltU) - n2 KnTItW)' The equation involving J,,- refers to HE nm modes while that with J,+ ,(U) refers to EH,, modes. The following alternative form which applies when n , x n2 was also first noted by SNYDER [1969a]
,
uJ , 7 2( U) -~
'
WK,T2(W) (6) JnT1W) = Knr I(W) In the limiting case when n, + n 2 , the form of equations (5) and ( 6 ) reveals
VII,
0 21
.
FIBRES WITH CORE AND CLADDING
335
6-
OI
5-
3 0
C
._ & al
L-
v
normalized frequency
Fig. 3. Eigenvalue U as a function of normalised frequency V. The solid lines are determined numerically from equation (5); the dashed lines are the approximate solution given by equation (T3.2) of Table 3.
that the modes HE, + l m and EH, - l m are degenerate. When V + co, W + co thus U satisfies Jnr ( V , ) = 0 in that limit. For propagating HE,, and EH,, modes, U lies between consecutive roots of the Bessel function J,* 1 ( U ) . Figs. 3 and 4 show U and W as functions of V while special cases of U and W are contained in Table 3. The behaviour of modes below cut-off will be deferred to the section on leaky modes. Table 3 contains approximate analytic expressions for U derived originally by SNYDER[19691 and GLOGE[1971a]. Equations (T3.1a), (T3.lb) and (T3.2), are very accurate while (T3.2) is useful for many calculations. When A = 0.05 equation (T3.1 b) giving U for the HE, l mode is accurate to within 0.5 % in the range 1.5 < V < 5 . . From the definitions of U and V given in equations (1) and (2), GLOGE [1971a] writes
336
5
h
t
b
OPTICAL F I B R E W A V E G U I D E S
e
Ib
-+ 3
VII, §
d
I
"s IB
21
FIBRES WITH CORE A N D C L A D D I N G
331
338
O P T I C A L F I B R E WAVEGUIDES
10-
8-
6W L-
2-
OO' Fig. 4. The eigenvalue Was a function of normalised frequency V for the lowest order modes of the fibre shown in the configuration of Fig. 2(b): 01 = HE,,; 1 1 = HE,,, TM,,; 21 = HE,,, EH,,; 02 = HE,,; 31 = HE,,, E H , , ; etc.
Thus with equations (T3.1) or (T3.2) and (7) 8, the normalised phase-change coefficient, can be easily evaluated as a function of V. Fig. 5 shows results due to GLOGE [1971a] who uses a linearly polarised modal notation and is discussed in section 2.7. The relationship to the conventional notation is explained in Fig. 4. 2.4. RAY INTERPRETATION
Evidently p increases from n2 to n, as V increases from the cut-off value V = U, towards infinity. Fig. 6 shows a ray interpretation which is based on an asymptotic evaluation of the Bessel function, whose dependence describes the transverse field within the core region. From Fig. 6 we see that 8 = n, cos 4 cos 6. (9)
VII,
4 21
FIBRES WITH C O R E A N D C L A D D I N G
339
V Fig. 5 . Normalised propagationcoefficientb defined by equation (8) as a function of normalised frequency V for optical fibre configuration of Fig. 2(b).
Equation (7) then leads to
and for small angles
where 8, is the complement of the critical angle for total internal reflection at the interface between core and cladding regions. We now see that as fl increases from n2 to n, the angle 8 which the ray vector n,k subtends with the generatrix of the cylinder, changes from 8, to zero. -.. - .
2.5. VALIDITY OF CORE A N D CLADDING MODE APPROXIMATIONS
Fig. 7 shows
fl as a function of
V for the configuration of Fig. 2(a) as
b
a
U plane
Fig. 6 . Rays in an optical fibre waveguide: (a) Dotted region corresponds to trapped modcs such as ray I . Ray 2 describes a leaky ray; ray 3 describes a refracting ray. (b) Section through plane containing ray 1. (c) Detailed view of trapped leaky and refracting ray regions, Locus A B C corresponds to increasing meridional angle 6, while 4 remains constant.
2.t
V", §
21
341
FIBRES WITH CORE A N D CLADDING
1.40
-
a I30 -
1.20-
I10 -
0
I0 0
1
2
V
3
4
I
I
I
I
1
2
3
4
V
Fig. 7. Normalised propagation coefficient Bas a function of normalised frequency V. Fibre parameters n, = 1.53; n2 = 1.50; r 2 / r 1= 5.
computed by CLARRICOATS and CHAN[1973]. Detailed examination of the results leading to the figure showed that the core mode approximation was exceedingly good above the core mode cut-off ( V > V,) while the cladding mode approximation was quite accurate at p values a few percent below the value p = n,, corresponding to the core mode cut-off. However, an interesting cross-over in the propagation curves occurs just below that value. The degeneracy at the cross-over is broken in the presence of finite losses and detailed examination of the fields show that continuity is maintained along the upper and lower branches of the propagation curves.
342
O P T I C A L FIBRE WAVEGUIDES
CVK
Ei
2
Further studies of cross-overs have been undertaken more recently by KUHN[1975] and YIP and HUANG[1975]. 2.6. GROUP DELAY AND PULSE DISPERSION IN THE ABSENCE OF MODE COUPLING
To determine the response of a fibre of length L, the delay z is found from the equation L dp c dk
7 = - - = -
L vg
(13)
where c = the velocity of light in vacuum and ug = the group velocity of the mode. From equation (7) and following GLOGE [I971a], we have
where it is assumed that kdn, kdn, N , = - - -dk - - N 2 = - dk
&,2.
The first term in equation (14) arises from the material dispersion and is independent of the waveguide mode, the second is due to waveguide dispersion and is mode dependent. By using the relation for dUjdV given in Table 3 U2 d(Vb) _ _- 1 - -[l-2un] dV V2 = 1+ =
c[lV2
i]
when V > 1
2u,
when U = V at cut-off.
(17)
From the limiting form of IC,when W + 0, one finds that at cut-off for HE,,, HE,,, TM,, and TE,, modes d(Vb)/dV = 0. While for HE,, and EH,d( Vb) dV
2m
modes
VIL §
21
343
F I B R E S W I T H CORE A N D C L A D D I N G
F J
0
2
6
8
10
12
V Fig. 8. d(Vb) /dV as a function of normalised frequency Y for optical fibre configuration of Fig. 2(b).
GLOGE [1971a] has plotted d( Vb)/dV as a function of V for the lowest order modes ofa multimode fibre as shown in Fig. 8. For fibres with large Vvalues, which support modes with large values of n, the spread in delay AT is obtained from equations (19) and (16) as
In the limit of large n, the spread in arrival time of the fastest and slowest modes, as predicted by equation (20), is precisely that obtained using a ray approach. Extreme skew rays cross the axis of the fibre at an angle 8, = (2A)* and for these, the time difference relative to the axial ray is given by the limiting form of equation (20). GAMBLING, PAYNEand SUNAK [1971], GAMBLING, DAKIN, PAYNE and SUNAK[1972] and ROSMAN [1972] have used the simple ray model to investigate the response of fibres and have reported good agreement with experiment. In these cases mode coupling must presumably have been small for, as discussed later, when present it leads to a dependence on length of the form L* rather than L.
344
OPTICAL FIBRE W A V E G U I D E S
Cvn, 0 2
For large V, AT x 5A10-6 s km-' thus, as an example, when A = 0.01 AT x 50 x lo-' s km-'. The above calculation is pessimistic since mode coupling reduces AT, also in practice, higher order modes are usually both less strongly excited and more strongly attenuated, which reduces the spreading. The effect of a finite cladding boundary on the group velocity has also been studied by CLARRICOATS and CHAN[1973] Fig. 9 shows v,/c as a function of Vfor a fibre with A = 0.02 and r z / r l = 5. We find that provided a mode is above core mode cut-off, vg corresponds closely to the value predicted on the basis of an infinite cladding. For the HE,, mode the approximation is good above V = 1.5.
Fig. 9. Normalised group velocity V, as a function of normalised frequency V. Fibre parameters as Fig. 7.
For a single mode waveguide, pulse dispersion is minimal due to the term d( Vb)/d Vin equation (14) and in practice the dominant pulse broadening effect is caused by material dispersion coupled with the finite linewidth
VII,
8 23
345
FIBRES WITH C O R E A N D C L A D D I N G
of the source, as noted by DYOTT and STERN[1970], KAPRONand KECK [1971], GLOGE[1971b] and others. For a multimode waveguide, the time spread caused by a light source of frequencyf and bandwidth B is given by
From equation (14)
0.3
I
I
I
I
I
I
I
0.4
0.5
0.6
0.7
08
0.9
1.0
ho* Pm Fig. 10. arjaf for HElI mode as a function of V and L for optical fibre comprising Schott K, and K, glasses as core and cladding materials. Dotted curves show dispersion in the constituent materials. Fibre parameters r J r , = 5.
For a single mode waveguide, the first term dominates and published values of dN/dk may be used to estimate pulse dispersion. To demonstrate * this point, Fig. 10 shows computed values of a~/ affo rthe HE,, mode in a glass fibre waveguide without approximations and including the effects
346
O P T I C A L FIBRE W A V E G U I D E S
CVIL
0
2
of finite cladding. Also shown are curves of kdNJdk and kdN2/dk as a function of V. For most purposes, the approximation described above gives an accurate description of the pulse broadening effect. In the example of Fig. 10, the dispersion caused by a source such as a GaAs laser, with 0.1 % relative linewidth and for a fibre with V = 2.4, is lO-'Os km-'. This is three orders of magnitude lower than that caused by modal dispersion in multimode waveguide. However, light emitting diodes (LEDs) have relative linewidths of up to 4% and with liquid core fibres, which have about three times the dispersion of glass, the effects of material and modal
600
700
800
900
1000
1100
1200
Fig. 1 1 . Bandwidth limit due to material dispersion: variation with wavelength for constant line width AI = 40 nm. Curves correspond to respectively silica and various glasses.
VII,
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FIBRES WITH CORE AND CLADDING
347
dispersion become more comparable. Many workers have investigated how fibre bandwidth is related to material dispersion. Fig. 11, due to DYOTT [1974], shows how fibre bandwidth depends on LED wavelength [1974a], shows for different glasses, while Fig. 12, due to TIMMERMAN maximum bit rate as a function of source bandwidth for a specificmaterial.
Gbitk.
,GaAs-laser
-2 -a Lz
'\\\
- \
He!&-laser
10-
a
E
1X
!i 0.1
1
I
In a multimode waveguide, where the second term in equation (22) may be significant, it is useful to express the incremental delay AT for a given mode in terms of the mode number A4 and the total number of modes MT which the fibre will support. To do so we must first relate the total number of modes to the V value of the fibre. Several methods have been used. For large V , most modes have U close to the asymptotic value U , . To obtain MT; MARCUSE [1974a] proposes that the roots of Jn(Um)= 0 then be counted with Urn< V and making use of the asymptotic form
with m = 1, 2, 3,
...
For constant Urn,the integer values of n and m satisfying equation (23) lie along parallel lines as in Fig. 13. Neglecting the term 3 in equation (23), all values of n and m lie within the triangle. Because there are two
348
O P T I C A L FIBRE W A V E G U I D E S
n
L
Fig. 13. n, m,mode number plane. Dashed curve indicates the location of points n + 2m = constant.
polarisations and two mode types (HE and EH) for each value of n and m, the total number of modes is four times the area of the triangle. As the maximum values of n and m are 2 V/nand V/nrespectively, the total number of (trapped) modes MT is given by
Fig. 14 shows that the above represents an underestimate. A more accurate value can be obtained by assuming that the beam width aM of all modes of a fibre with core of radius rl is 2 / n r , . Then if the associated solid angle is nu: and the acceptance solid angle no:, the number of modes accepted by the fibre is given by
which is seen in Fig. 14 to be quite accurate even at low V values. The above approach was first adopted by GLWE[1971]. An alternative method
FIBRES WITH CORE A N D C L A D D I N G
V, (cut-off
349
V value1
Fig. 14. Computed total number of modes MT as a function of cut-off V value. Curve B includes modes of the form HE,, and EH,, with both polarisations present. Curve A includes modes with only one polarisation present.
due to PASK,SNYDERand MITCHELL [1975] gives the same result. The use of a compound mode number by GLOGE [1972a] M = (n+2rn)
(26)
together with the knowledge that for all V, U z U, (the cut-off value of U ) allows us to write
u, = (2M)Q
(27)
then from equation (8)
and from equation (16) V d2(Vb)
-
dV2
_
-
2u2 V2
N
2M I v - - -
MT
(29)
350
OPTICAL FIBRE W A V E G U I D E S
CVII.
0
2
equation (22) may then be rephrased as
The first term of equation (30) is positive whereas the second term is negative. Thus, as proposed by GLOGE[1971b], the possibility exists for some compensation of material dispersion by waveguide effects in the case of higher order modes for which M w M T . The possibility of compensating material dispersion by appropriate choice of core and cladding glasses has been proposed by DYOTT and STERN [1970], SMITHand SNITZER[1973], also JURGENSEN [1975]. The latter author shows that no suitable glasses exist which will provide compensation in a monomode fibre at wavelengths below about 1.3 p. Also, so far, attenuation considerations have dominated the choice of materials for optical fibres and the limitations on bit-rate imposed by material dispersion have taken second place. 2.7. FIELDS
Table 4 presents the field components of the HE,, and EH,-, mode under the assumption that n , x n 2 . Fig. 15 shows computed transverse fields of the HE11 , and HE2 modes and the linearly polarised nature of these fields is evident. It may also be demonstrated as follows. If the Cartesian components of the transverse electric field are formed from the cylindrical components we have
when a modal combination is formed so that
E, = J v + ,(K, Y) sin (v + 1)O’ + J,- l(Kl r) sin (v - 1)O’
(32)
0
Jv(Kr)cos v0’
(33)
The integer v in equation (33) is associated with a quasi-degenerate modal pair corresponding to the HE,,. l)m mode and the EH,,- ,), mode. GLOGE [1971a] calls such a modal combination an LP mode choosing these
FIBRES W I T H CORE A N D C L A D D I N G
351
TABLE 4 Field components when n, = n2
Component
Function r < r1
r > r,
* Upper sign denotes HE,, mode, lower sign denotes EH,-,, mode. initials because the field is almost linearly polarised. The notation LP,, identifies the above mode pair. An extensive discussion of this topic has been presented by MARCUSE[1972b, 1 974a). If initially we assume that a artesian field, as expressed in equation (33), represents a solution for the configuration of Fig. 2(b) and provided n, z n2, then, following ARNAUD119741, continuity of Ey and aE@r at r = r1 leads directly to the characteristic equation
where the functions are defined with eq. (1). This solution corresponds to either and HE,,. l)m mode or an EH,,- l)m mode. The appropriate characteristic equations for HE,,,,, and EH,,, modes are then
With a little manipulation it can be shown that equation (35) is identical to equation (4) in the limit n , = n2. There is, however, considerable simplicity in the above approach, especially if an extension to a multilayer fibre is envisaged. Aside from the relevance to fibres with several boundaries, the technique is useful when fibres with continuously graded refractive index profiles are analysed.
352
O P T I C A L F I B R E W AVE G UI D ES
Fig. 15. Normalised field intensity as a function of radial position within cladded optical fibre corresponding to Fig. 2(a). Parameters as Fig. 7. 2.8. POWER FLOW
The power flowing in core and cladding regions is obtained from an integration of the longitudinal component of the Poynting vector over the appropriate cross-section. If the longitudinal fields exhibit a dependence 'OS n8' sin
w,§21
FIBRES W I T H CORE A N D CLADDING
353
On writing v = n+ 1 and recalling that HE,, and EH,... l m modes are nearly degenerate, the power in the core region for an LP,, mode is given by
If v = 0, the above expression is doubled. From a similar integration, the power in the cladding region of Fig. 2(b) is given by
K, is defined in Table 3. GLOGE [1971a] gives a related expression which is twice as large since it includes the power in both orthogonal polarisations. He also removes the constant a, by expressing the field at r = rI as
In this notation the total power PTis then given by
while Pcore
=
~
1-
PT
U2
-[l-Ky]
VZ
and Pclsd u2[l-K,]. ---
v2
PT
For large V, 'K, =
1-
1 -
V
then
and
Pclad -- -u2
P,
v3
354
O P T I CAL FIBRE WAVEGUIDES
V
Fig. 16. (power in cladding)/(total power) and (power in core)/(total power), as a function of normalised frequency V.
Fig. 16, due to GLOGE [1971a], shows the above ratios as a function of V. CLARRICOATS and CHAN[I9731 have produced similar results for the configuration of Fig. 2(a) and have found that the role of the outer cladding boundary is not significant other than near the core mode cut-off. The above figures show that for all modes above their core mode cut-off, their power flow is concentrated into the core. If equal power flows in every mode, as could be the case if excitation is by means of an LED source, the total power flow in the cladding for all modes is given by
2.9. ATTENUATION DUE TO A LOSSY LAYER
Consider now a cladding region which beyond r > r2 possesses a loss tangent tan6. The attenuation coefficient a of a mode which is not too near cut-off is then given by
VII,
0 21
355
FIBRES W I T H C O R E A N D C L A D D I N G
For large V we obtain the simple result
If the loss region has only a finite thickness dr and its inner boundary still lies at r = r 2 ,
CLARRICOATS and CHAN[19731 have computed the attenuation characteristics of a number of lower order modes of a cladded fibre with a layer of finite thickness, by using exact expressions for the fields. Their results, which are shown in Fig. 17, conform closely to the law expressed in equa-
\
lC01
\
Fig. i7. Attenuation coefficient as a function of normalised frequency for optical fibre configuration of Fig. 2(a). Parameters as for Fig. 7.
356
O P T I C A L FIBRE W A V E G U I D E S
[vn,
52
tion (48)over the linear range. The very substantial differential attenuation which exists between the HE,, mode and next higher order mode triplet, provided V does not approach 2.4, makes it possible to provide powerful suppression of all higher order modes in monomode fibres. This requires a careful choice of fibre parameters and the application of a lossy paint or a lossy plastic coating to the cladding boundary. KUHN[1975] has discussed this optimisation problem in detail and has included in his study
Fig. 18. Leaky mode attenuation coefficient as a function of ratio r J r , with Vvalue as parameter, for configuration of Fig. 2(a) with n 1 = 1-05; n2 = 1.025; n3 = 1.5.
VI1,
§ 21
351
FIBRES WITH CORE A N D CLADDING
the case when the outer lossy region (which is assumed to be semi-infinite) has a real part of refractive index greater than n, . Under these circumstances so-called leaky mode conditions prevail and the attenuation rises rapidly with increasing n, or decreasing V. CLARRICOATS and CHANG [I9721 have analysed the structure of Fig. 2(a) when n3 > n, > n, and Fig. 18 shows computed attenuation coefficients as a function of V, due to CHAN [1973]. Although the parameters are relevant to a microwave application the main features apply in optical waveguides. 2.10. LEAKY MODES
A very comprehensive study of leaky modes has been undertaken by SNYDER[1974] and his co-workers for the configuration of Fig. 2(b). Leaky conditions occur when I/ is less than the cut-off value V, = U, for a given mode and then equation ( 5 ) has complex roots. Fig. 19, due to SAMMUT and SNYDER[1975] shows the real and imaginary parts of U as a function of V, while Fig. 20 shows the leaky mode attenuation coefficient 101
I
I
I
I
1
2
3
1
I
1
I
I
I
5
6
7
8
9
9t
I
U
V Fig. 19. Complex eigenvalue U as a function of normalised frequency V for optical fibre configuration of Fig. 2(b).
358
O P T I C A L FIBRE W A V E G U I D E S
1
2
3
4
5
6
7
[VII.
8
§ 2
9
V
Fig. 20. Attenuation coefficient a as a function of normalised frequency V corresponding to leaky modes of HE,, class n = 2, 3, 4, 5, 6, 7.
a where
Below cut-off, excepting modes with n = 1, Ur provides an analytic continuation of U above cut-off and Ui gives rise to attenuation of the mode in the z direction. For modes with n = 1, there are no solutions just below cut-off, although for HE,, modes, m 2 2, solutions do exist when V is decreased further below cut-off. However, these solutions are not continuous with the bound modes. Expressions at V c - V = AV for U,- Uc = AUr and Ui are given by SNYDER[19741 as
AV n-I’
=-
n 2 3
VII,
0 21
FIBRES WITH CORE A N D C L A D D I N G
ui =
-
ZAV In 1AVI2 ’
359
n=2
For n > 1 the modes are only weakly attenuated compared to the below cut-off modes of a dielectric slab waveguide. A physical explanation is to be found in the ray diagram of Fig. 21. There we find that weakly leaky modes correspond to rays which satisfy 8 > 8, with 6 < 8,. These rays would appear to be trapped according to geometric optics since the angle made with the tangent plane 6 , is less than the critical angle for total internal reflection. However, as demonstrated by SNYDER[19731, MARCATILI [1973] and MARCUSE [1973b] for cylindrical fibres, it is the angle 0 which the ray makes with the generatrix of the cylinder which determines whether a ray is trapped or the corresponding mode is bound. When both 8 and 6 are greater than 8,, the corresponding ray is refracting and the leaky mode attenuation is large, see Fig. 20. Such modes are called leaky refracting modes by SNYDER[1974] and the associated rays are leaky refracting rays. An anatomy of modes on dielectric structures, due to SNYDER [1974], is contained in Table 5 . The field of a weakly leaky mode decays exponentially to a radius rtp whereafter propagating conditions are found. This behaviour is precisely analogous to that found in cylindrical antennas where a stored energy
Fig. 21. Ray diagram indicating that some leaky rays in the configuration of Fig. 2(b) are trapped in the configuration of Fig. 2(a). See also Fig. 6.
360
O P T I C A L FIBRE W A V E G U I D E S
[vir.
0
2
TABLE5 Anatomy of modes on dielectric structures Bound modes
I
Leaky modes
(1) Energy is unattenuated (1) Energy with the structure is attenuated due to radiation, i.e. as it propagates, ix., the the fields are made up by waves that undergo partial reflection. fields are made up by waves that undergo total internal reflection.
(2) Fields external to the (2) Fields oscillate (radiate) at finite distances from structure. structure are evanescent. (3) Characteristics found from the real roots of eigenvalue equation. (4)
modes gives bound energy of structure.
(3) Characteristics found from certain complex roots of the eigenvalue equation. On structures that support bound modes, leaky modes are the bound modes below their cutoff frequency. modes approximate the radiation field within and external to the structure far from the source.
Examples 1 Exist only on non-finite Exist an all dielectric structures structures with an index of ‘refraction greater then their Refracting leaky modes Tunnelling leaky modes surrounds. Examples: Cylinder or slab in free space Attributes
( I ) Radiation appears to (1) Radiation appears to originate at boundary of originate some distance the structure, i.e.,the fields from the boundary of the everywhere external to structure. The fields the structure have an external to, but immediately oscillatory behavior. adjacent structure are evanescent and oscillate at greater distances from the structure. (2) These modes are strongly leaky. As I, -P 0, the leakage is nonvanishing.
(2) These modes are weakly leaky. As I , -+ 0, the leakage vanishes.
a Examples
Exist on all dielectric structures.
Exist only on structures that have (a) an index of refraction higher than that of their surrounds and (b) have curved boundaries that are concave inwards. The dielectric sphere
vn, 0
21
FIBRES WITH CORE A N D C L A D D I N G
361
TABLE 5 (continued) Bound modes
Leaky modes
1
Bound or trapped rays The rays that form the bound modes undergo total internal reflection. Refracted rays
Tunnelling rays
These rays undergo partial reflection due to refraction at the boundary.
These rays undergo partial reflection due to tunnelling upon incidence at the curved boundary.
region precedes the domain of propagation as one moves outward from the cylinder boundary. The higher the azimuthal dependence of the mode, the greater rtp as indicated by
where
Qnm= U 2 - V 2 .
(55)
Fig. 22 shows the evanescent and propagating regions asso ia d with weakly leaky rays. As the energy appears to tunnel across the region rtP- Y, , Snyder designates such modes and rays as tunnelling modes and tunnelling rays respectively. Expressions for the attenuation coefficients of leaky rays are presented by SAMMUT and SNYDER [1975] while Fig. 23 shows ale, as a function of OlO,. Although leaky modes can play a significant role in the transport of power in the configuration of Fig. 2(b), especially if V is very large, it should be recognised that Fig. 2(b) is only an approximation to that of Fig. 2(a) and there leaky modes will not prove significant unless n3 > n,. This is evident in Fig. 21 which shows that a leaky ray of the configuration of Fig. 2(b) is trapped in that of Fig. 2(a). Because of the large difference between A = (n,- nz)/nl and A , = $[n; - l]f/n,, a very large number of modes which would be leaky in Fig. 2(b) are in fact trapped by the outer cladding boundary and these are the so-called cladding modes of the structure. However, if n3 is lossy, many of these modes will be highly attenuated.
362
[w § 2
OPTICAL FIBRE W A V E G U I D E S
Fig. 22(a). Geometrical interpretation of the fields inside and outside of optical fibre configuration of Fig. 2(b) under leaky ray conditions. Ci, denotes interior caustic; C,,, is the caustic beyond which radiation occurs. The unshaded regions correspond to rays i.e. oscillatory fields.
-2
Fig. 22(b). Field of a leaky mode for the configuration of Fig. 2(b). For r less than rtpr the field resembles a bound mode. For r greater than rip, energy is radiated. 0 is the direction of radiation in the far field.
FIBRES W I T H CORE A N D C L A D D I N G
363
Q/Qc
Fig. 23. Diagram showing the U plane for leaky (LR) and refracting (RR) rays.
2.1 1 . ATTENUATION DUE TO BENDS
If a lossless fibre is bent into an arc as in Fig. 24(a), energy in a mode which would propagate without attenuation when the fibre is straight, leaks from the fibre and the mode suffers attenuation. UNGER[1964], MARCATILI and MILLER[1969] were the first to consider the problem of attenuation in curved dielectric waveguides. MARCATILI and MILLER [19691 studied in particular, slab waveguides with uniform refractive indices and the relative behaviour of slabs with uniform and graded indices was taken up later by GLOGE[1972b]. The above authors used modal methods to obtain their attenuation coefticients, whereas SNYDERand LOVE [1975] determines attenuation by generalising a ray treatment. He extends the Fresnel transmission coefficient for plane media so as to account for the curvature and obtains a result for slab waveguides which is in accord with GLWE[1972b] (whose expression in reference equation (1 1) should be halved). SNYDER[I9741 has also derived an attenuation coefficient for a curved cylindrical fibre as an extension of the curved slab case ;an outline of his method follows. By considering the curved slab boundary, in Fig. 24(b), as a perturbation of a plane interface, a modification of the analysis which leads to Fresnel’s
364
O P T I CAL FIBRE W AVE G UI D ES "2 < "1
n 3 s n2
(b) Fig, 24. Configurations for curved slab and cylindrical optical waveguides. (a) Curved cylindrical fibre; (b) Curved slab waveguide.
laws yields a transmission coefficient T(e)= 1-
Power of reflected ray Power of incident ray
k' is the wavenumber for plane wav8propagation in the medium n, The angle between successive reflections of the r,ay, A& is given by .-> **
A+ =
2(Po -P A
Po9
.
The power of the mode which is contained in the slab q(0) is
VIL
§ 21
FIBRES W I T H C O R E A N D C L A D D I N G
365
Finally, the attenuation coefficient a (= 2pa' where a' is the coefficient used in SNYDER[19741) is given by
thus a = +k'(ef -
e2)exp -(+k'po(8:
- e2)t).
According to SNYDER [19741, for a circular fibre supporting many modes, only meridional rays which strike the boundary at a radius po suffer significant bending loss and their attenuation coefficient will be the same as for a slab waveguide, except for the factor q(8) which can be found from Table 3. Near cut-off we can neglect K in the expression for q(8), then on substituting for (U/V)' from equation (12)
Substitution of equation (62) into equation (60) yields the attenuation coefficient. This is dominated by the exponential term, the argument of which vanishes when 8 approaches 8,. Fig. 25, due to GLOGE[1972b],
Q/Qc Fig. 25. Curvature loss as a function of normalised ray angle for graded refractive index and stepped refractive index planar waveguides.
366
OPTICAL FIBRE W A V E G U I D E S
[VII,
§ 2
shows the curvature loss as a function of eje, for slabs of thickness t with uniform and graded profiles where - and 0 are related through e2
= (ez - 2t/p0).
2.12. MODE COUPLING DUE TO PERTURBATIONS
From the above description, we see that it is the energy in rays whick. are close to the critical angle in the straight fibre, which is lost first when the fibre is gradually bent. In modal terms, it is those modes which are closest to cut-off which first couple to the radiation modes and which give rise to attenuation. Only for a severe bend would low order modes in a multimode fibre, couple directly to the radiation field. Mode coupling due to perturbations in optical waveguides including bends, has been extensively considered by MARCUSE [1969, 1972, 1973a, [1969a, b]. 1974a) following an initial treatment of the problem by SNYDER GLOGE [1973] and MARCUSE [1972a] have shown how mode coupling influences pulse propagation while SNYDER[1969~1,also CLARRICOATS and CHAN[19731 have considered how isolated inhomogeneities couple power between modes and from a mode to the radiation field. When the core refractive index is perturbed by h ( r , 8',z) the coupling coefficient between the pth and qth modes is given by MARCUSE [1974a] as
where ep and e, are the linearly polarised vector fields associated with the modes of the unperturbed fibre as given by the equations of Table 4. 2.13. COUPLING DUE TO BENDS
For a fibre with a section of constant radius of curvature po and length L, the ratio of the power coupling coefficient between the pth and qth [1974a], modes (with v > 0) at L is, MARCUSE
where
P, iP,. Notice that the power oscillates between the modes with a period which
VIJ,
§ 21
361
F I B R E S W I T H CORE A N D C L A D D I N G
,guided modes
radiation modes
----
I
I
I
I
I
I
I
I
I
f
I
PN
I
l
,
l
4
c
P
depends on the beat wavelength A, see Fig. 26, where ’4 = 27Wp-8,8,) x 4r,p
far from cut-off, equation (64) reduces to
where 8 x ep x 8, is the ray angle within the fibre as defined in the previous section. Equation (66) shows that coupling between modes due to a bend is inversely dependent on the sixth power of the bend radius. The effect of random bends in a fibre has been considered by GLOGE [1972] using a ray treatment and by MARCUSE[1969, 1973a1,who uses modal methods to show that the power coupled between the pth and 4th modes Ppqis PjJq =
I ~,,I2F(P,- 8,).
(67’
In equation (67), the z dependence of K,, (see equation (63)) has beell removed leaving Kp4 with F(B, -p,) representing the power spectrum of the z dependent curvature function. Only the spectral component with wavelength A, given by equation (65) influences coupling between the pth and qth modes. For modes far from cut-off,
where ( ) denotes an ensemble average and the integration extends over the fibre length L. Fig. 27 due to GLOGE[1972a], shows how the output beam angle and
368
O P T I C A L FIBRE W A V E G U I D E S
[VII,
8
2
40-~ 8
0
0 0
- 300
C=7x10-5radZ’m
VI (u
0
E!
6N
r
-0
0
- 200
NL
a
n
E
0
0
C 3
0
0
- 100 2
I
0
2.5
I
5.0
I
7.5
I
I
10.0
12.5
15.0
length m
Fig. 27. The output beam angle and total number of modes as a function of length in a multimode fibre which propagates 700 modes. The straight line represents a best fit to a theory which assumes mode coupling arises due to random bends. C is the curvature spectrum function.
total mode number, varies with length along a fibre where coupling of neighbouring modes is due mainly to random bends. The straight line represents a best fit to theory in which the coefficient, labelled C by GLWE [1972aJ, is the curvature spectrum function. C is given by
where 6 is the amplitude of sine waves sin (2nz/A) used to model the random curvature and v] the number of undulations per unit length. The above results refer to a multimode fibre. In a single mode fibre, radiation loss can occur due to bends, core diameter fluctuations, or on account of inhomogeneities, while coupling to cladding modes can also occur. Fig. 28, due to CLARRICOATS and CHAN[1973], shows for a monomode fibre with finite cladding boundary, how the normalised power which is.both coupled into the cladding modes from the HE, mode, and is transformed into radiation, varies with the location of an isolated inhomogeneity. The curve labelled HE,, , shows the power coupled back into the HE,, mode by the inhomogeneitv. Following SNYDER[1969a, b], the valid assumption is made that equal amounts scatter in both forward and backward directions. Evidently, coupling to radiation predominates over coupling to the cladding modes. The latter varies with r/rl in a manner
"11,
§ 21
FIBRES W I T H C O R E AND CLADDING
369
rlr
Fig. 28. Normalised coupling coefficient between .HE,, mode and other modes of optical fibre, in the configuration of Fig. 2(a), as a function of the position of an inhomogeneity of volume V and permittivity difference A&. Also shown is the coupling into radiated power.
which is determined by the convolution of the field patterns of the HE,, mode and the coupled mode. The influence on system performance of multiple reflections due to scattering, was first analysed by DYOTT and STERN[1971] and an extended study has been made by HUBBARD[1972]. 2.14. MODE COUPLING AND PULSE DI~PERSION
Although, in general, one aims to minimise inhomogeneities during fibre manufacture, it transpires that provided coupling to the radiation field is not too great, intermodal coupling actually plays an advantageous role in multimode fibres. PERSONICK [1971: was the first to recognise that, when
370
OPTICAL FIBRE W A V E G U I D E S
CVK
02
mode coupling is present, pulse dispersion is dependent on L* rather than L, subsequently GLOGE[1972] and MARCUSE [1974a,b] studied the problem in depth and in the following section, we present a resume of the analysis of GLOGE [1973]. In a multimode fibre with large V,many modes propagate and associated rays are so densely packed that their distribution as a function of 8 can be considered continuous. The power distribution in the fibre P(0, z, t ) then satisfies
where the first term on the right-hand side arises from attenuation effects at the cladding and core cladding interface, which increase as 0’. The coefficient A is measured in m-l rad-’; 0 independent loss is omitted but this can be incorporated in the final solution. The second term is associated with the differential group delay according to ray optics. The third term arises from mode coupling between closely adjacent modes and takes the form of a diffusion process in the ray picture. D = (4/n2)C, see equation (69), is the coupling coefficient which is assumed in the following to be independent of 6. Gloge also discusses the influence of a 8 dependent coupling coefficient. Equation (70) may be solved using the Laplace transformation (71) then equation (70) becomes
0, = (4D/A)*
(77)
yrn = (4DA)*.
(78)
VI1,
6 21
FIBRES W I T H CORE A N D C L A D D I N G
371
For C.W.excitation, (s = 0), the angular width 0 (z, 0) changes monotonically from 0,to 0, as z increases. 0, characterises a distribution which propagates without change and with the minimum overall loss coefficient Y,. Closed form solutions of (74) exist for very small and very large values of y,z. For the former, if the Gaussian angular distribution at the input and at infinity are the same,
The denominator expresses the loss in the short distance z, while the terms in the numerator indicate that the distribution has been preserved and the delay is precisely that predicted by ray optics in the absence of coupling. By integratingp(8, z, s) with respect to 8 and taking the Laplace transform, the impulse response Q(z, t ) of the fibre is obtained as
For y,z zs- 1
Fig. 29 due to GLOGE[1973], shows the impulse response according to equation (81) normalised for equal peak values and plotted versus normalised time for different values of ymz. T is the 1 / width ~ which the impulse response Q would assume at z = l/y, if coupling were absent. Gloge has also calculated the delay 6,(0) and the half-width of the pulse t,(8), as a function of ymz, see Figs. 30 and 31. We see that for y,z 1, the input pulse (z = 0) propagates without broadening and merely suffers a mode-dependent delay nQ2z/2c.Then, with increasing ymz, pulses in the individual modes widen and overlap. When y,z > 1, the pulse width in all the modes increases mainly as T ( ~ , z ) This ~ . effect is explained by the energy exchange mechanism which ensures that there is a distribution between modes possessing different group velocities. Only the edges of the distribution correspond to propagation almost entirely in either fast or slow modes. Fig. 31 shows the relative pulse width as a function of y,z
O P T I C A L F I B R E WAVE G UI D ES
Fig. 29. Imuulse resuonse according to eauation (81). Normalised for eaual ueak values and plotted-as a function of norkalised time for different fibre iengths. 1.2-
c
1.0
-
0.8
-
cp L
0
t 0.610
0.4
u
-
0.2
04
0.6
0.8
1-0
1. 2
1.4
Y -2 Fig. 30. Delay and time spread of the fibre output on-axis and at an angle 0 = 0,.
1.6
FIBRES WITH CORE A N D C L A D D I N G
373
Y w2 Fig. 31. Relative width of the impulse response as a function of the normalised fibre length. The two straight lines show the asymptotic behaviour for very short and very long fibres.
and reveals the important and desirable result that in the presence of coupling in long fibres, the pulse dispersion increases only as L*. In the above treatment, it has been assumed that the steady state distribution at z = co is such that 0, < 8, ,i.e. that the mode angle corresponding to the 1/e point in the Gaussian power distribution is much less than the complement of the critical angle. If this is not the case, the steady state distribution is generally determined by a sharply rising loss term at 8, =8, rather than by the quadratic term A8’ as in equation (70). Although there will be an increase in overall attenuation, the pulse width can then be smaller by a factor of several times compared to that shown in Fig. 31.
374
OPTICAL FIBRE W A V E G U I D E S
[VII,
52
2.15. REDUCTION OF PULSE DISPERSION BY INTENTIONAL INHOMOGENEITIES
MARCUSE [1974b] has made a very detailed study of the compromise between pulse width reduction and the excess loss caused by mode coupling. He has also explored the possibility of providing a selective coupling mechanism which would prevent all but a minimal amount of power from escaping due to coupling between high order modes and the radiation field. This is possible, in principle, if refractive index fluctuations could be created with a spectrum function possessing a sharp cut-off at high values of (pp-fi4), i.e. at short beat wavelength A . Since power in modes with high azimuthal number v, is confined to regions of the core such that r > rmax= vmaxrl/V, the influence of the fluctuations on these modes can be further reduced if they are confined to the region Y < rmax. By assuming a flat spectrum between A = 00 and A = 0.13 cm; MARCUSE [1974b1 predicts a ten fold reduction in pulse width for a 1 km fibre length (compared to the case without coupling). For this calculation V = 40, A= [((An)2)]* = and rmax/rl= 0.8. Although such improvements are spectacular, they have not yet been achieved experimentally because of the difficulties associated with imposing controlled refractive index fluctuations during manufacture as discussed by MARCUSEand -BY [1975]. Differences in mode coupling in fibres with similar parameters are evident and these are probably associatedwith differences in the random fluc tuations which are always present in fibres. They occur for example, when fibres are pulled using the double crucible method of manufacture especially
01
1 10 100 200 modulus of drum surface in kg I mm
Fig. 32. Distortion loss as a function of drum surface modulus. Fibre diameter = 120 pm. Core diameter = 80 pm. Relative index difference = 0.5%. rms drum surface roughness = lpm. Correlation distance = 1 mm. Mean lateral force per unit length is the parameter.
VIL
5 21
FIBRES W I T H CORE A N D C L A D D I N G
375
when different equipment is used. GAMBLING, PAYNEand MATSUMURA [1974] have shown that the tension of a fibre wound on a drum influences mode coupling and attribute the effect to bending stresses. However, GARDNER [1975] shows that the surface upon which the fibre rests also influences mode coupling and this factor may have contributed to Gambling’s results. This latter factor is obviously significant when designing an encapsulation for a fibre and a thorough investigation of this problem has been undertaken by GLOGE[1975]. He relates the elastic properties of the surface which the fibres contacts, to the spectrum of fluctuations which cause loss. As an example, Fig. 32 shows how the properties of the winding drum influence loss and clearly suggest the choice of a plastic drum surface, such as polystyrene, and a low winding tension. 2.16. EXCITATION
2.16.1. Incoherent
The excitation of an optical fibre by an incoherent source was first considered by POTTER[1961] and more recently by PASKand SNYDER [1974]. The latter authors have also considered the influence of leaky ray attenuation on the power carried to the output of the fibre. If the source illuminates the whole end face of the fibre and the angular distribution of rays Z(8,O’) is given by
r(e, 0’) = z,
COS
e
(83)
corresponding to a Lambertian source, the power in the trapped rays satisfying 8 8, is
-=
PTR= n’r: I. sin2 8,.
(84)
The total power radiated is n2r:Zo thus the ratio PTR/PT = sin’ 0, x 8: = 24.
(85)
However, rays which, according to Fig. 6, satisfy 8 > 8,, S < 8, also transport power, only with increasing attenuation as 6 -+8,. For leaky rays,
Because the leaky rays are attenuated, only a percentage of the total power guided by the fibre at z = 0, remains after distance z. Fig. 33 due to PASK and SNYDER [1974] shows how the percentage varies with z/rl and I/. For a typical multimode fibre with V = 50, and 8, = 0.1, about 10% of the
376
OPTICAL FIBRE WAVEGUIDES
p!
L W , 9; 2
geometric optics
v- 200
zlr, Fig. 33. Fraction of incident power within fibre as a function of fibre length normalised to fibre radius. Parameter is the V value for the fibre.
power remains in leaky-modes at a distance of 1 km. However, if the cladding is made from high loss material both leaky- and higher-order bound modes will experience significantly higher attenuation than lower order bound-modes and this causes a more substantial change with distance in the power distribution among the rays as has been experimentally observed by KAISER,HARTand BLYLER[1975]. The case when the source and fibre are separated has been considered in depth by Y ANG and KINGSLEY [19751, who, together with ABRAM, ALLENand GOODFELLOW [1975] have also considered how a lens can improve coupling between an LED and a fibre when the source radius is less than the fibre diameter. A summary of results due to YANGand KINGSLEY [1975] is shown in Table 6. 2.16.2. Coherent
We have just seen that an incoherent source with illumination angle exceeding the acceptance angle of the fibre, excites all the bound modes with near uniform intensity. By contrast, a coherent source may be arranged to excite one mode efficiently although, in the case of a multimode fibre, this generally calls for the use of a lens. Insight into the general problem of coherent excitation is provided by considering the case of on-axis excitation by a parallel beam of monochromatic energy. This situation approximates very well a focussed gas laser and, with a compound-lens, that of a solidstate laser. The problem was first studied in the present form by SNYDER [1966] following earlier work on the coupling of fibres by JONES[1965].
vn, § 21
311
FIBRES W I T H C O R E A N D C L A D D I N G
TABLE 6 Summary of results of analyses for disk and strip geometries with and without a lens (YANG and KINGSLEY [19751)* Source geometry
Power into fiber
Disc, dia = D
s’B, TD’ sin2( 0 1 / 2 )
Fiber normal to junction
36pW
0.4 W
Focused disc, dia = 2F tan 0, Lens, dia = D and focal length = F. Lens and fiber coaxial with junction normal
21r2B,TF2 tan’ O1
~ J V F ’tan’ 0,
Rectangular strip
-I
Elec. power required JVD~
F2
+
x [ l - F2 D2/4
- n2 B , TD’ tan2 0, 4 % -
31 pW
0.031 W
2.16 B1 TD’n, tan ( 0 , / 2 )
D’ JV -no cot 0, 2
0.4 mW
6.2 W
3s B1 Tn, FD tan’ O1 2 (1-(2F tan O,/D)’)
F3 J V . 8no - tan2 0, D
0.26 mW
0.28 w
width = D
D length = no - cot O1 2 Fiber axis in center of strip Focused rectangular strip
width = 2F tan 0, 4n0 F’ tan 0, length = D(1-(2F tan OJD)’) Lens, fiber coaxial with center of strip
* The numerical values of the coupled powers and the electrical power input required assume a radiance of 30 W(ster. cm’) for a power input, J K of 2 x lo4 W/cm2, and a fiber having a numerical aperture of 0.14, or an acceptance half angle of 8”. T = transmission coefficient, B , = radiant power per unit sold angle. The inset to Fig. 34(a) shows the configuration and for the present, we consider the angle of incidence 8, to be zero and ri = rl. The field just inside the fibre, at z = 0, is assumed to have the same distribution as in the incident wave. We can then expand in terms of the normal modes and on applying orthogonality, obtain an expression for the amplitude coefficients of the modes:
E = cage4 4
where E = E,$, = 0,
H
=
Ca,h,. 4
r < rl r > r,
(87) (88)
378
O P T I C A L F I B R E WAVEGUIDES
Fig. 34. Excitation of an optical fibre in the configuration of Fig. 2(a). (a) Normal incidence rz/rl = 5, --- r z / r l = 15, ----- core excitation. Fibre parameters as Fig. 7. mode approximation. ~
Let
J.'px h:.
ds = sq
and
then aq
= sols,
FIBRES W I T H CORE A N D C L A D D I N G
10
*
bEfl /
379
(b)
angle of incidence rad
(b) Excitation efficiency as a function of angular incidence for cylindrical beam of uniform intensity incident at an oblique angle to the axis. V = 10; ri/ri = 1 . Fibre parameters as in Fig. I .
380
O P T I C A L FIBRE W A V E G U I D E S
[vn, 0 2
and the ratio, power in qth mode/incident power
SNYDER[1966] and Clarricoats [1970] have evaluated the above ratio without assuming that n, x n, , although an appreciable simplification follows from the assumption, which applies to all optical fibres used in communications applications. Then, if r2 = 00 (infinite cladding approximation),
4w2 --
u2v2'
-= 0,
for HE,, modes for EH1, modes.
For HE,, modes well above cut-off
w w
v,
then
As an example, when m = i, U , = 2.4 and P,/Pi -+ 69%. When the radius of the incident beam ri is not equal to r l
(
2url)2[(')z - _ri Kl(Wri/r1)]2 Pi VWri rl K,(W * For off-axis incidence, HE,, and EH,, modes are excited since the incident plane wave gives rise to a phase-variation about the azimuthal coordinate 0'. The above method can be applied, although the integrals are more [1966] gives a result when r2 = co and complicated to evaluate. SNYDER n, < = n, which leads to equation (99), while CHAN[1972] has undertaken a more general study including the case when r2 is finite corresponding to the inset to Fig. 34(a): p q
=
VII,
Q
31
F I B R E S WITH N O N - U N I F O R M R E F R A C T I V E I N D E X
38 1
8 is the ray direction associated with the qth mode in the fibre while 8, is the direction of the refracted incident ray within the fibre. When 8 =Or, equation (99) simplifies to give
We see from equation (100) and Fig. 34(b) thatthe peak excitation occurs when the refracted ray direction and modal ray directions coincide and that the intensity decreases with increasing n. Fig. 34 also quite accurately describes the radiation pattern of a fibre excited in the appropriate mode. The above results refer to a uniform intensity input beam. The Gaussian beam at normal incidence was first considered by STERNand DYOTT [1971] while the oblique incidence case has been studied by CHAN[1972], MARCUSE [1970] also IMAI and HARA [1975]. Although excitation efficiencies are in general slightly higher than for the uniform beam case, as the input and modal fields more closely correspond, the behaviour is otherwise very similar.
93. Fibres with Non-Uniform Refractive Index 3.1. INTRODUCTION
In the preceding section we have seen that single mode fibres possess a large bandwidth which is limited mainly by material dispersion. They suffer the disadvantage that they are difficult to join without appreciable coupling loss because the power carrying region is only a few microns in diameter. They are also completely unsuitable for excitation by light emitting diodes (LED’S). These disadvantages are largely overcome in multimode fibres but with a substantial bandwidth penalty arising mainly from modal dispersion. While mode coupling can limit this dispersion, controlled mode coupling, which provides optimum coupling without additional loss, has yet to be achieved experimentally. Efficient single mode propagation may be achieved through a potentially multimode fibre by using a lens to improve coupling from a much smaller laser aperture, but such a configuration may prove hard to produce in a fully engineered system. Several alternative possibilities to the above fibre types have been investigated with a view to overcoming most of the above disadvantages. Among those with circularly symmetric refractive index profiles, two main types have emerged, both 9s a result of initial development in Japan.
382
O P T I C A L FIBRE W A V E G U I D E S
CVIL
0
3
These are the so-called W-fibre, which has the profile displayed in Fig. l(c) and which can exhibit monomode properties. The other has a continuously graded core refractive-index as shown in Fig. l(e); this fibre exhibits multimode properties but with inherent delay equalisation properties. In ray terms, the fibre has self-focussing properties and the registered trade name SELFOC has been applied to such fibres. Fibres manufactured using different techniques have refractive-index profiles which differ from the near parabolic profile but these can have quite satisfactory bandwidth properties, an example is shown in Fig. l(Q. 3.2. THE W-FIBRE
The double-clad fibre with a low index inner cladding (named the W-fibre because of its refractive index distribution) has been extensively investigated by KAWAKAMI and NISHIDA [1974, 19751. In common with the single-clad fibre with finite cladding radius, see Fig. 2(a), the dominant mode transforms from a core mode to a cladding mode at a non-zero Vvalue. However,
a) i )
Fig. 35. Characteristics of the eigenvalues and fields in a doubly clad (W) fibre. (ai) Was a function of U.Parameter 6 = r2 - rl . (aii) W as a function of U,showing shift of higher mode cut-off. Note that for the HE, mode Wis substantially greater at higher mode cut-off value of V for W fibre as compared with singly clad fibre. The field is therefore more tightly confined to the inner core. (b) & (c) Transverse field distributions within singly clad (b) and doubly clad (c) fibres.
,
VII,
§ 31
FIBRES W I T H N O N - U N 1 F O R M REFRACT IVE I N D E X
383
with the introduction of an additional cladding region of lower refractiveindex, the core radius at which the next higher mode propagates is larger than for a singly-clad fibre. Also, the field of the dominant mode can be more tightly confined to the core as compared to the singly-clad fibre. KAWAKAMI and NISHIDA [1974,1975] have used a perturbation method to solve the characteristic equation for the doubly-clad fibre. Fig. 35 shows W as a function of U for both singly-clad and doubly-clad fibres. The construction shows that at the cut-off point for the next higher order mode, U for the HE,, mode has a value much closer to 2.4 (which is the first zero of J,(x)) in a doubly-clad fibre, as compared to a singly-clad fibre. Thus the field in the doubly-clad fibre is contained more closely to the the core than in the singly-clad case which reduces the attenuation if the cladding has a higher loss than the core; it also minimises bending loss. Fig. 35(a) shows that the value of V = U at cut-off for the TE,, mode (which is degenerate with the TMol and HEzl modes) occurs at a value of 3.5 for the W-fibre as compared to 2.4 for a singly-clad fibre with the same core and outer-cladding refractive indices. This implies that a larger core radius can be chosen for a W-fibre which eases the problem of coupling from a laser. 3.3. GRADED INDEX FIBRES
3.3.1. General remarks The properties of multimode dielectric waveguides with continuously varying core refractive-indices have been investigated by many authors. In general, two main approaches to solving the propagation equation have been adopted, these are : (1) wave equation solutions, (2) solutions based on the properties of rays. The first category can be subdivided into (a) analytical methods applied directly to the wave equation, and (b) methods based on a piecewise continuous representation of the continuous refractiveindex distribution. Those in subcategory (a) can be further divided between solutions in which a polynomial is found to represent the wave function and those in which a W.K.B. solution is found. The latter can be more readily extended to cases where the refractive-index profile is of the form
n = no(l - 24
(:>’>’,
n = n,(l - 241, as shown in Fig. 36.
r a
(102)
’
384
OPTICAL FIBRE WAVEGUIDES
radius r
[vn, 0 3
-
Fig. 36. Refractive index profiles according to equation (101).
Of particular importance has been the search for an optimum value of a which yields minimum pulse dispersion. We first treat the case where a = 2.
3.3.2. Fibres with parabolic index variation Recognition that a cylindrical dielectric with a near parabolic variation in refractive-index distribution would be self-focussing goes back at least to the study of FLETCHER, MURPHYand YOUNG[1954]. They demonstrated that, according to geometric optics, meridional rays (4 = 0 in Fig. 6) are brought to a point focus on axis if the refractive-index n(r) satisfies nz((r)= n: sech' (r/l) (103) =
ni( 1- (r/l)' + 3 r / o 4 + . . .).
(104)
Thus, provided r/l -=K1, the condition for all meridional rays to have the same time delay requires the refractive index variation to have a nearly
VIL
5
31
385
FIBRES W I T H N O N - U N I F O R M REFRACTIVE I N D E X
parabolic variation with r. However, KAWAKAMI and NISHIZAWA [19681 demonstrate by use of the eikonal equation for rays, that for helical rays to have the same longitudinal velocity the refractive index-distribution should satisfy
=
ni[1-(r/1)2+(r/2)4+.
. .].
(106)
On comparing equations (104) and (106) we see that, according to ray optics, the refractive-index distribution required to eliminate pulse dispersion for meridional rays differs from that required for helical rays. The above ray types correspond to limiting cases in modal terms. Meridional rays strictly correspond to modes with zero azimuthal dependence, i.e. n = 0. However, when n is small, as for example with HE,, modes, the corresponding rays are almost meridional. Helical rays correspond to the highest order azimuthally dependent modes, such as HE,, where n is a maximum. Intermediate values of rn and n correspond to general skew rays. A general study of rays in inhomogeneous guiding media has been conducted by STREIFER and PAXTON[1971] whose results support the above conclusions relating to the particular cases. The choice of optimum profile using ray methods, has been studied in depth by ARNAUD[1975] while CHECCACCI, FALCWand SCHEGGI [I9751 have related ray and modal methods. Modal analysis of graded refractive index cylindrical fibres was first [1968], also by KURTZand reported by KAWAKAMIand NISHIZAWA STREIFER [19691. Following Kawakami and Nishizawa, we assume that
n2(r) = nz [I-,,
(bT+6(2~I)~(:r]
where the constant I in equations (103) and (1 05) 'corresponding to the ray period, has been rewritten as a/(2d)*. Then, if a/(24)* B 1, the wave equation becomes
=
By separating the variables
0.
(108)
386
OPTICAL FIBRE WAVEGUIDES
and
where K 2 = o2&,p0ng-fl2 E
Po
p 2 = 2A02 L n ; , a2
Whend=Oandif*+Oasr+oo,
K 2 = 02eopOng-f12 = k2k
= 2 - (2A)*(2m
a
B2
+n + 1).
LL is a Laguerre polynomial and the functions t,4n,, form an orthogonal set with n, m = 0, 1, 2, . . .. First order perturbation theory then leads to K2
=
2k 6 -(2A) *(2m n+ 1)(2A)[6m2 +6m(n+ 1) (n l)(n 2)], a a
+
+ + +
(1 16)
equation (1 16) provides the dispersion relationship implicitly relating /? to o.For zero pulse dispersion, we require that the group velocity ao/afi of the n, mth mode has the value corresponding to the axial refractive index no, namely c / n o . The following condition must then be satisfied,
B2
=
k2-(2m+n+l)-
24 a2
(117)
which implies that
6=
(2m+n+1)’ 6m2+6m(n+l)+(n+l)(n+2).
(118)
Thus, the coefficient of the fourth order term in equation (107) for n(r), depends on the mode number, demonstrating that a dispersionless condition cannot be attained with the above form for n(r). We note that for a meridional ray for which m > n, 6 = 5, while a helical ray for which n B m, 6 = 1, precisely in accord with equations (104) and (106). For intermediate [19741has investigated values corresponding to skew rays 3 .c 6 c 1, MILLER a square fibre with generalised refractive index variations and has computed
VII,
P
31
FIBRES WITH
NON-UNIFORM R E F R A C T I V E
INDEX
387
the impulse response as a function of the indices governing the variation. He has determined optimum values for the constants and the influence of small departures from these optimum values. All of the above authors assume that material dispersion can be neglected, an assumption which
ooooo
V results due to Kirchoff results obtained by a five steps stair case approximation of the permittivity profile
Fig. 37. Normalised propagation coefficient as a function of normalised frequency V for a ~ 000 results due to graded refractive index fibre n , = 1.51; n2 = 1.50; A = 4 . 5 lo-’. Kirchhoff, results obtained by a five steps staircase approximation of the refractive index profile. ~
388
OPTICAL FIBRE WAVEGUIDES
[VIL
83
influences their specific conclusions regarding optimum fibre parameters. We shall return to this subject shortly. The solution to equation (108) has been found on the assumption that the medium is unbounded in the transverse direction. KIRCHHOFF [19703 has considered the more realistic situation in which the fibre refractive index changes to a value n2 = (1 -gd) at r = a, where g 2 1, i.e. he includes the possibility of a refractive index step at r = a. Figs. 37 and 38 show the normalised propagation coefficient and normalised group velocity of
I
I
1 0 0 0 0 0
I
1
I
I
I
I
1
L 5 6 7 0 V results due t o Kirchoff results obtained by a five steps stair case approximation of the permittivity profile
2
3
-
Fig. 38. Normalised group velocity as a function of normalised frequency V for graded refractive index fibre with parameters as for Fig. 37.
VII,
5
31
FIBRES W I T H N O N - U N I F O R M REFRACTIVE I N D E X
389
lower order modes as predicted by KIRCHHOFF [1970] and independently computed by CLARRICOATS and CHAN[1970] who have used a staircase approximation to the continuous parabolic refractive index variation. These authors have examined the number of steps required in order to represent practical variations with accuracy and have found that five steps are sufficient.The virtue of their numerical method lies in its generality. For example, index profiles associated with diffusion have been successfully treated. These authors have solved the characteristic equation of the multilayer cylinder exactly thus generating a rather cumbersome matrix equation and rather time consuming computations. A valuable simplification arises with the use of the scalar approximation described earlier in relation to equations (34) and (35). Another numerical method based on an earlier study of an inhomogeneous dielectric waveguide at microwave frequencies by VIGANTSand SCHLESINGER [1962], has been described by YIP and AHMEW[1974]. They have applied their method to the configuration considered by CLARRICOATS and CHAN[1970] extending their results by including a finite cladding boundary. 3.3.3. Impulse response of graded index fibres Although the above numerical methods can be used to obtain the propagation characteristics of a highly overmoded fibre with an arbitrary refractive index profile, the impulse response can only be obtained after considerable computation effort. An alternative approach which also provides valuable physical insight into the relation between modal and ray methods is based on the W.K.B. approximation. Following GLOGEand MARCATILI [1973a, b], we place @(r)= E'(') in equation (108). Then, neglecting a2u/ar2,which is reasonable as u varies slowly with r, the resulting quadratic equation can be solved for at@,
-au_-dr
1 2r
-
+-j [k2(r)-p2 -(n2 + + ) / r 2 ] f
where
k2(r) = k2n2(r). For given p and n we can find radii R1and R, at which the root in equation (1 19) vanishes as in Fig. 39(a). These radii define a ring shaped region in which equation (1 19) has an imaginary part and the field of the mode is periodic. The inner radius R1 represents the ray caustic, the outer radius R 2 , the turning point. Outside this ring shaped region the field of a bound mode is evanescent. As p decreases, we see from Fig. 39(b) that when
390
O P T I C A L F I B R E WAVEGUIDES
CVIL §
3
Fig. 39. Propagation regions for a graded refractive index fibre of radius a . (a) R , corresponds to the inner ray caustic R , the turning point for the ray. (b) Conditions corresponding to a leaky ray within a graded refractive index fibre. Propagating conditions occur between R , and R , and beyond R , . (c) Conditions corresponding to refracting rays.
p < p, = kn, another radially propagating region appears at infinite radial distance, if the cladding is infinite corresponding to the onset of leaky mode propagation. As p decreases further, the evanescent region in the cladding shrinks and finally disappears with the onset of refracting modes. We can use the expression for
in order to solve the total number of propagating modes MT for p, and finally for the group delay 7. In order to have radially decreasing fields
VII,
P 31
FIBRES W I T H N O N - U N I F O R M R E F R A C T I V E I N D E X
39 1
outside of the region R , < r < R,, the phase of the field within the region must sum to an (approximately) integer number of half. periods so that, rnn = JR2u(r)dr =
r2
F ( r )- B2 - r2
Rt
R1
where terms such as m + b , and nZ+$ have been replaced by m and n2 respectively. To obtain the total number of bound modes MTwe must sum m and n for all combinations up to their maximum values. The requirement for evanescent field in the cladding limits p to a maximum value p, = kn, which is reached at cut-off. The largest n with = p, occurs when m = 0 and the largest m, occurs when n = 0. Treating m and n as continuous variables, which is a reasonable assumption in a highly overmoded fibre,
The factor 4 in equation (121) arises because of the degeneracy of the modes and their two states of polarisation as discussed previously. A change in the order of integration leads to
4[[
(k2-8E18
M, = z o o
[k'-/?:-
gpndr
(122)
where a is the radius at which n(r) = n,. On integrating (122) with respect to n we obtain M, =
L
[k2(r)-P,"]rdr = k2
l
[n'(r)-n;]rdr.
(123)
For small index differences the integral represents the volume under the circularly symmetric refractive index distribution curve. Equation (123) can be rephrased so as to provide a relation between the number of modes M having propagation coefficients larger than p, the selected value of B and the refractive index profile defined by k(r),
where the upper limit in the integration, R2(0), is the radius at which kn(r) = B. For fibres with refractive index profiles of the form of equations (101) and (102) shown in Fig. 36,
392
OPTICAL FIBRE WAVEGUIDES
[VlI.
P 3
where
ki
= k2ni.
For j? = j?, the total mode number MT then becomes M T
a a+2
= __ a2kiA .
The special case a = 2, corresponding to a parabolic profile, yields
M, = $V2
(127)
which is one half the value for a uniform profile fibre (a = co). The propagation coefficient of the Mth mode is found from equation (124)
f i M = kn,
[
1-24 (A4)'"" ~
2)l+
The group delay z, of the mth mode is given by
On substituting for &, and differentiating with respect to k, OLSHANSKY and KECK[1975] give a-2-Y A 301-2-2y
(~r+"+
7, =C
+
2(a 2)
+ O(A3)]
(130)
where
In the limit that the derivatives of no and A vanish, equation (132) reduces to the form given originally by GLOGEand MARCATILI [1973al; we continue with that assumption for the moment. To obtain the impulse response if all modes are excited equally, we need to know the density of modes arriving during an element of time dt, where t is the incremental time associated with the Mth mode after removal
VII,
5 31
F I B R E S W I T H NON-UNIFORM R E F R A C T I V E I N D E X
393
of the delay Lnolc, thus G -1. t = __ Ln0
(133)
The impulse response is then 1 dM MT dt
--I_-/
a + 2 a+2 1 a a-2 A
(2fa)’a
(t)2’a,
except for a x 2
(134)
while O
a-2 A, a+2
except for a x 2
( 136)
Outside of this time interval the impulse response is zero. GLOGEand MARCATILI [ 1973bl have derived, using equations (1 36) and (137), the impulse response for various refractive index profiles as shown in Fig. 40. As can be seen, the impulse response becomes extremely narrow when a = 2 . In the absence of dispersion the optimum value for CI giving the narrowest impulse response is given by aoPr= 2 - 2 4 .
(138)
In this case, the modes of highest and lowest order both arrive at the same time t = 0; all other modes are faster, the fastest arriving at t = -A2/8.
The absolute tempera1 width of the received pulse t,,,
(139)
is given by
Ln, A 2 tmax= __ - . c 8
However, if the effects of material dispersion are included, OLSHANSKY and KECK[I9751 show that the r.m.s. pulse broadening is given by
+ + 4 4 A2(2a+2)*]1 (2a+ 1) (5a + 2)(3a+ 2)
4c, c2 d(a 1)
(141)
394
O P T I C A L FIBRE WAVEGUIDES
-7
-1/3A
A*
1/38
0
2i3A
A
relative delay, t
Fig. 40. The impulse response for graded refractive index fibres with refractive index profiles corresponding to Fig. 36.
where c1 =
a-2-y a+2
~
9
c2
=
3~ - 2- 2y 2(a+2) '
In Fig. 41 0 is shown as a function of a for a fibre having the dispersion characteristics of a TiO, doped silica core with a silica cladding and at a wavelength J = 0.9 p. When allowance for material dispersion is made, the optimum value of a satisfies aOpt= 2 + y - A
(4+ Y)(3+Y ) 5+2y
It can be seen that equations (142) and (138) are approximately equal when y = 0. The above authors have investigated how aOptvaries with
VII,
§ 31
FIBRES WITH N O N - U N I F O R M REFRACTIVE I N D E X
-$
:material limit I
a
-
m
-
2
1;
'
0.01
'source wavelength 900nm
11
E L
395
\I wave guide I
-
n.a.0.16
I
--1
,
1
.
1
#
1
1
,
1
.
1
.
1
,
Fig. 41. r.m.s. pulse width as a function of refractive index parameter u. _ _ result obtained assuming that the refractive index and refractive difference are independent of wavelength; _--__ a result obtained from equation (141) and for a fibre with dispersion characteristics of a TiO, doped silica core with a silica cladding at a wavelength of 0.9 p.
source wavelength and find for the above fibre, a monotomic decrease between 2.5 and 2.2 in the range 0.5-1.1 p. Also shown in Fig. 41 are the limits on pulse broadening imposed by the finite linewidth of the source for a typical LED and GaAs laser. Evidently there is a strong incentive to manufacture a fibre with near optimum a if a narrow linewidth source is to be used. STEINBERG [1974] has shown that for a source of finite bandwidth B and with neglect of material dispersion,
2. (144) In the above analysis, it has been assumed that all modes are excited equally. TIMMERMANN c1974b-J and JACOMME[1975] have considered the more general problem where a source with an arbitrary distribution of power as a function of solid angle excites the modes of the graded-index fibre. =
A~B,
a =
396
O P T I C A L FIBRE W A V E G U I D E S
CVK
03
3.4. FIBRES WITH RING-SHAPED REFRACTIVE INDEX PROFILES
Although fibres with refractive index profiles of the form denoted by equations (103) and (104) gve the least modal dispersion when CI is chosen appropriately it is not always possible to achieve such distributions in and MARCATILI practice. Other profiles have been investigated by GLOGE [1973a] and OSTERMAYER [1975]. Here we describe briefly, the impulse response for a ring shaped distribution of refractive index as shown in Fig. 42, in which the refractive index n(r) is given by n(r) = n,[1-d(r-Ro)Z/a2], = q(1-A)
Ro-a < r < Ro+a
(145)
elsewhere.
(146)
GLOGEand MARCATILI [1973a] use the W.K.B. method to obtain the impulse response, however, these authors as well as TIMMERMANN [1974b1 have shown the correspondence which exists between that method and that based on rays. We use the latter because of its simplicity. In contrast with the centred parabolic distribution discussed previously, skew rays in the fibre of Fig. 42, are confined to a ring shaped radial region in which the refractive index has a maximum value no at the centre of the ring. We can therefore anticipate that although meridional rays will experiencedelay compensation,just as they would in a graded slab geometry, the skew rays will not and these rays will be slowed appreciably, the more so the skew angle. To facilitate easy comparison with the analysis of GLOGE and MARCATILI [1973a] we use v to denote the azimuthal number of the mode.
radius r
Fig. 42. Refractive index profile for a ring shaped distribution.
VII,
P
31
FIBRES W I T H N O N - U N I F O R M REFRACTIVE I N D E X
397
If Ro > a and v > 1 the skew angle q is given by V
(147)
+=k,R, and
k, = kn,. The axial velocity of the ray is given by
'[
C
I:
u=-cos~=1-n1 n,
.
On substituting for 4 from equation (147), the time for the ray to progress through a length L of the fibre is then
[1973a1provided which agrees with the derivation of GLoGE and MARCATILI nz zs- The maximum value of 4 is (2A)* so
a.
v,:
=~A(/C,R~)~.
(1 50)
The incremental delay t = T - LnJc, is then given by
As anticipated highly skew rays are significantly delayed, in fact by as much as in a step-index fibre. To evaluate the impulse response, we must know the numbers of modes m which possess a given v. We assume that because of the parabolic refractive index variation about r = R,, they will all arrive at a time given by equation (149). We can then work out the power density at a given value of r by forming m(dv/dt). With reference to Fig. 6 consider a ray propagating with skew angle 4 corresponding to azimuthal number v. In general there exists a range of discrete values of 6 between 0 and 6,,, corresponding to mode numbers between 1 and mmax.The corresponding range of & d/? is given by dfl = k , cos $(l- cos d,,)
(151)
3ko(2A - $2).
(152)
-
We next assume that the difference in p arising from a change in meridional
398
OPTICAL FIBRE W A V E G U I D E S
[VIL
§ 3
mode number is the same as for a slab with parabolic refractive index profile namely Sp = (2A/a)*,then
mmax= =
dp k,a (24-4') - Sp - 2 (2A)t
(153)
~
(;Pea [
1-
($)I.
In passing we note that, on summing equation (154) between v = 0 and v = v,, , we obtain the expression for the total number of modes given by GLOGEand MARCATILI [1973a1, namely, (155)
&zkgR , A .
If allowance is made for degeneracy of the modes and the two states of polarisation the total mode number is four times the above value. Finally, we obtain the impulse response from equation (1 54) as
(
>'3
L i l y - __ ct mdv - kgR,ac [ ( __ (156) dt 2L0 n Ln1 The function has been plotted by GLOGE and MARCATILI [1973a] in Fig. 43, who have also obtained results for the case when Ro/ais not large. It is easily
I
0.2
I
04 relative delay
TC/A n1 L
Fig. 43. Impulse response for a graded refractive index fibre with profile corresponding to Fig. 42,
VII,
0
31
FIBRES WITH N O N - U N I F O R M R E F R A C T I V E I N D E X
399
shown that half the power in the output pulse arrives in a time t = 0.124~1, L/c (157) the remainder being drawn out into a tail of length An, L/c, caused by the higher order azimuthally dependent modes. 3.5. CONCLUSIONS
We have now examined the basic properties of optical fibres of different types and have found advantages and disadvantages among them all. The monomode fibre offers the greatest bandwidth but presents serious coupling and jointing problems. The graded index fibre with an optimum index has pulse dispersion comparable with that of a monomode fibre. However, the precise control of the refractive index profile represents a significant technological problem but one which appears to have been solved by a number of manufacturers. The fibre offers significant advantages in terms of coupling and jointing when compared to the monomode fibre. Even when the profile departs significantly from the optimum value there exists advantages compared to multimode step-index fibre. The latter may be suitable for short-haul applications where the penalty in terms of pulse dispersion is acceptable. If controlled mode coupling can be achieved in a simple manner the multimode step-index fibre may also find a place in mediumhaul applications again because of advantages offered in coupling and jointing. Since fibres have to be formed into cables before they can be used in systems the influence of the cable on the fibre is significant and has been investigated. Serious losses can occur due to leakage from the fibre if care is not taken in the choice of fibre environment and cable shell. However, these problems also show good prospect of solution. The theory of optical fibres now appears to be reasonably well understood. Both modal and ray methods have contributed about equally to our understanding, as have analytic and numerical techniques. By judicious use of approximations a number of problems have been solved which, a priori, appeared quite formidable. In conclusion, one may anticipate the time when the impulse response of a graded index fibre excited by an arbitrary source will be susceptible to computer simulation. Such a development should prove of real value to the fibre manufacturer and systems engineer.
400
OPTICAL FIBRE WAVEGUIDES
[VII
Acknowledgements
The author of this review is indebted to many for their helpful suggestions over the years on the subject of optical waveguides. He thanks especially Dick Dyott who gave considerable help to the author in the preparation of this review. He also thanks Leo Felsen, Charles Kao, John Midwinter, Murray Ramsay, Charlie Sandbank and Allan Snyder for their helpful comments. References ABRAM,R. A., R. W. ALLANand R. C. GOODFELLOW, 1975, The coupling of light emitting diodes to optical fibers using sphere lenses, to be published. ARNAUD, J. A,, 1974, Bell Syst. Tech. J. 53, 675. J. A,. 1975, Bell Syst. Tech. J., to be published; Electron. Letters, 11, 8. ARNAUD, G. and D. J. KINGSLEY, 1965, IEEE Trans. on Microwave Theory and Techniques BIERNSON, MTT-13, 345. CHAN,K. B., 1972, Electromagnetic wave propagation in optical waveguides, University of London Ph.D. Thesis. CHAN,K. B. and P. J. B. CLARRICOATS, 1973, Queen Mary College Technical Report EE/73/10. and A. M. SCHEGGI,1975, Resume of Colloquium on the CHECCACCI, P. F., R. FALCIAI Optics of Guided Waves, Paris. CLARRICOATS, P. J. B., 1960. Proc. IEE, Part C, Monograph No. 409E and 410E. CLARRICOATS, P. J. B., 1970, Lecture Coursenotes, University of Colorado. CLARRICOATS, P. J. B. and K. B. CHAN,1970, Electron. Letters 6. 695. CLAKRICOATS, P. J. B. and K. B. CHAN,1973, Proc. IEE 120, 1371. CLARRICOATS, P. J. B. and D. C. CHANG,1972. Proc. G-AP Intern. Symp., p. 296. COLLIN,R. E., 1960, Fleld Theory of Guided Waves (McGraw Hill). DYOTT,R. B. and J. R. STERN,1970, IEE Conf. Publication No. 71, pp. 176-181. DYOTT,R. B. and J. R. STERN,1971, Electron. Letters 7,624. DYOTT,R. B., C. R. DAYand M. C. BRAIN,1973, Electron. Letters 9, 288. DYOTT.R. B.. 1974. The disperslon limited bandwidth of fibre made from optical glasses (unpublished memorandum). ELSASSER. W. M.,1949. J. Appl. Phys. 20, 1193. ENOCH,J . M., 1961. J. Opt. SOC.Am. 51, 499. FLETCHER, A,, T. MURPHY and A. YOUNG,1954. Proc. Roy. SOC.223, 216. W. A., D. N. PAYNEand H. SUNAK,1971, Electron. Letters 7, 549. GAMBLING, and H. SUNAK,1972, Electron. Letters 8, 260. GAMBLING, W. A., J. P. DAKIN,D. N. PAYNE GAMBLING, W. A., D. N. PAYNEand H. MATSUMURA, 1974, Electron. Letters 10, 148. GARDNER, W. B., 1975. Bell Syst. Tech. J. 54, 457. GLOGE,D., 1971a, Appl. Optics 10, 2252. GLOGE,D., 1971b, Appl. Optics 10, 2443. GLOGE,D., 1972a, Bell Syst. Tech. J. 51, 1767. GLOGE,D., 1972b, Appl. Optics 11, 2506. GLOGE,D., 1973, Bell Syst. Tech. J. 52, 801. GLOGE,D. and E. A. 3. MARCATILI, 1973a, Bell Syst. Tech. J. 52. 1161. GLOGE.D. and E. A. J. MARCATILI, 1973b, Bell Syst. Tech. J. 52, 1563. GI.OGE,D., 1975, Bell Syst. Tech. J. 54, 243. HONDROS,D. and P. DEBYE,1910, Ann. Phys. 32, 465. HUBBARD, W. M., 1972. Appl. Optics 11. 2495.
VII]
REFERENCES
40 I
IMAI,M. and E. H. HARA,1974, Appl. Optics 13, 1893. IMAI,M. and E. H. HARA,1975, Appl. Optics 14, 169. JACOMME, L., 1975, Resume of Colloquium on Optical Guided Waves, Paris, p. 1.6. JONES,A. L., 1965, J. Opt. SOC.Am. 55, 261. JURGENSEN, K., 1975, Appl. Optics 14, 163. A. H. CHEIUNand A. D. PEARSON, 1972, Topical Meeting on InteKAISER,P., A. R. TYNES, grated Optics-Guided Waves, Materials and Devices (unpublished). KAISER,P., A. C. HARTand L. L. BLYLER, 1975, Appl. Optics 14, 156. KAO,K. C. and G. A. HOCKHAM, 1966, Proc. IEE 113, 1151. KAPANY, N. S., 1967, Fiber Optics (Academic Press). KAPANY, N. S. and J. J. BURKE,1972, Optical Waveguides (Academic Press). KAPRON,F. P., D. B. KECKand R. D. MAURER, 1970, Appl. Phys. Letters 17. 423. KAPRON,F. P. and D. B. KECK,1971, Appl. Phys. 10, 1519. KAWAKAMI, S. and J. NISHIWAZA, 1968, IEEE Trans. Microwave Theory and Techniques MTT-16, 814. S. and S. NISHIDA,1974a, Electron. Letters 10, 38. KAWAKAMI, S. and S. NISHIDA,1974b, IEEE J. Quantum Electronics QE-10, 879. KAWAKAMI, KAWAKAMI, S. and S. NISHIDA,1975, IEEE J. Quantum Electronics QE-11. KIELY,D. G., 1953, Dielectric Aerials (Methuen & Co.). KIRCHHOFF, H., 1970. ZEE Conf. Publication No. 71, p. 69. KUHN,M. H. and K. PETERMANN, 1975. Proc. Topical Meeting on Optical Fiber Transmission. Williamsburg. KUHN.M. H.. 1975. A.E.U. 29. 201. MARCATILI, E. A. J. and S. E. MILLER.1969, Bell Syst. Tech. J. 48. 2161. KURTZ,C. N. and W. STREIFER, 1969, IEEE Trans. Microwave Theory and Techniques. MTI-17, 1 I . MARCATILI, E. A. J., 1973, J. Opt. SOC.Am. 63, 1372. MARCATILI, E. A. J., 1974, Bell Syst. Tech. J. 53. 645. MARCUSE, D., 1969, Bell Syst. Tech. J . 48. 3233. MARCUSE, D.. 1970, Bell Syst. Tech. J. 49. 1695. MARCUSE, D.. 1972a, Bell Syst. Tech. J. 51, 1199. MARCUSE, D., 1972b, Light Transmission Oplics (Vim NO~II..III~. Reinhold). MARCUSE, D., 1973a, Bell Syst. Tech. J. 52. 81 7. D., 1973b, J. Opt. SOC.Am. 63, 1369. MARCUSE. MARCUSE, D., 1974a, Bell Syst. Tech. J. 53. 1795. MARCUSE, D., 1974b, Theory of Dielectric Optical Wnvcyuides (Academic Press). MARCUSE, D. and H. M. PRESBY, 1575. Bell Syst. Tcch. J 54. 3. MILLER, S. E., E. A. J. MARCATILI and TINGYE LI, 1973, Proc. IEEE 61, 1703.’ MILLER, S. E., 1974, Bell Syst. Tech. J. 53, 177. OGILVIE. G. J., R. J. ESDAILE and G. P. KCDD,1972, Electron. Letters 8. 533. OLSHANSKY, R. and D. 8 . KECK.1975. Proc. Topical Meeting on Optical Fiber Transmission. Williamsburg. F. W., 1975, Proc. Topical Meeting on Optical Fiber Transmission, WilliamsOSTERMAYER, burg. PASK.C. and A. W. SNYDER, 1974. Opto-Electronics 6, 297. and D. J. MITCHELL, 1975. Number of modes on optical waveguides. PASK, C., A. W. SNYDER to be published. 1972, Electron. Letters 8, 374. PAYNE,D. N . and W. A. GAMBLING, PERSONICK. S. D., 1971, Bell Syst. Tech. J. 50. 843. POTTER,R. J., 1961, J. Opt. SOC.Am. 51, 1079. RAMSAY, M., 1973, Opto-Electronics 5, 261. ROSMAN, G., 1972, Electron. Letters 8. 455. SAMMUT, R. and A. W. SNYDER, 1975, Characteristics of Leaky Modes on Circular Optical Waveguides, t o be published.
402
OPTICAL FIBRE WAVEGUIDES
[VII
SANDBANK, C. P., 1975, Proc. Topical Meeting on Optical Fiber Transmission, Williamsburg. S. A,, 1943, Electromagnetic Waves (Van Nostrand). SCHELKUNOFF, SMITH,L. and E. SNITZER,1973, Appl. Optics 12, 1592. SNITZER, E. and H. OSTERBERG, 1961, J. Opt. SOC.Am. 51, 499. SNYDER, A. W., 1966, J. Opt. SOC.Am. 56, 601. SNYDER, A. W., 1969a, IEEE Trans. Microwave Theory and Techniques MlT-17, 1130. SNYDER, A. W., 1969b, Ph.D. Thesis, University of London. SNYDER, A. W., 1969c, IEEE Trans. Microwave Theory and Techniques, MlT-17, 1138. SNYDER, A. W., 1973, Electron. Letters 9, 437. SNYDER. A. W. and D. J. MITCHELL, 1974, Electron. Letters 10, 11. SNYDER, A. W., 1974, Appl. Phys. 4, 273. SNYDER, A. W. and J. D. LOVE;1975, IEEE Trans. Microwave Theory and Techniques 23,134. STANDLEY, R. D. and W. S. HOLDEN,1974, Bell Syst. Tech. J. 53, 779. STEINBERG. R. A., 1974. Electron. Letters 10, 375. STERN,J. R. and R. B. DYOTT,1971, Electron. Letters 7, 52. STRATTON, J . A.. 1941, Electromagnetic Theory (McGraw Hill). STREIFER, W. and K. B. PAXTON,1971, Appl. Optics, 10, 769 and 116. C. C., 1974a, A.E.U. 28, 144. TIMMERMANN, TIMMERMANN. C. C., 1974b. A.E.U. 28, 186. TYNES, A. R., 1974, J . Opt. SOC.Am. 64, UNGER.H. G.. 1964, A.E.U. 19, 189. VIGANTS,A. and S. P. SCHLESINGER, 1962, IRE Trans. G-MTT, p. 375. 1975, Appl. Optics 14, 288. YANG,K. H. and J. D. KINGSLEY, YIP, G. L. and Y. H. AHMEW,1974, Proc. URSI Symp. on Electromagnetic Wave Theory, p. 272. YIP, G. L. and J. C. Y. HUANG,1975, Resume Colloquium on Optical Waveguides, Paris.
AUTHOR INDEX A ABRAM,R. A., 376,400 ADAWI.I., 278. 321 ADLARD COLES,K., 59,85 G. S., 259, 321 AGARWAL, AHMEW, Y. H., 389,402 ALFANO,R. R., 164, 175, 184, 192, 193 ALLAN,R. W., 376, 400 ALLEN,F. G.,263,300,302,313,315,321,323 ALLEN,L., 41, 44, 83, 86 ANDEREGG. M.. 259. 321 ANDERSON, D. E., 295. 324 ANDERSON. J. A,, 85 G. A,, 298, 321, 323 ANTYPAS, APPELBAUM, J. A., 263: 321 ARAKAWA, E. T., 260,281,288,289,308,310, 313, 321, 322 E., 28, 44 ARCHBOLD, ARNAUD, J. A,, 351, 385, 400 ARSENHAULT, H., 12, 13,46 ASAKURA, T., 19, 21, 22, 25, 41, 42, 43, 44, 45, 46 ASAWA,C. K., 146, 156 ASHCROFT,N. W., 261, 279, 324 J. C., 281, 324 ASHLEY, ASHLEY, K. L., 295, 298, 321 AUZEL,F., 97, 135, 151, 152, 156 AXE,J. D., 110, 113, 136, 156
B BABCOCK, H. W., 78, 85 BAER,A. D., 302, 319, 321 BAER,Y., 303, 304, 321 BAKER,J. M., 316, 321 BALKANSKI, M., 314, 315, 324 BALLANTYNE, J. M.. 274, 314, 321 BARAKAT, R., 16, 33, 44, 45 BARASCH, G. E., 96, 117. 156 P. B., 283. 324 BARNA, 403
BARNETT, B., 109, 115, 158 BARTELINK, D. J., 274, 321 BASTIE,P., 290, 292, 323 BATES, R. H. T., 77, 85 BEAVERS, W., 68, 85 BECKER, P. J., 110, 11 3, 156 BECKMANN, P. 5, 6, 27, 33, 45 BECQUEREL, J., 93, 156 BEDARD, G., 18, 45 BELL,R. L., 298, 321, 323, 325 BERGE,S., 313, 322 BERGLUND, C. N., 265, 321 BERNDTSSON, A., 304, 323 J. F., 307, 325 BERTHELEMY, BETHE,H., 93, 156 BEUTLER, H. G., 235, 242 BIERNSON, G., 331, 400 BIGUEURE, M., 265, 284, 286, 306, 325 BIRGENEAU, R. J., 135, 144, 156 BIRKHOFF,R. D., 288, 321 BIZOUARD, M., 290, 292, 323 BLANARU, D. L., 285, 306, 324 BLEANEY, B., 93, 156 BLECHSCHMIDT, D., 302, 303, 321 BLUM,E., 85 L. L., 376, 401 BLYLER, BOEHM,L., 109, 115, 158 BOKSENBERG, A,, 51, 63, 85 BOLMONT, D., 314, 315, 324 BONNEAU, D., 70, 73, 85, 86 BORGNINO,J., 53, 86 BORN,M., 19, 23, 45, 61, 85 BORTOLANI,V., 263, 321 BOTDEN,T. P. J., 97, 135, 156 BOURCET,J. C., 150, 156 E., 316, 325 BOURSEY, BOUSSUG~, 57, 69, 86 BOUTRY, G. A., 310, 324 BOWEN,I. S., 85 BOYER,W., 85 BRAIN,M. C., 331. 400
404
AUTHOR INDEX
BRAUNDMEIER, A. J., 288, 321 BRECKINRIDGE, J. B., 85 BRIGGS,H. B., 307, 323 BROER,L. J. F., 95, 156 BROIDA, H. P., 166, 192 BROWN,R. Hanbury, 49, 50, 83, 86 E.. 4, 45 BUCHWALD, A. D., 163, 192 BUCKINGHAM, A,, 79, 86 BUFFINGTON, BUKIETYNSKA, K., 113, 156 BURCH,J. M., 37, 45, 197, 242 BURCKHARDT, C. B., 1I, 45 BURGESS, T. J., 191, 192 BURKE,J. J., 401 BURTON, J. A., 286, 321 C
CAGNET, hl., 68, 69, 85 CALANDRA, C . , 263, 321 A., 260,289,308,310,313,315, CALLCOII,T. 316, 318, 321, 325 CAMPAGNA, M., 291, 321 CAMPILLO, A. J., 176, 192 W. T., 109, 110, 113, 156 CARNALL, CAKOLI, C., 279, 321 CARR,D. L., 295, 298, 321 CASHION, J. K., 316, 322 CASTLEBEKRY, D., 155, 157 CHABRIER, G., 280, 286, 287, 288, 290, 307, 308, 310, 313, 321, 322, 323, 325 CHAKRABORTY, A. K., 33,45 CHAMBERLAIN, J. R., 110, 111, 124, 156 CHAN,K. B., 332,341,344,354,355,357,366, 368, 380, 381, 389, 400 CHANDRASEKHAR, S., 6. 45 CHANG, D. C., 357, 400 CHANG, J. C., 18, 45 CHECCACCI, P. F., 385, 400 CHERIN, A. H., 329, 401 L. A,, 5, 45 CHERNOV, CHIAO,R. Y., 176, 192 CHOPPIN, G. R., 113, 156 CLARK, K. G., 214, 242 CLARRICOATS, P. J. B., 332,341,344,354,355, 357, 366, 368, 380, 389, 400 CODE,A. D., 79, 85 L. W., 191, 192 COLEMAN, COLLIN,R. E., 400 COLLINS, R. A., 295, 322, 323 M . A,, 16, 45 CONDIE, COQUET, E., 258,265,280,283,284,285,286, 306, 316, 322, 325
J., 238, 240, 242 CORDELLE, CORNAZ, J., 280, 286, 290, 321, 325 CRANE,R. B., 41, 45 CROWELL, C. R., 265, 271, 283, 322 CURRIE, D. G., 68, 69, 85
D DAINTY, J. C., 11, 12, 16, 19. 22, 24, 36, 37, 38, 40, 42, 45, 54, 58, 64. 85 DAKIN,J. P., 343, 400 DAS,N. C., 235, 243 D A U D ~A,, , 260, 322 DAVID,L. W., 295, 322 DAVIS,J., 83, 86 S. G., 262, 322 DAVISON, DAVYDOFF, A. S., 276. 322 DAY,C. R., 331,400 DEBYE,P., 333, 400 DEDERICHS, P. H., 276, 322 DE DOMINICIS, C. T., 279, 324 DE GENNES, P. G., 142, 156 DE HAAS,W. J., 4, 25, 45 DEKKER, A. J., 273, 323 DELSART, C., 110, 113, 124, 156 DEMARIA, A. J., 163, 192 DETRIO,J. A., 110, 156 DEUTSCHER, K., 275, 286, 288, 323 DEXTER, D. L., 96.97, 112, 118, 134, 135, 138, 139, 146, 149, 156, 157, 158 DIBARTOLO, B., 95, 156 DIEKE,G. H., 93,94,95,96,99, 100, 101. 117, 156, 157 DOLIZY,P., 288, 322 DONIACH, S., 255, 277, 322 DONLAN, V. L., 110, 159 DOWLEY, M. W.. 214, 242 G., 52, 55, 58, 86 DRYUEN, DUBRIDGE, L. A,, 247, 323 DUCKETT, S. W., 274, 302, 322 DuGuAY,M.A.,163,164,165, 167, 168, 173, 174, 175, 177, 178, 180, 181. 183, 186,188. 191, 192, 193 DYOTT, R. B., 331, 345, 347, 350, 369, 381, 400,402 DYSON,F. J., 79, 85 E
EASTMAN, D. E., 291, 292, 315, 316, 321. 322 EDEN,R. C., 298, 322 EDGE,I. A,, 295, 322 EDGECUMBE. J., 321, 323
AUTHOR INDEX
M., 32, 45 ELBAUM, ELIASSON, B., 4, 45 ELLIOT, J. L., 66, 85 ELSASSER, W. M., 400 EL’YASHEVICH, M. A,, 95, 157 J. G., 260, 262, 308, 309, 310, 313, ENDRIZ, 322 ENLOE,L. H., 11, 37, 45 ENNOS,A. E., 28,44 ENOCH,J. M., 331, 400 ENSTROM, R. E., 322 M., 315, 322 ERBUDAK, ERDUS,P., 136, 157 ESCHARD. G., 288, 322 J. S., 322 ESCHER, BDAILE, R. J., 329, 401 EVERITT, A. C., 110, 156 EXNER, K., 4, 45 F I. L., 170, 172, 192, 193 FABELINSKII, FABRE,D., 264, 324 FALCIAI, R.,385, 400 FAN,H. Y., 248, 312, 322 FEIBELMAN, P. J.. 267. 295, 322. 323 FEINLIEB, J., 79, 86 P. P., 149, 157 FEOFILOV, B., 259,3 I6,3 I 7,3 18,321,322 FEUERBACHER, FIELDS,^. R., 109, 110, 113, 156 W., 65, 85 FINSEN, FISCHER, D. G., 322 FISCHER, G. B., 266, 267, 280, 306, 325 FISCHER, R.,177, 192 T. E.,302, 314, 315, 322, 325 FISCHER, FITTON,B., 259, 316, 317, 318, 321, 322 FIZEAU, H., 62, 86 J., 227,229,238,240,241,242,243 FLAMAND, FLETCHER, A,, 384, 400 FLODSTR~M, S. A., 309, 322 FONG,F. K.,96, 118, 121, 124, 126, 132, 150, 156, 157 FONGER, W. H., 115, 158 J. Y., 308, 322 FONTENEAU, F~RSTER, T., 97, 134, 157 FORSTMANN, F., 259, 316, 322 FOWLER, R. H., 248, 322 FOWLER, W. B., 112, 157 FRANK, G., 295, 298, 322 FREED, K. F., 124, 157 FRIED,D. L.,51, 86 FUCHS,R., 259, 322 FUJII,H., 19, 21, 22, 25, 41, 42, 43, 44,45
405
C GABBE, D., 155, 157 GADZUK, J. W., 324 W. A,, 329, 343, 375,400,401 GAMBLING, W. B., 122, 131, 149, 157, 158 GANDRUD, GARBE,S., 295, 297, 298, 322 GARDNER, W. B., 375,400 GARMIRE, E., 176, 192 GARTLAND, P. 313,322 G., 109,157 GASHUROV, L., 285,322 GAUDART, GAVIOLA, E., 53, 86 GAY,J., 84, 86 GEORGE, N., 32,34,45, 173, 192 GERASIMOV, F. M., 235,243 GERMAN, K. R., 119, 129, 157 GERRITSEN,H. J., 14, 45 GERSTEN, J. I., 279, 322, 325 GESELL, T. F., 281, 288, 322 GEUSIC, J. E., 149, 152, 155, 157, 158 GEZARI,D., 51, 58, 70, 86 GIORDMAINE, J. A., 163, 192 GIRES,F., 163, 167, 192 GLASS, I. S., 66, 85 GLOGE,D., 335, 338, 342, 343, 345, 348, 349, 350, 353, 354, 363, 365, 366, 367, 368, 370, 371, 375, 389, 392, 393, 396, 397, 398,400 GOBELI,G. W., 263, 300, 302, 313, 315, 321, 323 GOLDFIXHER, L. I., 9,45 GOODFELLOW, R. C., 376,400 GOODMAN, J. W., 5, 9, 17, 25, 33, 34, 35, 41, 42, 43,45, 55, 78, 86 G~RLICH, P., 247, 313, 323 GORTER,C. J., 95, 156 J. P., 280, 286, 287, 288, 290, GOUDONNET, 308,310,313,321,322,323,325 GOUGH, P.T.,77,85 GOULD,R. D., 295, 322, 323 GOUT,C., 264,324 GRANT,W. J. C.,95,139, 140,144, 157 M., 32.45 GREENBAUM. W. D., 315, 322 GROBMAN, J. B., 110, 113, 157 GRUBER, GUGGENHEIM, H. J., 131, 151, 157 GUICHAR, G., 314, 315, 324 GURMAN, S. J., 263, 323 GUSKOVA, O., 64, 86
o.,
H HABER,H.. 234. 243
406
AUTHOR INDEX
HACKETT,W. H., 295, 323 HALL,C. K., 283, 323 HAMANN, D. R., 263,321 HAMM,R. N., 288, 321 H A ~ U RBROWN, Y R., 49, 50, 83, 86 HANEMANN, D., 305, 324 HANNAN, W. J., 14, 45 HANSEN, J. W., 163, 164, 167, 168, 173, 174, 175, 177, 183, 191, 192 HARA,E. H., 381,401 HARADA, T., 235, 243 HARDY,J. W., 79, 86 HARIHARAN, P., 4, 45 HARP,J. C., 51, 86 HARRISON, G. R., 197,243 HARRISON,M. J., 259, 324 HART,A. C., 376, 401 HARTMANN, P., 295, 324 HARVEY, J. W., 85 HEBB,M. H., 273, 323 HEDEN,P. F., 303, 304, 321 HEDMAN, J., 303, 304, 321, 323 HEINE,V., 263, 316, 322, 323 HELLWARTH, R. W., 173, 192 HELLWEGE, K. H., 95, 111, 157 HERMEKING, H., 279, 323 HmseL, A,, 231, 243 HEW, R. A,, 151, 153, 157 HEYNAU, H.. 163, 192 HICKMOIT,T. W., 295, 323 HIOKI,R., 9, 46 HIRAYAMA, F., 97, 134, 140, 157 HIRSCHBERG, K., 215,286, 323 Ho, P. P., 184, 193 HOCKHAM, G. A,, 329,401 HOFMANN, H. H., 275, 286, 288, 323 HOFMANN, J., 308, 323 HOGARTH, C. A., 295, 323 HOHN,D. H., 11,45 HOLDEN,W. S., 331, 402 H ~ L Z LJ., , 305, 323 HONDROS, D., 333, 400 HOOGSCHAGEN, J., 95, 156 HOPFIELD, J. J., 258, 259, 323 HOSHINA, T., 109. 110, 157 HOWARTH, L. E., 265, 271, 283, 322 HUANG,J. C. Y., 342,402 HLJBBARD, W. M., 369, 400 HLJDIS,J., 304, 305, 325 HUEN,T., 325 HUFNAGEL, R. E., 51, 86 HUGUES, A. L., 247, 323
HUNTER, W. R., 227,228, 236,237, 238,239, 243
HUTLEY, M. C., 220,221,227,230,231,232, 233, 243
I IMAI,M., 381, 401 INOKUTI,M., 97, 134, 140, 157 IPPEN,E.P., 163, 164, 166, 170, 171, 172, 173, 192, 193
IVES,H. E., 307, 323 J
JACOBI,K., 305, 315, 323, 324 JACOMME, L., 395,401 JAIN,A,, 32. 34. 45 JAKEMAN, E.. 4, 37, 39, 40, 41, 45,46 JAMES,L. W., 272, 296, 298, 321, 323 JANSSEN, J. M. L., 183, 192 JASPERSON, S. N., 260, 323 JAZZAR,M. S., 302, 325 JENNISON, R. C., 78, 86 JENSSEN, H. P., 131, 155, 157 JOHNSON, L. F., 131, 151, 155, 157 JOHNSON, R. L., 233, 234, 235, 238, 243, 244 JONES,A. L., 376, 401 JONES,D. G. C., 41,44 JONES,R. P., 164, 177, 185, 187, 193 J$RGENSEN, C. K., 111, 114, 157 JORTNER, J., 124, 157 JOURNET, A,, 84, 86 JUDD,B. R., 95, 107, 108, 111, 114, 157 JURGENSEN, K., 350, 401
K KAC, M., 15, 46 KAISER,P., 329, 376, 401 KA~~,E.0.,266,271,274,275,313,315,323 KANTER, M., 265, 293, 295, 323 KAo, K. C., 329,401 KAPANY, N. S., 329, 401 KAPRON,F. P., 329, 345, 401 KATRICH, G. A., 265, 283, 323 KAWAKAMI, S., 382, 383, 385, 401 KAYSER, H., 197, 243 KECK, D. B., 329, 345, 392, 393, 401 KELLEY, P. L., 176, 192 KELLY,M. J., 263, 321 KERR,J., 192 KIDD,G. P., 329, 401
AUTHOR INDEX
KIEL,A., 96, 118, 119, 129, 157 KIELY,D. G., 401 KING, M., 32, 45 UNGSLEY, D. J., 331, 376, 377, 400 J. D., 149, 151, 153, 157, 402 KINGSLEY, KIRCHHOFF,H., 388, 389,401 KISHI,M., 221, 243 KISLIUK,P., 120, 157 KITA, T., 235, 243 KLMSON, M., 303, 304, 321, 323 KL-EIN,W., 323 KLIEWER,K. L., 259, 322 KNAPP,S. L., 68, 69, 85 KNOX,K.T., 86 A. N., 86 KOLMOGOROFF, G. Y., 127, 131, 159 KOLODNYI, KOPF, L., 150, 158 KORFF,D., 52, 55, 58, 86 D. V., 64, 86 KOROLKOV, KOSHELEY, M. V., 235, 243 G. F., 108, 158 KOSTER, KOZLOV,Yu. G., 86 KRAMERS,H. A., 93, 157 KRASUTSKY, N., 150, I57 H., 298, 324 -EL, KRULIKOWSKI, W. F., 265,266,271,280,283, 303,322,323 KROPP,J. L., 131, 157 KRUPKE,W. F., 110, 112, 113, 157 KWNIWA, S., 109, 110, 157 KUHN,M. H., 342, 356, 401 KULAGIN,E. S., 68, 86 KURTZ,C. N., 385, 401 KUSHIDA,T.,95, 135,142,143,144,146, 147, 149, 157, 159 L LABATE,E. E., 265, 271, 283, 322 A., 19, 46, 51, 54, 58, 64, 69, 70, LABEYRIE, 13, 85, 86, 197,238,240, 242, 243 D. C., 274, 323 LANGRETH, C. J., 302, 319, 321 LAPEYRE, LAUDI?,L. D., 314, 315, 323 LAUER,H. V., 118, 121, 132, 157 LAWRENCE, R. S., 51, 86 LEASK,M. J. M., 134, 157 D., 279, 321 LEDERER-ROZENBLATT, LEE, R. W., 51, 86 LEGG,K. O., 295, 322 LEIGH,J., 185, 192 R. B., 86 LEIGHTON, LEVINE,J. D., 262, 322
407
LEWOWSKI,T., 290,292, 323 LI, Tingye, 331, 401 LIEBLICH,N., 109, 115, 158 LIEWER,K. M., 68, 69, 85 LINZ,A., 155, 157 LIU, Y.C., 17, 86 LIU, Y. Z., 298, 323 &WEN, E., 229, 243 LOH, E., 102, 109, 158 LOHMAJVN, A. W., 77, 86 LOVE,J. D., 363, 402 S., 12, 13,46 LOWENTHAL, J. L., 52, 86 LUMLEY, LYE, R. G., 273, 323 LYNDS,R., 73, 86
M MACEK,C. H., 289, 323 MAHAN,G. D., 267, 278, 279, 323 MAKINSON, R. E. B., 278, 323 MALLEY,M.M., 164,165, 175, 176, 177, 188, 190, 191, 192 MANDEL,L., 17, 18,45,46 W. J., 109, 158 MANTHEY, MARCATE], E. A. J., 331, 359,363,389,392, 393, 396, 397, 398, 400, 401 MARCOS,H. M., 149, 152, 158 D., 347, 351, 359, 366, 367, 370, MARCUSE, 374, 381,401 W., 25, 46 MARTIENSSEN, MARTIN,F., 53, 86 R. U., 298, 299, 323 MARTINELLI, MATSINGER, B., 110, 113, 159 H., 375,400 MATSUMURA, A. T., 177, 180, 181, 192 MATTICK, MAURER,R. D., 329,401 MAYER,G., 163, 167, 192 MAYER,H., 280, 284, 285, 306,307, 324 D., 229, 243 MAYSTRE, B. J., 77, 86 MCGLAMERY, MCKECHNJE, T. S., 7, 8, 11, 12, 32, 46 MCPHEDRAN, R., 229, 243 MCRAE,A. U., 316, 321 MEAD,C. A., 265, 295, 324 ME, C. H. B., 283, 323 MELNIK,P., 304, 323 A. R., 259, 324 MELNYK, MELVILLE, R. D. S., 34, 45 S.,307, 324 METHFESSEL, METZGER,P. M., 302, 322 S. W.,302, 324 METZGER, M m , N. I., 274, 321
408
AUTHOR INDEX
MICHELSON, A. A., 61, 62, 86, 197, 243 MIDDLETON, D., 6, 12, 46 MIKELSKIS, H., 231, 243 MIKFSELL,A. M., 52,86 MILLER,M. G., 52, 55, 58, 86 MILLER,M. M., 96, 118, 124, 126, 157 MILLER,R. H., 51, 73, 80, 86 MILLER, S. E., 331, 363, 386, 401 MITA,Y., 152, 158 MITCHELL, D. J., 349,401,402 K., 248, 262, 309, 324 MITCHELL, MIYAKAWA, T., 96, 118, 146, 158 MOLL,J. L., 265,272,274,282,283,296,298, 321, 322, 323, 325 MBNCH,W., 262, 324 MONIN,J., 310, 311, 312, 324 MOON,R. L., 321, 323 MOORE,C. A,, 120, 157 MOORE,G., 265, 285,286, 324 MOOS,H. W., 95, 96, 118, 122, 124, 125, 126, 129, 130, 131, 149, 150, 157, 158 MORIYAMA, S., 235, 243 MOTEGI,N., 150, 158 F. M., 4, 45 MOTTIER, MOUROU,G., 164,165,175,188,190,191,192 MULLER,R. A., 79, 86 K., 19, 44 MURATA, MURPHY, T., 384,400 MURTY,M. V. R., 235,240,243 N
S. L., 96, 118, 124, 126, 157 NABERHUIS, NAGATA, H., 221, 243 NAKAZAWA, E., 142, 143, 146, 149, 158 NAMIOKA, T., 235, 238, 240, 243 NAPIER,P. J., 77, 85 NELSON, H., 298, 324 NEMCHENOK, R. L., 285, 324 NETZEL,T., 185, 192 NEWMAN, D. J., 102, 158 C. W., 108, 158 NIELSON, NIEMANN, B., 238, 241, 243 NILSSON, R., 304, 323 NIQUET,G., 295, 324 NISENSON, P., 73, 86 NISHIDA, S., 382, 383, 401 NISHIWAZA, J., 385, 401 NODA,H., 235, 238, 240, 243 C.. 303, 304, 321, 323 NORDLING, NOZIERES, P., 279, 324
0
ODGERS, G. J., 79, 81, 86 OFELT, G. S., 95, 107, 158 OGILVIE, G. J., 329, 401 OHTSUBO, J., 43, 46 OLINER,A,, 231, 243 OLSEN,J. N., 165, 192 OLSHANSKY, R., 392, 393,401 A. M., 127, 131, 159 ONISHCHENKO, ORBACH, R., 96, 138, 158 ORTON,J. W., 110, 156 OSTERBERG, H., 331, 333, 402 OSTERMAYER, F. W., 149, 152, 157, 158, 396, 40 1 OTTO,A., 260, 289;323, 324 V. V., 149, 157 OVSYANKIN, OWEN,T. C., 191, 192 OWYOUNG, A,, 173, 192
P PAGE,J. L., 111, 124, 156 PAILLETTE, M., 167, 192 PALMBERG, P. W., 305, 324 PALMER, D. A., 197, 242 H. A., 52,86 PANOVSKY, PARRY, G., 18, 25, 26, 32, 46 W. D., 122, 124, I58 PARTLOW, PASK,C., 349, 375,401 PATTANAYAK, D. N., 259, 321 PAUTY,F., 280, 306, 325 PAUTY,M., 280, 306, 307, 308, 322, 325 PAXMAN, D. H., 111, 124, 156 PAXTON, K. B., 385, 402 PAYNE,D. N., 329, 343, 375, 400, 401 PEACOCK, R.D., 113, 114, 115, 158 PEARSON, A. D., 329,401 PEASE,F. G., 61, 86 PEDERSEN, H. M., 27, 29, 30, 31, 41,46 PEISAKHSON, I. V., 235, 243 PEISNER,J., 282, 283, 284, 288, 324 PELLETIER-ALLARD, N., 110, 113, 124, 156 PENDRY, J. B., 263, 322, 323 PEPPER,S. V., 280, 324 PERLMAN, M. L., 304, 305, 325 PERSONICK, S. D., 369, 401 PETER~ANN, K., 401 PETIT,R., 229, 243 PIEPENBRING, F. J., 284, 324 PIERCE,D. T., 286, 291, 292, 321, 324 F’IEUCHARD,G., 221, 238. 240. 242. 243 PLATT,J. R., 57, 86 E. W.. 316, 324, 325 PLUMMER,
AUTHOR INDEX
PONG,W., 265,285, 286,290, 291, 292, 324 POPE, M., 164, 184, 192 PORTER Jr., J. F., 149, 158 POTTER,R. J., 375,401 POUEY, M., 238, 243 F., 264, 324 PRADAL, H.M., 374, 401 -BY, PRIOL,M., 282,284, 288, 324 W. M., 52, 86 PROTHEROE, PWSEY, P. N., 37, 39, 40, 41, 45, 46
Q QUEMERAIS, A., 282, 284,288, 324 QUINN,J. J., 265, 280, 306, 324 R RAJNAK, K., 109, 110, 113, 156 RAMACHANDRAN, G. N., 4, 25,46 RAMAN,C. V., 4, 46 RAMBERG,E. G., 14, 45, 247, 325 RAMSAY,M., 401 RANKE, w., 315, 324 RAYLEIGH, L., 4, 37, 38,46 REEDJr., E. D., 130, 158 REED, I. S., 10, 23, 34, 46 REISFELD, R., 109, 113, 115, 131, 158 M., 163,164,176,177,185,187, RENTZEPIS,P. 192, 193 RHODES,78, 86 T. N., 279, 305, 324, 325 RHODIN, RICE,S. O., 13, 17, 46 RICH, T. C., 151, 157 RICHARD, J. C., 298, 324 E. H., 79, 81,86 RICHARDSON, M. C., 177, 182, 183, 192 RICHARDSON, RICHTER,H. J., 145, 150, 158 J. W. T., 305, 324 RWGWAY, RINCK,B., 111, 124, 158 RISEBERG,L. A., 96, 118, 121, 122, 124, 125, 126, 129, 130, 131, 150, 158 RITCHIE,R. H., 260, 281, 324 D., 197, 243 RITTENHOUSE, RlVlERE, J. c., 247, 324 ROBIN,S., 260, 282, 284, 288, 322, 324 M., 146, 156 ROBINSON, ROBOZ,P., 283, 324 ROCCA,A., 86 C., 52,68, 86 RODDIER, F., 52, 53, 86, 87 RODDIER, ROGSTADT, 86 ROMANO-MORAN, R., 295, 298, 321
409
J., 57, 69, 86 R~ SC H, ROWAN, G., 343,401 Ross, G., 9, 19,46 ROSSMANITH, R., 177, 192 ROULET,B., 279, 321 H. A., 235, 243 ROWLAND, RUDOLPH,D., 197, 204, 206, 208, 221, 228, 233,235,238, 241,243,244
S SAINT JAMES, D., 279, 321 Y.,235,243 SAKAYANAGI, SALA,K., 177, 182, 183, 192 SAMMUT,R., 357, 361, 401 C. P., 402 SANDBANK, SARBEI, 0. G., 265, 283, 323 J. F., 151, 153, 157 SARVER, K., 291, 321 SATTLER, SAUTER, F., 259, 324 A., 164, 186, 191, 192 SAVAGE, G. C., 164, 188, 193 SAVAGE, A., 260, 322 SAVARY, E. D., 295, 324 SAVOYE, H., 298, 324 SCHADE, SCHAICH, W. L., 261, 279, 324 A. L.,96, 159 SCHAWLOW, SCHEER, J. J., 299, 313, 325 SCHEGGI,A. M., 385,400 S . A., 402 SCHELKUNOFF, SCHIFF,L. I., 310, 324 S. P., 389,402 SCHLPSINGER, SCHMAHL, G., 197, 204, 206, 208, 221, 224, 228, 235, 238, 241, 243, 244 SCHNATTERLY, S. E., 260, 323 SCHUBIN, S., 325 ' SCHULMAN, J. H., 97, 135, 156 SCOTT, W. C., 96, 159 SCRIBOT, A. A., 16, 18, 46 SEAH, P., 305, 324 SEBENNE,C., 314, 315, 324 SEIBERT, M., 184, 193 SEYA,M., 235, 238, 240, 243 SHANK,C. V., 163, 164, 166, 170, 171, 172, 173, 192, 193 S. L., 163, 164, 166, 176, 180, 184, SHAPIRO, 192, 193 N. K., 220, 244 SHERIDON, F., 176, 177, 193 SHIMIZU, S., 142, 143, 144, 146, 147, 149, SHIONOYA, 150, 158, 159 D. A., 247, 324 SHIRLEY, A. R., 285,324 SHUL'MAN,
410
AUTHOR INDEX
SUNAK, H., 343, 400 SIEGERT, A. J. F., 15,46 S U R R AG. ~ , T., 110, 159 SIEGMANN, H. C., 286, 291,292, 321,324 L., 279, 325 SUTTON, SIMMONS, J. G., 295, 325 SWDAM,B. R., 176, 192, 193 SIMON, H., 247, 325 SUZUKI, T., 9,46 SINGH,K., 4, 46 SZE, S . M., 265, 282, 283, 322, 325 SKIBOWSKI, M., 302, 303, 321 B. J., 313, 322 SLAGSVOLD, T SLEPIAN, D., 15, 46 SMITH, J. A., 291, 292, 324 TAIT,R. H., 279, 325 SMITH,L., 350, 402 TAMM,I., 262, 325 N. V., 247,266,267,271,280,306,325 SMITH, TANIMOTO, O., 142, 159 SNITZER, E., 331, 333, 350, 402 SNYDER, A. W., 331, 334, 335, 349, 357,358, TARNG,M. L., 305, 325 V. I., 5, 46 359, 361, 363, 365, 366, 368, 375, 376, 380 TATARSKI, TENG,Y. Y., 231, 244 401,402 THOMAS,D. G., 258, 259, 323 SOMMER, A. H., 247,298, 299, 325 THOMAS, H., 280,284, 306, 307, 324, 325 SOVERS, 0. J., 109, 157 THOMAS, L. H., 310, 324 SPEER,R. J., 233, 235, 238, 243, 244 SPICER,W. E., 255, 262, 265, 266, 271, 275, THOMSON, B. J., 86 277, 280, 283, 290, 298, 300, 301, 303, 309, THORNBER, K. K., 279, 325 J. J., 298, 325 TIETJEN, 310, 315, 319, 321, 322, 323, 325 SPILLER, TIGANOV, E., 25,46 E. V., 170, 193 SPITZER, TIMMERMANN, W. G., 265, 271, 283, 322 C. C., 347, 395, 396, 402 SPIZZICHINO, TONG,S. Y., 279, 325 A,, 5, 6, 27, 33, 45. TOPP,M. R., 164, 177, 185, 187, 193 SPRAGLIE, R. A., 32,46 TOWNES, C., 84, 87, 176, 192 STACHNIK, R. V., 51, 58, 70, 73, 86 TRIBILLON, STANDLEY, R. D., 331, 402 G., 32, 46 STANLEY, N. R., 51, 86 TROTIMOV, A. K., 149, 157 J. F., 287, 308, 310, 313, 321 STARUNOY, V. S., 170, 193 TRUITARD, STEFFEN, H., 285, 306, 324 D., 235, 238, 244 TURNER, STEINBERG, R. A., 395, 402 R. R., 318, 325 TURTLE, R. G., 304. 305. 325 STEINHARDT, TWISS,R. Q.,50, 86 W., 253,289, 302, 303, 308, 321, TYNES, A. R., 329, 331, 401, 402 STEINMANN, 323, 325 TZOAR,N., 279, 322, 325 STEPHAN, H., 87 STERN,E. A., 231, 244 U STERN,F., 252, 325 STERN,J. R., 345, 350, 369, 381,400, 402 UEBBING, J. J., 298, 321, 325 STETSER,D. A., 163, 192 UNGER,H. G., 363,402 STEVENS, K. W. H., 93, 156 STOICHEFF, B. P., 176, 177, 193 V STRAKOVSKAYA, S. E., 285, 324 STRAITON,J. A., 402 VANDERZIEL, J. P., 149, 150, 152. 158 STREIFER, W., 385,401,402 VAN DE STADT, H.,84, 87 STROHBEHN, J. W., 5,46, 51, 86 VANLAAR,J., 299, 313, 325 STROKE, G. W., 197, 198, 244 VANUITERT,L. G., 97, 134, 149, 150, 152, STRUCK, C. W., 115, 158 155, 157, 158 STUART, R. N., 275,286,325 VANVLECK,J. A., 93,95, 158 STURGE, M. D., 121, 124, 158 VARITIMOS, T. E., 110, 113, 159 SUGANO, T., 265, 282, 283, 325 R. R., 295, 325 VERDERBER, SUHRMAN, R., 247, 325 VERNIER, P. J., 258, 259, 265, 279, 280, SUMI,K., 247, 323 283-288,290,295, 306, 307, 308,310, 313, SUMIDA, R., 265, 285, 286, 324 316, 321-325
AUTHOR INDEX
41 1
VERNIN,J., 52, 86, 87 VIGANTS,A., 389, 402 VUJOEN,P. E., 302, 325 VOGEL,G. C., 164, 186, 188, 193 VON FRAUNHOFER, I., 197, 243 VON LAUE,M., 4, 9,46
WLERICK, G., 57, 69, 86 WOLF,E., 19, 23, 45, 61, 85, 259, 321 WOOTEN, F., 275, 286, 325 WYANT,J. C., 79, 86 WYBOURNE, B. G., 91,101,108,110,113, 156, 159
W
Y
WACLAWSKI, B. J., 316, 325 YAKOVLEV, E. A., 235,243 WAGNER,L. F., 300, 301, 315, 325 YAMADA,N. S., 142, 144, 146, 147, 159 WATTS,R. K., 145, 149, 150, 151, 153, 154, YAMAGUCHI, I., 9, 22, 46 158 YANG,K. H., 376, 377,402 W ~ ~ ~ ~ , M . J . , 9 6 , 1 0 9 , 1 1 0 , 1 1 3 , 1 1 7 , 1 1 8 , 1 2YATSIV, 4, S., 96, 116, 159 127, 128, 130, 131, 132, 133, 135, 142, 149, YEN, W. M., 96, 159 158, 159 YIP, G. L., 342, 389, 402 WECHT,K. W., 163, 192 YOKOTA,M., 142, 159 WEHNER,G. K., 305, 325 YOUNG,A. T., 52, 87, 384, 400 WEISSLER,G. L., 312, 325 Yu, W., 175, 184, 193 WELFORD,W. T., 41,46 WELLER,P. F., 136, 156 2 WILLIAMS, B. F., 298, 322, 325 WILLIAMS, M. W., 288, 321 ZERNIKE, F., 87 WILSON,I., 229, 243 ZVEREV, G., 127, 131, 159 WINDSOR, M. W., 131, 157 ZWORYKIN, V. K., 247, 325
SUBJECT INDEX A
-, speckle, 29 axisymmetry, 67
Abbe sine condition, 235 aberration free lens, 8, 35 aberrations, 57 - in gratings, 202 - introduced by blank, 21 1 absorption coefficient, 282 actinides, 92 active phasing, 78 age theory, 273 Airy disk, 3, 54 et seq. alkali halides, 290 angular acceptance, 164 - dispersion, 198 anisotropic media, 258 annealing, 315 anomalies, 23 1 anomalous structure, 316 aperture shape, 56 apertures, large-, 60 aplanatic system, 235 Apollonius' circle, 240 apparent stellar diameters, 50 array of fibres, 187 - - photodiodes, 185, 188 array of telescopes, 49, 62, 79 et seq. aspherical waves, 241 astronomy, 20, 36, 49 et seq. asymmetrical groove profiles, 219, 231 atmosphere, 19 - optics, 51 - phase shift, 61 - scattering, 4 attenuation - coefficient, 365 et seq. - in lossy layers, 354 et seq. - length, 263 Auger electrons, 267, 279, 304, 315 autocorrelation, 4, 9 et seq., 59
B ballistic approximation, 270 band bending, 299 bandwith, spectral-. 55, 64 baselines, 73 beamsplitter, 66 et seq. beat wavelength, 367 Beckman-model, 6 et seq. bends in fibres, 363 et seq. binary stars, 20, 53, 57, 70 birefringence, 174 bit rate, 347 blaze waveleugth, 2 0 Bloch states, 254 et seq. Bohr radius, 138 Boltzmann statistics, 146 Born-approximation, 32 Born-Oppenheimer wavefunctions, 1 16 borosilicate fibres, 330 Bose-Einstein average, 120 branching ratios, 1 13 bright speckles, 59, 78 Brillouin zones, 254 et seq. Brownian motion, 52 C
carbon disulfide, 165 et seq. Cassegrain telescopes, 80 CdS, reflectivity of-, 258 Ce3+, 150 cell -, seeing, 54 -, phase, 55 central limit theorem, 6, 35 centrosymmetric objects, 3, 77 characteristic curve of photoresist, 21 5 412
SUBJECT I NDE X
characteristic equation for modes, 333 et seq., 351 et seq. charge transfer states, 1I5 cladded optical fibres, 329 cladding - boundary, 344 et seq. - mode approximation, 339 - modes, 361 closure relation, 107 coating, photoemission through a - layer, 290 coherence - parameter, 24 -time, 61, 82 hermitian - matrix, 34 -, mutual, 18 -, temporal, 25, 57, 213 -, spatial, 18 coherent array of telescopes, 62 - synthetic aperture, 49 collective motion of electrons, 253 - - - ions, 253 collinear gating, 174 comparison of holographic and classical gratings, 223 compound mode number, 349 conductivity, 268 configurations of fibres, 330, 332 construction of holographic gratings, 207 contrast - of fringes, 50, 60 et seq. - of speckles, 39 convergence, 6 convolution, 20, 60, 62 cooperative relaxation, 133 et seq. - luminescence, 149 core, liquid-fibres, 346 - mode approximation, 339 et seq. - - cut-off, 332, 341 et seq. cores of fibres, 329 et seq. correlated random variables, 16 - surface structure, 28 correlation area, 35 - matrix, 33 Coulomb repulsion, 264 coupling - due to bends, 366 et seq. - due to perturbations, 366 - of electrons, 278 -, selective, 374 -, surface, 316 Cr3+, 150 critical angle, 333 et seq. cross over in propagation of modes, 341
-.
41 3
cross section -, absorption, 108 et seq. -, stimulated emission, 1 13 et seq. crystal field theofy, 93, 101 et seq. crystalline Stark field, 98 curvature spectrum function, 368 curved slab boundary, 363 cut-off frequency, 123 CTCF, 258
D DAP, 250 et seq. damped excitation, 252 Debye temperature, 267 decay in rare-earths, 102 et seq. deconvolution, 60 degree of mutual coherence, 61 - - spatial coherence, 3 defocus, 37 delay, 61 delay in fibres, 342 et seq., 372, 393 - lines, 80 density of absorbed photons, 248 - - state, p(E), 265 depolarization, 33 depopulation, thermal-, 126 detection, post-, pre-, 49 Dexter model, 139 dichroic mirror, 175 dielectric constant, 252 diffraction gratings, 197 et seq. - interference, 58 - limited MTF, 62 diffuser, 6 et seq. -, white noise, 24 diffusion equation, 142, 272 - limited relaxation, 98 direct transitions, 253 directless condition, 386 dispersion - equation, 258 - of gratings, 198 -, material, 342 et seq. -, pulse, 329, 342 et seq., 369 -, reduction of pulse, 374 -, waveguide, 342 et seq. disturbances, 52 double-clad fibre, 382 dynamic crystal field interaction, 118 et seq.
414
SUBJECT INDEX
E echelle gratings, 203 echelon technique, 164, 185 et seq. ED, 106 er seq. EDD, EDQ, 136 efficiency, 228, 231 eigenvalues, 16 eigenvalues of modes, 333 et seq. Einstein relations, 105 electric displacement, 251 electric field amplitude, 25 1 electron - bombardment, 291, 304 - diffraction, 276 - hole pair creation, 253, 264 - hole recombination, 268 - spectroscopy, 247 one-approximation, 254 -, (quasi) free-gas, 253 -, scattered, 302 -, secondary, 302 electronographic photographs, 57 electrostatic interaction, 135 elongation of speckles, 31 energy density required, 216 - levels of rare-earths, 98 entropy, 7 EPR, 134 EQ, 109 EQQ, 139 Er, 113 ESCA, 247, 303 escape -. elastic-depth, 302 et seq. - depth, 248 et seq., 284, 288 - depth trends, 319 - probability, 248 et seq., 263 etalon, oven stabalized, 214 etendue, 242 - efficiency, 379 - of fibres, 375 Eu3, 143, 150 evanescent region, 361 - waves, 260 exchange interaction, 136 excitation in rare-earths, 102 excited state lifetime, 104 experiments on - - cooperative relaxation, 142 et seq. - - escape depth, 219 et seq. - - multiphonon relaxation, 124 et seq. - - radiative decay, 111 et seq.
-.
F fibre
-, double clad, 382 bandwidth, 347 et seq. optics, 84, 186, 329 et seq. - with uniform index, 331 et seq. - with nonuniform index, 381 et seq. -, square, 386 -, W, 382 field components in fibres, 350 et seq. filter, linear, 12 Fizeau aperture, 60 - interferometer, 64, 66 flight, light photographed in, 177 fluctuation -, density, 52 -, intensity, 10 -, partial pressure, 52 -, temporal, 4 fluorescence -, chromium, 149 - decay time, 164, 183, 191 - lamps, 151 - quencing, 134 Forster-Dexter theory, 134 Fourier transform, 6 et seq., 58 et seq. - space, 62 - synthesis method, 221 Fraunhofer plane, 5 et seq. Fredholm equation, 16 frequency, normalised, 333 et seq. Fresnel equations, 256 fringes, spacing o f , 202 frustrated total reflection, 260, 289 -
G GaAs, 295 et seq. GaAs laser. 346 gain, resolution, 60 gamma distribution, 16 gas laser, 376 gated picture ranging, 164, 180 et seq. gating -, collinear, 174 - in glass, 173 -, transverse, 176 er seq. Gaussian, 55 - distribution, 5 - scanning aperture, 14 et seq. - spread function, 60 ghosts, 226, 242
SUBJECT I N D E X
glasses, 131 Goodman model, 6 graded index fibres, 329, 351, 383 grass, 226 grating -, concave, 235 -, echelle, 203 - blank, 200 -equation, 198 - interferometer, 224 - with imaging, 235 -, holographic, 197 et seq. -, metallic, 260 -, special types Of, 239 -, tripartite, 236 grazing incidence, 219 Green functions, 274 groove profiles of gratings, 219, 229 grooves, nonuniform, 235 group delay, 342 et seq., 392 - theory, 101 -velocity, 174, 264, 342, 388
index variation, parabolic, 384 infrared, near, 57 - upconversion, 97 inhomogeneities, intentional, 374 intensity interferometer, 50, 82 ef seq. interference -,diffraction, 58 - fringe system, 201 - grating, 200 interferometer -, direct, 59 ef seq. -, heterodyne, 84 -, intensity, 82 et seq. -, triple, 78 internal modes of molecular groups, 123 intramultiplet transitions, 110 instrumental profile, 224 et seq. ion - ion energy transfer, 133 et seq. - lattice interactions, 91, 116 - pair energy transfer, 96 - phonon coupling, 119 et seq. isoplanatic patch, 57, 60 - spread function, 12
H Hamiltonian
-, crystal field, 117 -, ion pair coupling, 138 helical rays in fibres, 385 Hermitian coherence matrix, 34 heterogeneous wave, 252 heterodyning, 50, 84 Hg-line, hyperfine structure, 225 history of rare-earths, 93 H O ~ +125, , 154 holographic diffraction gratings, 197 ef seq. - - -, theory of, 198 et seq. - - _ , principles of, 200 et seq. honeycomb pattern, 57,62 host dependence, 129 ei seq. humidity, 52 hyperfne structure of Hg line, 225 hypersensitivity, 114 I
IH approach, 140 image intensifier, 69 imaging properties of gratings, 235, 242 impedance, 259 impulse response in fibres, 371, 387
J Judd-Ofelt theory, 106
K Kerr shutter, 162 er seq. Ketteler-Helmholtz formula. 253 knife-edge test. 53 Kramers-Kronig relations, 252
L La, 124, 129, 149 laminar profiles, 149, 219 lanthanides, 92 Laplace transformation, 370 Laportes’ rule, 106 laser, 95, 154 et seq, 346 -, display of ultrashort-pulses, 179 -, gas, 376 - excitation in rare-earths, 150 - speckle, 55, 56 lattice vibrations, 116 et seq. leaky modes, 357 ef seq.
415
416 Leed emission, 279, 315 level assignment in rare-earths, 93 limbdarkening, 70 linear dispersion, 198 linebroadening, 116 line density of gratings, 229 linewidth o f a spectral line, 116,213 Lippmann -Schwinger equation. 278 Lippmann plates. 197 liquid, 59 -, scattering by
M magnons, 134 Mahan theory, 278 many body effects, 275 mask aperture, 50, 60 material dispersion, 342 et seq. matrix element, 254 et seq. MD transitions, 109 er seq. MDD transitions, 136 meridional rays in fibres, 384 metal-insulator-metal, 294 metastable fluorescing levels, 117 Michelson interferometer, 64 er seq. migration in rare-earths, 135 - of resonant excitation, 97 Moire patterns, 209 monolithic telescope mirrors, 61 monomode fibres, 331 et seq., 356 mode -, cladding, 361 et seq. -, compound-number, 349 -, core, 339 et scq. -, leaky, 357 et seq. -, LP, 350 et seq. - coupling, 343, 369 et seq. -, perturbations in-coupling, 366 -, quasidegenerate-pair, 350 -, total number of, 347, 392 -, trapped, 333 er seq. molecular group vibrations, 130 monster waves, 59
SUBJECT INDEX
Monte-Carlo calculations, 267 MTF, 58, 59, 62 multi - alkali layers, 286 - aperture systems, 64 - beam splitter, 186 - channel sampling, 164, 185 - mode fibres, 329 et seq. - phonon relaxation, 96, 116 et seq - polar terms, 97 - - interactions, 140 - pole operator, 135 et seq.
N Nd in glass, 113 er seq. Nd3+ laser, 154 NEA, 268, 295 et seq. nearest neighbor coupling, 144 nitrobenzene, 169 non centro symmetric sites, 106 non direct transitions, 255, 318 nonlinear index, 176 nonradiative decay, 91, 132 - probability, 104 nonresonant transfer, 98 nonuniform index, 381 et seq. normal speckle pattern, 4 et seq. numerical aperture, 8, 22, 333 0
object -, general diffuse, 13 -, very rough, 22 -, white noise, 13 occupation probability, 254 ODs, 281 operator, electric dipole, 107 optical fibre waveguides, 329 et seq. Orbach process, 116 orbit-lattice interaction, 96, 120 et seq. organ array, 187 oscillator strength, 95, 108 et seq. OSO, 164, 186 et seq.
P pair creation, 302 paramagnetic ions in solids, 133
SUBJECT I NDE X
resonance, 103 partial coherence, 18 et seq. patch, isoplanatic, 57 PE, 246 et seq. PED, 248 et seq. Peppers’ formulation, 289 periodic grating errors, 200 perturbation - of free ion states, 101 - in higher orders, 1 I8 - in mode coupling, 366 -, phase and amplitude, 53 phase-cell, 55 phase-change coefficient, 388 et seq. phonon, 253 - induced relaxation, 96 - processes, 103, 116 et seq. - scattering, 267 et seq. -, stimulated emission, 120 photo - bleaching, 185 - cathodes, 247 et seq. - emission, PE, 246 et seq. - excitation coefficient, 249, 255 - multiplier, 66 - polymers, 216 - yield, Y, 248 et seg. - resist, 197, 214 et seq. photography - emulsions, 197 -, ultra high speed, 177, 182, 191 photoncounting, 51, 63, 68 et seq. picket-fence mask, 66 picosecond pulses, 163 et seq. - -, sub, 170 plasma, laser produced, 182 - oscillations, 260, 308 plasmons, 231, 253 et seq. plasmon scattering, 264 point spread function, 14 et seq. - symmetry, 101 Poisson equation, 300 - law, 63, 64 polarization, 6, 229 polychromatic speckles, 29 power flow, 352 et seq. Poynting vector, 260, 352 production of holographic gratings, 216 profiles of grating grooves, 218 et seq. propagation coefficient, 332, 388, 392 pulse -, display of laser, 179, 191 - broadening, 393 -
- dispersion, 329, 342, 369 -, reduction of - broadening, 374 pupil function, 12
Q quality of diffraction gratings, 200 quantum efficiency, true, 310 - noise, 63 quasi degenerate mode pair, 350 quenching -, fluorescence, 97, 134 -, self, 134
R radial structure of speckles, 25 radio waves, 50 radiative decay in rare-earths, 91, 106 - probability, 104 - quantum efficiency, 105 Raman processes, I16 random - bends in fibres, 367 - errors in gratings, 200 - k-approximation, 266 - process, 5 ranging, gated picture, 164 rare-earth - luminescence, 91 et seq. - spectroscopy, 93 -, trivalent, 93 rate equations, 151 ray interpretation for fibres, 338 et seq. Rayleigh, 55 - criterion, 198 - distribution, 56 - scattering, 52 - statistics, 59 reduced decay, 125 reduction of speckles, 32 refractive index parameter, 391 et seq. relaxation -, diffusion-limited, 149 - of 5d4f, 132 - phenomena in rare earths, 91, 95 -times, 164, 166 resolution, 49 - gain, 60 resolving power of gratings, 199 resonant transfer, 138
417
418
SUBJECT INDEX
ring-shaped refractive index, 396 rough surface approximation, 6, 41 Rowland circle, 233, 239 rubber telescope imaging, 51 ruling errors, 200 et seq. - -, progressive. 204 el seq. Russel-Saunders states, 101 S
sampling -,gated, 164 -, multichannel, 164, 188 et seq. - optical signals, 183 -oscilloscope, 164 sawtooth groove profile, 220, 242 scale, inner and outer, 52 scattered light, 226 et seq. scattered waves, 260 scattering -, diffuse, 226 -, instantaneous light, 178 -, inelastic phonon, 116 - intensity, 10 Schlieren effects, 53 screening effect, 248, 251 et seq. secondary electrons, 271, 302 seeing cell, 54 - compensation, 78 selection rules, 136, 255 et seq. selective coupling in fibres, 374 selffocussing, 175 et seq. SELFOC,382, 384 et seq. sensitized luminescence, 134 shadow patterns, 53 shear, lateral, 67 shearing, wavefront-interferometer, 79 signal-to-noise ratio. 14, 63, 228 sinusoidal grooves, 217 single pixel, 66 SIT-tube, 70 space, 49 spatial distribution, 258 SPBS,261 speckle, 3, 49 et seq. -, bright, 59, 78 -, fringed, 62 -, honeycombed, 62 -, laser, 55, 56 -, lifetime of, 57 -, shape of, 56 - interferometry, 51, 54
- structure, 53 spectral bandwidth, 55, 64 - resolution, 225 - yield distribution, 312 spectroscopic binary, 65 spectroscopy, time resolved, 191 spherical aberration, 235 - blanks for gratings, 233 - mounts, 82 spin-lattice relaxation, 116 spin polarization of photo-electrons, 291 - selection rule, 133 spontaneous multiphonon - emissiofi rate, 127 - emission probability, 108 spread function, 61 et seq. - -, Gaussian, 60 square fibre, 386 stability of laser light, 213 standing wave pattern, 68 Stark field in crystals, 98 - levels, 96, 116 et seq. statistical properties of specklepattern, 3 statistics -, first order, 5 et seq. -, second order, 8 et seq. - of measured intensity, 13 et seq. steady state distribution, 373 stellar diameters, 20 step, 3-model, 248 et seq. -, seperation in 3,249 et seq. stigmatic images, 240 structure of images, 53 subpicosecond pulses, 170, 192 surface -, PE from - states, 262 et seq. - absorption term, 260 - contamination, 262 - dependent speckles, 33 - photoexcitation, 261, 305 - plasmons, 231, 260 - roughness, 32, 260 - states, 299, 314 -, transmission by the, 268 symmetrical groove profiles, 217 symmetry properties, 67 synchrotron radiation, 315 synthetic aperture, 49, 50
T Tb, 142, 150, 155 telescopes, array of, 49, 62, 79 et seq.
S U B J E C T INDEX
temperature dependent relaxation, 119, 124 et seq.
temporal speckles, 83 393 theory of - - cooperative relaxation, 135 et seq. - - escape probability, 270 et seq. - - multiphonon relaxation, 117 et seq. - - photoemission, 250 et seq. - - radiative decay, 106 et seq. thermal convection, 52 - light sources, 4, 375 thermalization, 256 thermoinjection, 294 thin films, 257 et seq. timeresolved spectroscopy, 191 timeresponse, 174 TiO,, 394 top-hat aperture, 14 toroidal blanks for gratings, 234 transfer function, 12 transition layer, 259 - probability, 254,269 transmission - cross coefficient, 23 - of Kerr cells, 165 et seq. - properties of fibres, 332, 364 temporal frequency spectrum, 53 trialkali photocathodes, 247 triangle rule, 123 tripartite gratings, 236 triple interferometer,78 tunnel effect, 294 turbulence -, interference, 52 -, wake 52 -, windshear, 52 turbulent-behaviour of the atmosphere, 51 - width,
-.
U ultraviolet, near, 57 uniformity of gratings, 219 upconversion of phosphors, 97, 135, 149, 151 et seq. utilization of rare-earth metals, 91
- sidebands, 116 vibronic intensities, 1I3 visibility, 60 et seq. -, improvement of, 180 - modulus determination, 61 - phase problem, 76 vision through the skin, 181 volume scatterers, 4
W wave deformation, 53 wave, monster, 59 wave, surface, 333 et seq. wavefront interferogram, 224 waveguide -, optical fibre, 329 et seq. -, slab, 363 - dispersion, 342 et seq. Weisskopf-Wigner theory, 276 W-fibre, 382 et seq. white noise object, 13 Wiener spectrum, 9 et seq. wind - flow, 57 - shear turbulence, 52 - speed, 53 winding tension in fibres, 375 W.K.B. approximation, 389, 396 Wood anomalies, 231, 260 work function, 269 et seq.
X X-ray -, diffraction, 77 -, gratings, 232 et seq., 242 -, image intensifying screens, 151 -, soft - region, 219 -, ultrashort - pulses, 164 X-rays, 247,279,291, 303
Y Y,12’7, 132 Yb, 143, 149, 151 Youngs’ fringes, 50, 61 et seq.
V Z
variation of temperature and pressure, 214 Vega, 74 vibrational energy, 96
Zeeman levels, 1 16 zone plate, 206, 235, 241
419
CUMULATIVE INDEX - VOLUMES I-XIV 11, 249 ABEL~S, F., Methods for Determining Optical Parameters of Thin Film VII, 139 ABELLA, I. D., Echoes at Optical Frequencies AGARWAL, G. S., Master Equation Methods in Quantum Optics XI, 1 IX, 235 AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion IX, 179 ALLEN,L., D. G. C. JONES, Mode Locking in Gas Lasers IX, 123 AMMANN, E. O., Synthesis of Optical Birefringent Networks ARMSTRONG, J. A., A. W. SMITH,Experimental Studies of Intensity Fluctuations in Lasers VI, 21 1 ARNAUD, J. A,, Hamiltonian Theory of Beam Mode Propagation XI, 245 BALTES, H. P., On the Validity of Kirchhoff s Law of Heat Radiation for a Body in a XIII, 1 Nonequilibrium Environment R., The Intensity Distribution and Total Illumination of AberrationBARAKAT, I, 67 Free Diffraction Images XII, 287 S., Beam-Foil Spectroscopy BASHKIN, VI, 53 BECKMANN, P., Scattering of Light by Rough Surfaces BLOOM, A. L., Gas Lasers and their Application to Precise Length Measurements IX, 1 IV, 145 BOUSQUET, P., see P. Rouard BRYNGDAHL, IV, 37 O., Applications of Shearing Interferometry XI, 167 BRYNGDAHL, O., Evanescent Waves in Optical Imaging BURCH,J. M., The Metrological Applications of Diffraction Gratings 11, 73 CHRISTENSEN, J. L., see W. M. Rosenblum XIII, 69 CI ARRICOATS. P.J. B.. Optical Fibre Wiiveyuides A Review X I V . 327 COHEN-TANNOUDJI, C., A. KASTLER, Optical Pumping v, 1 CREW, A. V., Production of Electron Probes Using a Field Emission Source XI, 221 CUMMINS. H. Z., H. L. SWINNEY, Light Beating Spectroscopy VTII, 133 XIV. 1 DAINTY. J. C.. The Statistics of Speckle Patterns DECKER Jr., J. A,, see M. Harwit XII, 101 DELANO, E., R. J. PEGIS,Methods of Synthesis for Dielectric Multilayer Filters VII, 67 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 DUciuAY. M. A.. The U l t r a f d Optical Kerr Shutter XIV. 161 EHERLY.J. H.. Interaction of Very Intense Light with Free Electrons VTI. 359 FIORENTINI, A., Dynamic Characteristics of Visual Processes 1, 253 FOCKE, J., Higher Order Aberration Theory IV, 1 FRANCON,M, S. MALLICK,Measurement of the Second Order Degree of Coherence VI, 71 FRIEDEN, B. R., Evolution, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions 1X. 311 FRY,G. A., The Optical Performance of the Human Eye v111, 51 GABOR,D., Light and Information I. 109 420
CUMULATIVE INDEX
42 I
GAMO,H., Matrix Treatment of Partial Coherence 111, 187 GHATAK, A. K., see M. S. Sodha XIII. 169 V. L., see V. M. Agranovich GINZBURG. IX, 235 GIOVANELLI, R. G., Diffusion Through Non-Uniform Media 11, 109 K., J. PETYKIEWICZ,Applications of Optical Methods in the Diffraction GNIADEK, IX, 281 Theory of Elastic Waves GOODMAN, J. W., Synthetic-Aperture Optics VIII, 1 GRAHAM,R., The Phase Transition Concept and Coherence in Atomic Emission XII, 233 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 Refractive Index V, 247 JACQUINOT, P., B. ROIZEN-DOSSIER, Apodisation 111, 29 JONES, D. G. C., see L. Allen IX, 179 KASTLER, A., see C. Cohen-Tannoudji v, 1 KINOSITA, K., Surface Deterioration of Optical Glasses IV, 85 KOPPELMAN, G., Multiple-Beam Interference and Natural Modes in Open Resonators VII, 1 KOTTLER,F., The Elements of Radiative Transfer 111, 1 KOTTLER. F., Diffraction at a Black Screen, Part I: Kirchhoffs Theory IV, 281 KOTTLER.F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory VI. 331 KUBOTA.H., Interference Color I, 211 LABEYRIE. A,. High Resolution Techniques in Optical Astronomy XIV, 47 LEAN,E. G., Interaction of Light and Acoustic Surface Waves XI, 123 v1, 1 LEITH,E. N., J. UPATNIEKS, Recent Advances in Holography VIII, 343 LEVI,L., Vision in Communication LIPSON,H., C. A. TAYLOR, X-Ray Crystal-Structure Determination as a Branch of Physical Optics V, 287 MALLICK, S., see M. Frangon VI, 71 11, 181 MANDEL, L., Fluctuations of Light Beams MANDEL, L., The Case for and against Semiclassical Radiation Theory XIII, 27 MARCHAND, E. W., Gradient Index Lenses XI, 303 VIII, 373 MEHTA, C. L., Theory of Photoelectron Counting MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation V11, 231 MNAMO~ K.,, Wave Optics and Geometrical Optics in Optical Design I, 31 MURATA,K., Instruments for the Measuring of Optical Transfer Functions v , 199 VIII, 201 MUSSET,A,, A, THELEN,Multilayer Antireflection Coatings VII, 299 Oom, S.,The Photographic Image PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 PEGIS.R. J.. see E. Delano VII, 61 PERSHAN, P. S., Non-Linear Optics V, 83 IX, 281 PETYKIEWICZ, J., see K. Gniadek PICHT,J., The Wave of a Moving Classical Electron V, 351 RISEBERC, L. A,, M. J. WEBER.Relaxation Phenomena in Rare-Earth Luminescence XIV. 89 VIII, 239 RISKEN,H., Statistical Properties of Laser Light ROIZEN-DOSSIER, B., see P. Jacquinot 111, 29 ROSENBLUM, W. M., J. L. CHRISTENSEN, Objective and Subjective Spherical Aberration Measurements of the Human Eye XIII, 69 IV, 145 ROUARD,P., P. BOUSQUET,OpticaI Constants of Thin Films IV, 199 RUBINOWCZ, A., The Miyamoto-Wolf Diffraction Wave
422
CUMULATIVE INDEX
XIV. 195 R~II)OI.PH D.. . see G. Schmahl VI. 259 SAKAI, H., see G. A. Vanasse XIV. 195 SCHMAHL. G.. D.RUWLPH,Holographic Diffraction Gratings SCHULZ,G., J . SCHWIDER, Interferometric Testing of Smooth Surfaces XIII, 93 SCHWIDER, J., see G. Schulz XIII, 93 SCULLY, M. O., K. G. WHITNEY, Tools of Theoretical Quantum Optics X, 89 X, 229 SITTIG,E. K., Elastooptic Light Modulation and Deflection SLUSHER, R. E., Self Induced Transparency XII, 53 VI, 211 SMITH,A. W., see J. A. Armstrong X, 165 SMITH,D. Y., D. L. DEXTER, Optical Absorption Strength of Defects in Insulators x , 45 SMITH,R. W., The Use of Image Tubes as Shutters V. K. TRIPATHI, Self Focusing of Laser Beams in PlasSODHA,M. S., A. K. GHATAK, mas and Semiconductors XIII, 169 V, 145 STEEL,W. H., Two-Beam Interferometry IX, 73 J. W., Optical Propagation Through the Turbulent Atmosphere STROHBEHN, G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution STROKE, Spectroscopy 11, 1 SVELTO,0,, Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams XII, 1 VIII, 133 H. H., see H. Z. CLJMMINS SWINNEY, V, 287 TAYLOR, C. A., see H. Lipson VII. 231 TER-MIKAELLAN, 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 TSUJIUCHI, J., Correction of Optical Images by Compensation of Aberrations and 11, 131 by Spatial Frequency Filtering UPATNIEKS, J., see E. N. Leith VI, 1 VI, 259 VANASSE, G. A., H. SAKAI,Fourier Spectroscopy I. 289 VAN HEEL.A. C. S.. Modern Alignment Devices VERNIER, P., Photoemission XIV, 245 WEBER,M. J., see L. A. Riseberg XIV, 89 WELFORD, W. T., Aberration Theory of Gratings and Grating Mountings IV, 241 WELFORD, W. T., Aplanatism and Isoplanatism XIII, 267 K. G., see M. 0. Scully x , 89 WHITNEY, WOLTER,H., On Basic Analogies and Principal Differences between Optical and Electronic Information I, 155 X, 137 WYNNE,C. G., Field Correctors for Astronomical Telescopes YOSHINAGA, H., Recent Developments in Far Infrared XI, 77 VI, 105 YAMMI,K., Design of Zoom Lenses YAMAMOTO, T., Coherence Theory of Source-Size Compensation in Interference Microscopy VIII, 295 YOSHINAGA, H., Recent Developments in Far Infrared XI, 17
