Progress in Optics, Volume 12

PROGRESS IN OPTJCS V O L U M E XI1

EDITORIAL ADVISORY BOARD M. FRANCON,

Paris, France

E. INGELSTAM,

Stockholm, Sweden

K. KINOSITA,

Tokyo, Japan

A. LOHMANN,

Erlangen, Germany

W. MARTIENSSEN,

Frankfurt am Main, Germany

M. E. MOVSESYAN,

Ereuan, U.S.S.R.

A. RUBINOWICZ,

Warsaw, Poland

G. SCHULZ,

Berlin, Germany ( G . D . R . )

W. H. STEEL,

Sydney, Airstralia

G. TORALDO DI FRANCIA, Florence, Italy W. T. WELFORD,

London, England

PROGRESS I N OPTICS VOLUME XI1

EDITED BY

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

Contvibutors

0. S V E L T O , R. E. S L U S H E R ,

M. H A R W I T , J . A. D E C K E R Jr., K. H. D R E X H A G E , R. G R A H A M , S. B A S H K I N

1974 NORTH-HOLLAND P U B L I S H I N G COMPANY

- AMSTERDAM

*

LONDON

0 NORTH-HOLLAND

PUBLISHING COMPANY

- 1974

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CONTENTS O F VOLUME I ( 1 9 6 1 )

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I.

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THE MODERNDEVELOPMENT OF HAMILTONIAN OPTICS. R . J PEGIS WAVE OPTICS AND GEOMETRICAL OPTICS I N OPTICAL DESIGN.K . MIYA-

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MOT0

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1-29 3 1-66

THEINTENSITY DISTRIBUTION A N D TOTAL ILLUMINATION OF ABERRATIONFREE DIFFRACTION I~UGES. R . BARAKAT. . . . . . . . . . . . . . 67-1 08 IV . LIGHTAND INFORMATION. D GABOR. . . . . . . . . . . . . . . . 109-153 V ON BASICANALOGIES AND PRINCIPAL DIFFERENCES BETWEEN OPTICAL A N D ELECTRONIC INFORMATION. H WOLTER. . . . . . . . . . . . . 155-210 VI . IWTERFERENCE COLOR. H KUBOTA. . . . . . . . . . . . . . . . . 21 1-25] VII DYNAMIC CHARACTERISTICS OF VISUALPROCESSES. A . FIORENTINI . . . . 253-288 VIII MODERN ALIGNMENT DEVICFS. A C . S . VAN HEEL. . . . . . . . . . 289-329 I11

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CONTENTS O F VOLUME I 1 (1963) 1.

. 111. IV . I1

V.

VI .

RULING.TESTING A N D USEOF OPTICAL GRATINGS FOR HIGH-RESOLUTION SPECTROSCOPY. G W STROKE . . . . . . . . . . . . . . . . . . . 1-72 THE METROLOGICAL APPLICATIONS OF DIFFRACTION GRATINGS.J . M. BURCH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73-108 DIFFUSION THROUGH NON-UNIFORM MEDIA.R . G . GIOVANELLI . . . . 109-129 CORRECTION OF OPTICAL IMAGES BY COMPENSATION OF ABERRATIONS AND BY SPATIAL FREQUENCY FILTERING. J. TSUJIUCHI . . . . . . . . . . . 131-180 FLUCTUATIONS OF LIGHTBEAMS. L. MANDEL . . . . . . . . . . . . 181-248 METHODS FOR DETERMINING OPTICALPARAMETERS OF THINFILMS. F ABEL~ . S. . . . . . . . . . . . . . . . . . . . . . . . . . . . 249-288

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CONTENTS O F V O L U M E 111 (1964) I. I1

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111

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THEELEMENTS OF RADIATIVE TRANSFER. F KOTTLER. . APODISATION. P. JACQUINOI' A N D B ROIZEN-DOSSIER .. MATRIX TREATMENT OF PARTIAL COHERENCE. H GAMO.

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1-28 29-186 187-332

CONTENTS OF VOLUME I V (1965) I. I1 .

111.

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IV V. VI

VII .

HIGHERORDER ABERRATION THEORY.J. FOCKE. . . . . . . . . . . APPLICATIONS OF SHEARING INTERFEROMETRY. 0. BRYNGDAHL . . . .. SURFACE DETERIORATION OF OPTICALGLASSES. K K~NOSITA . . . . . OPTICAL CONSTANTS OF THIN FILMS. P . ROUARD A N D P BOUSQUET ... THEMIYAMOTO-WOLF DIFFRACTION WAVE.A . RUBINOWICZ .... . ABERRATION THEORY OF GRATINGS AND GRATING MOUNTINGS. W. T . . . WELFORD .......................... F. DIFFRACTION AT A BLACKSCREEN.PARTI: KIRCHHOFF'S THEORY. . . KOTTLER. . . . . . . . . . . . . . . . . . . . . . . . . .

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1-36 37-83 85-143 145-197 199-240 241-280 281-314

CONTENTS O F VOLUME V (1966) I

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11.

I11. IV .

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1-81 OPTICAL PUMPING. C . COHEN-TANNOUDJI AND A KASTLER. . . . . . NON-LINEAR OPTICS.P . S. PERSHAN. . . . . . . . . . . . . . . . 83-144 TWO-BEAM INTERFEROMETRY. W H STEEL . . . . . . . . . . . . . 145-197 INSTRUMENTS FOR THE MEASURING OF OPTICAL TRANSFER FUNCTIONS. K. MURATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 199-245

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LIGHTREFLECTION FROM FILMS OF CONTINUOUSLY VARYING REFRACTIVE 247-286 INDEX.R . JACOBSSON ....................... X-RAYCRYSTAL-STRUCTURE DETERMINATION AS A BRANCH OF PHYSICAL OPTICS.H . LIPSONAND C . A . TAYLOR . . . . . . . . . . . . . . . 287-350 THEWAVEOF A MOVISGCLASSICAL ELECTRON J . PICHT . . . . . . . 351-370

V. VI .

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C O N T E N T S O F V O L U M E VI (1967)

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RECENT ADVANCES I N HOLOGRAPHY. E N . LEITHAND J . UPATNIEKS . SCATTERIKG OF LIGHTBY ROUGHSURFACES. P. BECKMANN . . . . . . OF THE SECONDORDERDEGREE OF COHERENCE. M 111 MEASUREMENT FRANFON AND S MALLICK .................... DESIGN OF ZOOMLEKSES. K . YAMAJI. . . . . . . . . . . . . . . . IV OF LASERS TO INTERFEROMETRY. D R . HERRIOTT . . V. SOMEAPPLICATIONS EXPERIMENTAL STUDIES OF INTENSITY FLUCTUATIONS IN LASERS. J. A. VI ARMSTRONG AND A W . SMITH. . . . . . . . . . . . . . . . . . VII FOURIER SPECTROSCOPY. G . A . VANASSE AND H . SAKAI. . . . . . . . VIII . DIFFRACTION AT A BLACK SCREEN.PART11: ELECTRO~MAGNETIC THEORY. F. KOTTLER ........................... 1.

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1-52 53-69 71-104 105-170 171-209 21 1-257 259-330 33 1-377

C O N T E N T S O F V O L U M E VII (1969) I

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MULTIPLE-BEAM INTERFERENCE AND NATURAL MODESI N OPENRESONAG . KOPPELMAN ....................... 1-66 METHODS OF SYNTHESJS FOR DIELECTRIC MULTILAYER FILTERS. E . DELANO AND R J . PEGIS. . . . . . . . . . . . . . . . . . . . . . . . . 67-137 ECHOES AT OPTICAL FREQUENCIES. I D . ABELLA. . . . . . . . . . . 139-1 68 IMAGE FORMATION WITH PARTIALLY COHERENT LIGHT.B. J THOMPSON 169-230 QUASI-CLASSICAL THEORY OF LASERRADIATION. A . L MIKAELIAN AND M . L . TEK-MIKAELIAN ...................... 231-297 THEPHOTOGRAPHIC IMAGE.S . OOUE. . . . . . . . . . . . . . . . 299-358 INTERACTION OF VERYINTENSE LIGHTWITH FREEELECTRONS. J. H 359415 EBERLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . TORS.

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111

1v. V

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VI VII

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C O N T E N T S OF V O L U M E V III (1970)

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1-50 SYNTHETIC-APERTURE OPTICS.J . W . GOODMAN. . . . . . . . . . . 51-131 THEOPTICAL PERFORMANCE OF THE HUMANEYE.G . A FRY . . . . . 133-200 LIGHTBEATING SPECTROSCOPY. H Z CUMMINS AND H L. SWINNEY. MULTILAYER ANTIREFLECTION COATINGS. A . MUSSETAND A . THELEN . . 20 1-237 STATISTICAL PROPERTIES OF LASER LIGHT.H . RISKEN. . . . . . . . . 239-294 COHERENCE THEORY OF SOURCE-SIZE COMPENSATION IN INTERFERENCE MICROSCOPY. T. YAMAMOTO. . . . . . . . . . . . . . . . . . . 295-341 VISIONIN COMMUNICATION. L LEVI . . . . . . . . . . . . . . . . 343-372 VIII THEORY OF PIiOTOELECTRON COUNTING. c. L MEHTA . . . . . . . . 373-440 1

. 1v. V. V1. VII . I1 111.

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I

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C O N T E N T S O F V O L U M E I X (1971) I

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11

GAS LASERS AND THEIR APPLICATION TO PRECISE LENGTH MEASUREMENTS. A.L.BLooM . . . . . . . . . . . . . . . . . . . . . . . . . . 1-30 31-71 PICOSECOND LASERPULSES. A . J . DEMARIA. . . . . . . . . . . . .

OPTICALPROPAGATION THROUGH THE TURBULENT ATMOSPHERE, J. W. ... .. . . .. . . .. STROHBEHN . IV. SYNTHESIS OF OPTICALBIREFRINGENT NETWORKS, E. 0.AMMANN . V. MODELOCKING IN GASLASERS, L.ALLENAND D. G . C. JONES . VI. C R Y ~ AOPTICS L WITH SPATIAL DISPERSION, v. M. AGRANOVICH AND V.L.GINZBURG . . . . . .... VII. APPLICATIONS OF OPTICAL METHODS IN THE DIFFRACHON THEORY OF ELASTIC WAVES,K. GNIADEK AND J. PETYKIEWICZ. . . . . VIII. EVALUATION, DESIGNAND EXTRAPOLATION METHODSFOR OPTICAL ON U S E OF THE PROLATEFUNCTIONS, B. R. FRIEDEN .. SIGNALS, BASED 111.

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73-1 22 123-177 179-234 235-280 281-310 3 11407

C O N T E N T S O F V O L U M E X (1972)

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45- 87

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89-135 137-164

BANDWIDTH COMPRESSION OF OPTICAL IMAGES,T. S. HUANG. . THEU S E OF IMAGE TUBES AS SHUTTERS, R. w. SMITH . . . . . . . . TOOLS OF THEORETICAL QUANTUM OPTICS,M. 0. SCULLY AND K. G. ., , . . W H I T N E Y . .. . , . . . . . . . . . . . . . . . CORRECTORS FOR ASTRONOMICAL TELESCOPES, c. G . WYNNE. IV. FIELD V. OPTICALABSORPTION STRENGTH OF DEFECTS IN INSULATORS, D. Y. SMITHand D. L. DEXTER . . . . ... . .. . . LIGHTMODULATION AND DEFLECTION, E. K. SITTIG . . . VI. ELASTOOPTIC VII. QUANTUM DETECTION THEORY, C. W. HELSTROM . . . . ... . . I.

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1- 44

165-228 229-288 289-369

C O N T E N T S O F V O L U M E X I (1973) MASTER EQUATION METHODS I N QUANTUM OPTICS,G. S. AGARWAL. . 1- 76 RECENTDEVELOPMENTS IN FARINFRARED SPECTROSCOPIC TECHNIQUES, H. YOSHINAGA . . . . . . . . . . . . . . . . . . . . . . . . . 77-122 111. INTERACTION OF LIGHTA N D ACOUSTIC SURFACE WAVES, E.G. LEAN . . 123-166 IV. EVANESCENT WAVES I N OPTICAL IMAGING, 0.BRYNGDAHL . . . . . . 167-221 V. PRODUCTION OF ELECTRON PROBESUSINGA FIELDEMISSION SOURCE, A. V. CREWE. . . . . . . . . . . . . . . . . . . . . , . . . 223-246 VI . HAMILTONIAN OF BEAM MODEPROPAGATION, J. A. ARNAUD. . 247-304 THEORY VII. GRADIENT INDEX LENSES, E. W. MARCHAND. . . . . . . . . . . . 305-337

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PREFACE The term “optics” has always been interpreted in this series in a broad sense, and the range of topics covered by the articles in this volume reflect this fact once again. Two of the articles, those on self-focussing and selfinduced transparency, deal with subjects that owe their origin largely to the laser. Both concern the behavior of the polarization of a medium through which light passes and, particularly, the way in which it depends upon properties characteristic of laser light. The growing influence of laser theory on other aspects of physics is exemplified by the article on the phase transition concept and coherence in atomic emission. Considerable advances have also occurred in experimental techniques in optics and in spectroscopy. Indicative of such advances are the articles on modulation techniques in spectrometry and on beam-foil spectroscopy. In the same spirit, another article in this volume describes a relatively new and beautiful technique, employing monomolecular dye layers, for studying a variety of phenomena involving the generation and propagation of light waves. It is, as ever, a pleasure to welcome to the pages of this series a new group of authors, each active and well known in his own field.

EMILWOLF Department of Physics and Astronomy, University of Rochester, N.Y., 14627 July 1974

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CONTENTS

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I SELF.FOCUS1NG. SELF.TRAPPING. AND SELF-PHASE MODULATION O F LASER BEAMS by 0. SVELTO (Milano. Italy) ............................ 2. PHYSICAL MECHANISMS OF NONLINEAR POLARIZATION . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Molecular-orientation Kerr effect . . . . . . . . . . . . . . . . . .

I.

~NTRODUCTION

2.3 Nonlinear electronic distortion . . . . . . . . . . . . 2.4 Nonlinear polarization due to molecular interactions . . 2.5 Nonlinear polarization due to electrostriction and heating

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. . . . . . . . . . . . . . 3 . WAVEPROPAGATION .......................... 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stationary self-focusing (theory) . . . . . . . . . . . . . . . . . . . 3.3 Stationary self-focusing (experimental results) . . . . . . . . . . . . . 3.4 Non-stationary case: Self-phase modulation and self-steepening . . . . . 3.5 Non-stationary case: Self-focusing of nanosecond pulses (theory of moving foci) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Non-stationary self-focusing: picosecond excitation . . . . . . . . . . 3.7 The “filament” diameter . . . . . . . . . . . . . . . . . . . . . .

.. 4 . CONCLUSIONS

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

3 12 12 14 20 21 23 25 25 27 32 36 41

44 47 48 50

I1. SELF-INDUCED TRANSPARENCY by R . E . SLUSHER (Murray Hill. N.J.)

1 . INTRODUCTION ..

.......................... 2 . COHERENT OPTICAL PULSEI N A TWO-LEVEL ABSORBER ........... 2.1 Coherent pulse and absorber . . . . . . . . . . . . . . . . . . . . 2.2 2.3 2.4 2.5 2.6 2.7

Schroedinger’s equation and the polarization vector Coupling to Maxwell’s equation . . . . . . . . lnhomogeneous absorber and the area theorem . Sharp-line absorber and the pulse shape . . . . . Pulse delays . . . . . . . . . . . . . . . . . Incoherent relaxation and the critical length . . . .

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XI

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55 58 58

60 63 64 68 70 72

CONTENTS

XI1

3. EXPERIMENTS . . . . . . . . . . . . . . . . 3.1 Ideal conditions . . . . . . . . . . . . . 3.2 Laser-absorber systems . . . . . . . . . . . . . . . 3.3 H g laser-Rb absorber system

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . VARIATIONS OF SIT . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Off-resonance effects and self-focusing . . . . . . . . . . . . . . . . 4.3 Degeneracy effects . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Chirping and On pulses . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 76 78 80 86 86 88 91 95 99

I11. MODULATION TECHNIQUES IN SPECTROMETRY by M . HARWIT (Ithaca. N.Y.; Concord. Mass.) and J . A . DECKERJ r . (Concord. Mass.) 1 . INTRODUCTION

. . . . . . . . . . . . . . . . . . . . . . . . . . . . A SPECTRUM? . . . . . . . . . . . . . . . . . . . . . . 2 . WHYMODULATE 2.1 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.3

103 105

107

The high throughput advantage . . . . . . . . . . . . . . . . . . . Dispersing spectromodulators . . . . . . . . . . . . . . . . . . . .

109

110

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3 SPECTROMODULATORS A N D INTERFEROMETERS: ANALOGOUS INSTRUMENTS . . . . 112 3.1 Simple spectromodulator . . . . . . . . . . . . . . . . . . . . . . 112 3.2 Golay’s static multislit spectrometer . . . . . . . . . . . . . . . . . 113 3.3 Golay’s dynamic multislit spectrometer . . . . . . . . . . . . . . . . 115 3.4 Mertz’s Mock interferometer and Girard’s grill spectrometer . . . . . . 117 3.5 Hadamard-transform spectrometers . . . . . . . . . . . . . . . . . 118 MODULATION TECHNIQUES BE USED - A N D 4 . WHENSHOULD SPECTRAL THEY NOT?. . . . . . . . . . . . . . . . . . . . . . . . 4.1 Sources of noise . . . . . . . . . . . . . . . . . . 4.1.1 Photon noise . . . . . . . . . . . . . . . . . . 4.1.2 Background noise . . . . . . . . . . . . . . . . 4.1.3 Modulation or scintillation noise . . . . . . . . . 4.1.4 Detector noise . . . . . . . . . . . . . . . . . 4.2 Optimum operation . . . . . . . . . . . . . . . . . 4.3 Cooled and uncooled instruments . . . . . . . . . . .

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5 THE( ~ R OF Y THE HADAMARD TRANSFORM SPECTROMETERS . . . 5.1 Codes for Hadamard transform spectrometry . . . . .

W H E N SHOULD

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5.2 Estimation of spectral shape for singly encoded instruments . . . . . . . 5.3 Choice of masks . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Optical arrangement . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Doubly encoded (or multislit multiplex) Hadamard transform spectrometers 5.6 Theoretical analysis of a doubly multiplexed spectrometer . . . . . . . 5.6.1 The basic equation . . . . . . . . . . . . . . . . . . . . . . 5.6.2 The detector noise . . . . . . . . . . . . . . . . . . . . . . 5.6.3 The estimation problem . . . . . . . . . . . . . . . . . . . . 5.6.4 Using N 2 measurements . . . . . . . . . . . . . . . . . . . 5.6.5 2 N - 1 measurements are enough . . . . . . . . . . . . . . . . 5.7 Comparisons with other grating spectrometers. . . . . . . . . . . . . 5.7.1 Remarks on the wide aperture advantage . . . . . . . . . . . . 6. SPECTROMETRIC IMAGING

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121 122 122 122 123 123 123 126 126 126 127 129 130

I30 131 132 132 133 133 134 134 135 137

XI11

CONTENTS

7. A SURVEY OF BINARY-ENCODED MULTIPLEX SPECTROMETERS . . . . . . . . . 7.1 Decker and Harwit 19-slot H T S spectrometer . . . . . . . . . . . . . 7.2 D e Graauw and Veltman 255-slot pseudo-random binary multiplex spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Decker 255- and 2047-slot spectrometers . . . . . . . . . . . . . . . 7.4 Phillips and Harwit 19 x 19-slot doubly-multiplexing spectrometer . . . . 7.5 Hansen and Strong high resolution spectrometer . . . . . . . . . . . . 7.6 Commercial binary encoded multiplex spectrometers . . . . . . . . . . 7.6.1 HTS-19-1 airborne astronomical spectrometer . . . . . . . . . . 7.6.2 HTS-255-15 analytical infrared spectrometer . . . . . . . . . . . 7.7 Harwit’s imaging spectrometer . . . . . . . . . . . . . . . . . . . 7.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140 141 144

145 146 146 147 150 151

8. SOME PRACTICAL CONSIDERATIONS IN COMPARING INTERFEROMETERS A N D SPECTRO-

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151

10. CONCLUSIONS . .

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154

. . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS

154

MODULATORS

152

SPECAPPENDIX A . SOME PROPERTIES OF CYCLIC CODES FOR HADAMARD-TRANSFORM TROMETRY

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APPENDIX B. CYCLICCODESFOR HADAMARD-TRANSFORM SPECTROMETRY . .

. . APPENDIX C . CODEOPTIMIZATION REFERENCES .

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160

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161

IV . INTERACTION OF L I G H T WITH MONOMOLECULAR DYE LAYERS

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by K H . DREXHAGE (Marburg/Lahn. Germany)

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1 ~NTRODUCTION

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OF MONOLAYER SYSTEMS. . . . 2. PREPARATION 2.1 Deposition of cadmium-arachidate layers . 2.2 Preparation of monomolecular dye layers .

165

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3 . OPTICALPROPERTIES OF CADMIUM-ARACHIDATE LAYERS. . . . . . . . . . . 3. I Determination of the refractive indices . . . . . . . . . . . . . . . . 3.2 Determination of layer thickness . . . . . . . . . . . . . . . . . .

174 174 176

SCATTERING AT A D Y EMONOLAYER . . . . . . . . . . . . . . . 4 . COHERENT 4.1 The phase of the scattered wave . . . . . . . . . . . . . . . . . . . 4.2 The reflectivity of dye layer systems . . . . . . . . . . . . . . . . . 4.3 The absorption of dye layer systems . . . . . . . . . . . . . . . . . 4.4 The orientation of the dye chromophores . . . . . . . . . . . . . .

180 181

184 187 188

5 . STANDING LIGHT WAVES

190

. . . . . . . . . . . . . . . . . . . . . . . . 6. EVANESCENT LIGHTWAVES . . . . . . . . . . . . . . . . . . . . . . . 6.1 The decay of evanescent waves . . . . . . . . . . . . . . . . . . . 6.2 Evanescent waves in a birefringent medium . . 6.3 Emission of evanescent waves . . . . . . . . .

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194 194 197 198

XIV

CONTENTS

7. THENEARFIELD OF A RADIATING MOLECULE. . . 7.1 Kuhn’s concept of energy transfer . . . . . . 7.2 Energy transfer between dye monolayers . . . 7.3 The near field of an electric-quadrupole source

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199 i99 202 204

.

8 THERADIATION PATTERN OF A FLUORESCING MOLECULE I N FRONT OF A MIRROR 206 8.1 The radiation pattern in case of a single mirror and an isotropic medium . 206 8.2 Observation of radiation patterns with the monolayer technique . . . . . 211 8.3 Fluorescing molecule between two mirrors . . . . . . . . . . . . . . 213 8.4 The influence of the layer birefringence . . . . . . . . . . . . . . . . 215 DECAYTIMEOF A MOLECULE I N FRONT OF A MIRROR . . . . . 9. FLUORESCENCE 9.1 Electric-dipole oscillator in front of a single mirror . . . . . . . . . . 9.2 Dielectric interface as mirror . . . . . . . . . . . . . . . . . . . . 9.3 Electric-dipole source between two mirrors . . . . . . . . . . . . . . 9.4 Competing non-radiative processes . . . . . . . . . . . . . . . . . .

216 217 220 223 226

APPENDIX . SOMEOPTICALPROPERTIES OF UNIAXIAL CRYSTALS . .

226

. . . . . . . ACKNOWLEDGMENTS ........................... REFERENCES ...............................

229 229

V . T H E PHASE TRANSITION CONCEPT A N D COHERENCE IN ATOMIC EMISSION by

R . GRAHAM (Stuttgart. Germany)

1 . INTRODUCTION.

........................... 2. MEANFIELD THEORY OF LASERMEDIA . . . . . . . . . . . . . . . . . . 2.1 Introductory remarks . . . . . . . . . . 2.2 Thcmodel . . . . . . . . . . . . . . . 2.3 Descriptive analysis . . . . . . . . . . . 2.4 Mean field theory without fluctuations . . 2.5 Mean field theory with fluctuations . . .

3 . BEYOND THE MEANFIELD THEORY . .

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

. . . .

. . . .

. . . .

. . . .

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

3.1 Introductory remarks . . . . . . . . . . . . . . 3.2 Equations for the slowly varying fields . . . . . . 3.3 Mode continua in three-dimensional isotropic media 3.4 Mode continua in one-dimensional media . . . . .

4. DISCRETE MODESPECTRA .. 4.1 Introductory remarks . 4.2 Single mode lasers . . 4.3 Two mode lasers . . .

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

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

.......... .......... .......... 5 . DISCRETE MODESI N NONLINEAR OPTICS. . . . .

. . . .

. . . .

.........

. . . . . . . . . .

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

5.1 Introductory remarks . . . . . . . . . . . . . . . 5.2 Equations of motion . . . . . . . . . . . . . . . 5.3 Mean field approximation . . . . . . . . . . . . . 5.4 Probability distributions for the slowly varying fields 5.5 Results in the limit N + 03 . . . . . . . . . . . . .

. . . . . . . .

APPENDIX 1

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

APPENDIXI1

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

235 242 242 242 245 241 250 254 254 255 251 261 263 263 264 269 272 272 273 275 276 218 280 282

CONTENTS

.

. 6 NOTEADDEDIN PROOF

........................ ACKNOWLEDGMENTS ........................... REFERENCES ...............................

xv 282 283 283

.

VI BEAM-FOIL SPECTROSCOPY by S . BASHKIN (Tucson. Arizona) 1. INTRODUCTION

.................... 2. SPECTRAL LINESHAPES. . . . . . . . . . . . . . . . . . . . . . . . . 3. WAVELENGTH STUDIES AND ENERGYLEVELSCHEMES. . . . . . . . . . . . 4 . DOUBLY-EXCITED LEVELS. . . . . . . . . . . . . . . . . . . . . . . . 5. METASTABLE ONE-ELECTRON LEVELS . . . . . . . . . . . . . . . . . . . 6. MEAN-LIFE MEASUREMENTS ....................... 7. APPLICATIONS OF LIFETIME DATA. . . . . . . . . . . . . . . . . . . . . 8. COHERENCE AND ALIGNMENT ....................... 8.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Quantum beats . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Alignment and an external non-oscillatory magnetic field . . . . . . . . 8.5 Coherence with a non-oscillatory electric field . . . . . . . . . . . . . 8.6 Oscillating external fields . . . . . . . . . . . . . . . . . . . . . . 9 . CHARGE-STATE IDENTIFICATION . . . . . . . . . . . . . . . . . . . . . 10. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ACKNOWLEDGMENTS ........................... REFERENCES . . . .. . . . . . . . . . . . . . . . ........... ............................ SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . CUMULATIVE INDEX . VOLUMES I-XI1 . . . . . . . . . . . . . . . . AUTHOR INDEX

289 290 292 304 310 312 323 325 325 326 326 321 332 334 336 339 339 340 345 353 3 60

This Page Intentionally Left Blank

E. WOLF, PROGRESS IN OPTICS XI1 Q NORTH-HOLLAND 1974

I SELF-FOCUSING, SELF-TRAPPING, AND SELF-PHASE MODULATION OF LASER BEAMS BY

0.SVELTO Laborarorio d i Fisica del Plasma ed Elettronica Quantistica, Reparto di Elettronica Quantistica, lstituto di Fisica del Politecnico, Milano, Italy

CONTENTS

PAGE

$ 1 . INTRODUCTION

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

3

$ 2. PHYSICAL MECHANLSMS OF NONLINEAR

POLARIZATION

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

$ 3 . WAVE PROPAGATION

25

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

48

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

50

$ 4. CONCLUSIONS

REFERENCES

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

12

0 1. Introduction Among the nonlinear optical effects that have been investigated in recent years, an important place is occupied by the so-called "self-action" effects of powerful light beams. To understand what we mean by "self-action", let us express the ith component of the nonlinear polarization vector P("')(t) of the medium in the following way:

where E j ( t ) is the component of the electric field alongj-axis. If all fields have the same frequency o,the self-action effects are the result of those terms in (1.1) which are oscillating at frequency w . It then follows from (1.1) that we will only be concerned with the even-rank tensors X i j k l , X i j k l m n , etc. We will also assume that these tensors are real, so that effects such as two-photon, four-photon absorption etc. will not be considered. Unlike other nonlinear optical effects, such as harmonic generation and parametric processes, where waves interact at several very different frequencies, during the process of self-action the wave remains quasimonochromatic. The selfaction effect then results in a change in the form of the amplitude (selffocusing, self-steepening), in a change in the phase (self-phase modulation) and in the state of polarization of the wave. In this report, we will review these self-action effects with particular attention payed to the processes of self-focusing, self-trapping, and self-phase modulation. Since the wave is supposed to be quasi-monochromatic, it is convenient to express its electric field E(r, t ) in the medium as E(r, t ) = +[A(r, t ) exp i(w, t - k r ) + c.c.], where the complex amplitude A ( r , t ) is a slowly varying function [compared with exp i (mot-k r ) ] of both space, r, and time, t . To describe in a simple way some of the self-action effects, let us consider a linearly polarized wave in an isotropic medium. In this case, for reasons of symmetry, the induced nonlinear polarization vector P("') will be along the field direction. We will further assume that, among the even-rank tensors of (1.1), only X i j k l is 3

4

LASER BEAMS

[I,

P1

significant (a medium of this type is called a cubic medium). Under these assumptions the effect of the nonlinear polarization can be expressed in terms of a dielectric constant E (c.g.s. units will be used in this work) E = Eo+E2(E2)

= &,+(&2I~41~/2).

(1.3)

We assume that E is a real quantity so that we neglect the effects of both lineal and nonlinear absorption. We can also describe the medium in terms of an effective index of refraction n

n

=

no+&,

(1.4)

where no is the linear part of the refractive index, and where the nonlinear part 6n, for a cubic and isotropic medium, can be written as*

6n Since E

=

=

n2(~2= > nzJ~J2/2.

(1.5)

n2, from (1.3) and (1.4) it follows: n, = n , ~ , / 2 ~ = , ~,/2n,.

(1.6)

One should also notice that, according to eq. (1.4), the third-order nonlinear polarization can be expressed as n0 6 n E , . Pi3’ = 271

After these considerations we can proceed to describe in a simple way the self-focusing and self-trapping phenomena. If a beam with a bell-shaped transverse intensity profile is entering a medium of the type described by (1.4) and (1.5), and if n, > 0, the central and most intense part of the beam will produce an effective refractive index n, which is larger than that in the wings of the beam. The medium will then act as a positive lens which focuses the beam on its axis. This is called the self-focusing phenomenon. It will obviously be counteracted by the beam diffraction. It is therefore possible that, for sufficiently intense fields, the two effects compensate each other and the wave propagates without spreading. This is the self-trapping phenomenon. An elementary analysis of the self-trapping condition can be made by considering the behavior of the light rays at the boundary of a cylindrical beam that has a constant intensity distribution (Fig. 1). The refractive index outside the beam is no and, inside the beam, it is no + 6n, where 6n is a constant

* It is worthwhile to emphasize that both E and 6n are defined in this report, in terms of the time average < E z ) of the square of the field. Some authors define 6n as 6n = nzlAIZ. Care must therefore be taken in comparing the results of different papers since the definition of n2 in the two cases obviously differs by a factor two.

1. §

11

INTRODUCTION

5

Fig. 1. Illustration of a simplified derivation of the expression for the critical power of self-trapping. The shaded area is occupied by the beam.

along the beam. A ray incident at the boundary of the beam from the inside, moves from a medium which is optically denser to a medium of lower optical density. Consequently, at a sufficiently large angle, (b, total internal reflection is possible. The critical angle corresponds to a ray whose inclination 0, to the beam axis is such a that

n

2= cos eo no + 6 n If 6n P,,, the focusing effect will overcome diffraction, and the beam will come to a focus (Fig. 2a). If we neglect diffraction effects, the focusing distance z, can be roughly calculated by establishing the condition that the two optical path lengths BF and AF of Fig. 2b are equal. Here point B is chosen as the half-intensity point of the beam. Accordingly we can write: (no

+6nA)zf= (no + Sn,)[r: + zf2]+,

(1.14)

where 6nA and 6n, are the nonlinear refractive indices at A and B. When (6nA,6n,) wo - W 0 ) A j 44 .

(2.3)

%

Note that, although spatial dispersion can usually be neglected, it is not generally possible to neglect the temporal dispersion in (2.1). This is because (usually) the complex amplitudes A j ( r , t ) are not slowly varying in time compared to the medium reaction. We will consider this point further on. The tensor x i j k l has non-zero components for all groups of point symmetry and isotropic media. Its non-zero components, as determined by the symmetry properties of the medium, can be found for all classes of crystals (BRISS[1962]). In particular, for an isotropic medium (MAKER and TERHUNE [1965], CLOSE,GIULIANO, HELLWARTH, HESS,MCCLUNGand WAGNER [19661): x i j k l = 6cl 1 2 2 6 i j 6 k l + 3 c 1 2 2 t 6 i 1 6 j k , (2.4) where the two coefficients c1122= c1122(-coo, wo, wo, -coo), and c122t= ct221 (-ao, coo, wo, -ao), are characteristic of the medium being tested. For a linearly polarized wave, from (2.4), (2.3) and (1.7) we get

6n

=

(Wno) [6c,,22+c122tII~l2,

(2.5)

while for an elliptically polarized wave, from (2.4), (2.3) and (1.22), 2n 6n+ = - [ 6 ~ 1 1 2 2 1 ~ + l 2 + ( 6 ~ 1 1 2 2 + 6 ~ 1 2 2 t ) I ~ - I 2 1 no

(2.6a)

From (1.23) we then see that the rotation of the polarization ellipse is

For a circularly polarized wave (e.g. right handed) from (2.6a) [ A - = 01

an+

=

(271hO) [ 6 c 1 t 2 2 l l A + J 2 .

(2.8)

The two coefficients cllZ2and c1221have different physical meanings. In

14

[I,

LASER BEAMS

s2

particular c1 2 2 describes the “isotropic” part of the nonlinear polarization. This term alone in fact, for a linearly polarized beam, would give a nonlinear refractive index change which is the same in any direction (i.e. no birefringence). The coefficient c~~~~ describes the “anisotropic” part of the nonlinear polarization. It is, in particular, responsible for the rotation of the polarization ellipse [see (2.7)]. Note also that the comparison of (2.5) and (2.8) shows that the nonlinear refractive index produced by a linearly polarized wave is generally different from that due to a circularly polarized wave of the same intensity (unless the anisotropic coefficient clZz1 is zero). Note finally that, in terms of c1122 and c1221,the coefficient n2 defined in (1.5) is given as n2 = ( 4 7 h ) [ 6 ~ ~ ~ 2 2 + 3 ~ 1 2 2 ~ 1 . (2.9) As a conclusion to this section, we can say that, in an isotropic medium, the self-action effects are fully described by two coefficients (cllZ2and clZz1) which are characteristic of the medium. Their calculation together with the calculation of the associated relaxation time is the purpose of next sections. 2.2. MOLECULAR-ORIENTATION KERR EFFECT

One of the causes of nonlinear polarization, which only applies to anisotropic molecules, is due to the tendency of the alignment of a molecule in the optical field of an e.m. wave. According to BREWER, LIFSITZ,GARMIRE, CHIAOand TOWNES[1968] calculations, let us assume, for the sake of simplicity, that we have a symmetric top molecule (e.g. CS2) with principal polarizabilities ctll and ctI (rx,, > N J along and perpendicular to the symmetry axis. The general case, where all three principal polarizabilities are different, is similar in its general characteristics (see CLOSE,GIULIANO, HELLWARTH, HESS,MCLUNGand WAGNER[1966]). The calculation of the nonlinear refractive index induced by the E-field of an e.m. wave can be obtained from Lorenz-Lorentz’s relation [eq. (1.26)] by calculating the average molecular polarizability (ct). Let us assume that the symmetry axis of the molecule makes an angle 8 with the E-field. Projection of the induced moments of all and rxI into E, produces the average polarizability in the E direction (2.10)

The average is over all angles 8 in the statistical distribution. Hence cos2 e exp ( U cos2 e) sin e de

exp ( a cos2 e) sin e de, (2.11)

1,

§ 21

M E C H A N I S M S O F N O N L I N E A R POLARIZATION

15

where a = (a,,-u,)(E:)/2kT and isothermal behavior is assumed. The quantity E, is the local field and E; is understood to be averaged over an optical period. From (1.26) and (2.10) it then follows:

~2 - 1_ _n;-1 _ - _$np(Ax)[(cos’ @-+I, n2+2

ni+2

(2.12)

where Au = ull-01, and where, in zero field, the index is no and (cos’8) = 1 3. Putting in (2.12) n = no+& and neglecting terms proportional to (6n)2, the induced nonlinear change 6n can expressed in the form

6n

=

@ n s a l [ ( ~e) ~~ -31, Z

(2.13)

where 6nsat =

{(ni + ~ ) ’ / ~ ~ O > ( % P ) ( A ~ > -

(2.14)

The meaning of tin,,, is immediately obtained from (2.13) by letting E, -+ 03. Then (cos28) = 1 (complete alignment) and 6n = 6nsal. Therefore, 6nsa, is the value of nonlinear refractive index change due to complete alignment of the molecules.

4

12

8

16

a

Fig. 3. Molecular orientation Kerr effect. Normalized behavior of the nonlinear refractive index change. 6n vs. field parameter u. The slope of the curve at the origin is indicated as a dashed line.

The quantity (cos20) is obtained from (2.11) and can be expressed in terms of a tabulated function (Dawson’s integral). Accordingly, from (2.13) one can calculate 6n. The variation of 6n with the field parameter a is shown in Fig. 3. The behavior immediately before saturation is best explained by expanding the two exponentials in (2.11) and integrating term by term. This gives 2 -&a3.. ., (2.15) (cos’ e) = ++&a+&a

16

LASER BEAMS

[I,

9: 2

and from (2.13) 6n

=

36nsat[ Q 4 5a + - K9 -4a5z - - L 1 4 1 7 s a 3 +

*-*I*

(2.16)

Since, for a Lorentz cavity ( E : ) = [+(n;+2)I2 ( E ' ) and using (1.2), eq. (2.16) may be alternatively written as 6n = 3[nzlA12+n41A14-n61A16],

(2.18)

where in particular (2.19)

To give a numerical example, let us consider the classical case of liquid CS2. In this case tinsatE 0.58 and n2 w 1.3 x lo-" e.s.u. From Fig. 3 we also see that the saturation effect is quite evident and becomes pronounced for a > 2. Again for liquid CS2, when a = 2 the electric field is -4 x lo7 V/cm. So far, only the nonlinear refractive index induced in the direction of the applied field has been calculated*. In particular, a specific expression for n2[eq. (2.19)] has been obtained. This is not enough, however, to describe the nonlinear properties of the medium to third order. We have seen in fact in the previous section that an isotropic medium is characterized by two constants. Following the same line as above, we could now calculate the nonlinear refractive index change in a direction orthogonal to the applied field (CLOSE, GIULIANO, HELLWARTH, HESS,MCLUNGand WAGNER [1966]). From this calculation it follows that, for the orientational Kerr effect, one has C 1 2 2 1 / C l 1 2 2 = 6. (2.20) From (2.5) and (2.8) it then follows that: 6n = 46n,.

(2.21)

The nonlinear refractive index produced by a linearly polarized wave is four times greater than that of a circularly polarized wave of the same intensity. Since now specific expressions for n2 [eq. (2.19)] and the ratio of c1221 to c1122 [eq. (2.20)] have been derived, from (2.9) we see that both coefficients c1122 and clZz1are obtainable. Note finally that it can also be shown (see CLOSE,GIULIANO, HELLWARTH, HESS,MCLUNGand WAGNER [1966]) that 6n,

=

-t6nl,,

(2.2 1a)

* Actually the calculation has been carried out to orders higher than IAI*. This corresponds in ( 1 . 1 ) to consider even-tensors of orders higher than fourth-rank.

1 9 5

21

MECHANISMS OF N O N L I N E A R P O L A R I Z A T I O N

17

where 6n, and 6nll are respectively the nonlinear refractive index changes perpendicular and parallel to an applied field. So far an instantaneous response of the medium has been assumed, i.e. we have assumed that the complex field amplitudes A j ( r , t ) are slowly varying in time compared to x(tl, t z , t , , rl , r z , r , ) appearing in (2.11. In order to take into account the finite response time of the medium, it is usual to modify eq. (1.5) to z6ri+6n = *nzlAIZ, (2.22) where T is a suitable relaxation time which is a characteristic of the medium. Let us see how eq. (2.22) can be justified for the case of the molecularorientation Kerr effect. According to (2.13), to study the dynamic behavior of 6n(t), one needs to study the dynamic behavior of (cos28(t)) -+. To do so, we need to know the time behavior off(f2, r, t ) , wherefdf2 is the probability that a molecule at position r and at time t will ba found with its axis lying within an incremental range of solid angle df2 about the direction f2. This is given by the equation (DEBYE [1945] and HERMAN [1967])

V,'f

+ v,

*

{f V,[ U(S2, r, t ) / k T ] } = szaj-pt,

(2.23)

where the angular gradient operator is

a v,=a-+4-a8

a sin ea4

(2.24)

4 being spherical polar and azimuthal angles with respect to the field direction. The quantity z is a suitable relaxation time to be discussed later on. Finally U(f2, r, t ) is the potential energy function for a molecule located at position r in the external field. It is given by

8 and

u = -+[a+(alI-sLJ(cosz

o-+)](E:),

(2.25)

where Ix = +(aII +2a,).

(2.26)

At steady state (af/at = 0), the solution of (2.23) is f cc exp( - U/kT)and this solution has been used in (2.1 1). If now, at t = 0, the e.m. field is suddenly removed, then, for t > 0, U = 0 and (2.23) reduces to the diffusion equation v,'f = 6zaflat. (2.27) Sincef depends only upon 8, (2.27) has the general solution W

f ( 8 , t ) = C ~ , P , ( C O S 8)exp [-l(l+l)t/6z],

(2.28)

I=O

where I is an integer and PI are the Legendre polynomials. The average

18

[I,

LASER BEAMS

82

value of cos2e(t)-f is then

Thelatterexpressionin (2.29)followsfrom the fact thatP,(cos 0) cc cos28-i. Substituting (2.28) into (2.29) we see that, owing to the orthogonality of Legendre polynomials, only the term in (2.28) which is proportional to P,(cos 0) will determine the time behavior of (cos20(t) -f). Since the time behavior of this term is proportional to exp[-1(1 + l)t/6z] = exp[-t/z], it then follows that both (cos%(t)-f) and Sn(t) will have the same time behavior, i.e. sn(t) = Gn(O)exp[-t/z]. Since this same behavior would be obtained by (2.22), this last equation may be in this way justified. We now want to discuss the relationship between the relaxation time z here introduced, and the relaxation time zD of the dielectric constant of a polar molecule (Debye relaxation time). This dielectric constant is proportional to the average component ( p E ) of the permanent moment p o of the molecule along the applied field, which is PE

= pO(cos

e> = ~ O ( p l ( c O s

e)>,

(2.30)

where

From (2.28) we now see that the time behavior of (cos8) is that of PI polynomial, which is proportional to exp - [r/3z], so that zD =

32.

(2.32)

If we liken the molecule to a sphere of radius a in a liquid of viscosity q, then the Debye relaxation time is (DEBYE [1945])

zD = 4nqa31kT.

(2.33)

From (2.32) and (2.33) one can calculate z. Equation (2.32) may be qualitatively understood as follows: The orientational relaxation time is in general defined by Gn(t = z)/Sn(O) = l/e, which, because of the P,(cosO) spatial dependence, is the average time taken to make a rotation of -39" [i.e. P,(cos 39") x l/e]. In a similar fashion, the dielectric relaxation, having P,(cosO) dependence, has a relaxation time equal to the average time taken for a rotation of 68" [i.e ,PI (cos 68") w l/e]. Since the dielectric relaxation time corresponds to a greater angular rotation, it is clear that zD > z. Because of the diffusional character of the reorientation process, the time to reorientate 68", is a factor 3 greater than the time to reorientate 39". The discussion

1,521

M E C H A N I S M S OF N O N L I N E A R P O L A R I Z A T I O N

19

above is applicable only for a random walk process in which each individual step represents an angle which is small compared to 39" (so that (2.27) is applicable). IVANOV [1964] has considered a jump diffusion model in which one assumes that the molecule spends a time z, in a given orientation and then jumps in a time less than z, into a new orientation. He shows that, for large angular jumps, eqs. (2.32) and (2.33) should be substituted by t = t D = 7,.

(2.34)

It seems that in some liquids the diffusion model is applicable, whereas for other liquids the jump model is a more realistic one (PINNOW,CANDAU and LITOVITZ [1968]). We finally wish to discuss the connections between the molecular-orientation Kerr effect, and the depolarized Rayleigh-wing scattering. It is well known (see FABELINSKI [1968]) that, for anisotropic molecules, the spectrum of the depolarized scattered light is, roughly speaking, made of two parts (Fig. 4). The first part (near wing) comes from fluctuations of molecular orientation and gives a Lorentzian spectrum i.e. g(o)cc 1/ [1+ ( A W ~ ) ~ ] ,

Fig. 4. Typical spectrum of the depolarized Rayleigh-wing scattered intensity for an asymmetric non-polar molecule.

where t is the same relaxation that we have introduced in (2.22) (see PINCANDAU and LITOVITZ[1968]). This part gives the shaded portion of the spectrum in Fig. 4. The second part is sometimes called the far-wing of the Rayleigh line (see Fig. 4) and it is generally understood as being caused by a kind of cooperative scattering, i.e. a scattering arising from molecular interactions. It will be discussed further on in this work. It can be further shown (see for instance KIELICH[1960]) that the total integrated intensity due to fluctuation of molecular orientation is proportional to NOW,

20

LASER BEAMS

[I,

02

V ~ ( d a )where ~ , V is the scattering volume. This integrated intensity (area of shaded region of Fig. 4) is then proportional to n2 (see also HELLWARTH [1970]). The measurement of the depolarized spectrum is then able to give both the relaxation time z and n 2 . 2.3. NONLINEAR ELECTRONIC DISTORTION

A second cause of nonlinear polarization, which applies to any molecule, is of electronic origin, and it arises from a nonlinear distortion of the electron orbits around the nuclei, i.e. it arises from a molecular “hyperpolarizability”. Its relevance to self-focusing and self-trapping was pointed out by BREWER and LEE [1968]. This cause is associated with a very fast relaxation time, roughly of the order of the period of a Bohr orbit (27cah/e2 z sec). Since this phenomenon will respond to an optical cycle, it will also be responsible for third-harmonic generation from the same substance. The corresponding value of n, can then be estimated from third-harmonic generaand tion experiments. One then gets values of n2 ranging between e.s.u. If the frequency of the applied field is much less than the lowest electronic absorption frequency, the tensor xijkldue to electronic contribution is symmetric in all suffixes and equal to its static value (MAKERand TERHUNE [1965]). For an isotropic medium it then follows that: c1221/c1122

= 1,

(2.35)

to be compared with (2.20) which applies for molecular-orientation Kerr effect. From (2.5) and (2.8) it then follows that:

6n

=

1.56n+.

(2.36)

It can also be shown (see MAKERand TERHUNE [1965]) that, in this case,

6n, = +6nII,

(2.37)

where 6n, and 6nll are respectively the nonlinear refractive index changes perpendicular and parallel to an applied field. Although the saturation properties of the nonlinear refractive index due to electronic origin are not well known, the calculation done by BREWER and MCLEAN[1968] for CO molecule indicates that saturation should set in at field strengths comparable with that of ionization (- 109V/cm). The saturation, in this case, should therefore occur in a much stronger field than is the case of molecular-orientation Kerr effect (4x lo7 Vjcm in the cited example of CS,). It is finally worthwhile to point out that this cause of the change in value

1,

s 21

21

M E C H A N I S M S OF N O N L I N E A R P O L A R I Z A T I O N

of the nonlinear refractive index has no analogous counterpart in the depolarized Rayleigh-wing spectrum (see HELLWARTH [19701). The contribution to the refractive index change due to nonlinear electronic distortion does not seem to be important for liquids made of anisotropic molecules such as CS, (see HELLWARTH [1970]). On the other hand, it seems to be one of the predominant mechanisms in liquids made of symmetric molecules such as Ar (MCTAGUE, LIN,GUSTAFSON and CHIAO[1970], ALFANOand SHAPIRO11970 b]), and CC14 (HELLWARTH, OWYOUNG and GEORGE[1971]). It seems also to be the predominant mechanism in glass (ALFANOand SHAPIRO[1970 a] and OWYOUNG,HELLWARTH and GEORGE [1972]). 2.4. NONLINEAR POLARIZATION DUE TO MOLECULAR INTERACTIONS

A third contribution to nonlinear refractive index change comes from molecular interactions. Generally speaking we can say that this contribution comes from the fact that the effective field E’ on a given molecule is given by

E’ = E + F ,

(2.38)

where E is the external field and F is the field due to the surrounding molecules, i.e. due to molecular interaction. So far the field F has been taken into account through the Lorentz correction factor. A more careful study of the physical situation shows that the problem is more complicated, and that the proper consideration of F leads to new contributions to the nonlinear refractive index. Possibly, the simplest example of this type of mechanism is a low pressure gas made of spherical molecules (e.g. Ar gas). At sufficiently low pressure we can think of molecular interaction in terms of bimolecular collisions. When two molecules are interacting (colliding), the electronic distribution of the colliding atoms is distorted by interatomic interaction producing a polarizability in the colliding complex which differs fiom the sum of the polarizability of the separated atoms (MCTAGUEand BIRNBAUM[1971]). Let us consider a pair of interacting atoms placed at distance r, and call all(r) and a,(r) the parallel and perpendicular polarizabilities of this pair of atoms which are considered as one axially symmetric molecule. For this molecule we can repeat the calculation made in sec.2.2. The resulting n2 can be immediately obtained from (2.19) by substituting the number density p with the number density of interacting pairs of atoms (+pz V ) :the number of interacting pairs is in fact +(pV)’, where V is the volume of the sample.

22

LASER BEAMS

[I,

52

The quantity Aa2 is then substituted with (Aa2(r)), where (2.39) Here g(r) is the radial distribution function, which for a gas at low density is given by (2.40) g(r) = exp[- U ( r ) / W , where U(r) is the intermolecular potential. From (2.19) we then get (2.41) It is interesting to notice that n, is proportional to p2, i.e. to the square of the pressure of the gas. Here again the nonlinear refractive index change is closely related to the depolarized Rayleigh-wing spectrum. From the work of MCTAGUEand BIRNBAUM [1971] we can see that n, of (2.41) is again proportional to the total integrated depolarized scattered intensity. Since the scattered spectrum is now more similar to an exponential rather than to a Lorentzian curve, a relaxation equation of the type as in (2.22) cannot be accepted in this case. We can nevertheless associate a relaxation time z to this phenomenon roughly given by the inverse of the width of the Rayleigh line. This time then approximately results as: 7 = r,lU,hr (2.42) where ZJ, is the thermal velocity and ro is the range of the interaction, which, for long-range interactions, is approximately equal to the kinetic diameter. sec, independently Putting ro z 18, and U,, x lo5 cm/sec we get z x of pressure. We can finally ask the question of up to what pressure value the model of bimolecular collision applies. Obviously it applies up to when n, (and therefore the integrated intensity) still varies as p2. At high densities (i.e. at high pressures), the interaction of more than two particules in fact causes the integrated intensity (and n,) to very slower than p2 (THIBEAU, OKSENGORN and VODAR[1968]). From the work just mentioned we then see that the bimolecular collision model, for Ar gas, can be safely applied up to a pressure of 100 atm. The next example that we will consider is that of a gas made of nonsymmetrical molecules. The calculation, in this case, can be made by taking into account the reorientation and local spatial redistribution of molecules

-

1,621

MECHANISMS OF N O N L I N E A R P O L A R I Z A T I O N

23

(HELLWARTH [1966]). It can be shown that the corresponding expression for n2 [see eq. (41) of the work just mentioned] contains both (2.19) and (2.41) as particular cases. Since this expression also includes terms which are proportional to powers of p higher than p2, we can suppose that it is of more general validity. The expression is, however, not valid at a density value of the order of that of a liquid (HELLWARTH [1967]). For the case of a solid or a liquid, therefore, no one of the previous considerations can be safely applied. In this case one can use a new model: the libration (or rocking) model. It was suggested by STARUNOV [I9651 to account for the far-wing of the depolarized Rayleigh-wing line (Fig. 4), and its relevance in self-focusing and self-phase modulation was pointed out by POLLONI, SACCHI and SVELTO [1969]. It can be shown (CUBEDDU, POLLONI, SACCHI and SVELTO [1970]) that this model leads to a dynamic behavior of 6n of the type z;6ii+zl 6ri+6n = $n21A12, (2.43) where both the characteristic times z1 and z2 and the coefficient n2 can be obtained from the Rayleigh-wing spectrum. For the case of C S 2 , for instance, we can estimate (CUBEDDU, POLLONI, SACCHIand SVELTO[1970]) 72 w 71 w 0.2 psec and (n2)libr./(n2)Kerr x 0.2. 2.5. NONLINEAR POLARIZATION DUE TO ELECTROSTRICTION AND HEATING

As previously mentioned, both electrostriction and heating change the number density p. From Lorenz-Lorentz relation (1.27) we then get the corresponding refractive index change

6n =

(no' - l)(n:

+2) 6p -

6n0

Po

(2.44)

The contribution to 6 p due to electrostriction (SHEN[1966]) can be determined by solving a wave equation having on its right-hand side an additional term due to the electrostriction pressure (2.45) Here u is the velocity of the acoustic wave, u = (pS)-*, where fi is the compressibility, 2 r / u is the damping of the acoustic wave, and y = 2n0p0an/ap. In general, this type of nonlinear effect differs from those previously considered for the nonlocal character of the nonlinear response. It is not advantageous, then, to use the concept of nonlinear susceptibility, and one

24

LASER BEAMS

[I, §

2

should rather solve eq. (2.45). For the case of an unmodulated plane wave, however, eq. (2.45) can be readily solved to give

6n = + K , A ~ I = A ~i n~, l ~ 1 ~ ,

(2.46)

where K p = y2fi/8~~oA, can be called the electrostriction coefficient. For a wave of finite width a, we can also think of associating a relaxation time zI (where zI = u/u and where u is the sound velocity) with 6n. Notice finally that the result of (2.46) does not depend on the type of wave polarization, hence, in particulai it results that 6n = 6n+. In terms of the c-coefficients previously considered, this implies that the anisotropic part c1221in this case is zero [see (2.5) and (2.8)]. Accordingly, (2.7) shows that electrostriction does not lead to any rotation of the polarization ellipse. The electrostriction mechanism is very important and perhaps predominant in the case of solids. It is however a relatively slow mechanism (z = 10-8-10-9 sec), so that it may play an essential role only for light pulses whose time duration is appreciably longer than the time considered above. The contribution to 6n due to heating (LITVAK[1966]) can simply be obtained from the equation 6n = (dn/dT)6T, (2.46) where the temperature changes 6T satisfies the heat-conduction equation pcP d6Tldt = Q+kV26T.

(2.47)

In (2.47) k is the coefficient of thermal conductivity, pcp the specific heat per unit volume, and Q is the heat source. It has the form (2.48) where u is the absorption coefficient and c the light velocity in the medium. Here again, as for electrostriction, the response of the medium has a nonlocal character. Equation (2.47) shows that we can associate with this phenomenon a relaxation time z equal to the diffusion time i.e. z = ro2 Pcplk,

(2.49)

where ro is the beam radius. The order of magnitude of z may range in practical cases (ro w 1 mm) between 0.1-1 sec. For a time duration of the incoming light source much shorter than z, heat conduction may be neglected and (2.46) and (2.47) give (2.50)

WAVE PROPAGATION

25

Estimates of LITVAK[1966] indicate that n2 may be now typically n2 x l O - I 3 t e.s.u. where t is in nanosecond. For times much longer than z, eq. (2.50) is approximately valid provided that the time t is substituted by the relaxation time z (CARMAN, MOORADIAN, KELLEY and TUFTS[1969]).

0 3. Wave Propagation 3.1. INTRODUCTION

As already indicated in sec.1, self-action effects lead to a change in the angular spectrum of the propagating quasi-monochromatic wave (i.e. selffocusing and self-trapping). It also leads to a change in frequency spectrum (self-phase modulation and self-steepening), and (for an elliptically polarized wave) to a rotation of the polarization ellipse. In general, all the phenomena above indicated occur simultaneously. In what follows, however, in order to simplify matters, we will consider particular, though very im; portant cases, in which only one of the above phenomena predominates. The results will then make it possible to discuss the problem in its entirety. For the sake of simplicity, we will limit ourselves to the case of a linearly polarized wave. Accordingly, the case of the rotation of the polarization ellipse will not be further considered. Maxwell's equations using a source term due to the nonlinear polarization P("'),will be our starting point. Under the assumption V - E = 0, these equations reduce to the well known wave equation

where c is the phase velocity and c0 is the dielectric constant of the medium. For a cubic and isotropic medium, the relationship between P("')and E is given by (1.7) i.e.

and the specific expressions for 6n have been considered in previous sections. For a quasi-monochromatic wave propagating along z-direction, the electric field E(Y, t ) can be written as [see (1.2)]

where k = oo/c, and the complex amplitude A(r, t ) is a slowly varying

26

[I,

LASER BEAMS

63

function [compared with exp {i (mot-kz)}] of both space, P, and time, t . Upon substitution of (3.3) and (3.2) into (3.1), and by neglecting second spatial (i.e. aZ/dz2)and temporal derivatives of A (slowly varying envelope approximation) we get

=AIA+k

26n -A.

(3.4)

110

Here AI is a two-dimensional Laplace operatoi in the plane perpendicular to the beam axis z. When 6n = 0, eq. (3.4) reduces to the parabolic equation which is well known in the approximate theory of diffraction. Thus, eq. (3.4) is sometimes also called the quasi-optics equation. Such an equation has been considered and thoroughly discussed by KELLEY119651, TALANOV [1965], AKHMANOV, SUKHORUKOV and KHOKHLOV [1966, 19671. It should be noted that, when dispersion of the medium is considered, eq. (3.4) still remains approximately valid, provided that the phase velocity c is substiTARAN,HAUS,LIFSITZand tuted by the group velocity u (GUSTAFSON, KELLEY 119691). Let us express the complex amplitude A as A = A , exp (- iks),

(3.5)

where the real quantities s (the eikonal of the wave) and A , , are both functions of P and t . By substituting (3.5) into (3.4) and by considering an axially symmetric beam, we get, using cylindrical coordinates, the following eqs. (3.6a) and (3.6b): 2

-as+ - 1- as

[az

at.

zSri+6n = tn,A:.

(3.6~)

A dynamic equation for the medium in its simplest form, as in (2.22) has also been added. Equations (3.6), which were worked out by AKHMANOV, SUKHORUKOV and KHOKHLOV [1967], constitute the fundamental equations which will be used in the following sections. Equation (3.6~)correctly describes either the molecular-orientation mechanism, or the electronic (z w 0) or librational mechanisms [if a term proportional to 6ii is added, see (2.43)]. If more than one of the previous mechanisms is present, we should write 6n = Sn, and then write dynamical equations for the single terms Sn, (BLOEMBERGEN [1971]).

I,

B 31

21

WAVE PROPAGATION

3.2. STATIONARY SELF-FOCUSING (THEORY)

The stationary case corresponds to all time derivatives in (3.6) equal to zero. Then (3.6) reduces to (3.7a) (3.7b) The self-trapped solution is obtained from (3.7) by letting dA/dz = &/az Equation (3.7b) then gives ds/dr = 0 and (3.7a) reduces to

- -n 2 A0' no

=

_1_ k2Ao

[3+ 1 "1

= 0.

,

r dr

from which the trapped solutions Ao(r) have to be found. Without solving the equation it is immediately apparent that, if Ao(r) is a solution, also A,-(r') = TAo(Tr), with r being any arbitrary number, is also a solution. This means that the solution (or the solutions) can be scaled to any arbitrary remains constant size. The power of the beam, however, P a j:A;r'dr', (critical power for self-trapping). The radial behavior for the self-trapped solutions has been computed by CHIAO,GARMIRE and TOWNES [1964] for the lowest order, and by HAUS[1966] for the higher order solutions. The radial behavior of the lowest order solution is shown as a solid curve in Fig. 5. The corresponding critical power is*

which should be compared with (1.13) which was obtained with a simple physical calculation. It is instructive to try in (3.8) a Gaussian solution for Ao(r). We therefore assume Ai(r) = E ; exp [ -(r/a)'].

(3.10)

It is immediate to verify that (3.10) is not a self-trapped solution i.e. it is not a solution of (3.8). If however, on the left-hand side of (3.8), Ai(r) is expanded in power series of r up to r 2 , we then see that (3.8) is satisfied. We can then say that (3.10) is an approximate solution of (3.8) which is valid for r/a P,, in fact, the self-focusing effect overcomes the diffraction and the beam keeps focusing. To study this case, we will assume that a beam with a Gaussian radial profile [as in (3.10)] is entering the nonlinear medium at z = 0. In this case we look for a solution of (3.7) of the type (3.11a) s(r, z )

= 4(z)+trZP(z),

(3.11b)

i.e. we look for a solution, in terms of a Gaussian beam of variable beam width [given byf(z)] and of spherical wavefront, with variable radius of curvature. The reason for this choice is that this solution satisfies the parabolic equations (3.6) when nz = 0, i.e. in the linear diffraction case. When n2 # 0, (3.1 1) is still a solution of (3.7) only if the term n,AZ,/n, in (3.7a) is expanded in power series of r up to r 2 . This means that (3.11) is now an approximate

I,§

31

29

WAVE P R O P A G A T I O N

solution of (3.7) which only holds for r/a (P,,),, the intensity on the axis keeps increasing while the beam propagates, and (3.17) shows that the beam reaches a sharp focus at z = zfd. From (3.15), (3.13) and (3.11) we get (3.19) According to this approximate theory we see that, for a Gaussian beam, l/zf and l/zd combine quadratically [see (3.15)] rather than linearly [see (1.19)]. This difference implies a similar difference between (1.20) and (3.19). More generally, if the incoming wave has a radius of curvature R, then (df/dz),=, = j(0) = 1/R and (3.16) gives two focal spots, namely

_1_' -- -1zfl

1

(3.20)

Zfd

(3.21) and the second solution (3.21) obviously exists only if R < 0 (converging and 1/R combine wave) and IRI < zfd. Equation (3.20) shows that linearly. This property has been shown here to apply for a Gaussian beam. TALANOV [19701 has shown, however, that it is of general validity for cubic media. So far we have been concerned with an approximate solution of type of (3.11). In this case, when f -+ 0, the whole beam concentrates to a single point (sharp focus). For this reason the above theory is also called the aberrationless theory of self-focusing. We recall, however, that the previous results only apply in the proximity of the beam axis. The solution of the whole problem is a much more complicated task and it has been done by a computer (DYSHKO,LUGOVOI and PROKHOROV [1967, 19701, and MARBURGER and DAWES[1968]). The computer results indicate that: (i) not the whole beam focuses to a single point, but it acquires an annular structure and several focal spots are obtained; (ii) Each spot comes from the focusing

1,031

WAVE PROPAGATION

31

of an appropriate annular region and it is of finite dimensions (the intensity at the focal position is finite, i.e. no sharp focus). A geometrical optics representation of this case is depicted in Fig. 6 . The position z,, of the first focal spot vs. the power of the incident beam is plotted as curve 3 in Fig. 7 (an incoming beam with transverse Gaussian profile was assumed). For comparison, in the same figure, the approximate analytical relations (1.20) and (3.19) are also plotted as curves 1 and 2 respectively. Notice that the

0

Fig. 6. Geometrical optics representation of the self-focusing phenomenon in a real beam, LUWVOIand PROKHOROV showing formation of multiple focal spots (after DYSHKO, [1967]).

Fig. 7. Normalized plot of the focusing distance zfdvs. the power of the incident beam P. The quantities P,,and a are defined in the text [eqs. (3.9) and (3.10)]. The curve 3 is taken LUGOVOI and PROKHOROV from the behavior of the first focal spot as computed by DYSHKO, [1967]. Approximately the same curve is also obtainable from the computer study of DAWNESand MARBURGER [1969]. Curves 1 and 2 represent the approximate expressions given by eqs. (3.22) and (3.19) respectively. Note that the computer calculated curve (which assumes a Gaussian intensity profile for the incoming beam) falls somewhat between the two curves 1 and 2.

32

LASER B E A M S

[I,

53

quantity r t in eq. (1.20) has been identified with $a2, since in this way a better agreement is reached between the approximate expression (1.20) and the computer results. Consequently, eq. (1.20) is now written as

(3.22) where a is defined according to (3.10). Since the beam is so unstable its evolution in the nonlinear medium must strongly depend upon the inhomogeneity of the incoming beam. This problem has been studied by BESPALOV and TALANOV [1966]. They showed that transverse inhomogeneities of the beam with an optimal diameter should focus first. This optimum diameter A is given (in our notation) by = IoCi/(327TnzI)a,

(3.23)

where I is the intensity of the incoming beam. The power carried on by is seen from (3.23) to be approxthis transverse instability (P x &l’Z) imately equal to the critical power P,,. For a ruby laser beam of 1 MW power with a 2 mm diameter entering a cell containing CS, (n2 = 1.3 x lo-” e.s.u., I, = 0.6943 pm), eq. (3.23) gives A x 180 pm. 3.3. STATIONARY SELF-FOCUSING (EXPERIMENTAL RESULTS)

In this section we will discuss a few experimental results which support the theoretical considerations on self-focusing developed in the previous section. Strictly speaking these results have not been obtained with truly stationary beams. Since, however, the time duration of laser pulses (Qswitched lasers, 10-100 nsec) is much longer than the relaxation time of the medium, we can think of the corresponding results as typical of a stationary beam. The first thorough and convincing experimental analysis of the self-focusing phenomenon has been reported by GARMIRE, CHIAOand TOWNES[1966]. They established that the initially unfocused light beam, with sufficiently homogeneous amplitude and phase front, becomes compressed into a thin filament of -50 pm diameter in a cell containing CS2. The experimental set-up is shown in Fig. 8. A diffraction limited beam of 0.5 mm diameter and 10-100 KW power from a ruby laser, was incident on a cell containing CS, . The spatial evolution of the beam was observed by immersing microscope slides every two centimeters along the beam in order to reflect a small fraction of it out of the cell. Fig. 8 shows the magnified images of the beam profile as it reflects off the glass plates. The power of the input beam at which the beam was self-focused was P,, = 25 kW. This value should be

I,

531

WAVE PROPAGATION

33

Fig. 8. Evolution of beam focusing in CSz. Left: without dashed cell; right: dashed cell adds 25 cm path length. (a) Gas laser control; (b), (c) and (d), beam focusing at increasing power, (e) 1-mm pinhole (from GARMIRE, CHIAOand TOWNES 119661).

compared with the theoretical value of 14 kW which is obtainable from curve 3 of Fig. 7. The approximate formula for self-focusing given by (1.20) was verified by WANG[1966a, 1966bl. He used the following expression for zfd (KELLEY [1965]) [compare with (3.22)] N

zfd

= ~no(a2/f)[co/n213 [p'-p$rI-',

(3.24)

wheref is the ratio of the radius of the beam, a, to a characteristic transverse radius of curvature of the laser intensity. One should have f = 1 for a Gaussian beam. Equation (3.24) may be rewritten as (3.25)

.I:[-

U2

a=ln

3

(3.26)

O f

Equation (3.25) gives the threshold power for self-focusing in a column of liquid of a given length 1 = z,, . The plot of P* vs. (1/1) for a given material should yield a straight line. The point of interception on the ordinate of this straight line should give the value of P,,,whereas the slope should be proportional to CI, i.e. to a2. These two predictions have both been verified by WANG[1966a]. In Fig. 9 the experimental behavior Pt vs. (1/1) for three liquids is reported. The critical powers determined from these plots were 64 kW for benzene, 19 kW for nitrobenzene and 55 k W for toluene. Using

34

LASER BEAMS

5-

o

nLtrobenzene-

0

u

0

1

2

3

4

5

cell length

6

7

8

x 102Wn-1)

Fig. 9. Plot of the square root of threshold power for self-focusing vs.'the inverse of cell length for benzene, toluene and nitrobenzene (after WANG [1966a]).

I

2

I 4

I

I

6

8

cell length

I I 10 12 14 -1 I -2 cm (10

Fig. 10. Plot of the square root of threshold power far self-focusingin CS2 as a function of the inverse of the cell length with both linearly and circularly polarized beam (after WANG [1966b]).

WAVE PROPAGATION

35

a similarprocedure, WANG[I 966b] determined the ratio of the critical powers for circularly and linearly polarized beams (Fig. 10) in CS,. As seen from the diagram, the critical powers differ by only a factor 2. Since for CS2 the molecular-orientation Kerr effect predominates (for nanosecond excitation), according to (2.21) one should expect this ratio to be 4. It must be noted, however, that this conclusion only applies when the circularly-polarized wave becomes self-focused as a whole. A more detailed analysis shows however that, in a nonlinear medium, a circularly polarized wave is unstable, and its self-focusing results in wave channels with linear rather than with circular polarization. In this case the critical power is only twice the critical power for a linearly polarized beam (see CLOSE,GIULIANO, HELLWARTH, HESS,MCCLUNGand WAGNER[1966], and CHABAN[1967]). We finally wish to mention the work of MAIER,WENDLand KAISER [ 19701 in which the exit beam diameter, after a given length of liquid CS2, was experimentally measured and compared with the theoretical result which was computed from the parabolic equation by taking into account both the divergence and time behavior of the actual input beam. A good agreement was found between experimental and theoretical results. According to what was discussed in the previous section and according to the above experimental results, we can now say that the self-focusing phenomenon in cubic media appears to be well understood. Actually, as it was already mentioned in 0 1, a closer inspection of the beam at the exit face of the trapping cell revealed the existence of the so-called large-scale (50-100 pm) and small-scale ( - 5 pm) filaments. It is not yet clear as to whether the two classes of filaments really refer to two different physical situations. For a smooth TEMoo mode, it rather seems that the beam gradually and smoothly collapses into (usually) only one small-scale filament (LOYand SHEN[1969], MAIER,WENDLand KAISER[1970]). In this case the term large-scale filament does not seem to be appropriate. For an incoming inhomogeneous beam, however, it seems that, whiIe the beam propagates into the nonlinear medium, it first develops some large scale (50-100pm) inhomogeneities which then break into a large number of small-scale filaments (Fig. 11) (CHIAO,JOHNSON,KRINSKY, SMITH, TOWNES and GARMIRE [1966], BREWER and LIFSITZ [1966], and KOROBKM and SEROV[1967]). It is quite possible that the large scale inhomogeneities correspond to the optimal diameter of instability given by (3.23). It has been in fact convincingly shown by ABBIand MAHR119711that the small-scalefilaments are originated by the inhomogeneities (spatially and temporally) of the incoming beam. Further convincing and nice experiments on this subject have recently been done by CAMPILLO, SHAPIRO and SUYDAM [1973, 19741.

36

Fig. 11. (a) Image of a ruby laser beam emerging from a 50 cm cell of CS2 and exhibiting large-scale and small-scale filaments. The bright central portion is the large-scale filament, while the many small bright filaments demonstrate the small-scale trapping. (b) Stokes KRINSKY,SMITH, radiation under conditions similar to (a). (From CHIAO,JOHNSON, TOWNES and GARMIRE [1969].)

Although the so-called large-scale filaments, in this way may be understood, the occurrence of a limited diameter small-scale filament still remains to be explained. Some possible explanations will be discussed in sec. 3.7. 3.4. NON-STATIONARY CASE: SELF-PHASE MODULATION AND SELFSTEEPENING

While in sec. 3.2 we had considered the simplified case of Ao(r, t ) and s(r, t) being independent oft, we will dually consider in this section the case in which Ao(r, t ) is independent of r (plane wave). The cubic nonlinearly produced in the former case a stationary self-focusing. In the latter case we will see that it leads to self-phase modulation and self-steepening of the beam. When both A , and s are independent of r eqs. (3.6) reduce to (3.27a) dA,/dz'

= 0,

zd6n/at'+ 6 n = $nz A ; .

(3.27b) (3.27~)

I,

o 31

WAVE PROPAGATION

Here, a new coordinate system, z’ and

1’

such that

,

z = z

r’

=

37

r-(z/V),

(3.28a) (3.28b}

has been used. From (3.27b) we now see that A, = A,(t’) = A,[r-(zlu)], i.e. that the pulse amplitude travels without being deformed. Equation (3.27~)then shows that Sn is function of only [t-(zlu)]. Consequently from (3.27a), we get (3.29) where the boundary condition, s = 0 for z = 0, has been assumed. Since the nonlinear part of the phase is ks, we then see that (3.29) is coincident with the expression (1.24) which was established by a simple physical argument. Let us consider, as a particular example, the case of a smooth light pulse for A , ( [ ) (GUSTAFSON, TARAN,HAUS,LIFSITZand KELLEY [1969]). This pulse, after traveling a distance z into the material, will acquire a phasemodulation such that its Fourier spectrum will look like that shown in Fig. 12. The appearance of a quasi-periodic structure in the Fourier spectrum may be understood in a simple way. Let us assume that the amplitude and phase q5 = ks of the light pulse, after traveling a distance z in the medium, are as indicated in Fig. 13 [4(t)is delayed compared to A , ( t ) due to the

Fig. 12. Typical behavior of the spectral intensity of a wave whose amplitude and phase are as shown in Fig. 13. The frequency w’ = w o - w , is the difference between the laser frequency wo and the actual frequency of the spectrum. The region w’ > 0 thus corresponds to the Stokes side.

38

LASER BEAMS

A

/

-vAoit' \

\ \

I I I

\

\

\

I

A01

I I

t'

t"

t

Fig. 13. Typical time behavior of amplitude A&) and phase $ ( t ) of light pulse after traveling a given distance in a nonlinear material (self-phase modulation). If the peak value of the phase $p is appreciably greater than n,the Fourier spectrum of this pulse will show a frequency broadening predominantly due to the phase rather than amplitude modulation and the resulting spectrum will be as shown in Fig. 12.

non-instantaneous response of the medium, see (3.27c)l. Since the instantaneous frequency wi of the wave is wi = w o - $ ,

(3.30)

>
0) and falling (4 < 0) parts of when $ 2 0 then wi 4(t) will therefore contribute to the Stokes and anti-Stokes sides of the spectrum respectively. From the same argument it is apparent that the maximum extensions of the spectrum at the Stokes (ws) and anti-Stokes sides (mas) (see Fig. 12) are given by 0 s = was

=

(1401

(3.31a)

(1$1>2,

(3.3 1b)

where the time derivatives are calculated at the two inflection points 1 and 2 of 4(t).The amount of spectral broadening to be expected is in this way calculated. The quasi-periodic structure can now be explained by considering two times t f and t", for instance or the rising part of 4(t) (Stokes side) (Fig. 13). If t' and t" are such that $(t') = 4(t") = 4, the two points will

I,

8 31

WAVE PROPAGATION

39

both contribute to the Fourier spectrum at frequency w o- 4. Their contribution can be obtained by calculating the Fourier spectrum by the stationary phase method (CUBEDDU,POLLONI,SACCHIand SVELTO[1970]). It turns out that the two contributions are two complex numbers whose phases, as expected, are respectiveIy the phases 4(t’)and 4(t”) of the wave at the two points considered. If 4(t”)-4(t’) = n, the two contributions will be 180” out of phase and they will tend to cancel each other out. The first minimum of the spectrum (see Fig. 12) will thus result. All successive minima and maxima can be understood in the same way. From this argument it is then apparent that the value of the phase at the peak c$p is given by

4p = (2m-l)q

(3.32)

where m is the number of minima of the spectrum of Fig. 12 at the Stokes side. Since the same argument can be repeated at the anti-Stokes side, we see that the number of minima at the Stokes and anti-Stokes sides are the same. Other two useful relationships which can be readily obtained are (CUBEDDU, POLLONI, SACCHI and SVELTO [ 19701): (1$1)1

O.36(AwJ3

(3.33a)

O.36(AwaJ3

(3.33b)

= =

where the third derivatives are calculated at the inflection points 1 and 2 and where Aws and Awas are indicated in Fig. 12. Equations (3.31b(3.33) establish very useful relationships between the Fourier spectrum and the time-behavior of the phase 4(t).If the spectrum is known, then (3.31)-(3.33) yield 4pand the first and third derivatives of 4(t) at the two inflection points. In this way the time behavior of 4(t) can be determined fairly accurately. This method is particularly useful when A , ( t ) is an ultrashort light pulse: in this case, in fact, measurements in the frequency domain (frequency broadenings being large) are easier to make than measurements in the time domain (time durations being small). The measurement of 4(t)is a very useful tool for investigating the dynamic response of the medium. If the dynamic response of the medium is as in (3.27c), from the measured spectrum we can measure the relaxation time T (POLLONI, SACCHI and SVELTO 119691). In fact, from (3.27~)and (3.29) we get (4 = ks) .rd+c$ = *kzn, A i l n o . (3.34)

If we consider (3.34) at the two inflection points 1 and 2 and take the ratio

40

LASER BEAMS

[I,

53

of the two expressions we get: (3.35) where A , , and A,, are the field values at the times corresponding to the inflection points (Fig. 13). They are related to the corresponding intensity of the spectrum at the Stokes, Z, , and anti-Stokes, I,, extrema (see Fig. 12) bv (3.36) Since $ ( t ) can be measured from the spectrum, (3.35) and (3.36) allow the measurement of T. Once T is known, from (3.34) we can obtain the time behavior of A: i.e. the time behavior of the incident beam. If the pulse energy and the propagation distance z are also known, we can also get a measurement of n2. Hence, both n2 and T (i.e. the dynamic behavior of the medium) and the time behavior of the incident beam can be obtained from the measurement of the self-phase modulated spectrum. Self-phase modulation of ultrashort pulses in small-scale filaments was

Fig. 14. Typical spectral broadening of small-scale filaments as obtained with picosecond excitation (from POLLONI, SACCHIand SVELTO [1969]).

1,

B 31

WAVE P R O P A G A T I O N

41

first observed and correctly interpreted by SHIMIZU [1967]. Later works in this field have been done by CHEUNG,RANK,CHIAOand TOWNES[1968], GUSTAFSON, TARAN,HAUS,LIFSITZ and KELLEY [ 19691, DENARIEZ-ROBERGE and TARAN[1969], POLLONI, SACCHI and SVELTO [1969], CUBEDDU, POLLONI, and SVELTO [ 19701, CUBEDDU, POLLONI, SACCHI and ZARAGA [1971], SACCHI and CUBEDDU and ZARAGA [1971]. Examples of self-phase modulated spectra in small-scale filaments are shown in Fig. 14. Although in this case the beam is far from being plane, the general features of the experimental results can be understood in terms of what has been discussed in this section. For a more careful study of the actual situation the radial dependence of the light beam in the filament should be taken into account (see sec. 3.6). It is worthwhile to mention that even an amplitude modulated wave would give a spectrum which resembles that in Fig. 11, with some additional sub[1968]). structure in it, however (CHJXJNG,RANK,CHIAOand TOWNES We see from (3.27b) that, within the approximations so far made, the light pulse travels without being deformed. The phenomenon of self-steepening is, in this way, neglected. Within the slowly varying envelope approximation we set a2P"'/8t2w cozP"' in (3.1), so that we neglected the time variation of the amplitude (ccGnA) of P. If this variation is not neglected, then, as a next higher order approximation, eq. (3.27b) transforms to

A = - __ i -(&Ao), a aAo + a0 c at aZ n o c at

-~

(3.37)

which together with the material equation ( 3 . 6 ~ describes )~ self-steepening. From what has been discussed above, we see that self-steepening is a higher order effect compared to self-phase modulation. This is further confirmed by the fact that, at the shock distance z, [eq. (1.24a)], the spectral broadening on the Stokes side, according to (3.31a) and (3.29) (4 = ks) would be w, = +coo. The broadening due to self-phase modulation has here already reached a very large value, actually comparable to the central frequency coo. Accordingly, a correct treatment of self-steepening phenomenon cannot be done without considering the dispersion of the medium. So far, the selfsteepening effect has not been experimentally observed. 3.5. NON-STATIONARY CASE: SELF-FOCUSING OF NANOSECOND PULSES (THEORY OF MOVING FOCI)

We have so far considered the case of A(r, t ) being either time independent (sec. 3.3), or radially independent (sec. 3.4). We wish now to consider the somewhat more general case in which both radial and temporal dependence is taken into account. In this section we will limit our discussion to the case

42

LASER BEAMS

[I,

s3

in which the time duration of the light pulse is much longer than the dynamic response time of the medium. In this case, the dynamic equation (3.6~)can be approximately written as 6n w i n 2 A i . If this expression is substituted for 6n in (3.6a), and the new coordinate system (3.28) is used, then (3.6a) and (3.6b) reduce in form to the stationary solutions (3.7a) and (3.7b) in which the coordinate z is substituted by z‘. Thus any solution of the stationary case involving the power P (or the intensity) of the incoming beam at z = 0, can be converted to a time-dependent solution by replacing P with P[t-(zlu)], where P ( t ) is the time behavior of the incident pulse (MARBURGER and WAGNER [1967]). For this reason, this case is also called quasistationary. In particular, since the input power changes with time, the positions of multiple foci of Fig. 6 will change in time. and PROKHOROV [1968] suggested Following on from this fact, LUGOVOI that the small-scale filaments are simply the tracks of foci moving in accordance with the time variation of the input laser power. The idea is essentially that, when this power varies with time, the foci will in their movement (eventually) cross the exit face of the trapping cell to form what appears to be a “filament”* (if a time integrated picture at the exit face is taken). This idea, which initially caused some controversy, appears now to be well proved experimentally when the input beam is made of a light pulse of nanosecond SEROVand duration (Lou and SHEN [1969], KOROBKIN,PROKHOROV, SUCHELEV [1970], LIPATOV, MANENKOV, PROKHOROV [1970], and LOYand SHEN[1970, 19731). Because of the relevance of this moving foci picture, it is interesting to calculate the focal movement for a prescribed time behavior of the input beam. This movement is easiest to calculate in the case of aberrationless (i.e. only one focus) self-focusing.In this case eq. (3.22) can be approximately used, and, according to what has been said above, the position of the focal spot as a function of time [i.e. zfd = zfd(t)]can be obtained from the equation (LOYand SHEN[1970]) (3.38a) where P (t) is the time behavior of the incoming beam. Eq. (3.38a) can be expressed in the dimensionless form (3.38b) where zc w (2.3)nOa2/1,.The solution of zfd = Zfd(t) from (3.3813) may be

* From this point on we will use the word “filament” to indicate both a true light f i b ment or a track of a movig focus (i.e. a dielectric filament).

1 . 8 31

WAVE PROPAGATION

43

more conveniently obtained in a graphical way as shown in Fig. 15 (SHEN and LOY [1971]). The light beam of point 1 of Fig. 15b will enter the cell at time t , , will then move with light velocity (dashed line), and it will focus at point A whose zrdis obtained from (3.38b) by having P = PI.

( b!

t (arbitrary units1

Fig. 15. Graphical way of calculating the movement of the focal spot (a) once the time behavior of the imput beam P ( t ) is known (b).

For the above quasi-stationary picture to be acceptable, one should also consider the fact that, when the beam is focusing, its on-axis intensity is sharpening in time (MARBURGER and WAGNER[1967], and MCALLISTER, MARBURGER and DESHAZER [1968]). If the pulse then gets shorter than the relaxation time of the medium, one could not neglect the time derivative of 6n in (3.6~)and the quasi-stationary picture would be inadequate at and beyond the first focal point. Actually, if the light would come to a sharp focus (aberrationless case), the pulsewidth should in that point be zero (MARBURGER and WAGNER[19671) and the quasi-static approximation would cehainly be inadequate. Computer calculations however, indicate that, in reality, the focus has a finite Airm longitudinal dimension, so that the timewidth At, of the light intensity pulse at the focus is also finite (Atm = Azm/v, where u is the velocity of the focal spot, see Fig. 15). According to the

44

[I,

LASER BEAMS

B3

estimate of LUGOVOI and PROKHOROV [1968], one has At,,, z 50 x lo-’’ sec, which is appreciably longer than the relaxation times of the medium which may be involved (see Table 1). According to what has been discussed in this section, we can therefore reach the conclusion that, for nanosecond excitation, the picture of moving foci appears to be well established both from the theoretical and experimental sides. Actually, the observed limited size of the “filaments” i s not explained by the moving foci picture, and this problem will be deferred to sec. 3.7. For picosecond excitation, however, the experimental results (DENARIEZROBERGE and TARAN[1969], CUBEDDU, POLLONI,SACCHI,SVELTOand ZARAGA [1971]) seem to be in disagreement with the moving foci picture and, rather, seem to be in agreement with a waveguide picture (true filament). This is not surprising however, since, when the pulse gets as short as the relaxation time, the quasi-stationary picture is not valid any more, and the whole problem should be reconsidered. This will be done in the next section. Before ending this section, however, we wish to mention that, even for nanosecond excitation, not all the results are, as yet, completely clear. Besides the problem of “filaments” diameter, which will be discussed further on, we will mention the case where a light beam is strongly focused to a point inside the material by an external lens (ASKARYAN, DIYANOV and MUKHANADZHANOV [1971I). In this case the small-scale filaments observed at the exit face of the material do not seem,to be consistent with the moving foci picture. 3.6. NON-STATIONARY SELF-FOCUSING: PICOSECOND EXCITATION

For exciting pulses of duration comparable to the relaxation time of the medium, the equations (3.6) must be considered in their entirety. In this case, approximate analytical solutions (AKHMANOV, SUKHORUKOV and KHOKHLOV [I 9671) and computer calculations (SHIMIZUand COURTENS [1973], and FLECK and CARMAN [1972]) both indicate that a quasi-waveguide tends to be formed in the medium. Let us consider first the approximate analytical solution. To this purpose, by using the coordinate system of (3.28), eqs. (3.6) transform to

=o za6n/dt‘+ 6n = i n 2 A ; .

(3.39b) (3.39c)

1,131

45

WAVE PROPAGATION

In analogy with (3.1 1) we now look for a solution of the following type

[ [

A;(Z', r, t ' ) = Eg(t') exp -

f s(zl, r,

t')

2 ( z r ,t ' )

= +(z', t')+p(z',

r2

11

(3.40a)

azf2(z1,t ' )

t')r2,

(3.40b)

where E i ( t ) is the time behavior of the incoming light pulse. If we confine ourselves to consider the section of the beam near the axis, from (3.39) and (3.40) we get [compare with (3.12)l: 1 a2f 1 f d z r 2 k2a4f4

---exp 2n0a2 z o

f4

t:") ~

dy,

(3.41a)

(3.41b)

SUKHORUKOV and KHOKBy approximately solving eq. (3.41a) AKHAMANOV, HLOV [I9671 were able to work out forf(z', t') (i.e. for the beam diameter evolution) an approximate solution which indicated that the beam acquires, during propagation, a horn-shaped form as schematically indicated in Fig. 16. The general physical significance of the solution depicted in Fig. 16 can be understood as follows. When the light pulse enters the medium, its leading edge will not experience any nonlinear refractive index change, since, according to (3.39c), it takes some time ( W T ) for this change to come about.

Fig. 16. Trumpet-shaped quasi-trapped solution which is possible for an incoming pulse of time duration of the order of the relaxation time of the medium.

Accordingly, the first part of the beam (unshaded part of Fig. 16) keeps diffracting according to the usual linear diffraction theory. The cumulative effect of this part, will however leave in the medium a nonlinear index change. The trailing edge of the pulse can then get trapped in the medium as a result of this index change (shaded part of Fig. 16). We can also say

46

LASER BEAMS

[I,

§3

that the diverging part of the beam creates in the medium a temporary waveguide under which the trailing part of the beam gets trapped (see also SHENand LOY[1971]). Since the leading edge of the pulse keeps expanding by diffraction, the trailing edge (shaded region of Fig. 16) cannot remain trapped for a very long distance. The distance below which this edge remains trapped is however much greater than that predicted by the linear diffraction theory ( -ka2 where a is the filament radius). We can therefore think of the shaded region of Fig. 16 as a quasi (or leaky) waveguide. For the trapped part of the pulse, f ( z ‘ ,t ’ ) is almost independent of z’ and t’. This has immediately two consequences: (i) from (3.40a) we see that a steady-state pulse with constant diameter is present in the trapped region; (ii) from (3.41b) we see that /3 w 0, which, according to (3.40b), indicates that the phase of the beam is independent of r. These two properties have actually been verified in trapping experiments with picosecond pulses (CUBEDDU, POLLONI, SACCHI and ZARAGA[1971]). It is also worth mentioning that, in these experiments, the transverse profile of the light intensity in the filament was found to be Gaussian, in accordance with the assumption of (3.40a). We wish finally to notice that (3.41~)describes how self-phase modulation develops in the “filament”. When f is independent of z’ and t‘ (Le. trapped portion of the beam of Fig. 16), thenf‘ can be taken out from the integral of (3.41~)and eq. (3.41~)transforms to (3.42) where 6n(0, t ) is the nonlinear refractive index change at r = 0. Except for the constant factor l/k2a2f2 (phase shift due to the waveguide effect) eq. (3.42) now coincides with (3.27a) which is valid in the case of a plane wave. The above approximate calculation is not completely satisfactory, however, since its starting point [eqs. (3.40)] neglects the aberrations of the selffocusing process. In the stationary case we have in fact seen (see Fig. 6) that the aberrations are quite pronounced. One cannot therefore completely trust the results obtained from (3.41). It is interesting however to notice that detailed computer calculations using the parabolic equations, exactly confirm the presence of a quasi-trapped waveguide, and all the other results which have been indicated above (SHIMIZU and COURTENS [1973], and FLECK and CARMAN[1972]). Under these conditions we can say that also the case of picosecond excitation is now reasonably understood, and that this case seems to lead to a quasi-trapped solution (i.e. a true “filament”).

I , § 31

WAVE P R O P A G A T I O N

41

3.7. THE “FILAMENT” DIAMETER

One of the problems which still waits for an adequate solution is that of the diameter of the small-scale “filaments”. This diameter is typically of a few microns (e.g. 5 pm for CS2, 10 pm for nitrobenzene) both in the case of nanosecond and picosecond excitation. The theories which have been so far discussed, i.e. either the moving foci theory of sec. 3.5 and the nonstationary theory of 3.6, are both unable to explain the existence of a limiting diameter. Although in fact it is likely that, in a real case, the beam (due to aberration) will not reach a sharp focus but rather a focus of finite dimensions, these dimensions cannot be typical of a given material. The parabolic equation (3.4) and the material equation (3.6~)admit in fact a scaling law, so that any solution can be scaled to any arbitrary size diameter. Therefore, in order to explain the filament diameter, some new physical mechanisms in the nonlinear response of the material must be included. During the focusing process in fact, the intensity on the beam axis keeps increasing, and it appears reasonable to assume that, at sufficiently high intensities, new phenomena not included in the material equation (3.6~)should be considered. The phenomena which have been so far considered are : (i) saturation of the nonlinear refractive index change, (ii) losses due to a suitable nonlinear optical effect (e.g. multiphoton absorption or stimulated Raman scattering). In the case of saturation of the nonlinear refractive index change, the steady state trapped solution is obtained from (3.6a): (3.43) where the functional relationship between Sn and A: must be specified (PIEKARA[1966, 19681) according to the physical mechanism which is being considered. Notice that, for the case where there is no saturation (i.e. Sn = 3n&) eq. (3.43) reduces to (3.8). If the molecular orientation Kerr effect is considered, then the saturation of 6n can be represented by the curve of Fig. 3 and, approximately, by eq. (2.18). Computer results indicate however that the saturation of molecular orientation Kerr effect would give a filament diameter of -0.2 pm for CS2 (GUSTAFSON, KELLEY, CHIAO and BREWER[1978], and MARBURGER, HUFF, REICHERTand WAGNER [1969]), which is more than an order of magnitude smaller than the observed value. This saturation does not therefore seem to play any important role in filament stabilization. One could not either invoke the saturation of the nonlinear electronic distortion, which, as previously mentioned, seems to occur at field intensities which are -2 orders of magnitude higher then those of the molecular orientation Kerr effect. Recently (GUSTAFSON and

48

LASER BEAMS

[I,

04

TOWNES [1972]) a new saturation mechanism has been considered due to steric effects and the compressibility of molecules. The idea here is to consider a moiecuie placed in the field of neighbouring ones. Since all molecules tend to be aligned in the same direction by the field, as some molecular alignment occurs, a field-dependent stress develops, which tends to prevent further molecules from aligning along the same direction. This phenomenon can therefore be considered as a possible saturation mechanism for the librational model. Estimates indicate a theoretical filament diameter of 1.5 pm for CS,, which is in better agreement with the experimental results. Among the nonlinear loss mechanisms for beam stabilization, the stimulated Raman scattering phenomenon has been considered in some detail (RAHNand MAIER[1972]). Reasonably good agreement has been obtained in this case between the theoretically calculated diameter and the experimentally observed one in CS, , toluene, benzene and benzene-CS, mixture under nanosecond excitation. Although losses due to stimulated Raman scattering may account for the filament diameter under nanosecond excitation, it does not seem to be the filament stabilizing mechanism for picosecond excitation. In this case, in fact, filaments of the limited size (e.g. 5 pm in CS,) are often observed to occur without any presence of stimulated Raman light (CUBEDDU, POLLONI, SACCHIand SVELTO[1970]). A second nonlinear loss mechanism which may be responsible for beam stabilization has been proposed to be the avalanche ionization due to the optical field of the e.m. wave (YABLONOVITCH and BLOEMBERGEN [1972]). A crude calculation shows that this phenomenon can indeed produce a beam stabilization at a diameter of a few micrometers. According to what has been discussed above, we can say that the existence of a limited “filament” diameter does not appear to be as yet completely understood. N

5 4. Conclusions In this review the self-action effects of a powerfuland quasi-monochromatic light wave which propagates in a nonlinear isotropic medium have been considered. Among these self-action effects, particular emphasis has been given to the processes of self-focusing, self-trapping and self-phase-modulation. The experimentaland theoretical material here presented offers evidence that much progress has been made in this field in the last few years. The main physical effects have been predicted and observed experimentally, and a mathematical framework has been developed and makes it possible to

1,141

CONCLUSlONS

49

trace, at least qualitatively, the main features of these self-action effects. At the same time a number of important points are still to be investigated in the future. In particular we would like to mention: (i) Investigations into the mechanisms which are responsible for self-action effects in different materials and under different experimental conditions (e.g. long pulse or short pulse excitation); (ii) The problem of “filament” diameter; (iii) An investigation into the role of dispersion. For picosecond pulses, in fact, the self-phase-modulation phenomenon may give rise to spectral broadenings more than 1000A wide. In this case, the dispersive properties of the medium cannot be neglected. Dispersion may be responsible for a partial conversion of FM to AM, thus producing a subpicosecond structure on the incoming pulse; (iv) Unification of the self-focusingand stimulated scattering theories. Such a treatment would make it possible to reveal the mutual influence of the indicated effects, that is, the action of self-focusing on stimulated scattering, and the not less important reaction which can, probably, explain some features of the behavior of self-focusing and self-trapping. As a conclusion, the importance and impact of self-action effects in the field of quantum electronics should be stressed. The propagation of a powerful light wave in a nonlinear medium in fact is by itself a very interesting case of nonlinear diffraction. Moreover, the study of self-action effects has led to a considerably greater understanding of the physical mechanisms of nonlinear polarization, i.e. of the nonlinear dynamic behavior of matter. Actually one makes use of these self-action effects to measure physically interesting parameters of this behavior. The relevance of self-focusing and self-trapping to stimulated scattering phenomena (Brillouin, Raman, Rayleigh) should also be stressed. In material with high nonlinear coefficient n, ,self-focusingand self-trapping effects usually preceed, and hence strongly influence the behavior of these stimulated phenomena. Actually self-focusing and self-trapping can also themselves be considered stimulated phenomena, since, according to CHIAO,KELLEYand GARMIRE [1966] they can be looked upon as a self-pumped version of light-by-light scattering. We wish finally to point out that self-action effects are not only of considerable scientific, but also of considerable practical importance. Self-focusing and self-phase modulation will in fact ultimately limit the amount of power which can be sustained by a given material. This is, for instance, of relevance to highpower solid state amplifiers. The problem of nonlinear defocusing (if n2 < 0) due to thermal effects will in turn limit the amount of power which can be propagated over large distances in the atmosphere.

50

LASER BEAMS

References ABBI, S. C. and H. MAHR,1971, Phys. Rev. Lett. 26, 604. ALFANO, R. R. and S. L. SHAPIRO, 1970a, Phys. Rev. Lett. 24, 592. ALFANO,R. R. and S. L. SHAPIRO,1970b, Phys. Rev. Lett. 24, 1217. AKHMANOV, S. A., A. P. SUKHORUKOV and R. V. KHOKHLOV, 1966, Zh. Eksp. i Teor. Fiz. 50, 1537; Sov. Phys. JETP 23, 1025. S. A., A. P. SUKHORUKOV and R. V. KHOKHLOV, 1967, Zh. Eksp. i Teor. Fiz. AKHMANOV, 51, 296; Soviet Phys. JETP 24, 198. S. A., A. P. SUKHORUKOV and R. V. KHOKHLOV, 1968, Soviet Phys. Uspekhi AKHMANOV, 10, 609. ASKARYAN, G. A., 1962, Zh. Eksp. i Teor. Fiz. 42, 1567; Sov. Phys. JETP 15, 1088. ASKARYAN, G. A., KH. A. DIYANOV and MUKHAMADZHANOV, 1971,Zh.ETF Pis. Red. 14, 452; JETP Lett. 14, 908. BESPALOV, V. I. and V. I. TALANOV, 1966, Zh.ETF Pis. Red. 3, 471; JETP Lett. 3, 307. BLOEMBERGEN, N., 1971, Picosecond nonlinear optics, Esfahan Laser Symp. 1971, to be published. BREWER, R. G. and J. R. LIFSITZ,1966, Phys. Lett. 23, 79. BREWER, R. G., J. R. LIFSITZ,E. GARMIRE, R. Y. CHIAOand C. H. TOWNES, 1968, Phys. Rev. 166, 326. BREWER, R. G. and C. H. LEE,1968, Phys. Rev. Lett. 21,267. BREWER, R. G. and A. D. MCLEAN,1968, Phys. Rev. Lett. 21,271. BRISS,R. R., 1962, Proc. Phys. SOC.(London) 79, 946. CAMPILLO, A. G., S. L. SHAPIRO and B. R. SUYDAM, 1973, Appl. Phys. Lett. 23, 628. CAMPILLO, A. G., S. L. SHAPIRO and B. R. SUYDAM, 1974, Appl. Phys. Lett., in press. CARMAN, R. L., A. M. MOORADIAN, P. L. KELLEY and A. TUFTS,1969, Appl. Phys. Lett. 14, 136. CHABAN, A. A., 1967, Zh.ETF Pis. Red. 5, 57; JETP Lett. 5, 45. CHEUNG, A. C., D. M. RANK,R. Y. CHIAOand C. H. TOWNES, 1968, Phys. Rev. Lett. 20, 786. CHIAO,R. Y., E. GARMIRE and C. H. TOWNES, 1964, Phys. Rev. Lett. 13, 479. CHIAO,R. Y., P. L. KELLEY and E. GARMIRE, 1966, Phys. Rev. Lett. 17, 1281. CHIAO,R. Y., M. A. JOHNSON, S. KRINSKY, H. A. SMITH,C. H. TOWNES and E. GARMIRE, 1966, IEEE J. Quant. Electr. 2, 467. L. D. HESS,F. J. MCCLUNG and W. G. CLOSE, D. H., C. R. GIULIANO, R. W. HELLWARTH, WAGNER, 1966, IEE J. Quant. Electr. QE-2, 553. 1970, Phys. Rev. A2, 1955. CUBEDDU, R., R. POLLONI, C. A. SACCHI and 0. SVELTO, CUBEDDU, R., R. POLLONI, C. A. SACCHI and F. ZARAGA, 1971, Phys. Rev. Lett. 26,1009. CUBEDDU, R. and F. ZARAGA, 1971, Optics Comm. 3, 310. DAWES,E. L. and J. H. MARBURGER, 1969, Phys. Rev. 179, 862. DEBYE,P., 1945, Polar Molecules (Dover Publications, New York) Chap. 5. DE MARTINI, F., C. H. TOWNES, T. K. GUSTAFSON and P. L. KELLEY, 1967, Phys. Rev. 164, 312. DE NARIEZ-ROBERGE, M. M. and J-P. E. TARAN,1969, Appl. Phys. Lett. 14, 205. DYSHKO, A. L., V. N. LUGOVOI and A. M. PROKHOROV, 1967,Zh.ETF Pis. Red. 6,655; JETP Lett. 6, 146. DYSHKO, A. L., V. N. LUGOVOI and A. M. PROKHOROV, 1970, Sov. Phys. Doklady 14, 976. FABELINSKI, I. L., 1968, Molecular Scattering of Light (Plenum Press, New York). FLECK,J. A. and R. L. CARMAN, 1972, Appl. Phys. Lett. 20,290. GARMIRE, E., R. Y. CHIAOand C. H. TOWNES, 1966, Phys. Rev. Lett. 16, 347. GIULIANO, C. R. and J. H. MARBURGER, 1971, Phys. Rev. Lett. 27, 905. GRISCHKOWSKY, D., 1970, Phys. Rev. Lett. 24, 866. GUSTAFSON, T. K., P. L. KELLEY, R. Y. CHIAOand R. G. BREWER, 1968, Appl. Phys. Lett. 12, 165.

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REFERENCES

51

GUSYAFSON, T. K., J-P. E. TARAN, H. A. HAUS,J. R. LIFSITZand P. L. KELLEY, 1969, phys. Rev. 177, 306. GUSTAFSON, T. K. and C. H. TOWNES, 1972, Phys. Rev. A6, 1659. G. and G. MAYER,1965, Compt. Rend. 261, 4014. HAUCHECORNE, HAUS,H. A., 1966, Appl. Phys. Lett. 8, 128. R. W., 1966, Phys. Rev. 152, 156. HELLWARTH, HELLWARTH, R. W., 1967, Phys. Rev. 163,205. HELLWARTH, R. W., 1970, J. Chem. Phys. 52,2128. 1971, Phys. Rev. A4,2342. HELLWARTH, R. W., A. OWYOUNG and N. GEORGE, HERMAN, R. N., 1967, Phys. Rev. 164,200. IVANOV, E. N., 1964, Sov. Phys. JETP 18, 1041. 1966, IEEE J. Quantum Electr. QE-2, 470. JAVAN,A. and P. L. KELLEY, JOENK, R. J. and R. LANDAUER, 1967, Phys. Lett. 24A, 228. KELLEY, P., 1965, Phys. Rev. Lett. 15, 1005. S., 1960, Acta Phys. Polon. XIX, 149. KIELICH, KOROBKIN, V. V. and R. V. SEROV,1967, Zh.ETF Pis. Red. 6, 642; JETP Lett. 6, 135. KOROBKIN, V. V., A. M. PROKHOROV, R. V. SEROV and M. YA. SUCHELEV, 1970,Zh.ETF Pis. Red. 11, 153; JETP Lett. 11, 94. LALLEMAND, P. and N. BLOEMBERGEN, 1965, Phys. Rev. Lett. 15, 1010. N. I., A. A. MANENKOV and A. M. PROKHOROV, 1970,Zh.ETF Pis. Red. 11,444. LIPATOV, LITVAK,A. G., 1966,Zh.ETF Pis. Red. 4, 341; JETP Lett. 4,230. LOY,M. M. and Y. R. SHEN,1969, Phys. Rev. Lett. 22,994. LOY,M. M. and Y. R. SHEN,1970, Phys. Rev. Lett. 25, 1333. LOY,M. M. and Y. R. SHEN,1973, IEEE J. Quantum Electr. QE-9, 409. V. N. and A. M. PROKHOROV, 1968,Zh.ETF Pis. Red. 7,153; JETP Lett. 7, 117. LUGOVOI, MAIER,M., G. WENDLand W. KAISER,1970, Phys. Rev. Lett. 24, 352. and C. M. SAVAGE, 1964, Phys. Rev. Lett. 12, 507. MAKER,P. D., R. W. TERHUNE MAKER, P. D. and R. W. TERHUNE, 1965, Phys. Rev. 137, 801. J. H. and W. G. WAGNER, 1967, IEEE J. Quantum Electr. QE-3, 415. MARBURGER, J. H. and E. L.DAWES,1968, Phys. Rev. Lett. 21, 556. MARBURGER, MARBURGER, J. H., L. HUFF,J. D. REICHERT and W. G. WAGNER, 1969, Phys. Rev. 184,255. MCALLISTER, G. L., J. H. MARBURGER and L. G. DESHAZER, 1968, Phys. Rev. Lett. 21,1648. and R. Y. CHIAO,1970, Phys. Lett. 32A, 82. MCTAGUE, J. P., C. H. LIN,T. K. GUSTAFSON 1971, Phys. Rev. A3, 1376. MCTAGUE, J. P. and G. BIRNBAUM, 1972, Phys. Rev. B5, 628. OWYOUNG, A., R. W. HELLWARTH and H. GEORGE, PIEKARA, A. H., 1966, IEEE J. Quantum Electr. QE-2, 249. PIEKARA, A. H., 1968, Appl. Phys. Lett. 13, 225. N. F. and A. R. RUSTAMOV, 1965, Zh.ETF Pis. Red. 2, 88; JETP Lett. 2, 55. PILIPETSKII, PINNOW,D. A., S. J. CANDAU and T. A. LITOVITZ, 1968, J. Chem. Phys. 49, 347. 1969, Phys. Rev. Lett. 23,690. POLLONI, R., C. A. SACCHI and 0. SVELTO, RAHN,0. and M. MAIER,1972, Phys. Rev. Lett. 29, 558. SHEN,Y. and Y. SHAHAM, 1965, Phys. Rev. Lett. 15, 1008. SHEN,Y., 1966, Phys. Lett. 20, 378. SHEN,Y.R. and M. M. LOY,1971, Phys. Rev. 3A, 2099. F., 1967, Phys. Rev. Lett. 19, 1097. SHIMIZU, SHIMIZU,F. and E. COURTENS, 1973, in: Fundamental and Applied Laser Physics, Proc. of the Esfahan Symp. (John Wiley, New York) pp. 67-79. STARUNOV, V. S., 1965, Opt. i. Spektroskopiya 18, 300; Opt. Spectr. (USSR) 18, 165. TALANOV, V. I., 1964, Izv. Vysshikh Uchebn. Zavedenii, Radiofiz. 7, 564. V. I., 1965, Zh.ETP Pis. Red. 2, 218; JETP Lett. 2, 138. TALANOV, TALANOV, V. I., 1970, Zh.ETP Pis. Red. 11, 303; JETP Lett. 11, 199. THIBEAU, M., B. OKSENGORN and B. VODAR,1968, J. Phys. (Paris) 29, 287. WANG,C. C., 1966a, Phys. Rev. Lett. 16, 344. WANG,C. C., 196613, Phys. Rev. 152, 149. 1972, Phys. Rev. Lett. 29, 907. YABLONOVITCH, E. and N. BLOEMBERGEN

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E. WOLF, PROGRESS IN OPTICS X1I @ NORTH-HOLLAND 1974

SELF-INDUCED TRANSPARENCY BY

R . E. SLUSHER Bell Telephone Laboratories, Murray Hill, N . J . 07974, U S A

CONTENTS

Q 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . .

55

Q 2. COHERENT OPTICAL PULSE IN A TWO-LEVEL ABSORBER . . . . . . . . . . . . . . . . . . . . . . .

58

Q 3 . EXPERIMENTS . . . . . . . . . . . . . . . . . . . . .

76

Q 4. VARIATIONS OF SIT . . . . . . . . . . . . . . . . . .

86

REFERENCES

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

99

6 1. Introduction Lasers now produce UV, visible and infrared electromagnetic waves In the coherent and controllable form previously attainable only at radio and microwave frequencies. Laser radiation can now be stabilized in frequency to a few parts in 10' even without locking to an atomic or molecular line. Dye lasers are making tuning over wide ranges possible. A laser oscillator can be pulsed or a CW laser output can be gated to form short (pulse lengths, rp > lo-'' sec) intense bursts of coherent radiation. Self-induced transparency (SIT) phenomena originated from the study (MCCALLand HAHN[1967]) of the propagation of these intense coherent laser pulses in a resonantly absorbing medium. SIT phenomena are quite spectacular and include the following: (a) A highly absorbing medium suddenly becomes completely transparent as the intensity of an incident optical pulse rises above a well defined threshold ; (b) The pulse velocity in the lossless propagation region can be many orders of magnitude slower than the velocity of light, c; (c) Above the transparency threshold-intensity the pulse shape is drastically modified, and the pulse can break into a series of pulses. These dramatic effects are a result of the coherent coupling of the electromagnetic wave fields to the wave functions of the absorbing system. Any change in the state of the absorber that is incoherent with the optical pulse will degrade the SIT phenomena listed above. Common incoherent processes which place an upper limit on rP for SIT are spontaneous radiation from the excited state of the absorber, collisions in gases, and phonon or defect coupling to absorbers in a solid. In addition to this basic limitation in pulse length a number of other conditions must be fulfilled by both optical pulse and absorber which will be discussed below (see Table 3.1). This review will first summarize the basic theory of SIT using a plane wave incident optical pulse with uniform intensity. The absorber discussed here will be a simple two level absorbzr, e.g., the s to p transition of an Rb atom 55

56

SELF I N D U C E D T R A N S P A R E N C Y

“I, §

I

shown in Fig. 1.1. The propagation of a coherent optical pulse at resonance with such a simple absorber is described as a self consistent solution of STAT E

ENERGY

Q

--hw0 2

5s-

__-__

+=

lo>+

VECTOR MODEL

/b>

a-

ICI=lb>

Fig. 1 . l . Energy levels, eigenstates and vector model for SIT using a two-level Rb atom for an absorber.

Maxwell’s and Schroedinger’s equations as shown in Fig. 1.2. The semiclassical treatment of the radiation field is adequate for SIT because the optical pulse contains many photons for the intensities and beam areas used in most experiments. The transparency threshold is attained when the incident optical pulse becomes intense enough to cause the state of the absorber to change coherently from the ground state, Ib), to the excited state, la), and back to the ground state. Radiation absorbed during the first half of the pulse (absorber state changing from Jb) to la)) is re-emitted in phase with the last half of the pulse as the absorber state changes back to Jb). The process occurs entirely coherently in contrast to saturation bleaching of an absorber which can be treated by rate equations for the populations of ground and excited states since the absorption rate is of the same order of magnitude as incoherent relaxation from excited to ground states. The coherent SIT process is highly nonlinear compared to the linear absorption region where only very small admixtures of the upper state are excited. In the linear absorp-

11,

§ 11

57

INTRODUCTION

t

SCHROEOINGER EQ.

t

I

I

SPECTRAL AVERAGE I

-co

t

M A X W E L L ‘ S EP. I

1

1

t SELF-CONSISTENT E ’ = E FIELD

Fig. 1.2. Schematic of self-consistent solution describing coupling of radiation field to resonant absorber, Cycle of this type is used for computer simulations of SIT.

tion region Beer’s law applies and the intensity of the radiation decreases exponentially with distance, z, into the absorber, i.e.,

I ( z ) = I(o)e-”

(1.1)

where I is the intensity of radiation, and cc is the absorption coefficient. When a number of atoms within a cubic wavelength of resonant radiation are in the same coherent superposition of ground and excited states, the state of the ensemble is “super-radiant”. This coherent state was first discussed by DICKE [1954] and leads to radiation from the system at a rate proportional to the square of the number of atoms in a coherence volume as a result of the radiated fields adding in phase. This state for optical transitions was demonstrated in photon echo experiments (KURNIT,ABELLAand HARTMANN [1964], also for a review see ABELLA[1968]). Photon echoes arise when a simple “thin” (clL > l), L is much greater than the wavelength of the exciting radiation, and the coherent states of the absorber are excited by a single propagating coherent pulse of radiation at the resonant frequency of the absorber. SIT and echo phenomena occur simultaneously in thick absorbers excited by multiple pulses. Other aspects of super-radiance phenomena are discussed in a review article by ARECCHI, [1969], and in recent articles by REHLER and MASSERINI and SCHWENDIMANN EBERLY[1972], and EBERLY[1972]. The third section of this review summarizes the experimental SIT results. Several laser-absorber systems (e.g., ruby laser-ruby absorber, CO, laser-SF, or NH,D absorber, Hg laser-Rb absorber) have been used to verify theoretical predictions. Although the SIT phenomena have not found many applications to date, the availability of dye lasers which can be tuned to many atomic resonances may increase the use of SIT phenomena in shaping short pulses in both time and frequency. In principle, SIT could be used as a delay, or pulse shaping component of integrated optical circuits. Several of the possible applications involve effects of focussing the radiation in the absorber, off resonance effects, degeneracy effects, and especially for picosecond pulses an extension of the theory to include systems with many absorbing levels and frequency modulations of the pulse. Recent progress on these topics is summarized in the fourth section of this review. Primary emphasis in this review is placed on the interpretation of experiments and experimental problems and possibilities. Several important theoretical topics such as the stability of analytical SIT solutions are omitted here. Several other excellent reviews of self-induced transparency which emphasize theoretical aspects are now available (LAMB[1971] and a review that includes amplifying media and experiments by KRYUKOV and [ 19701). LETOKHOV § 2. Coherent Optical Pulse in a Two-Level Absorber 2.1. COHERENT PULSE A N D ABSORBER

Self-induced transparency in its simplest form requires several conditions of both absorber and excitation pulse. Although these conditions are often difficult to obtain in realistic experiments, the theory of the simple case offers great insight into the problem and can be extended to more complex experiments (see 8 4). The absorber for the simplest SIT analysis has a single pair of nondegenerate levels. It is assumed for simplicity (SIT is

11,

B 21

C O H E R E N T PULSE I N A T W O - L E V E L A B S O R B E R

59

basically unchanged for moderate thermal excitation) that the thermal excitation from ground to excited state is very small, i.e., kT 1. If eq. (2.2) is not satisfied large back scattering is expected. Recent work on the region where eq. (2.3) does not hold is discussed in section 4.4. The phase, $(z, t), is assumed to depend on both time and position; however, the time dependence will be dropped to obtain the simplest SIT results. The time dependence of $ becomes important if the incident pulse is chirped, for off-resonance cases (0:a,), or if the assumption in eq. (2.3) is relaxed. The absorber is assumed to be unifornily distributed in space at a random set of positions with density N per cm3. The conditions that the incident wave is uniform and plane, that c t l > T:) pulses or a CW beam. For z, < T fthe linear absorption is less than that in eq. (2.45) and the effectivelinear absorption coefficient is then of the order of zp/Tf smaller than for the broadline case (7, > Tf), but the area theorem given by eqs. (2.45) and (2.46) is stillvalid as long as T f 0) is considered here.

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[II,

A

37T

277

77

$2

!.O

0.8

> W

LT

w

z w

5

0.6

a

z_ \

> W

IL

0.4 W

t3

a

t 3

0

0.2

0

L

1

0.2

I

I

I

I

I

I

0.6 0.8 0.4 T , INPUT ENERGY (ARB. U N I T S 1

Fig. 2.3. Computer solutions for the output-input energy ratio for SIT pulses as a function of input energy and area; (a) assumes uniform plane wave and no losses, (b) Gaussian spatial profile plane wave with no losses, (c) experimental input pulse shapes, TI and T i losses for a Rb vapor absorber described in 5 3 and uniform plane wave. All curves assume aL = 5 . Points on (c) are the only computed points; (a) and (b) used closely spaced input areas because of simplicity of computer program.

2.5. SHARP-LINE ABSORBER AND THE PULSE SHAPE

The area theorem and computer solutions in the previous section lead one to search for lossless propagating pulse shapes for a 2n excitation pulse. An analytical solution for a lossless 27c pulse can be obtained by considering the sharp-line limit (7, T I , Ti) or inhomogeneously (TT < TI,Ti) broadened. For the sharp-line homogeneous absorber the derivation (eqs. (2.40) to (2.45)) of the area theorem is not possible because of the incoherent relaxation terms in the Bloch equations, ti = vAw-u~T,’, i, =

-uA~-K~&w~~-v~T,’

(2.52) (2.53)

and

FV = v&’w-(W-

Wo)/Tl.

(2.54)

In this case there is no time between T and t > To where simple solutions exist for u ( A o , z, t)(compared to eq. (2.41)) and dA(z)/dz must depend on TI and T i . Although an area theorem does not insure evolution to a stable pulse for the homogeneous absorber, the simplest analytic solutions can be found for SIT in the sharp-line limit for either inhomogeneous or homogeneous broadening. An analytic solution for a propagating 2n pulse can be obtained by assuming TI = T; = T ; +. co and an incident pulse in exact resonance with the sharp-line absorber (Am = 0). Then eqs. (2.53), (2.54) and (2.26) yield u(0, z, t ) = Np sin q(z, t )

(2.55)

and W ( 0 ,z, t )

=

w,cos q(z, t ) .

(2.56)

Assuming a traveling wave solution with area of 2n and using g(Aw) = 6(0) for the sharp-line limit, eqs. (2.39) and (2.55) give

(2.57) where V is the velocity of the assumed propagating pulse. Then using eqs. (2.26) and (2.57) one obtains &(z, f) = (2/rcz) sin &p(z, t ) ,

(2.58)

and a2q/at2 = (l/z2) sin q(z, t).

(2.59)

The propagation velocity is found from 1/ v = 27cKONpz2/qc+ q/c.

(2.60)

The requirement of finite pulse energy leads to the unique 27( hyperbolic

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SELF I N D U C E D TRANSPARENCY

IIL62

secant solution obtained by MCCALLand HAHN[1967], LAMB[1967] and MCCALLand HAHN [1969], (2.61)

lcz

Eq. (2.59) is the same as the equation which governs the motion of a pendulum where tp is its angular position. As will be shown in section 4.2, eq. (2.61) is also a solution off resonance for a sharp-line absorber and thus also is a solution for a broad-line inhomogeneous absorber although the uniqueness of the solution is not shown for these cases. Computer solutions in Fig, 2.2 show a 1.1.n pulse evolving to this hyperbolic secant form. 2.6. PULSE DELAYS

The propagation velocity for the 2.n hyperbolic secant (27ch.s.) pulse can be much less than the velocity of light in vacuum. For a sharp-line one obtains the inverse velocity or delay time per centimeter from eqs. (2.60) and (2.461, 1/ V , = &eff

7

+V / C ,

(2.62)

where aeffis an effective absorption constant which is less than the a for a CW beam because the Fourier transform of the pulse is broader than the absorber spectral width. For the homogeneous sharp-line case (2.63) where for the homogeneous line width, g H ( A 4 = G/{.nC1+

( W T m

(2.64)

and a(0) is the absorption constant measured for a weak (linear absorption limit) CW optical beam at exact resonance. The sharp-line inhomogeneous case gives an effective absorption, Eeff

= az/T,*,

(2.65)

where the exact expression depends on the spectral shape of the inhomogeneous broadening. For a homogeneous atomic absorber (atomic beam) with CI = 100 cm-’, z = 5 nsec, Ti = 50 nsec, and q = 1, the first term in eq. (2.62) dominants and V, = c/750. This very slow propagation velocity expressed in spatial 1.5 m long in free space forms an terms means the 5 nsec-pulse which is excitation region 2mm long in the absorber. As shown in Fig. 2.4, a

-

-

II,O

21

COHERENT PULSE I N A TWO-LEVEL ABSORBER

71

nl

n

Fig. 2.4. Pictorial description of one rotation of the macroscopic pseudopolarization vector P by a 222 hyperbolic secant pulse as the pulse is transmitted through an ctL = 1 absorber. Pulse is delayed one pulse length for each absorption length as the pulse is coherently absorbed and re-emitted.

pulse is delayed one pulse length for every effective absorption length since for the 2n pulse it requires the duration of the pulse to coherently excite the absorber and return it to its original ground state. The spatial extent of the excitation region in the absorber along the direction of propagation is always of the order of rxsj if the pulse velocity is much less than c. A 2n h.s. pulse is also a lossless solution for the inhomogeneous broadline absorber and a delay time per centimeter, l/VB = +az+q/c

(2.66)

is obtained. The absorption constant, a, in this broad-line limit is the same as for a weak CW beam since all Fourier components of the pulse experience the same absorption. Expressed in terms of a time delay in the absorber eq. (2.66) or eq. (2.62) gives 7D = taeff7L (2.67)

12

SELF I N D U C E D T R A N S P A R E N C Y

[II,

52

in the limit that V T I , T;). Initial portions of an incoherent pulse can equalize the ground and excited state populations allowing the latter portions of the pulse to be transmitted withrelatively less attenuation. This incoherent bleaching process can also give an effectivemaximum in delay as a function of input intensity. The important distinction between SIT delays and those in incoherently saturated absorbers described by rate equations is that for SIT the delayed pulse intensity is greater than the intensity in the latter portions of the initial pulse. Experimentally generated pulses often have extended “tails” and one can be certain of coherent SIT effects only if the delayed output pulse has an intensity greater than the input pulse tail. Dispersion effects can also cause pulse delays if the incident pulse is not at exact resonance with the absorber. SZOKEand COURTENS [1968] show that SIT and linear dispersion theory merge if the exciting pulse is far from resonance and apparent delays result in both cases. Pulse reshaping and apparent delays may also result from “hole burning” where a portion of the spectrum of an inhomogeneously broadened absorber is saturated. In summary, pulse dealys alone are often not a crucial experimental test of coherent SIT effects. Pulse delays are associated with other interesting phenomena. For example, COURTENS [1968] has shown that large Faraday rotations of linearly polarized light incident on a Zeeman split absorber should be excepted. In this case off-resonant phase differencesor different delays for the two components of circular polarizations in a linearly polarized input beam can cause pulse-breakup and drastic changes in the polarization of the output pulse. 2.7. INCOHERENT RELAXATION A N D THE CRITICAL LENGTH

Incoherent relaxation effects inevitably limit both the possible pulse length for SIT and the length of absorber through which a 27~h.s. pulse can propagate without appreciable loss. Any transition with an electric dipole

11,

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COHERENT P U L S E I N A T W O - L E V E L ABSORBER

73

moment relaxes, by spontaneously emitting a photon, from the excited state at a rate 1/Tab = pi/(3hC3). (2.68) The spontaneous relaxation time is typically of the order of 30 nsec for alkali atoms ( p o w 5D, oox 5 x lof5), and 0.3 sec for vibrational transitions in molecules ( p o x 0.05D,oox 5 x lox4). This spontaneous relaxation contributes to both Tl and T i , i.e.,

(see SLUSHER and GIBBS[1972], Appendix B). Although spontaneous relaxation is a fundamental limit, collisions, interaction with other systems (e.g. phonons in a solid or collisions with a buffer gas), and relaxation and coupling to other levels are often important limiting effects in experimental situations. By considering the energy loss due to incoherent relaxation, it is possible to determine how many absorption lengths a 2n h.s. pulse can traverse before being absorbed. It is assumed that the incoherent processes are such as to maintain a constant number of absorbers in the two level system. For the sharp-line case (both homogeneous and inhomogeneous) it has been shown by MCCALLand HAHN [1959] (section IV) that to first order in TITs for a 2.n h.s. input pulse, dF,/dz

=

-4Nho~/3T,,

(2.70)

where the pulse energy Fs is given by eq. (2.48) and

1/T, = 2 / T , + l / T . ,

(2.71)

where Tl and TZ)are the effective relaxation times in eqs. (2.52) to (2.54) for the W , u and u components of the pseudopolarization. For a 2n h.s. pulse 27 = 2c/(nnlC%)

(2.72)

dF,/dz = - 8Nhoc/(3n~’T,FJ.

(2.73)

and eq. (2.70) becomes

As F decreases because of incoherent losses, z must increase in order to maintain the 2n h.s. condition, and for the sharp-line case this means an increase in tbe effective absorption coefficient. For the broad-line inhomo-

14

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62

(2.74) (2.75) In both sharp line (homogeneous and inhomogeneous) and broad-line absorbers a 2n pulse loses energy and z increases. PIERCE and HAHN[1972] have introduced a “critical length”, Z , , at which z has increased to become equal to Tq.For z > 2, the pulse will lose energy much more rapidly because incoherent decay processes dominate for z 2 T,. Intergrating eq. (2.74) and setting z(ZJ = T, and ~ ( 0=) zo, one obtains 2, = 3~,/(2a,,,7,)

(2.76)

for a sharpline absorber and similarly 2, = 3 ~ , 1 ( 4

(2.77)

for a broad inhomogeneous absorber. Since as z increases the propagation velocity decreases, the decaying excitation region in the absorber will slow down while remaining the same width ( E a-1 in the z direction) until it finally decays by linear absorption. Computer calculations of the relaxation effects on pulse shapes are shown in Fig. 2.5 for the broad-line case. Note that the effective delay time increases when losses are added. Similar results are obtained in the sharp-line limit except that the evolution toward the stable pulses is highly dependent on the input pulse shape. Evolution of the pulse energy and area are shown in Fig. 2.6 as a function of az. These computer results are in good agreement with the estimates in eqs. (2.76) and (2.77). As the pulse length approaches TI and T; and the pulse area decreases to the region near n the effective delay time must begin to decrease since in the incoherent linear limit the propagation velocity will approach c/v. This incoherent relaxation effect on pulse delay can be seen in computer calculations by HOPFand SCULLY[1970a] which include only T;processes. If the excited state relaxes to a level other than the ground state the effective density of absorbers will vary. This effect can be included in computer calculations by adding a fourth differential equation coupled to eqs. (2.52) to (2.54)(see SLUSHER and GIBBS[19721). Off-resonance transitions also can influence pulse propagation through terms in Maxwell’s equations that are dropped from eqs. (2.33) and (2.34). SLUSHER and GIBBS[1972] discuss some of these effects for alkali atoms.

11,

5 21

COHERENT PULSE IN A TWO-LEVEL ABSORBER

75

TIME (nsecl

Fig. 2.5. Evolution of a 2 n pulse in an inhomogeneously broadened absorber with experimental input shape (SLUSHERand GIBES[1972]). Computer solutions show evolution to 2n hyperbolic secant pulse (a) with no incoherent decay processes and to a broaded and attenuated pulse (b) with the incoherent decay times TIand Ti appropriate for Rb vapor absorber. Parameters are: (b) TI = 33.6nsec, T i = 56nsec (a) T I = 10 000 nsec, Ti = 20 000 nsec aL EolEr Area UL EolEr Area 6.28 0 I .O 6.28 0 1.o 6.24 2.5 0.928 5.89 2.5 0.74 5.52 5.0 0.54 6.25 5.0 0.922 0.921 4.85 7.5 0.34 6.26 7.5 3.14 10.0 0.14 6.21 12.5 0.920 0.12 12.5 0.02 aL is labelled on each pulse. Energy ratio, Eo/E, decreases in (a) because of reshaping losses as 2 n h.s. is formed.

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ABSORPTION

[II,

c, 3

LENGTH ( a z )

Fig. 2.6. Decrease of pulse area and energy as a function of the effective absorption depth, uz, in a Rb vapor absorber with incoherent decay, TI = 33.6 nsec and T ; = 56 nsec. Solid curves are for the inhomogeneous broad-line absorber (pulse shapes in Fig. 2.5). Dashed curves are for an inhomogeneous sharp-line absorber ( T : z 12 nsec > t,,z 3 nsec) where z t u ( 0 ) (see GIBBSand SLUSHER[1972]). Arrows indicate the estimated critical lengths described in the text.

8 3. Experiments 3.1. IDEAL CONDITIONS

Theoretical assumptions necessary to analyze self-induced transparency severely restrict the parameters of the optical pulse and absorber used for an experimentaltest of the predicted phenomena. These restrictions are listed in Table 3.1 along with a list of how they were satisfied for a Hg laser -Rb absorber system. Several other laser absorber systems are discussed below but the most complete experimental check of the simplest form of SIT to date used the Hg-Rb system. The optical pulse must be frequency and phase stable so that the c$ = 0 assumption (see section 2.3) is satisfied. This requires that d, be at least less than l/z, or $ < 100 MHz for zp w 10 nsec. This stability is usually achieved in a single cavity mode of a laser if the gain and effective background index do not vary. Recently these stability requirements have been met by tunable dye lasers which will greatly expand the possible absorbers for SIT. A dye laser can also be locked to an atomic vapor or beam resonance to avoid cavity drift problems.

11,s

31

77

EXPERIMENTS

TABLE 3.1 Ideal SIT experimental parameters Ideal

A. Optical pulse 1. Frequency and phase stability (Av 2. Spatial homogeneity is important on a volume scale of order la-’. In order to exclude statistical variations, the total number of absorbing atoms or molecules per absorption volume (a-’ times the beam area) should be greater than lo4. For example, in the Hg-Rb system with a uniform beam diameter of 50 p and a 100 cm-’, the total number of atoms is approximately lo4. Degeneracy of the ground or excited state is often a complication since various excited transitions have different dipole moments and are excited at different rates. These effects are often encountered in alkali atoms and moleccular transitions because of nuclear spin and rotational quantum numbers Iarger than one (see section 4.3). Diffraction effects are quite difficult to include in the theory. In order to avoid them the diameter of the input beam, d, should satisfy ud’/A > 1. Near an input area (fc~!?~z~) of 7c for the central beam, diffraction may be particularly important since the sides of the beam at lower intensity will be absorbed and stripped away.

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3

3.2. LASER-ABSORBER SYSTEMS

Self-induced transparency experiments in the optical region to date have involved primarily (1) a laser in the visible and an atomic or impurity ion transition, and (2) lasers near l o p and vibrational-rotational transitions of simple molecules. SIT effects have also been observed for phonons resonant with paramagnetic centers by SHIREN[1965 and 19701. Since a highly occupied ground state is needed for large a in SIT, most laser transitions do not make good absorbers because for lasing transitions the ground state needs to be easily depleted to maintain population inversion. Coincidences between the thousands of strong atomic absorption lines and fixed laser lines in the visible are quite rare. A CW dye laser and using rhodiamine 6G dye and the simple sodium D, line transition now seems a feasible SIT system. Ruby was the first SIT system used for both absorber and incident laser pulse by MCCALLand HAHN[1967]. The ruby laser operated at liquid nitrogen and the passive absorbing ruby was at liquid helium temperature. This temperature difference allows a frtquency match between the E(2A) c, 4A2(f$) lasing levels and the 4A2(++) c-f E(2E) absorbing levels (see MCCALLand HAHN[1969], fig. 10) of the C r f 3 ion. Incoherent relaxation, by phonons from the E(2A) to the 2A(2E) level, limits the pulse times to zp 1) must be described by SIT and the area theorem.

11,

§ 31

EXPERIMENTS

79

The next experimental results in SIT involved molecular transitions at resonance with lop laser transitions. SIT phenomena where observed using a c02laser and a vibrational rotational transition of SF, by PATELand SLUSHER[1967] and PATEL [1970]. The excited transition in SF, lies in a very complicated and uninterpreted region of lines (see HINKLEY [1970]). The question was raised by RHODES,SZOKEand JAVAN [1968] as to why the rotational degeneracies should not modify the SIT phenomena for SF6 and they showed that many interesting variations of SIT are expected for the degenerate case (see section 4.3). Several COz laser lines are absorbed by SF6 and RHODES and SZOKE[1969] have observed effects on some lines that behave as if they were highly degenerate. Molecular transitions involving an angular momentum quantum number, J, for the ground state and J' for the excited state have the unique dipole moment required in the analysis of 0 2 only for J and J' less than 3 (see section 4.3). Several near coincidences between such simple transitions and strong fixed laser lines near lop have been found, so that Stark tuning can be used to tune to exact resonance. BREWER,KELLEYand JAVAN[I9691 have Stark tuned the v2 vibration [u2 = 0 t, 1, 404(a) -+ 5,,(s)] of Ni4H2D to the P(2O)CO2 laser line at 944.191 cm-'. BREWER and SHOEMAKER [1971] have used a Ci3H3F, v3 band transition (J, K = 4, 3 3 5, 3) which overlaps the P(32) C 0 2 laser line at 1035.474 cm-'. Using this C13H3F absorber and a Stark shift to bring it to exact resonance they have observed photon echoes and optical nutation (see HOCKER and TANG[1968]) in the aL < 1 limit. Tunable high power spin-flip Raman lasers are also avialable in this frequency range (see PATEL and SHAW[1971]). Dye lasers have been the great hope for coherent resonant experiments because of their tunability. Preliminary SIT results were obtained for potassium vapor by BRADLEY, GALEand SMITH[1970] but a well controlled SIT experiment using a dye laser has not yet been accomplished. At present, a prism or grating in an Ar laser pumped CW dye laser cavity yields a lasing line width from 0.1 to 111. An etalon in the cavity can reduce this width to 5 0.001 A or less than 100 MHz, i.e., near the line width required to do an interpretable SIT experiment with a pulse length less than 10 nsec. Similar line widths have been achieved with pulse excited dye lasers. CW dye lasers have been mode-locked by IPPEN,SHANK and DIENES[1972] to achieve pulse lengths as short as 1.5 psec which are tunable from 5900A-6100A. ARTHURS,BRADLEYand RODDIE[1972] have mode-locked pulsed dye lasers to obtain 3 psec pulses. These short mode-locked pulses require peak intensities of the order of MW/cm2 to achieve SIT threshold for alkali atoms which seems presently attainable by focusing the incident pulse. Chirping

80

SELF I N D U C E D T R A N S P A R E N C Y

[II,

B3

may be appreciable (c$ 2 l / ~for ~ these ) pulses which may cause interesting SIT modifications (see section 4.4). SIT experiments may provide useful methods for shaping (see section 4.1) or measuring the properties of these ultrashort pulses. The SIT theory for this range of pulse lengths is complicated by the importance of both Maxwell equations (eqs. (2.33) and (2.34)) and the weakness of the slowly varying approximations. 3.3. Hg LASER-Rb ABSORBER SYSTEM

As an example of many of the predicted SIT effects, the results obtained by GIBBS and SLUSHER [1970,1972] using a pulsed "'Hg ion laser and a R b vapor ( N 10" atoms/cm3) absorber will be described. All the requirements for comparison with the theory listed in Table 3.1 are satisfied by this system. The experimental apparatus is shown in Fig. 3.1. A heated tube containing "'Hg is positioned in a cavity formed by mirrors MI and Mz ,and pulsed N

I

I

LASER OUTPUT

I I I

POCKELS CELL OUTPUT

Ir

"!jet

I

I

t \

I

Fig. 3.1. Schematic diagram of Hg-Rb SIT apparatus (see SLUSHER and GIEBS119721). MI is a 3-m totally reflecting mirror; V, is a 1-psec voltage pulse causing I-psec singlemode laser pulse; A, is the aperture to select TEMoo mode; SMS is single-longitudinalmode selector; V, is the modetuning voltage across the piezoelectric transducer; M2 is a 4 % transmission flat output mirror; PI is the Gaussian transverse intensity profile; PC is the Pockels-cell gating 5-10 nsec portion of the laser pulse; LP and QW are linear and circular polarizers; S is a superconducting solenoid; L1, Lz and L3 are imaging lenses; E is the magnetic field (= 74.5 kG); C is the Rb vapor cell; P1 is the stripped Gaussian profile after SIT interactions in the Rb cell; Aa is the limiting aperture used to observe a uniform transverse intensity; and D is an avalanche photodiode or cross-field photomultiplier detector.

11,

B 31

81

EXPERIMENTS

to give a 1 psec laser burst at 7944.66 A. An aperture Al limits the transverse mode to the Gaussian profile TEMoo mode. A thin metal film (-50 A) mounted on a piezoelectric translator in the cavity (SMS) restricts the laser to a single longitudinal mode. The output from this laser was frequency stable (drift rate was approximately 50 MHz per hour) when care was taken to thermally and mechanically isolate the laser cavity. A 7 nsec portion of'the laser pulse was gated by the Pockels cell (PC) and a quarter wave plate selected the sense of circular polarization absorbed by the Rb. A uniform region of the pulse at the center of the Gaussian beam was selected by aperture A2 in order to satisfy the requirement A2 in Table 3.1. A fast avalanche photodiode (D) time resolved the transmitted pulse shapes. The Rb levels excited are shown in Fig. 3.2. A 74.5 kOe magnetic field tuned the transition to resonance with the Hg laser. This Zeeman splitting

5P

2F;/2"Rb ABSORPTION AT 14.5 kG COINCIDES WITH THE 202Hg LASER

-a

i'

3 . l

0

m r-

214 GHZ 3/2

5s

I

H

\ '-l/2

"Rb

ENERGY

LEVEL

3/2

DIAGRAM

Fig. 3.2. Diagram of the relevant energy levels of "Rb as a function of magnetic field strength. The Zeeman interaction at 74.5 kOe lifts the low-field degeneracy and increases the absorption frequency to coincide with the Hg laser emission frequency.

82

III,

SELF I N D U C E D T R A N S P A R E N C Y

B3

eliminated the degeneracy (encountered in all alkali atoms) caused by nuclear spin. The limiting spontaneous emission time between the two excited levels was T,,, = 42 nsec and a weaker 84 nsec decay time to a third level, c. SIT results have been obtained in both the inhomogeneous broad-line (TZ 0.8 nsec c zp 3 nsec C Ti 55 nsec, TI 33.6 nsec) and sharp-line (T, 2 nsec < T,* 10 nsec < T i , TI) limits. The broad-line limit is obtained by the Doppler width of the isotropic vapor and the sharp-line limit with an atomic Rb beam (GIBBSand SLUSHER [1972]). In the broad-line limit the ratio of output energy to input energy, shown in Fig. 3.3, exhibits the sharp rise predicted near the transparency threshold area of n. The experimental points (black dots) are in good agreement with computer calculations for the uniform plane wave case using the previously measured 4.35 D dipole moment for Rb. Note that the onset of transparency is slower when the A2 aperture does not limit the observed portion of the beam to the uniform central region. The experimental and computed

--

-

N

-

-

COMPUTER INPUT AREA

05

10

!

20

15

10

'

1

477

d'

INPUT ENERGY PER U N I T A R E A

23 6r

'

~e

(erg cm2)

Fig. 3.3. SIT nonlinear transmission in Rb vapor with aL = 5. Solid curve is a uniform plane-wave computer solution. Solid dots are data taken with 200-pm output aperture to approximate uniform plane wave. Triangles are data with no aperture corresponding to a plane-wave with Gaussian intensity profile. The pulse shapes for the circled points are shown in Fig. 3.4.

11,

B 31

83

EXPERIMENTS

pulse shape at points [a(a')] through [e(e')] are shown in Fig. 3.4. Computer solutions include T I = 33.6 nsec and Ti = 56 nsec and the small effect of the decay to the third level (see SLUSHER and GIBBS[I9721 for details of the computer solution). The 2n pulse is in the region near the experimental

... ,......,.......,.....,' __. "

0

5

10 15 TIME (nsec)

...a'..

.L.%%.I.. L.l

20

rr

TIME (nsec)

Fig. 3.4. Input and output pulse shapes for aL z 5 and uniform plane wave conditions. Sampling scope output and input pulses (obtained by rotating quarter wave plate through 90") are shown in (a) through (e) corresponding to the gun-barre1 points in Fig. 3.3. Computer generated output pulses are shown in (a') through (e') corresponding to the peacesymbol points of Fig. 3.3. Input areas for a' through e' are 6.28, 8.7, 10.5, 17.5 and 23. Many more experimental curves were taken as indicated in Fig. 3.3 showing the continuous transition from no delay or reshaping to a 6 n input pulse. For input intensities much less than that in (a) no delay was observed. The maximum delay of about 8 nsec occurred just below (a), which has an area of nearly 2n. Point (b) is just above 2n with low loss and slight peaking. This almost lossless propagation through an absorber with linear transmission of 0.7 % is in excellent agreement with the theory of SIT. Points (c), (d) and (e) show pulse breakup and peaking for inputs ofjust under 3 n and 5n and approximately 672.The areas given here are for a sech input pulse; the actual areas are larger because of the nonhyperbolic secant nature of the input, particularly the long tail. The slightly smaller experimental delays likely result from an actual experimental aL of slightly under the measured value of 5 . The change in aL required for complete agreement is well within the experimental uncertainty.

84

S E L F INDUCED T R A N S P A R E N C Y

[II,

53

point at (a) since an area of 27r was used for the theoretical curve labelled (a’). The measured 2x pulse delay is zD = 8 nsec. Using the measured aL x 5 and z x 3.5 nsec (T for a hyperbolic secant pulse is the FWHM pulse width divided by 1.76), +uLs = 8.5nsec is the theoretically predicted delay from eq. (2.67) so that the observed and predicted values of pulse delay are in good agreement. The increase in delay expected because of incoherent energy loss is small for the crL of 5 used here as seen from Fig. 2.5. The observed delay corresponds to a velocity in the absorber of 42400. For input areas greater than 271 a narrowing and peak amplification are observed near 371 (b and c in Fig. 3.4). This effect can be qualitatively understood using the argument shown in Fig. 3.5. The atoms are excited to the

lr

2n

t

31r

Suggests : 1 G r e a t e r energy loss f o r 3n t h a n 2n pulse 2 Peak amplification f o r 3n pulse

T

Fig. 3.5. Simple comparison of 2 z and 3n sech input pulses. Peaking occurs for the 3n case because the excited atoms are stimulated to reemit in the short time t , to t2, whenthe input is maximum. Since the output pulse has area 2n with a width not much longer than t2,-t,, its peak must be higher than the input peak which occurs in the t , to t Z ntime interval in which the input area is only n.

XI,

5 31

EXPERIMENTS

85

upper state during the initial portion of the pulse and forced to reemit back to the ground state during a shorter period by the strong fields near the peak of the 371 pulse. The last n of the pulse is absorbed leaving a narrowed 2n propagating pulse. The loss of the last n of pulse area explains the small energy output ratio decrease near 3n in Fig. 3.3. Input areas near 472 and 6n cause the pulse to break up into first two and then three separate 2n pulses. Again the agreement between theory and experiment is quite good considering that the experimental trace is an average of several thousand pulses. The shapes of the multiple pulses after break-up depend on the incident pulse shape. For the experimental situation shown in Fig. 3.4 the first 2n pulse after break-up is shorter and thus travels at a higher velocity than the second (see eq. (2.66)). This means the two 2n pulses formed after break-up of the 4n pulse would continue to separate if uL were increased. Incoherent relaxation is included in the computer calculations shown in Figs. 3.3 and 3.4. For Rb in the broad inhomogeneous limit the critical length is 15.2 ( D L Z= ) ~3~ ~ = 15.2 (3.1) 3 ~

from eqs. (2.77) and (2.75) and since uL = 5 for this experiment there are relatively small losses. However, the energy ratio at 2n in Fig. 3.3 does not reach unity primarily because of spontaneous relaxation. Losses to the third level (levelc in Fig. 3.2) were also included in the computations but only made small fractional changes in the results. The sharp-line inhomogeneous limit has also been studied for Rb vapor by using an atomic beam perpendicular to the incident optical pulse beam (GIBBSand SLUSHER [1972]). This configuration resulted in a reduced Doppler width of 15 to 30 MHz compared to the natural line width (TIand T;processes) of 6 MHz. The Fourier transform of the incident pulse was 120 MHz so that the inhomogeneous sharp-line limit was obtained (7, x 2 nsec < T,*x 8-15 nsec < T I = 33.6nsec, T; = 56 nsec). For an uL = 6 (measured by a CW laser) results similar to those in Fig. 3.4 were obtained for input areas from n to 6n. Pulse reshaping and break-up in this sharp-line limit are quite similar to broad-line limit except that the effective uL was reduced to 1.5 by the factor of z , / T t . ESTES, ETESON and NARDUCCI [1970] have shown by computer calculations that pulse reshaping in the homogeneous sharp-line limit is also expected to be similar the broad-line inhomogeneous limit.

86

SELF I N D U C E D T R A N S P A R E N C Y

0 4. Variations of SIT 4.1. FOCUSING

A number of interesting variations in SIT phenomena occur when the conditions other than the simple ones in Table 3.1 are obtained. First consider the effect of focusing the incident pulse into the absorber at resonance. In this way a spherical wave front is obtained in contrast to the plane wave assumption (A2 in Table 3.1). For the Hg-Rb experiments described in the previous section the absorber was at the focal region of the lens (L,in Fig. 3.1) where the wave fronts are plane. Now we consider the geometry shown in Fig. 4.1. For a uniform spherical wave Maxwell's equation bscomes

where r is the radial distance from the focal point. The € / r term accounts for the increase in intensity of the beam because of focusing. Note that since it is assumed that w is at exact resonance there is no refractive change in the propagation direction of the pulse at the incident or exit face of the absorber. It was suggested by MCCALLand HAHN[1969] that the increase in intensity due to focusing could be used to compensate for incoherent losses; thus, maintaining a 271 h.s. pulse at constant area and pulse width. In addition, a quite interesting and potentially useful effect discussed by MCCALL[19691 occurs because of focusing when the input pulse area is near 3n. As described in section 3.3 a plane wave 3n pulse experiences peak amplification, narrowing and an energy loss of ( n / 3 ~ w ) ~0.1. By focusing the input beam, the narrowing and peaking process can be continued throughout the absorber. The area lost by a 3n pulse in a reshaping length L, = R/a [ R is typically 2 to 5 depending on the input pulse shape];

(aA/ar),,, w - A / ~ L ,

(44

can be offset by a pulse area gain due to focusing; Fig. 4.1. Optical pulse compression by focusing. (a) Optical pulse compressor. L1 is a 13 mm minimum aberration single-element lens. The top and bottom of the absorption cell were about 6.6 and 1.6 mm above the focal point, respectively. The aperture A shown at the cell exit was a 25 pm aperture placed a t the image plane of an output magnifying telescope consisting of 2.5 cm and 31 cm lenses. (b) Comparison of experimental data with computer simulation. The solid and dashed curves are the experimental and theoretical output pulses with CLLw 25 as viewed through the 25 pm aperture. The dotted curve is the pulse detected through the aperture with crL % 0 (obtained simply by converting the input polarization from absorbable right circularly polarized to nonabsorbed left circularly polarized light). The input pulse area of 3 . 5 7 ~

87

V A R I A T I O N S OF S I T

I I I I

I

I

I1 II II

COMPUTER I+-OUTPUT

I1

1 I II II I1

I

I

II

I1 I1

'I

74.5 kOc

(01

I\

II 11 11

II

'I

II I1

II I1 EXF! OUTPUT

* INPUT

*..I..

0

5

10

"1 I IS I

.

- 0 .

PO

assumed in the uniform-profile computer calculation agrees well with the experimental calibration of the area by self-induced transparency without focusing. In the computer calculation the pulse intensity is assumed to increase by focusing as the inverse square of the distance to the theoretical focal point. The inclusion of diffraction, laser amplitude fluctuations, and time jitter would broaden and attenuate the output pulse.

88

SELF I N D U C E D TRANSPARENCY

(aA/ar)Focusx A/3LR or a&'/& x 6/3L,,

111,

§4

(4.3)

where A is the pulse area defined by eq. (2.27). The gain from the spherical wave formed by a simple lens is the &/r term in eq. (4.1). From eq. (4.2) the optimum pulse compression occurs when r is approximately constant and equal to 3L,. These conditions were closely approximated by GIBBSand SLUSHER [1971] in the experimental arrangement shown in Fig. 4.1. A 1.3cm focal-length lens focused a 0.8 mm-diameter laser beam with pulse area near 37c through a 87Rb absorption cell 5 mm long. The optical pulse was generated by Hg I1 laser shown in Fig. 3.1 and a 74.5 kOe field was used to tune to exact resonance. The output pulse in Fig. 4.1 has been compressed by nearly a factor of ten due to the continuous reshaping of the 37c pulse in the absorber. Reasonable agreement is obtained between the experiment and the computer simulation considering that the experiment is an average over several thousands of pulses. Pulse compression displayed in Fig. 4.1 was obtained in the broad-line limit. If this compression technique is to be applied to pulses inthepicosecond range, one must consider the sharp-line limit if an atomic absorber is used (z, : .... .. ....... ............... . . ............ .. I..*...

~

1 l'+#*-.

~

: , row of W is obtained by shifting the ithrow cyclically one place to the right. This means that a mask like the one drawn in Fig. 7b can be used with the spectrometer. This particular mask corresponds to a matrix of mask elements 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 w = 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1

i.e., a spectromodulator with seven exit slit positions, where 1 describes an open position, and 0 a closed position. Such a mask has two advantages. First, it can be self-supporting and therefore permits the construction of a spectrometer which requires no transmission materials. In operation the mask is stepped one slit width along the length of the mask - i.e., in the direction of dispersion - for each successive encoding position. More important, perhaps, cyclic codes enable us to avoid the tedious construction of N masks each having N slots, for a total of N 2 slots. Instead we construct only one mask with 2N- 1 slots.The costof maskconstruction is reduced by +N and the design of the advance mechanism is considerably simplified since the bulk weight of the masks also decreases as N $N. Codes of this type are equally useful when both the entrance and exit apertures of a spectromodulator are encoded.

-

5.2. ESTIMATION OF SPECTRAL SHAPE FOR SINGLY ENCODED INSTRUMENTS

SLOANE, FINE,PHrLLrPs and HARWIT[I9691 have developed the following theory: The optical spectrum whose shape is to be estimated is spatially dispersed and the band of interest partitioned into N channels. The average channel, after a selected integration time, is denoted by energy in the jth Y j . The measurement process consists of observing the spectrum through M masks, the energy thlough the ith mask being ~ y = l w i j Ywhere j Wi = (oil,. . ., w i N )is the ith vector of mask elements. The photodetector adds a random noise v i to the signal o i j Y j and yields a measurement

cy=l

128

M 0 V U L AT1 0 N T E C H NI Q U ES I N S P E C . l R O M E T R Y

[Ill,

!i5

N

qi = v i +

2 oijYj,

i = 1, 2 , . . ., M .

(10)

j= I

The noise v i has the following properties: ( vi) = 0; vi is independent of the signal; ( v : ) = a ' ; successive measurement noises are assumed to be uncorrelated ((vivj) = 0 if i # j ) , and ( ) represents an ensemble average. I n order to estimate { Y j } by { we need at least as many measurements M as there are unknowns N . Furthermore, at least N distinct masks { W i } are needed to estimate the spectral shape. Hence, assume vectors of observations q = ( q , , . . ., qM), channel energies Y = ( Y , , . . ., Y,), measurement noises v = ( v , , . . ., v w ) , and a matrix of masks W = (W:, . . ., W z ) = ( o j i )(The . T stands for transpose.) With this notation q = Y W + v . An estimate 3 of Y is a function of the observations, Y(q), expected to lie close to Y . As a measure of the accuracy of the estimate wc adopt the mean square 4 error criterion: We minimize E = ( ( Y - Y ) ( $ - ~1'). For purposes of computational convenience we agree to restrict Y to be a linear function of the observations, Y = q A , for some matrix A ; in the absence of more detailed statistical knowledge concerning the anticipated spectral shape or the photodetector noise characteristics, this seems appropriate. Before an essentially uniquely best experimental design can be derived, however, we still need to make the assumption that the estimator is unbiased; i.c., ( 3 ) = Y . This assumption can be defended on the following grounds: 1) we need an estimator which, on the average for a large number of applications, yields the true value; 2 ) the unbiased estimator can be shown to be desirable when there is a large unccrtainty in Y relative to our prior knowledge of the spectrum and the measurement noise power. We must now select those matrices A , W that minimize E subject to the constraint we have set. First we see that

sj>

h

4



=

< q A ) = (rt)A.

(11)

Furthermore, ( q ) = ( Y ) W + ( v ) = (Y)W.

(12)

Hence, unbiasedness requires that WA = I, the identity matrix. If M = N , Wis square, and A = W-'. If Wis not square but is N x M ( M > N ) , we can use the generalized inverse A = w'r( w wT)-

1.

Having solved for A in terms of W we now select W to minimize ((Y-Y)(!$-Y)T). W e note that

3-Y

=

YWA+vA-Y = vwT(WWT)-'.

(13) E

=

1 1 1 , ~51

H A D A M A R D TRANSFORM SPECTROMETERS

By assumption (v'v)

= 02Z.

c/o' =

129

Thus

Trace [W'(WWT>-'(WWT)-'W].

If M = N , this simplifies to c/o2 = Trace [W-'(W-l)T]. The optimum experimental design will be completed if we can find that W-subject to w i j = 0 or 1 or - 1 - that has the minimum c/oz. 5.3. CHOICE OF MASKS

We just showed that in the case M = N , when there are as many measurements as unknowns, the matrix W = (aij)of mask elements should be chosen so that for oij= 0 or 1 or -1, Tr [ W - ' ( W - ' ) T ] is as small as possible. Three possible choices for the matrix Ware given here. Appendix A gives some characteristics of Hadamard matrices needed to understand the following results: If H is an N x M normalized Hadamard matrix, and G is the ( N - 1) x ( M - 1) matrix obtained by deleting the first row and column of H , and S is obtained from G by replacing + 1's by 0's and - 1's by l's, then three possible choices for the matrix W of mask weights, together with the corresponding values of Tr [ W - ' ( W-')'I, are as follows:

+

TABLE 1 Matrix W

Trace

+1,

-1

G'

2--2/N

+1,

-1

' S

4-8/N+4/N2

+I,

HT

1

Elements

0

We may note that (a) Another choice for W is RT, where R is the matrix obtained from G by replacing - 1's by 0's. This gives a trace, however, which is slightly but uniformly worse than that from S'. (b) All the matrices given in Table 1 are superior to the single slit case (where W = I). In the single slit case, Trace = N. (c) NELSON and FREDMAN [I9701 have shown that the Hadamard matrix describes an optimum mask. A proof due to PHILLIPS[1972] is given in Appendix C .

130

MODULATION TECHNIQUES I N SPECTROMETRY

[III,

55

5.4. OPTICAL ARRANGEMENT

A multislit spectrometer using the STcode works as follows: Radiation which passes through the entrance aperture is rendered parallel and directed towards the dispersive element. The dispersed radiation is collimated and focused upon the multislit mask at the exit plane of the instrument. The radiation transmitted by the mask passes through the post-optics of the system and impinges on the detector. We then obtain a spectrum by sequentially stepping M masks at the exit plane and recording the detector output for each mask. The inversion procedure described in section 5.2 enables us to recover the spectrum. In some instances, depending upon instrumental design and the choice of code, it is advantageous to make use of both the reflected and transmitted radiation of a mask. This scheme would be particularly useful for masks which utilize the HT and GT matrices. Here the +l’s would represent reflecting slots and the - 1’s would represent transmitting slots. For the HTmatrix all elements of the first column would be +l’s and the masks corresponding to each row of HT would reflect the first spectral element at all times. The remaining N-1 slots would be stepped in the usual fashion. Such instruments have not yet been constructed. Infrared spectral measurements normally require that the radiation be chopped. One can then realize codes with 0’s and 1’s by chopping between the transmitted radiation and a standard source. The codes with 1’s and - 1’s could be realized by chopping between transmitted and reflected radiation.

+

5.5. DOUBLY ENCODED (OR MULTISLIT MULTIPLEX) HADAMARD TRANSFORM SPECTROMETERS

In the instrument analyzed here, an initial stage of multiplexing is introduced at the entrance of the spectrometer and a second multiplexing stage follows at the instrument’s exit plane. The signal-to-noise ratio advantage which can be achieved over an ordinary grating instrument is shown to be comparable to that of the Michelson instrument. That this should be so can be made intuitively understandable as follows: In order to describe the intensity of N spectral elements, a Michelson interferometric spectrometer will generally need to make determinations of 2N different mirror separations. As noted before, under certain conditions this number can be reduced to N . In a similar way, we will show that there are N determinations required for a doubly multiplexed grating instrument. While N measurements suffice to reconstruct the spectrum of radiation passing through an instrument having N entrance and N exit slit positions,

111,

8 51

HADAMARD TRAXSFORM SPECTROMETERS

131

we will here actually evaluate the performance of the system on the assumption that N 2 measurements are taken; i.e., that light is successively passed through the N exit mask positions for each of the N entrance mask positions. There is no loss of generality involved here because the various instruments will be compared on the basis of: (a) identical total observing time T during which all spxtral measurements must b: taken; (b) constant radiant energy density incident on unit area of the spectrometer entrance aperture; (c) constancy of the number N of unknown spectral elements to be determined during time T; and (d) identical photodetectors. The reason for presenting the argument on the basis of N 2 measurements, rather than N , lies in the greater generality of the mathematical treatment oftheerror analysis. A need for N Z measurement is not always convenient for large N , and there is no compelling reason why this should be required in practical situations. It is interesting however that the use of N Z measurements can give additional information about the difference in the spectral distribution of light reaching the instrument from differing portions (strips) of the source. The multislit spectrometer whose performance we analyze makes use of an encoding mask placed at the entrance aperture of a conventional grating spectrometer. This mask is N slit widths wide. In any given mask position, light from the source is permitted to pass through about 3 of these N slits and is bIocked from passage through the others. In general, there are M different mask positions in which different combinations of slits are permitted to pass light from the source into the spectrometer. The encoding process then consists of the successive use of each of the M mask positions to pass light through M different preselected combinations of entrance slit locations. A second encoding mask, Fig. Ic, is placed in the exit focal plane of the spectrometer. This mask has M’ different positions, each position passing light through some of the N‘ exit slit locations in a preselected way. By making measurements of the intensity of radiation passing through different combinations of entrance and exit mask positions, the radiation and HARWIT[1969] spectrum can be recovered. SLOANE,FINE,PHILLIPS have given a discussion for M = N = N’ = M ’ . This is the derivation reproduced here. 5.6. THEORETICAL ANALYSIS OF A DOUBLY MULTIPLEXED SPECTROM-

ETER

The following theory has been developed by HARWIT,PHILLIPS,FINE and SLOANE [1970]: Let E = ( E ~ , ) be the N x N matrix describing the entrance mask, where cir = 1 or 0, according as the rfh entrance slit position

132

M 0 D U L A T I 0 N TE C I4 N I Q U E S I N S P E C T R 0 M E T R Y

[Ill,

a5

5

N,

is open or closed when the entrance mask is in position i ( 1 5 i 1 5 r 5 N ) . Similarly let x = ( z ~ , describe ~) the exit mask.

5.6.1. The basic equation When the entrance mask is in position i and the exit mask is in position j , the detector measures N

N

where Yr,s is the sth spectral component of the signal that will enter the rthentrance slit and exit from the 3''' exit slit, if the slits are open; and vi, is the detector noise for measurement (i, j ) . We assume that the irradiation of the entrance aperture is homogeneous, and that spectrometer optics are arranged so that the spectrum produced by the rth entrance slit acting alone is a shift by r places of the spectrum produced by the first entrance slit acting alone. Thus there are 2 N - 1 unknown spectral components: Y - ( N - l ) , . . .,

v-1,

Yo, Y , ,

. . ., YN-1

given by Yr,s= Yr-,. Then (14) may be written in matrix form as,

.-I -

Y-1 . . . Y - N + , Y o . . . Y-N . . . . . . . . . . . . . . . .. . . YN-, YN-2... Yo

1

Yo Yl

5.6.2. The detector noise We assume that vi, in the (i,j)fhmeasurement has the following characteristics: (i) the expected value of vi, is 0, i.e., E(vi.j ) = 0; (ii) thc noise obtained in different measurements is uncorrelated; i.e., E(\li.j\vk.l) = a2Si,k6j,l where 0 is the root mean square noise level. For purposes of comparison we observe that if M measurements are made in time T, then o2 =

KM/T

where K is a constant depending upon the photodetector.

(16)

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HADAMARD TRANSFORM SPECTROMETERS

133

5.6.3. The estimation problem

We have the possibility of making up to N 2 measurements yli, i(i,j = 1, Yis. Given these measurements, we must then decide how to estimate the Yr’s. As justified in section 5.2 for a singly multiplexed spectrometer, an unbiased estimator for Y t will be used which is a linear function of the measurements, ,

. ., N ) to estimate the 2 N - 1 unknown

y r =

C at, i,

j

Vi,j

i, j

where the coefficients a,, i, are to be determined. Unbiasedness means that E$, = Y,. When all N Z measurements are used, this implies that the a’s must satisfy

N

N

C C

u+N

ut,i,j i=l j=1

C

Ei,rXj,r-u

=

dt,u

for u < 0.

(18)

As a measure of the accuracy of the estimate we adopt the mean square error criterion: minimize Nt=

N- 1

1

C -N+

0 : 1

C

E(P,-Y,)’

t=-N+l

=

c 2 x~ t , i, i

: i , ~ .

(19)

The estimation problem is then to choose the masks, E , x, the number of measurements to be made, and the a’s, so as to satisfy (17) and (18) and to minimize (19). Unfortunately, we do not know how to perform this minimization, we can only evaluate a given scheme once it is presented. In the next paragraph we evaluate 0: for a scheme where E = x, N 2 measurements are made, and where the u’s are given by equations (20) and (21). 5.6.4. Using N 2 measurements

N 2 measurements qi, are made; i.e., one measurement is made for every combination of entrance and exit mask positions. Then Q

= E-lyl(XT)-l

would equal Y if there were no noise. In the presence of noise each element ar, of the tth diagonal of Q - counting the main diagonal as the Oth, etc. is an unbiased estimate of Y,; and since each has the same mean ( Y t )and can be shown, at least in the case of the S matrix discussed below, to have the same variance (for fixed t), we use as the estimate of Y, an equally weighted, linear combination of the

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We can now treat the special case where E and x are both taken to be the matrix S defined as follows: Let H be a normalized ( N + 1) x ( N + 1 ) Hadamard matrix of 1’s and - 1’s with the first row and column consisting of +l’s. S is obtained by deleting the first row and column of H and replacing + 1’s by 0’s and - 1’s by 1’s. Using the properties of the S matrix, one can show in this case that

+

where t

=

- ( N - I), . . ., ( N - 1).

5.6.5. 2N- I measurements are enough

If E and x have rank N , it is possible to obtain unbiased estimates of the 2 N - 1 unknown Yt’s by measuring only 2 N - 1 q i ,j’s. However, there seems to be no simple rule for finding which q i , j to use, nor for finding the a’s. 5.7. COMPARISONS WITH OTHER GRATING SPECTROMETERS

To make a fair comparison, we assume: (1) a fixed total measuring time T ; (2) constant slit widths; and (3) an equal number N of unknown spectral elements to be estimated. For the doubly encoded system there are initially 2N-I unknowns Y - ( N - l ) , . . . Y o , . . ., Y N - , , but for t h s comparison we will suppose we only wish to estimate the N central elements y(N-

1)/2

7

.

*>

9

..

*)

y(N-

1)/2

(taking N odd for convenience). Of course, we can still obtain some estimate of the ends, but these errors will not appear in the comparisons. As a measure of performance we take, as before, the total meansquare error in all the unknowns:

Table 2 compares three different grating spectrometers. The first column is for a single entrance and exit slot. N measurements are made in time T, with a mean square error 0’ in each. The second column is for a singly

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135

multiplexed instrument with an exit mask S. The last column is for the doubly encoded system using eq. (22) for,:rc and has - by eq. (16) - been multiplied by a factor of N to allow for having to make N Z measurements in time T . TABLE 2 Comparison of total mean square error for three grating spectrometers in estimating N

unknowns

_ - _ _ _ ---- - --__--- -----___ N Conventional Singly encoded Doubly encoded -__ _ _ _ - - _ ---------~

2.56~~ 2.0002 1102 1.5102 1902 0.98~~ NuZ ( 2 2 . 2 / N ) d for N large* _ _ _ _ - _ _ _ _-------_ _ - _- - -__ --(8is the mean square noise in a single measurement made in time TIN; u2= (constant) N / T . )

3 7 11 19 N

2.2502 3.06~~ 3.36~~ 3.610’ [2-2/(N T - I ) ] ~ u ~

30 2

702

-

*

This number becomes 1 6u2/ W when W slots are used to detect N spectral elements and W >> N .

A more accurate expression for the error in the double mask case is provided by 2 22.18 40.72 cr,o,a, = 0 -- - - N N2 +’)3:-( ’

5.7.1. Remarks on the wide aperture advantage By using more entrance slits than are necessary, a doubly encoded spectrometer may be used to obtain a total mean square error for N unknowns that is much less than the figure (22.2/N) cr2 of Table 2. Suppose that N spectral elements are to be estimated. A doubly encoded spectrometer is used with W entrance and exit slots, where W is greater than N , and W 2measurements are made. There are now 2 W- 1 unknowns, but (20) and (21) were used to estimate only the N elements that we are interested in. The total mean square error is now, from (16) and (22),

where cr:

= W 2 a 2 / NTherefore . C J : ~ , ~N ,

16cr2/W for W ?> N ,

(26)

which is much less than (22.2/N)a2. It represents the wide aperture advantage, but it provides only a partial advantage since the root mean square

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M O D U L A T I O N T E C H N I Q U E S I N SPECTROMETRY

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SNR improves only as the square root of the increase in aperture instead of linearly with aperture. This indicates that the codes used for doubly encoded spectrometers are not yet quite optimum. We would expect a theoretical performance similar to that of the mock interferometer, but actually attain one only as great as that of the Girard grill spectrometer. Improved coding for pairs of masks should be sought.

Spectral element

Fig. 9. Operation of a doubly encoded instrument (a) to give spectral information for individual entrance slots (top 19 traces); (b) to give a summed spectrum only - no spatial and HARWIT [1971]). information (bottom trace) (PHILLIPS

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SPECTROMETRIC IMAGING

137

Equation (26) states that W should always be made as large as possible, the limitations being the deterioration of resolving power with increasing total aperture width. This is an engineering limitation discussed in 9 7. Fig. 9 illustrates the relationship between a purely spectral form of presenting the data obtained in the N x N measuring mode and the form in which some spatial information is also presented. PHILLIPSand HARWIT [1971] first reduced their data to show the spectrum of radiation for each entrance slit of an instrument having an aperture 19 slits wide, both at the entrance and at the exit. These spectra are shown in the first 19 traces of the figure, and represent spectra for a mercury arc source, near 1.7 microns. The bottom trace is a sum of the spectra obtained for individual entrance slit positions. As is to be expected, the summed spectrum appears to be less noisy.

Q 6. Spectrometric Imaging GOTTLIEB[1968] has pointed out that two-dimensional images can be analyzed in terms of the one-dimensional cyclic codes used for encoding the Hadamard transform spectrometers. For example, in dealing with a picture which could be divided into a 15 x 17 array of spatially resolved elements, a 255 element sequence of 1’s and 0’s could be used. The 255 elements are subdivided into shorter segments of 17, the first 17 covering the first row of the picture; the second 17, the second row; and so on. If the 255 element sequence is the first row of the 255 x 255 cyclic S matrix, we can obtain a data point giving the total brightness of radiation passing through the mask when the picture is imaged onto the mask. We then use the second row of the S matrix - suitably divided into 17 element segments, to obtain a second intensity reading; and this is repeated until 255 brightness readings have been obtained. By going through a data reduction mathematically identical to the reduction of the data obtained with a singly encoded spectrometer, we obtain a reproduction of the initial picture. All that this process does is subdivide a picture into strips. When these strips are placed end to end, the brightness distribution along the total length of this array can then be analyzed by a one-dimensional method. This has suggested to HARWIT[1971] that a spectrometric imager should be feasible.* Such an instrument focuses a scene onto a two-dimensional encoding mask placed at the entrance aperture of a spectrometer. At the exit aperture a one-dimensional mask is used. The two-dimensional mask again is a folded S-matrix of the kind specified by Gottlieb. Such two-di* Covered by United States Patent No. 3, 720, 469; and must not be used without written permission of Spectral Imaging, Inc.

Fi

138

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M O D U L A T I I O N T E C H N Q U E S IN SPECTROMETRY

06

........... .. .:. ...... .................. .......... :::..: ...... . . . . ........ .:. :. :-.. .: ......... ............. . .-............ . ::.;:..:. ::

ri

F! pipi . .:

..

. -

..

....

.. .... ..:

. . . ......

... . .

..

...

..

. . .

..

..

....

............ ........... ............ ....... ...... ........... .:: : : ....... .. ...... . . . . . . .

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

>:.:.

:*,.*

:::.:::::.::

..

...

. ..-. . ..

Fig. 10. A black-and-white reproduction of the color transparency used for testing is shown. It represents a color slide of the Ring Nebula in Lyrae originally obtained at Mount Wilson. In addition, fifteen reconstructed pictures are shown. Only two of these are bright; the others represent weak colors and/or noise. As explained in the text, only seven of the pictures represent a homogenous wavelength. Frames having a vertical line show right and left halves obtained at different wavelengths. Pictures corresponding to adjacent wavelengths lie below each other in columns. The array of pictures is to be read downwards, with successive columns going from left to right. If this procedure is followed, the first complete picture - the third frame in the first column - represents the picture a t shortest wavelengths. The first and third frames of the second column appear roughly to correspond to yellow and red, the strongest colors present in the original. For the divided frames, the left half represents a long wavelength color and the right half a short wavelength - see text. Despite the coarse resolution, the elongated doughnut shape of the original is clearly discerned (HARWIT[1973]).

mensional arrays can also be made cyclic so that there is no need for making up physically new masks for each separate entrance mask position. In fact, if the entrance array has m x n spatial elements, it suffices to make a single mask having (2m- 1) x (2n- 1) encoded positions. Since the encoding at the entrance and exit apertures of this instrument

S P E CTR 0METRIC I M A G I N G

139

Fig. 1 1 . The top pattern shows the two-dimensional mask used for encoding spatial position. The bottom pattern gives the one-dimensional mask used for spectral encoding. These masks respectively show 125 and 29 open or blocked positions. In any one of the 63 two-dimensional mask positions, a 7 x 9 element pattern is exposed through a blocking mask. In any one of the 15 one-dimensional mask positions, a 15 element array is exposed (HARWIT[1973]).

is mathematically one-dimensional, the data reduction is fully equivalent to that of the doubly encoded spectrometer and presents no new difficulties. HARWIT[1973] has constructed such an instrument. Fig. 10 shows a reconstruction obtained when a color slide of the “Ring” Nebula in Lyrae was placed into the instruments’s entrance aperture, next to the two-dimensional encoding mask, and illuminated with white light. There are 9 x 7 spatial resolution elements. The figure shows the information displayed in terms of 15 different panels, each obtained at a differentwavelength. Because of end effects, only the central seven panels represent monochromatic radiation. The others, which have a vertical dividing line, present different color views on the right and left sides of the panel. Since there are 15 exit aperture slits, and since the scene was 9 slits wide, this reduction displays 23 different colors, although only the seven central panels show the same color throughout. Both the one-dimensional and the two-dimensional arrays used in this instrument are displayed in Fig. 11. An alternate form of display would be to show the 15 element spectra for each of the 63 spatial resolution elements.

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MODULATION TECHNIQUES IN SPECTROMETRY

“11,

57

Just as in Fig. 9, the wavelength range of these spectra would not be identical for the left and right positions of the display. However, by making the exit aperture wide enough, the spectral overlap for the elements can always be made as large as desired, subject only to a degradation of spectral resolution if the width becomes too great. The source of this degradation is briefly mentioned in § 8 below.

0 7. A Survey of Binary-Encoded Multiplex Spectrometers We present here a brief survey, in rough chronological order, of most of the binary-encoded dispersive multiplex spectrometric systems known to the authors. We will not describe the theory or operation of any of these systems in depth, but will detail those features that appzar to us most salient, referring the reader to the published literature for more detailed descriptions of each individual system. 7.1. DECKER AND HARWIT 19-SLOT HTS SPECTROMETER

To the best of our knowledge, the first operating prototype binary coded multiplex spectrometer was that described by DECKER and HARWIT[ 19691. This instrument was a modified commercially-available 0.25-meter EbertFastie monochromator, fitted with a 19-slot Hadamard transform coding mask at its exit focal plane. The encoding mask used a periodicity-I9 cyclic S matrix code, with 0.1-mm-wide slots. It was used with a 295-line/mm grating, to produce a limiting resolution of about lOA, operating in second order. The mask was manually moved by means of a micrometer-driven two-axis translation stage. A conventional room temperature lead sulfide (PbS) infrared detector was used to record spectra i n the 1 . 5 region ~ of the infrared. The electronic system was completely conventional; data was recorded on punched paper tape for later decoding on a time-shared computer. While this instrument did not use “dedispersion”, and therefore could not be used to verify the signal-to noise ratio gains theoretically predicted for Hadamard-transform spectrometers, published spectra obtained with it did verify the mode of operation of the instrument and demonstrate that no loss of resolution was involved in binary modulation multiplex spectrometry. 7.2. DE GRAAUW AND VELTMAN 255-SLOT PSEUDO-RANDOM BLNARY MULTIPLEX SPECTROMETER

DEGRAAUW and VELTMAN[I9701 reported the design and construction of a 255-slot multiplex spectrometer based on the use of what theycalledpseudorandom binary sequence multiplex codes. These codes are essentially

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BINARY-ENCODED MULTIPLEX SPECTROMETERS

141

identical to the S-matrix codes described by SLOANE,FINE,PHILLIPS and HARWIT [1969]. This instrument was operated in the visible region of the spectrum (in the area of the magnesium b triplet) and hence did not have the multiplex advantage; deGraauw and Veltman’s interest in multiplexing was primarily as a convenient method for obtaining high time resolution, as they were using the instrument for time-resolved solar eclipse observations. This instrument is nonetheless of considerable interest, first because it was the first binary-encoded multiplex grating spectrometer to actually be used for an observation (as distinct from laboratory measurements of the spectrometer’s performance), and secondly for unique aspects of its construction. The instrument was a high resolution Litrow type grating spectrograph operated in the field by a heliostat and utilizing a large-diameter long-focal length objective lens (25 cm diameter, 340 cm focaI length). While the 255-slot pseudo-random code used was cyclic, the actual encoding mask used 255 different mask patterns (in the correct cyclic order), fabricated by photoresist techniques in a copper film on the outside of a precision glass cylinder. This cylinder was then rotated to obtain theappropriatebinary modulation; a flat mirror inside the cylinder directed the light to a beamsplitter and thence through two lenses onto two photomultipliers. An exit slit parallel to the direction of dispersion served as a field stop limiting the field of view to one mask pattern at a time, as well as defining the field of view on the solar disk. Each mask slot was 50pm wide, which corresponded to 0.1 A of spectral resolution; the simultaneously viewed spectral bandpass was then 25.5 A. Data was taken on a 7-channel analog magnetic tape recorder; the 7 channels were used in conjunction with the two separate photomultipliers to accommodate the large dynamic range inherent to eclipse observations while retaining the accuracy desired. The spectra were reconstructed by means of a hard-wired shift register decoding system described in some detail in the paper. This instrument was used by the Netherlands Solar Eclipse Expedition to Mexico in 1970 to obtain measurements of line profiles of the photosphere/chromosphere transition region, and as such is (to the authors’ knowledge) the first dispersive multiplex system to actually be used for a field observation. 7.3. DECKER 255- A N D 2047-SLOT SPECTROMETERS

The first experimental verification of the “multiplex advantage” (in signalto-noise ratio) for binary-modulated dispersive multiplex spectrometers was

C o r r e c t e d Czerny-Turner

-A-

A -

2047-matrix mask, a t e x i t focal plane

t o p r e a m p l i f i e r and d a t a r e c o r d e r

150-groove/nun g r a t i n g , b l azed f o r 2.0:.

k -

/ 1 - - -- - -- --- - --

Field stop s l i t s

I

vl m

c

3 0

x

N P

111,

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BINARY-ENCODED MULTIPLEX SPECTROMETERS

143

obtained by DECKER[ 1971bl using a 255-slot 0.5-meter Hadamard-coded spectrometer. This instrument, whose schematic is given in Fig. 12, was built around a commercially-available 0.5-m fully corrected Czerny-Turner spectrograph specifically designed for wide-exit-focal-plane applications. It was equipped with a bilaterally-variable entrance slit and a 150 groove/mm grating blazed for a first order wavelength of 2 . 0 ~in ; this configuration it had a first order resolution of about 18A (4.5 cm-I at 2p) with a 0.185-mm wide entrance slit (i.e., a resolving power of around 1100). The Hadamard-transform encoding mask used a periodicity-255 cyclic S-matrix code of 509 total mask slots, computed using the shift-register logic given by BAUMERT [1964]. To eliminate the additional detector noise that would have been caused if a larger detector had been used to collect the dispersed radiation, a “dedispersion” system was used. This cancelled the grating dispersion and allowed the encoded “exiting” beam to be brought back to a focus on approximately the same size detector as would have been used if the optical system had been operated as an equivalent monochromator. This was accomplished, in the manner shown in Fig. 12, by mounting the encoding mask at the exit focal plane of the spectrometer bisecting a 90” corner-reflector which returned the exiting encoded radiation back through the spectrometer in the reverse sense. The encoded radiation was detected by a 1-mm square lead sulfide (PbS) detector mounted on the upper portion of the entrance slit above a tuning fork chopper used to modulate the entering light. The encoding mask was moved sequentially by a stepping-motor driven belt-drive controlled by an electronic indexing circuit. The post-detector electronics was completely conventional, and data was recorded on punched paper tape for later decoding on a time-shared computer, which also automatically plotted the output spectra on an x-y-plotting Teletype unit. A FORTRAN IV decoding program was used, which took account of the cyclic binary nature of the coding matrix, but did not use any of the “fast transform” algorithms; the decoding computation required approximately 1.5 seconds of computer time. Theoretical estimates for the signal-to-noise gain of this multiplex instrument, as compared to an optically-equivalent scanning monochromator observing the same spectra in the same total observing time, were generated using the theory of SLOANE, FINE,PHILLIPS and HARWIT[1969]. Comparison spectra of the 1.5 to 1 . 7 ~emission structure of a mercury spectral lamp were taken operating the instrument in both the Hadamard and scanningmonochromator modes; typical comparison spectra were given in Fig. 8. Signal-to-noise ratio gains measured for these comparison runs agreed with

144

MODULATION TECHNIQUES IN SPECTROMETRY

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07

the theoretical estimate to well within the standard deviation of the measurements. This same instrument was later modified for operation with a 2047slot HTS code (DECKER [1971c]). This modification differed from the earlier 255-slot system only in the code mask used and in the manner of decoding the data. The periodicity-2047 cyclic S-matrix code mask is shown in Fig. 13; it contained a total of 4093 0.038-mm-wide slots used 2047 at a time. The operation of the system was as described above, except that the order-ofmagnitude increase in the quantity of data to be digested per spectrum required the use of an on-site (i.e., at the central computer center) high-speed paper tape reader for data input and an on-site high speed Calcomp plotter for plotting the output spectra. The total computer time required to read-in the data tapes, decode the 2047 x 2047 multiplexing matrix and plot a normalized and calibrated spectrum was about 7 to 8 minutes, of which approximately 1.5 minutes was taken by the FORTRAN IV matrix manipulation program (the rest was accounted for by input/output). Unfortunately, funding difficulties terminated work with this more advanced instrument prior to the completion of quantitative signal-to-noise comparisons; however, published spectra (DECKER [1971c]) verified the basic operation of the instrument. We should acknowledge here that work with both the 255-slot and 2047slot instruments was conducted at Comstock and Wescott, Inc., Cambridge, Massachusetts and was supported there by the Cambridge Research Laboratory, USAF Systems Command, Bedford, Massachusetts.

7.4. PHILLIPS A N D HARWIT 19 X 19-SLOT DOUBLY-MULTIPLEXING SPECTROMETER

The first experimental operation of a doubly multiplexing dispersive spectrometer - that is, a binary-modulated multiplex instrument employing multislit arrays at both the entrance and exit - was reported by PHILLIPS and HARWIT [1971]. Their instrument was a small Litrowf/8 spectrometer employing 19-slot S-matrix code-masks at both entrance and exit. The slot width used was 0.625 mm (by 3.5 mm high) resulting in a total aperture length of 12.lnim - which shows the extremely large apertures possible with even simple instruments using this technique. Both entrance and exit masks were manually moved using micrometer driven translation stages.

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MULTIPLEX SPECTROMETERS

145

Fig. 9 shows the 1.7-p emission spectra of a mercury spectral lamp taken by this instrument. The first 19 spectra give the spectral distribution ofradiation entering each of the 19 entrance slots. The individual spectra are not identical, but successive spectra are shifted by one resolution element, as explained by HARWIT,PHILLIPS,FINEand SLOANE119701. The bottom trace is a spectrum with greatly enhanced signal-to-noise ratio (SNR) of the entire source obtained by averaging identical spectral elements of the first 19 spectra. The authors note the instrument’s dual capability to perform either onedimensional “spectral imaging” or alternately, to give a spectrum with a high signal-to-noise ratio spatially averaged over the entire sourcc. These authors also reported that they had obtained laboratory spectra with an instrument having 3 entrance slots and 19 exit slots, and therefore apparently operated the first assymetric doubly-multiplexing Hadamard spectrometer. 7.5. HANSEN A N D STRONG HIGH RESOLUTION SPECTROMETER

HANSEN and STRONG[I9721 reported the construction and successful operation of a high resolution astronomical spectrometer employing a 127-slot Hadamard-transform binary multiplexing code. This instrument is of interest for its unique optical design as well as for the highperformanceit achieves. It is distinguished by the use of an echelle grating, used in the Litrow configuration, in conjunction with a long focal length (1.6 meters) alkali halide achromatic lens which is used both as the camera and “collimating” lenses in the spectrometer as well as the collimating and field lenses in a reversepath dedispersion system. The binary multiplexing mask used the periodicity-I27 S-matrix Hadamard code and was mounted at the exit focal plane of the spectrometer bisecting a corner mirror in much the same manner as used earlier by DECKER[1971b]. The long focal length and high precision of the optics allowed a resolution of approximately 0.1 cm-’ (in the 6p region) to be achieved. Other interesting features of this instrument are the use of a cut-out in the center of the corner-mirror prism as an entrance slit, the use of between-path chopping to minimize the effect of scattered radiation, and the use of an “image slicer” to reassemble the long thin dedispersed slit image into a more-nearly-square image for focussing onto the detector. Although no rigorous signal-to-noise comparison has been made, these authors have published several scanning monochromator/HTS comparison spectra which qualitatively show the b:neficial effects of spectral multiplexing.

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7.6. COMMERCIAL BINARY ENCODED MULTIPLEX SPECTROMETERS

The next instruments to appear, historically, were a family of Hadamardtransform spectrometers produced by Spectral Imaging, Inc., of Concord, Massachusetts. We will briefly describe the first of their instruments and also a current design aimed at the analytical spectrochemistry market. 7.6.1. HTS-19-1 Airborne Astronomical Spectrometer This instrument (DECKER [1973]) was designed for airborne infrared astronomical observations - specifically, for airborne observation of the 2 . 8 to ~ 3 . 5 ~water-of-hydration absorption band in the reflection spectrum of Mars. It flew as one of the three primary experiments onboard NASA's Convair 990 Observatory Aircraft during the 1971 Mars Opposition, and, as such, it represented the first operational use of a commercially-available Hadamard transform spectrometer. This instrument, which is shown in Fig. 14, was designed primarily to be

Fig. 14. Spectral Imaging, Lnc. IITS-19-1 airborne astronomical spectrometer (DECKEH [ 19731).

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B I N A R Y - E N CO D E D

M U LT I P L E X

S P E CT R 0 M E T E R S

147

lightweight and rugged enough to operate successfully in the airborneobservatory environment (the primary problem area was vibration resistance, especially during takeoff and landing). Additionally, the very short time available for the design and construction of the system dictated the almost total use of off-the shelf components. The system was designed around a small commercially-available 250-mm f/3.5 Ebert spectrometer, fitted with 19-slot cyclic S-Matrix HTS coding masks at both entrance and exit focal planes. The entrance aperture was approximately 6.3 mm wide by 1.0 mm high; since the focussed image of Mars was expected to be only 0.338 mm in diameter, the instrument was actually operated with all but one entrance slot blocked. The exit coding mask was traversed by means of a stepping-motor driven translation stage and data was recorded on punched paper tape for later digital computer decoding. Two 64 x 64 mm2 diffraction gratings were used in a back-to-back quick-change mounting; this gave the user a choice of observing a total wavelength span of approximately 2 . 1 3 ~ at a resolution of approximately 78cm-’, or to observe the shorter wavelength span of 0 . 7 ~ at a resolution of approximately 26 cm-’. This instrument was used in a series of observation flights in late July and SHAACK,SAGANand DECKER [1973]) early August 1971 (HOUCK,POLLACK, when it was mounted at the focal point of one of the three 30cmf/22 Cassegrain telescopes of NASA’s Convair 990 “Galileo” airborne observatory aircraft. Approximately eight hours of Mars data was collected at altitudes ranging from 1I .9 to 12.5 kilometers (39 000 to 41 000 feet) and definitely established the existence of water-of-hydration in the Martian surface rocks.

7.6.2. HTS-255-15 Analytical Infrared Spectrometer This instrument (DECKER [ 1972bl) is a high-performance moderate-resolution infrared spectrometer designed specifically for analytical spectrochemistry and operation by unskilled personnel in industrial environments. It covers the 2 . 5 to ~ 1 5 . 0 ~wavelength band ( 4 000 to 666cm-‘ -the “fingerprint” region of the analytical infrared) simultaneously in five grating orders at an average resolution of better than 3.5cm-’. The optical schematic of this system is shown as Fig. 15. It is built around a coma-corrected assymetric 500-mm, f/6.9 Czerny-Turner grating spectrograph equipped with a 64mm - square 10-groove/mm grating blazed for a wavelength of lop. It uses the 255-slot S-matrix coding mask shown on Fig. 16 at the exit focal plane for spectral discrimation, and the 15-slot S-matrix code mask shown in Fig. 17 at the entrance focal plane for wide-aperture multislit

3

I

01 fr

qs

4

m

n

vl

c

m

Input

X = 2.5 t o L

er 2.

v = 666 t o

E

1 5 - s l o t HTS coding mask, a t entrance focal plane, 2 8 . 5 x 0 . 1 5 min2 s l o t s

Order-sorting and focussing system, one mirror/dichroic/detector per each of f i v e g r a t i n g orders used

-

B I S A R Y E N C 0 D E D M U L 1.1 P L E X S P E C T R 0 M ET E R S

149

Fig. 16. 255-slot exit (spectral encoding) mask of HTS-255-15.

Fig. 17. 15-slot entrance (widc-aperture) mask of HTS-255-15.

operation. The HTS-255-15 is operated as a doubly-encoded multislit-multiplex spectrometer, spectrally multiplexes 749 individual wavelength resolution elements, and has an effective entrance aperature 2.2mm wide by 28.5mm high, for a throughput of 1.3 mm2 str. This instrument also uses “reverse-path’’ dedispersion and an image-slicer to change the slit images to the more-nearly-square shapes appropriate to the detectors used. Order-sorting is accomplished by a cascaded set of dichroic-mirror filters; five separate detectors are used (one for each of the grating orders), operated near their maximum sensitivity wavelengths. The “standard” instrument uses “room temperature” infrared detectors; pyroelectric detectors for wavelengths beyond 5p and thermoelectrically-cooled lead seIenide (PbSe) and lead sulfide (PbS) detectors for shorter wavelengths; optional liquid nitrogen-cooled detectors (mercury-cadmium telluride (Hg-CdTe), gold doped germanium (Ge: Au), indium antimonide (InSb), lead sulfide (PbS) and indium arsenide (InAs)) are also available.

150

MODULATION TECHNIQUES IN SPECTROMETERY

[In,0 7

This system has approximately a factor of 200 to 50 000 greater efficiency than the equivalent scanning-monochromator spectrometer, when detector sensitivities are taken into account. It also has approximately a factor of 30 higher efficiency than the nearest-comparable commercially-available Fourier transform spectrometer.* A significant element in this instrument’s design is the emphasis on fully-automatic operation and suitability for use by untrained personnel “in the field” and in industrial environments. Both code masks are operated by motor-driven translation stages controlled by automatic indexing and sequencing systems. While the system is designed for use with a built-in minicomputer, it is normally supplied for use with an experimenter’s own data system and computer, and hence is fully self-contained. This instrument represents the first application of commercial industrial automation standards to multiplex dispersive spectrometers. 7.7. HARWIT’S IMAGING SPECTROMETER

We mentioned earlier that the use of a two-dimensional binary encoding mask at the entrance of a multiplex spectrometer gave it a sputiuf imaging capability, as well as its normal spectral discrimination capability (HARWIT [197 1 I). One essentially obtains a three-dimensional array of information (two spatial dimensions, one spectral dimension) about an object - one has the spectrum of every point in an image or, conversely, an image of the object at each of the wavelengths of its spectrum. HARWIT [1973] has recently experimentally verified the operation of this device. In his simple bench test instrument, a suitably-chopped light source is focussed through the “scene” (a color transparency) and the two-dimensional entrance (imaging) mask, dispersed by a prism, spectrallyencoded by a one-dimensional exit mask, and, finally, sensed by a single detector. The 63-spatial-element entrance encoding mask and the 15spectral-element exit mask are shown in Fig. 11. Both masks were moved manually by micrometer driven translation stages. Figure 10 gives a typical output of that instrument: the second element of the second row is a blackand-white photograph of the subject transparency (the Ring Nebula in Lyrae); the remaining I5 elements are monochromatic images of that transparency in 15 adjacent wavelength elements. The original transparency had primarily two colors, which appear in the frames above and below the photograph. We should note that the spectral shift from frame to frame, * This is almost entirely due to its more-efficient five-detector operation, as compared to the FTS system’s pyroelectric detector.

111, §

81

INTERFEROMETERS A N D SPECTROMODULATORS

151

which was also shown in the 19 x 19 slot spectra of Fig. 9 (PHILLIPS and HARWIT[1971]), appears in the “spectral imager” also: the frames containing a vertical line actually contain data points representing a different color on each side of the line, due to these end effects. 7.8. EPILOGUE

It should be obvious that this field has grown extremely quickly: from very simple 19-slot exit-coding systems using manually-moved masks to multislit-multiplex systems involving hundreds of spectral resolution elements and automatically-stepped masks at both entrance and exit of very sophisticated optical spectrometers, in a period of only three or four years. It is equally apparent that the field is growing, both technicaIly and in absolute numbers, at least as rapidly while this survey is written. The instruments described above are those known to the authors as of mid-1972; we are also aware of at least three separate developments pushing beyond any of the instruments mentioned above. Essentially, these new developments are aimed at increasing the aperture of doubly-multiplexed systems (as the current program at Cornell University), at developing systems for ever-more-sophisticated and adverse environments and at the further development of the imaging spectrometer concept (primarily for use in space applications: earth resources monitoring and planetary astronomy). One unifying feature of all three developments is the use of an ever-increasing number of simultaneously viewed elements per multiplexed scene: from the several hundred elements per scene typical of current designs, through the several thousand elements/scene of the designs now on the drawing board, to hundreds of thousands or millions of spectral and/or spatial elements per scene for systems proposed for use within the next few years.

0 8. Some Practical

Considerations in Comparing Interferometers and Spectrornodulators

A number of practical factors may favor the construction of spectromodulators rather than interferometers - or vice versa - under different circumstances. These will be briefly summarized here: Design tolerances: In the interferometer, construction tolerances usually involve dimensions and motions that have to be maintained to within fractions of wavelengths. For spectromodulators, the corresponding tolerances are fractions of a slit width, and these tolerances are normally two orders of magnitude more relaxed. Spectromodulators should therefore be

152

MODULATION TECHNIQUES IN SPECTROMETRY

[III

9: 9

intrinsically more suitable for rugged applications, and usually less costly to construct. Computations: Intrinsically the Fast Hadamard Transform is faster than the Fast Fourier Transform, as mentioned in section 3.5. Eventually this may give the Hadamard-transform instruments an edge over interferometers, even though at the present stage, there is greater general familiarity with Fourier transforms among scientists and engineers. Resolution and bandwidth: The spectromodulators ale still too new to be properly judged in terms of ultimate relative performance. STRONG [ 1971 ] and HANSENand STRONG[1972] have been particularly concerned about the quality of imaging in these systems. It may be that optical imaging considerations will not permit instruments of the highest resolution which also have both a high throughput and a wide spectral coverage, since the imaging becomes more difficult if a wide spectrum must be covered in one stroke. There may, however, be ways in which the computer can make allowances for some of the wavelength-dependent magnification effects and some of the aberrations so that ultimate performance is not compromised. Nevertheless, since there are no prospects for meter-sized diffraction gratings or prisms, it is unlikely that resolution equal to that of the highest resolution interferometers will be realized by spectromodulators in the foreseeable future. The high resolution Michelson interferometers do make use of path differences of the order of meters. Combined instrumentation: There seem reasonably good prospects for combining some of the image encoding properties of spectromodulators with interferometric techniques in instruments that might combine the best features of both classes of apparatus. It seems to us that this is a promising avenue along which t o progress.

9 9. Some Unsolved Problems in Transform Spectrometry Problem 1. A set of spectrometric measurements is begun. Part way through these observations, atmospheric emission becomes variable. At the end of the run, conditions again are good, but part of the observations needed t o obtain the spectrum are compromised. What is the best strategy for making effective use of the data? The corresponding problem does not arise in exactly the same way in conventional spectroscopy, because there we would have good spectral data for part of the spectrum, and we could discard (or repeat the observations on) the less reliable parts if necessary. Problem 2 . I n Hadamard-transform spectrometers encoded for both Fell-

111,

8 91

SOME UNSOLVEI)

PKOBLEMS

153

gett’s and at least the partial wide aperture advantage, the highest SNR has been obtained if the full set of n x N data points is used in analyzing the data. Here n and N refer to the number of entrance and exit slits of the instrument. Since only n + N - 1 spectral elements are involved, a spectrum can be obtained if no more than n N - 1 data points are measured. We do not, however, know whether this mode of operation uniformly leads to poorer spectral SNR than the more extensive set of n x N data points - for similar total observing time intervals. We also do not know which set of n + N - 1 measurements to choose nor what codes are optimum in this more modest mode of operation (which suppresses all spatial information and produces only a spectrum). Problem 3. Spectromodulators depend on proper imaging of entrance slits onto exit slits. For high throughput instruments this imaging becomes increasingly difficult as various aberrations become important. To what extent can computational techniques correct these aberrations to yield a better spectrum, particularly if the aberrations actually are known in advance - have been measured - for the specific instrument? Computational enhancement of the spectrum should be possible wherever aberrations distort, but do not diffuse, the exit image along the direction of dispersion. Problem 4. In Appendix C we show that a Hadamard code provides optimum encoding for a singly encoded instrument. When there are two or more successive stages of encoding, are Hadamard codes still optimum? The arguments of section 5.7 indicate that they are not. Improved encoding techniques should therefore be sought. Problem 5. The spectrometric imagers designed to date have all made use of a two-dimensional mask constructed by folding a one-dimensional array. Are there general classes of two-dimensional arrays which uniformly provide pictorial data with higher signal-to-noise-ratio? Can any of these be used for constiucting a cyclic two-dimensional mask?

+

[Note:After this manuscript had bzen completed, HARWIT, PHILLIPS, KING and BRIOTTA[1974] finished a study which goes a considerable distance toward solving Problem 2. They treat the special case of spectrometers in which a transmission filter restricts radiation to a total of ( N - n + 1) contiguous spectral elements. A simple solution for integral values of ( N - n + 1)/n is then derived and shown to provide a spectrum with S N R equivalent to that obtained in n x ( N - n + I ) measurements. In their instrument the length of the cyclic code is ( N - n + 1). However the exposed portion of the exit mask also contains ( n - I ) repeated elements of the code, and is therefore N elements long. A particularly interesting case involves N + 1 = 2v. In that

154

MODULATION TECHNIQUES IN SPECTROMETERY

[III,

0 10

case the data analysis is surprisingly simple and can be done by inspection. In this sense the instrument bears a resemblance to Girard’s grill spectrometer.]

0 10. Conclusions Modulation techniques were first introduced into spectrometry in order to provide spectra with improved signal-to-noise-ratio. Frequently this permitted use of noisy detectors for measurements that were not possible using standard spectrometers. More recently incorporation of several stages of modulation has permitted the construction of instruments that analyze radiation into spatial as well as spectral components, essentially to provide a color picture of a scene at wavelengths /or which photographic techniques do not exist. The Reed-Muller codes based on Hadamard matrices have played a particularly important role in the development of such instruments. With continuing improvements in detector technology and in cryogenics, increasing numbers of spectrometric measurements are becoming photon noise limited. In such measurements modulation techniques no longer lead to a multiplex advantage. However, the throughput advantage still can be obtained, and of course spatial information can also be encoded. If present trends continue, arrays of low noise detectors will become available within the next few years. In that case, the iole of modulation techniques will again shift. We can, for example, expect instruments in which vidicons are combined with spectromodulators or interferometers to yield data similar to those provided by spectrometric imagers. In essence, the role of modulation in such instruments would then be the addition of one or more extra dimensions - e.g., spatial, spectral, or polarization - to the data being gathered. We can therefore expect a continuing and changing role for modulation techniques in spectrometry. Acknowledgmeiits At Cornell University, recent work on Hadamard-transform spectrometers has been supported by a grant from the Laboratory Director’s Fund at AFCRL (Contract F19628-71-C0183). We acknowledge many useful discussions with Drs. J. R. Houck, L. King, P. G. Phillips, and G. Vanasse.

PROPERTIES OF CYCLIC CODES

I l l , APP. A]

155

Appendix A

Some Properties of Cyclic Codes for Hadamard-Transform Spectrometry

A Hadamard matrix H of order N (cf. HALL[1967]) is an N x N matrix of +I’s and -1’s such that HH’

=

NI.

(All

H may always be normalized so that the first row and column consist entirely of 1’s. Let G denote the remaining ( N - 1) x ( N - 1) matrix, thus:

+

1...1 1

H =

‘

G

I Let row i and r o w j be any two distinct rows of H other than the first row. Then it is easily shown that

row i has

+ 1 and r o w j has + 1 in + N places

........+ 1 . . . . . . . . . . . . - 1

.......... . . . . . . . . - 1 . . . . . . . . . . . .+ 1 . . . . . . . . . ....... - 1 . ...........-1 . . . . . . .

643)

Thus N must be a multiple of 4. (It is conjectured that Hadamard matrices exist wheneoer N is a multiple of 4.) Further, if N is a multiple of 4, and at least one of the following conditions is also satisfied: (i)

N =I.’+1,

p prime

(ii) N = p(p+2)+ 1, p and p + 2 prime (iii) N

= 2”

then G can be chosen to be a cyclic matrix, that is, a matrix in which the (i+ 1)s‘ row is obtained by shifting the ith row cyclically one place to the right. For example, when N = 8, G may be taken to be

156

M 0D L 'L A T I 0 N T F. C I i N I Q U

-

+ +

+

-

-

-

where

++-++ + - + G = + - - - + + - + - - - + + + - + - - - + ++-+---

[III, APP. A

E S I N S P EC T R 0 M E T R Y

+ standsfor + I

and - for -1.

- - - -

645)

From (A3) we calculate the dot product of any two rows of H or of G : InH: r o w i - r o w j =

(0, i z j \N, i = j .

In G : row i . row j

--I,i#j \ N - I , i = j.

=

Also, each row of G contains ( 4 N - 1 ) + 1's and ) N - 1's. The first choice for W is the matrix H'. From ( A l ) we obtain H-' = H T / N , and Tr [ ( H - ' ) ' H - ' ] = Tr [ H H T / N 2 ]= Tr [ N I ] = 1. The second choice for Wis the matrix GT. From (A6) GGT = NZ- J where J is an ( N - 1 ) x ( N - 1) matrix of 1's and

+

GJ

JGT

=

so G - ' = ( G ' - J ) / N . Tr [ ( C - l ) T G - l ]

=

=

[(+N-l)++N(-l)]J

=

-J

Then

Tr

[(G-J)(GT-J)] N2

=

1

Tr - - ( I + J ) N

=

2 2 - -. N

The last choice for W is S', where S is the matrix obtained from G by replacing 1's by 0's and - 1's by + 1's. Clearly each row of S contains ( I N - I ) 0's and j N l's, and from (A3):

+

+

In S: row i . row j

=

\aN, i \;N, i

z

j

= j.

We can show that SST = a N ( I + J ) SJ = JST = i N J S-' = 2(2ST-J)/N Tr [(S-')TS-l] = 4 - 8 / N + 4 / N 2

=

(2-2/N)2.

Appendix B contains a table giving the first row of S matrices for some of those values of Nfor which we currently know that a cyclic S matrix exists.

111, A P P . B ]

CODES FOR H A D A M A R D - T R A N S F O R M SPECTROMETRY

157

Appendix B Cyclic Codes for Hadamard-TransformSpectrometry (After BAUMERT' [ 19641)

-_

n

Sequence

3 7 11 15 19 23 31 35

101 01011 11011 10001 11001 11111 00001 01011

43 47

11001 01001 11011 11100 10000 01000 11010 110 01111 01111 00101 01110 01001 10110 00101 01 100 00100 00

59

01011 10101 00100 11101 llI00 11111 00000 01000 11011 01010 0010

63

00000 11111 10101 31100 11011 10110 10010 01110 00101 11100 10100 01100 001

67

01001 01001 10001 11101 01111 11001 00010 11101 10000 00101 00001 11001 10101 10

71

01111 11011 10100 11011 10001 10101 10100 01 110 10010 10011 10001 00110 10001 00000 0

79

01101 10011 11010 01011 11110 11000 01100 01010 10101 11001 11100 10000 00101 10100 00110 0100 01011 00101 11100 01100 01010 11111 11010 01 110 11001 00011 01000 00001 01011 10011 10000 10110 010 01101 00111 00011 11111 00010 11011 10111 01010 01000 01001 10100 11011 11011 01010 00100 01001 01110 00000 01110 00110 100 01011 00001 11111 01001 00010 10101 10011 11101 11101 00111 00110 01100 01101 00001 00000 11001 01010 11101 10100 00001 11100 10

83

103

107

10 10001 00110 11101 01011 01011 10001

0

10111 01000 00110 10110 11110

0110 01010 000 11001 10100 1 10000 10101 10010

11000

I58

M O D U L A T I O N TECHNIQUES I N S P E C 1 ROMETRY -

n 127

00000 00011 11110 10000

-

-

-

-

Sequence

-

~-

[Ill, A P P . B

~-

-

. _ _ ~

01111 11101 01010 01100 11101 11010 01011 01111 01101 01101 10010 01000 11100 00101 01010 11100 11010 00100 11110 00101 00001 01 -

~-

~

~-

A frequently used code is

255

00000 01001 01011 01001 00101 01101 01001

OOlOI 11110 11110 01101 11010 01000 00011

~

511

00000 01001 10110 00110 00111 11010 01110 11101 11111 11111 01100

11101 01110 01101 00111 11011 11101

00001 ooO01 11011 00111 11100 10010

11111 10001 10010 10001 00110 01001

11001 01011 10100 10110 10011 10000

ooO01 00110 10100 oO010 01011 00111

-~

~

00010 00010 00110 00010 01110 01010 10110 00011 01111 10111 00100 01010 00010 10110 10011 11110 11001 00100 1 1 1 1 I 10010 01 101 01001 10011 o0000 001 10 00110 01010 10010 1 1 1 1 1 11010 00101 10001 11010 11001 01100 11110 11011 10100 00011 01011 01101 11011 00oO0 10110 10111 10101 00000 01010 01010 11110 01011 10111 OOOOO 01110 10010 01111 01011 10101 00010 01000 01100 11100 00lOI 10110 01101 00001 11011 11000 01111 1 1 1 1 1 00000 11110 00010 11100 11001 00000 10010 10011 10110 10001 11100 00110 11000 10101 00100 01110 00110 11010 10111 oO010 01000 1 -~

1023

10001 10101 11110 00011 11110 00100 10001

-

~

-~

~~

OOOOO 00001 00000 01001 00010 00001 10010 01101 00001 00101 01000 01111 01011 10101 10110 11000 00000 11000 00110 11001 10000 10101 10101 11000 11011 11110 00100 01111 00111 10110 11010 00000 01010 00010 11010 10100 01 11 1 10111 10010 0101 1 00000 10011 00100 01010 00110 11011 10000 00111 10001 11011 11111 00100 00110 00101 10111 01000 01101 01011 00111 10010 11011 00100 00010 00100 10011 00000 01011 00010 10011 10110 01110 00101 1 1 1 1 1 01010 00101 11011 01011 00001 10011 01101 01000 00111 01001 11101 00110 10100 10011 10000 01111 10011 I O O l l 01111 01000 10101 01101 11110 00010 01110 10001 11010 11111 01101 00100 00100 00101 00101 01100 01110 01111 11101

111, A P P . B ]

( ' O D E S FOR H A D A W A R D - T R A N S F O R M S P E C T R O M E T R Y

-

-

.~

__

-_

__

- -_

-~

-

10000 10001 10100 11100 10011 I1011 11101 00100 10100 00001 00010 00101 10011 01001 01001 10111 00101 01110 01110 11101 11001 01000 10011 01100 01000 10010 10101 00111 11100 11000 00100 10111 00001 01111 01010 01011 11000 10101 11101 11010 11111 10000 00011 10000 11111 11101 01100 10110 01001 001 2047

--

Sequence

n

-

~-

~-

- __

11000 01101 IOIOO 01100 00011 0000I 11001 10011 OlIlO 01011 11010 11110 10101 11111

-_

159 ~

-

--___ 11011 00011 00011

IOlIl OIOOl OllOI

11011 10101 11100 01101 11010 IOOIl 01110 01000 10111 00010 01111 -

00000 11110 11001 00110 01010 Ill00 01111 10001 10011

01111 01110 10100 01101 00001

-

00000 00000 10000 00001 01000 00010 00100 00101 01010 01000 000OI 10100 00011 10010 00110 11101 01110 10100 01010 00010 10001 00100 01010 IlOlO 10000 11000 01001 11100 10111 00111 11101 11001 00101 01110 11000 01010 11100 10000 10111 00101 00110 11OOo 11110 11101 10010 10101 11100 ooO01 00010 I I 1 1 1 00100 10001 11011 0101 1 0101 I 0001 1 0001 I 10110 10100 10110 00011 00111 0 0 1 1 1 11101 11100 00101 UOllO 01000 1 1 1 1 1 10101 10000 10001 11001 01011 01110 00011 01011 00111 00011 11101 10110 00101 10111 01001 10101 0 0 1 1 1 10000 1 1 100 11001 101I 1 11111 10100 00000 10010 00001 01101 00010 01100 10101 1 1 1 1 1 00001 00001 10010 10011 11100 01110 00110 lIOll 01110 11011 01010 11011 00000 11011 10001 11010 11011 01000 1101I 00101 11011 11001 01010 01110 00001 11011 00011 01011 10111 00010 10101 10100 00001 10010 00011 11101 00110 00100 1 1 1 1 1 01011 10001 00010 11010 10100 11000 00011 I 1 100 001 10 001 10 01 1 1 1 01 11 1 11001 01000 01 I10 00100 11011 01011 11011 00010 01011 10101 10010 10001 11100 01011 00110 1001 1 I I 1 10 01 110 0001 I 1 I01 1 001 10 0101 I I I I I 1 10010 00000 11101 000OI 10100 10011 10011 01110 11111 01010 10001 00000 01010 10000 10000 01001 01000 10110 00101 00111 01000 11101 00101 lOI00 I1001 I001 1 1 1 I I 1 1 I 1 10 00000 0001 I 00000 001 I I 10000 01100 11000 1 1 1 1 1 11101 10000 00101 11000 01001 01100 10110 01111 00111 11001 11100 01111 00110 11001 11110 11111 00010 10001 10100 0101 I 10010 10010 I 1 100 01 100 10110 I 1 11 1 00110 10001 11110 01011 00011 10011 10110 11110 10110 10010 001 10 01 101 01 1 1 1 1 1 100 01000 001 10 10100 01 110 00010 I101 1 00100 I101 I 1101I I1010 01010 01001 10001 1011 I I101 1 10100

00101 01001 001 10 10111

160

MODULATION TECHNIQUES I N SPECTROMETRY

[ I I I , A P P. C

Sequence

n

01010 10010 10000 01100 01000 11110 10101 10010 00001 11101 00011 00100 10111 11011 00100 01011 11010 10010 01000 01101 10100 11101 10011 10101 11110 10001 00010 01010 10101 10000 00001 11000 00011 01100 00111 01110 01101 01011 11100 00010 00110 00101 01111 01000 01001 00100 10110 11011 00110 11011 11110 11010 00010 11001 00100 11110 11011 10010 11010 11100 11000 10111 11101 00100 00100 11010 01011 11001 10010 01111 11101 11000 00101 01100 01000 01110 10100 11010 00011 11001 00110 01110 11111 11010 10000 01000 01000 10100 10101 00011 00000 10111 10001 00100 11010 11011 11000 11010 01101 11001 11101 01111 00100 01001 11010 10111 01000 00101 00100 01000 11010 10101 11000 00001 01100 00010 01110 00101 11011 01001 01011 00110 00011 11111 00110 00001 11111 00011 00001 10111 10011 10100 11110 10011 10010 01110 11101 11010 10101 01

Appendix C Code Optimization

NELSON and FREDMAN [1970] have shown that the minimum value of for a singly multiplexed instrument is a2. PHILLIPS [1971] has given a somewhat different proof based on the following four matrix relationships [1970]): (cf. FINKBEINER

E

1. If A is a real, non-singular n x n matrix, then the determinant

Det A 2 k"n 'jZ where k 5 A i j for all elements A i j of A . 2. If C = AAT, then C is real, symmetric, and positive definite. 3. Tr [C] 5 n[Det C]"".

4. For non-singular A, Det A

=

[Det

We can then state that Tr AAT 7 n[Det AAT]'/", and since AAT is positive definite, Det AAT = [Det AATl = IDet A Det A T / . For an encoding matrix W = A-' we then have lDet ( W - ' ) Det (W-')T[ = [[Det W Det WT[]-' Hence Tr (AAT) 2 n[[Det W[IDet WT[]-'In.

=

[IDet W[IDet WTI]-'.

1111

161

REFERENCES

But from the first relation above, JDet WI S k"n"'l, and since all elements of Ware 1,0 or - 1, we can set k = 1. We then see that JDet WllDet WTI I n", so that Tr [(W-')(W-')T]

2 1.

But our evaluation leading to Table 1 showed that for the Hadamard code, Tr [(W-')(W-')T] = 1. This type of encoding therefore is optimal.

References BAUMERT, L. D., 1964, Generation of Specified Sequences, in: Digital Communication with Space Applications, ed. S. W. Golomb (Prentice-Hall, Inc., Englewood Cliffs, New Jersey) pp. 17-32, 169. J. W. and J. W. TUKEY, 1958, Math. of Comp. 19,297. COOLEY, DECKER Jr, J. A., 1970, Appl. Optics 9, 1392. DECKER Jr., J. A., 1971a, Appl. Optics 10, 510. DECKER Jr., J. A., 1971b, Appl. Optics 10, 24. DECKER Jr., J. A., 1971c, Appl. Optics 10, 1971. Jr., J. A., 1972a, Analytical Chemistry 44, No. 2, 127A. DECKER DECKERJr., J. A., 1972b, Hadamard-Transform Analytical Spectrometer, in: Analysis Instrumentation, Vol. 10, eds. R. L. Chapman, G. A. McNeill and A. M. Bartz (Instrument Society of America, Pittsburgh, Pa.) pp. 49-54. Jr., J. A., 1973, Appl. Optics 12, 1108. DECKER DECKER Jr., J. A. and M. HARWIT,1968, Appl. Optics 7, 2205. DECKER Jr., J. A. and M. HARWIT,1969, Appl. Optics 8, 2552. 1970, Appl. Optics 9, 2658. DEGRAAUW, T. and B. P. T. VELTMAN, FELLGETT, P. B., 1951, Doctoral Thesis, Cambridge University. FELLGETT, P. B., 1958, J. de Phys. et la Rad. 19,187. FELLGETT, P. B., 1967, J. de Phys., Colloque C2, 28, p. C2-165. FINKBEINER, D. T., 1960, Introduction to Matrices and Linear Transformations (W. H. Freeman and Co.) pp. 97, 191. GIRARD,A., 1963, Appl. Optics 2, 79. GOLAY,M. J. E., 1949, J. Opt. SOC.Amer. 39, 437. GOLAY,M. J. E., 1951, J. Opt. SOC.Amer. 41, 468. GOTTLIEB, P., 1968, IEEE Trans. Inform. Theor. IT-14, 428. J. F., 1965, Univ. of Hull (England) Reports, Contract AF61 (052)-751 (unGRAINGER, published), Annual Summary Report. J. F., 1966, Univ. of Hull (England) Reports, Contract AF61(052)-75 I , GRAINGER, Final Scientific Report. GRAINGER, J. F., J. RINGand J. H. STELL,1967, J. de Phys., Colloque C2,28, p. C2-44. HALLJr., M., 1967, Combinational Theory (Blaisdell Publishing Co.) p. 204. 1972, Appl. Optics 11,502. HANSEN, P. and J. STRONG, HARWIT,M., 1971, Appl. Optics 10, 1415. HARWIT,M., 1973, Appl. Optics, 12, 285. HARWIT,M., P. G. PHILLIPS,T. FINEand N. J. A. SLOANE,1970, Appl. Optics 9, 1149. HARWIT,M., P. G . PHILLIPS,L. KINGand D. A. BRIOTTA Jr., to be published. D. SCHAACK, C. SAGANand J. A. Decker Jr., 1973, Icarus 18, HOUCK,J., J. B. POLLACK, 470.

162

MODULATION TECHNIQUES IN SPECTROMETRY

[III

IBBETT,R. N., D. ASPINALL and J. F. GRAINGER, 1968, Appl. Optics 7, 1089. JACQUINOT,P., 1954, J. Opt. SOC.Amer. 44, 761. JACQUINOT,P., 1960, Rep. Prog. Phys. 23, 267. MERTZ,L., 1965, Transformations in Optics (John Wiley and Sons). MORET-BAILLY, C. MILANand J. CADOT,1970, Nouvelle Revue D’Optique Apliquee 1, 137. NELSON,E. D. and M. L. FREDMAN, 1970, J. Opt. SOC.Amer. 60, 1664. PHILLIPS, P. G., 1968, M. S. Thesis, Cornell University. PHILLIPS, P. G., 1972, Thesis for Ph. D., Cornell University. PHILLIPS, P. G. and M. HARWIT,1971, Appl. Optics 10, 2780. PRATT,W. K., J. KANEand H. C. ANDREWS, 1969, Proc. IEEE 57,58. SLOANE, N. J. A., T. FINE,P. G. PHILLIPS and M. HARWIT,1969, Appl. Optics 8, 2103. STRONG, J., 1971, Appl. Optics 10, 1439. TINSLEY, B. A., 1966, Appl. Optics 5, 1139. VANASSE,G. A. and H. SAKAI,1967, Progress in Optics, Vol. VI, ed. E. Wolf (Amsterdam, North-Holland) p. 261.

E. WOLF, PROGRESS IN OPTICS XI1 0 NORTH-HOLLAND 1974

INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS BY

KARL H. DREXHAGE" Physikalisch-Chemisches Institut der Universitat, MarburglLahn, Germany

*

Now with Eastrnan Kodak Research Laboratories, Rochester, N.Y., USA.

CONTENTS

PAGE

(i 1 . INTRODUCTION

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

165

Q 2. PREPARATION OFMONOLAYERSYSTEMS . . . . . . 166

5 3. OPTICAL PROPERTIES OF CADMlUM-ARACHlDATE LAYERS . . . . . . . . . . . . . . . . . . . . . . . . S; 4 . COHERENT SCATTERING AT A DYE MONOLAYER

174

. . 180

Q 5. STANDING LIGHT WAVES . . . . . . . . . . . . . . . 190 S; 6. EVANESCENT LIGHT WAVES . . . . . . . . . . . . .

194

Q 7 . THE NEAR FIELD OF A RADIATING MOLECULE . . . 199

Q 8 . THE RADIATION PATTERN O F A FLUORESCING MOLECULE I N FRONT O F A MIRROR . . . . . . . . . . . 206 (i 9 . FLUORESCENCE DECAY TIME O F A MOLECULE I N

FRONT OF A MIRROR . . . . . . . . . . . . . . . . . 216 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . .

226

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . 229 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

229

0 1. Introduction Although light is the most familiar form of electromagnetic radiation, some aspects of its generation and propagation have not been as carefully studied as the corresponding behavior of microwaves or radio waves. The reason for this can be traced to the short wavelength of light, which is small compared with the dimensions of most physical apparatus. Recently a new investigative technique has become available that avoids the difficulties traditionally associated with the short wavelength of light. This development was initiated by ZWICKand KUHN [1962], who were looking for a novel method of studying experimentally the distance dependence of energy transfer from an excited donor molecule to anearby acceptor molecule. The idea was to separate physically donor and acceptor by a well defined distance and then to examine the fluorescence of donor or acceptor as a function of separation. They attempted this with layers of adsorbed dyes separated by means of monomolecular fatty acid layers, as was first suggested by AUGENSTINE [1960]. Stimulated by these experiments, F. P. Schafer suggested the possibility of using monomolecular layers of fluorescent dyes to study optical phenomena such as standing light waves and the influence of a mirror on the light emission by molecules. After an improved monolayer technique had become available, these ideas were realized experimentally (DREXHAGE [ 19661, BUCHER,DREXHAGE, FLECK,KUHN,MOBIUS, SCHAFER, SONDERMANN, SPERLING, TILLMANN, WIEGAND [19671). The new technique has found a great variety of applications other than in optics, which have been reviewed recently by KUHNand MOBIUS[1971], KUHN[1972] and MOBIUSand BUCHER[1972]. In this article we shall concentrate on the optical studies, emphasizing those phenomena that have been examined for the first time. We shall begin by describing the preparation of monolayer systems and the determination of their optical constants. The experiments with dye layers which are reviewed in the later sections may be divided into two classes: (1) The dye acts as a weakly absorbing probe for the light field and its light emission (fluorescence) serves merely as a measure thereof. ( 2 ) The absorption of light by the dye only provides the energy for 165

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the fluorescence, which is the phenomenon to be examined. Clearly, the t w o situations are reciprocal, and often both types of experiments can be made with the same sample.

5 2. Preparation of Monolayer Systems The technique of built-up films on which the work to be discussed in this article is based, was founded by BLODGETT[1934, 19351 and BLODGETTand LANGMUIR [1937]. An important improvement was introduced by SHERand CHANLEY [1955], who replaced the “piston oil” used by Blodgett and Langmuir for the compression of the film, by a floating barrier. For general reviews of the monolayer technique the reader is referred to TRURNIT 119451, GAINES [1966] and MOBIUSand BUCHER119721. In this section we summarize the preparation of built-up films from fatty-acid and dye monolayers only insofar as it is relevant to the work dealt with in the subsequent sections of this article. 2.1. DEPOSITION OF CADMIUM-ARACHIDATE LAYERS

When a solution of a long-chain fatty acid, e.g. arachidic acid (C,,H,, COOH), in an organic solvent like benzene or chloroform is dropped onto a clean water surface, the solvent evaporates quickly and leaves the arachidic acid, which is insoluble in water, at the surface. The fatty-acid molecules are attached to the water with the hydrophilic carboxyl group, whereas the other end of the molecule is hydrophobic and tends to point upward. Depending on the surface area and the concentration of the arachidic-acid solution one has to add a certain numbzr of drops to the water surface in order to cover it completely with a monomolecular film of the acid. This

Fig. 1. Deposition of monomolecular layers with the Blodgett technique.

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Fig. 2. Apparatus used by the author to deposit monomolecular layers on a glass slide. The trough and floating barrier are made from plexiglass. The trough is 30 cm wide, 60 cm long and 8 cm deep. The width of the float (length 8 cm) is by about 2 mm smaller than the inside of the trough. The sides of the float as well as the rim of the trough are polished and parallel to allow a smooth, piston-like movement. The adjustable stop at the bottom of the dipping device carries a counter. (From DREXHACE [1970b], used with permission of Scientific American.)

point is easy to perceive since the drops no longer spread, but remain unchanged in form of small lenses floating on the surface. The solvent would evaporate slowly from these droplets and leave the acid behind in crystalline form. This is avoided, however, by moving a float backwards (Figs. 1 and 2), so that the surface area available to the fatty-acid film is increased and the droplets spread immediately. By means of a weight-and-pulley arrangement (Fig. 1) one now applies a small force to the float, in order to exert a surface pressure on the film. Under the pressure the arachidic-acid molecules form a densely packed monomolecular film covering the water surface in front of

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the float, the thickness of which is equal to the length of the arachidic-acid molecules. For preparation purposes the film is kept under a surface pressure of about 30 dyne/cm corresponding to a weight of about 1 g per 30 cm float width. As the arachidic-acid film is nearly incompressible between 5 and 60 dyne/cm, small pressure changes, which may occur during the preparation process, do not affect the packing of the molecules in the film. Under the right conditions (see below) the film can be transferred to almost any substrate. If, for instance, a glass plate is lowered through the surface, on withdrawal one layer of arachidic acid is deposited on the hydrophilic glass. The carboxylic end groups of the arachidic acid attach themselves to the glass surface (Fig. 3), and the slide emerges dry from the water. Because the terminal CH,-groups of the arachidic acid point outward, the surface of

Fig. 3. Schematic picture of the molecular arrangement in perfect CdCZo layer-systems.

the slide is now hydrophobic, and on the next dipping two additional layers are deposited. This process may be repeated many times until the layer system has the desired thickness, which is given by the number of deposited arachidic-acid layers times the thickness of a single layer. The deposition of the monolayer can be monitored by watching the float moving forward under the influence of the weight as the film is removed from the water surface. When the film has been used up, the float is removed and a hydrophobic barrier is moved across the surface of the water. Thereby the remainder of the film is pushed toward the rear end of the trough, where it can be removed by a strip of filter paper, and the main area of the water surface is now clean again and can be covered with a fresh monomolecular film. The deposition of fatty-acid layers is aided by the presence of bivalent metal ions in the water upon which the film is formed. Thereby the fatty acid is converted into its salt to a degree which depends on the ion concen-

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tration and the p,-value of the water. Barium chloride and stearic acid (C,,H,,COOH) have been most often used to build monolayer systems (BLODGETT[1935]). The work we are concerned with in this article, however, has been done almost exclusively with monolayers of arachidic acid (C,,H,,COOH) formed on water which contains cadmium chloride. Although the technique appears to be very simple, the successful preparation of monolayer systems depends on a number of peculiarities of method that are not self-evident. Thus it may be worthwhile to consider here some important details of the preparation of cadmium-arachidate layers (DREXHAGE [19663). The experimental set-up used successfully by the author is shown in Fig. 2. After thorough cleaning of the trough it is filled to the brim with distilled (once is enough) water, and about one gram of cadmium chloride (CdCI, .2.5 HzO; analytical grade) is dissolved molar. in the water, in order to obtain a cadmium-ion concentration of about 5 x With this concentration cadmium-arachidate (abbrev.: CdCzo) layers are best deposited at a p,-value of about 5.5, which coincides with the acidity of air-saturated water and therefore maintains itself automatically, and no buffering or other additives to the water are called for. By comparison, barium-stearate layers are best deposited at a p,-value of 8.5 (BLODGETT [1935]), which is more difficult to keep constant due to the continuous absorption of carbon dioxide from the air. The good optical quality of cadmium-arachidate [1939], and the writer found them superior t o layers was already noticed by BLODGETT fatty-acid monolayers formed on water that contained BaZ+,W + ,C a Z + ,M n 2 + , C o z + or ZnZ+-ionsat various concentrations. The arachidic acid is applied to the water surface from a solution containing -1.5 mg per ml of solvent. If the concentration is too high, evaporation produces saturation before the monomolecular film is fully developed, so that some crystalline arachidic acid is left behind, contaminating the film. On the other hand, a much smaller concentration of arachidic acid is not recommended, because a large quantity of solvent has to be transferred to the surface. Thus an unnecessarily large amount of non-volatile impurities present in the solvent is incorporated into the film. It is advisable to use only purified solvents that evaporate without any noticeable residue, which is usually the case after a thorough distillation over a column. The arachidic-acid solution should be handled only with extremely clean, surfactant-free glassware, the cleanliness of which can be ensured by rinsing it before use with distilled chloroform. Any contact of the solution with materials like rubber, cork, plastics etc. is to be avoided. Because chloroform attacks plexiglass, some caution is required on application of the solution to the surface. A watchglass on the bottom of the trough catches droplets that may fall through the water surface. The most serious problem encountered in the preparation of high-quality layers concerns the purity of the arachidic acid, because any impurities invariably become incorporated into the film and may ruin its homogeneity. Arachidic-acid samples that are commercially available, are often not sufficiently pure, even if labeled “purissimum”*. This usually causes a foggy appearance of layer systems with more than about 50 CdCzo monolayers, and the glass plates tend to emerge wet from the trough, after about 100 CdCZO monolayers have been deposited. With pure arachidic acid, however, the plates emerge entirely dry even after deposition of more than 500 monolayers, and scattering of light is almost

* The author expresses his gratitude to Dr. H. Lange, Henkel and Cie. GmbH, Dusseldorf, Germany for a valuable gift of highly pure arachidic acid. - Recently arachidic acid of good quality has become available from Analabs, Inc., North Haven, Connecticut.

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undetectable by the naked eye. Although the quality of such layers can be considered excellent for many applications, it is still by far inferior to the optical quality of thin films prepared by modern evaporation techniques. Before the actual film deposition begins, the surface is covered with a scavenging arachidic-acid film. Impurities in the trough solution that would otherwise contaminate the operational film enter the scavenging film and are swept away upon its removal. After scavenging has been repeated several times, the trough solution can be considered clean and is now ready for reproducible film deposition. The same solution can be used unchanged for several weeks. The microscope slides to be used as substrates for the monolayer systems are cleaned preferably with an ultrasonic cleaner containing an alkaline cleansing agent. Without an alkaline treatment, most glasses tend to shed the cadmium-arachidate layers after a few dippings. Subsequently the slides are rinsed thoroughly with distilled water and dried in a stream of hot air. Multilayer deposition on the hydrophilic glass surface yields an odd number of layers. Evaporated gold or silver films on glass are hydrophobic, and consequently an even-numbered set of monolayers is obtained. It is advisable to watch closely the behavior of the meniscus at the glass surface during the dipping, which is preferably done rnanuaZ1y with an apparatus like the one shown in Fig. 2. In case the operator observes any irregularities, he can stop the deposition at any moment. The dipping speed is not critical and is typically 10 cm/min; only the first monolayer may require more time for dry deposition. To ensure reproducible results, a film that has been sitting more than about five minutes on the trough solution should be replaced by a fresh one. In this way, the accumulation of dust particles from the atmosphere is kept at a negligible level.

A more refined deposition technique for CdC,, and other monolayers has been reported by BUCHER,ELSNER,MOBIUS,TILLMANN, WIEGAND[1969] {see also MOBIUSand BUCHER[1972]), in which a novel trough construction is used. The surface of the water is cleaned by sucking the film from the surface with an aspirator, and the CdC1, solution lost during this procedure is automatically replenished from a reservoir. In addition, the surface pressure is monitored during the deposition process, which can be helpful with monolayers whose properties depend sensitively on the surface pressure. For the deposition of cadmium-arachidate layers, however, this apparatus is unnecessarily sophisticated. 2.2. PREPARATION O F MONOMOLECULAR DYE LAYERS

The cadmium-arachidate layers, which are transparent in the visible and near ultraviolet part of the spectrum, serve in the work of this article as spacers to separate dye layers from each other or from metal mirrors or dielectric interfaces. Various types of thin layers of fluorescent dyes have been employed in studies of optical phenomena by several authors, e.g., DRUDEand NERNST[1892], SELBNYI[1911, 19381, FREED and WEISSMAN [ 19411, SHKLYAREVSKII, MILOSLAVSKII, GOLOYADOVA [19641, KOSSEL [19581. In most of this work the resolution was limited by the thickness of the dye layer, which was only slightly smaller than the wavelength of light. For a

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higher precision this thickness would have to be reduced with the ultimate goal of a dye layer of molecular cross section, i.e. a monomolecular layer. The first attempt to prepare such monomolecular layers of fluorescent dyes was reported by ZWICKand KUHN[1962]. They observed that certain dyes were adsorbed, when a glass plate coated with barium-stearate layers was immersed into an aqueous solution of the dye. This technique was applicable to many dyes, mostly of the cyanine type, and from the light absorption it was concluded that the amount of adsorbed dye corresponded in many cases roughly to a monolayer coverage (DREXHAGE, ZWICK,KUHN [1963], DREXHAGE [1964]). However, the dye layers, prepared by this technique, were not truly monomolecular in the sense that all dye molecules are located at the same clearly defined distance from the surface. Some of the dye molecules apparently diffuse during the adsorption process, which occurs by ion exchange with protons (DREXHAGE [1966]), into the underlying fattyacid layers, to an extent which depends particularly on the composition of the adsorbing layer system (barium stearate, barium arachidate or cadmium arachidate), as was determined by energy-transfer experiments (DREXHAGE

Fig. 4. Molecular structures of arachidic acid (a), N,W-distearyloxacyanineiodide (b), and a europium dibenzoylmethane complex (c).

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[19661, BARTH,BECK,DREXHAGE, KUHN, MOBIUS,MOLZAHN,ROLLIG, SCHAFER, SPERLING, ZWICK[1966]). The irregularities in these dye layers can extend over distances of more than 100 A, which renders them unsuitable for experimental testing of energy-transfer theories. Very regular monomolecular dye layers can be prepared by a technique analogous to the deposition of CdCzo monolayers (DREXHAGE [1966], DREXHAGE and KUHN[1966], BUCHER,DREXHAGE, FLECK, KUHN,MOBIUS, SCHAFER,SONDERMANN, SPERLING,TILLMANN, WIEGAND[ 19671). The special dyes developed for this purpose (SONDERMANN [1971]) carry one or more stearyl substituents (Fig. 4) and hence have material properties very similar to long-chain fatty acids: When spread on a water surface, they stick the hydrophilic chromophore into the water, while the hydrophobic stearyl chains tend to point upward. Because all the work discussed in the later sections has been conducted with dye layers of this type, some details of their preparation will be reviewed. A compound representative of the class of cyanine dyes is the oxacyanine derivative the structure ot which is given in Fig. 4. Mixed monolayers of this dye and cadmium arachidate (molar mixing ratio of dye and CdCzo 1 : 10 or smaller) are obtained by spreading a chloroform solution that contains the components in the desired ratio on the same submolar CdCI, solution) as is used for the preparation of cadmium-arachiphase (5 x date layers. Such films have properties very similar to CdCZ0films and can be deposited in exactly the same fashion. In one dipping process a hydrophobic surface is covered with two monomolecular layers, whose chromophores are in close contact. Because the chromophores of the two layers are located practically in the same plane, such a bilayer can be regarded for many purposes as a single monomolecular dye layer. Many other dyes of this type can also be used to form mixed layers with cadmium arachidate, thus providing a wide range of wavelengths in absorption and fluorescence.

The spectra of absorption and fluorescence are rather broad with most organic dyes (Fig. 5 ) . The peak absorption of a monomolecular layer is of the order of a few percent and decreases with decreasing mixing ratio between dye and CdC,, . The chromophores in cyanine-dye layers are oriented parallel to the surface, and the transition moments for absorption and fluorescence have this same orientation (see Q 4). An important factor in fluorescence studies is the quantum yield of fluorescence. It varies considerably, even in a class of closely related dyes, and is quite high (nearly 100 %) in some monomolecular dye layers considered in this article. For instance, the blue fluorescence of a deposited oxacyanine monolayer (dye :CdC,o = 1:10) is intense enough to be visible in a partly darkened room, when excited with an ultraviolet lamp. Besides the cyanine dyes, an entirely different fluorescent compound has been used extensively: a europium complex whose structure is also depicted

I",

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Wavelength, nm Fig. 5. Absorption (solid) and fluorescence (dashed) spectrum of mixed double layers of N,W-distearyloxacyanine and cadmium arachidate (dye : CdCzO= 1 : 10); dye chromophores face to face.

in Fig. 4.* It has a strong absorption band with a maximum at 370 nm, which can be ascribed to the organic ligands. The energy absorbed in this band is rapidly transferred intramolecularly to the europium ion, which then fluoresces in a number of relatively narrow bands near 600 nm (see 0 8). The decay time of the fluorescence is about 1 msec and therefore very convenient to measure. The transition moments of the fluorescence are randomly oriented, as would be expected from the highly symmetrical environment of the europium ion. Monolayers of the pure europium complex are extremely viscous and for this reason cannot be deposited: The slide simply slips through the film without picking up a monolayer. However, on admixture of cholesterol in a molar ratio of 1 : 1 and under an increased surface pressure of 35 dyne/cm, smooth monolayer deposition on a hydrophobic [ 19661). Monolayers showing particularly reproducible surface is achieved (DREXHAGE fluorescence were obtained on admixture of tripalmitin (molar ratio of dye and tripalmitin 1 : 3) under a surface pressure of 40 dyne/cm or of heptadecyl methyl ketone (molar ratio 1 : 2) under 30 dynelcm. Since the viscosity of these films decreases with increasing temperature, it is sometimes useful to raise the water temperature by a few degrees above room temperature in order to ensure smooth deposition of the monolayer.

* The europium-dibenzoylmethane complex was prepared by Dr. W. Sperling, University of Marburg, with a procedure similar to the one of BAUER,BLANC,Ross 119641.

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The deposition of these monolayers requires much more attention than the preparation of CdCzo or mixed cyanine-dye layers. Because the europium complex is sensitive even to weak acids, its monolayers decompose slowly due to the influence of the free arachidic acid in the underlying and covering CdCZolayers, as can be judged from the gradual disappearance of the red fluorescence. Mixed layers with tripalmitin or heptadecyl methyl ketone are much more stable in this respect than the layers prepared with cholesterol. In contrast to the CdCzo and mixed cyanine-dye layers, the films of the europium complex are formed from a solvent mixture consisting of benzene and acetone in the volume ratio 4 : 1 on pure water without addition of a salt, whosep,-value is maintained between 6.5 and 8.0 through the addition of a few drops of sodium-hydroxide solution. Even if prepared most carefully, the monolayer often is inhomogeneous in that some areas fluoresce more efficiently than others. Furthermore, fluorescence measurements on the deposited monolayers of this compound must be done under carefully controlled conditions, since, e.g., the fluorescence decay time depends on the temperature and on the oxygen content of the atmosphere surrounding the layer system.

From the method of preparation of these dye layers it is expected that the chromophores form a single layer of molecular cross-section. This hypothesis is supported by all optical experiments, in which the characteristic distance is about lOOOA or larger. However, in experiments on energy transfer (critical distance about 100 A) some differences between theory and experiment are observed (see 9 7), which are most likely caused by irregularities in the dye-layer systems.

8 3. Optical Properties of Cadmium-Arachidate Layers In the work to be discussed in following sections, the monomoIecuIar dye layers are embedded in a system of transparent cadmium-arachidate spacer layers. Knowledge of the thickness and the optical constants of this dielectric is the basis for the quantitative evaluation of the experiments. 3.1. DETERMINATION OF T H E REFRACTIVE INDICES

It has been discovered by BLODGETT and LANGMUIR [I9371 that multilayers of barium stearate are birefringent. They behave like positive uniaxial crystals with the optic axis oriented normal to the surface. This birefringence, which is obviously caused by the parallel alignment of the fatty-acid molecules within the film, is also observed with the cadmium-arachidate layers with which this article is primarily concerned. To determine the ordinary refractive index no of a layer system,* the above authors coated glass plates of different refractive index with systems whose optical thickness was chosen t o be a quarter wavelength. Whether

* Throughout this article the following notation is used: no for the ordinary and n, for the extraordinary refractive index of a layer system; n 3 and n , for the refractive indices of the supporting glass and of air or a n immersion liquid, respectively.

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the intensity of the light, reflected by the sample, is decreased or increased by the 1/4-layer, depends upon the refractive indices of both the layer system and the substrate (BORNand WOLF[1970] p. 63). For normal incidence the reflectivity of one side of the sample is given by

where n, and n3 are the refractive indices of air and glass, respectively. Through the selection of glasses with appropriate refractive indices, the ordinary refractive index of the layers can be determined visually with a precision of three decimal places. With this method it was observed that no = 1.491 at 1= 589 nm for barium-stearate layers (BLODGETT and LANGMUIR [ 19371). The ordinary refractive index of cadmium-arachidate layers, determined by the same technique, was found to be no = 1.526 for green light (DREXHAGE [1966]), whereas BLODGETT [1939] reported the value 1.54. The accuracy of this technique can be slightly increased by a quantitative measurement of the reflectivity. In this case it would suffice to make measurements on only one sample, because the value of no can be calculated from R with the help of equation (3.1). However, the accuracy of the method is limited in any case by the ubiquitous thin layers of unknown refractive index on the glass surface, which influence the amplitude and phase of waves reflected at the layer-substrate boundary in an unpredictable way. For the determination of the extraordinary refractive index n, BLODGETT and LANGMUIR [1937] employed three different methods. In the first method a sequence of multilayer steps was deposited, in staircase fashion, on glass of high refractive index, and the reflectivity of the sample was examined in dependence on the angle of incidence. Due to interference between the waves, reflected at the air-layer and layer-glass boundaries, the reflectivity of the steps varies periodically with their optical thickness. The contrast between the diffeient steps depends on the reflection coefficients at the boundaries involved and vanishes if one of these coefficients becomes zero. Thus, for light with the electric vector in the plane of incidence, the contrast disappears if the plate is viewed at Brewster's angle (IBr.*Using the independently determined value of no, the extraordinary index n, can then be calculated (see Appendix) with tan O,,

= ( n , / n , ) [ ( n z - n:)/(n:

- n:)]+.

(3.2)

The second method also employs a series of steps, deposited on high-index

* Throughout the article the angle of incidence on the side of the air or immersion liquid is denoted 0; inside the layer system the angle of incidence is called 3~ for the wave normal and for the ray (see Appendix).

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glass or on chromium. Again the light reflected by the step sequence is investigated; however, now at a larger angle of incidence, where the contrast is high. The number 15 of monolayers that marks the difference in thickness between one step of minimum reflectivity and the next, is given by

mnd, cos a = +A.

(3.3)

Here d , is the thickness of a single layer, and LY is the angle of incidence of the wave normal inside the layer system, which is related to the angle of incidence 8 on the air side by Snell's law,

n sin a = n , sin 0.

(3-4)

The refractive index n of the layers depends, in the case of light polarized parallel to the plane of incidence, on the angle a (see Appendix):

n - , = n i 2 cos2 a+nC-, sin' a.

(3.5)

Thus the value of n, can be calculated from a measurement of 37 and 0 and an independent determination of no and d , . The third method makes use of an optical compensator to determine the relative phase retardation of the ordinary and extraordinary rays. It is somewhat awkward, because a total of about 5000 deposited monolayers was needed for the measurement. Nevertheless, the results obtained by all three methods were in good agreement, and the value n, = 1.551 (2. = 589 nm) was given by BLODGETT and LANGMUIR [I9371 for barium-stearate layers. Entirely different methods, which involve fluorescent dye layers, have also been used to determine the refractive indices no and n, of CdC,, layers. The values no = 1.522 and n, = 1.59 (both at 1= 612 nm) were observed by FLECK [1969] and n, = 1.60 (3. = 405 nm) by FORSTER [1967]. Details of these measurements are given in 5 8 and 5 6, respectively. The remarkably strong birefringence (n,-no = 0.07) of the CdC,, layers can not be ignored in quantitative experiments with monomolecular dye layers, as will become evident in the later sections. 3.2. DETERMINATION OF LAYER THICKNESS

All optical methods for the measurement of the layer thickness make use of multiple-beam interference in one way or another and involve a comparison of the thickness with the wavelength of the measuring light. We mention here only the three most important methods, which have yielded accurate values of the layer thickness. The oldest technique (BLODGETT [ 19351) employs plates of high-index glass, which are coated with a stair-like succession of fatty-acid layers. The

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reflectivity of such a plate varies periodically with the optical thickness of is determined visually. The optical the layer system, and the value of thickness nodl of a single layer then follows from equation (3.3). In order to establish a sufficiently high contrast between the steps, the reflection coefficients at the air-layer and layer-substrate interface should be comparable, which is the case, if high-index glass is used as support for the layers. Instead of such glass a metal of low reflectivity, e.g., chromium or tantalum, can be used. In this case a particularly good contrast is obtained at large angles of incidence, where the reflection coefficient of the air-layer interface is relatively high (BLODGETT and LANGMUIR [1937]). Using the independently determined refractive index no = 1.491, these authors found the value d , = 24.40 8, for the thickness of a barium-stearate monolayer. Another now widely applied method, called ellipsometry, was introduced to the study of built-up layers by ROTHEN[1945]. Recently it was used in the case of CdC,, layers by ENGELSEN[1971] and STEIGER [1971]. In this method monochromatic light, plane polarized at 45" to the plane of incidence, is reflected from a metal surface at an oblique angle. Since the components in and perpzndicular to the plane of incidence experience different changes in phase and amplitude, the reflected light is elliptically polarized. If a thin transparent layer is deposited on this metal surface, multiple-beam interference affects phase and amplitude of the two reflected components to a different degree, because the reflection coefficient of the layer-air interface is different for the two components. Obviously, the interference depends on the optical thickness of the applied layer, which is measured by the change in ellipticity of the reflected light. With a fairly simple apparatus the measurement can be made to an accuracy of +0.1 A. The relation between the parameters of the ellipse and the layer thickness is rather complex, and it is practical to calibrate the apparatus with layers of known thickness and make relative measurements. In the case of birefringent layers the theory becomes even more complicated, because both the ordinary and the extraordinary ray are involved (ENGELSEN[1971]). Nevertheless, the technique has proven useful in many applications (ROTHEN[1968]). It has also been extended to transmission measurements (ENGELSEN[ 19721). A third method, conceptually very simple, is closely related to the work of Blodgett and Langmuir, the main difference being a quantitative measurement of the light intensities in place of visual observation (DREXHAGE [1966], DREXHAGE and KUHN[1966],* BUCHER,DREXHAGE, FLECK,KUHN,MOBIUS,

m

In this paper Fig. 2b is incorrect. The correct plot is given by BUCHER,DREXHAGE. FLECK, KUHN,MOBIUS, SCHAFER, SONDERMANN, SPERLING, TILLMANN, WIEGAND[ 19671 and in Fig. 7 of this article.

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I N T E R A C T I O N OF L I G H T W I T H M O N O M O L E C U L A R D Y E L A Y E R S

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3

SCHAFER, SONDEKMANN, SPERLING, TILLMANN, WIEGAND[1967]). A microscope slide is coated on one side with an evaporated semitransparent silver mirror. A certain number N of the monolayers to be studied (total thickness d = N d , ) is then deposited on top of the mirror, and the light transmission is measured, preferably with reference to another silver-coated slide of the

t Silver Fig. 6. Scheme for determining the thickness of fatty-acid layers. The transmission is modified by multiple interference within the layer system.

same initial transmission. The change in transmission that occurs due to multiple-beam interference within the layer is related to its optical thickness (Fig. 6), and the relative transmission T, of the sample is given by the Airy function T, = G~'t;,/[l + p 2 r ; l -2pr21 cos (4-S)]; (3.6)

4

= 4nNnd, cos

a/A.

(3.7)

Here z and t,, are the amplitude transmission coefficientsfor a light wave 'entering the layer system through the silver mirror and passing the layer-air interface, respectively. Similarly, p and rz denote the amplitude reflection coefficients at the layer-silver and layer-air interface, whereas 6 is the phase shift occurring on reflection at the layer-silver interface. The factor G takes into account the presence of the uncoated surface of the slide and the transmission T, of the reference slide: G = t ~ 3 / T r ( l - r ~ l R 3 The ) . quantity t i 3 is the transmission coefficient for light passing the air-glass interface, and r31 is the reflection coefficient for light incident from the glass onto this interface. Although the reflectivity R , of the system mirror + layers varies

,

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P R O P E R T I E S OF C A D M I U M - A R A C H I D A T E

179

LAYERS

with its thickness (see 5 4), the value of G varies only slightly and may be considered constant in most practical cases. According t o equations (3.6) and (3.7) the relative transmission T, is a periodic function of the number N of monolayers. Hence the thickness dl of a single cadmium-arachidate layer can be determined from a plot of T, versus N (Fig. 7) if the refractive index n of the layers is known. With no = 1.526 at 1 = 546 nm and perpendicuIar incidence (cos a = 1) the value d, = 26.4 A+O.l A was obtained from the data given in Fig. 7. Applying Fresnel's formulas one can calculate the quantities r Z 1 ,t , , , r31 and t , , ,

I

O

00

Oo

ODo

0

0

0

0

0 0

0

0 0

0 0

0

0

0

0 0 0

3 c

L o

1.2,

z lo0

0

0 0

0

0 0

0

0 0 0

0

0

0

0

0

oo

OO

ooooo

Number

0 -

0

0

OO

I

0

0

0000

of monolayers

1

,

I

N

Fig. 7. Plot of relative transmission T. versus number N of cadmium-arachidate layers prepared according to section 2.1. I = 546 nm; a = 0; T,= 0.240. All values were obtained on the same sample by increasing N in steps of two at a time. (From DREXHAGE [1966].)

whereas p, z and 6 are dependent on the density of the silver coating and can be determined from the plot in Fig. 7. Values of T,,calculated from equation (3.6) with the parameters so determined, agreed with those observed experimentally within 0.5 %, indicating a very good reproducibility in the deposition of the CdCzo layers. It must be borne in mind, however, that all optical methods described here yield a layer thickness averaged over the irradiated area. Thus any microscopic irregularities, randomly incor-

180

INTERACTION O F LIGHT WITH MONOMOLECULAR D Y E LAYERS

[IV,$4

porated into the layer system, would be averaged out and can not be detected with these methods. The sensitivity of this method depends on the value of the product p r , , (equation (3.6)). With a silver mirror of high reflection coefficient ( p > 0.8) the relative transmission T,(N) can vary by more than the factor 2 (Fig. 7). Hence fairly accurate measurements can be accomplished even with commercially available photometers. It is particularly easy to make relative thickness determinations if monolayers of known thickness, e.g., CdC,, layers, are used for a calibration. The change in transmission that occurs on deposition of the layers of unknown thickness is compared with that produced by a certain number of CdC,, layers assuming the same refractive index for both. Furthermore the above technique can be used to determine the extraordinary refractive index n, of layer systems, if the angle of incidence at which the relative transmission T, becomes independent of N is measured (Brewster's angle, r , , = 0). With the otherwise determined ordinary index no the extraordinary index n, is then obtained from equation (3.2). It is to be mentioned here that there is a long-standing discrepancy between the thickness of monolayers measured by optical and by X-ray techniques, which has not yet been explained satisfactorily. The X-ray measurements yielded a markedly larger value of the thickness of barium-stearate [1938], layers than did the optical techniques described above (HOLLEY BERNSTEIN[1940]). A similar discrepancy in the case of CdC,, layers can be inferred from X-ray data reported by MA", KUHN,SZENTPALY [1971], who found d , = 28.0 8, (at 20 "C).* Moreover, these authors report a temperature dependence of the X-ray determined thickness (d, =28.7 A at -35 "C), which has not been noticed by other workers. The optical value for CdC,, layers has been confirmed recently by BUCHER[1970], who measured d , = 26.5 8,with a variation of the transmission method described above.

4 4. Coherent Scattering at a Dye Monolayer According to elementary dispersion theory a dye molecule can be considered as a damped electron oscillator, which carries out forced vibrations under the influence of the alternating electric field of a light wave. The oscillating electron is the source of a secondary wavelet which is coherent with the exciting light. In the case of many such molecules, i.e. bulk matter, * Contrary to these findings an X-ray determined value dl STEIGER [1971].

=

26.8 A is mentioned by

IV, §

41

COHERENT SCATTERING A T A DYE MONOLAYER

181

all the secondary wavelets interfere with each other and with the exciting wave, thus giving rise to the known phenomena of absorption, reflection, refraction and dispersion. In fact, all optical properties of matter can be explained on the basis of such atomistic considerations (BORNand WOLF [1970] p. 84). Because of the huge number of molecules in a three-dimensional medium the treatment of the interference of primary and secondary waves is rather complicated. The often considered plane layer of atomic oscillators is realized experimentally by a monomolecular dye layer (DREXHAGE [1966]). 4.1. THE PHASE OF THE SCATTERED WAVE

When a plane wave of light impinges normally on a monomolecular layer of dye molecules the oscillators are excited into forced vibrations that occur in all molecules with the same phase. The phase is, however, dependent on the frequency v of the exciting light and on the eigenfrequency vo of the oscillators. In the case v > vo in opposite phase. All the secondary wavelets emitted by the vibrating electrons are in phase and combine to form two plane waves propagating with and against the direction of the exciting wave (Fig. 8). It can be shown that the

Fig. 8. Scattering at a layer of oscillators excited by a plane wave (solid). The spherical wavelets arising from the oscillators add up to plane waves (dashed) moving in backward and forward direction. I n the case shown (v = y o ) the phase shift S has the value z, and maximum attenuation of the exciting wave results.

phase of these scattered plane waves lags behind the phase of the oscillators by +T (KUHN [1933] p. 340, KAUZMANN[I9571 p. 600). The scattered plane waves have therefore a phase shift* of $71 for v > vo with respect to the incident wave. This phase shift is directly related to the phase of the oscillators themselves.

*

As usual the term phase shift denotes a forward shift of the phase.

182

INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS

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The forward-scattered wave interferes with the exciting wave, which thus is attenuated in the region of resonance (Fig. S), but merely shifted in phase at frequencies that are remote from the eigenfrequency. The backwardscattered, i.e. reflected, wave is of extremely small intensity (about times the incident intensity in the region of resonance). However, its amplitude is only about ten times smaller than the amplitude of a wave reflected at the layer-air interface, which makes simple interference experiments feasible. In order to determine the phase shift of the reflected wave, a glass plate is coated with a monomolecular dye layer,* which then is covered with a number of (non-absorbing) CdCzo layers (DREXHAGE [1966]). The transmission of such a layer system (Fig. 6 with dye layer in place of the silver film) is in analogy to equation (3.6) given by

It is assumed here, that the refractive indices of glass and CdCzo layers are identical. The quantities z and p are here the amplitude transmission and reflection coefficients of the dye layer, whereas 6 is the phase shift of the wave reflected at the dye layer. If both sides of the slide are coated with

x 0

0

0

0

408nm 395nm

A

380nm

0

? A A

A

A

U

20

40

Number of CdC,,

60

L

60

cover-layers

Fig. 9. Relative transmission T. of a glass plate coated with a dye double layer on both sides as function of the number of CdCZOcover-layers for three different wavelengths within the absorption region of the dye (normal incidence). Values obtained on the same sample by increasing N in steps of four at a time. Dye double layers (chromophores face to face) consist of N,N'-distearylthiacyanine and cadmium arachidate (dye : CdClo = 1 : 10). (From DREXHAGE [1966].)

* Here and elsewhere it is understood that the glass plate is coated with, say, five CdC20 layers. before the dye monolayers are deposited. Dye monolayers, deposited on the hydrophilic glass surface, often show erratic optical properties.

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COHERENT SCATTERING A T A D Y E MONOLAYER

183

identical layer systems, the transmission T, of the sample relative to an uncoated slide (transmission T,) is described by

T, = T2/[T,(1-R:)]. The quantity R , denotes the reflectivity of the layer system for light incident from the glass. In all practical cases Ri is so small that it can be neglected in equation (4.2).

Wavelength X,nm Fig. 10. Relative transmission T. of the sample of Fig. 9 without additional CdCzo coverlayers as function of the wavelength I (normal incidence). (From DREXHAGE [1966].)

-

0

2

0

o?mooo00

-

0 0

0 0 00 0

T -

I

0

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I N T E R A C T I O N OF L I G H T W I T H MONOMOLECULAR DYE LAYERS

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04

According to equations (4.1), (3.7) and (4.2), the transmission T, is expected to be a periodic function of the number of CdC,, cover-layers. Experimentally this was indeed observed for a number of different wavelengths (Fig. 9). It is obvious from this figure that the phase shift 6 varies with the wavelength of the exciting light, and the plot of 6 as a function of the wavelength il (Fig. 1 1 ) exhibits the expected variation throughout the absorption band of the dye (Fig. 10). The shoulder in the plot of 6 versus 1 reflects a shoulder in the absorption spectrum. From such data as shown in Fig. 9 the reflection coefficient p of the dye monolayer can also be determined. At the wavelength at which 6 = 7c (Fig. 8) p is expected to be 1-7. A reasonably good agreement was observed in both cases studied (DREXHAGE [1966], ELSNER119691). 4.2. THE REFLECTIVITY OF DYE LAYER SYSTEMS

The wave, scattered backward by a dye monolayer, affects not only the transmission, but also the reflectivity of a monolayer system. In fact, the influence upon the reflectivity is the more pronounced, as was shown by DREXHAGE [1966] and BUCHER[1970]. The basic features of the effect become clear from an inspection of Fig. 12. Depending on the phase shift 6 and on the distance d, the interference between the wave, reflected at the dye layer, and the wave, reflected at the layer-air interface, may be constructive or destructive. Accordingly, the reflectivity of the layer-air interface is enhanced or reduced by the incorporated dye layer. In order to calculate the reflectivity of such a thin film all partial waves [1956]), and are added in the usual manner (MAYER[1950] p. 144, WOLTER for the reflectivity R, (light incident from the air) and R3 (light incident from the glass) the following expressions are obtained (DREXHAGE [1966]): R , = [ p 2 + r i l -2pr2, cos ( 4 - 6 ) ] / [ 1 + p 2 r i l -2pr2, cos (4-41,

(4.3)

R3 = p 2 + r ; 1 ( ~ 4 + p 4 - 2 ~ 2cos p 2 26)+272pr21cos ( 4 + 6 ) - 2 p 3 r 2 , cos (4-6) 1+ p 2 r i l -2prz1 cos (4-6)

(4.4)

Since the uncoated side of the plate gives rise to multiple reflections, the reflectivities R,, and R,, of the sample with reference to an uncoated glass plate (reflectivity R,) are given by RSi = [Ri + 6 i ( T 2 - R i R ~ ) I / [ R ~ ~ - ~ I R ~ ) (4.5) I

and Rs3

=

[ r i i + R 3 ( 1 - 2 r i i ) ] / [ R ~ ( l - r ~R3)I. i

(4.6)

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S 41

COHERENT SCATTERING A T A DYE MONOLAYER

185

b) 6 = r , n d = X / 4 :

-;-..I?-*d ) 6 = 3 ~ / 2nd , =3x/8 :

-..._.._..._....,

,_,........ .......

______. ..............___

7 " . . . C.__._.... .'

&--

~

,

\..-/'

0

4.-

0

R, < rz:

&&

Fig. 12. Influence of a monomolecular dye layer on the reflectivity of the interphase between air and CdClo layers (light incident from the air). Cases a) and b): wavelength of light corresponding to absorption maximum of dye, 6 = n;cases c) and d): wavefength of light much greater, 6 = tn.Reflectivity is enhanced (cases a) and c)) or decreased (cases b) and d)) depending on the optical path nd. Amplitudes of reflected waves are not to scale.

Here it is again assumed that the refractive indices of glass and CdCzo layers are identical. The experimentally observed reflection spectra Rsl (A) and Rs3(A)(Fig. 13) were found in good agreement with equations (4.5) and (4.6), if ~ ( 2was ) taken from the absorption spectrum. It is noteworthy that the dye monolayer has an influence on the reflectivity even at wavelengths where it does not show any absorption (+&curves in Fig. 13). This reflects the fact that the electron oscillators still vibrate with a significant amplitude even at these frequencies remote from resonance.

186

I N T E R A C T I O N OF L I G H T W I T H MONOMOLECULAR DYE LAYERS

[IV, $ 4

I .02

-

ul

1.00 x

._ > .c

c

V

aJ I l l

cL 0.9E

0.9E

I

I

I

350

400

450

Wavelength X ,

500

nm I

+12Cd C2o

0.96

I

350

W 400

I 450

500

Wavelength A, nm

Fig. 13. Reflectivity of a glass plate carrying on one side a single dye layer (NJV”’disteary1thiacyanine : CdCzo = 1 : 10) relative to an uncoated glass plate; light incident a) from the air and b) from the glass. Solid curve: no additional CdCzo layers. The addition of 12 or 24 CdCzo layers corresponds to an optical path of -)A and -42, respectively. (From BUCHER [1970].)

However, since the phase shift 6 is 37r or 3n, no attenuation of the incident wave occurs. In the case of a three-dimensional medium this corresponds to the familiar situation that, while there is no light absorption, the wavelength in the material is different from vacuum (refractive index greater or smaller than unity).

1v.

I41

C O H E R E N T S C A T T E R I N G A T A D Y E MON OLA Y ER

187

More complicated interference systems, involving a dye monolayer, have been treated by BUCHER[1970]. He showed theoretically as well as experimentally that the transmission of an interference filter, which contains a monomolecular dye layer between two metal mirrors, can only be understood, if the reflection at the dye layer is taken into account. This is essential for a correct interpretation of studies on electrochromism, in which the absorption of the dye layer is changed under the influence of an electric field (BUCHER[1970], BUCHERand KUHN[1970]*). Because in several related papers (BUCHEK,WIEGAND,SNAVELY,BECK, KUHN [19691, SCHMIDT, REICH,WITT [1969, 19711, KLEUSERand BUCHER[1969], SCHMIDTand REICH[1972a, b]) this difficulty was not appreciated, the interpretation of the experimental results given there is in doubt. The absorption and reflection spectra of monomolecular dye layers have been measured with sensitive single-beam photometers, which were described by DREXHAGE [1964], BUCHER,ELSNER,MOBIUS,TILLMANN, WIEGAND [1969] and BUCHER[1970]. 4.3. THE ABSORPTION OF DYE LAYER SYSTEMS

We have seen that the light response of a monomolecular dye layer, embedded in a system of CdC,, layers, can be understood by describing the dye molecules as elementary oscillators, which coherently emit secondary waves of the same frequency as the exciting light. This must not be confused with the emission of fluorescence, which takes place after the absorption of energy by the oscillators and which is not coherent with the exciting radiation and generally of lower frequency. Some details of fluorescence emission will be covered in later sections. It was shown that the transmission T and the reflectivities R , and R3 of a dye layer system can be calculated by considering the waves scattered by the dye layer. The absorption, i.e. the fraction of the incident light energy, absorbed in the dye layer, can then be found from the identities A,+R,+T

= 1;

A 3 + R 3 + T = 1.

(4.7)

The value of T is independent of the direction of irradiation. However, in unsymmetrical layer structures like the one shown in Fig. 6, the reflectivity is generally dependent on the direction of incidence, as is evident from equations (4.3) and (4.4). Therefore the amount of light energy absorbed by the * The expression for the transmission of a dye layer between metal films, used in this paper, is incorrect, since it is based on the presumption that absorption and transmission are complimentary. However, the error thus made is only about 15 % because of the low reflectivity of the metal films and is smaller than the experimental errors.

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I N T E R A C T I O N OF L I G H T WITH M O N O M O L E C U L A R D Y E L A Y E R S

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$4

dye layer depends on the direction of irradiation, and we must distinguish between the quantities A , (light incident from the air) and A , (light incident from the glass), which can be calculated from equations (4.1), (4.3), (4.4) and (4.7). If the absorption is very weak, as is usually the case with monomolecular dye layers, it is obtained for normal incidence (DREXHAGE [1966]) A , z A(l-r:l); A , z A [ 1 + r ~ , + 2 r 2 , cos (4nn0d/A)].

Here A denotes the absorption of the dye layer embedded in an extended isotropic medium, i.e. n, = no = n 3 . The relations (4.8) and (4.9) can be immediately understood, if it is realized that for small absorptions the absorbed energy is proportional to the square of the local field amplitude. When the light is incident from the side of the glass, a standing light wave (see 5 5 ) is formed in front of the layer-air boundary, whose amplitude Ed is given by Ed = E, [ 1 + r:, +2rz1 ~os(4nn,d/1)]~. In the other case (light incident from the air) the dye layer is not within a region of standing light waves, and thus the absorption A , is independent of the distance d. The marked variation of A , with distance d (by a factor of 2.3 in the systems considered here!) can not be ignored in quantitative monolayer studies. In order to circumvent the distance dependence of the absorption, it is sometimes advisable to irradiate dye layer systems from the side of the air. The quantity A can be obta.ined from measurements of the transmission T as a function of d = N d , (Fig. 9). For small values of A (e.g., A O.l), it is only necessary to determine the maximum and minimum values of the [1966]): function T ( d ) (DREXHAGE A

=

1-+(Tmax+Tmin).

(4.10)

It is important to note that the quantities A , A , and A , are, in general, different from each other and from 1 -T. In particular, 1 -T is not to be confused with one of the absorptions, as is done quite often in the published literature, including earlier papers by this author. If both sides of the glass are coated with identical laye1 systems, the absorption A of one dye layer can be obtained in analogy to equation (4.10) from (4.11) 4.4. THE ORIENTATION OF THE DYE CHROMOPHORES

Because the chromophores of most organic dyes are anisotropic, it is to be expected that a non-random orientation of the dye molecules in the

IV, 5

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COHERENT SCATTERING A T A DYE MONOLAYER

189

monolayer gives rise to dichroism, i.e. a dependence of the absorption on the polarization of the light. In fact, this phenomenon has been used to [19661, TILLMANN determine the orientation of the chromophores (DREXHAGE [19661, BUCHER,DREXHAGE, FLECK, KUHN,MOBIUS, SCHAFER, SONDERMANN, SPERLING, TILLMANN, WIEGAND[I 9671). The slide coated with the dye layer was irradiated at an angle 6J # 0, and the transmission was measured with light, polarized perpendicular and parallel to the plane of incidence. Due to the refraction at the air-layer boundary the light traverses the dye layer at the angle c1 (equation (3.4)), and for perpendicular polarization the absorption A, is given by A,

=

~ / c o as = An,/(n,t-sin2

e)+.

(4.12)

The increase of the absorption by the factor I/cosu is caused by the increased number of dye molecules seen by the light and occurs for both polarizations.* In the case of parallel polarization, however, the absorption A l l depends in addition on the particular orientation of the chromophores. We may distinguish the following cases: a) transition moments randomly oriented, =

~ / c o as = ~ n , / ( n , -sin2 2

e)+;

(4.13)

b) transition moments randomly oriented in layer plane, (4.14)

A ~= , A cos c1 = A(n,2 -sin2 e)+/n,;

c) transition moments perpendicular to layer plane, =

~ ‘ s j antan a

=

A’ sin2 O/[no(n,t-sin2

e)+].

(4.15)

Since A = 0 in case c, the symbol A’ is used here for the absorption. For the dichroic ratio D = A,/Al, we obtain from the above relations in case a D = 1, in case b D = ng/(ng-sin2e) and in case c D = 0. At the (often used) angle of incidence 0 = 45” and with n, = 1.53 we obtain from equations (4.12) to (4.15) A , = 1.13 A and in casea: A l l = 1.13 A, case b: All = 0.89A, case c: All = 0.24A’,

D = 1; D = 1.27; D = 0.

For the practical determtnation of A , and All one has to proceed, as was described in section4.3 for the quantity A , i.e. one measures the transmission of the sample as function of the distance d and calculates A , and A l l in

*

For the sake of simplicity we neglect here the birefringence of the layers and assume

n = no for both polarizations.

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I N T E R A C T I O N O F L I G H T W I T H M O NO M O L E C U LA R D Y E LA Y ER S

i

0.061

.

I

I

I

Perpendicular polarization

I

I

I

20

40

60

[IV, 5 5

1

1

Number of Cd Cz0 cover - layers Fig. 14. Relative transmission T, of a glass plate coated with a dye double layer on both sides as function of the number of CdClo cover-layers for 3, = 408 nm and oblique inci[1966].) dence (0 = 45"). Same dye layers as in Figs. 9, 10 and 1 1 . (From DREXHAGE

analogy to equation (4.10) or (4.11). In case of the cyanine-dye monolayers (Fig. 14) considered in sections 4.1 and 4.2, it was observed experimentally at the wavelength of maximum absorption (A = 408 nm): A , = 1.13 A and D = 1.26 (DREXHAGE [1966]). From the latter value it could be concluded that the transition moments are oriented according to case b. It is to be noted that a qualitative observation of a dichroism in transmission measurements does not allow the above conclusion to be drawn, as was erroneously stated, e.g., in the article by BUCHER,DREXHAGE, FLECK, KUHN, MOBIUS,SCHAFER,SONDERMANN, SPERLING, TILLMANN, WIEGAND[19671. Even in case of random orientation of the transition moments (case a, A , = All) the transmission is different for the two polarizations. The agreement between the experimentally determined values of A , and A II and the theory given above is of particular interest, as it shows that the laws governing reflection and refraction at a dielectric boundary apply even at a distance as small as about 25 A. I

0 5. Standing Light Waves When two waves of the same frequency, travelling in opposite directions, interfere with each other, a so-called standing wave is formed. In the case

IV,

8 51

191

STANDING LIGHT WAVES

of light waves this phenomenon was first demonstrated by WIENER[I8901 with a thin layer of photographic emulsion, and somewhat later by DRUDE and NERNST [1892] with a fluorescent film. By using a monomolecular dye layer as a probe standing light waves can be studied with greatly enhanced [1966], DREXHAGE and KUHN[ 19661, BUCHER,DREXresolution (DREXHAGE HAGE, FLECK, KUHN,MOBIUS, SCHAFER, SONDERMANN, SPERLING, TILLMANN, WIEGAND[19671, DREXHAGE [1970bl). In these experiments a microscope slide was coated with a highly reflecting silver or gold mirror and then covered with a stairlike succession of CdC20 layers. This was done by making successive dips, each of which was a few millimeters shorter than the previous one. On top of these CdCz0 stairs a monomolecular layer of a fluorescent dye was deposited. Thus the dye molecules are separated from the mirror by a stepwise increasing distance, which is determined by the number of underlying CdCIO layers. If the slide is irradiated with monochromatic light polarized perpendicular to the plane of incidence, the incident and reflected wave interfere, and the resultant electric-field amplitude Ed is given by

Ed = Eo[l

+p2 +2p COS (4xn0d COS

- d)]’.

(5.1)

Here we neglect the reflection at the air-layer boundary and assume nearly perpendicular incidence on the mirror, whose amplitude reflection coefficient and phase shift are called p and 8, respectively. At large angles of incidence and with polarization perpendicular to the plane of incidence, the reflection at the air-layer boundary can not be ignored; on the other hand, with parallel polarization the standing-wave phenomenon is more complicated owing to the fact that the electric vectors of incident and reflected wave have different directions (WOOD [I9341 p. 210, BORNand WOLF [1970] p. 277). For weak absorptions the energy absorbed by the dye layer is proportional to the square of the local electric-field amplitude E d . The fluorescence intensity is proportional to the absorbed energy and is expected to be extremely small at the nodes of the standing wave, which occur at distances di,given by the condition 4xn0di cos a/).-S = ( 2 i - 1 ) ~ ; i = 1,2, 3 . . ..

(5.2)

Between these nodes the absorption and thus the fluorescence intensity should vary according to equation (5.1) and reach maxima at distances dj (antinodes), given by 4nn,d,cos a l l - 6 = 2 ( j - 1 ) x ;

j = 1,2, 3 . . ..

(5.3)

It was found that the experimentally observed position of the nodes agreed

192

I N T E R A C T I O N OF L I G H T W I T H M O N O M O L E C U L A R DYE LAYERS

[IV,45

very well with equation (5.2) (DREXHAGE [ 19661). However, the fluorescence intensity on the steps between the nodes does not follow equation (5.1), and, moreover, the fluorescence color depends on the distance from the mirror (Fig. 15). This phenomenon is caused by an influence of the mirror on the fluorescence which will be discussed in $ 8 . This side effect makes it very difficult to verify equation (5.1) quantitatively in the particular case described. It should be possible to eliminate this difficulty, if instead of a metal mirror a dielectric reflector were used that has a high reflectivity for the wavelength of the exciting light, but low reflectivity for the wavelengths of the fluorescence. The position of the nodes of the standing wave depends on the phase shift 6 (equation (5.2)), which is dependent on the mirror material. This can be demonstrated with a slide coated with adjacent films of silver and gold, before the whole is covered with an identical layer system. The relative displacement of the standing-wave patterns in front of the silver and gold mirror, as seen in Fig. 15b, is a direct manifestation of the difference in phase shifts between the two metals. Furthermore, it can be seen from Fig. 15b, that the fluorescence intensity at the antinodes is smaller in front of the gold mirror than in front of the silver mirror. This is readily explained in terms of the reflection coefficient p , which is smaller for gold than for silver (equation (5.1)). As was mentioned in section 4.3, the standing wave in front of a layer-air boundary can have a marked influence on the amount of light absorbed by a dye monolayer. This can be shown with a slide that is coated on one side with a monolayer of a fluorescent dye and then covered with a stair-like succession of CdC,, layers (DREXHAGE [1966]). If the exciting light is incident from the side of the air (Fig. 16, top), the dye layer is not within the region of a standing wave, and therefore the fluorescence intensity appears uniform (no dependence on the distance d ) . If, on the other hand, the layer Fig. 15. Standing light waves in front of metal mirror, probed by a monomolecular layer of a fluorescent dye. Wavelength of illumination 366 nm, angle of incidence 0 = 30", spacing d increasing from left to right in steps of 6 CdClo layers (a) and of 4 such layers (b), silver mirror (a and b, upper half) and gold mirror (b, lower half). Mixed dye layers (double layers, chromophores face to face) consist of N,N'-distearyloxacyanine, N , N distearyloxacarbocyanine and tripalmitin (a), and of N,N'-distearyloxacyanine, N,N'distearylthiacyanine and cadmium arachidate in the ratio 1 : 1 : 20 (b); ultraviolet exciting light absorbed by the oxacyanine, fluorescence in the green and yellow emitted by the second dye following energy transfer. Plate a viewed at two different angles. High speed Ektachrome. - No fluorescence on steps with dye at a node of standing wave, moderate to strong fluorescence at antinodes. Intensity and color of fluorescence depend on spacing d and on viewing angle (0 8). Differences in phase shift and reflectivity between silver and gold mirror become apparent in plate b.

This Page Intentionally Left Blank

IV, o 51

STANDING LIGHT WAVES

193

Fig. 16. Standing light waves in front of layer-air boundary, probed by a fluorescent dye layer. Identical glass plates coated on one side with a monomolecular dye layer and covered with a stair-like succession of CdCro layers; spacing d increasing from left to right in steps of 4 such layers; same dye layers as in Fig. 15b, wavelength of illumination 366 nm, angle of incidence 0 = 30". Top: irradiation of the layer system from the air, dye nof within region of standing waves; bottom: irradiation from the glass side, absorption and fluorescence intensity vary with distance from boundary.

system is irradiated from the glass side (Fig. 16, bottom), the dye layer probes the standing wave in front of the layer-air interface, and thus the absorption varies with the distance d according to equation (4.9). The fluorescence intensity in this case appears at small distances (steps at the very left) much stronger than in Fig. 16, top, at any distance. While the effect is partly due to the described variation in the absorption, it is further enhanced by the influence of the reflectingboundary on the fluorescence emission (4 8). The absorption of all the dyes that have been used in monolayer studies so far, is due to electric-dipole transitions. Therefore their absorption is proportional to the square of the local electric-field amplitude and, in case of a standing light wave, shows minima at the distances dl that are given by equation (5.2). However, at these same distances the magnetic-field amplitude has maxima, which would give rise to maximum absorption, if a dye were used, whose absorption is due to a magnetic-dipoletransition. Similarly,

194

INTERACTION OF LIGHT W I T H MONOMOLECULAR DYE LAYERS

[IV,

86

a dye with an electric-quadrupole transition would probe the electric-field gradient in front of the mirror and would show a pattern like the magneticdipole absorber.

8 6. Evanescent Light Waves When a light wave is incident on the boundary between two media of different refractive index at an angle greater than the critical angle, it is totally reflected. Nevertheless, one expects a wave in the second (less dense) medium, whose amplitude decays exponentially with increasing distance from the surface. It is, therefore, called a surface or evanescent wave. While this phenomenon has been studied quantitatively with microwaves long ago (SCHAEFER and GROSS[1910]), it is very difficult to examine in the case of light, because it is confined to a narrow region of only a few wavelengths in thickness. Compared with earlier qualitative demonstrations of the effect (WOOD [1934] p. 418), the monolayer technique has made feasible a very precise investigation of the properties of evanescent light waves. In these experiments a monomolecular layer of a fluorescent dye was placed by means of CdC,, layers at a well defined distance from the boundary, and the fluorescence excited by the evanescent wave was measured. Because the absorption of such a dye layer is very small (< O.Ol), the inevitable disturbance of the phenomenon of total reflection is negligible. In a related experiment the emission of evanescent waves by fluorescent molecules has also been studied. 6.1. THE DECAY OF EVANESCENT WAVES

In order to examine the decay law of the evanescent wave a glass plate is coated on one side with a monomolecular layer of a fluorescent dye, which then is covered with a stepwise increasing number of CdCzolayers (FORSTER [1967], DREXHAGE and FORSTER [1967], DREXHAGE [1970b]). The plate is immersed in a liquid of a refractive index n, , which is higher than no and n,, and irradiated with a beam of monochromatic light at an angle B greater than the critical angle of total reflection (Fig. 17). The evanescent wave occurring in the layer system is detected through the fluorescence emitted by the dye molecules. Because the distance d between the surface of the layer system and the dye molecules is established by the monomolecular CdCzo layers covering the dye layer, the decay of the evanescent wave is probed with high precision by measuring the relative fluorescence intensity on the various steps of the sample.

IV. § 61

195

E V A N E S C E N T LIGHT W A V E S

Optically dense liquid

-

-

-

- -

..

-

..

.

-

- -

Fig. 17. Evanescent wave accompanying total reflection can be probed by monomolecular layers of fluorescent dyes. Intensity of fluorescence is proportional to the square of local field amplitude.

Theoretically the electric-field amplitude E , of the evanescent wave is given by (BORNand WOLF [1970] p. 47)

E,

-

exp [ -(2xd/A)(n: sin' O -TI:)*].

(6.1)

Here we confine ourselves to the case of polarization perpendicular to the plane of incidence. The energy absorbed by the dye molecules is proportional to E:. The intensity of the fluorescence, emitted by the dye layer and measured according to Fig. 17, is affected by the layer-liquid boundary. As will be discussed in 0 8, the fluorescence intensity I d emitted at the angle an with the normal to the boundary is given by (FORSTER[1967]) Id

=

1,[1 +r:l + 2 r 2 , cos (47rn0dcos ctfl/if,)],

(6.2)

where r Z 1is the Fresnel reflection coefficient of the layer-liquid boundary and A,, the wavelength of the fluorescence. Because the fluorescence was observed at an angle Of, = +n-O, the refraction at the back of the slide must be taken into account by applying Snell's law (no sin cl,, = n,sin Of,). According to equation (6.2) the quantity I,, which is proportional to Ef, is modified considerably (up to about 10% in the described experiment) with variation of the distance d. It must be calculated from the measured fluorescence intensity Id with equation (6.2). A logarithmic plot of I, as function of the distance d was found to agree very well with the exponential decay, expected from relation (6.1), for several angles of incidence 6 (Fig. 18). Because the CdClo layers are attacked by organic solvents, an aqueous solution must be used as liquid of high refractive index. In the experiments discussed in this section a saturated solution of a 1 : 1 mixture of thallous formate and thallous malonate (Clerici's solution) in water was chosen, whose refractive index nl z 1.7, although somewhat de-

196

I N T E R A C T I O N O F L I G H T W I T H M O N O M O L E C U L A RD Y E L A Y E R S

[IV.5 6

pendent on the temperature, is high enough to provide a large angular region of total reflection. However, this liquid usually shows some fluorescence of its own, which interferes with the fluorescence of the dye layer. In order to reduce this background fluorescence, dye layers containing a mixture of two cyanine dyes have been employed successfully (FORSTER [1967]). Here one dye serves as absorber for the exciting light ( I = 405 nm), z 580 nm), after an energy transfer between the and the second dye emits fluorescence dyes has taken place. The fluorescence of the dye layer is thus shifted toward wavelengths differentfromthe fluorescence of the Clerici solution and can be isolated by means of filters.

H O ZI

r - 1

l -

._ c

Lo C

W

c

C .-

Number of CdCm cover - layers

Fig. 18. Evanescent waves: logarithmic plot of fluorescence intensity f, as function of the number of CdC2, cover-layers for varying angles of incidence (perpendicular polarization). Experimental points obtained with eq. (6.2) from measured values f,,. ?. = 405 nm. no = 1.526, nl = 1.728; fluorescence maximum 580 nm. Mixed dye layers (double layers, chromophores face to face) consis! of N,N’-distearylthiacyanine, N,N’-distearylindocarbocyanine and cadmium arachidate in the ratio 4 : 1 : 110. Lines represent theoretical decay and FORSTER [1967].) calculated from eq. (6.1). (From DREXHAGE

Whereas in the above experiments the angle of incidence was held constant and the distance between dye layer and the boundary was varied, the MANDEL,DREXHAGE opposite was done in related work by CARNIGLIA, [1972]. Here the fluorescent dye layer was separated from the layer-liquid boundary by a fixed number of 16 CdC,, layers providing a distance d = 450 A. The s’ample was immersed in Clerici’s solution and irradiated with monochromatic light (A = 476 nm), polarized perpendicular to the plane of incidence. The fluorescence intensity was recorded as function of the angle of incidence 8. Because the angle a,, , at which the fluorescence was observed, was held constant, the effect described by equation (6.2) was constant throughout the experiment and could be neglected. The experimental plot of I,, versus the angle of incidence 8 was found to be in good agreement with the theory, both below and above the critical angle (Fig. 19).

EVANESCENT LIGHT WAVES

Angle of incidence

197

8

Fig. 19. Evanescent waves: fluorescence intensity Id (-lo) as function of the angle of incidence 8 (perpendicular polarization); 8, critical angle. 16 CdCzo cover-layers (d = 450 A), 1 = 476 nm, n,/n.= 1.10. Mixed dye layers (double layers, chromophores face to face) consist of N,N'-distearyloxacarbocyanine, N,N'-distearylindocarbocyanine and tripalmitin in the ratio 1 : 1 : 16. Experimental values (dots) and theoretical curve (solid). [1972].) (From CARNIGLIA, MANDEL, DREXHAGE 6.2. EVANESCENT WAVES IN A BIREFRINGENT MEDIUM

When the exciting light is polarized perpendicular to the plane of incidence, as was the case in the above experiments, the electric vector of the evanescent wave vibrates perpendicular to the optic axis of the CdC,, layers, which then behave like an isotropic medium with refractive index no. On the other hand, if the incident light is polarized parallel to the plane of incidence, the birefringence of the CdC,, layers must be taken into account. The eleclric field of the evanescent wave can be derived in a manner analogous to the way in which relation (6.1) is commonly derived. Into the equation for a homogeneous wave, which has been refracted at the boundary between two isotropic dielectrics, are inserted the ray index s of the CdC,, layers and the ray refraction angle p. The transition to the case of total reflection is then made in the usual way. With equations (A.3) and (AS) (see Appendix) the electric-field amplitude (component parallel to surface*) of the evanescent

* The chromophores in the cyanine-dye monolayers, employed in these experiments, are oriented with their transition moments parallel to the surface and respond only to this component of the light field (see 5 4).

198

INTERACTI ON OF L I G H T W I T H M O N O M O L E C U L A R D Y E L A Y E R S

[rv, $ 6

wave is then given by

E,,

-

exp [-(2nd/il)(n: sin' 8-nS)'non,3/(nt+(n,2-n~)n: sin2 el].

(6.3)

This expression is distinctly different from relation (6.1), to which it reduces for n, = no. It was observed experimentally that for the same angle of incidence 8 the decay of Ellwith the distance d was slower than the decay of E, (FORSTER [1967]). This shows directly that the extraordinary refractive index n, of the CdCzo layers is greater than the ordinary index no. A comparison of the experimental data with relation (6.3) allows the determination of the extraordinary index, and thus the value n, = 1.60+0.01 is obtained for il = 405 nm. 6.3. EMISSION OF EVANESCENT WAVES

Light emitted from a point source is refracted at the boundary to a more dense medium in such a way that the part entering the dense medium is confined to a cone, the aperture of which is given by the critical angle of total reflection, and no light is found in the region of larger angles. However, if the light source is very close to the boundary, this is no longer true, as was first suggested by SELBNYI [1913]. By applying the principle of reciprocity he expected that in this case some light should enter the normally dark region just as well, as on reversal of the propagation direction the evanescent wave would cause an absorption by dye molecules that are near the boundary. This phenomenon has been examined quantitatively with a monomolecular layer of fluorescent dye molecules, separated from the boundary by a certain number of CdC,, layers (CARNIGLIA, MANDEL, DREXHAGE [1972]). The experiment was conducted in a manner very similar to the studies of absorption in evanescent waves, which were discussed above. In this case the dye layer was irradiated from the glass side at a constant angle of incidence, and the intensity of the fluorescence emitted into the high-index liquid was recorded as function of the angle 8. In analogy to relation (6.1) one expects for the fluorescence intensity lL, emitted into the dense medium at an angle 8 greater than the critical angle

I,

exp [ - (4nd/il)(n: sin2 8 -no")'].

The experimental data showed that indeed fluorescence was emitted into the region beyond the critical angle in agreement with relation (6.4).

IV, 0 71

THE NEAR FIELD OF A R A D I A T I N G MOLECULE

199

9 7. The Near Field of a Radiating Molecule As was mentioned in the introduction, the investigation of radiationless energy transfer between dye molecules was the starting point for the development of the monolayer technique discussed in this article. This phenomenon, which is very important in the field of molecular physics, can be looked upon as absorption in the near zone of an oscillating dipole (the excited molecule). The near field can be probed with a monomolecular layer of another dye in a way similar to Q 5 and Q 6, where standing and evanescent light waves were examined with such a probe. We will concentrate here on the basic features of the phenomenon and refer to the review articles by KUHNand MOBIUS [1971] and KUHN[1972] for more details and applications. 7.1. KUHN’S CONCEPT OF ENERGY TRANSFER

If the excited dye molecule is assumed to be a classical oscillating dipole of frequency v and dipole moment p = p,, cos 2nvt, it will create an electromagnetic field like a radio antenna. At distances r no equations (8.27) and (8.30) are invalid at those angles 0 at which emission of evanescent waves occurs (5 6).

IV, S 81

215

RADIATION P A T T E R N I N FRONT OF A MIRROR

If the difference n, -no is very small, the reflection coefficient r t , is also small, and thus the multiple-beam interference is negligible. In this case the dielectric interface causes only a refraction of the pattern Pl(a),* which can be utilized to determine the refractive index of the layer system. For this purpose the position of a fluorescence maximum (preferably at large angle 0 ) is measured in an immersion liquid with a refractive index n, very close to no and the result is compared with the position to be expected from equation (8.5). Making use of Snell’s law the unknown index no can then be calculated. At larger differences n , -no equation (8.27) must be used. With this method FLECK[1969] obtained for CdCzo layers at 1 = 612 nm the value no = 1.522 and similarly for the extraordinary index the value ne = 1.59 (see section 8.4). 8.4. THE INFLUENCE OF THE LAYER BIREFRINGENCE

The angular patterns of the fluorescence polarized parallel to the plane of incidence are inevitably affected by the birefringence of the CdC,, layers. It is not possible t o match the refractive index of the layers with an immersion liquid for all angles of incidence. If, e.g., the refractive index n, of the liquid is chosen to match the ordinary index no of the layers, a marked mismatch will occur at large angles of incidence, and the radiation patterns observed experimentally deviate considerably from those expected according KUHN, MOBIUS,SCHAFER, to section 8.1 (BUCHER, DREXIIAGE, FLECK, SONDERMANN, SPERLING, TILLMANN, WIEGAND[ 19671, FLECK[ 19691). To account for the influence of the layer-liquid interface one can proceed in a manner analogous to section 8.3 and obtains for the angular pattern P,,(O)of the fluorescence polarized parallel to the plane of incidence

PI!@)= ~.,(p)t\22/[t+ p :

-2p,, r:, cos (41~,2(d+d’)cos

T-;:

p/Si-a:.)].

(8.30) The reflection coefficient pII and the phase shift SI1 of the underlying mirror are functions of the angle of incidence p of the ray. The function F , , ( p )is obtained from equations (8.2), (8.4) etc., if one replaces the factors cos2a, sin2a, etc. by cos2p, sin2j3, etc. and the quantity I$ by 4nntd cos p / s L For instance, in the case of randomly oriented electric-dipoles one finds so from equation (8.6) F,,(B)= 1

+p f +2p,, cos 28 cos ( 4 4 d cos p/si. -s,,).

* The effect of slight variations in the refractive index PI(=)is negligible at large angles a.

no o n the fluorescence pattern

216

IhIERACTlON O F L l C H r WITH M O N O M O L E C U L A R D Y E L A Y E R S

[IV,$9

The reflection coefficient r2', and the transmission coefficient t l k of the dielectric interface are given by equations (A.9) and (A.lO). The ray index s is connected with the angle p by s2 = nt c o s 2 / ~ + nsin2P, ~ and the refraction is expressed by n,s sin 0 = n: sin /? (see Appendix). It follows from equation (8.30) that the pattern P (0) depends in any case on the total thickness d t d ' of the layer system, i t . in practice on the number of CdC,, layers covering the fluorescent dye layer. This is in contrast to the case of perpendicular polarization, where the influence of the layer surface can be eliminated completely through the proper choice of the immersion liquid. The above treatment was found to be in good agreement with fluorescence patterns measured on the europium complex by FLECK [1969]. A typical example for the influence of the birefringence is shown in Fig. 27, where the patterns P,(LY),calculated with equation (8.6), are compared with the experimentally observed patterns P (0). 8:O"

8-3"

--

-_ . ....,

. I .

x ,

30'

',

-....

..'-.

'-.. 30"
> 1 and cannot be applied to cases like, e.g., the reflecting interface between two transparent media*. Whereas the values of TA, so obtained, are in general comparable with those, given by equation (9.7), there is a serious discrepancy between the values of calculated with equations (9.6) and (9.1 1). The energy transfer to the absorbing mirror is naturally included in this approach. Both this treatment and the one by MORAWITZ[1969] predict also a shift of the oscillator frequency near the mirror, which has not yet been observed experimentally.

TL

9.2. DIELECTRIC INTERFACE AS MIRROR

The influence of a reflector on the radiation resistance of a fluorescing molecule, as treated in section 9.1, is expected to occur with any kind of reflecting surface. If its optical properties are known, the decay times T; and 7; can be calculated from equations (9.6) and (9.7). In the case of an interface between transparent dielectrics of different refractive index the reflection coefficient is given by Fresnel’s formulas (BORN and WOLF[I9701 p. 40): (9.13) = ( n cos a - n , cos 0)/(. cos = + ! I , cos 0); n

rZ1

=

( n cos e - n , cos a)/(. cos o

* H. Kuhn, private communication.

+ ~ cos , .).

(9.14)

IV,

0 91

F L U O R E S C E N C E D E C A Y TIME

22 1

These equations apply at angles of incidence a smaller than the critical angle of total reflection. The phase information is contained in the sign of rk, and r ; l . In the range of total reflection, however, the phase shift is given by" = ( n 2 sin2 a-n:)*/n cos a ; (9.15) tan tan

[+(s:,+n)]

=

n(n2 sin2 a-nf)*/n: cos a.

(9.16)

Here n denotes the refractive index of the layers, which are assumed to be isotropic, and n , is the refractive index of the second dielectric. The angles a and l3 are connected by Snell's law, n sin a = n, sin 6. The decay times TL, T Aand &, calculated withequations (9.13) to (9.16) for n = 1.54 and n , = 1 (air), are shown as functions of the distance d i n Fig. 29 (DREXHACE[I~~O~]).

-

0

I

2000

I000

Distance d,

A

Fig. 29. Fluorescence decay time of europium complex as function of the distance d between dye and layer surface. Experimental data (circles) obtained on a mixed double layer of Eu-complex and tripalmitin in the ratio 1 : 3 with varying number of CdCZ0coverT calculated with layers. Theoretical plots of (dotted), T&/T (dashed), and T ~ / (solid) eqs. (9.6), (9.7), (9.10) and (9.13) to (9.16) using n = 1.54 and nl = 1. (From DREXHAGE [ 1970al.)

In the case of electric dipoles located directly at the interface (d = 0) the integration in equations (9.6) and (9.7) can be carried out in closed form,

* For the method of calculation see BORNand WOLF[I9701 p. 49. The formulas given there are slightly different owing to different definitions of the phase shifts.

222

INTERACTION OF LIGHT WITH MONOMOLECULAR DYE LAYERS

and it is obtained with n/n,

=

[IV,

59

a (DREXHAGE [1972])

TIT$

- a 3 + 2 a.2. .- 2..2 a - t - 3 a 2 (a 2 + 1.) - f In [ a -.. ~ .+ u ~ - ' + ( a - ' -. .t ) ( ~ ~ ~ + ~ ) ~ ] 7 5 3 a + a - a --a

(9.17) and TlTh

a' -

+ $2- 2u4+ 2u3- $ a 2 - 1 +3a4(a2+ 1)-_ . _

4 In [ _a _- 1 a + a - a 3 --a .-

7

+ a - +( a

-

. .

+I ) ~ ]

- t)(a2-

5

(9.18)

These equations, which are valid for any value of a,* describe the radiation rate of atoms or molecules adsorbed at an interface, a situation of particular practical importance. By substituting lla for a it is found that T A is identical for a radiating molecule on either side of the interface, whereas T ; changes by the factor a4,as the molecule crosses the interface from the medium with index n , into the medium with index n.** Both results might be expected, since the tangential component of the electric field and the normal component of the electric displacement must be continuous across the interface (BORNand WOLF [1970] p. 4). The change in decay time isexpected to be most pronounced, if the fluorescing molecule is directly at the interface. We obtain, e.g., with a = 1.54 the values = 3.60 and T ~ / T= 1.08, and with a = 2.00 the values .$IT = 9.28 and ~ $ / 7= 1.05. The above derivation of the decay time for any distance d is entirely based on thefar field of oscillating electric dipoles. The validity of this procedure may be doubted for small distances d. However, identical expressions for the decay time are obtained by application of the reciprocity theorem, where such objections do not apply, because the behavior of standing light waves is well understood for any distance d from the reflector (Q 5). A theoretical treatment, entirely different from the above, considers the induced dipoles in the material surrounding the excited molecule and the field produced by these dipoles at the site of the oscillator (TEWS,INACKER, KUHN[1970]). Depending on the phase relation between the induced field

* In the case n c: n , the emission of evanescent waves must be taken into account in equations (9.6) and (9.7) (see section 6.3). ** At large distance from the mirror the radiative decay time T o f an electric-dipole oscillator in a medium of index n equals t,.,/n, where T,,, is the decay time in vacuo (compare equation (7.3)).

IV, 9:

91

223

FLUORESCENCE DECAY TIME

and the motion of the oscillator it is accelerated or slowed down, and thus the decay time will be shortened or increased. The interaction between induced dipoles is neglected, and the following expressions are obtained:

-T - = I + -

-

-

8n2 2n:+n2

TM

sin z

4 cos z

Z

Z2

+ 4 sin z +z(si(z)-+n)] -

-

z3

(9.19)

and T

. 1.

+

-z 3

+z(~i(z)-fn)].

TM

(9.20)

These results differ considerably from equations (9.6) and (9.7), applied to a dielectric interface as mirror, in particular for an oscillator oriented parallel to the mirror normal. For instance, at the distance d = 0 one obtains with n = 1.54 and n, = 1.00 the value &/T = 1.98 and with n = 2.00 and n, = 1.00 the value T ~ / T= 1.88. The discrepancies are most likely due to the approximations on which equations (9.19) and (9.20) are based. Experimentally only the interface between the layer systems and air has been studied as reflector. A monolayer of the europium complex, deposited on a glass plate on top of, say, 5 CdC,, layers, was covered with avarying number of CdC,, or tripalmitin layers, and the decay time of the red fluorescence was measured as a function of the distance between the dye layer [1970a], TEWS,INACKER, KUHN [1970]). The and the interface (DREXHAGE results (Fig. 29) d o not show a particularly pronounced distance dependence of the decay time and agree with equations (9.6) and (9.7) assuming a quantum yield q = 0.7 and with equations (9.19) and (9.20) under the assumption q = 1 .O (section 9.4). 9.3.

ELECTRIC-DIPOLE SOURCE BETWEEN TWO MIRRORS

The gradual decrease of the fluorescence decay time with the distance d observed in the studies of section 9.2 does not prove an influence of the reflector on the decay time, although it was found to be in agreement with the predictions. Such a decrease of the decay time, in principle, could be caused equally well by an enhancement of competing non-radiative processes due to the cover layers. Likewise, an immersion liquid might reduce the decay time through a chemical attack on the very sensitive europium complex. But a very pronounced and unambiguous mirror effect has been observed using a highly reflecting metal mirror, which was covered only with a varying number of CdC,, layers and a monolayer of the europium com-

224

INTERACTION O F LIGHT WITH MONOMOLECULAR DYE LAYERS

[IV,

59

plex, avoiding any additional treatment. Unfortunately, the interface between the layers and air constitutes an additional reflector, which needs to be accounted for in the theoretical treatment. The theory, discussed in section 9.1, can be extended in a straightforward way to the case of an emitter between two mirrors. It is presumed as before that the metal mirror (reflection coefficients p L , p . ; phase shifts 6,, a,,) acts upon the oscillator like a nonabsorbing mirror with the transmission coefficients ( 1 - p:): and ( I - pf)*. Then the radiation pattern on either side of the layer system can be calculated, taking into account the multiplebeam interference as was outlined in section 8.3. The total energy, radiated per unit time, is obtained in analogy to equation (9.4), and with equation (7.3) the following expressions for the decay-time ratios are found:

The reflection coefficients r;,, ril and the phase shifts Sil, S:, of the dielectric interface are given by equations (9.13) to (9.16), whereas those quantities for the metal mirror can be calculated from the optical constants of the mirror material in the usual manner (BORNand WOLF[I9701 p. 615). The quantity z‘ stands for 4rnd‘ll, where d’ denotes the distance between the fluorescing molecule and the dielectric interface. Furthermore it is assumed n > n, so that no emission of evanescent waves occurs. Equations (9.21) and (9.22) reduce to equations (9.6) and (9.7) for n, = n (r& = r i , = 0). Of particular interest is again the decay time ,;I‘ which is obtained with equation (9.10). It has been calculated as a function of distance d for the experimentally relevant case of a silver mirror and the parameters n = 1.54,

IV,

S 91

225

FLUORESCENCE DECAY T I M E

n, = 1, d' = 0 and ;C = 612 nm by DREXHAGE [1970a, b]*. The ratio of the decay time &(d) so obtained and the decay time &(oo), which is calculated from equations (9.17), (9.18) and (9.10), is shown for a gold mirror in Fig. 30. The layer-air interface, acting as a second reflector, gives rise to a generally more pronounced variation of the decay time than is caused by the gold mirror alone (compare Fig. 28). This is also true for other mirror materials, e.g. silver, copper or aluminium.

r-

O

L .~

0

.-

. _ -

_.

-

-

..-:

-

.

..

-

-

-

--

.-

- -

--

-.

4000

2000

-

6(

Distance d , A

Fig. 30. Fluorescence decay time of europium complex between gold mirror and layer surface as function of the separation d from the mirror. Experimental data: circles; same dye layers as in Figs. 28 and 29; no cover layers (d' = 20 A). Theoretical plots of ~ ; ( d ) / ~ L ( c m ) for q : 1.0 (solid) and q = 0.7 (broken) calculated with eqs. (9.21), (9.22), (9.17), (9.18) and (9.10) using for the gold mirror the optical constants v 0.505 and I ' K = 3.66. (From DREXHAGE [1966, 1970al.) 7

The experimental decay times, measured on mixed monolayers of the europium complex and tripalmitin under N, at 0 "C (DREXHAGE [1966]), show a less pronounced variation with the distance d than the theoretical plot of ~",d)/.rR(m)(Fig. 30). However, if one assumes for the quantum

* I n these articles it was inadvertently omitted that the theoretical and experimental data referred to systems with both a silver mirror and a dielectric interface (n = 1.54; n , - I ; d' 0). Likewise preliminary data, quoted by KUHN[I9671 and taken on mixed layers of the europium complex and cholesterol, involved besides the metal mirror (gold) a dielectric interface (n, 1; d' - 0).The theoretical curve given there was calculated by neglecting the dielectric interface, omitting the contribution described by equation (9.3) and assuming q =- I . 7

226

I N T E R A C T I O N O F L I G H T WIT11 M O N O M O L E C U L A R D Y E L A Y E R S

[IV, A P P .

yield a value q = 0.7 (see section 9.4), the agreement becomes quite good. Whereas the severe discrepancies a t distances between 0 and 300A are caused by energy transfer, the slight deviations at larger distances can be attributed possibly to the birefringence of the CdC,, layers, which has been neglected in the above treatment. 9.4. COMPETING NON-RADIATIVE PROCESSES

In the above discussion we have neglected any thermal deactivation processes that might occur in the excited molecule. If they are fast enough to compete with the rate of fluorescence 1/7, they must be taken into account. This is usually done in terms of the quantum yield q = Lf/(Lf+L,) (equation (7.4)). If we assume that the quantity L,is independent of the distance from the mirror, it follows that the quantum yield qM of the molecule near the mirror is given by (9.23) q i l = 1 + ( T M / T ) ( q - ' - 1). Thus the quantity qM varies with the distance d from the mirror. Furthermore we find for the ratio of the real decay times T' = q.r and T$ = qM?, T'/T$

=

1+q[(r/rM)-1].

(9.24)

Hence the decay time ratios, as given by equation (9.6) etc., describe only those cases in which the thermal deactivation can be neglected (q = I). If the quantity q has a value smaller than one, the variation of the decay time with distance is less pronounced, as shown for q = 0.7 in Fig. 30 (DREXHAGE [ I970al). This effect can be utilized to determine the quantum yield q of the emitting FLECK, KUHN, state, provided that a reliable theory for 7Mexists (DREXHAGE, SCHAFER, SPERLING [1966], DREXHAGE, KUHN,SCHAFER [ 19681). The method is much more sensitive than the determination of q from the critical distance of energy transfer (equation (7.5)). While, e.g., the curves of 7&' are distinctly different for q = 1 and 0.7 (Fig. 30), the value of the critical distance do is reduced only by the factor 0.7* = 0.91 in case of q = 0.7, which is within the limits of experimental error.

Appendix: Some Optical Properties of Uniaxial Crystals The strong birefringence of the CdCzo layers has a distinct influence on most optical phenomena studied with these layer systems, which behave

like a uniaxial crystal with the optic axis parallel t o the layer normal. The angles ct for the wave normal and p for the ray, being identical in isotropic

IV, A P P . ]

O P T I C A L P R O P E R T I E S OF U N I A X I A L C R Y S T A L S

227

media, must be distinguished here for light polarized parallel to the plane of incidence (the extraordinary wave)*. They are related with the indices no and ne by no2 tan ci = n,‘ tan p, (A. 1) i.e. the angle of incidence /3 of the ray is smaller than the corresponding angle ct in case of CdC,, layers (n, > no). Likewise we must distinguish between the velocity of light propagation along the wave normal and along the direction of energy flow (the ray), which may be expressed in terms of the indices n and s. The refractive index n for the wave is given by n-’ = no- 2 cos2 ci+ne-’ sin’ and the ray index s is related to the angle s’ = no2 cos’

ci,

(A4

p of the ray by

p+nl

sin’ p.

(A.3)

The refraction between the layer system and an isotropic medium with refractive index n, (angle of incidence 0) is governed by Snell’s law

n sin

2 =

n, sin 0,

64.4)

which can also be expressed as nf sin fi = n, s sin O .

(A4

The ray index s can be eliminated with equation (A.3), and one obtains relations describing the refraction of the ray: s i n p = n o n l sin 0/[n:+(n,2-nf)n: sin o

=

n,” sin p/n, [n:

sin’

el*;

+(nf - n,’) sin’ p] ’.

(A4

(A.7)

Because the electric field of the light wave is perpendicular to the ray direction (and not to the wave normal), it is often preferable to trace the rays, in particular, if one deals with absorption or emission by electric-dipole oscillators. The reflection of light at the interface between birefringent materials has [I9281 p. 715 in the most general form. It can be been treated by SZIVESSY shown that his rather complicated formulas agree with Fresnel’s formulas, if one introduces the ray index s and the angle of incidence j? of the ray. * For the theory of light propagation in anisotropic media the reader is referred to the article by RAMACMANDRAN and RAMASESHAN (19611 p. 54. A very UsefUl account, which emphasizes graphical representations, is found in the book by WAHLSTROM [19691 P. 197.

228

I N T E R A C T I O N O F L I G H T W l . r H M O N O M O L E C U L A R D Y E 1.AYERS

[IV, A P P .

Thus one obtains for the reflection coefficients ri2, r i l and the transmission coefficients t : * , t i , at the surface of the layers r',

=

,

( n cos p - s cos O ) / ( n I cos p + s cos O ) ,

r : , = (scosO-n,

c o s ~ ) / ( s c o s U + i i ,cosp),

(A4 (A.9)

t i l = 2n, cos O / ( n , cos p + s cos 0)

(A.10)

a).

(A.11)

and t i , = 2s cos p/(s cos O + n l cos

From equations (A.8), (A.3) and (A.6) one finds for the Brewster-angle o,, (r12 = 0 ) ~ (A.12) tan o,, = ( n C / i i l ) [ ( n-nf)/(nz-r~;)]i. With the values no = 1.52, tz, = 1.59 and n, = I , e.g., it is calculated O,, = 55.8", which also would be found, if the layers were isotropic with an index n = 1.47. The Brewster-angle Per on thc side of the layer system is found similarly: tan par = (n,/n,)[(nf -nf)/(n,2 - n 3 ] 3 . (A.13) With no = 1.52, n, = 1.59 and n, = 1 it follows BRr = 30.2", which would be obtained in case of isotropic layers of index n = 1.72. It may be noted here that the direction of a ray reflected at the angle O,, is not perpendicular to the direction of the refracted ray nor to the normal of the refracted wave, nor to the electric polarization of the uniaxial medium. This casts some doubt on the validity of the common intuitive derivation of the Brewster-angle, as suggestive it may be (see, e.g., SOMMERFELD [1954] p. 25, BORNand WOLF [I9701 p. 43). In case of n, > n, the critical angle O,, of total reflection is given by sin O,,

=

n,/nl ,

(A.14)

-=

whereas in the opposite case n , no,n, the critical angle p,, is found from equation (A.6) as sin B,, = n o n , / [ n f + ( n f - r i f ) n f l f . (A.15) For instance, with the values no = 1.52, 11, = 1.59 and n , = 1 we calculate = 37.7", a value also obtained, if the layers were isotropic with an index n = 1.63. This and the examples given above show that it is of questionable value to simplify the treatment of birefringent layers by assuming them to be isotropic with some average refractive index n. Equation (A.9) can be extended to the case of an absorbing medium 1 in the usual manner (BORX and WOLF [I9701 p. 615), and thus the reflection coefficient p,, and the

p,,

IVI

REFERENCES

229

phase shift all are obtained for reflection at an underlying metal mirror (section 8.4). The radiation of an electric dipole is also influenced by its anisotropic environment. If, e.g., the dipole axis is oriented parallel to the optic axis of the layers, the electric field E D in the far zone (Y >> A) is given by KUEHL [I9621 as E D = (4z2po/A2r)(no n:/s3) sin p, (A.16) and the total energy L,, emitted by the oscillator per unit time, is obtained in analogy to section 9.1 : L, = ~ / z / 2 ' r E ~ r zsin pdfid$ = 16x4cn0&/3A4. 872 o o

(A.17)

The total radiation field of an electric dipole, including the near zone, has been derived by CLEMMOW [1966] p. 159. Acknowledgments

The author expresses his gratitude for the cooperation and support which he has received from many colleagues at the University of Marburg during the course of his work there. In particular, I wish to thank Prof. H. Kuhn and F. P. Schafer for many stimulating discussions and helpful suggestions. I am also grateful to Prof. H. Wolter, whose excellent lectures have inspired my interest in optics. The valuable suggestions regarding the manuscript by H. Biicher and F. C. Strome Jr. of the Eastman Kodak Laboratories are greatly appreciated. References AUGENSTINE, L. G., 1960, in: Comparative Effects of Radiation, Conf. Proc., Puerto Rico 1960, eds. M. Burton, J. S. Kirby-Smith and J. L. Magee (John Wiley, New York) p. 322. BARTH,P., K. H. BECK,K. H. DREXHAGE, H. KUHN,D . MOBIUS,D . MOLZAHN, K. ROLLIG, F. P. SCHAFER, W. SPERLING and M. M. ZWICK,1966, Optische und elektrische Phanomene an rnonornolekularen Farbstoffschichten, in: Optische Anregung organischer Systerne, Conf. Proc., Schloss Elmau 1964, ed. W. Foerst (Verlag Chernie, Weinheimi Bergstr.) p. 639. BAUER,H., J. BLANCand D . L. Ross, 1964, J. Arner. Chern. SOC.86, 5125. BERNSTEIN, S., 1940, J. Amer. Chem. SOC.62, 374. BLODGETT, Katharine B., 1934, J. Amer. Chern. SOC. 56, 495. BLODGETT, Katharine B., 1935, J. Arner. Chern. SOC.57, 1007. BLODGETT, Katharine B. and I. LANGMUIR, 1937, Phys. Rev. 51, 964. BLODGETT, Katharine B., 1939, Phys. Rev. 55, 391. BORN,M. and E. WOLF,1970, Principles of Optics, 4th ed. (Pergamon Press, Oxford).

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[IV

BUCHER,H., K. H. DREXHAGE, M. FLECK,H. KUHN,D. MOBIUS,F. P. SCHAFER,J. SONDERMANN, W. SPERLING, P. TILLMANN and J. WIEGAND, 1967, Mol. Cryst. 2, 199. and P. TILLMANN, 1967, BUCHER,H., H. KUHN,B. MANN,D. MOBIUS,L. v. SZENTPALY Photogr. Sci. Eng. 11, 233. and J. WIEGAND,1969, Z. Phys. BUCHER,H., 0. v. ELSNER,D. MOBIUS,P. TILLMANN Chem. (Frankfurt am Main) 65, 152. B. B. SNAVELY, K. H. BECKand H. KUHN,1969, Chem. Phys. BUCHER,H., J. WIEGAND, Letters 3, 508. BUCHER,H., 1970, Dissertation, University of Marburg, Germany. BUCHER,H. and H. KUHN,1970, Z. Naturforsch. B 25, 1323. CARNIGLIA, C. K., L. MANDELand K. N.DREXHAGE, 1972, J. Opt. SOC.Amer. 62, 479. CLEMMOW, P. C., 1966, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press, Oxford). D. L., 1953, J. Chem. Phys. 21, 836. DEXTER, F. W., 1938, Phys. Rev. 53, 420. DOERMANN, F. W. and 0. HALPERN,1939, Phys. Rev. 55,486. DOERMANN, DREXHAGE, K. H., M. M. ZWICKand H . KUHN,1963, Ber. Bunsenges. Phys. Chem. 67,62. DREXHAGE, K. H., 1964, Dissertation, University of Marburg, Germany. DREXHAGE, K. H., 1966, Optische Untersuchungen an neuartigen monomolekularen Farbstoffschichten (Habilitations-Schrift, University of Marburg, Germany). 1966, Ber. DREXHAGE, K. H., M. FLECK,H. KUHN, F. P. SCHAFERand W. SPERLING, Bunsenges. Phys. Chem. 70, 1179. K. H. and H. KUHN,1966, Optical and Electrical Phenomena on MonomolecDREXHAGE, ular Layers, in: Basic Problems in Thin Film Physics, Conf. Proc., Clausthal-Gottingen 1965, eds. R. Niedermayer and H. Mayer (Vandenhoeck-Ruprecht, Gottingen) p. 339. and H. KUHN,1967, Ber. Bunsenges. Phys. Chem. 71, 915. DREXHAGE, K. H., M. FLECK DREXHAGE, K. H. and H. FORSTER,1967, Quantitative Untersuchung der bei der Totalreflexion am optisch diinneren Medium auftretenden Grenzflachenwelle, paper presented a t the Westdeutsche Chemiedozententagung, Saarbriicken, Germany, April 12. 1968, Ber. Bunsenges. Phys. Chem. 72, DREXHAGE, K. H., H. KUHNand F. P. SCHAFER, 329. K. H. and M. FLECK,1968, Ber. Bunsenges. Phys. Chem. 72, 330. DREXHAGE, K. H., 1969, Long Range Energy Transfer Involving Higher Order Transitions, DREXHAGE, paper presented at the Fifth Intern. Conf. on Photochemistry, Yorktown Heights, New York, September 2. K. H., 1970a, J. Lurninesc. 1, 2, 693. DREXHAGE, DREXHAGE, K. H., 1970b, Scientific American 222, 108. K. H., 1972, Spontaneous Emission Rate in the Presence of a Mirror, Third DREXHAGE, Rochester Conf. on Coherence and Quantum Optics, Rochester, New York, June 21. DREXHAGE, K. H., 1974, Photogr. Sci. Eng. (to be published). DRUDE,P. and W. NERNST,1892, Wiedem. Ann. Phys. u. Chem. 45,460. 0. v., 1969, Dissertation, University of Marburg, Germany. ELSNER, D. DEN,1971, J. Opt. SOC.Amer. 61, 1460. ENGELSEN, ENGELSEN, D. DEN, 1972, J. Phys. Chem. 76, 3390. FLECK,M., 1969, Dissertation, University of Marburg, Germany. T., 1946, Naturwissenschaften 33, 166. FORSTER, T., 1948, Ann. Phys., 6. Folge 2, 55. FORSTER, FORSTER, H., 1967, Diplomarbeit, University of Marburg, Germany. FREED, S. and S. I. WEISSMAN, 1941, Phys. Rev. 60, 440. GAINESJr., G. L., 1966, Insoluble Monolayers at Liquid-Gas Interfaces (Interscience, New York). HALPERN, 0. and F. W. DOERMANN, 1937, Phys. Rev. 52, 937. and L. STRYER, 1969, Proc. Nat. Acad. Sci. U.S. 63,23. HAUGLAND, R. P., J. YGUERABIDE

IVI

REFERENCES

23 1

HOLLEY, C., 1938, Phys. Rev. 53, 534. INACKER, O., H. K U H NH. , BUCHER, H . MEYER and K. H. TEWS,1970, Chem. Phys. Letters 7, 213.

KAUZMANN, W., 1957, Quantum Chemistry (Academic Press, New York). D. and H. BUCHER,1969, Z. Naturforsch. B 24, 1371. KLEUSER, KOPPELMAHN, G., 1969, Multiple-Beam Interference and Natural Modes in Open Resonators, in: Progress in Optics, Vol. 7, ed. E. Wolf (North-Holland, Amsterdam) p. 1. KOSSEL,D., 1958, Praxis der Naturwissenschaften 7, 44. KUEHL,H. H., 1962, Phys. Fluids 5, 1095. KUHN,H., 1967, Naturwissenschaften 54, 429. KUHN,H., 1968, On Possible Ways of Assembling Simple Organized Systems of Molecules, in: Structural Chemistry and Molecular Biology, eds. A. Rich and N. Davidson (W. H. Freeman, San Francisco) p. 566. KUHN,H., 1970, J. Chem. Phys. 53, 101. KUIIN,H. and D. MOBIUS,1971, Angew. Chem. 83, 672; internat. Edit. 10, 620. KUHN,H., 1972, Spectroscopy of Monolayer Assemblies, Part 1, Principles and Applications, in: Physical Methods of Chemistry, Part IIIB, eds. A. Weissberger and B. W. Rossiter (Wiley-Interscience, New York) p. 579. KUHN,W., 1933, Theorie und Grundgesetze der optischen Aktivitat, in: Stereochemie, ed. K . Freudenberg (F. Deuticke, Leipzig) p. 317. 1971, Chem. Phys. Letters 8,82. MANN,B., H. KUHNand L. v. SZENTPALY, MAYER,H., 1950, Physik dunner Schichten, Part I (Wiss. Verlagsges., Stuttgart). MOBIUS,D., 1969, Z. Naturforsch. A 24, 251. MOBIUS,D. and H. BUCHER,1972, Spectroscopy of Monolayer Assemblies, Part 11, Experimental Procedure, in: Physical Methods of Chemistry, Part IIIB, eds. A. Weissberger and B. W. Rossiter (Wiley-Interscience, New York) p. 650. MORAWITZ, H., 1969, Phys. Rev. 187, 1792. RAMACHANDRAN, G. N. and S. RAMASESHAN, 1961, Crystal Optics, in: Handbuch der Physik, Vol. 2 5 / l , ed. S. Flugge (Springer, Berlin) p. 1 . ROTHEN,A., 1945, Rev. Sci. Instrum. 16, 26. ROTHEN,A., 1968, Surface Film Techniques, in: Physical Techniques in Biological Research, 2nd ed., Vol. 2, Part A, ed. D. H. Moore (Academic Press, New York) p. 217. SCHAEFEK, C. and G. GROSS,1910, Ann. Phys., 4. Folge 32, 648. SCHMIDT, S., R. REICHand H. T. WITT, 1969, Z. Naturforsch. B 24, 1428. SCHMIDT, S., R. RtlCH and H. T. W i n , 1971, Naturwissenschaften 58, 414. SCHMIDT, S. and R. REICH,1972a. Ber. Bunsenges. Phys. Chem. 76, 599. SCHMIDT,S. and R. REICH,1972b, Ber. Bunsenges. Phys. Chem. 76, 1202. SEL~NY P.,I ,191 1, Ann. Phys., 4. Folge 35, 444. S E L ~ N YP., I , 1913, C. R. Acad. Sci. 157, 1408. SELBNYI, P., 1938, Z. Physik 108, 401. S E L ~ N P., Y I ,1939, Phys. Rev. 56, 477. SHER,1. H. and J. D. CHANLEY, 1955, Rev. Sci. Instrum. 26, 266. SHKLYAREVSKII, I. N., V. K. MILOSLAVSKII and V. I. GOLOYADOVA, 1964, Opt. Spectrosc. 17, 413; Russ.: 17 (1964) 765. A., 1954, Optics (Academic Press, New York). SOMMERFELD, SONDERMANN, J., 1971, Liebigs Ann. Chem. 749, 183. S T ~ I G E R., R , 1971, Helv. Chim. Acta 54, 2645. STRYER, L. and R. P. HAUGLAKD, 1967, Proc. Nat. Acad. Sci. U.S. 58, 719. SZENTPALY, L. v., D. M ~ B I Uand S H. KUHN,1970, I. Chem. Phys. 52,4618. SZIVESSY, G., 1928, Kristalloptik, in: Handbuch der Physik, Vol. 20, eds. H. Geiger and K. Scheel (Springer, Berlin) p. 635. TEWS,K. H., 0. INACKER and H. KUHN,1970, Nature (London) 228, 276; erratum: 228, 791.

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1NTERACTION O F L I G H T W I T H M O N O M O L E C U L A R D Y E L A Y E R S

[IV

P., 1966, Dissertation, University of Marburg, Germany. TRURNIT, H. J., 1945, Uber monomolekulare Filme an Wassergrenzflachen und iiber Schichtfilme, in: Fortschritte der Chemie organischer Naturstoffe, Vol. 4, ed. L. Zechmeister (Springer, Berlin) p. 347. WAHLSTROM, E. E., 1969, Optical Crystallography, 4th ed. (John Wiley, New York). WIENER, O., 1890, Wiedem. Ann. Phys. u. Chem. 40, 203. WOLTER,H., 1956, Optik dunner Schichten, in: Handbuch der Physik, Vol. 24, ed. S. Fliigge (Springer, Berlin) p. 461. WOOD,R. W., 1934, Physical Optics, 3rd ed. (Macmillan, New York). Z W I C KM. , M. and H. KUHN,1962, Z. Naturforsch. A 17, 411.

TILLMA",

E. WOLF, PROGRESS I N OPTICS XI1 0 NORTH-HOLLAND 1974

V

THE PHASE TRANSITION CONCEPT AND COHERENCE IN ATOMIC EMISSION BY

R.GRAHAM Institut fur theoretische Physik der Universitiit Stuttgart, Germany

CONTENTS

PAGE

5 9 6 5

1 . INTRODUCTION

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

235

2. MEAN FIELD THEORY OF LASER MEDIA . . . . . . . 242 3 . BEYOND THE MEAN FIELD THEORY . . . . . . . . . 254

4. DISCRETE MODE SPECTRA . . . . . . . . . . . . . .

263

4 5. DISCRETEMODESINNONLINEAROPTICS . . . . . . 272 APPENDIX I . . . . . . . . . . . . . . . . . . . . . . . .

280

APPENDIX I 1 . . . . . . . . . . . . . . . . . . . . . . . .

282

9 6. NOTE ADDED I N PROOF . . . . . . . . . . . . . . . . 282 ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . 283 REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

283

0 1. Introduction Optics is quite usually described in macroscopic terms. Its equations of motion are the macroscopic Maxwell equations. These are obtained from the basic microscopic theory by averaging over regions in space which contain many “atoms” but are still small compared to the optical wavelength. Nevertheless, a theory of optics based on these equations cannot be considered to be a macroscopic theory in the same sense as, e.g., the Navier Stokes equations give a macroscopic theory of fluids. The latter equations have been averaged over time intervals also (or, equivalently for ergodic systems, over a local equilibrium ensemble) and, hence, are only valid for large distances and long time intervals. The optical equations were averaged over space only. In this sense optics can be regarded as a macroscopic theory in space; it is still a microscopic theory in time, however. In order to derive macroscopically observable results from it, averages of the results of the theory over time or an ensemble are still required. Prior to the invention of the laser, the only ensembles which had to be used in practice, were thermal ensembles. Coherence properties of electromagnetic fields in such ensembles were investigated by BOURRET[ 19601, and WOLF[1965]. BERAN and PARRENT [1964]; see also the review of MANDEL If the average would be performed in the Maxwell equations with such ensembles, all field strengths in these equations would average to zero, since they consist of randomly phased thermal noise. Only expressions quadratic in the field strengths could give nonzero averages. This fact can easily be understood: The average number of thermal quanta in some mode of the electromagnetic field is, according to Planck’s law, always very small for the optical part of the electromagnetic spectrum at practically attainable temperatures. On the other hand, in order to give the value of the phase cp of a mode amplitude some meaning, we would have to form wavepackets involving states with quantum numbers large compared to 1. Such states are not available for optical modes in a thermal ensemble. As a result the phases cp remain unobservable (MANDEL [1967]) and the mode amplitudes average t o zero. Macroscopic, “hydrodynamic” equations for the electro235

236

THE PHASE TRANSITION CONCEPT

[v.

0

magnetic fields do not exist in pre-laser optics (although they exist for microwave fields, e.g.) simply because the electromagnetic fields are not “hydrodynamic” quasi-conserved quantities in that frequency range. With the appearance of the optical laser, this picture changed completely. At the laser threshold the amplitudes of some modes change their time dependence and become slowly varying, quasi-conserved quantities. These mode amplitudes also become “macroscopic quantities”, in the sense that their mean occupation numbers (intensities) become large and increase with the size of the system. Their phases acquire a physical meaning, since wave packets containing states with large quantum numbers can easily be formed. Therefore, a long time “hydrodynamic” description of laser action above threshold becomes possible. Indeed the theories of HAKENand SAUERMANN [1963a, b] and of LAMB [1964], which are successful in describing most features of laser action, are based on such a “hydrodynamic” description. The “local equilibrium ensemble” on which these theories are based (although not mentioned explicitly in the above quoted papers) is a “restricted ensemble” in which the phase of the modes above threshold is held fixed (though arbitrary) whereas the phases of the modes below threshold are completely at random. For a discussion of restricted ensembles see BOGOLJUBOV [1962a, b, c], HOHENHERG and MARTIN[1965], WAGNER[1966]. Therefore, the amplitudes of all modes below threshold have to be put equal to zero in the macroscopic, ensemble averaged equations of motion, in agreement with the remarks made before, whereas the modes above threshold have finite amplitudes with fixed phases. After the appearance of the laser, the construction of a whole class of new light sources became possible, all of which are based on some kind of new stimulated emission process. Well known examples are furnished by stimulated Raman and Brillouin emission and parametric emission. Each of these light sources shows a characteristic threshold behaviour which occurs when the induced emission process begins to dominate over the corresponding spontaneous emission process. Similarly to the laser case, these thresholds are characterized by the appearance of new slowly varying, quasi-conserved quantities, which usually are the amplitudes of the electromagnetic field strength at certain frequencies. The theories, developed for these processes (see BLOEMBERGEN [1965]) are therefore similar to the Haken-Sauermann and Lamb theories of laser action in that they assign zero amplitudes to the modes below threshold and finite amplitudes with fixed but arbitrary overall phases to all modes above threshold. Systems which are described by macroscopic, averaged Maxwell equations

v,g 11

INTRODUCTION

237

(like lasers or related sources of coherent light or like low frequency electromagnetic fields) and systems containing a condensate wave function (like superfluid He or superconductors) exhibit a number of analogies in their long range space-time behaviour. MARTIN[ 19651 gave a detailed exposition of rhis analogy. The basic common feature of these systems is the existence of a coherent macroscopic wave function whose phase is a well defined object, which is slowly varying in space and time. It gives rise to space dependent interference and quantization effects like flux and vortex quantization in superconductors and He 11, respectively (see, e.g., ANDERSON [1965]), or the mode structure of macroscopic, coherent, electromagnetic fields, e.g., in wave guides or resonators. Similarly it leads to time dependent interference effects like Josephson oscillations (JOSEPHSON [1962]) or like beat notes between electromagnetic fields. The formal similarities between such systems include, e.g. the use of restricted ensembles and broken symmetries in both cases. Now, in He at sufficiently large temperature, like in thermal ensembles in optics, the phase of the condensate mode looses its meaning, since it can no longer be defined by forming large quantum number wave packets for that mode. The critical point, at which the change between microscopic occupation (independent of the system's size) and macroscopic occupation (increasing with the system's size) occurs, is, of course, the I-point of the superfluid phase transition in He. It is equally well characterized by the appearance of a new hydrodynamic variable (the phase of the condensate wave function or, equivalently, the superfluid velocity). The physical reason for the changed hydrodynamic behaviour is, of course, the change of symmetry occurring with the transition, which also accounts for the use of different local equilibrium ensembles on both sides of the transition. The new hydrodynamic mode is just the Goldstone mode associated with the broken symmetry. In view of these very detailed analogies, it is quite natural to enquire whether the threshold behaviour of lasers and similar sources of coherent light can be likened to systems undergoing phase transitions. Remarks and allusions to analogies between laser threshold and phase transitions can be found quite frequently in the literature. WAGNER and BIRNUAUM [I9611 compared their theory of the laser threshold to Bose Einstein condensation. CUMMINGS and JOHNSTON [ 19661 made an attempt t o interpret the ).-transition of He in anaIogy to a simple laser theory. KOREKMANN [1966], in his application of Green's function theory to lasers, showed that long range order requires infinite field energy, i.e. the thermodynamic limit. Remarks to the analogy may further be found e.g. in LAX

238

THE P H A S E TRANSITION CONCEPT

[v, Q 1

[1968] and HAKENand WEIDLICH [1969]. The first detailed discussion in terms of space-time dependent laser fields was given in GRAHAM and HAKEN [1968b]. Later the striking formal analogy between laser theory and the Ginsburg-Landau theory was demonstrated by GRAHAM and HAKEN [1970], and SCULLY see also HAKEN[1970b, 19721. At the same time DEGIORGIO [1970] gave a detailed account of the analogy between single mode laser theory and the mean field theory of a ferromagnetic phase transitions. The full appreciation of this analogy then led to the development of a phenomenological theory of threshold phenomena in lasers and nonlinear optics (GRAHAM and HAKEN [1971a, b], GRAHAM [1972, 1973a, b], GROSSMANN and RICHTER [1971a, b], RICHTER and GROSSMANN [1971,1972]). Furthermore some techniques of laser theory were applied to ferromagnets (GOLDSTEIN, SCULLYand LEE[1971]). Apart from the explicit remarks to the phase transition analogy in the literature on lasers, the analogy is implicated in most papers on this topic, without being mentioned explicitly. Indeed, if today the development of laser theory and of phase transition theory is viewed in the light of this analogy, one can appreciate that many steps in the one theory have had their independent counterparts in the other and vice versa. This is not to say, of course, that the development of one theory was after or even dependent on that of the other. In fact, both theories have and had to cope with problems all of their own. E.g. one prominent feature of phase transition theory, that it deals with systems having an infinite number of relevant degrees of freedom, was absent in the main part of laser theory, where, due to the mode structure of the resonators used, it has been sufficient to consider the degrees of freedom of a few modes only (cf. HAKEN[1966] for a theoretical foundation of the latter point; laser fields with infinite number of degrees of freedom where considered by GRAHAM and HAKEN [1968b, 19701 when the phase transition analogy was appreciated). On the other hand, lasers are systems far from thermal equilibrium which are driven by quantum mechanical fluctuations. These two features, and the problems arising from them, are absent from phase transition theory. A very substantial part of the literature on laser theory was devoted to these two classes of problems. In this literature, new and beautiful techniques were developed, not having counterparts in phase transition theory, by means of which the statistical dynamics of systems far from thermal equilibrium can be formulated in a quantum theoretical way, and can even be mapped on the dynamics of equivalent classical processes (see the review by HAKEN[1970] for a detailed exposition of these techniques and their application).

v, 5 11

INTRODUCTION

239

The present review will not deal with those developments. The point in discussion here is the fact that the underlying similarity of the physics of laser thresholds and phase transitions has lead to the invention of a number of similar techniques and approximation schemes, and it is on those that we shall focus our attention. The basic theories of HAKENand SAUERMANN [1963a, b] and of LAMB [ 19641were mean field theories in the terminology of phase transition theory. As was mentioned already, their procedure amounts to the use of a restricted ensemble for the amplitudes of the modes above threshold. In addition, they replaced the mean values by the most probable values. In the next improvement upon this procedure, the actual fluctuations of the mode amplitudeswere taken into account in a linear fashion (HAKEN[1964, 1965, 19661, LAX [1966a, b], RISKEN,SCHMIDand WEIDLICH [1966a, b]). This procedure has also its counterpart in phase transition theory, where it is used, e.g., to calculate susceptibilities within the mean field approach via the fluctuation dissipation theorem (cf., e.g., the review of KADANOFF, GOTZE,HAMBLEN et al. [1967]). The conductivity of superconductors has [1968, 19701. This approach been calculated along these lines by SCHMIDT reveals a well known internal inconsistency of the mean field idea: the fluctuations, which are treated as small and unimportant within mean field theory, turn out to diverge at the transition. In the next improvement of the theory, the threshold behaviour of single mode lasers was analyzed by means of “master equations” or Fokker-Planck equations (for the numerous references see, e.g., HAKEN[1970], RISKEN [ 19701, LAX [ 19681) avoiding all linearization approximations. One of the central results of these approaches is an expression for the probability density of the laser mode amplitude (RISKEN[1965], LAX and HEMPSTEAD [1966], SCULLY and LAMB[I9661 and others), or the corresponding photo-count distribution. These expressions show a remarkable correspondence to the probability densities of order parameters obtained from the phenomenological Landau theory of phase transitions (cf. DEGIORGIO and SCULLY[1970], GRAHAM and HAKEN[1970], GRAHAM[1973]). The mean field approximation in laser theory has been overcome in a similar way as in phase transition theory, namely by calculating fluctuation intensities as moments of the probability densities. In phase transition theory such calculations are very difficult and have only recently become possible through the application of renormalization group techniques (WILSON [1971a, b]). For mode continua in lasers the calculation of moments of the distribution function would present a similarly formidable task. However, for lasers with a few modes only, and in particular for single mode lasers,

240

THE PHASE TRANSITION CONCEPT

[v.

01

the moments can be easily evaluated. These approaches show that no singularities arise at the threshold of a single mode laser. Instead a smooth change from below t o above threshold occurs in a very narrow threshold region whose width is proportional t o I/JN where N is the number of “atoms” participating in laser action. The results of these theories were later confirmed by experiments in the threshold region. The smoothness of the transition has to be expected on general grounds if the analogy t o phase transition theory is taken seriously. It is well known that broken symmetries and phase transitions in finite systems with reasonable interactions will not occur. If such systems are put into a state with broken symmetry, there will always exist a finite probability for the occurrence of a fluctuation which restores the full symmetry. In lasers, the slow temporal diffusion of the phase of the electromagnetic field accomplishes this task, and accounts for the small residual linewidth of the laser light above threshold. A similar remark applies t o the work in which laser fields propagating in one-dimensional, infinite media were considered (GRAHAMand HAKEN [1968, 19701). In such media, (like in one-dimensional superconductors, cf. HOHENBERG [1967], RICE[1965]), the symmetry, which appears to be broken in the mean field approximation, is restored by a spatial diffusion of the phase of the laser field. A new argument has t o be used, when laser action in three-dimensional infinite media is considered, as we will briefly d o in this review. We will show that a phase transition for this case can again be ruled out due to the fact that a whole continuum of modes, equal in wave numbers, but differing in spatial directions, pass the threshold simultaneously. These modes lie on a spherical surface in k-space, the radius being given by the inverse laser wavelength. The density of the number of modes in the vicinity of such a spherical surface in k-space does not go t o zero in three dimensions (as it does, of course, in the vicinity of a single k-value), which makes a condensation into the modes on that surface impossible. These results and arguments make it clear, that the analogy between threshold phenomena in optics and phase transitions can be strictly used only when a discrete set of modes is considered, and then only if we take the “thermodynamic limit” N + 00, I/ + cc, N / V = const. while keeping the number of modes fixed. The latter constraint implies, of course, that the linear size of the volume is small compared to the coherence length, which is the typical range of the interaction between the laser atoms via the electromagnetic field. Since this length is quite large (of the order lo2 cm) this limit is not completely unrealistic. The same limit was considered

v, P 1 1

INTRODUCTION

24 1

recently by HEPPand LIEB[I9721 in a discussion of the thermal properties of a single mode interacting with two-level atoms. They showed that for very strong coupling a phase transition occurs in that limit even in thermal equilibrium. HAAKE and GLAUBER [ 19721 studied transienl cooperative effects (“superradiance”) of two-level atoms coupled to one mode in the same limit. DOHM[ 1972a, b] also considered this limit in detail in a certain laser model, whose stationary distribution hc could calculate exactly for finite N . The singularities, which appear at the laser threshold when the width of the transition region goes to zero, are of the mean field type, as one expects from phase transition theory. In this sense, the original results of LAMB [1964] and HAKENand SAUERMANN [1963a, b] are exact in the limit N -, 00. Hence, in order to perform experiments within the threshold region, one has to chose lasers, in which the number of atoms, N , participating in laser action, is as small as possible. Since most laser theories treat the level structure of the laser atoms and the pumping scheme on a very phenomenological level, these theories usually do not give N in terms of other parameters but rather treat it as a known, predetermined quantity. Some experiments in the threshold region (ARECCHI, RODARIand SONA [1967]) indicate that N in many lasers may indeed not be as large as one might have expected ( N z lo7 seems possible in the quoted experiment), which accounts for the fact that deviations from the critical behaviour due to the finiteness of the system are, in fact, readily observable. In the present review we want to present various approaches to the theory of lasers and similar coherent sources from the view point of phase transition theory. We start in 8 2 with a review of the mean field approach. We will apply it, in particular, to the case of mode continua in one and threedimensional isotropic media and exhibit the various singularities which arise. As an example of how fluctuations are treated within mean field theory, we investigate how coherence and long range order spreads among the laser atoms as we approach threshold from below. In 8 3 mean field theory is left behind and expressions for the probability densities of the slowly varying, “hydrodynamic” fluctuations are obtained. We discuss the analogies and differences of these results to results of phase transition theories. In particular we show that no phase transitions are obtained in the thermodynamic limit for the case of mode continua, and that for afinite number of modes phase transitions indeed occur in that limit. In 9 4 we consider the thermodynamic limit of one and two mode lasers in some detail and discuss analogies to phase transitions in the static and dynamic behaviour of such lasers. Our discussion up to and including 0 4 is solely concerned with a discussion of laser emission. Ln 8 5 we will finally show how our discussion can be ex-

242

THE PHASE TRANSITION CONCEPT

[v, ii 2

tended to other stimulated emission processes which are considered in non. linear optics. In particular we will treat stimulated parametric scattering and Raman scattering from our point of view. Provided the number of modes is kept finite in the limit N + 00, V + co, N / V = const., singularities of the mean field type appear at the threshold of stimulated emission.

5 2. Mean Field Theory of Laser Media 2.1. INTRODUCTORY REMARKS

In this section we will consider a system of pumped two-level atoms interacting with the electromagnetic field. We look at this problem from the point of view of many particle physics. The two-level atoms are treated as a system of particles interacting with each other by emitting and absorbing quanta of the electromagnetic field. We are able to vary the strength of this interaction by appropriate pumping of the system. Gradually turning on the pumping, it will be our aim to study the transition of the system from a region where the particles move independently and at random to a region where a collective motion takes over. This transition will be identified with the laser threshold. After introducing the model in 6 2.2 and describing qualitatively some of its features in 62.3, we investigate this transition in the frame of mean field theory in 3 2.4. In 6 2.5 we reintroduce fluctuations into mean field theory in the spirit of linear fluctuation theory. 2.2. THE MODEL

As the most simple model of laser emission, we will consider a system of N atoms (embedded in some otherwise uninteresting medium) with a homogeneously broadened two-level transition (transition frequency v, homogeneous linewidthy,). It is assumed that the atoms interact with each other only via the electromagnetic radiation which they emit. This radiation travels in the background medium with the phase velocity c without appreciable dispersion and with a linear loss, described by some loss constant ti (sec-I). We couple all the atoms to an external pump. If we could simply switch off the interaction with the electromagnetic field, the pump would bring the population inversion per atom to a fixed value oo in some characteristic time 7,;'. Since we will assume that each two-level atom has only one electron we have - I 2 oo 5 + 1 . This model is well known in laser theory (cf. HAKEN[1970]). Its dynamical variables are: ( I ) the electron flip operators a s , a, which raise and lower the electron in the atom at point x,, (2) the operator of the population inversion oJl,

v,

D 21

M E A N FIELD

243

THEORY OF LASER M E D I A

(3)

the creation and annihilation operators b:,, h k , of the electromagnetic modes with wave vector k and polarization 1,. The operators x P , z; , crI1 have the algebraic properties of spin-4 operators (see appendix I) and commute at equal time with each other for different p , and with the h,,, b&. The mode operators satisfy the usual Bose commutation relations. bl, is connected with the negative frequency part of the electric field strength by

where the ek, are the two polarization vectors of each mode with 1 satisfying (k *

ekl)

=

0,

(ek].

'

ekl')

= 81,';

= 1,

2,

(2.2)

V is the volume of the system. We assume cyclic boundary conditions. The Hamiltonian of our model takes the form

describe dissipation and pumping processes The parts Hbath, and Hsy,, in a standard fashion, which we don't discuss explicitly here (see HAKEN [1970] for an extensive discussion). The coupling constants between the atomic dipoles and the modes are given by -

gzip

=

-i(Pp

'

ek,)\/2nv/h

(2.4)

where p , is the atomic dipole matrix element at x,. In eqs. ( 2 . 3 ) and (2.4) we already introduced the dipole approximation and neglected the nonresonant terms in the interaction (rotating wave approximation). Furthermore, in eq. ( 2 . 4 ) we replaced wk, = clkl by the resonance frequency v, anticipating that only mode frequencies close to this frequency will be considered. We assume that JpJ = p is independent of the atomic position and that the directions of the dipole moments are distributed at random, independent of the dynamics of the system. Then we can use the relation ~

( g z i . p gk'/.'p) -

=

2 (.kk

' ek'/.')

(2.5)

where (. . .) indicates the average over the orientations of the dipole moments. We will always take this average in the following, without further explicit indication. g 2 is defined by

244

Lv, 0 2

THE PHASE 'TRANSITION CONCEPT

In the following we will find it useful to introduce nonlocalized mode operators also for the atomic variables by the transformation

0,

=

1 C 0 k exp (-ikx,). Nk

-

Note that (2.9)

=

(E-kl)',

(Tk

=

6-k.

Their commutation relations follow from the spin-4 properties of a,, a,', cp in a straightforward fashion (see appendix I). Our basic working equations, derived from the Hamiltonian (2.3) by perturbation theory with respect to the heatbath coupling and the Markoff approximation in a standard way (cf. HAKEN[1970]), then take the form

b,&

=

+ i g + (iwk, - Kkl)bkl+ -= c(kl

>,&:

=

(iV-71)

Jv

+

Nkl-

ig

1 (ekl

---Z

Jv

+ t) + Fki( '

(2.10) (2.1 1)

ek'l')b:l,6k_k'+rk+(t)

k'l'

-(ek'l'

*

ek+kfA,')bkfl* c(L+k'l")+

@k(t).

(2.12)

In eqs. (2. lo), (2.12) we introduced the total population inversion Do

= o0 *

N

(2.13)

and the linear loss constants i c k l . The quantities ~ & ( t ) rll(t), , @k(t) are rapidly "fluctuating operator forces" which turn the Heisenberg equations

v, 0 21

MEAN FIELD THEORY OF LASER MEDIA

245

of motion (2.10)-(2.12) into a set of operator Langevin equations. They are required in all Heisenberg equations with dissipation by quantum mechanical consistency arguments (cf. e.g. the papers by SENITZKY [1960, 1961, 19671, LAX [1963], HAKENand WEIDLICH[1966], the recent review of SCULLY and WHITNEY [1972] and the earlier quoted reviews in laser theory) and can be introduced into classical equations of motion with dissipation by thermodynamic arguments (cf. LANDAU and LIFSHITZ[ 19581). While quantum theory allows to determine the commutators of these operator forces, thermodynamics gives their correlation functions. (Of course, both properties are related by fluctuation dissipation theorems in equilibrium.) We refer the reader to the extended literature on this subject. The properties we need to know for our present purposes are listed in appendix I. The eqs. (2.10)-(2.12) define our model. They describe the motion of the atomic dipole moments under the influence of dissipation and pumping. They further describe the mutual interaction of the atomic dipole moments via the electromagnetic fields, as well as their random motion. The random motion is induced by the dissipation mechanisms (thermodynamic properties of fluctuating forces) and quantum effects, notably spontaneous emission (operator properties of fluctuating forces). We will investigate the competition of random motion and collective motion in these equations from various viewpoints. First we will give a simple descriptive analysis of some features of our model. 2.3. DESCRIPTIVE ANALYSIS

We first try to get a clear intuitive idea of what happens in the laser medium as we approach the laser threshold. For this purpose we adopt a semiclassical picture; i.e. we describe the individual laser atoms by the occupation numbers of their two energy levels and we visualize them at the same time as small radiating dipole moments p,a; which have some amplitude and some phase. As a reference phase against which we measure the phases of the individual radiating dipoles we chose the corresponding local phase of the electromagnetic field. Provided some atoms are in their excited state we can have spontaneous transitions to the ground state accompanied by the emission of electromagnetic radiation. In this case, the phase of the semiclassical atomic dipole moment precedes by Acp = ++T the phase of the emitted electromagnetic field". Another atom in its ground state may absorb the electromagnetic field and go to its excited energy level. In this

* The phase relations which we discuss in this section are exhibited by eqs. (2.10)-(2.12) if we use (2.1) and P(-)(xt)= C,,p,u,+6(x-xP) to introduce the electric field strength and the local atomic polarization, respectively.

246

THE PHASE TRANSITION CONCEPT

s

[v, 2

case, the phase of its dipole moment is shifted by Acp = -An compared to the electromagnetic field; in other words, it emits an electromagnetic field which is shifted by +ITcompared to the incoming field so that the latter is annihilated by destructive interference. On the other hand, an atom in its excited state may make an induced transition, which amplifies the incident electromagnetic field conserving its phase. The corresponding atomic dipole moment then has to radiate with Acp = +$n; it therefore has a well defined phase relationship with the original emitting dipole moment. In summary, the electromagnetic field is the carrier of an interaction between the atomic dipole moments. This interaction favours a “parallel” in phase alignment of the atomic dipole moments, provided the atoms are pumped into their upper energy levels. Obviously, the strength of the external pump gives a handle on the strength of the aligning interaction. The range of this interaction is given by the distance which the electromagnetic field can travel before it gets absorbed in the host material or before its emitting dipole moment has relaxed to zero, whatever pi-ocess is the faster. In order to get a net alignment of the atomic dipole moments, the aligning interaction has to overcome their random motion which comes from spontaneous emission processes. In this respect, the present system may be likened to a system of spins, whose interaction favours a parallel alignment, which the random thermal motion tends to destroy. If the thermal motion is reduced by cooling the system, a collective alignment of the spins occurs. Similarly, if the intensity of the external pump is strong enough, a coherent collective motion of the atomic dipole moments is made poss,ble. In the latter case, the net alignment is limited by the following saturation effect. There is a certain probability that an atom which has already participated in an induced emission process will reabsorb some of the emitted electromagnetic radiation. In this case, a new component with Acp = -An will be added to the original dipole moment of this atom with Acp = +in, in other words, its original dipole moment will be reduced. The amount of this reduction will depend critically on the relative size of the lifetime of the electromagnetic radiation in the host material, IC- and the relaxation time of the atomic dipoles, y; We first look at the case IC >> yI. This condition is usually not fulfilled in laser media, but it is fulfilled, and even the defining condition, in so called superradiant systems, cf. e.g. BONIFACIO, SCHWENDIMANN and HAAKE[197 I a, b], which have recently been realized experimentally, by SKRIBANOWITZ, HERMAN, MACGILLIVRAY and FELD[1973]. In this case, an appreciable part of the electromagnetic radiation will have decayed before it is reabsorbed by the atoms, while the size of the atomic dipole moment with Acp = +in

‘.

’,

v, k 21

247

M E A N FIELD THEORY OF LASER MEDIA

will have changed much less during the same time. Consequently, the new componenr of the atomic dipole moment with A q = -+n will be much smaller than the original component with A q = ++n, so that a large net alignment with A q = ++n remains. Saturation effects are therefore not very important in this case, in which collective long range effects in the medium will mainly show up in the motion of the atomic dipole moments, while the electromagnetic field will be rather heavily damped. In the opposite case K Din at least for one value of ( k , A), the trivial solution becomes unstable. In addition, a number of nontrivial solutions exist which can be classified by their characteristic wavenumber k , . k , is determined from the condition that the population inversion is equal to the threshold inversion corresponding to k = k , ,

D

= DiOn.

(2.19)

Eq. (2.19) is obtained from eqs. (2.10), (2.11). It can be satisfied for several modes only if these have equal DiOn.Dial, defined by eq. (2.18), depends on w k i , and on x k A , which in turn are determined by the properties, the shape and the boundary conditions of the medium. If, e.g., the losses in one preferred spatial direction are significantly lower than in the other directions (as in aone-dimensional, pencil shaped medium), eq. (2.19) cannot

v,

8 21

249

M E A N F I E L D .THEORY OF LASER MEDIA

be solved for several modes at the same time and an effective selection of one mode takes place. For media and boundary conditions which are completely isotropic, eq. (2.19) only fixes Ikol, i.e., all modes with equal wavenumbers but different directions of k enter the solution in a completely symmetrical fashion. Let us first look at a medium in which one direction is distinguished by a very small K ~ Eqs. ~ . (2.10), (2.1 1) are t h w easily solved by 1)

(2.20)

(iQk>.t )

(2.21)

h,: = b L ( 0 )eXp (iszk, a:,

a;l(0)

= a L ( 0 ) exp =

-iJv-

+ +

(2.22)

V)/(Kki+YI)

(2.23)

Kki --

(Kkj. 71 i( 1’- wkA))hk+(O)

Kkl+YI RkA

= (;I1w k l +

Kkl

for k = k , and zero otherwise. Eq. (2.21) describes a coherent collective motion of the atomic dipoles, while eq. (2.20) gives the radiation emitted due to this motion. Eq. (2.23) is the well known result for the frequency pulling in homogeneously boadened lasers, TOWNES [1961]. From eq. (2.12) with k = 0 we obtain for the amplitude of the coherent motion of the atomic dipole moments

(2.24) for k’ = k , , and zero for k‘ # k , , where we assumed isotropy with respect to the directions of polarization 1.. Eq. (2.24) shows that akol N (note that D o , D i n N ) . Therefore, we obtain in the limit N -+ co

-

-

- d3(k-

k,)

(2.25)

which describes a condensation into one mode with wave vector ko in the preferred direction. In the case of an isotropic medium, eqs. (2.20)-(2.23) are still valid, but the amplitudes are now non-vanishing for all k with Ikl = Ikol. From eq. (2.12) we obtain then instead of eq. (2.24) (2.26) Assuming isotropy with respect to 2.’ and the directions of k’, we find from

250

THE PHASE TRANSITION

eq. (2.26) in the limit N

-+

coxcEP.r

[v,

82

co

Contrary to eq. (2.25), our result (2.27) d o s no longer predict a condensation into one mode. It still does predict long range order or long range coherence of the motion of the atomic dipole moments. We can easily verify this by calculating ( a , ' ~ , . )from (2.27), obtaining

While this spatial correlation function has infinite range, it cannot be factorized for (x,,-x,,l -+ x,. Hence, in the isotropic medium, (2,') vanishes in the limit V - + CO; the symmetry, which predicts an arbitrary phase of (a,'), is not broken. The result (2.25), in the one-dimensional case, would imply such a broken symmetry. The order parameter of this broken symmetry is ( a n o A / V )or, by eq. (2.22), ( h k o A / JV). Note that the critical index E of this order parameter, defined by (2.29)

has its classical mean field value c = t . In summary, we can state that, within the mean field approximation, we obtain long range order of the atomic dipole moments regardless of the dimensionality of the system. Since laser media have a very large coherence length to,the mean field approximation may be expected to give quite good results, apart from a very narrow region around threshold, whose width will depend on the number of atomic dipoles within one coherence volume. For lasers, operating with a finite number of modes, this sccms in fact to be the case. Foi such lasers our procedure above is essentially equivalent to the theories of HAKENand SAUERMANN [1963a, b] and LAMB[I9641 which describe experiments outside the threshold region quite well. 2.5. MEAN FIELD THEORY WITH FLUCTUATIONS

Correlation functions can be calculated within the mean field approach by linearizing the equations of motion around the mean field value of the variables (cf., e.g., KADANOFF,GOTZE,HAMBLEK et al. [1967]). In laser theory this approximation was the next step following the development of the

v,

3 ’I

MEAN FIELD THEORY O F LASER MEDIA

25 1

theory without fluctuations (cf. HAKEN[1964, 1965, 19661, LAX [1966a, b], SAUERMANN [1966]). We will apply this procedure to show how the long range order, obtained in the preceding section, is gradually established as the pump intensity Do approaches the threshold value Di,,. Since, within this approximation, eqs. (2.10), (2.11) are still decoupled for different k, and are also still linear, we can solve them in a simple way by treating F A and r:, as inhornogeneities. Eq. (2.12), in the same approximation, is completely decoupled from the rest and has the steady state solution d t eXp { - y l l ( f - T ) ) @ k ( T ) .

(2.30)

The steady state solution of eqs. (2.10), (2.1 1) has the same structure, but contains two different exponentials, corresponding to the two coupled equations. We represent the two exponents by two complex frequencies Q:!2. Their real parts give the frequencies, their imaginary parts give the linewidths of the electromagnetic and the atomic modes. We obtain with 6;. exp ( i P r )

-

For Do -+ Dii, the imaginary part of Q:’ passes through zero, indicating first a slowing down of the respective electromagnetic and atomic modes and finally a break down of the present analysis for Do > D i O A since , instability occurs. The real parts of the frequencies (2.31) remain finite for Do + Df,, and match with the result (2.23) for Do = DiOA. The explicit solutions of the linearized eqs. (2. lo), (2. I 1) can now easily be written down. We list them in appendix 11. The expectation values and correlation functions of the operators ak,, h k , are obtained as (zkl)

= -Do)/Ds]-+;

to

= C/(K+YI).

(2.39)

In the integral, the term with p = p‘ has to be excluded; this term was only introduced by our replacement of &J,by the c-number Do (cf. eq. (A. 1.1)). For p = p’, the expectation value ,.’;

=

N2IN

(2.40)

is given correctly by the first term in eq. (2.38). We may extend the lower boundary of the integration in eq. (2.38) to - co, since the integrand is peaked near y = vt/c. We obtain then

v, B 21

253

M E A N F I E L D T H E O R Y OF L A S E R M E D I A

x ((~sin-Ix,-x,,l+cosV C

V lx,-xf,,l C

This correlation function consists, therefore, of a short range part, which takes care of the spin operator properties, and a long range part, which is due to the interaction of the atomic dipole moments via the electromagnetic field. The range of the latter contribution is determined by the coherence length (, given by eq. (2.39). The coherence length diverges at threshold within our present approximation. The value of its critical exponent (eq. (2.39)) is characteristic of a mean field theory. Far below threshold, the interaction between the atomic dipoles is screened to within a length to, by the absorption of the electromagnetic field in the medium N K , and by the damping of the dipole fluctuations yI, whatever is stronger. The long range coherence in the atomic system cannot be observed directly; however, it is reflected in a long range coherence of the electromagnetic field. This is shown in eq. (2.34) where the same quantity Ck, appears as in eq. (2.33). This quantity, which diverges at threshold (eq. (2.36)), introduces the long range order for atoms and field. It appears as a result of the coupled motion, which makes it impossible to decide whether the atoms or the field are the genuine source of long range order. However, because of the appearance of the ratio I C ~in,eq. / ~ (2.33), ~ in comparison to 1 in eq. (2.34), it is possible to decide whether the long range coherence will be more pronounced in the atomic system or the field. If ~ / y ,is large (like it is, e.g., in superradiant systems cf. BONIFACIO, SCHWENDIMANN and HAAKE [1971a, b] and others quoted there) the coherence will show up mainly in the atomic system, since the electromagnetic field is heavily damped. If ~ / y ,is small, like it is in most conventional lasers, even if they are operated without mirrors, the electromagnetic field will be the main carrier of long range coherence, while the motion of the atomic dipole moments will be damped strongly and will contain only a small admixture of the collective motion. Together with long range coherence in space there appears a similar COherence in time, as can be seen by looking at the correlation function

-*

+k,(W)

I d . eior(tlA(t

+

.)uk,(t)).

(2.42)

Close to threshold this quantity is dominated by the slowest relaxation

254

[v,

THE PHASE TRANSITION CONCEPT

03

process and takes the approximate form (2.43)

It shrinks to a &function in w for Do -+ D i d , since Im(S2y) 0 in this limit. In the same limit the mean field approach must, of course, break down. This is seen best by a comparison of the result (2.41) with the corresponding result (2.28). While the former predicts a divergence of the correlation function at threshold due to the increase of fluctuations, the latter asserts that the same quantity should be zero at this point. This inconsistency in the treatment of fluctuations near the transition, which is characteristic to all mean field approximations, can only be overcome by abandoning this approximation. This will be done in the next section. We will see then that the analogy between the laser threshold and a phase transition, which was so pronounced within the mean field approach, will become much weaker. --f

Q 3. Beyond the Mean Field Theory 3.1. INTRODUCTORY REMARKS

It is now our aim to investigate the real status of the various singularities which arise at the laser threshold within the mean field approach. In phase transition theory one would, at this point, start from the canonical equilibrium distribution in order to evaluate the partition function, from which all (static) macroscopic properties of the system can be derived in a rigorous fashion. In our present case we cannot use the canonical ensemble, since our system is not in thermal equilibrium. Therefore, our first goal must be to calculate a probability density which appropriately describes the steady state of our system and which replaces the canonical distribution in the ensuing discussion. Clearly it will be sufficient for our purpose to analyze only that part of the electromagnetic and atomic fields which are varying slowly in space and time, apart from the main propagation at optical frequencies and wavelengths. In fact, we have seen in the preceding section that just these parts of the fields are connected with long range coherence. Tn particular, since the commutation relations of atoms and fields are a local property, as was illustrated by our result (2.38), they are not expected to contribute anything of significance to these slowly varying “hydrodynamic” components of the fields. Therefore, we will be satisfied with a classical theory. Fluctuations will be taken into account, however (of necessity, if we want to avoid the mean field approximation), by keeping the correlation

V,

s 31

BEYOND THE MEAN FIELD THEORY

255

functions of the fluctuating forces in eqs. (2.10)-(2.12). Only the commutators of both the fields and the fluctuating forces will be disregarded.* After deriving from eqs. (2.10)-(2.12) approximate equations of motion for the slowly varying parts of the fields in section 2, we will derive from them and discuss in section 3 expressions for the probability distribution of these slowly varying field amplitudes. In particular we will discuss whether singularities at the laser threshold can occur in the case of mode continua in the thermodynamic limit. 3.2. EQUATIONS FOR THE SLOWLY VARYING FIELDS

By splitting off the main time dependence of the field amplitudes bLA,ak:, with = a&(t) exp (iQkA r, bk+(f) = bzA(t) exp (iQkA t ) , where Qki is given by eq. (2.23), we obtain a set of field amplitudes

fiz2(f),

& ( t ) which vary slowly in time. Depending on the size of the ratio

iiki./yI,

either the electromagnetic amplitudes btL(t)(if iikl O

1

( 1 ) -ix x f - v

Eb-’(x)exp i

-

x

2lTvh

=

(ekAb&.) exp (- ikx),

(3.28)

(e,,b:l)exp(-ikx)

(3.29)

1/

-

k < O l

where we replaced c ( k ( by v in view of the quasi-monochromatic nature of the field. In terms of these field strengths the result (3.27) takes the form

-G

; ~ ? E-i - ) ( x ”) + , ~E;-’(x) -

i

2 0 , ~ V dx d~ ! - N 2 DTy 2nVh x [(IE:-’(x)lz + ]Eb-)(x)12)’+21E:-’(x) . Eb+’(x)12]\ I

- I

-

I

(3.30)

where F is the beam cross section perpendicular to the preferred x-direction. In the fourth order term of eq. (3.30) the spatrdlly rapidly varying beat term 2 Re(E,(-)(x). Eb(+)(x))’ is omitted, since its contribution is negligible for the long spatial distance we consider. The result (3.30) is similar to the result obtained by GRAHAM and HAKEN [1970], who considered fields travelling in the forward direction only. It shows the very close formal analogy which exists between the present description of coherence in laser media and the well known Ginzburg Landau

V,

s 41

IJISCREIE M O D E SPECTRA

263

theory of the macroscopic wave function in superconductors. Ln fact, if we consider only fields travelling in the forward direction (i.e. put E i * ) = 0 in eq. (3.30)), eq. (3.30) is completely analogous to the Ginzburg Landau probability density functional for the macroscopic wave function $(x) of a one-dimensional superconductor with coherence length to in a magnetic field equal t o zero. We only have to make the replacements E,'+'(x)C* $(x), ( D o- D,)/D, e, (T,- T)/T,, to++ to.Spatial correlation functions of such a superconductor (and such a laser) can be evaluated from (3.30) without further approximations, by functional integration methods, cf. GRUENBERG and GUNTHLR [1972], SCALAPINO, SEARSand FERRELL [1972]. Their results show how the range of coherence increases, if Do approaches D,from below. The scale of the coherence length is given by (7, N , F ~ ; / K V ) +and can become very large, if this quantity is large. Nevertheless the coherence length remains finite in principle, even at Do 2 D,, due to the one-dimensional nature of the system, HOHENBERG [1967]. We can conclude, therefore, that a divergence of the coherence length and phase transition like singularities will not appear a t threshold, if the lascr atoms can interact with each other via the entire continuum of electromagnetic modes, regardless of the number of spatial dimensions of the system; however, the threshold transition may be quite sharp in view of the largeness of the coherence length e0 in lasers". The case of discrete mode spectra, constructed by means of optical resonator cavities, will be considered in the next section.

9 4. Discrete Mode Spectra 4.1. INTRODUCTORY REMARKS

Ln the case of discrete mode spectra, the limit N -+ a,V --t 03, N / V = const. has a quite different meaning than in the case of mode continua considered before, since the number of modes is kept fixed. This new limit is not as artificial as it may seem at first glance, because of the very large coherence length to = C / ( K + Y ~ ) . Typical values of to are of the order of I m. Therefore, the limit N + 03, V + 03, but all lengths N ) exp ( f N a Z ) ( l+ 4 , ( ~ ' 4 N a ) )

(4.6)

is the error integral, and are obtained as usual by (4.7) (4.8)

We can evaluate Z in the limit of large N and obtain

z-+

1

for a < 0

--

IalN

z

-+

fi):'(

(4.9) for a > 0.

exp

Eqs. (4.7), (4.8) then give the results (P')

2

-+

Nil (7") - ( 7 2 ) Z

-+

(4.10)

for a > 0.

(4.1 1)

(:if

(i2)

--f

(?4)-(?2)2

for a < 0,

a

2 N

-+ -

266

1 H E PHASE TRANSITION CONCEPT

[v,

s4

These limiting results show that singularities and a phase transition of the mean field type occur for a = 0. If a goes to zero, the limiting results depend on the way in which we put N + 03, a -+ 0. If we let a go to zero like a = 51J N 4 0, E = const., we obtain a threshold region, where neither (4.10) nor (4.1 1 ) hold. We get instead (4.12)

(?*)-(?)

(4.13)

+

From (4.10)-(4.13) we obtain

for a < 0 4

2 Nu2

for a > 0

Of course, if we let N -+ co first, and then put a 4 0, the width of the thresI I J N goes to zero.* hold region The phase transition thus shows up in the "static" properties of the single mode laser in the sudden increase of the intensity (4.1 1) and in the discontinuity of the quantity (4.14) at threshold. In thermodynamic systems the intensity corresponds to a first order derivative of the free energy, which is continuous at the critical point of a second order phase transition whereas (4.14) would correspond to a second order derivative like a specific heat which is discontinuous at the critical point. The dynamical behaviour of single mode lasers near threshold can be analyzed by solving for the Green's function of eq. (4.3). This task was achieved numcrically by RISKENand VOLLMER [ 1967a, b] and HEMPSTEAD and LAX [1967]. We are interested in the correlation times ( A v ) - l , (Am)-' of the quantities (6+(?+?) 6(7)) and (6+(7)6'(7+?)6(7+ ?)6(1)) respectively, in the limit N + 03. We define these times by the equations N

+ 7

-=

d?

(6 +(?)5(0 ) )

(6'(0)6(0))

(4.15)

* From the approximatc form of eqs. (4.2) it is clear that the width of the threshold region of (&-O,)/& is given by ( ~ N / y : l D , ~ ) k .

V, (i 41

DISCRETE MODE SPECTRA

t

267

(4.16) They can be obtained either from the numerical results quoted above or directly from eq. (4.3) by means of scaling considerations*. Let us use the latter approach. For a < 0 (i.e. below threshold) we put Fz

= x 2I / N

(4.17)

(cf. eq. (4.10)). Then eq. (4.3) is reduced to a linear Ornstein-Uhlenbeck process (UHLENBECK and ORNSTEIN [19301, WANCand UHLENBECK [19451) after letting N -, co. We obtain for Av, Aw in this region (4.18) Note that the time is still scaled by eq. (4.2) for this and all results below. For a = 0 we can make eq. (4.3) independent of N by the new scaling transformation i=JNT~. F~ = x,Z/JN, Therefore, amplitudes and correlation times are independent of N on the new scale introduced with x2 and T ~ On . the original scale of F2 and 7 we have then, apart from numerical factors of the order (N)', (F2)

AV Aw

- I/JN - 1/JN, -

(cf. eq. (4.12)), for a

=

0.

(4.19)

l/JN,

For a > 0 we make the transformation

r- 2 = a+x,/JN

(4.20)

according to eq. (4.1 1). Then, we find from eq. (4.3) that the probability density factorizes with respect to x 3 and ij up to order N - ' . The equation for the phase part alone is a simple diffusion equation with the diffusion constant D? = aN. This part, which determines the correlation time of the phase dependent quantity (b"+(i+?)b"(i)), can be made independent of N by the transformation i=NT~. (4.21) Hence Av scales like

* A very elegant method for evaluating eqs. (4.15), (4.16) numericallywas described by SMITH [ 19741.

268

THE PHASE TRANSITION CONCEPT

Av

-+

I/aN

for a > 0

[v,

Q4

(4.22)

and goes t o zero in the “thermodynamic limit”. The vanishing of Av indicates the appearance of long range order in time, since in the same limit (bf(t+r)h(t))

-+

( b + ( t + ? ) ) ( b ( t ) ) # 0.

(4.23)

The average ( b ) , taken with the distribution (4.5) vanishes, of course, due to the phase symmetry of (4.5), even in the limit N -+ 00. If, however, a small symmetry breaking term like hi: exp(i@)+h*F exp( -iq) is introduced into the exponent of eq. (4.5), whose amplitude 1/1 we let go to zero only after N -+ cc, a non zero result for ( b ) is obtained. By such a procedure we avoid t o average over a whole ensemble of lasers with different, fixed phases 4 (cf. BOGOLJUBOV [1962a, b, c], HOHENBERG and MARTIN[1965], WAGNER [ 19661). The laser mode, in the N -+ 03 limit, has therefore a fixed though arbitrary phase, quite analogous to the fixed, symmetry breaking phase of the condensate wave function in a superfluid. However, a Goldstone mode, whose frequency vanishes as its wave number approaches the laser mode wave number, is not connected with this broken symmetry, since by construction no such modes in k-space are available near the single mode of the laser (it should be remembered that Ak = z / L >> < , I ) . To put it differently: the zero linewidth mode in the present case is not the Ak 0 limit of a continuous mode spectrum. The correlation time ( A m ) - ‘ , associated with the fluctuations of l6(?)I2, can be obtained from the amplitude part of eq. (4.3) after the transformation (4.20). Again we find a linear Ornstein-Uhlenbeck process in the limit N -+ 03, with the correlation time (Am)-’ approaching -+

Am+a

fora>O

(4.24)

for N 00. From (4.18) and (4.24) we learn that the laser intensity has a finite correlation time, except at threshold, a = 0, where it experiences a critical slowing down for N -+ 03. The singularities of the linewidths at threshold are again of the mean field type. The properties of the electromagnetic mode have their counterpart in the properties of the polarization of the laser medium. From eq. (3.3) we obtain in the single mode case -+

V,

o 41

DISCRETE MODE SPECTRA

269

The polarization amplitude has, therefore, a part which is rapidly fluctuating in time due to the uncorrelated spontaneous emission of the atoms. However, it has also a slowly varying "hydrodynamic" part which is connected with the slowly varying electromagnetic field by a nonlinear transformation. In the limit N -+ 00 we obtain from (4.25), using eqs. (4.23). (4.1 l ) , (4.2) in the limit N + co

().; ($))--gv (--)

for a < 0

0

V

b'

N

JN

fora>O.

(4.26)

The broken symmetry in the limit N -+ 00 appears, therefore, in the atomic system and the electromagnetic mode simultaneously. Whether the polarization amplitude or the electromagnetic amplitude is taken as an order parameter is merely a matter of taste. Since for the present example we assumed that K > y,, y,: can be treated quite analogously. Physically, this case would correspond to a system in which the screening length of the atom-atom interaction is determined by a very short field absorption length. While it seems difficult to fulfill such a relation for optical fields, it might be fulfilled for atoms or spins interacting via phonons, which are more heavily damped. If enough energy is fed into such a system, a coherent stationary motion of the atoms or spins should occur. The field, mediating the interaction between the atoms, would reflect this coherent motion in a linear fashion, according to eq. (3.4). The single mode case can be reduced to the form (4.3) by the scaling transformation*

(4.28) (4.29) 4.3. TWO MODE LASERS

A procedure which is similar to the one given in the preceding section can

*

The width of the threshold region of (Do-DD,)/D.is now given by ( y ~ N ~ / @ ~ * ) i .

270

THE PHASE TRANSITION CONCEPT

rv, 5 4

be applied to multi-mode lasers, as long as the total number of modes remains finite in the limit N + MI. We consider the two mode laser as a simple example. For recent treatments of this problem see GROSSMANN and RICHTER [1971b], GRAHAM and SMITH [1973]. We consider the case of two standing waves, i.e. we put in eq. (3.3) 6: = - b f k , and bA = -b+k2 and take all the other mode amplitudes equal to zero. We will assume further that the frequency difference of the two modes is much larger than yil, although it is still small compared to yI, a condition which eliminates the possibility of mode locking. Confining ourselves to this case we obtain from eq. (3.3)

(4.30)

and a corresponding equation for b l , where we have put

(4.32)

After scaling time and amplitudes by 4

'

2

6 D o N g - bi

=

,,@Pi

exp {i+i-i(wi-Awi)f)

(4.33)

we obtain the Fokker-Planck equation

(4.34)

which was discussed by GRAHAM and SMITH[1973]. The stationary distribution of eq. (4.34) is obtained as

w

-

exp [--+N(+F:+JF:-U~ F : - u ~~5++7:7:)]

(4.35)

v, $ 4 1

271

DISCRETE MODE SPECTRA

since eq. (4.34) again satisfies detailed balance. From eq. (4.35) we obtain 03: in the limit N -+

For a,, a, < 0,

(F;)- 0, a, < : a , , (Ff>

(F;)-(F;y

--f

+

(Fi)

a,:

2

L 4

N b , - 3a I I '

(F;)-(?y

;

- -

N

4

+

-

.

N z ( a 2 - ~ a , ) 2'

(F;F;)-(F;)(F;}

-+

-8 3N2(a2-ja,)2

(4.37)

'

For a , > 0, u2 > +a,,

We see that singularities (discontinuities in the derivatives d ( P ) / d a ) appear at the threshold of the first mode, a , = 0, and at the (shifted) threshold of the second mode, a, = + a , . At the latter threshold, singularities occur in both modes. The correlation times (Avi)-', ( A m i k ) - ' of the amplitude and intensity fluctuations, defined as in the single mode case, can be obtained in the N + 00 limit by scaling considerations quite similar to those in the preceding section. The only difference is introduced by the mode-mode coupling. We simply list the results, which again describe the appearance of long range order in time, and slowing down effects of the mode intensities. For a , , a, < 0,

A v ~ fAwi -+

-+

(4.39)

212

THE PHASE TRANSITION CONCEPT

For a , > 0, a2 < +a,,

A U J ~+ 2 [ ~ 2 - $ ~ ,;[ Am, 2la,l; A w , ~+ 4aIla2-+0-a,l/($a, - ~ 2 ) . +

(4.40)

For a, > 0, a, > $ a 1 ,

A v ~-+ 5/{9N(~,- + ~ 2 ) ) ;

Av2

+

5/{9N(~2-5~1)}.

(4.41)

If a, > 0, u2 > +a,,the correlation times (Amik)-' in the limit N -+ 00 are given by lengthy algebraic expressions since the two laser modes are then mixtures of the two normal modes of the system. However, the correlation times (AA,, 2 ) - ' of these two normal modes are given by simple expressions in the limit N + KI which take the form _.

AA1, 2

-+ &(a1

~-

~

+ a2-tJ(a,+ 0 2 ) ~-20(a,

- 3 ~ 2 ) ( ~ -+a,)). 2

(4.42)

Since ( A & - ' diverges at the second threshold (aZ = :a,), the associated normal mode dominates the long time behaviour of both laser modes if that threshold is approached from above and the intensity fluctuations of both laser modes will experience a critical slowing down. This line narrowing due to mode-mode coupling can be expected to occur in multimode lasers at the thresholds of all modes whose intensities are coupled by the laser medium. Analogous effects in thermodynamic systems are well known (cf. FORD, LANGLEY and PUGLIELLI [1968]) and have been dealt with theoretically by mode-mode coupling theory (cf. KADANOFFand SWIFT[1968a, b], KAWASAKI [1970]). For finite N , the slowing down does not become critical, but should be observable, if (D&-Df,)/DZ, is sufficiently large compared to 1 (cf. GROSSMANN and RICHTER[1971b], GRAHAM and SMITH[1973]). Furthermore, the discontinuity of Aml across the second threshold is smoothed out for finite N .

8 5. Discrete Modes in Nonlinear Optics 5.1. INTRODUCTORY REMARKS

Up to now we have only considered resonant induced atomic emission in laser media in which a population inversion is created by an external, incoherent pump. Now we will look at induced non-resonant multi-photon emission processes in systems, which are pumped coherently by a laser source (parametric emission). In such media, a long range order of the atomic polarization field

V,

8 51

DISCRETE MODES IN NONLINEAR OPTICS

273

at the frequency wp of the pumping field is introduced from the outside. Due to the occurrence of induced parametric emission, coherence and long range order at lower frequencies w i and wave numbers k i (with x i w i = wp, x i k i = k , ) will appear, if the intensity of the pump exceeds some threshold value. All frequencies considered are usually far from a n atomic resonance. The threshold behaviour of the atomic polarization and electromagnetic fields is dominated by the competition between the local spontaneous emission processes and the collective induced emission processes. Therefore, an analysis similar to the one given for laser media can be applied. In the following we consider the case in which parametric emission from a single excited mode into two discrete standing waves of a resonator takes place. Such a model has been considered by many authors. It was realized experimentally by GIORDMAINE and MILLER [1965, 19661. A linear three mode model was discussed by LOUISELL,YARWand SIEGMAKN [1961], GORDON, LOUISELLand WALKER [ 19631, LOUISELL [ 19641, MOLLOW and GLAUBER [1967a, b J who all neglected dissipation effects. WAGNER and HELLWARTH [1964] included dissipation in the linear model. The nonlinear model with dissipation, which we consider in the following, was investigated in detail in the papers of GRAHAM and HAKEN[1968], GRAHAM [1968a, b, 1970a, b], WHITEand LOUISELL[1970]. This model describes optical parametric emission*, Raman-Stokes emission or Brillouin emission, depending on whether the second of the two modes, into which emission occurs, is a photon, an optical phonon, or an accoustical phonon. 5.2. EQUATIONS OF MOTION

The electric field strength inside the resonator at the threc frequencies involved, can be written as E ! - ) ( x , r) = - i

E (, - ) (x. r)

=

4nhwi

v-

--

e,b+ sin k j x

(5.1)

-

if we assume standing waves for the generated fields and a running wave for the field at the pump frequency. A similar decomposition can be made for the atomic polarization inside the medium. The polarization at frequency o1is given by P:-’(r) . e l ,Ji sin k , x = x$(wl

= w j -w,)~by’(x, t ) ~ , , ( x , t )

(5.3)

* The case of degenerate parametric oscillation was discussed by LANDAUER and WOO [ I 9 7 1 1 and G R A H A M [1973b].

274

THE

PHASE TRANSITION

CONCEPT

[v,

85

(cf. BLOEMBERGEN [1965] p. l l ) , where f ‘ is the third rank tensor of the nonlinear susceptibility. From eq. (5.3) we obtain with (5.1), (5.2)

P,-)=

- .-

- n h J w , opZ”!f*(AkL)h: h, ,

(5.4)

. ~ ” ‘ ( w=, o,+a2): e 2 e , )

(5.5)

where = (e,

%“I

f(AkL) Ak

=

=

exp (iAkL)- 1 iAkL

k,-k,-kl.

(5.7)

A corresponding expression holds for P i - ’ . The polarization at the pump frequency is obtained as Pb-’ = + f ’ * . nh zf(AkL)h:b: V . x I E.,- ( - I

+

JW,

(5.8)

where 2‘ is the linear dielectric susceptibility of the medium and EL-) the negative frequency part of the pumping laser field. In eq. (5.8) we made use of the permutation symmetry relations, satisfied by the tensor x“’ (BLOEMBERGEN [I9651 p. 11). The equations of motion of the three mode is now amplitudes have the form of eq. (2.10), but the polarization determined by the preceding equations. We obtain therefore

6:

=

(io,- K , ) b :

-

h

--=-f * b ; b , + F : ( t )

Jv

15.9)

a corresponding equation for b l and

6,’

=

(iw,-K,)h,f

+

h* f h ,+ b,+ +jVFI-’exp(iw,r)+F,+(t) (5.10)

Jv

with

-

h

=

-

f i J 2 n 3 h ~a2 , (I),,

F ~ - ) e x p ( i w , r )=

1/

(5.1 1)

-..

%EL (-) .

2 X ~ p1

(5.12)

The properties of the fluctuating forces F:, F,’ in these equations are the same as given in eq. (A. 1.4). The equations of motion (5.9), (5.10) can be derived from an effective Hamiltonian (see GRAHAM [1968a]), the 6, 6’ being boson operators of the various modes. They can, therefore, be taken as a microscopic “rigorous” starting point for the theory which follows.

v,

0 51

D I S C R E T E M O D E S IN N O N L I N E A R O P T I C S

275

Then we have an alternative way to formulate the dynamics of the system by using the Schrodinger picture. Moreover, we can write the equation of motion in terms of aquasi-probabilitydensity function (cf. GRAHAM [1968a]). Using the Wigner distribution function W, introduced by WIGNER[1932] and discussed in detail by MOYAL[1949], we obtain instead of eqs. (5.9), ( 5 . lo) the generalized Fokker-Planck equation aW- -at

a-

\ah,

+ 1

((iw,+Cl)bl+

Ki(G,(T)+f)-

i=1.2,p

h*f

Jv

d2W

ahidbF

b,b:)

W+

+ 41 Jh*fV --

-2

a 6b2

(I -2)W

d3W -

-

1 +{c.c.>.

6b,*db,db2j

(5.13)

The b’s are now c-number representations of the mode amplitudes. In the classical limit W reduces to a classical probability distribution of the mode amplitudes. We will be only interested in this classical limit in the following. The equation of motion may as well be formulated in terms of the P-representation, GRAHAM [1968a], introduced by GLAUBER [1963a, b] and SuDARSHAN [1963], or the diagnoal matrix element with respect to coherent states (GRAHAM [1970a]), discussed by GLAUBER [1964] and by SMITH [1969]. 5.3. MEAN FIELD APPROXIMATION

The mean field approximation of the equations of motion is again obtained by discarding the fluctuating forces in eqs. (5.9), (5.10) (cf. 9: 2.2). The time independent solution of the resulting equations is easily obtained (cf. GRAHAM and HAKEN[ 19681, YARIV and LOUISELL [ 19661). We give here only the resulting expressions for the mode intensities. For (5.14)

(5.15)

we obtain

describing an undepleted transmission of the pump field through the system,

276

THE PHASE TRANSITION CONCEPT

without any parametric scattering. For

lFL-)12 > lFL-)’l2

(5.19)

we obtain lbpl2 -

I F (L- ) s 1 2

VlK2 P

(5.20)

K,lb,I2 = K21b212

The intensities 1b1I2,Ib212 of “signal” and “idler” are non-zero above threshold, and the transmitted pump intensity is depleted to the threshold value. The intensities of all three modes, as a function of the pump intensity, have discontinuous derivatives at threshold. The singularities are again of the mean field type, if we put into correspondence

(I FL- )I 2 - IFL- ’”2)/ IFL-

)SIZ

(5.22)

(T, - T ) /T,

c,

( b l ) , (b2), ( P i - ’ ) , ( P i - ) ) c,order parameter. 5.4. PROBABILITY DISTRIBUTIONS FOR THE SLOWLY VARYING FIELDS

We will first consider the case Am = 0, f c l = x 2 = K which describes optical parametric oscillation. Afterwards we discuss the case Aw = 0, I C > ~ > K ~ more , appropriate to Raman oscillation, in which the phonons are heavily damped (with K ~ compared ) to the photons. Furthermore we will assume that K~ >> K which is, e.g., fulfilled if mirrors are used, reflecting at w1 and w 2 but not at wp. With this assumption we may neglect I compared to IK,S;~ in eq. (5.10) and obtain

bzl

(5.23) where we have neglected 1F‘;l compared to IJvFL-’l. Putting this back into eq. (5.9) we obtain

and a corresponding equation for

6:

=

exp(-iwit)b+;

b l , with

FT = exp(-iwit)F’

(i

=

1,2,p).

(5.25)

V,

B 51

277

DISCRETE MODES IN NONLINEAR OPTICS

We can eliminate most system parameters by the scaling transformation

(5.26)

From the Langevin equations for the mode amplitudes we obtain the Fokker-Planck equation

(5.27) if we use symmetrized correlation coefficients of the fluctuating forces, and neglect thermal fluctuations compared to quantum fluctuations (iij(T) us, its maximum is shifted towards finite amplitudes 16,12 = 16,12 = llal-a”l. If the damping constants K ~ K,, are very different, K , >> I C ~(as, e.g., in Raman oscillators) we may eliminate the second mode as rapidly varying. In that case we obtain from eq. (5.9) for b, (5.29) We can now solve eqs. (5.29), (5.23) for 6, and 6, in terms of 6,. Restricting l~ is consistent with our preourselves to the case 1611z> lhfI2/V) and keeping only the dominant terms containing the fluctuating forces, we obtain the equation

278

THE PHASE TRANSITION CONCEPT

[v, P 5

(5.30) With the scaling transformation

(5.31)

and using symmetrized correlation coefficients as before, we obtain again the Fokker-Planck equation (4.3), GRAHAM [I 968b], WHITEand LOUISELL [1970] already derived for single mode lasers. We will now investigate the results of this section in the limit N -, co in order to show that a phase transition of the mean field type occurs in that limit. 5 . 5 . RESULTS IN THE LIMIT N - + co

N

The moments of the distribution (5.28) are easily evaluated in the limit + OO.*We obtain for la1 < d

* We consider the classical limit again, hence the quasi-probability density can be replaced by a classical probability density.

V,

9 51

D I S C R E T E M O D E S IN N O N L I N E A R O P T I C S

279

quite analogous to the results for the single mode laser, and in agreement with the mean field results. The dynamics of the system in the limit N 00 can be evaluated from the Fokker-Planck equation. For la1 < as we put -+

6; = i$IJN

(5.34)

which reduces eq. (5.13) to a linear Ornstein-Uhlenbeck process for the si if we let N -P 00 (cf. UHLENBECK and ORNSTEIN[1930], WANG and UHLENBECK [ 1945I). Since the two modes are coupled even for la1 < a, and even in the limit N co, their correlation times, defined as in the laser case, will contain contributions from both normal modes of the system. The relaxation times I.;,; of the two normal modes are simply given by -+

=

(5.35)

a'flal.

For la( -+ as (la1 < a s ) the slow normal mode with 1, 0 will dominate both electromagnetic modes, and they will both slow down. For la1 > aswe put -+

(5.36)

Eq. (5.13) then separates into an amplitude part and a phase part if N -P 00. The phase part describes a pure diffusion process of the phase difference d l -g2 with the diffusion constant

(5.37) which goes to zero for N -+ co. The coherence times A v l ' , A v i ' of the amplitude fluctuations which are dominated by this diffusion process, go therefore also to zero like Av,

=

Av2

1 -+ -

(5.38)

(Nlal --a5)*

ASin the laser case, the vanishing of A v l , Av2 in the limit N -+ co indicates the appearance of long range order in time and of a broken symmetry. The coupled intensity fluctuations will also show a slowing down as in the laser case. The correlation times I.;,: of their normal modes above threshold are

280

[“

T H E P H A S E ’I R A N S I T I O N C O N C E P T

obtained from the amplitude part of eq. (5.13) for N

+

co as

_.

j . , . 2= lalfJla12-4a*(lal-a’).

(5.39)

For la1 + a, the slow normal mode with vanishing I., will dominate the intensity fluctuations in both modes and they will both slow down. Therefore all effects in the laser which become critical as N + 30, do have a counterpart in the present example. This is also true for the case of unequal damping, K~ >> K ~ A. discussion of the results for that case is superfluous, however, since after the scaling transformation (5.31) all results are identical to that obtained for the single mode laser.

Appendix I ln this appendix we list some of the formal relations, which appear in our model in 9: 2.1. 1. Relations satisfied by 2-level operators = +(l+a,,)

, . ;z

2

TI,

=

., +2

= j(1 - o p )

z,;.

(A.I.1) =

o; = 1.

0

2. Commutation relations of non-local atomic operators in a system with random orientation of dipole moments [.k,??

+ ’

bk,i.,J

=

-(ekL

’ ek’l’)‘k+k’

(A. 1.2)

3. First and second order correlation functions of the fluctuating forces in eqs. (2.10)-(2.12) (cf. SEN~TZKY [1960, 19611, LAX[1966a, b], HAKEN and WEIDLICH [1966]).

Field

3. Coherent excitation also occurs. By alignment, we mean that the level population for a given J > 3 is a function of lm,l. (The term “oriented” is sometimes used as we use “alignment”, and “alignment” is elsewhere taken to mean that the population of level m, is not equal to that of -m,.) By coherence, we mean that the phase factors in the wave functions which describe the m, sublevels for a given spectroscopic term are fixed in a given experiment. The existence of alignment is indisputable, but its precise origin is not known; in particular, one cannot predict what the population imbalance will be for an arbitrary experiment. The origin of coherence is better understood: it has to do with the short impact time during which excitation occuts. When alignment and coherence are present simultaneously, one finds “quantum beats”, which can be used to measure Land6 g-factors, fine structure and hyperfine-structure separations, and, also, level lifetimes without the complication of cascade repopulation of the level of interest. Work using the coherence and alignment features is reviewed. In a source where many different stages of ionization are produced simultaneously, it is not always easy to associate a spectral line with the spectral order to which it properly belongs. Methods have been developed to make the charge assignment; we treat them briefly.

Q 2. Spectral Line Shapes The contributions to line broadening which are peculiar to the beam-foil light source come entirely from the Doppler effect. Figure 1 illustrates several of the ways in which the Doppler effect manifests itself. The wavelength, A, which is detected when a photon of wavelength A, is emitted from a moving source whose velocity is at angle 6 relative to the direction of observation, is given by

A

=

A,(1

-p

4-

cos 0)

’

where p is the ratio of the speed of the emitter to the speed of light. It therefore follows that any detector, even if it has an infinitesimal acceptance angle, will respond to a range in wavelengths because (a) there is an intrinsic velocity spread in the beam, (b) there is an intrinsic angular divergence of the beam, and (c) there is scattering in the foil. If one takes account of the

VI,

s 21

SPECTRAL LINE SHAPES

SLIT-

291

v

-

Fig. I . Arrangements for observing the 5-ions in beam-foil experiments. The points A and B define a beam segment, 0 is a representative angle between the c-ion velocity and the light which is accepted by a spectrometer, F is a (shielded) Faraday cup, L, and L2 are field lenses, and the slits indicate entrances into spectrometers. Lenses are not always used. The Doppler effect arises because of the finite size of the aperture which collects the light and because of the change in 0 between points A and B.

finite acceptance angles of real detectors, two other factors enter: (d) from a given point on the beam, 8 is not constant over the acceptance aperture, and (e) if light is collected over the beam segment from A to B (Fig. l), there is, again, a variation in 8 for the photons which reach the slit. Nobody has yet made a complete analysis of the above factors, although different parts have been treated by JORDAN [1968], STONER and RADZIEMSKI [1970, 19731, STONERand LEAVITT[1971a, b], and LEAVITT,ROBSONand STONER[1973]. One cannot write a simple expression which describes the situation accurately for the ranges of [-ions, [-ion energies, foils, etc., which are used in beam-foil experiments. However, some representative numbers can be given. Thus, lines at 5000 A from nitrogen at 1 MeV could easily FINK,MALMBERG, MEINEL and TILFORD have widths of 5-10 A (BASHKIN, [1966]); such widths are intolerable if serious spectroscopy is to be done. A great improvement can be made by recognizing that the principal contributor to the line width is apt to be the finite acceptance angle of the spectrometer. The refocusing method (STONER and LEAVITT[1971b]), in which the position of the grating or exit slit is adjusted to compensate for the variation of I over the length of the observed beam segment, permits one to reduce the line width to 5 1.0A. It has also been shown (STONER and LEAVITT [1971a]) that, where lenses may be used, the finite width of the entrance slit need not broaden the lines to any significant degree. Thus, at the cost of increasing a line width from 1.1 to 1.2 A, the detected intensity was raised by a factor of 20. The influence of scattering in the foil may be gauged from a comparison of the above with the line width obtained for the case of 200-keV Ar par-

-

292

BEAM-FOIL SPECTROSCOPY

[VI,

$3

ticles excited by collisions with CO, vapor (STONER and RADZIEMSKI [1972]). These particles are moving only one-fourth as fast as 1-MeV nitrogen atoms, so that the Doppler effect is substantially reduced by that factor alone. However, the scattering rises sharply because of the larger charges in Ar and CO, than in N + C . It rises also because scattering varies roughly as I/,?,, but is reduced because of the smaller energy loss in CO, relative to the carbon foil. (Unfortunately, one cannot circumvent these complications.) The empirical finding (STONERand RADZIEMSKI [1972]) is that line widths of 0.1 A at 5000 A can be obtained. Aside from refocusing, efforts to reduce the finite-aperture effect have been made by JORDAN[I968], BAKKEN and JORDAN119701, and by DUFAY, GAILLARD and CARRE[1971], who employ field lens L, (see Fig. I ) , or an equivalent mirror, which gives a large Doppler shift to each line but minimizes ;i./iXl and dA/l;p; by KAY and LIGHTFOOT [1970], who choose L, to have a short focal length and place L, close to the beam, so that the beam velocity of significance to the spectrometer is reduced by the demagnification of the optics; and by CARRIVEAU, DOOBOV, HAYand SOFIELI) [1972], who place the field lens L , one focal length in front of the entrancz slit, thercby ensuring that the principal illumination of the grating comes from light which leaves the beam at 90" to the particle velocity. The use of a lens precludes using these last mcthods in the vacuum ultraviolet; in some instances, a mirror may be used in place of a lens so as to permit work in the vacuum ultraviolet. In addition, observations at 0" are experimentally more awkward than those at 90" because the beam stop gets in the way. As a practical matter, most beam-foil experiments view the beam at 90". T o secure small line widths remains an important goal. At the present time, line-blending interferes with the proper identification of the parent levels, which in turn renders suspect a number of measurements of level lifetimes. For elements like the rare earths, where the level density is high, the prospect of using the beam-foil source is not attractive simply because of the line-width difficulty. The ultimate limit on linewidth will be set by the scattering in the foil. One of the reasons for using carbon is to keep the scattering small. The use of beryllium would reduce the scattering by a factor of two, but the health hazard has been a serious deterrent. There's no point in trying boron because boron foils are hard to prepare and have little mechanical strength or life under the beam as compared with carbon.

-

9 3. Wavelength Studies and Energy-Level Schemes The principal function of the wavelength, for our purposes, is to identify

ij

1 5 MeV

Oi

i

t

-

t-

cn Z W I-

Z -

900

800

700 WAVELENGTH

600

(A)

Fig. 2. Part of the vacuum U.V. beam-foil spectrum of chlorine. Reproduced from BASHKIN and MARTINSON [1971] by kind permission of the authors and the journal.

294

BEAM-FOIL SPECTROSCOPY

IVI,

§3

the level, and the characteristics thereof, from which the light comes. This is especially interesting because beam-foil spectra frequently include spectral lines not previously reported. To date, these lines have been identified as having either of two origins, namely, C-ions whose structures had been incompletely explored in prior work, and multi-electron levels the radiative decay of which cannot be seen in other kinds of spectral sources. An example of the former is given by the beam-foil studies of C1 (BASHKIN and MARTINSON [ 19711, MARTINSON, BASHKIN, BROMANDER and LEAVITT [1973]). One of the spectra is illustrated in Fig. 2 (BASHKIN and MARTINSON [1971]),while Fig. 3 (MARTINSON, BASHKIN, BROMANDER and LEAVITT [1973]) gives the level structure of CI VII as deduced from the data. Of the lines shown in Fig. 3, only the two resonance transitions were known (BOWEN and MILLIKAN [1925], PHILLIPS [1938]) prior to the beam-foil work, although levels up to 6f had been seen (PHILLIPS [1938]) via resonance transitions of shorter wavelength.

ns

I 4001

--

nD

nd

nf

na

nh

ni

nk

nl

nm

i1

Fig. 3. Energy-level diagram for C1 VII. Wavelengths are given in Angstroms and level positions are in kilokaysers or cm- ’. Reproduced from MARTINSON, BASHKIN, BROMANDER 119731 by kind permission of the authors and the journal. and LEAVITT

o

VI, 31

NAVELENGTH S T U D I E S A N D ENERGY-LEVEL SCHEMES

,

1

2700

1

1

1

1

1

2600 WAVELENGTH

1

1

1

1

295

1

2500

(A)

Fig. 4. Part of the U.V. beam-foil spectrum of Mn. The incident 5-ion beam was M n + with an energy of 249 keV. Reproduced from CURTIS,MARTINSON and BUCHTA[I9731 by kind permission of the authors and the journal. + +

Fig. 5. Part of the beam-foil spectrum of Tm. The incident 5-ion beam was T m + + +with [I9731 by kind permission of the an energy of 249 keV. Reproduced from MARTINSON author and the journal.

296

B E A M - F O I L SPECTROSCOPY

[v, 0 3

Given the limited resolving power of BFS, the fine-structure of the higher n-levels cannot be detected by ordinary spectral analysis. Many of the transitions associated with large n-values can be treated as hydrogenic (BASHKIN and MARTINSON[1971], MARTINSON, BASHKIN,BROMANDER and LEAVITT [1973], LENNARD, SILLSand WHALING[1972], BROMANDER [ 19731, LENNARD and COCKE[1973]). The versatility of the beam-foil source is demonstrated by Figs. 4-6 which illustrate spectra of three different elements. The data on Mn (CURTIS, MARTINSON and BUCHTA[1973]) and Tm (MARTINSON [1973]) were taken in Stockholm, and the P spectrum (MAIOand BICKEL[1973]) was obtained at Arizona. These results also show that high-energy accelerators are not necessary. Even such systems as Si IV (BERRY,BICKEL,BASHKIN,DESESQUELLES and SCHECTMAN [1971a]), P V (CURTIS,MARTINSON and BUCHTA [1971]), G a 111, Ge IV, HS V, and Se VI (SPIRENSEN [1973]), Br VII (ANDERSEN, NIELSEN and SPIRENSEN [1973b]), Pb I v and Bi v (ANDERSEN, NIELSEN arid SPIRENSEN [ 19721) have been studied successfully with particle

450 -

Phosphorus (Moss 311 E = 5 MeV loop Slits i=5pA

400 -

350

-

300 -

I-

z

"z v,

2501

200

d) Fig. 6(a).

VI,

I 31

W A V E L E N G T H STUDIES A N D ENERGY-LEVEL SCHEMES

-

r

I

I

500

n

I

I

I

I

I

I

297

!

Phosphorus (Mass 31)

I

!

450 -

i

400 -

-

0

n

N I

0

N0

350 -

4 N

cn 300-

M a

2

ln

I-

3

"

250-

:k

200

-

50

I

700

750

I

1

1

I

1

800

850

900

950

1000

I

1

1100

1150

I

A(%

Fig. 6(b). Fig. 6. Part of the beam-foil spectrum of phosphorus as obtained by MAIOand BICKEL [1973] with a bombarding energy of 5 MeV. Tentative identifications are given for a few of the lines in (b), but it is clear that many prominent lines are unidentified. The slit widths refer to the entrance slit of a 1-meter normal-incidence McPherson vacuum spectrometer. Permission from the authors and the journal to reproduce these results is gratefully acknowledged.

energies below 500 keV or so. On the other hand, high-energy machines are useful for BFS. Tandem Van de Graaffs have been used to generate optica or electron spectra up to Ar XVI (BROMANDER [1973], SELLIN, PEGG,BROWN, SMITHand DONNALLY [1971], SELLIN,PEGG,GRIFFINand SMITH[1972], SELLIN[1973], DONNALLY, SMITH,BROWN,PEGGand SELLIN[1971], PEGG, SELLIN,GRIFFINand SMITH[19721, COCKE,CURNUTTE and MACDONALD [ 1973]), while linear accelerators operating at 1 MeV/nucleon (DUFAY, DENISand DESESQUELLES [ 19701, BUCHET, BUCHET-POULIZAC, Do CAOand DESESQUELLES [1973]), or I0 MeV/nucleon have produced levels in [-ions

298

[VI, 0 3

BEAM-FOIL SPECTROSCOPY

with [ as large as (2-1) for (C, N, 0, Ne) (DUFAY, DENIS and DESES[19701, BUCHET,BUCHET-POULIZAC, DOCAO and DESESQUELLES [1973]), and (Si, S, and Ar) (SCHMIEDER and MARRUS[1970a, b], MARRUS and SCHMIEDER [1970, 19721, MARRUS[1973]).

QUELLES

- 30MeV -

-

7u

0

I

m I

l

-

I

0

m I

s

a

-

m

VI,

B 31

WAVELENGTH STUDIES A N D ENERGY-LEVEL

SCHEMES

299

U

12 MeV

c I

i

) I

I

j

i

I I 1 I

2000

m

I

I

I I I 3000

I I I I

1 I

I

I I I I

I

I I I 1 1 1 1 I;;$&

4000

Fig. 7(b). Fig. 7. The energy-dependence of spectral line intensity is clearly displayed in these beamfoil spectra of oxygen. Note the hydrogenic identification given to many of the lines. These data are from a paper by LINDSKOG, MARELIUS,HALLIN, PIHL,SJODINand BROMMANDER [1973]. We acknowledge the authors’ kind permission to use this figure.

As the particle energy is raised, there is a general tendency towards c-ions of increasing 5. There are three general consequences. The first, which is illustrated in Fig. 7, is that the relative intensities of spectral lines from

300

BEAM-FOIL SPECTROSCOPY

[VI, 8 3

various stages of ionization are altered. Secondly, the wavelengths of the transitions move towards the soft X-ray region, so that the techniques of detection and analysis of the radiations differ from the conventional optical ones. Thus, in the studies of transitions in He-like or Li-like C1 (COCKE, CURNUTTE and MACDONALD [1972, 19731, and of H-like and He-like Si, S and C1 (SCHMIEDER and MARRUS[1970a, b], MARRUS and SCHMIEDER [1970, 19721, MARRUS[1973]), a solid-state Si(Li) crystal was employed, the photons having energies of 1-3 keV. Thirdly, the character of the transitions changes in that the familiar electric dipole decays which dominate the spectra of [-ions of small [ become relatively less probable, while “forbidden” magnetic or two-photon radiations occur with appreciable rates (SCHMIEDER and MARRUS[1970a, b], MARRUSand SCHMIEDER [1970, 19721, MARRUS [1973]), and multi-electron levels with highly forbidden decay modes also [1971], become common (SELLIN,PEGG,BROWN,SMITHand DONNALLY SELLIN, PEGG,GRIFFIN and SMITH[1972], SELUN[1973], DONNALLY, SMITH, BROWN,PEGGand SELLIN [1971], PEGG,SELLIN,GRIFFIN and SMITH[1972], COCKE.CURNUTTE and MACDONALD [1972, 19731).

Fig. 8(a). Fig. 8. Beam-foil spectra of carbon, nitrogen, and oxygen, illustrating the similar spectral features in a given isoelectronic sequence. The results are reproduced from BUCHET, BUCHET-POULIZAC, DO CAOand DESESQUELLES [1973]with the kind permission of the authors and the journal.

W A V E L E N G T H S T U D I E S A N D ENERGY-LEVEL SCHEMES

Fig. 8(b).

30 1

302

B E A M - F O I L SPECTROSCOPY

tVI, 5 3

This is a good place in which to mention that the determination of line intensity as a function of [-ion energy is complicated by the need to normalize to some quantity which is either independent of energy or whose variation with energy is well known. There is no such quantity, although the mean charge, 5, of a beam transmitted through carbon has been reasonably well measured for a wide range of (-ions and particle energies (WITTKOWER and BETZ [1973]). Hence, by collecting the beam in a Faraday cup and making suitable corrections for ( ( E ) one can normalize with an uncertainty of perhaps a few percent. Other factors, such as the energy-dependent change in light collection efficiency or the appearance and disappearance of blending lines must also be considered. It is a curious fact that some lines persist over a broad energy range while others, presumably from the same stage of ionization and even nearby energy levels, wax and wane rather sensitively with energy. ANDERSEN, BICKEL,BOLEU,JESSEN and VEJE[1971a], in studying He and Li, show that excitation cross sections may bear little resemblance to charge-state distributions and that the energy dependence of spectral lines from a given element depends on both the charge state and on whether one-electron or multi-electron levels are involved. Because much of the light in the beam-foil source comes from [-ions of large 5, it is often difficult to identify the connecting levels. Representative examples are found in the cases of Na (BROWN,FORDJr., RUBINand TRACHSLIN [1968]) and P (MAIOand BICKEL[1973]), where numerous lines remain unclassified. The difficulty is worsened by the poor wavelength determinations (+ 1-3 A being usual), but lessened by recourse to studies of neighboring (-ions. The use of isoelectronic sequences in the analysis of spectra dates to the very earliest times in classical spectroscopy (for example, see E D L ~ [1964]), N and assumes an important role in beam-foil spectroscopy. We see an illustrative case in Fig. 8, where BUCHET,BUCHET-POULIZAC, Do CAOand DESESQUELLES [I9731 have compared the spectral patterns from C , N, and 0. The spectra show three prominent lines of C V and N VI (Fig. 8b) and 0 VII (Fig. 8a). Some of the other lines from N are also mirrored in the oxygen data. Thus it seems likely that the comparable lines correspond to the same transitions in (-ions of the same 5. In Fig. 9, we see another kind of analysis (BUCHET,BUCHET-POULIZAC, Do CAOand DESESQUELLES [1973]), according to which lines of the same wavelength should appear in [-ions which have lost the same number of electrons, as, for example, C VI, N VI, 0 VI and Ne V1. Figure 9 shows several such correlations. There is a problem in applying the above methods for assigning lines to transitions. In principle, the relative intensities of the corresponding lines

Nitrogen

Fig. 9. Beam-foil spectra of carbon, nitrogen, oxygen, and neon, illustrating the similarity of spectral features in emitters which have lost the same number of electrons. These results are reproduced from BUCHET,BUCHET-POULIZAC, D o CAO and DESESQUELLES [I5731 with the kind permission of the authors and the journal.

304

BEAM-FOIL SPECTROSCOPY

[VI,

04

should be similar. Look, however, at Fig. 9, where lines at 535 A appear in N VII and 0 VII. The corresponding line in Ne VLI is missing. A possible source of this vacancy is that the accelerator in use had a fixed energylamu and did not lend itself to the excitation of Ne’6 ions. You will note, also, that 1727 8, is assigned to Ne VIII, whereas the lines at 1729 8, are attributed to C VI, N VI and 0 VI. Presumably the general sparsity of Ne VII lines led the authors to conclude that 1727 A had to originate in a different ionization stage. A similar remark applies to the failure to see an 0 VIII line at 1727 8,; 0 VIII was not produced efficiently and only the relatively strong transition at 1633 8, was seen. Thus the line identifications are often complicated. One might remark that a good theory of the beam-foil interaction would allow us to understand the circumstances under which some levels are highly populated and others not. The identification of transitions is aided by compilations such as we see in Fig. 10, where there is a plot (MARTINSON, private communication) of the term value, E, per “effective charge” ([+0.4) versus 2 for several spectroscopic terms. Any line which is believed to involve one of the levels must place that level on the pertinent smooth curve.

35

il--

=

=

-

-

3pp ID 2

o

i

Fig. 10. Energy per “effective charge” versus 2 for several spectroscopic terms. The ordinate is in kilokaysers or cm-’. These data were kindly provided by I. Martinson.

5 4. Doubly-Excited Levels Doubly-excited levels seem to be created in profusion in the beam-foil source; we restrict our discussion of such levels to beam-foil work. Those levels, which may be arranged in a spectroscopic hierarchy of their own,

VI,

5 41

DOUBLY-EXCITED

305

LEVELS

have a term of lowest energy which lies well above the ionization energy, at least for the [-ion of smallest 5. (As one proceeds along an isoelectronic sequence in the direction of larger (, the energies of the multi-electron levels become lower relative to the ionization energy, and ultimately become negative.) Thus autoionization is the expected mode of decay, with level lifetimes of lo-'' sec or less. However, the selection rules for autoionization (AJ = 0, An = 0 and Table 1) sometimes lead to metastability because the continuum does not contain states to which transitions can occur with TABLE 1 Autoionization selection rules for LS coupling. AS

AL Coulomb Spin-orbit Spin-spin

0

0

0,

0, 3 1

0, & I , * 2

0, & I , &2

appreciable probability. For example, the continuum of 1-electron levels coupled to a singlet ground term (e.g., Li I 1s 2s('S,)nZ 2L) does not include quartets. Consequently,radiative decays of levels such as Li I Is 2s np 4P0lead to the lowest such term, whence electron emission or forbidden radiative transitions must occur. Both the radiative and autoionization processes have been studied with the beam-foil source. Consider, first, the radiative work, which has been reported for He (BERRY,MARTINSON, CURTISand LUNDIN [1971d], BERRY,DESESQUELLES and DUFAY[1972, 19731, KNYSTAUTAS and DROUIN[1973]), Li (MARTINSON[1970, 19731, BERRY,DESESQUELLES and DUFAY[1971, 19731, BICKEL,BERGSTROM, BUCHTA,LUNDINand MARTINSON [1969a1, BICKEL,MARTINSON,LUNDIN,BUCHTA,BROMANDER and BERGSTROM [1969b], BUCHET, DENIS,DESESQUELLES and DUFAY [1969], GAILLARD, DESESQUELLES and DUFAY[1969], BERRY,BROMANDER and BUCHTA[1970], ANDERSEN, BICKEL,CARRIVEAU, JENSEN and VEJE[1971b], ANDERSEN, BOLEU, JENSEN and VEJE [1971c], BERRY,BROMANDER, MARTINSON and BUCHTA [1971b], BERRY,PINNINGTON and SUBTIL[1972]), Be (MARTINSON [1970], BERGSTROM, BROMANDER, BUCHTA,LUNDINand MARTINSON [1969], BERRY, BROMANDER, MARTINSONand BUCHTA [1971b], ANDERSEN,JESSEN and S0RENSEN [ 1969a1, HONTZEAS, MARTINSON,ERMANand BUCHTA[1972, 19731, MCCAVERT and RUDGE[1972]), B (MARTINSON[1970], BERGSTROM, BROMANDER, BUCHTA,LUNDINand MARTINSON [19691, MARTINSON, BICKEL and ~ L M E[1970]), C (MARTINSON [19701, BERGSTROM, BROMANDER, BUCHTA, LUNDINand MARTINSON [1969]), N (BERRY,BICKEL,BASHKIN,DESESQUEL-

306

BEAM-FOIL SPECTROSCOPY

[VI, B 4

and SCHECTMAN [1971a]), and CI (COCKE, CURNUTTE and MACDONALD [1972, 19731). These experiments were stimulated by the attempts to classify beam-foil lines which do not fit into the ordinaiy energy-level schemes, and by previous experimental and theoretical work which, dating back as far as 1928 (COMPTON and BOYCE[1928], Wu [1934], FENDER and VINTI[1934]), suggested the occurrence of discrete multi-electron, autoionizing levels.

LES

I

I I

I

I

I

I

I

I I

I

I

I

I

I

I

I

I

I

I

I

I

W

ml

tn

I

0

m

0 1

.

M 290

I40

WAVELENGTH (%)

Fig. l l a . Beam-foil spectra of He (bottom) and Li (top), reproduced with the kind permission of BERRY,DESESQUELLES and DUFAY119731.

VI, 0 41

D O U B L Y - E X C I T E D LEVELS

201

307

I 100 keV

He'

-+

C

*

V

0

'=,

101

c

c

0

V

Y

Fig. 11b. Beam-foil spectrum of He in the same wavelength range as in Fig. 1l a (bottom). Reproduced with the kind permission of KNYSTAUTAS and DROUIN [1973].

Typical spectra for He (see Fig. 11) show numerous lines which cannot be fitted into the normal level diagrams for He I and He 11. Unfortunately, different assignments have been proposed by the two reporting groups. This illustrates the problem which results from relatively low resolving power, since these may be as many as ten lines within 1 A (MCCAVERT and RUDGE [1972]). Another example is given in Fig. 12, where we show the level scheme (HONTZEAS, MARTINSON, ERMAN and BUCHTA [ 19731) for two-electron terms in Be 11. Figure 12 shows a satisfying internal consistency, and the assignment of the levels is strengthened by the report (HONTZEAS, MARTINSON, ERMAN and BUCHTA[1973]) of lines which connect to bound systems. The fact that some levels have been identified in four or more members of an isoelectronic sequence gives further corroboration to the level schemes. However, the analysis of the spectrum is made awkward by the simple experimental facts that (1) a variety of spectrometers is needed to cover the entire spectral range, (2) the detection efficiency for a given spectrometer arrangement may be strongly dependent on wavelength, (3) lifetime effects

308

BEAM-FOIL SPECTROSCOPY

[VI,

Q4

A

ev l&O-

2sns

2pns

LS

w

2snp

2;np

2snd

/.Po

P w

'D

2pnd

w

135-

130-

125-

120-

115-

Fig. 12. Quartet system of autoionizing two-electron levels in Be. Reproduced from HONTZEAS, MARTINSON, ERMANand BUCHTA[1973] with the kind permission of the authors and the journal.

enter into the picture, and (4) there are ever-present limitations imposed by resolving power. The discrepancy noted above in the helium assignments illustrate the need for caution in dealing with the multi-electron levels. There are, of course, similar problems for ordinary levels. We have mentioned that the beam-foil source liberates electrons as well as light. These electrons, many of which have discrete energies, have been PEGG,GRIFFIN and SMITH[1972], DONNALLY, studied for 0 and F (SELLIN, SMITH,BROWN,PECGand SELLIN[1971], BERRY[1972]) and C1 (PEW, GRIFFIN, SELLIN and SMITH[1973]). They have paid particular attention to C-ions of Li-like and Na-like character, for example, Cl14+ and C16+.The experimental method is to direct a beam of foil-transmitted [-ions down the axis of a cylindrical electrostatic analyzer. Electrons ejected from the beam

DOUBLY-EXCITED

309

L EV ELS

8 - c m FOIL POSITION

ELECTRON ENERGY SPECTRA 70-220 eV 5 MeV CHLORINE BEAM

3-crn FOIL POSITION

70

90

110 130 150 170 ELECTRON ENERGY ( e V )

190

210

Fig. 13. Beam-foil electron spectra of chlorine at two different distances from the foil. Reproduced from PEGG,GRIFFIN, SELLINand SMITH [1973] with the kind permission of the authors and the journal.

in a small solid angle are energy-analyzed. A spectrum so obtained is illustrated in Fig. 13. As discussed by DONNALLY, SMITH, BROWN,PEGGand SELLIN[1971], an experimental problem in measuring the electron energies is that they vary over the finite width of the entrance aperture, quite analogous to'the Doppler broadening for the light. A significant energy spread is also contributed by scattering. Total spreads are a few eV per keV. Despite these difficulties, the electron energies have been well-enough determined to associate them with autoionizing, but metastable, levels in [-ions of three or eleven charges. The metastability arises from the large spin of the electronic arrangements such as 1s 2s 2p "P+ in Cl'"' or perhaps 1s' 2s' 2p5 (nZ)(n'Z')and 1s' 2s 2p6 (nl)(n'Z'), with n, n' 5 3, for C16+.What is surprising is that there appear to be many metastable, autoionizing levels in a great variety of [-ions, and they are populated prolifically in the beam-foil source,

310

BEAM-FOIL SPECTROSCOPY

IVI,

I5

0 5. Metastable One-Electron Levels The transitions which are most commonly observed in spectroscopy satisfy the selection rules AL = & 1, AJ = 0, 5 1, An = 1, with 0 4 0 transitions forbidden and, in L-S coupling, AS = 0. The selection rule for AS arises because the electric-dipole operator commutes with the spin and thus cannot connect levels of different spin. In point of fact, L-S coupling is not perfect, and so-called "intercombination lines", for which AS = & 1, do occur, especially in heavy elements; for example, there is the strong resonance line 12537 A in Hg (6s' IS, - 6s 6p 'P;). The selection rules characterize electric dipole radiation, in which the photon carries away one unit of angular momentum and a unit of parity. Thus, in the cases of hydrogen and helium, for which some of the low-lying levels are drawn in Fig. 14, we see that some direct decays to the ground state are forbidden.

2 'P

? ' S O T

I I

I

1

/ I

I

I I I I

H

I I I

/

He

Fig. 14. Low-lying energy levels of hydrogen and helium. For He-like C-ions with 5 > 2, the 'P levels are inverted. Allowed transitions are indicsted by solid arrows, forbidden ones by dashed arrows.

VI,

o 51

METASTABLE ONE-ELECTRON

LEVELS

311

If one considers transitions of higher order, such as magnetic dipole, electric or magnetic quadrupole, or double electric dipole, one finds that they occur with rates which are strongly increasing functions of Z . Thus, in DONNALLY and FAN [1968], a beam-foil study of He-like oxygen, SELLIN, SELLIN, BROWN,SMITH and DONNALLY [19701, observed the intercombination line 1s’ ‘So-ls 2p 3P, (see Fig. 14); they also measured the mean life of the upper level. Using the heavy-ion linear accelerator at Berkeley, MARRUS and SCHMIEDER [1970,1972], SCHMIEDER and MARRUS [1970b], MARRUS [1973] showed that the “forbidden” decays illustrated as dashed lines in Fig. 14 actually occur with substantial rates in Si, S, and Ar. Indeed, the Z dependence of the forbidden transitions may be greater than that of allowed electIic-dipole transitions. At Z 18, for example, the M2 decay 2 3P2 -+ 1 ’ S o dominates the El decay 2 3P, + 2 %, (see Fig. 15). It is worth mentioning that

-

FORBIDDEN DECAYS IN He

FORBIDDEN DECAYS IN H

z=I T ~ ~ = ~ XZ I O3 ’ e TZEl=

o 12~

c

3 e c

rMI=45x1O5 Z-”sec

5~10’sec

Z=18 14x10-’sec

012 sec

353x 10-’sec

4 5 ~ 1 sec 0~

126xIO-’sec

I

Is,

rZE,(He)=O5 ti)

Fig. 15. Illustration of forbidden decays in hydrogen and helium. The lifetimes are taken from MARRUS[1973].

GABRIEL and JORDAN [1969a], well before the experiments of Marrus and Schmieder, proposed to account for certain solar coronal lines in terms of just such forbidden transitions in [-ions of high [, and used the line intensities to deduce the electron density in the solar corona (GABRIEL and JORDAN [1969a, b, c]). These experiments on H-like and He-like [-ions show generally good agreement with calculations of the level positions and decay rates. An interesting problem is to measure the Lamb shift for one-electron [-ions of high (; so far, such data have been published for systems up to hydrogenic

312

BEAM-FOIL SPECTROSCOPY

IVI, S 6

oxygen (LEVENTHAL, MURNICK and KUGEL[1972], LAWRENCE, FANand BASHKIN [1972], LEVENTHAL [1973]), but they aren't quite good enough to test the predictions of quantum electrodynamics.

6 6. Mean-Life Measurements Probably the most important achievement of BFS is its making possible the measurement of the mean lives of excited levels in (-ions with arbitrary values of [. Moreover, BFS is not restricted to resonance terms or to those which can be reached by optical pumping. The early experiments were, naturally, devoted to the light elements, and especially to elements which make gaseous compounds. More recently, attention has been devoted to the metals and rare earths. Of the first 37 elements in the periodic table, only Co and Zn have so far escaped study in BFS, while Cd, In, Sn, Sb, Pd,Te, Xe, Hg, T1, Pb, Bi, Gd, Tb, Dy, Tm, and U are other elements for which some work has been reported. Needless to say, no one has made a thorough study of all the levels of any [-ion. However, a given level has often been studied for many members of an isoelectronic sequence. Thus, MARTINSON [1973] notes that the 3s-3p resonance transition has been measured in eight [-ions, from Na I to Ar VIII. Such measurements provide a valuable check on theoretical calculations of transition probabilities and on level assignments. Recent developments on the theoretical side have taken ever more detailed account of configuration mixing, and the lifetime data are a particularly sensitive test of the validity of the theory. In Fig. 16 we see a comparison (MARTINSON [1973]) of the experimental and theoretical results forthe 3s' 3p 'P0-3s 3p2 3 D transition in the A1 I sequence. The importance of the lifetime experiments is such that a great deal of effort has gone into methods of reducing the uncertainties in the measurements. Three main problems can be treated separately. First of all, the reciprocity between distance and time obviously involves the particle speed. More exactly, it involves the speed of the particles after their penetration through the foil. The best way of finding that velocity is, naturally, by making a direct measurement. In practice, that is virtually never done, but it sometimes is (BICKEL,BERGSTROM, BUCHTA,LUNDIN and MARTINSON [ 1969a1, ANDERSEN, M0LHARE and SBRENSEN[ 19721). Rather, one usually calculates the energy loss experienced by the particle as it goes through the foil. This leads to serious problems, especially for medium-weight and heavy elements at low energy, partly because foil thickness is usually not well known, and largely because the calculations are not always satisfactory.

VI,

0 61

MEAN-LIFE

Ar C l S

f

0.08

A1

313

MEASUREMENT

P

Si

A1

I sequence

3 s2 3p 2Po-3s 3p2 2D

0.07 0.06

0.05

Hofmann (emission 1 .Curtis et a1 ( B F S ) A B e r r y et a l ( B F S ) r B a s h k i n and M a r t i n s o n (BFS) x L i v i n g s t o n e t a1 (BFS) o F r o e s e Ftscher (theory1 oBeck and Sinanog'Lu (theory)

0.04 0.03

O

0

+

t

0.02

0 0

0.01

0

0 0

0

0.04

0.06

0.081/z

Fig. 16. Oscillator strength versus Z - ' for a transition in the Al I sequence. The open symbols are from theory, the others from experiment. Reproduced from MARTINSON [1973] with the kind permission of the author and the journal.

Foil thicknesses are sometimes measured in terms of the energy loss sustained by alpha particles, and sometimes by an optical transmission method which is based on an absolute determination of the carbon content of a foil (STONER [19691). Most often, however, only the manufacturer's estimate is available and this may be grossly uncertain. (Many laboratories make their own foils; they rarely describe how they determine foil thicknesses.) Even if the foil thickness is well known, the translation into an energy loss depends SCHARFF and SCHIOTT [1963]. As pointed out, on the theory by LINDHARD, for example, by ANDERSEN [19731, the calculations overestimate the energy loss, especially for heavy C-ions at low energy. CARRIVEAU (private communication) and his associates have found that the actual energy loss by Kr at 200 keV is less than half that calculated from Lindhard's theory. Thus if one uses calculated energy losses, the effect is to make the beam-foil mean lives appear longer than they are. The other two main complications in the lifetime work are line blending and cascades. We will see later that these effects can be reduced or even

314

[VI,§ 6

BEAM-FOIL SPECTROSCOPY

eliminated in some cases; however, the requisite method is not yet in general use. The most common treatment of lifetime data is to fit the observed decay curve by a sum of exponentials. A recent example (ROBERTS, ANDERSEN and SBRENSEN [1973b]) appears in Fig. 17, where the data and the decomposition into two exponential decays are given.

i

2945.5A T i U e4G 300 keV

\

T,,, = 3.0 ns

\ ,

I

1

10 20 30 40 DISTANCE FROM FOIL ( m m )

Fig. 17. A decay curve for 12945.5A from 4Gin Ti 11. Note the decomposition into a cascade contribution and the main decay. Reproduced from ROBERTS, ANDERSEN and S0RENSEN [1973b] with the kind permission of the authors and the journal.

In Fig. 17, we see that the decay curve was followed over a factor of 30 in intensity. However, one sees from Fig. 17 that there is still appreciable slope to the curve at the longest observed times. It would be better practice to extend the region of observation to the place where the signal is lost in background plus noise. This is often hard to do, since the lifetimes may be long, necessitating movement of the foil over awkward distances. For instance, a medium-weight (-ion (say 20) with a moderate energy (say 1.5 MeV) moves 3.8 mm per nanosec. For a mean life of 15 nsec, this gives a mean decay distance of 5.7 cm, so to follow a decay curve over even 3 mean decay lengths requires moving the foil over a straight track 17 cm long. Not only is such a length hard to incorporate into a target chamber, but there

VI, 3 61

MEAN-LIFE MEASUREMENTS

315

is an associated difficulty not yet discussed. In order for the signals from different points to have any physical significance, they must be normalized to some common denominator. The usual practice is to normalize to the charge which is collected in a Faraday cup (see Fig. 1). The big problem is that the beam is scattered by the foil so that the cup collects a decreasing fraction of the 6-ions as the foil-to-cup distance increases. Generally speaking, one cannot move the cup with the foil, for one quickly obscures the spectrometer’s view of the light. The best one can do, other than going to the limit of measuring the scattering and correcting for it directly, is to make the cup large in diameter. Often this dictates the minimum distance from the beam to the entrance slit of the spectrometer, especially for work in the vacuum u.v., where lenses cannot be used. An alternative normalization is to use the intensity of some spectral line or wavelength band as the common denominator. A filter-photo-multiplier combination which sees the light from some point close to the foil is relatively simple to arrange. Of course, this apparatus must be moved with the foil. However, one needs a window as long as the distance over which the foil is to be translated, and the whole must be in a light-tight enclosure. This method is not in general use. Two other problems are present for either of the above normalizations. One is that scattering at the foil affects the fraction of the beam which the spectrometer sees. Of course, one tries to use a long slit so that particles, even though scattered, can still emit their light into the spectrometer, but it is easy to see that the efficiency of colIection of light is degraded when the foil is far from the point of observation. The other problem is that foil characteristics change under bombardment. They may thicken or become thinner. They may develop pinholes, or disintegrate. All of these changes affect the relative amount of a given kind of light that is generated at the foil, and also the amount of scattering. If one is using charge normalization, the foils (pinholes and breakage not considered) may produce charge equilibrium so that thickness changes don’t alter the common denominator, but the likelihood is that the light output is not constant, so that the numerator suffers. The result of all this is that one must make a value judgment about the quality of the lifetime data and analysis. This, of course, comes down to a matter of opinion; quantitative assessments can be made only by recourse to laborious studies. Every measurement includes a contribution from background and noise. In addition, the level of interest may be populated by cascades as well as by the direct interaction at the foil. If we have a single cascade, we may write

316

s

N

Iv1, $ 6

BEAM-FOIL SPECTROSCOPY

s0+2u- " 5 2 -71

N;exp

(-) : u52

(- 5)sinh A 72-71

x exp

-,

(6.1)

V 71

where, S = signal; So = background plus noise; u = beam speed; z1 = lifetime of level of interest; z2 = lifetime of higher level which cascades into level of interest; d = distance downstream from foil at which S is detected; 24 = observed length of beam; NP, N i = initial populations of levels 1 and 2. So great a variety occurs of relative values of z1 and T ~ and , of So, N;/rl, and N ~ / that T ~no simple rules can be given to relate the quality of the data to the parameters in eq. (6.1). Note, however, from Fig. 18 that 24, the observed length of the beam, depends on the distance between the beam and the entrance slit, assuming that no field lens is used. That distance is determined by the need to collect the beam in a shielded Faraday cup, and possibly also by physical interference from the detector. In one fairly typical case, that distance is 6.6 cm, and 24 is 6.6 mm.

Fig. 18. Arrangement for measuring lifetimes. The foil is mounted on a movable frame, driven either manually or with a stepping motor. The spectrometer views the finite beam segment of length 24.

Now consider a beam with a speed of 3 x lo8 cmjsec, and a level with a mean life of sec. The mean decay distance is 3 mm. However, the smallest value d can have is 3.3 mm, for otherwise one is looking at the foil.

VI,

8 61

MEAN-LIFE MEASUREMENTS

317

Thus the level population has declined to 33 % of its initial value before the first measurement has been made. Suppose, further, that there is a cascade from a level with 7 = 10 nsec. At the first observation point, this level population is still 90 % of its initial value. If, then, the upper level has an initial population equal to that of the level of interest, the cascade intensity is 25 % of the total, and it grows rapidly for successive points downstream. From Fig. 18, one sees also, that the bulk of the desired information is contained in that edge of 24 which is closer to the foil, and the light from that edge illuminates only a small part of the grating. The rest of the grating is illuminated by light from the cascading level and by the background. Therefore the quality of the data is reduced by using the entire grating, and a mask should be employed (LIVINGSTON, IRWINand PINNINGTON [1972]). Unfortunately, a mask reduces the resolving power of the grating. Consequently, when spectral lines are close together, masking the grating may cause them to overlap, which again causes the lifetime data to be unsatisfactory. Only for the cases of isolated lines, low background and noise, insignificant cascading, and lifetimes 2 5 nsec can the customary measurements be considered satisfactory. We feel that lifetimes less than 1 nsec should be treated with reserve unless an especially good argument is presented to show the 201

I

I

I

I

1

-

'50

10 20 30 40 DISTANCE DOWNSTREAM FROM EXCITER FOIL (rnrn)

Fig. 19. Decay curve for 13714 from a doubly-excited level in Li. Reproduction from

BICKEL, BERGSTROM, BUCHTA,LUNDINand MARTINSON [1969a] with the kind permission of the authors and the journal.

318

BEAM-FOIL SPECTROSCOPY

[VI,

B6

correctness of the experiment. Merely following the customary method and extracting a decay constant is inadequate. Differing experimental arrangements can easily cause substantial discrepancies in lifetimes. Consider the data of Fig. 19, from BICKEL, BERGSTROM, BUCHTA,LUNDIN and MARTINSON [1969a]. They follow an unmistakable exponential decay,

J Fig. 20. Level diagram for multi-electron levels in Li. Reproduced from BERRY, DESESQUELLES and DUFAY 119731 with the kind permission of the authors and the journal.

VI, § 61

MEAN-LIFE MEASUREMENTS

319

from which a mean life of 6.4 nsec was deduced for the doubly-excited level 1s 2p2 4P of Li I. However, now examine Fig. 20, where we show the corresponding energy level diagram (BERRY,DESESQUELLES and DUFAY[19731). It is seen that the level in question is populated by no fewer than eight cascades! How then, can the decay be characterized by a single exponential? This kind of apparent inconsistency can be very frustrating, especially since the lifetime and energy level determinations often appear in separate publications from separate laboratories. We do not know how to account for the cited results, which are not merely an isolated case for which some fortuitous values of lifetimes and initial populations could lead to a strictly exponential decay despite the numerous cascades. This type of peculiarity occurs fairly frequently. The cascade situation would be materially simplified if the cascade lines were themselves specifically identified in the papers which more-or-less casually invoke cascades in explanation of the lifetime curves. This, again, is rarely done, partly because the cascade lines are in so different a wavelength region from the main line under investigation that a single spectrometer is not suitable for the work. Moreover, the judgment that one line is the precursor of another, even when both are seen, is based only on inference from an energy-level diagram, whereas what is really needed is a coincidence measurement. Delayed-coincidence experiments have been reported (MASTERSON and STONER [1973]), but the technique seems hard to use, requiring very long running times even where both members of a known simple cascade appear with high intensity. For the general case, where the participating transitions may have quite different wavelengths and relative intensities, and where the sequence of related transitions is not known, the time-coincidence method will not permit an easy deciphering of the cascade scheme. One can measure level lifetimes by several other methods. One, which is applicable when the lower state is so short-lived that it has appreciable width, is based on the fact that the non-instrumental part of the width of a spectral line comes from the combined natural widths of the initial and final levels. This approach has been used (BERRY, DESESQUELLES and DUFAY[1971,1972, 1973]), in studies of autoionizing levels. Unfortunately, there are no independent experimental verifications of the lifetimes so reported, and calculated values are generally inapplicable because the experiments cited above could not resolve closely-spaced levels of widely different theoretical lifetimes. The trouble one has in making a realistic assessment of lifetimes measured with the foregoing methods may be appreciated as follows. From the work of BERRY,SCHECTMAN, MARTINSON, BICKELand BASHKIN[1970b], we de-

320

BEAM-FOIL SPECTROSCOPY

[VI,4 6

duce anf-value of 0.38 for the (multiplet) transition 3s %-3p 'P in S VI, whereas SBRENSEN [1973] quotes unpublished work by himself and Andersen as giving 0.53. Since f i k = 1.5 x

,Iz (gk/gi)Aki>

wherefik is the absorption oscillator strength, I is the wavelength (in Angstroms) of the transition, g is a statistical weight, and A k i is the transition probability, the discrepancy reduces to one of lifetime. BERRY,SCHECTMAN, MARTINSON, BICKELand BASHKIN[1970b] resolved the two components of the transition array and made separate lifetime measurements using each line, obtaining 1.05 and 1.07 nsec for the P, and P, levels, respectively. The details of the Serrensen-Andersen measurement are not available. We believe that this particular problem may be related to the short mean life and the corresponding experimental difficulty in observing over several mean decay lengths. A different situation is found in oxygen, where the mean life of 0 111 3d 5 D is reported as 1.44nsec, 2.12nsec, and 2.84 nsec (DRUETTA and POULIZAC [1969]); as 25.2 nsec (LEWIS,ZIMNOCH and WARES[1969]); as 15 nsec (PINNINGTON [1970]); and as 7.0 and 8.6 nsec (DRUETTA, POULIZAC and DUFAY[1971]). The three numbers given by DRUETTA and POULIZAC [1969] were based on three separate lines in the transition array 3p 5D0-3d 5D;a cascade correction was included in the treatment of the 2.84 number. DRUETTA and POULIZAC [1969] and PINNINGTON [1970] used a line from 3p 5P0-3d 5D, since they were unable to detect the lines from 3p 5D0-3d 5D. DRUETTA, POULIZAC and DUFAY[1971] use both transition arrays, but report that the line from 3p 5P0-3d 5 D was blended with a line from 0 I1 or 0 IV. No indication is given as to why the value given by DRUETTA, POULIZAC and DUFAY[1971] is so different from those of DRUETTA and POULIZAC [1969], nor does the latter paper account for the spread of a factor of 2 in its listings. Unfortunately, the confusion which is generated by these conflicting ieports is widespread, and the reader can only occasionally make a value judgment as to which, if any, of the lifetimes approximates the truth. When different particle energies are employed, relative line intensities are apt to differ. Blends are an ever-present possibility. The common suggestion that cascades occur is rarely substantiated by anything other than the shape of the decay curve, and that could be severely influenced by, for example, normalization procedures. This last could also be affected by foil thickness, because of scattering, and, as we have noted previously, foil thickness is notoriously uncertain. Further on the matter of cascades, we have already

VI.

5 61

MEAN-LIFE MEASUREMENTS

32 1

commented that there is no clear understanding as to when a line will or won't exhibit a cascade contribution, and it would be most helpful if authors would address themselves to the question of whether it is sensible to invoke the cascade mechanism. To be specific, the 0 111 3d 5D term referred to above lies 398000 cm-' above the ground state. The ionization level is only slightly higher, namely, at 443000 cm-l. MOORE[I9491 lists but two terms (4p 5D0at 438000 cm-' and 4p 'Po at 439000 cm-') which could decay by fully allowed transitions to 3d 5D, and the oscillator strengths for those transitions are almost certainly vanishingly small (they are not listed in WIESE,SMITHand GLENNON [1966]). Thus the present author finds it perplexing that cascades, emphasized by LEWIS,ZIMNOCH and WARES[1969] and DRUETTA, POULIZAC and DUFAY[1971], could play a significant role in the decay of 0 I11 3d 5D.It is our belief that authors would perform an important service were they to give some physical argument to buttress the claim of cascade influence. The striking discrepancies we have cited can, we believe, scarcely originate in cascades. It is far more likely that the problems are (a) line blending, (b) improper normalization, and/or (c) poor signalto-background ratio; these factors should be given a thorough analysis in every lifetime paper. 33L9.L A T i I1 Z4G'

1 0 0 ~ ''

'

' 1" 0 " " 20 ' " 30 ' ' ' ',40 DISTANCE FROM FOIL (mm) "

Fig. 21. Decay curves for A3349.4A from 2 4G" in Ti 11, taken at two different bombarding energies. There is no evidence of cascades over the wide range in intensities. ANDERSEN and SBRENSEN [1973a] with the kind permission Reproduced from ROBERTS, of the authors, Astrophysical Journal, and the University of Chicago Press. "Copyright 1973 by the American Astronomical Society. All rights reserved".

322

BEAM-FOIL SPECTROSCOPY

[VI,

06

Although we have stressed the uncertainty that sometimes makes lifetime data difficult to interpret, one must recognize that, overall, the lifetime experiments have contributed a wealth of information which is quantitatively satisfying. We have already cited the work by ROBERTS,ANDERSEN and SORENSEN [1973a], from which we reproduce our Fig. 21, showing intensity data for a line in Ti 11. Note particularly that the intensity was followed over a range of 100 or so, that the error bars indicate reasonable attention was paid to the physical significance of the experimental details, and that, from the size of the error bars at the largest distance, one can conclude that the line was followed well into the noise. Moreover, branching ratios were also measured. ROBERTS,ANDERSEN and S~RENSEN [1973a] yielded some 350 absolute oscillator strengths for transitions in Ti 11, along with others for T I I, 11, and IV. It is instructive to compare the results with those of a shock-tube experiment by WOLNIKand BERTHEL[1973], which appeared at about the same time. Luckily both groups examined many of the same transitions; the common transitions are listed in Table 2. The agreement between these entirely independent measurements is gratifying. TABLE 2 Absolute values of log (gf) for levels in Ti I1 Multipleta

4'Q

Beam-foilb

Shock-tubec

~~

20 32 34 39 49 50 51 60 61 70 82 86 87 87 92 93 I04 113 I14

4344 4341 3932 4583 4709 4590 4408 458 1 439 1 5227 4530 5129 4028 4054 4780 4375 4387 5027 4874

MOORE[1949]. ROBERTS, ANDERSEN and SQRENSEN [1973a]. ' WOLNIKand BERTHEL[1973]. a

-2.03 -2.22 -1.90 - 2.73 -2.63 -1.73 -2.14 -2.73 -2.79 - 1.24 -2.15 -1.51 -1.12 -1.33 - 1.37 -1.62 -1.41 -1.23 -1.01

-2.01 -2.10 -1.77 -2.74 -2.40 -1.85 -2.55 -2.89 -2.33 - I .44 -1.82 - 1.49 -1.26 - 1.38 - 1.42 - 1.60 -1.06 -1.12 -0.96

VI,

8 71

A P P L I C A T I O N S OF L I F E T I M E D A T A

323

0 7. Applications of Lifetime Data The lifetime data have had two principal applications, namely to astrophysics and atomic theory. The connection with astrophysics stems from the fact that determinations of relative element abundances in astronomical sources depend directly on the oscillator strength for the observed transition. The mean life of level k may be related to the total decay probability by

r;' = CA,.. ick

From eqs. (6.2) and (7.1), we see that a measurement of z, gives an upper limit to the oscillator strength for a particular transition. If one couples the measurement of z with another of the branching ratio of the transition of interest to the total transition intensity, one can find the oscillator strength itself, rather than just an upper limit thereto. Moreover, for cases where a single transition is dominant, the measurement of z is adequate to determine f. Of all the elements whose abundance one might wish to know, iron is outstanding in its significance for astrophysics. The reason is that there appears to be (CAMERON [1968]) a strong local maximum in the abundances of elements in the immediate neighborhood of iron, with iron itself at the peak. This apparently universal feature of the element abundances has given rise to extensive calculations (TRURON[1972]) on, for example, the mechanism whereby such a distribution could have been produced. Since the death throes of stars are involved (BURBIDGE, BURBRIDGE, FOWLER and HOYLE[1957]), the subject has singular fascination for the inquiring mind, and an enormous effort has been exerted on the problem. The several beam-foil measurements (WHALING, KINGand MARTINEZGARCIA[ 19691, WHALING,MARTINEZ-GARCIA, MICKEYand LAWRENCE [19701, ANDERSEN and S0RENSEN [19711, LENNARD and COCKE [19731, LENNARD, WAHLING, SILLSand ZAJC[1973]) of lifetimes in Fe I and Fe 11, along with numerous independent investigations (HUBERand TOBEY [1968], GARZand KOCK[1969], SEASDOLEN, HUBERand PARKINSON [1969], BRIDGES and WIESE[1970], WOLNIK, BERTHEL and WARES [1970, 19711, WIESE[1970], and PARKINSON [1972], BELLand UPSON[1971], KLOSE[1971], HUBER ASSOUSA and SMITH[1972]) have shown that oscillator strengths derived from earlier work (CORLISSand BOZMAN[1962], CORLISSand WARNER [1964, 19661, CORLISS and TECH[1968]) were seriously in error. On the basis of the more recent studies, the photospheric solar abundance of iron has been revised upwards (GARZ,KOCK,RICHTER,BOSCHEK, HOLMEYER and UNSOLD[1969a1, GARZ,HOLMEYER, KOCKand RICHTER[1969b1, GARZ,

324

BEAM-FOIL SPECTROSCOPY

[VI, I 7

RICHTER, HOLMEYER and UNSOLD[1970], BOSCHEK, GARZ,RICHTER and HOLMEYER [1970], COWLEY [1970], NUSBAUMER and SWINGS[1970], GREVESSE, NUSSBAUMER and SWINGS[1971], FOY [1972], Ross [1973]) by approximately a factor of ten, with a possible uncertainty of a factor of two (FOY [1972]). In the absence of mechanisms which cause alteiations in the surface abundances, one assumes that photospheric abundances are representative of the interior constituents of the sun. Thus the enhancement of the iron abundance is taken to indicate the iron complement of all of solar matter. The immediate consequence of the new iron abundance is to raise the opacity (BAHCALL [1964]), which, in turn, demands an increase in the central temperature so as to account for the observed radiant flux from the sun. The effect of the increased temperature on the thermonuclear events which presumably generate a star’s luminous power output has been studied by Bahcall and his associates (BAHCALL [1966], BAHCALL, BAHCALL and ULRICH [1969], BAHCALL and ULRICH [1971]). The result is that the calculated neutrino flux expected at the earth is 5 or more timesegreater than the experimental upper limit (DAVIS[1964], DAVIS,HARMER and HOFFMAN [1968], DAVIS,RADEKAand ROGERS[1971]), giving rise to a variety of speculations (BAHCALL, CABIBOand YAHIL[1972], BAHCALL [1973]) concerning such things as the particle stability of the neutrino, the existence of additional resonances in thermonuclear reactions, the long-term stability of the sun, and so forth. One can summarize this extensive activity on the nature of the sun, and, presumably, on all stars, by saying that reliable lifetime data have strengthened the idea that iron-group elements are more abundant than their neighbors and have raised serious questions as to how to interpret this abundance. Measurements on lifetimes of other elements in the iron group such as nickel (BRAND,COCKE and CURNUTTE [1973]), chromium (COCKE, CURNUTTE and BRAND[1971]), and manganese (CURTIS, MARTINSON and BUCHTA [1973]) also cause a revision upwards in their solar abundances. On the other hand, the abundance of thulium must apparently be reduced (CURTIS, MARTINSON and BUCHTA[1973]). Thus one sees that beam-foil lifetime data are critically important to the fundamental problem of ascertaining the relative abundance of the elements. Another use of lifetime data has been to assist in the refinement of calculations of transition probabilities and oscillator strengths. DALGARNO I19731, SINANO~LU [1969, 19731, WESTHAUS and SINANO~LU [I9691 have recently summarized the theoretical advances made by themselves and others in calculating oscillator strengths. The Z-expansion method treated by Dal-

VI, 0 81

325

COHERENCE A N D ALIGNMENT

garno shows that the regularities which others had discovered (WIESE[1968a, b, 19701, WIESEand WEISS[19681, SMITHand WIESE[19711, SMITH,MARTIN and WIESE[1973]) infversus 2 have a good theoretical basis. At the same time, the refinements of Sinanoglu's non-closed shell, many-electron theory, in which electron correlation is specifically included for both spectroscopic terms linked in a transition, have given oscillator strengths with uncertainties of a few percent. Since beam-foil experiments can also be that good, the comparison between theory and experiment becomes significant. An example appears in Fig. 22. I

I

I I I

I

I

I

I

I

I

)

BORON SEQUENCE 2s' 2 p 2po- 2s 2p' 'D A

o

0

0

NCMET

A PHASE

1

A

SHIFT

HANLE EFFECT

NBS

0.05

0.10

0.15

0.20

112-

Fig. 22. Comparison of theoretical and experimental values of oscillator strength vs. Z-' for a transition in the B I isoelectronic sequence. Reproduced from SINANO~LU [1973] with the kind permission of the author and the journal.

Many other transitions remain to be studied. Sinanoglu has called attention to the failure of certain critical experiments to give consistent values and to the need for further experimental work on doubly-excited systems, metastable states, and transitions in (-ions of large (-

5 8. Coherence and Alignment 8.1. COHERENCE

If levels of the same n, Z, s are collisionally populated in a time At Jj-are not equally populated, although the rate of production of MJ is the same as that of - M J . When there is alignment, the light emitted in a given direction is polarized; then there are coherence-generated oscillations, called quantum beats. 8.3. QUANTUM BEATS

When the beam-foil light source is used, At is certainly no longer than the 3 x lo-'' sec. transit time of the c-ions through the foil, a matter of Hence levels with energy separations of an electron volt or so may be coherently excited. The time-dependence of the level population is transformed into a spatial dependence, so one looks for the quantum beats in the variation of the intensity of a given kind of polarization as a function of distance between the foil and the point of observation. Although a number of experimental papers have been published (BASHKINand BEAUCHEMIN [1966], ANDRA[1970a1, LYNCH,DRAKE,ALGUARD and FAIRCHILD [19711, BURNS and HANCOCK [1971]) on this phenomenon, the first full study of coherent excitation with the beam-foil method was made by WITTMAN,TILLMANN and ANDRX[1973], who studied n = 3,4 in hydrogen and the 3 3P term of He I. Consider the latter case (see Fig. 23).

-

COHERENCE A N D ALIGNMENT

3p

327

‘P, 185564.9466cm-’

’ y 185564.6760 % ’ 185564.6540

2s a s ,

-$

159850.38

Fig. 23. Part of the triplet levels of He I and the geometry used in observing quantum beats.

For our purposes, the three components of A3889 A are indistinguishable. Let the direction of the beam be z and the direction of observation y . The intensity of the light which can be detected is then

I = I,+I,.

(8.3)

The relative magnitudes of these polarized components depend on time because of the oscillatory nature of the probability density of the 3P term. Suppose the situation at some time were such that I, = 0. We would then detect I,. Sometime later, the system would emit part of its light as I, and some as Iy , the latter being unobservable. The result is that the total detected intensity would fluctuate in time with a frequency, or set of frequencies, characteristic of the energy intervals in the upper term. Analysis of the frequency pattern can then give those intervals. Andrii’s first work (ANDRL[1970a]) was necessarily crude; the observed beam length was 1 mm (0.25 nsec), and the total observation time was only 6 nsec. Nonetheless the quantum beats were clearly displayed, as well for n = 3 and 4 in H as for the 3 3Pterm in He. In a later experiment (WITTMA”, TILLMANN and ANDRA [1973]), a striking improvement had been achieved; the results for He 1 3 3P are reproduced in Fig. 24. While Fig. 24 (top) shows the beat pattern for a time of 65 nsec, Fig. 24 (middle) illustrates the detailed features of a single period. In Fig. 24 (bottom), we see a similar study, but for 3He. (Note the different time scales for the three curves). It is thus clear that this quantum-beat technique is suitable for the determination of the energy separations of fine and hyperfine terms. WITTMAN, TILLMANN and ANDRA[1973], BERRYand SUBTIL[1973], and ANDRA,WITTMANN and GAUPP[1973a] have BERRY,SUBTIL,PINNINGTON, also applied the method to several terms in 6”Li. 8.4. ALIGNMENT AND A N EXTERNAL NON-OSCILLATORY MAGNETIC FIELD

The alignment means that the excited system behaves Iike a magnetic dipole which can be coupled to an external magnetic field, H . This coupling

328

BEAM-FOIL SPECTROSCOPY

3 37-3p2 BEATS 658 MHZ

HELIUM-4

10

I

I

I

10

20

QUANTUM-BEATS

I I

I J F 0 112-

L

I

0

3

4

L

A

1

2

3

4

2

I

I

I 30

OF THE 33P STATE OF 4He

0

la

I

5

6 (nsec)

I - I . ,

7

6 (nsec)

1

(ns)

VI, §

81

COHERENCE AND ALIGNMENT

329

creates new MJ levels out of the initial configurations, so one may consider that the field generates coherence among the Zeeman levels. Coupling between states of different parity does not occur. Let the geometry be as in Fig. 23 with a variable magnetic field in the x-direction. A short segment of the beam is observed at a distance d from the foil. The dipole precesses in the y-z plane, so that the intensity detected in the y-direction is a function of H, . If H, is itself varied linearly in time,

I,(H,) = A(I + B cos 20.1, t)e-" = A(I

+B cos 201, d/~)e-~'"

(8.4a) (8.4b)

where oLis the Larmor (circular) frequency,

thus

B is a measure of the alignment, and the other symbols are familiar. From the period of the sinusoidal intensity variation, one can determine the Land6 g,-factor (LIu, BASHKIN, BICKELand HADEISHI [1971], LIU and CHURCH [1971], CHURCH, DRUETTA and LIU [1971], CHURCH and LIU [1972, 1973b], GAILLARD, CARRE,BERRYand LOMBARDI [1973]). The attractive feature is that those numbers can be obtained for levels in [-ions of a wide range of 5, and for levels which do not connect to the ground term by allowed decays. The general result is that G S coupling is a good approximation in nearly all cases studied. A few exceptions have been found, especially for the n 3P term of Li I1 (GAILLARD, CARRE,BERRYand LOMBARDI [1973]). Furthermore, if the c-beam includes a level of known g, any unknown g; can be and LIU obtained in terms of the ratio (LIU and CHURCH[1971], CHURCH [1973bl) g; = g p / R ' , (8.7) Fig. 24. Quantum beats in He 1V and He 111. The top figure shows data extending over a time of 65 nsec. The middle figure shows the detailed structure of one of the periods in the top figure. The bottom figure shows the particulars for the same transition but in He 111. The arrows in part a of the bottom figure illustrate the transitions and their amplitudes which are contributors to the pattern of part c. Part b is a computer fit to the data using the frequency distributions of part a. These figures are reproduced from WITTMANN, TILLMANN and ANDRA[1973] with the kind permission of the authors and the journal.

330

BEAM-FOIL SPECTROSCOPY

[VL

88

where R and R’ are, respectively, the mean changes in magnetic field to produce one oscillation for the known and unknown values of gJ. An important application of this method is to check on the validity of the transition assigned for a given spectral line. For example, LIUand CHURCH [1971] note that 12778A, ascribed to Ne I11 3s’ 3D:-3p’ 3D3 (STRIGANOV and SVENTITSKII [1968]), shows no oscillations, a condition generally expected only for J = 0 or 4.Hence it is suggested that the line in question be reexamined. Several variations of the above technique have appeared. In one variation (LIu, BASHKIN, BICKEL and HADEISHI [1971], CHURCH,DRUETTA and LIU [1971], CHURCHand LIU [1972]), a long beam segment is observed. This gives an experiment similar to a Hanle-effect study in which, however, the finite mean life of a level under study may require a modification of the and LIU [1971]) appropriate equation. One finds (CHURCH,DRUETTA

where z is the mean life of the level and oL = g J ( p B / h ) His the Larmor precessional frequency. If the g-value is known, perhaps from quantum-beats work, this method enables one to find z, about which more is said later on. Results for g-values have been given for levels in Ne I, 11,111 (LIu, BASHKIN, BICKEL and HADEISHI [1971], LIU and CHURCH[1971]), 0 11, TI1 (CHURCH and LIU [1973b]), and Ar I, 11, I11 (CHURCH,DRUETTA and LIU [1971], CHURCH and LIU [1972]). From eqs. (8.6) and (8.8), it is seen that N and d play the same role as regards the oscillatory behavior of I. Advantage has been taken of this fact in another variation of the basic method (LIUand CHURCH[1972], CHURCH and LIU [1973c], LIU, DRUETTA and CHURCH[1972]). In this, the magnetic field is held constant, and the foil is moved backwards and forwards. This v)

L

C

0

V

a

I Chonnel number

7

Fig. 25. Quantum-beat data from which lifetimes can be determined without the influence of cascades. Reproduced from CHURCH and LIU 11973~1with the kind permission of the authors and the journal.

VI, 8 81

COHERENCE A N D ALIGNMENT

33 1

gives the exponential-type decay, on which is superimposed the oscillatory pattern. If, now, one sweep of the foil (with field on) is followed by a sweep with the field off, and the two signals added with opposite signs, the exponential behavior is eliminated and the oscillatory function is clearly displayed. One such result is shown in Fig. 25, the level in question being 0 I1 3p' 'F;. The advantage of this work is explained by the following equations (LIU and CHURCH [ 19721, CHURCH and LIU [ 1973~1).We represent the intensity, in the absence of a magnetic field, as

I(t)

=

+ ~ ( to)+D, ,

Ae-r/r

(8.9) where D is the background and C arises from cascades. When the magnetic field is on, we have I'(t)

=

e-'lr(A+B cos 2m,t)+~(t, H,)+D,

(8.10)

whence 1'-I

=

B~-"'(COS~~,~-~)+C(~,H)-C(~,O). (8.11)

If, as will now be argued, C(t, H )

= C(t, 01,

(8.12)

it follows that z can be determined independent of cascades. It is immediately obvious that the equality implied in eq. (8.12) holds when the cascading levels have J < 5, for such levels cannot exhibit alignment effects. Moreover an extensive study was made (LIu, DRUETTA and CHURCH[1972], LIU, GARDINER and CHURCH [1973], CHURCH and LIU [1973a]) on levels which are known to be subject to strong cascades from others with J > +, but no cascade influence was found. and LIU [1973c]) obtained by this and In Table 3 we list results (CHURCH the standard beam-foil method for lifetimes of levels in 0 11. The variety of numbers from a given reference is due to the use of several different transitions from the upper term. Since the lifetime is the same for all members of a multiplet, the spread illustrates that the standard method is subject to difficulties.What is more, note that the cascade lifetime, which, in the standard method affects the decay curve for 3p' 'F;, is deduced by DRUETTA, POULIZAC and DUFAY[1971] to be 4.0 nsec. However, the direct measurements on the upper level of prime importance (3d' 'G,) are 5.1 to 9.5 nsec. This illustrates the difficulty of ascertaining a cascade mean life from a standard decay curve. It appears that these problems are largely avoided by the method of LIU and CHURCH [1972], CHURCH and LIU [1973a, cl LIU, DRUETTA and CHURCH [1972], LIU,GARDINER and CHURCH[19731, and that the important matter of level lifetimes is perhaps best investigated in this manner.

332

PI, § 8

BEAM-FOIL SPECTROSCOPY

TABLE3 Lifetimes of some 0 I1 levels by two methods t

Transition

(upper level) (nsec)

T

(cascade) (nsec)

CHURCH and LIU [1973c]

Other

CHURCH and LIU [1973c]

Other

13.7h0.4

14.0b 12.8' 9.9 10.7d 12.0d 14.0d 8.7"

None

4.0b

3p"Fo4-3d"G3

6.8*0.2

9.P 6.1b 5.1"

None

2Sb 20b

3s 2P3-3p 'P"+

6.8hO.I

7.1b 6.0'

None

None

~

3s' 'D3-3p'

a

'Fa%

PINNINGTON and LIN [1969b]. DRUETTA, POULIZAC and DUFAY[1971]. DRUETTA and POULIZAC [1969]. KERNAHAN, LIN and PINNINGTON [1970a].

8.5. COHERENCE WITH A NON-OSCILLATORY ELECTRIC FIELD

In the previous examples of coherence, only levels of the same parity were connected. However, when an electric field is applied, either directly or by sending the c-ions through a transverse magnetic field, there is Stark coupling of levels of opposite parity. This coupling manifests itself as time- (or space-) dependent oscillations in the intensity of a spectral line. Measurements have been reported on several Lyman and Balmer lines in H and He 11. The early experiments on the visible lines in H (BASHKIN, BICKEL,FINK and WANGSNESS [1965], SELLIN, MOAK,GRIFFINand BIGGERSTAFF [1969a]), and He I1 (BICKELand BASHKIN[1967]) clearly established that the rate of radiation was an oscillatory function of time when the field was applied but the use of photographic recording precluded precise determinations of the intensity patterns. Later work, using photomultipliers (BICKEL[1968], ANDRA[1970b], ALGUARD and DRAKE[1973], PINNINGTON, BERRY, DESESQUELLES and SUBTIL[1973]) or a Geiger counter (SELLIN, MOAK,GRIFFIN and BIGGERSTAFF [1969b] for Ly a), was more satisfactory in that such intensity variations as occurred were properly recorded. Despite this improvement, certain confusions were introduced as a consequence of the several field geometries which were employed. We believe these confusions

VI, 8 81

C O H E R E N C E A N D ALIGNMENT

333

can be eliminated, essentially on the basis of AndrP’s work (ANDRA[1970b]) as follows. What Andra showed is that the oscillations can be seen only if the electric field is parallel to the beam. Most of the experiments used fields normal to the beam, but still generated oscillations; AndrP argued that only the fringing part of the field, which contained a component parallel to the beam, was effective. Experimental corroboration was provided by PINNINGTON, BERRY, DESESQUELLES and SUBTIL[I9731 in a study of Stark modulations in He I1 and Li 111. They used geometries as illustrated in Fig. 26b, c, while Fig. 26a F i N K and WANGSNESS [1965], shows the geometry used by BASHKIN, BICKEL, BICKELand BASHKIN[1967], and BICKEL[1968]. Geometry 26a gave oscillations but 26b did not. The reason seems to be that 26a has a significant longitudinal field, while 26b doesn’t. Again, 26c produces oscillations, as does the case where foils perpendicular to the beam serve as the field plates.

(a 1

(b)

(C)

Fig. 26. Various geometries used in investigating the effect of small external electric fields on emissions from hydrogen and He 11.

The observed patterns are Fourier-analyzed so as to deduce the component frequencies, these being related in turn to the energy separations between the levels the interaction of which generates the oscillations. For a simple model, consider the n = 2 term in H. The 2s level cannot decay to the ground state, whereas the 2p levels can. If, then, the external field causes a periodic transformation between the 2s and 2p level systems, the intensity of the radiated light (Ly a) will be high when the 2P character dominates the excited system, low when 2s is dominant, and, generally, oscillatory with a period dependent on the 2S-2P energy spacing. Other levels can be expected to give more complicated patterns because of the greater number of interfering states. One of the peculiarities is that only certain frequencies, none predicted in advance of the experiments, seem to occur. Consider the results of PINNINGTON, BERRY,DESESQUELLES and SUBTIL[1973]. The n = 6 term in He I1 was observed by means of the decay to n = 3. Figure 27 illustrates the level scheme and the dominant frequency which was detected. PINNINGTON, BERRY,DESESQUELLES and SUBTIL[1973] attribute their result to the requirement that, as described in our model, one interacting level should be

334

[VI,

BEAM-FOIL SPECTROSCOPY

08

long-lived, and the other short-lived, a condition which, they say, is best met by the f;-gg pair. They particularly note that the g-state cannot decay t o n = 3. 8.206

h"2/-

8.192

9%-

8.1704,

f,

f2

d5,

6,

8.134

Fig. 27. Levels for n

=

~

b

8.170 8.134

a2 dJ+8.062 ~

426697.862 If2

/2

8.192

h9/2-

~

7.845

"2

6 in He 11. The level positions are given in cm-'. The arrow indicates the pair which beat together.

We are not convinced that the argument is valid. In the first place, the g-h combination better satisfies the above condition than f-g, but the corresponding frequency was not seen. Secondly, the f+--g; pair is the same as the f,-g, pair as regards relative lifetimes, but the former set seems not to contribute to the pattern. Thirdly, PINNINGTON, BERRY,DESESQUELLES and SUBTIL[1973] stress that the g-states cannot decay to n = 3. While that is true, it seems to us that what really counts is the fact that the constituents of n = 6 are interacting with each other. Whether a particular state participates directly in the transition one chooses to detect should not be important. Thus, in the present instance, the instantaneous populations of the 6g-states should be affected by the decays to n = 4 and 5, and those of the 6f-states by the decays to n = 3, 4, and 5, as well as by mixing with all the other Mates. The effects of these several interactions should show up in every transition from n = 6 downward, but the corresponding frequencies are apparently not present. Our conclusion is that the phenomenon of Starkmixing of the hydrogenic levels is not yet adequately explained. 8.6. OSCILLATING EXTERNAL FIELDS

Two different kinds of experiments have been done with oscillating fields applied to the (-ion beam. In one kind (FABJAN and PIPKIN[1972], LUNDEEN, YUNGand PIPKIN[1973]), a foil-excited beam of 20-30 keV H-atoms was sent through an rf cavity, and the long-lived 3S, state was quenched by tuning the rf to match the S,-P, level separation. Such work has given the n = 3 Lamb shift to -0.015 % (FABJAN and PIPKIN[1972]). In a second type of work, the rf was generated by electrodes which, connected to d.c. power supplies, were constructed with a periodic geometry.

VI,

5 81

COHERENCE A N D ALIGNMENT

335

Thus, HADEISHI, BICKEL,GARCIAand BERRY[1969] used the arrangement of Fig. 28 to induce transitions between the 3S, and 3P, levels in hydrogen. The magnetic field separated the Zeeman studies to the value appropriate to the rf seen by the beam. This experiment demonstrated that a periodic potential barrier could be used in the study of fine-structure. MAGNETIC FIELD

*

Recorder

Fig. 28. Arrangement used in the application of rf to a beam of hydrogen. A variable d.c. potential difference is applied to the electrodes; the c-ions see this as rf with a frequency dependent on the C-ion speed and the spacing of the “teeth”. The magnetic field introduces a Zeeman splitting of the levels. Reproduced from HADEISHI, BICKEL,GARCIA and BERRY[I9691 with the kind permission of the authors and the journal.

AND^ [197Oc] sent a beam of H-atoms down the common axis of two helices which were connected to a d.c. power supply. An axial magnetic field was also applied. To the H-atoms, it appeared as though they were being irradiated with circularly-polarized light with a frequency dependent on the beam speed and the pitch of the helices. Andra observed the resonant decay of Ly cx as a function of the magnetic field. HADEISHI, BICKEL,GARCIA and BERRY[I9691 and ANDRA[197Oc] dealt with attempts to measure the Lamb shift in hydrogen, and they achieved a modest success. A somewhat different problem was studied by LIU, BASHKIN, BICKEL and HADEISHI[1971]. They sent Ne’ ions through a foil and then down the axis of a row of copper rings so arranged that direct current circulated circumferentially,but in opposite directions, in successive rings. This produced an oscillating axial magnetic field, H, with an intensity proportional to the current through the rings. By adjusting H, resonance transitions were induced between Zeeman sublevels of the 2p, level of Ne I. These resonance transitions affected the intensity with which light, polarized either parallel or normal to the magnetic field, was radiated in a given direction. Thus detection of that light which was transmitted through a linear polarizer

336

BEAM-FOIL S P E C T R O S C O P Y

[VI,

I9

indicated the resonance condition. If the polarizer is normal to the magnetic field, the appropriate expression, at resonance, reduces to (LIu, BASHKIN, BICKELand HADEISHI [19711)

I

-

(y~)~/[4y~~~+r~],

(8.13)

where y is the gyromagnetic ratio and r the decay constant for the level. Thus we see that this application of rf to the [-ion beam leads to the determination of y and r.

Q 9. Charge-State Identification One of the fortunate features of the beam-foil light source is that C-ions with a wide range of [-values can be excited. One of the unfortunate features is that three or four different values of [ appear in the same foil-transmitted beam. Thus it is a bit of a problem to decide to which particular %-iona given observed spectral line actually belongs. Of the methods in use, two derive from the particular properties of the beam-foil source. These are: 1. Measuring the line intensity as a function of [-ion energy and associating the line with that value of C which exhibits a similar dependence on [-ion energy (BASHKIN and MARTINSON [19711, WITTK~WER and BETZ[19731, ANDERSEN, DESESQUELLES, JESSEN and S0RENSEN [19701, ANDERSEN, BICKEL, BOLEU,JESSEN and VEJE [1971a], ANDERSEN, BICKEL,CARRIVEAU, JESSEN and VEJE[1970b], BERRY,DESESQUELLES and DUFAY[ 19711, ANDERSEN, JESSEN and S0RENSEN [1969a, b], MARTINSON, BICKELand ULME [1970], BERRY,MARTINSON, SCHECTMAN and BICKEL[1970a1, BERRY,SCHECTMAN, MARTINSON, BICKEL and BASHKIN [1970b1, BERRY, BICKEL, BASHKIN, DESESQ U E L L E ~and SCHECTMAN [1971a], KERNAHAN, LINand PINNINGTON [1970b1, KAY[1965], BASHKIN and MALMBERG [1966], DENIS,DESESQUELLES, DUFAY and POULIZAC [1968], DENIS,DESESQUELLES and DUFAY [1969], DENIS[1969], DENISand DUFAY[1969], DENIS,CEYZERIAT and DUFAY[1970], DRUETTA, POULIZAC and DESESQUELLES [19701, POULIZAC, DRUETTA and CEYZERIAT [ 19711, MARTINSON, BERRY,BICKELand OONA[19711, PINNINGTON and DUFAY[19711, ANDERSEN, ROBERTS and S0RENSEN [197I 1, POULIZAC and BUCHET[1971], BUCHET,POULIZAC and CARRE[1972], BERRY,BUCHETPOULIZAC and BUCHET [1973b]). 2. Deffecting the [-ions in an external field and carrying out a spectral BICKELand analysis on each of field-separated particle beams (MARTINSON, ~ L M E[1970], MALMBERG, BASHKIN and TILFORD [1965], FINK [1968a, b], Jr., RUBINand TRACHSLIN [1968], CARRIVEAU and BASHKIN BROWN,FORD [ 19701, BASHKIN, CARRIVEAU and HAY[19711.

o

VI, 91

337

CHARGE-STATE IDENTIFICATION

Each method has favorable and unfavorable aspects; the former, as one might guess from the relative bibliographies, is the easier to use. Its great virtue is experimental simplicity; all one need to do is observe the intensity of a line from some point along the beam and vary the [-ion energy. In general, what is observed is that the line intensity grows with energy until a rather broad maximum is reached, after which a monotonic decline occurs. If several lines exhibit a similar energy dependence which differs from that of another group of lines, it is natural to attribute the groups to different stages of ionization. There are two main problems with such an approach, namely, not all lines from the same stage of ionization behave the same way with C-ion energy and it is often impossible to be sure of the effect of line blends on the energydependence. Sometimes blends ate suspected as the cause of the former problem, but not always. For example, ANDERSEN, BICKEL,BOLEU,JESSEN and VEJE[1971a] obtained data some of which are reproduced in Fig. 29. The top curve originates from the transition Li I 2p 'P0-ls 2s 3s 4S and the middle one from a transition from an autoionizing level in Li I. Thus they both come from Li I, but their energy dependencies are quite different. In

X - 413.3nm Li Ia

z

I

'

1

I I I I 1 1 10 20 30 40 50 60 70 80 keV BEAM ENERGY IN THE LABORATORY SYSTEM

Fig. 29. Intensity variations of three different spectral lines as a function of particle energy. The two top curves are for transitions in Li 1, the bottom from a transition in Li 11. Adapted from ANDERSEN, BICKEL,BOLEU,JESSENand VEJE[I971 a] with the kind permission of the authors and the journal.

338

BEAM-FOIL SPECTROSCOPY

[VI, 0

9

fact, the bottom curve, from Li I1 3d 3 D 4 f 3P0,looks rather like the middle curve, although the charge states of the parent [-ions differ by one. In the above case, one can argue that the similarity between the two lower curves is to be expected because the metastable levels involved in Li I 1s 2s 2p 4P0-ls 2s 3s 4S are well above the ionization limit for Li I so that Li 11-like behavior is not unlikely. However, the argument is ad hoc: the force of the charge-state identification rests on the spectroscopic analysis of the source of A2934, and is both independent and contradictory of the association with the energy variation of charge states. This situation is not restricted to lines from autoionizing levels. Thus LIVINGSTON, IRWIN and PINNINGTON [1972], in a study of argon beam-foil spectra, assert that there are several intense lines from Ar IX or higher although, at the [-energy of 1.4 MeV which they used, less than 10 % of the beam has a net charge of 6 and higher. ANDERSEN, ROBERTS and SBRENSEN [I9711report strong differences in the energy-dependenceof A1 I1 lines from 3s nf 3F levels. The problem of line blending seems extremely hard to solve as regards charge identification. Numerous papers, among them those by LIVINGSTON, IRWIN and PINNINGTON [1972] and POULIZAC, DRUETTA and CEYZERIAT [1971], claim that the failure of certain lines to follow patterns established for others is rooted in line blends from transitions in two or more charge states, and this could well be the case. However, it does not result in confidence in charge assignments. It is our feeling that the correlation of line intensity with charge fraction is merely suggestive of the charge of the emitting [-ion, that it always depends on some quite independent information for calibration of the set of charges to be assigned, and that it is apt to be misleading if complete reliance is placed on it, particularly if there is any suspicion that blends are present. The second method has involved static electric fields which are applied transverse to the I-ion beam. Observations may be made normal to the plane of the split beams (MALMBERG, BASHKIN and TILFORD [1965], FINK[1968a]), in which situation a spectral decomposition is carried out on each of the separated (-ion beams, or in that plane (BROWN,FORDJr., RUBINand [19681, CARRIVEAU and BASHKIN[19701, BASHKIN, CARRIVEAU TRACHSLIN and HAY [1971]), in which case one makes use of the fact that there is a Doppler shift which differs with the charge of the emitter. The former approach gives completely unambiguous - and correct results. Unfortunately, the parabolic beam paths tend to produce rather poor images in the focal plane of the spectrometer. The latter has the draw-

VI,

B 101

CONCLUSION

339

back that transition arrays from different charge states move through one another, making it difficult to decipher the patterns. Both ways of examining the beams fail for short-lived emitters since decay then occurs before adequate spatial movement or Doppler shift has developed. If 7 = 3 x lo-’ sec for a level in N3+,the transverse displacement is 0.5 mm for a 1-MeV beam passing through a field of 50 kV/cm. The Doppler shift at the end of that time is only 4 A (at 4000 A) and of course, the line intensity has fallen by e-l. One gains by using the highest possible deflecting field. A practical upper limit is in the neighborhood of 75 kV/cm. The [-ions, on reaching a collector, release electrons which can give rise to electrical discharges in the target chamber. The discharge problem can be reduced by maintaining a pressure 5 torr and by slotting the electrodes so that ions (and electrons) are not collected in the region of high field. A word of caution might be in order. The energetic electrons do generate X-rays and care should be exercised to prevent their causing unnecessary irradiation of laboratory personnel.

-

0 10. Conclusion We have omitted a number of interesting topics. These include the question, What is the relative population of the levels of a given n and also of the magnetic substates of a given I? The literature contains quite a few quite different answers to these questions. Closely related is the fundamental problem of determining precisely what happens when the [-ions pass through the foil. Whether volume or surface effects are dominant has been much debated, but no one has yet presented a detailed theory of the beam-foil interaction. Still other matters untouched in the present review are prospective developments, such as laser stimulation of the foil-excited [-ions or the application of pulse techniques to the beam of [-ions. In this review, we have adopted a critical attitude towards beam-foil spectroscopy. The reason is simply that the successes in this field are well known, while some of the handicaps often go unrecognized. It is our hope that calling attention to the present drawbacks will lead to their early elimination.

Acknowledgments It is a pleasure to record my thanks to my colleagues for their critical comments, and to many people who have permitted me to reproduce figures from their papers. Preparation of this paper was supported by NSF, ONR and NASA.

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w.

AUTHOR INDEX A ABBI,S. C., 35, 50 ABELLA, I. D., 57, 78, 99 AHMAD,F., 96, 99 AKHMANOV, S. A., 11, 12, 26, 44,45, 50 ALFANO,R. R.,2 1, 50 ALGUARD, M. J., 326,332, 340, 343 N., 296,302,305, 336,337,340 ANDERSEN, ANDERSEN,T., 305,312,314, 322,323,336, 338, 343 P. W., 237, 283 ANDERSON, ANDRA,H. J., 326, 327, 329, 332, 333, 335, 340, 341, 344 H. C., 119, 162 ANDREWS, ARECCHI,F. T., 58, 96, 97, 99, 241, 284 E. G., 79, 99 ARTHURS, ARZT,V., 255,284 ASHER,I. M., 78,99 G. A., 10,44, 50 ASKARYAN, ASPINALL, D., 113, 162 ASPNES,D., 250,285 ASSOUSA, G. E., 323, 340 L. G., 165, 229 AUGENSTINE, B BAHCALL, J. N., 324, 340 N. A., 324, 340 BAHCALL, G. S., 292, 340 BAKKEN, BARTH,P., 172, 229 S., 291, 294, 296, 312, 319, 320, BASHKIN, 326, 329, 330, 332, 333, 335, 336, 338, 340-343 BAUER,H., 173, 229 L. D., 143, 157, 161 BAUMERT, G. J., 326, 340 BEAUCHEMIN, BECK,K. H., 172, 187,229 BELL,R. A,, 323, 340 BERAN,M., 235,284

BERGSTROM, J., 305, 312, 318, 340, 341 S., 180, 229 BERNSTEIN, BERRY,H. G., 296, 305, 306,308,319, 320, 327, 329, 332-336, 340, 342, 343 R. 0.. 322, 323, 344 BERTHEL, V. I., 32, 50 BESPALOV, BETZ, H. D., 302, 336, 344 BICKEL,W. S., 296,302,305,312,318-320, 329,330,332,333,335-337,340,342,343 J. A., 332, 344 BIGGERSTAFF, G., 22, 51,237, 286 BIRNBAUM, J.-E., 284 BJORKHOLM, BLANC,J., 173, 229 BLOCH,F., 60, 62, 99 Katharine B., 166, 174,177,229 BLODGETT, N., 11, 26, 48, 50, 51, 236. BLOEMBERGEN, 274, 284 N. N., 236,268,284 BOGOLJUBOV, BOLEU,R.,302, 336, 337, 340 R., 246, 253, 257, 284 BONIFACIO, BORN,M., 181, 191, 195,208,214,220,222, 224, 228, 229 BOSCHEK, B., 323, 324, 341, 342 R. C., 235, 284 BOURRET, BOWEN,I. C., 294, 341 BOYCE,T. C., 306, 341 W. R., 323, 341 BOZMAN, BRADLEY, D. J., 79, 99 BRAND,J. H., 324, 341 R. G., 11, 14,20, 35,47, 50, 79, 99 BREWER, J. M., 323, 341 BRIDGES, Jr., D. A., 153, 161 BRIOTTA BRISS,R. R., 13, 50 BROMANDER, J., 294, 296, 297, 299, 305, 340, 341, 343 BROUT,R., 248,284 BROWN,L., 302, 336, 338, 341 BROWN,M., 297,300,308,309,3 11,343,344 BUCHER,H., 165, 166, 170, 172, 177, 180. 187,189-191,202,204,206,207,230,231

346

AUTHOR INDEX

BUCHET, J. P., 297, 298, 302, 305, 336, 341, 343 M. C., 297, 298, 302, BUCHET-POULIZAC, 336, 341 BUCHTA,R., 296, 305, 307, 312, 318, 324, 340-342 BULLOUGH, R. K., 96, 99 F. M., 323, 341 BURBIDGE, BURBRIDGE, G. R., 323, 341 BURNS,D. J., 326, 341 C CABIBBO, N., 324, 340 CADOT,J., 188, 162 A. G. W., 323, 341 CAMERON, A. G., 35, 50 CAMPILLO, CANDAU, S. J., 19, 51 CARMAN, R. L., 25, 44, 46, 50 CARMICHAEL, H. J., 283,284 CARNIGLIA, C. K., 198, 230 CARRE,C., 292, 329, 342 CARRE,M., 336, 341, 342 G. W., 292, 305, 313, 336, 338, CARRIVEAU, 340, 341 P., 336, 338, 342, 343 CEYZERIAT, CHABAN, A. A., 35, 50 CHANLEY, J. D., 165, 231 A. C., 41, 50 CHEUNG, CHIAO,R. Y., 10, 11, 14, 21, 27, 32, 35, 41, 47, 49, 50 CHURCH, D. A., 329-331, 341, 343 CLEMMOW, P. C., 202, 229, 230 CLOSE,D. H., 13, 14, 16, 35, 50 COCKE, C. L., 296,297,300,306,323,324, 341, 342 A., 78,99 COMPAAN, COMPTON, K. T., 306, 341 COOLEY, J. W., 120, 161 CORLISS, C. H., 323, 341 E., 44, 46, 51, 72, 97, 99, 100 COURTENS, COWLEY, C., 324, 341 CRISP, M. D., 96, 98, 99 R., 23, 39, 41, 44, 46, 48, 50 CUBEDDU, CUMMINGS, F. W., 237, 284 B., 297, 300, 306, 324, 341 CURNUTTE, CURTIS,L. J., 296, 305, 324, 340, 341 D DALGARNO, A., 324, 341 DAVIDOVICH, L., 97, 99 DAVISJr., R., 324, 341

DAWES,E. L., 30, 50, 51 DEBEYE, P., 17, 50 DECKER Jr., J. A., 113, 119, 120, 125, 140, 143-147, 161 V., 96, 99, 238, 264, 284 DEGIORGIO, T., 120, 140, 161 DEGRAAUW, DE MARTINI,F., 8, 9, 50 DE NARIEZ-ROBERGE, M. M., 41, 44,50 DENIS,A., 297, 298, 305, 336, 341, 342 J., 296, 297, 298, 302, 305, DESESQUELLES, 306, 319, 332-334, 336, 340-343 L. G., 43, 51 DESHAZER, DEXTER, D. L., 205, 230 DICKE,R. H., 89, 99 DIELS,J. C., 98, 99 DIENES,A., 79, 100 Kh. A., 44, 50 DIYANOV, DOCAO,G., 297, 298, 302, 341 DOERMANN, F. W., 207,230 DOHM,V., 241, 264, 284 DONNALLY, B., 297,300,308,309,311,342, 344 M. H., 292, 341 DOOBOV, DRAKE, C. W., 326, 332, 340, 343 DREXHAGE, K. H., 165, 166, 171-173, 175, 177, 181, 182, 184, 187-192, 194, 196, 198, 201-205, 208, 209, 211, 213, 215, 216,218, 221-223, 225,229, 230 DROUIN,R., 305, 307, 342 DRUDE,P., 170, 230 M., 320, 321, 329-331, 336, 338, DRUETTA, 341, 343 DUFAY, M., 292, 297, 298, 305, 306, 319-321, 331,336,340-343 A. L., 30, 50 DYSHKO, E EBERLY, J. H., 58,96, 97, 99, 100 EDLBN,B., 302, 342 ELSNER,O., 170, 184, 187, 230 ENGELSEN, D. DEN,177,230 P., 305, 307, 342 ERMAN, ESTES,L. E., 85, 99 ETESON, D. C., 85, 99

F FABELINSKI, I. L., 19, 50 FABJAN,C. W., 334, 342 FAIRCHILD, C. E., 326, 343 FALK,J., 284 FAN,C. Y., 311, 312, 342, 344

AUTHOR INDEX

FELD,M. S., 246, 286 FELLGETT,~. B., 113, 118, 125, 161 FENDER, F. G., 306, 342 R. A., 263,286 FERRELL, FEYMANN, R. P., 60, 99 FINE,T., 117, 118, 127, 131, 141, 143, 145, 161, 162 FINK,D., 291, 332, 333, 338, 340 FINK,U., 336, 342 FINKBEINER, D. T., 160, 161 FISHER,M. E., 260, 284 FLECK, J. A., 44,46, 50 FI.ECK,M., 165, 172, 176, 177, 180-191, 202,203,208,209,211-216,218,226,230 FORD Jr., N. C., 272, 284 FORD Jr., W. K., 302, 336, 338, 341 FORSTER, H., 176, 194, 195, 198, 230 FORSTER, T., 201, 230 FOWLER, W. A., 323, 341 FOY,R., 324, 342 M. L., 120, 129, 160, 162 FREDMAN, FREED,S., 170, 230 R., 78, 99 FRIEDBERG, G GABRIEL, A. H., 3 11, 342 F., 305, 342 GAILLARD, M., 292, 305, 329, 342 GAILLARD, GAINESJr., G. L., 165, 230 GALE,G. M., 79, 89 GARCIA,J. D., 335, 342 GARDINER, C. W., 283,284 R. B., 331, 343 GARDINER, E., 10, 11, 14, 27, 32, 35, 49, 50 GARMIRE, GARZ,T., 323, 324, 341, 342 GAUPP,A., 327, 341 H., 284 GERHARDT, GEORGE, H., 21, 51 GEORGE, N., 21, 51 GIBBS,H. M., 64,73, 74, 80, 82,83, 85, 88, 89, 94, 99, 100 GIORDMAINE, J. A., 273, 284 A., 118, 161 GIRARD, C. R., 10, 13, 14, 16, 35, 50 GIULIANO, GLAUBER, R. 5.,241,257,273,275,284-286 GLENNON, B. M., 231, 344 GOLAY,M. J. E., 113, 115, 117, 161 J., 98, 99 GOLDHAR, J. C., 238, 284 GOLDSTEIN, V. I., 170, 231 GOLOYADOVA, J. P., 91,99, 273, 284 GORDON, GOTTLIEB, P., 119, 137, 161

347

GOTZE,W., 239, 250, 285 R., 238-240, 258, 262, 264, 270, GRAHAM, 272-275, 277, 278, 284 J. F., 112, 113, 161, 162 GRAINGER, GREVESSE, N., 324, 342 GRIFFIN,P. M., 297, 300,308,332,343,344 GRIENEISEN, H. P., 98, 99 D., 10, 50, 91, 99 GRISCHKOWSKY, GROSS, G., 194, 231 S.,238,264,270,272,284,286 GROSSMANN, GRUHL,Th., 94, 95, 100 GRUENBERG, L. W., 263, 284, 285 GUNTHER, L., 263, 285 T. K., 8, 9, 21, 26, 37, 41, 47, GUSTAFSON, 50

A., 284 GUTTNER,

H HAAKE,F., 241, 246, 253, 257, 284, 285 T., 329, 330, 335, 336, 342, 343 HADEISHI, HAHN,E. L., 5 5 , 59, 63, 66, 67, 70, 73, 78, 86, 91, 98-100 HAKEN,H., 236, 238-245, 250, 251, 255, 258,262, 264,273, 275, 280, 284,285 HALLJr., M., 161 HALLIN,R., 299, 343 O., 207,230 HALPERN, D., 239,250,285 HAMBLEN, W. H., 326, 341 HANCOCK, HANSEN,P., 145, 152, 161 D. S., 324, 341 HARMER, HARTMANN, S. R., 57, 78, 99, 100 HARWIT,M., 113, 117-119, 120, 127, 131, 137, 139-141, 143, 145, 150, 161, 162 G., 11, 51 HAUCHECORNE, HAUGLAND, R. P., 203,230, 231 HAUS,H. A., 26, 27, 37, 41, 51 HAY,H. J., 292, 336, 338, 340, 341 HECHT,T., 250, 285 R. W., 13, 14, 16,20,21, 23, HELLWARTH, 35, 50, 51, 60, 99, 273, 285, 286 HEMPSTEAD, R. D., 239, 264,266,285 HEPP, K., 241, 247, 282, 283, 285 HERMAN, I. P., 246, 286 R. N., 17, 51 HERMAN, HESS,L. D., 13, 14, 16, 35, 50 HINKLEY, E. D., 79, 100 HIOE,F. R., 283, 286 HOCKER, G. B., 79, 100 HOFFMAN, K. C., 324, 341 HOHENBERG, P. C., 236,240,260,263,268, 285

348

AUTHOR INDEX

HOLLEY, C., 180, 231 HOLMEYER, H., 323, 324, 341, 342 HONTZEAS, S., 305, 307, 342 HOPF, F. A., 67, 72, 74, 78, 91,92, 94, 98-100 HOUCK,J., 120, 147, 161 HOYIX,F., 323, 341 HUBER,M., 323, 342, 343 HUBER,M. C. E., 323, 342 HUFF,H. J., 47, 51 I IBBETT,R. N., 113, 162 O., 206,222, 223,231 INACKER, IPPEN,E. P., 79, 100 IRWIN,D. J. G., 317, 338, 343 E. N., 19, 5 1 IVANOV,

J JACQUINOT, P., 110, 162 JAVAN, A., 10, 51, 79, 91, 97-100 JENSEN, K., 305, 340 K., 302, 336, 331, 340 JESSEN, JESSEN, K. A., 305, 336, 340 JOENK,R. J., 8, 51 JOHNSON, M. A., 11, 35, 50 JOHNSTON, J. R., 237, 284 C., 311, 342 JORDAN, JORDAN, J. A., 291, 292, 340, 342 JOSEPHSON, B. D., 237, 285

K KADANOFF, L. P., 239, 250, 272, 285 KAISER, W., 35, 51 KANE,J., 119, 162, 250, 285 W., 181, 208, 209, 231 KAUZMANN, KAWASAKI, K., 272, 285 KAY,L., 292, 336, 342 KELLEY, M. J., 79, 99 P., 11, 26, 51 KELLEY, KELLEY, P. L., 8, 9, 10, 25, 26, 37, 41, 47, 49-5 1 KERNAHAN, J. A., 336, 342 R. V., 11, 12,26,44,45, 50 KHOKHLOV, KIE, L. K., 78, 99 KIELICH,S., 19, 51 KING,L., 161 KING, R. B., 323, 344 KLEUSER, D., 187, 231 KLOSE,J., 323, 342 KNYSTAUTAS, E. J., 305, 306, 342

KOCK,M., 323, 342 G., 213, 231 KOPPELMANN, KORENMANN, V., 237, 285 V. V., 35, 42, 51 KOROBKIN, KOSSEL, D., 170, 213, 231 L. B., 285 KREUZER, KRINSKY, S., 11, 35, 50 KRYUKOV, P. G., 58, 100 KUEHL,H. H., 231 KUGEL,H. W., 312, 343 KuHN,H., 165,171,172,177,180,181,187, 189-191, 199, 201-206, 208, 211, 213, 215, 216, 218, 220, 222, 223, 225, 226, 229-232 KUHN,W., 231 KURNIT,N. A., 57, 98, 99, 100

L LALLEMAND, P., 11, 51 LAMBJr., G. L., 58, 70, 98, 100 LAMB,W. E., 239, 250, 285, 286 LANDAU, L. D., 245, 285 R., 8, 51, 273, 285 LANDAUER, LANGLEY, K. H., 272,284 I., 166, 174-177, 229 LANGMUIR, G. M., 323, 344 LAWRENCE, G. P., 312,342 LAWRENCE, LAX,M., 237, 239, 245, 251, 255, 265,266, 280,285 LEAVIIT,J. A., 291,294,296, 342-344 LEE,C. H., 20, 50 LEE,P. A., 238,284 W. N., 296, 323,342 LENNARD, V. S., 58, 100 LETOKHOV, M., 312, 343 LEVENTHAL, LEWIS,E. A. S., 250, 285 LEWIS,M. R., 321, 343 LIEB,E. H., 241, 247, 282,283,285 LIFSHITZ,E. M., 245, 285 LIFSITZ,J. R., 11, 14, 26, 35, 37, 41, 50, 51 B., 292, 342 LIGHTFOOT, LIN, C. C., 336, 342, 343 LIN, C. H., 21, 5 1 LINDHARD, J., 313, 343 J., 299, 343 LINDSKOG, N. I., 42, 51 LIPATOV, T. A., 19, 5 1 LITOVITZ, LITVAK,A. G., 24,25, 51 LIU, C. H., 329-331, 336, 341, 343 LIVINGSTON, A. E., 317, 333, 343 M., 329, 330, 342 LOMBARDI, W. H., 255, 273, 275, 277, 284, LOUISELL, 285,286

AUTHOR INDEX

Lou, M. M., 35, 42, 43, 51 LUGOVOI,V. N., 30, 42, 44, 50, 51 S. R., 334, 343 LUNDEEN, LUNDIN, L., 305, 312, 318, 340, 341 LYNCH,D. J., 326, 343

M MACDONALD, J. R.,297, 300, 306, 341 MACEK,J., 326, 343 MACGILLIVRAY, J. C., 246, 286 MAHR,H., 35, 50 MAIER,M., 35, 48, 51 MAIO,A. D., 296, 302, 343 MAKER,P. D., 8, 13,20, 51 MALMBERG, P. R., 291, 336, 338, 340, 343 MANDEL,L., 196, 198, 230, 235, 285, 286 A. A., 42, 51 MANENKOV, MANN,B., 180, 204, 230, 231 J. H., 10, 30,43,47, 50 MARBURGER, A., 299, 343 MARELIUS, R., 298, 300, 311, 343 MARRUS, MARTH,R. E., 97, 100 MARTIN,G. A., 325, 344 MARTIN,P. C., 236, 237, 268, 285, 286 MARTINEZ-GARCIA, M., 323, 344 MARTINSON, I., 294,296,304,305,307,312, 318-320, 324, 336, 340-343 MATULIC,L., 96, 100 G. L., 58, 99 MASSERINI, K. D., 319, 343 MASTERSON, MAYER,H., 184, 231 MAYER,G., 11, 51 G. L., 43, 51 MCALLISTER, MCCALL,S. L., 55, 59, 63, 66, 67,70,73, 78, 86, 91, 99, 1 0 0 MCCAVERT, P., 305, 307, 343 MCCLUNG,F. J., 13, 14, 16, 35, 50 MCLEAN,A. D., 20, 50 MCTAGUE,J. P., 21, 22, 51 MEINEL,A. B., 291, 340 MERMIN, N. D., 260, 286 MERTZ,L., 162 MEYER,H., 206, 231 MICKEY,D. L., 323, 344 MILAN,C., 118, 162 MILFORD,F. J., 285, 286 MILLER,R. C., 273, 284 R. H., 294, 341 MILLIKAN, V. K., 170,231 MILOSLAVSKII, MOAK,C. D., 332, 344 M ~ B I U SD., , 165, 166, 170, 172, 177, 187, 189- 191, 199, 202-204,2 15, 229-23 I

349

MBLHARE, L., 312, 340 MOLLOW,B. R., 273,286 D., 172,229 MOLZAHN, MOORE,C. E., 322, 343 MOORADIAN, A. M., 25, 50 H., 218: 220, 231 MORAWITZ, MORET-BAILLY, 1 1 8, 162 MORI,H., 286 MOYAL,J. E., 275, 286 MUKHAMADZHANOV, 44, 50 MURNICK,D. E., 312, 343 MURRAY, J. E., 284 N

NARDUCCI, L. M., 85, 99 NELSON,E. D., 120, 129, 160, 162 NERNST,W., 170, 230 NICKEL,B. G., 260, 284 NIELSEN,A. K., 296, 340 H., 324, 342, 343 NUSSBAUMER, H. M., 283, 286 NUSSENZWEIG,

0 ~ L M E A., , 305, 336, 343 B., 22, 51 OKSENGORN, OONA,H., 336, 343 ORNSTEIN, L. S., 267, 279, 286 A., 21, 5 1 OWYOUNG,

P PALCIANSKAS, V. V., 250,285 W. H., 323, 342, 343 PARKINSON, PARRENT Jr., G., 235, 284 PATEL,C. K. N., 79, 91, 99, 100 PEGG,D. J., 297, 300, 308, 309, 342-344 PHILLIPS,L. W., 294, 343 PHILLIPS,P. G., 117-120, 127, 129, 131, 137, 141, 143, 145, 150, 160, 162 PIEKARA, A. H., 47, 51 PIERCE,T. M., 100 PIHL,J., 299, 343 N. F., 11, 51 PILIPETSKII, PINNINGTON, E. H., 305,317,327,332-334, 336, 338, 341-343 PINNOW,D. A,, 19, 51 PIPKIN,F. M., 334, 342, 343 J. B., 120, 147, 161 POLLACK, POLLONI,R.,23,39-41,44,46,48, 50, 51 POULXZAC, M. C., 320, 321, 331, 336, 338, 341-343 PRATT,W. K., 119, 162

350

AUTHOR I N D E X

PROXHOROV, A. M., 30, 42,44, 50, 51 PUGVIELLI, V. G., 272, 284

R RADEKA, V., 324, 341

RADZIEMSKI Jr., L. J., 291, 292, 344 ~ H N O., , 48, 51 G. N., 227, 231 RAMACHANDRAN, RAMASESHAN, S., 227, 231 RANK,D. M., 41. 50 RAYL,M., 250, 285 N. E., 58, 100 REHLER, REICH, R., 187, 231 REICHERT, J. D., 47, 51 RHODES, C. K., 79, 91, 92, 94, 95, 97-100 RICE,J. M., 240, 286 J., 323, 324, 341, 342 RICHTER, RICHTER, P. H., 238,264,270,272,284,286 RING,J., 112, 161 H., 239,255,264-266,284,285,286 RISKEN, ROBERTS, J. R., 314,322,336,338,340,343 ROBSON, J. W., 291, 342 RODARI, G. S., 241, 284 RODDIE,A. G., 19, 99 ROGERS, L. C., 324, 341 ROLLIG,K., 172, 229 Ross, D. L., 173, 229 Ross, J. E., 324, 343 ROTHEN, A., 177, 231 RUBIN,V., 302, 336, 338, 341 RUDGE,M. R. H., 305, 307, 343 A. R., 1 1 , 51 RUSTAMOV,

S SACCHI, C. A., 23,3941,44,46,48, 50, 51 SAGAN, C., 120, 147, 161 SAKAI,H.. 109, 162 SALAMO, G. J., 94, 99 H., 236, 239, 241, 250, 251, SAUERMANN, 255, 284-286 SAVAGE, C. M., 8, 51 D. J., 263, 286 SCALAPINO, D., 120, 147, 161 SCHAACK, SCHAEFER, C., 194, 231 SCHAFER,F. P., 165, 172, 178, 189-191, 202, 203, 208, 211, 215, 217, 218, 226, 229, 230 SCHARFF, M., 313, 343 SHAW,E. D., 79, 100 SCHECTMAN, R. M., 296,306,319,320,336, 340 SCHIOTT,H. E., 313, 343

SCHLOSSBERG, H. R., 98,99 C., 239, 284, 286 SCHMID, SCHMIDT, H., 239, 286 SCHMIDT, S., 187, 231 SCHMIEDER, R. W., 298, 300, 311, 343 P., 58, 99, 246, 253, 257. SCHWENDIMANN, 284 SCULLY,M. O., 72, 74, 78, 98-100, 238. 239, 245, 264, 284, 286 SEARS,M., 263, 286 G. L., 323, 343 SEASDOLEN, SELENYI, P., 170, 198, 207, 213, 231 SELLIN, I. A., 297, 300, 308, 309, 311, 332. 342-344 J. R., 245, 280, 286 SENITZKY. SEROV,R. V., 35,42, 51 SHAHAM, Y.,11, 51 SHANG-KENG MA, 260, 284 SHANK, C. V., 19, 100 SHAPIRO, S. L.. 21, 35, 50 SHAW, E. D., 79, 100 SHEN,Y.,11, 51 SHEN,Y.R., 42,43, 51 SHER,I. H., 166, 231 SHIMIZU, F., 41, 44, 46, 51 SHIREN,N. S., 78, 99, 100 I. N., 170, 231 SHKLYAREVSKII, SHOEMAKER, R. L., 79, 99 A. E., 273, 285, 286 SIEGMANN, SILLS,R. M., 296, 323, 342 SINANOGLU, O., 324, 344 SJODIN, R., 299, 343 N., 246, 286 SKRIBANOWITZ, N. J. A., 117, 118, 127, 131, 141, SLOANE, 143, 145, 161, 162 SLIJSHER, R. E., 64, 73, 74, 79, 80, 82, 83, 85, 88, 89, 91, 99, 100 SMITH,H. A., 11, 35, 50 SMITH,P. D., 79, 99 SMITH,M. W., 325, 344 SMITH,W. A., 267, 270,272, 275, 284,286 SMITH, W. H., 323, 340 W. W., 297, 300, 308, 309,311, 321, SMITH, 342-344 SNAVELY, B. B., 187, 230 C. J., 292, 341 SOFIELD, C. G., 96, 99 SOMEDA, A., 228, 231 SOMMERFELD, SONA,A., 241,284 SONDERMANN, J., 165, 172, 178, 189-191, 202, 203, 215, 230, 231 SBRENSEN, G . , 296, 305, 314, 320, 322, 323, 336, 338, 340, 343, 344

AUTHOR INDEX

SPERLING, W., 165, 172, 189-191,202,203, 208,211,215,217, 218, 226,229, 230 STARUNOV, V. S., 23, 51 STATZ,H., 94, 100 STELL,J. H., 112, 161 R., 177, 231 STEIGER, STONERJr., J. O., 291, 292, 313, 319, 342-344 R. L., 258,286 STRATONOVICH, STRIGANOV, A. R., 330, 344 J., 145, 152, 161, 162 STRONG, STRYER, L., 203,230,231 SUBTIL,J. L., 305, 327, 332-334, 341, 343 SUCHELEV, M. Ya., 42, 51 E. C. G., 275,286 SUDARSHAN, A. P., 11, 12,26,44,45, 50 SUKHORUKOV. SUYDAM, B. R., 35, 50 SVELTO, O., 23, 39-41,44,48, 50, 51 SVENTITSKII, N. S., 330, 344 SWIFT,J., 250, 272, 285 SWINGS,J. P., 324, 342, 343 L. v., 180, 204, 230,231 SZENTPALY, G., 227, 231 SZIVESSY, SZOKE,A.,72, 79, 91, 92, 94, 95, 97, 99, 100

T TALANOV, V. I., 10, 11, 26, 32, 50, 51 TANG,C. L., 79, 94, 100 TARAN, J.-P. E., 26, 35, 41, 50 TECH,J. L., 323, 341 R. W., 8, 13,20, 51 TERHUNE, TEWS, K. H., 206, 222, 223, 231 THIBEAU, M., 22, 51 TILFORD, S. G., 291,336, 338,340, 343 TILLMANN, K., 326, 327, 329, 344 P., 165, 170, 172, 178, 187, TILLMANN, 189-191,202-204, 215, 230,232 TINSLEY, B. A., 118, 162 TOBEY Jr., F. L., 323, 342 W. J., 91, 99 TOMLINSON, C. H., 8-11, 14, 27, 32, 35,41, 47, TOWNES, 50,249, 286 TRACHSLIN, W., 302, 336, 338, 341 TRLJRNIT,H. J., 165, 232 J. T., 323, 344 TRURON, TUFTS,A,, 25, 50 TWKEY, J. W., 120, 161

U UHLENBECK, G. E., 267, 279, 286 ULRICH,R. K., 324, 340 UNSOLD, A., 323, 324, 342

~

351

UPSON, W. L., 323, 340

V VANASSE, G. A., 109, 162 VEJE,E., 302, 305, 336, 337, 340 B. P. T., 120, 140, 161 VELTMAN, VERNON,F. L., 60,99 VINTI,J. P., 306, 342 VODAR,B., 22, 51 VOLLMER, H. D., 265, 266,286

W WAGNER, H., 236, 260,268,286 WAGNER, W. G., 13, 14, 16, 35, 43, 47, 50, 51, 237, 273, 286 WAHLSTROM, E. E., 227, 232 L. R., 273,284 WALKER, WALLS,D. F., 283, 284 WANG,C. C., 33, 51 WANG,C. H., 91, 99 WANG,M. C., 267,286 WANG,Y. K., 283,286 R. K., 332, 333, 340 WANGSNESS, WARES,G. W., 321, 323, 343, 344 B., 323, 341 WARNER, WEIDLICH,W., 238, 239, 245, 255, 280, 284-286 WEISS,A. W., 324, 344 S. I., 170, 230 WEISSMAN, H., 284 WELLING, WENDL,G., 35, 51 WESTHAUS, P., 324, 344 WHALING,W., 296, 323, 342, 344 WHITE,D. R., 273, 278, 286 K. G., 245,286 WHITNEY, WIEGAND,J., 165, 170, 172, 178, 187, 189-191,202,203,215,230 WIENER,O., 191,232 WIESE,W. L., 321, 323, 325, 341, 344 WIGNER,F. P., 275, 286 WILSON,K. G., 239, 286 WITT,H. T., 187, 231 WITTKOWER, A. B., 302, 336, 344 WITTMANN, W., 326, 327, 329, 341 WOLF,E., 181, 191, 195,208,214,220,222, 224, 228, 229, 235, 286 WOLNIK,S. J., 322,323, 344 WOLTER,H., 184,232 Woo, J., 273,285 WOGD,R. W., 191, 192,232 Wu, T. Y.,306, 344

352

AUTHOR INDEX

Y

Z

YABLONOVITCH, E., 48, 51 YAHIL, A., 324, 340 YARIV, A,, 273, 275, 285 YGUERABIDE, J., 203,230 YUNG, Y. L., 334, 343

ZARAGA, F., 41,44, 46, 50 ZAIC,W. A., 323, 342 ZEMBROD, A . , 94,95, 100 ZIMNOCH, F. S., 321, 343 ZWICK, M. M., 165,171,172,201,203,204, 229,230,232

SUBJECT INDEX A

- encoded multiplex spectrometers, 140 et

abbreviation, 252 absorption of dye layers, 187 et seq., 191 abundances of elements, 323, 324 acceptor dipole, 199 - molecule, 165 acid layers, 165 acoustic wave, damping of, 23 - -, velocity of, 23 Airy function, 178 aligning interaction, 246 alignment, 290, 326 et seq. - and a non-oscillatory magnetic field, 327

birefringence of a layer system, 174, 202, 206,212, 215, 217, 226 bivalent metal ions, 168 Bloch’s equation, 62, 69 Bohr orbit, 20 Bose-Einstein condensation, 237 - commutation relations, 243 boson operators, 274 Brewster’s angle, 175, 180, 228 Brillouin emission, 236, 273 - scattering, 49

et seq.

alkaline treatment, 170 anisotropic chromosphores, 188 annihilation operator, 243 Ar molecule, 21 arachidic-acid solution, 166 et seq. area theorem, 64 et seq., 98 atomic field, 256 - polarization field, 272, 273 autoionization, 305, 309, 319 auxiliary optics, 105 axial quadrupole, 205

B

seq.

C

cadmium-archidate layers, 166 et seq., 213

- - - _ , birefringence of, 174, 206 - _ -,_ optical properties of, 174 et seq. - _ - ,_optical quality of, 169 - _ - _ , reflectivity of, 175 - _ - ,-refractive index of, 175, 185

canonical ensemble, 254 cascades, 290, 313, 315, 319, 320, 321, 331 CC14 molecule, 21 charge-state identification, 336 et seq. chirping effects, 95 et seq. chromosphores, orientation of, 172, 188 e t seq.

bandwith, 104, 152 barium chloride layers, 169 - stearate layers, 171 beam-foil spectroscopy, 289 et seq. - - - -of C1,294 - - - interaction, 304 electron spectra, 309 Beer’s law, 57 bimolecular collisions, 21 binary digital encoding, 113

- anisotropic,

188 Clerici’s solution, 195 CO molecule, 20 CO, laser, 79 code optimization, 160 coherence, 273, 290, 325 et seq. - length, 252, 253, 263 - properties of an electromagnetic field, 256 - with a non-oscillatory electric field, 332 e t seq.

354

S U B J E C T INDEX

coherent excitation, 326 - optical pulse, propagation of, 56 -wave function, 237 coincidence measurements, 3 19 commutation relations, 268 complete alignment of molecules, 15 complex frequency, 251 condensation, 260, 264 correlation function, 250, 253 et seq., 263 - - of fluctuating forces, 268 -times, 266 et seq., 271, 279 creation operator, 243 critical length, 72 et seq. - distance between donor and acceptor, 200,204,205 cross-talk effect, 113 CS,-cell, 32 et seq., 47 - molecule, 14, 16 coupling constants, 243 Czerny-Turmer grating spectrograph, 147 - - -spectrograph, 143

dispersing spectromodulators, 110 dispersion effects, 72 donor molecule, 165, 199, 204 - -, decay of, 202 Doppler broadening, 60, 64 - effect, 290 et seq., 309 - shift, 338, 339 doubly-excited levels, 304 et seq. dye lasers, 58, 76 dye layers, absorption of, 187 et seq., 191 - -, absorption and fluorescence spectra of, 173, 187 - -, coherent scattering at, 180 et seq. - -, fluorescenceof, 165,171,191,194,198, 21 1 - -, irregularities in, 172, 203, 204 - -, preparation of monomolecular, 170 et

D

Ebert-Fastie monochromator, 140 -_spectrometer, 116, 147 effective absorption, 70 - dipole moment, 95 electric dipole - - decay, 300 - - interaction, 60 - oscillator, 193, 199, 206, 217, 223, 227 - - _ , field of, 199, 207, 217 - - -, orientation of, 208, 218 electric field amplitude, 191, 193 - - - of an evanescent wave, 195, 197 - - - of an electric dipole oscillator, 207 electric quadrupole, 194 - -, energy loss of, 205 - -, field of, 204 et seq. - -, orientation of, 209 electrochromism, 187 electromagnetic field, 256 electron flip operator, 242 electrostriction, 23 - coefficient, 24 ellipsometry, 177 encoding mask, 110 et seq. - -, choice of, 129 - -, matrix of, 127 energy density of a two level absorber, 62 - level schemes, 294 et seq. energy transfer between dye monolayers, 165,202,205,220 - - experiments, 171, 203

damped electron oscillator, 180 damping constants, 277 Dawson’s integral, 15 Debye relaxation time, 18 - unit, 60 Decker spectrometer, 141 - and Harwit spectrometer, 140 dedispersion, 121, 140, 143 degeneracy effects, 91 et seq. - on photon echoes, 91 DeGraauw and Veltman spectrometer, 140 delay time, 71 demagnification, 292 density matrix, 63 depolarized spectrum, 20 design tolerances, 151 detailed balance, 258, 261, 277 detection efficiency, 307 detector, 106 dichroism, 189 discrete mode spectrum, 263 et seq. - - in nonlinear optics, 272 et seq. dielectric constant, 4 - reflector, 192, 220 - relaxation time, 18 diffusion constant, 267, 279 - equation, 17 - process, 279 dipole approximation, 243 et seq.

seq.

dynamic equation of a medium, 26,42

E

-

SUBJECT I N D E X

- - measurements, 204 equations of motion, 273 - - -,mean field approximation of, 275 equilibrium phase transition, 283 error integral, 265 europium complex, 172, 217, 219, 223 - -, absorption and fluorescence of, 173 - -,emission of, 212 - -, optical experiments with, 174 - -, preparation of, 173 evanescent light waves, 194 et seq. - - -, decay of, 194 - - -, electric field amplitude of, 195, 197 - - -, emission of, 198, 224 _ - -in a birefringent medium, 197 et seq. evaporation techniques, 170 excitation cross section, 302 F Fabry-Perot interferometer, 1 15 Faraday cup, 291, 302, 315, 316 - rotation, 72 fatty-acid layers, 168 Fellget’s advantage, 118, 124, 152 field amplitudes, 255 et seq. filament, 11, 35, 40 -, diameter of, 47 -, stabilization of, 47 filter-photomultiplier combination, 315 fine structure, 290, 296, 335 finite-aperture effect, 292 fluctuation-dissipation theorem, 239 fluctuating forces, 244, 247, 257, 274,277 - -, correlation function of, 251 fluorescence, 165, 171, 191, 194, 198, 200, 202, 205, 21 I -, by a molecule, 213 - decay time, 216 el seq., 221,223 -, quantum yield of, 172 fluorescing molecule, radiation pattern of, 206 foci, theory of moving, 41 et seq. focusing, 86 foil characteristics, 31 5 - thickness, 313, 320 Fokker-Planck equation, 239, 258, 264, 270, 275, 277, 278 forbidden decays, 300, 31 1 forced vibrations, 180 et seq. Forster’s energy-transfer theory, 201 Fourier series, 28 - spectrum, 37

355

Fresnel‘s formulas, 179, 214, 220, 227 coefficient, I95 - transform, 118 - zone plate, 118

- reflection

G Gaussian beam, 28 et seq. of incident field, 67 Geiger counter, 322 Girard‘s grill spectrometer, 117, 154 Ginzburg-Landau theory, 238, 263 glass, 21 Golay’s dynamic multislit spectrometer, 115 - static multislit spectrometer, 113 Goldstone mode, 237, 268 grating spectrometer, 106 et seq., I34 Green’s function, 237, 266

- distribution

H Hadamard matrix, 127 er seq., 155 115, 118 et seq., 143 - _ - , cyclic codes for, 155 et seq. - - -,doubly encoded, 130 - _ -, theory of, 126 ez seq. Haken-Sauermann theories, 236, 239, 241, 250 Hamiltonian, 60, 243, 244, 274 Hanle-effect, 330 Hansen and Strong spectrometer, 145 Harwit’s imaging spectrometer, 150 heat-conduction equation, 24 heating, 23 Hekenberg representation, 61 - equations, 244 Hg laser, 76 R b absorber system, 80, 88 HTS-spectrometers, 146 el seq. hydrodynamic equations, 235 hyperfine structure, 290

- transform spectrometer,

--_

I incoherent bleaching process, 72 time, 65, 72 identifications of transitions, 302, 304 inflection points, 40 inhomogeneous static fields, 64 intensities of spectral lines, 299, 302 - of corresponding lines, 303

- relaxation

356

SUBJECT INDEX

intercombination lines, 3 10 interferometer, 107 iron peak, 323 isoelectric sequence, 289, 302, 307, 312 isotropic medium, 13 iteration procedure, 255

J Josephson oscillations, 237 jump diffusion model, 19 K Kerr effect, 9, 14 et seq., 20, 35, 47 Kuhn's concept of energy transfer, 199 L Lamb shift, 311, 334 -theories, 236,239, 241, 250 Landau theory, 239 Lande g-factors, 290, 329 Langevin equations, 245, 264, 277 Laplace operator, 26 laser absorber systems, 78 -, descriptive analysis of a, 245 - mode amplitude, 239 - model of a, 242 -, single mode, 264 et seq., 245 -, threshold of a, 237, 239, 242, 245, 259, 266, 271 -, two mode, 269 et seq. latent image formation, 204 layer thickness, determination of, 176 et seq.

,- - with photometers, 180 - -, - - with X-ray techniques, 180 Legendre polynomial, 17, 18 level assignment, 312 libration model, 23 lifetime data, applications, 323 et seq. - effects, 308 - of electromagnetic radiation, 246 light collection efficiency, 302 Lindhard's theory, 313 line blending, 292, 302, 313, 320 - broadening, 290 et seq. - narrowing, 272 linear absorption coefficient, 66 - accelator, 297, 31 1 linearized equations, 251 Littrow grating spectrograph, 141 - mirror, 115 _-

- spectrometer, 144 local equilibrium ensemble, 236 long range order, 260, 273 Lorentz cavity, 16 - correction factor, 21 Lorentzian spectrum, 19, 22 Lorenz-Lorentz relation, 9, 14, 23 low temperature assumption, 59 luminescence bands of dyes, 213

M Mach-Zehnder interferometer, 107 magnetic dipole, 193, 205, 206 - orientation of, 208 - radiation pattern of, 209 Markoff approximation, 244, 283 master equations, 239 Maxwell's equation, 25, 56, 63 et seq., 74, 86,235 Maxwell-Schroedinger equation, 64 e f seq. mean decay distance, 316 mean field theory, 238 - - - of laser media, 242 et seq. - - - with fluctuations, 250 et seq. - - - without fluctuations, 247 et seq. mean life, 289 - - measurements, 312 et seq. Mertz's mock interferometer, 117 metastability, 305, 309, 338 metastable one electron levels, 310 et seq. Michelson interferometer, 107, 117, 130 - throughput of, 110 microphotometry, 105 mode, condensation into a, 249 - continua in one dimensional media, 261 - continua in three dimensional media, 257 - intensities, 275 - locking, 270 - - mode locking, 271, 272 - operator, 243, 244 -, phase of a, 235 et seq. -, polarization vector of, 243 modulation mask, 110 - -, discontinuously stepping of, 113 monolayer system, 165 - -, preparation of, 166 monomolecular film, 167 moving foci, 41 et seq. multi-electron levels, 294 - mode laser action, 258 multiphoton absorption, 47 multiple-beam interference, 176 el seq., 213 - exit slits, 120

SUBJECT INDEX

N Navier-Stokes equation, 235 neutrino flux, 324 noise, background, 122 - components, 104, 124 -, detector, 123, 132 -, modulation or scintillation, 123 -, photon, 122 nonlinear electronic distortion, 20 --polarization, 3, 12, 21, 23 - - refractive index, 4, 9, 14 et seq. - - susceptibility, 23 - radiative deactivation, 200, 223, 226 - symmetrical molecules, 22

0 off-resonance effect, 88 optical pulse compression by focusing, 86 - compensator, 176 optimum operation, 123 et seq. orientation factor, 205 - of donor and acceptor, 200,205 orientational relaxation time, 18 Ornstein-Uhlenbeck process, 267, 268, 279 oscillating dipole, 199, 217 - external fields, 334 et seq. - -, electric field of, 214 - - in a unaxiaI medium, 202 oscillator strength, 320, 322, 324

P P representation, 275 parabolic equation, 26 parametric emission, 236, 242, 272 parity, 326, 332 partition function, 265 Pauli spin operator, 60 pendulum equation, 89 phase of a scattered wave, 181 - retardation of ordinary and extraordinary rays, 176 - shifts, 97, 208, 214, 221 -, time behavior of, 39 - transition, 235 et seq., 260,264,266, 283 Philips and Harwit spectrometer, 144 phonons, 78,269,277 phosphorence of dyes, 213 photo-chemical reaction, 204 - detector, 127 - electrons, 104 - multipliers, 332

357

photon echo experiments, 57 Pockels cell, 81 Planck’s law, 235 polarizability, 14, 21 polarization, 189, 212, 215, 274 - amplitude, 269 ellipse, rotation of, 13, 24 - induced, 61, 63 population inversion operator, 242, 248 power, critical, 5 , 27, 30 et seq. pressure, 21 prism spectrometers, 110 probability density, 258 er seq., 267, 326, 327 distributions, 276 et seq. pseudopolarization vector, 61, 89 density, 62 pseudorandom codes, 115 pump intensity, 248, 251 pulse area, 62 - break up, 55, 78, 95 - delay, 70 - energy, 66, 73 - envelope, 59 shape, 68

-

-

-

Q Q branch transitions, 92, 94 quadrupole, 205, 206 quantization effects, 237 quantum beats, 290, 326, 331 yield, 201, 204, 226 quasi-monochromatic wave, 25 - optics equation, 26

-

R radial distribution function, 22 radiation pattern, 206 et seq., 213, 215 radiative decay, 305 Raman emission, 236, 242, 290 - oscillation, 276, 277 - scattering, 47 - Stokes emission, 273 rank, tensor, 274 ray index, 227 Rayleigh line, 19 - wing scattering, 19, 21, 22 Rb atom, 56, 64 - absorber, 78, 80 reciprocity theorem, 198, 210, 222 Reed-Muller codes, 154

358

SUBJECT INDEX

reflection coefficients, 208, 214, 228 - spectra, 185 reflectivity of a layer system, 175, 183, 184 et seq., 187, 221 refocusing method, 291 refractive index, 4,9,185,205,215,221,227 relaxation time, 17, 24, 39, 279 - of atomic dipoles, 246 resolution, 104, 152 resonance, 257,273 response time, 17 restricted ensemble, 236 ring nebulae in Lyrae, 139, 150 Ronchi grid, 117 rotating wave approximation, 243 ruby laser, 78 S

scaling transformation, 264, 267, 269, 277, 278 scattering, 291, 309, 315 Schroedinger equation, 56, 60 et seq. - picture, 275 screening length of the atom-atom interaction, 269 selection rules, 310 self-action effects, 3 et seq., 12, 48 self-focusing, 6, 27 et seq., 88 - -, aberrationless theory of, 30, 42 - - distance, 31, 33, 42 - -, experimental results, 32 - -, geometrical representation of, 31 - - of nano-second pulses, 41 et seq. - - of pico-second pulses, 44 et seq. - -, theory of, 27 et seq. self-induced transparency, phenomena, 55 - - - experimental results, 58, 76 et seq. - - _ ideal experimental parameters, 77 - _ - nonlinear transmission, 82 self-phase modulation, 8, 36 et seq. - - -, Fourier spectrum of, 37 - - _ of ultrashort pulses, 40 - steepening, 8, 36 el seq. - trapping condition, 4 et seq. SF6 absorber, 19 sharp line absorber, 68 signal-to-noise-ratio, 104, 121 single beam photometers, 187 single mode laser Hamiltonian, 282 - - _theory, 238, 239, 264 e f seq. slowly varying approximation, 59 - - fields, equations for, 255

Snell’s law, 176, 195, 214, 221, 227 solar coronal lines, 3 11 solid state devices, 105 - - - microminiaturized, 105 spatial operator, 60 - degeneracy effects, 92 spectral line shape, 289, 290 et seq. - -, energy dependence of, 302 - range, 106 - shape, 127 spectrometer, 103 et seq. -, cooling of, 126 -, doubly multiplexed, 131 et seq. -, multislit, 131 -, unsolved problems of a transform, 152 spectrometric imaging, 137 et seq. spectromodulators, 103, 112 et seq. -, essential elements of, 105 spectrum, Fourier transform of, 111 standing light waves, 165, 188, 190 el seq., 222 _ - _ , antinodes of, 192 _ - - , nodes of, 191 Stark modulations, 333 statistical weight, 320 stearic.acid layers, 169 stearyl substituents, 172 steady state solution, 251, 282 superconductors, 237, 263 -, conductivity of, 239 -, mean field theory of, 283 -, nonlinear, 274 superfluid He, 237 superradiance, 241, 246 superradiant damping, 78 - - state, 57 symmetric top molecule, 14

T tandem Van de Graaff, 297 thermal ensembles, 235 thermodynamic limit, 240, 247 - - of a resonator, 273 three mode amplitudes, 274 - - model, 273 threshold condition, 248,259 - inversion, 248, 252 throughput, 110, 121 tipping angle, 62, 98 transition moments, orientation of, 189 - probabilities, 312, 320, 324 transmission coefficients, 214, 228

SUBJECT INDEX

- of an interference filter, 187 - of a layer system, 178, 182, 183 transparency threshold, 56 two-level absorber, 58 - - - atom, 242 - mode lasers, 269 e t seq. - - operator, 268 - standing waves, 270

U uniaxial cristals, optical properties of, 226 e l seq.

Utrecht solar eclipse expedition, 141

v versatility of beam-foil sources, 296 vidicons, 105, 154 vortex quantization, 237

359

W

wave equation, 23

- function, 326 - propagation, 25 et seq. wide aperture advantage, 135, 153 Wigner distribution function, 275, 277 Z

2 expansion method, 324 Zeeman degeneracies, 91 - levels, 329 - split absorber, 72 - splitting, 82 - studies, 335 -transitions, 92, 335 zeta-ion, 289, 291, 294, 308, 311, 312, 334, 336 - - of large zeta, 299, 302, 325

CUMULATIVE INDEX - VOLUMES I-XI1 ABELES,F., Methods for Determining Optical Parameters of Thin Films 11, 249 VII, 139 ABELLA, I. D., Echoes at Optical Frequencies AGARWAL, G. S., Master Equation Methods in Quantum Optics XI, 1 AGRANOVICH, V. M., V. L. GINZBURG, Crystal Optics with Spatial Dispersion IX, 235 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, 211 ARNAUD, J. A., Hamiltonian Theory of Beam Mode Propagation XI, 245 BARAKAT, R., The Intensity Distribution and Total Illumination of AberrationFree Diffraction Images I, 67 XII, 287 BASHKIN, S., Beam-Foil Spectroscopy BECKMANN, P., Scattering of Light by Rough Surfaces VI, 53 BLOOM, A. L., Gas Lasers and their Application to Precise Length Measurements IX, 1 IV, 145 BOUSQUET, P., see P. Rouard BRYNGDAHL, O., Applications of Shearing Interferometry IV, 37 XI, 167 BRYNGDAHL, O., Evanescent Waves in Optical Imaging BURCH,J. M., The Metrological Applications of Diffraction Gratings 11, 73 COHEN-TANNOUDJI, C., A. KASTLER, Optical Pumping v, 1 CREWE, A. V., Production of Electron Probes Using a Field Emission Source XI, 221 CUMMINS, H. Z., H. L. SWINNEY, Light Beating Spectroscopy VIII, 133 XII, 101 DECKER Jr., J. A., see M. Harwit 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 X11, 163 EBERLY, J. H., Interaction of Very Intense Light with Free Electrons VII, 359 I, 253 FIORENTINI, A., Dynamic Characteristics of Visual Processes FOCKE, J., Higher Order Aberration Theory IV, I FRANCON, M., S. MALLICK, Measurement of the Second Order Degree of CoVI, 71 herence FRIEDEN, B. R., Evoluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions IX, 31 1 FRY,G. A., The Optical Performance of the Human Eye VIII, 51 GABOR,D., Light and Information I, 109 GAMO,H., Matrix Treatment of Partial Coherence 111, 187 IX, 235 GINZBURG, V. L., see V. M. Agranovich 11, 109 R. G., Diffusion Through Non-Uniform Media GIOVANELLI, Applications of Optical Methods in the DiffracGNIADEK, K., J. PETYKIEWICZ, tion Theory of Elastic Waves IX, 281

CUMULATIVE INDEX

361

GOODMAN, J. W., Synthetic-Aperture Optics VIII, 1 GRAHAM, R., The Phase Transition Concept and Coherence in Atomic Emission X11, 233 HARWIT,M., J. A. DECKER Jr., 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: Kirchhoff’s Theory IV, 281 VI, 331 KOTTLER, F., Diffraction at a Black Screen, Part 11: Electromagnetic Theory I, 211 KUBOTA, H., Interference Color LEAN,E. G., Interaction of Light and Acoustic Surface Waves XI, 123 VI, I 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. Francon VI, 71 11, 181 MANDEL, L., Fluctuations of Light Beams XI, 303 MARCHAND, E. W., Gradient Index Lenses VIII, 373 MEHTA,C. L., Theory of Photoelectron Counting MIKAELIAN, A. L., M. L. TER-MIKAELIAN, Quasi-Classical Theory of Laser Radiation VII, 231 I, 31 MIYAMOTO, K., Wave Optics and Geometrical Optics in Optical Design V, 199 MURATA, K., Instruments for the Measuring of Optical Transfer Functions VIII, 201 MUSSET, A., A. THELEN, Multilayer Antireflection Coatings VII, 299 OOUE,S., The Photographic Image PEGIS,R. J., The Modern Development of Hamiltonian Optics 1, 1 PEGIS,R. J., see E. Delano VII, 67 PERSHAN, P. S., Non-Linear Optics V, 83 IX, 281 PETYKIEWICZ, J., see K. Gniadek v , 351 PICHT,J., The Wave of a Moving Classical Electron VIII, 239 RISKEN,H., Statistical Properties of Laser Light ROIZEN-DOSSIER, B., see P. Jacquinot 111, 29 IV, 145 ROUARD, P., P. BOUQUET,Optical Constants of Thin Films IV, 199 RUBINOWICZ, A., The Miyamoto-Wolf Diffraction Wave VI, 259 SAKAI, H., see G. A. Vanasse X, 89 SCULLY, M. O., K. G. WHITNEY, Tools of Theoretical Quantum Optics 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 D. L. DEXTER, Optical Absorption Strength of Defects in Insulators X, 165 SMITH,D. Y., x, 45 SMITH,R. W., The Use of Image Tubes as Shutters v, 145 STEEL,W. H., Two-Beam Interferometry IX, 73 STROHBEHN, J. W., Optical Propagation Through the Turbulent Atmosphere

362

CUMULATIVE INDEX

STROKE, G. W., Ruling, Testing and Use of Optical Gratings for High-Resolution Spectroscopy 11, 1 SVELTO, O., Self-Focusing, Self-Trapping, and Self-Phase Modulation of Laser Beams XII, 1 VIII, 133 SWINNEY, H. H., see H. Z. Cummins V, 287 TAYLOR, C. A., see H. Lipson VII, 231 TER-MIKAELIAN, M. L., see A. L. Mikaelian VIII, 201 THELEN, A., see A. Musset THOMPSON, B. J., Image Formation with Partially Coherent Light VII, 169 J., Correction of Optical Images by Compensation of Aberrations TSUJIUCHI, and by Spatial Frequency Filtering 11, 131 UPATMEKS, J., see E. N. Leith VI, 1 VANASSE, G. A., H. SAKAI,Fourier Spectroscopy VI, 259 VANHEEL,A. C. S., Modern Alignment Devices I, 289 WELFORD, W. T., Aberration Theory of Gratings and Grating Mountings IV, 241 WHITNEY, K. G., see M. 0. Scully X, 89 WOLTER, H., On Basic Analogies and Principal Differences between Optical and I, 155 Electronic Information WYNNE, C. G., Field Correctors for Astronomical Telescopes x , 137 YOSHINAGA, H., Recent Developments in Far Infrared XI, I1 VI, 105 YAMAJI, K., Design of Zoom Lenses T., Coherence Theory of Source-SizeCompensation in Interference YAMAMOTO, Microscopy VIII, 295 YOSHINAGA, H., Recent Developments in Far Infrared XI, 77